infinite horizon optimal control of stochastic delay...
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Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 791786 14 pageshttpdxdoiorg1011552013791786
Research ArticleInfinite Horizon Optimal Control of Stochastic Delay EvolutionEquations in Hilbert Spaces
Xueping Zhu1 and Jianjun Zhou2
1 School of Astronautics Northwestern Polytechnical University Xirsquoan Shaanxi 710072 China2 College of Science Northwest AampF University Yangling Shaanxi 712100 China
Correspondence should be addressed to Jianjun Zhou zhoujj198310163com
Received 6 September 2012 Accepted 16 December 2012
Academic Editor Juan J Trujillo
Copyright copy 2013 X Zhu and J Zhou This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The aim of the present paper is to study an infinite horizon optimal control problem in which the controlled state dynamicsis governed by a stochastic delay evolution equation in Hilbert spaces The existence and uniqueness of the optimal controlare obtained by means of associated infinite horizon backward stochastic differential equations without assuming the Gateauxdifferentiability of the drift coefficient and the diffusion coefficient An optimal control problem of stochastic delay partialdifferential equations is also given as an example to illustrate our results
1 Introduction
In this paper we consider a controlled stochastic evolutionequation of the following form
119889119883119906
(119904) = 119860119883119906
(119904) 119889119904 + 119865 (119904 119883119906
119904) 119889119904
+ 119866 (119904 119883119906
119904) 119877 (119904 119883
119906
119904 119906 (119904)) 119889119904 + 119866 (119904 119883
119906
119904) 119889119882 (119904)
119904 ge 119905
119883119906
119905= 119909
(1)
where
119883119906
119904(119897) = 119883
119906
(119904 + 119897) 119897 isin [minus120591 0] 119909 isin 119862 ([minus120591 0] 119867) (2)
119906 is the control process in a measurable space (119880U)and119882 is a cylindrical Wiener process in a Hilbert space Ξ 119860is the generator of a strongly continuous semigroup ofbounded linear operator in another Hilbert space 119867 andthe coefficients 119865 and 119866 defined on [0infin) times 119862 ([minus120591 0]119867)are assumed to satisfy Lipschitz conditions with respect toappropriate norms We introduce the cost function
119869 (119906) = 119864int
infin
119905
119890minus120582119904
119892 (119904 119883119906
119904 119906 (119904)) 119889119904 (3)
Here 119892 is a given real function 120582 is large enough and thecontrol problem is understood in the weak sense We wish tominimize the cost function over all admissible controls
The particular form of the control system is essentialfor our results but it covers numerous interesting cases Forexample in the particular cases119880 = 119867 and 119877(119905 119909 119906) = 119906 theterm 119906(119905)119889119905 + 119889119882(119905) in the state equation can be consideredas a control affected by noise
The stochastic optimal control problem was consideredin 1977 by Bismut [1] The optimal control problem forstochastic partial differential equations in the framework ofa compact control state space has been studied in [2ndash5]Buckdahn and Rascanu [6] considered an optimal controlproblem for a semilinear parabolic stochastic differentialequation with a nonlinear diffusion coefficient and theexistence of a quasioptimal (nonrelaxed) control is showedwithout assuming convexity of the coefficients In [7ndash11] theauthors provided a direct (classical or mild) solution of theHamilton-Jacobi-Bellman equation for the value functionwhich is then used to prove that the optimal control isrelated to the corresponding optimal trajectory by a feedbacklaw In Gozzi [10 11] the existence and uniqueness of amild solution of the associated Hamilton-Jacobi-Bellmanequation are proved when the diffusion term only satisfiesweak nondegeneracy conditions The proofs are based on
2 Abstract and Applied Analysis
the corresponding regularity properties of the transitionsemigroup of the associated Ornstein-Uhlenbeck process
The main tools for the control problem are techniquesfrom the theory of backward stochastic differential equations(BSDEs) in the sense of Pardoux and Peng first consideredin the nonlinear case in [12] see [13 14] as general referencesBSDEs have been successfully applied to control problemssee for example [15 16] and we also refer the reader to[17ndash20] Fuhrman and Tessiture [19] considered the optimalcontrol problem for stochastic differential equation in thestrong form assuming Lipschitz conditions and allowingdegeneracy of the diffusion coefficient under some structuralconstraint on the state equation Existence of an optimalcontrol for stochastic systems in infinite dimensional spacesalso has been obtained in [21ndash27] In [21] Fuhrman and Tes-sitore showed the regularity with respect to parameters andthe regularity in the Malliavin spaces for the solution of thebackward-forward system and defined the feedback law byMalliavin calculus Finally the optimal control is obtained bythe feedback Appealing to the Malliavin calculus comparedwith Fuhrman et al [23] the existence of optimal controlfor stochastic differential equations with delay is proved bythe feedback law Fuhrman and Tessiture [24] dealt with aninfinite horizon optimal control problem for the stochas-tic evolution equation in Hilbert space and the optimalcontrol is showed by means of infinite horizon backwardstochastic differential equation in infinite dimensional spacesand Malliavin calculus In Masiero [25] the infinite horizonoptimal control problem for stochastic evolution equationis also studied by means of the Hamilton-Jacobi-Bellmanequation In Fuhrman [26] a class of optimal control prob-lems governed by stochastic evolution equations in Hilbertspaces which includes state constraints is considered andthe optimal control is obtained by the Fleming logarithmictransformation Masiero [27] studied stochastic evolutionequations evolving in a Banach space where 119866 is a constantand characterized the optimal control via a feedback law byavoiding use of Malliavin calculus Since there is a lack ofregularity of 119865 and 119866 Malliavin calculus is not available inthis case the method in [27] also cannot be used as 119866 isnot a constant but we can prove a theorem similar to [26Proposition 32] which will be used to define the feedbacklaw
In the present paper we study the infinite horizon optimalcontrol problem for stochastic delay evolution equations inHilbert spaces and by usingTheorem 10 the optimal controlis obtained Since we do not relate the optimal feedback lawwith the gradient of the value function and do not considerthe associated Hamilton-Jacobi-Bellman equation we candrop the Gateaux differentiability of the drift term and thediffusion term
The plan of the paper is as follows In the next sectionsome notations are fixed and the stochastic delay evolutionequations are considered with an infinite horizon in particu-lar continuous dependence on initial value (119905 119909) is proved InSection 3 we give the proof of Theorem 10 which is the keyof many subsequent results The addressed optimal controlproblem is considered and the fundamental relation betweenthe optimal control problem and BSDEs is established in
Section 4 Section 5 is devoted to proving the existence anduniqueness of optimal control in the weak sense Finally anapplication is given in Section 6
2 Preliminaries
We list some notations that are used in this paper We usethe symbol | sdot | to denote the norm in a Banach space 119865with a subscript if necessary Let Ξ 119867 and 119870 denote realseparable Hilbert spaces with scalar products (sdot sdot)
Ξ (sdot sdot)
119867
and (sdot sdot)119870 respectively For fixed 120591 gt 0 C = 119862([minus120591 0]119867)
denotes the space of continuous functions from [minus120591 0] to119867endowed with the usual norm |119891|
119862= sup
120579isin[minus1205910]|119891(120579)|
119867 Let
Ξlowast denote the dual space of Ξ with scalar product (sdot sdot)
Ξlowast and
let 119871(Ξ119867) denote the space of all bounded linear operatorsfrom Ξ into 119867 the subspace of Hilbert-Schmidt operatorswith the Hilbert-Schmidt norm is denoted by 119871
2(Ξ119867)
Let (ΩF 119875) be a complete space with a filtration F119905119905ge0
which satisfies the usual condition By a cylindrical Wienerprocess with values in aHilbert spaceΞ defined on (ΩF 119875)we mean a family 119882(119905) 119905 ge 0 of linear mappings Ξ rarr
1198712
(Ω) such that for every 120585 120578 isin Ξ 119882(119905)120585 119905 ge 0 is areal Wiener process and 119864(119882(119905)120585 sdot 119882(119905)120578) = (120585 120578)
Ξ In
the following 119882(119905) 119905 ge 0 is a cylindrical Wiener processadapted to the filtration F
119905119905ge0
In this section and the next section F
119905119905ge0
will denotethe natural filtration of 119882 augmented with the family Nof 119875-null of F The filtration F
119905 119905 ge 0 satisfies the usual
conditions For [119886 119887] [119886infin) sub [0infin) we also use thefollowing notations
F[119886119887]
= 120590 (119882 (119904) minus 119882 (119886) 119904 isin [119886 119887]) orN
F[119886infin)
= 120590 (119882 (119904) minus 119882 (119886) 119904 isin [ainfin)) orN
(4)
By P we denote the predictable 120590-algebra and by B(Λ) wedenote the Borel 120590-algebra of any topological space Λ
Similar to [24] we define several classes of stochasticprocesses with values in a Banach space 119865 as follows
(i) 1198712P(Ω times [119905infin) 119865) denotes the space of equivalenceclasses of processes 119884 isin 119871
2
(Ω times [119905infin) 119865) admittinga predictable version 1198712P(Ω times [119905infin) 119865) is endowedwith the norm
|119884|2
= 119864int
infin
119905
|119884(119904)|2
119889119904 (5)
(ii) 119871119901P(Ω 119871
119902
120573([119905infin) 119865)) defined for 120573 isin 119877 and 119901 119902 isin
[1infin) denotes the space of equivalence classes ofprocesses 119884(119904) 119904 ge 119905 with values in 119865 such that thenorm
|119884|119901
= 119864(int
infin
119905
119890119902120573119904
|119884 (119904)|119902
119889119904)
119901119902
(6)
is finite and 119884 admits a predictable version(iii) K119901
120573(119905) denotes the space 119871
119901
P(Ω 119871
2
120573([119905infin) 119865)) times
119871119901
P(Ω 119871
2
120573([119905infin) 119871
2(Ξ 119865))) The norm of an element
Abstract and Applied Analysis 3
(119884 119885) isinK119901
120573is |(119884 119885)| = |119884|+ |119885| Here 119865 is a Hilbert
space(iv) 119871119901
P(Ω 119862([119905 119879] 119865)) defined for 119879 gt 119905 ge 0 and
119901 isin [1infin) denotes the space of predictable processes119884(119904) 119904 isin [119905 119879]with continuous paths in 119865 such thatthe norm
|119884|119901
= 119864 sup119904isin[119905119879]
|119884 (119904)|119901
(7)
is finite Elements of 119871119901P(Ω 119862([119905 119879] 119865)) are identified
up to indistinguishability
(v) 119871119902P(Ω 119862
120578([119905infin) 119865)) defined for 120578 isin 119877 and 119902 isin
[1infin) denotes the space of predictable processes119884(119904) 119904 ge 119905 with continuous paths in 119865 such that thenorm
|119884|119902
= 119864 sup119904ge119905
119890120578119902119904
|119884 (119904)|119902
(8)
is finite Elements of 119871119902P(Ω 119862
120578(119865)) are identified up
to indistinguishability(vi) Finally for 120578 isin 119877 and 119902 isin [1infin) we
defined Hq120578(119905) as the space 119871119902
P(Ω 119871
119902
120578([119905infin) 119865)) cap
119871119902
P(Ω 119862
120578([119905infin) 119865)) endowed with the norm
|119884|H119902
120578
= |119884|119871119902
P(Ω119871119902
120578([119905infin)119865))
+ |119884|119871119902
P(Ω119862120578([119905infin)119865))
(9)
For simplicity we denote 119871119901
P(Ω 119871
119902
120573([0infin) 119865)) 119871119902
P(Ω
119862120578([0infin) 119865)) H119902
120578(0) and K
119901
120573(0) by 119871119901
P(Ω 119871
119902
120573(119865)) 119871119902
P(Ω
119862120578(119865))H119902
120578 andK
119901
120573 respectively
Now for every fixed 119905 ge 0 we consider the followingstochastic delay evolution equation
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119905 119883119904) 119889119904 + 119866 (119904 119883
119904) 119889119882 (119904)
119904 isin [119905infin)
119883119905= 119909 isin C
(10)
We make the following assumptions
Hypothesis 1 (i)The operator119860 is the generator of a stronglycontinuous semigroup 119890
119905119860
119905 ge 0 of bounded linearoperators in the Hilbert space119867 We denote by119872 and 120596 twoconstants such that |119890119905119860| le 119872119890120596119905 for 119905 ge 0
(ii) The mapping 119865 [0infin) timesC rarr 119867 is measurable andsatisfies for some constant 119871 gt 0 and 0 le 120579 lt 1
10038161003816100381610038161003816119890119904119860
119865 (119905 119909)10038161003816100381610038161003816le 119871119890
120596119904
119904minus120579
(1 + |119909|119862)
10038161003816100381610038161003816119890119904119860
119865 (119905 119909) minus 119890119904119860
119865 (119905 119910)10038161003816100381610038161003816le 119871119890
120596119904
119904minus1205791003816100381610038161003816119909 minus 119910
1003816100381610038161003816119862
119904 gt 0 119905 isin [0 +infin) 119909 119910 isin C
(11)
(iii) 119866 is a mapping [0infin) timesC rarr 119871(Ξ119867) such that forevery 119907 isin Ξ the map 119866119907 [0infin) times C rarr 119867 is measurable
119890119904119860
119866(119905 119909) isin 1198712(Ξ119867) for every s gt 0 119905 isin [0infin) and 119909 isin C
and
10038161003816100381610038161003816119890119904119860
119866 (119905 119909)100381610038161003816100381610038161198712(Ξ119867)
le 119871119890120596119904
119904minus120574
(1 + |119909|119862)
10038161003816100381610038161003816119890119904119860
119866 (119905 119909) minus 119890119904119860
119866 (119905 119910)100381610038161003816100381610038161198712(Ξ119867)
le 119871119890120596119904
119904minus120574
(1003816100381610038161003816119909 minus 119910
1003816100381610038161003816119862)
119904 gt 0 119905 isin [0 +infin) 119909 119910 isin C
(12)
for some constants 119871 gt 0 and 120574 isin [0 12)We say that119883 is amild solution of (10) if it is a continuous
F119905119905ge0
-predictable process with values in 119867 and it satisfies119875-as
119883 (119904) = 119890(119904minus119905)119860
119909 (0) + int
119904
119905
119890(119904minus120590)119860
119865 (120590119883120590) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883120590) 119889119882 (120590) 119904 isin [119905infin)
119883119905= 119909 isin C
(13)
To stress dependence on initial data we denote the solutionby 119883(119904 119905 119909) Note that 119883(119904 119905 119909) is F
[119905119904]measurable hence
independent ofF119905
We first recall a well-known result on solvability of (10)on bounded interval
Theorem 1 Assume that Hypothesis 1 holds Then for all119902 isin [2infin) and 119879 gt 0 there exists a unique process 119883 isin
119871119902
P(Ω 119862([119905 119879]119867)) as mild solution of (10) Moreover
119864 sup119904isin[119905119879]
|119883 (119904)|119902
le 119862(1 + |119909|119862)119902
(14)
for some constant C depending only on 119902 120574 120579T 120591 L 120596 andM
By Theorem 1 and the arbitrariness of 119879 in its statementthe solution is defined for every 119904 ge 119905 We have the followingresult
Theorem 2 Assume that Hypothesis 1 holds and the process119883(sdot 119905 119909) is mild solution of (10) with initial value (119905 119909) isin
[0infin) timesC Then for every 119902 isin [1infin) there exists a constant120578(119902) such that the process 119883
sdot(119905 119909) isin H
119902
120578(119902)(119905) Moreover for a
suitable constant 119862 gt 0 one has
119864supsget119890120578(119902)1199021199041003816100381610038161003816119883119904
1003816100381610038161003816119902
119862+ 119864int
infin
119905
119890120578(119902)1199021199041003816100381610038161003816119883119904
1003816100381610038161003816119902
119862119889119904 le 119862(1 + |119909|
119862)119902
(15)
with the constant 120578(119902) depending only on 119902 120574 120579 120591 119871 120596 and119872
4 Abstract and Applied Analysis
Proof We define a mapping Φ from H119902
120578(119905) times [0infin) times C to
H119902
120578(119905) by the formula
Φ(119883sdot 119905 119909)
119904(119897) = 119890
(119904+119897minus119905)119860
119909 (0) + int
119904+119897
119905
119890(119904+119897minus120590)119860
119865 (120590119883120590) 119889120590
+ int
119904+119897
119905
119890(119904+119897minus120590)119860
119866 (120590119883120590) 119889119882 (120590)
119904 isin [119905infin) 119897 isin [minus120591 0] 119904 + 119897 ge 119905
Φ(119883sdot 119905 119909)
119904(119897) = 119909 (119904 + 119897 minus 119905)
119904 isin [119905infin) 119897 isin [minus120591 0] 119904 + 119897 lt 119905
(16)
We are going to show that provided 120578 is suitably chosenΦ(sdot 119905 119909) is well defined and that it is a contraction inH119902
120578(119905)
that is there exists 119888 lt 1 such that10038161003816100381610038161003816Φ (119883
1
sdot 119905 119909) minus Φ (119883
1
sdot 119905 119909)
10038161003816100381610038161003816H119902
120578(119905)
le 119888100381610038161003816100381610038161198831
sdotminus 119883
2
sdot
10038161003816100381610038161003816H119902
120578(119905)
1198831
sdot 119883
2
sdotisinH
119902
120578(119905)
(17)
For simplicity we set 119905 = 0 and we treat only the case 119865 = 0the general case being handled in a similar way We will usethe so called factorization method see [28 Theorem 525]Let us take 119902 gt 1 and 120572 isin (0 1) such that 1119902 lt 120572 lt (12) minus120574 and let 119888minus1
120572= int
s120590
(119904 minus 119903)120572minus1
(119903 minus 120590)minus120572
119889119903By the stochastic Fubini theorem
Φ(119883sdot 0 119909)
119904(119897) = 119890
(119904+119897)119860
119909 (0)
+ 119888120572int
119904+119897
0
int
119904+119897
120590
(119904 + 119897 minus 119903)120572minus1
(119903 minus 120590)minus120572
times 119890(119904+119897minus119903)119860
119890(119903minus120590)119860
119889119903119866 (120590119883120590) 119889119882 (120590)
= 119890(119904+119897)119860
119909 (0) + Φ1015840
(119883119904) (119897)
119904 isin [0infin) 119897 isin [minus120591 0] 119904 + 119897 ge 0
Φ(119883sdot 0 119909)
119904(119897) = 119909 (119904 + 119897)
119904 isin [0infin) 119897 isin [minus120591 0] 119904 + 119897 lt 0
(18)
where
Φ1015840
(119883sdot)119904(119897) = 119888
120572int
119904+119897
0
(119904 + 119897 minus 119903)120572minus1
119890(119904+119897minus119903)119860
119884 (119903) 119889119903
119884 (119903) = int
119903
0
(119903 minus 120590)minus120572
119890(119903minus120590)119860
119866 (120590119883120590) 119889119882 (120590)
(19)
Since supminus120591le119897le0
|119890(119904+119897)119860
119909(0)| le 119872119890120596119904
|119909|119862 the process 119890(119904+sdot)119860
119909(0) 119904 ge 0 belongs to H119902
120578provided 120596 + 120578 lt 0 Next we
estimate Φ1015840
(119883sdot) where
10038161003816100381610038161003816Φ1015840
(119883sdot)119904(119897)10038161003816100381610038161003816le 119888
120572int
119904+119897
0
(119904 + 119897 minus 119903)120572minus1
119872119890(119904+119897minus119903)120596
|119884 (119903)| 119889119903
(20)
setting 1199021015840 = 119902(119902 minus 1) so that
11989011990212057811990410038161003816100381610038161003816Φ1015840
(119883sdot)119904
10038161003816100381610038161003816
119902
le 119888119902
120572119872
119902 supminus120591le119897le0
119890119902120578119904
(int
119904+119897
0
(119904+119897minus 119903)120572minus1
119890120596(119904+119897minus119903)
|119884 (119903)| 119889119903)
119902
le 119888119902
120572119872
119902 supminus120591le119897le0
(int
119904+119897
0
(119904 + 119897 minus 119903)120572minus1
119890((120596+120578)119902
1015840
)(119904+119897minus119903)
times119890((120596+120578)119902)(119904minus119903)
119890120578119903
|119884 (119903)| 119889119903)
119902
le 119888119902
120572119872
119902 supminus120591le119897le0
(int
119904+119897
0
119890(120578+120596)(119904+119897minus119903)
(119904 + 119897 minus 119903)(120572minus1)119902
1015840
119889119903)
1199021199021015840
times int
119904+119897
0
119890(120578+120596)(119904minus119903)
119890119902120578119903
|119884 (119903)|119902
119889119903
le 119888119902
120572119872
119902
(int
119904
0
119890(120578+120596)119903
1199031199021015840
(120572minus1)
119889119903)
1199021199021015840
times int
119904
0
119890(120578+120596)(119904minus119903)
119890119902120578119903
|119884 (119903)|119902
119889119903
(21)
Applying the Young inequality for convolutions we have
int
infin
0
11989011990212057811990410038161003816100381610038161003816Φ1015840
(119883sdot)119904
10038161003816100381610038161003816
119902
119889119904 le 119888119902
120572119872
119902
(int
infin
0
119890(120578+120596)119904
1199041199021015840
(120572minus1)
119889119904)
1199021199021015840
times int
infin
0
119890(120578+120596)119904
119889119904int
infin
0
119890119902120578119904
|119884 (119904)|119902
119889119904
(22)
and we conclude that10038161003816100381610038161003816Φ1015840
(119883sdot)10038161003816100381610038161003816119871119902
P(Ω119871119902
120578(C))
le 119888120572119872|119884|
119871119902
P(Ω119871119902
120578(119867))
times (int
infin
0
119890(120578+120596)119904
1199041199021015840
(120572minus1)
119889119904)
11199021015840
times (int
infin
0
119890(120578+120596)119904
119889119904)
1119902
(23)
If we start again from (20) and apply theHolder inequality weobtain
10038161003816100381610038161003816119890120578(119904+119897)
Φ1015840
(119883sdot)119904(119897)10038161003816100381610038161003816le 119888
120572119872(int
119904+119897
0
119903(120572minus1)119902
1015840
119890(120596+120578)119903119902
1015840
119889119903)
11199021015840
times (int
119904+119897
0
119890120578119903119902
|119884 (119903)|119902
119889119903)
1119902
10038161003816100381610038161003816119890120578119904
Φ1015840
(119883sdot)119904
10038161003816100381610038161003816le 119888
120572119872(int
119904
0
119903(120572minus1)119902
1015840
119890(120596+120578)119903119902
1015840
119889119903)
11199021015840
times (int
119904
0
119890120578119903119902
|119884(119903)|119902
119889119903)
1119902
(24)
Abstract and Applied Analysis 5
So we conclude that10038161003816100381610038161003816Φ1015840
(119883sdot)10038161003816100381610038161003816119871119902
P(Ω119862120578(C))
le 119888120572119872|119884|
119871119902
P(Ω119871119902
120578(119867))
times (int
infin
0
119903(120572minus1)119902
1015840
119890(120596+120578)119903119902
1015840
119889119903)
11199021015840
(25)
On the other hand by the Burkholder-Davis-Gundy inequal-ities for some constant 119888
119902depending only on 119902 we have
119864|119884 (119903)|119902
le 119888119902119864(int
119903
0
(119903 minus 120590)minus212057210038161003816100381610038161003816119890(119903minus120590)119860
119866 (120590119883120590)10038161003816100381610038161003816
2
1198712(Ξ119867)
119889120590)
1199022
le 119871119902
119888119902119864
times (int
119903
0
(119903 minus 120590)minus2120572minus2120574
1198902120596(119903minus120590)
(1 +1003816100381610038161003816119883120590
10038161003816100381610038162
119862) 119889120590)
1199022
(26)
which implies that
[119864|119884 (119903)|119902
]2119902
le 1198712
1198882119902
119902int
119903
0
(119903 minus 120590)minus2120572minus2120574
times 1198902120596(119903minus120590)
[119864(1 +1003816100381610038161003816119883120590
1003816100381610038161003816119862)119902
]2119902
119889120590
(27)
so that
1198902120578119903
[119864|119884 (119903)|119902
]2119902
le 1198621int
119903
0
(119903 minus 120590)minus2120572minus2120574
1198902(120596+120578)(119903minus120590)
1198902120578120590
119889120590
+ 1198622int
119903
0
(119903 minus 120590)minus2120572minus2120574
1198902(120596+120578)(119903minus120590)
times 1198902120578120590
[1198641003816100381610038161003816119883120590
1003816100381610038161003816119902
119862]2119902
119889120590
(28)
for suitable constants 1198621 119862
2 Applying the Young inequality
for convolutions we obtain
int
infin
0
119890119902120578119903
119864|119884 (119903)|119902
119889119904le 1198621(int
infin
0
119904minus2120572minus2120574
1198902(120596+120578)119904
119889119904)
1199022
int
infin
0
119890119902120578119904
119889119904
+ 1198622(int
infin
0
119904minus2120572minus2120574
1198902(120596+120578)119904
119889119904)
1199022
times int
infin
0
119890119902120578119904
1198641003816100381610038161003816119883119904
1003816100381610038161003816119902
119862119889119904
(29)
This shows that |119884|119871119902
P(Ω119871119902
120578(119867))
is finite provided we assumethat 120578 lt 0 and 120596 + 120578 lt 0 so the map is well defined
If 1198831
sdot 119883
2
sdotare processes belonging to H119902
120578and 1198841 1198842 are
defined accordingly the entirely analogous passages showthat100381610038161003816100381610038161198841
minus 119884210038161003816100381610038161003816119871119902
P(Ω119871119902
120578(119867))
le 1198711198881119902
120572
100381610038161003816100381610038161198831
sdotminus 119883
2
sdot
10038161003816100381610038161003816119871119902
P(Ω119871119902
120578(C))
times (int
infin
0
119904minus2120572minus2120574
1198902(120596+120578)119904
119889119904)
12
(30)
Recalling the inequalities (23) and (25) and noting that themap Y rarr Φ
1015840
(Xsdot) is linear we obtain an explicit expression
for the constant 119888 in (17) and it is immediate to verify that119888 lt 1 provided 120578 lt 0 is chosen sufficiently large We fixsuch a value of 120578(119902) The first result is a consequence of thecontraction principle The estimate (15) also follows from thecontraction property ofΦ(sdot 119905 119909)
For investigating the dependence of the solution119883(119904 119905 119909)on the initial data 119909 and 119905 we reformulate (13) as an equationon [0infin) We set
119878 (119904) = 119890119904119860
for 119904 ge 0 119878 (119904) = 119868 for 119904 lt 0 (31)
and we consider the equation
119883(119904) = 119878 (119904 minus 119905) 119909 ((0 and (119904 minus 119905)) or (minus120591))
+ int
119904
0
119868[119905infin)
(120590) 119878 (119904 minus 120590) 119865 (120590119883120590) 119889120590
+ int
119904
0
119868[119905infin)
(120590) 119878 (119904 minus 120590)
times 119866 (120590119883120590) 119889119882 (120590) 119904 isin [0infin)
1198830(120579) = 119909 ((minus119905 + 120579) or (minus120591)) 120579 isin [minus120591 0]
(32)
Under the assumptions of Hypothesis 1 by Theorem 2 it iseasy to prove that equation (32) has a unique solution 119883 and119883sdotisin H
119902
120578(119902)for every 119902 isin [2infin) It clearly satisfies 119883(119904) =
119909((119904 minus 119905) or (minus120591)) for 119904 isin [minus120591 119905) and its restriction to the timeinternal [119905infin) is the unique mild solution of (10) From nowon we denote by119883(119904 119905 119909) 119904 isin [0infin) the solution of (32)
We need the following parameter-depending contractionprinciple which is stated in the following lemma and provedin [29 Theorems 101 and 102]
Lemma3 (ParameterDependingContraction Principle) Let119861119863 denote Banach spaces Let ℎ 119861times119863 rarr 119861 be a continuousmapping satisfying
1003816100381610038161003816ℎ (1199091 119910) minus ℎ (1199092 119910)1003816100381610038161003816 le 120572
10038161003816100381610038161199091 minus 11990921003816100381610038161003816 (33)
for some 120572 isin [0 1) and every 1199091 119909
2isin 119861 y isin 119863 Let 120601(119910)
denote the unique fixed point of the mapping ℎ(sdot 119910) 119861 rarr 119861Then 120601 119863 rarr 119861 is continuous
Theorem 4 Assume that Hypothesis 1 holds true Then forevery 119902 isin [1infin) the map (119905 119909) rarr 119883
sdot(119905 119909) is continuous
from [0infin) timesC toHq120578(q)
Proof Clearly it is enough to prove the claim for 119902 large Letus consider the map Φ defined in the proof of Theorem 2 In
6 Abstract and Applied Analysis
our present notationΦ can be seen as a mapping fromH119902
120578times
[0infin) timesC toH119902
120578as follows
Φ(119883sdot 119905 119909)
119904(119897) = 119878 (119904 + 119897 minus 119905) 119909 (0)
+ int
119904+119897
0
119868[119905infin)
(120590) 119878 (119904 + 119897 minus 120590) 119865 (120590119883120590) 119889120590
+ int
119904+119897
0
119868[119905infin)
(120590) 119878 (119904 + 119897 minus 120590)
times 119866 (120590119883120590) 119889119882 (120590)
119904 isin [0infin) 119897 isin [minus120591 0] 119904 + 119897 ge 119905
Φ(119883sdot 119905 119909)
119904(119897) = 119909 ((119904 + 119897 minus 119905) or (minus120591))
119904 isin [0infin) 119897 isin [minus120591 0] 119904 + 119897 le 119905
(34)
By the arguments of the proof of Theorem 2 Φ(sdot 119905 119909)is a contraction in H119902
120578uniformly with respect to 119905 119909
The process 119883sdot(119905 119909) is the unique fixed point of Φ(sdot 119905 119909)
So by the parameter-depending contraction principle(Lemma 3) it suffices to show that Φ is continuous fromH119902
120578times [0infin) times C to H119902
120578 From the contraction property
of Φ(sdot 119905 119909) mentioned earlier we have that Φ(sdot 119905 119909) iscontinuous uniformly in 119905 119909 Moreover for fixed 119883
sdot it is
easy to verify that Φ(119883sdot sdot sdot) is continuous from [0infin) times C
toH119902
120578 The proof is finished
Remark 5 By similar passages we can show that for fixed119905 Theorem 4 still holds true for 119902 large enough if the spaces[0infin) times C and H119902
120578are replaced by the spaces 119871119902(ΩCF
119905)
and H119902
120578(119905) respectively where 119871119902(ΩCF
119905) denotes that the
space of F119905-measurable function with value in C such that
the norm
|119909|119902
= 119864|119909|119902
119862 (35)
is finite
3 The Backward-Forward System
In this section we consider the system of stochastic differen-tial equations 119875-as
119883 (119904) = 119890(119904minus119905)119860
119909 (0) + int
119904
119905
119890(119904minus120590)119860
119865 (120590119883120590) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883120590) 119889119882 (120590) 119904 isin [119905infin)
119883119905= 119909 isin C
119884 (119904) minus 119884 (119879) + int
119879
119904
119885 (120590) 119889119882 (120590) + 120582int
119879
119904
119884 (120590) 119889120590
= int
119879
119904
120595 (120590119883120590 119884 (120590) 119885 (120590)) 119889120590 0 le 119904 le 119879 lt infin
(36)
for 119904 varying on the time interval [119905infin) sub [0infin) Asin Section 2 we extend the domain of the solution setting119883(119904 119905 119909) = 119909((119904 minus 119905) or (minus120591)) for 119904 isin [minus120591 119905)
We make the following assumptions
Hypothesis 2 Themapping 120595 [0infin)timesCtimes119870times1198712(Ξ 119870) rarr
119870 is Borelmeasurable such that for all 119905 isin [0infin)120595(119905 sdot) Ctimes119870 times 119871
2(Ξ 119870) rarr 119870 is continuous and for some 119871
119910 119871
119911gt 0
120583 isin 119877 and119898 ge 11003816100381610038161003816120595 (119904 119909 1199101 1199111) minus 120595 (119904 119909 1199102 1199112)
1003816100381610038161003816
le 119871119910
10038161003816100381610038161199101 minus 11991021003816100381610038161003816 + 119871119911
10038161003816100381610038161199111 minus 11991121003816100381610038161003816
1003816100381610038161003816120595 (119904 119909 119910 119911)1003816100381610038161003816 le 119871 (1 + |119909|
119898
119862+10038161003816100381610038161199101003816100381610038161003816 + |119911|)
⟨120595 (119904 119909 1199101 119911) minus 120595 (119904 119909 119910
2 119911) 119910
1minus 119910
2⟩119870ge 120583
10038161003816100381610038161199101 minus 119910210038161003816100381610038162
(37)
for every 119904 isin [0infin) 119909 isin C 119910 1199101 119910
2isin 119870 119911 119911
1 and 119911
2isin
1198712(Ξ 119870)We note that the third inequality in (37) follows from the
first one taking 120583 = minus119871119910but that the third inequalitymay also
hold for different values of 120583Firstly we consider the backward stochastic differential
equation
119884 (119904) minus 119884 (119879) + int
119879
119904
119885 (120590) 119889119882 (120590) + 120582int
119879
119904
119884 (120590) 119889120590
= int
119879
119904
120595 (120590119883120590 119884 (120590) 119885 (120590)) 119889120590 0 le 119904 le 119879 lt infin
(38)
119870 is a Hilbert space the mapping 120595 [0infin) times C times 119870 times
1198712(Ξ 119870) rarr 119870 is a given measurable function 119883
sdotis a
predictable process with values in another Banach space Cand 120582 is a real number
Theorem 6 Assume that Hypothesis 2 holds Let 119901 gt 2 and120575 lt 0 be given and choose
119902 ge 119898119901 120578 gt120575
119898 (39)
Then the following hold
(i) For 119883sdotisin 119871
119902
P(Ω 119871
119902
120578(C)) and 120582 gt minus(120575 + 120583 minus (119871
2
1199112))
(38) has a unique solution in Kp120575that will be denoted
by (119884(119883sdot)(119904) 119885(119883
sdot)(119904)) 119904 ge 0
(ii) The estimate
119864sup119904ge0
(119884 (119883sdot) (119904))
119901
119890119901120575119904
+ 119864(int
infin
0
11989021205751205901003816100381610038161003816119884(119883sdot
)(120590)10038161003816100381610038162
119889120590)
1199012
+ 119864(int
infin
0
11989021205751205901003816100381610038161003816119885 (119883sdot
) (120590)10038161003816100381610038162
119889120590)
1199012
le 119888(1 +1003816100381610038161003816119883sdot
1003816100381610038161003816119898
119871119902
P(Ω119871119902
120578(C))
)119901
(40)
Abstract and Applied Analysis 7
holds for a suitable constant 119888 In particular 119884(119883sdot) isin
119871119901
P(Ω 119862
120575(119870))
(iii) The map 119883sdotrarr (119884(119883
sdot) 119885(119883
sdot)) is continuous from
119871119902
P(Ω 119871
119902
120578(C)) toK119901
120575 and 119883
sdotrarr 119884(119883
sdot) is continuous
from 119871119902
P(Ω 119871
119902
120578(C)) to 119871119901
P(Ω 119862
120575(119870))
(iv) The statements of points (i) (ii) and (iii) still holdtrue if the space 119871119902
P(Ω 119871
119902
120578(C)) is replaced by the space
119871119902
P(Ω 119862
120578(C))
Proof The theorem is very similar to Proposition 311 in [24]The only minor difference is that the mapping 120595 [0infin) times
Ctimes119870times1198712(Ξ 119870) rarr 119870 is a givenmeasurable function while
in [24] the measurable function 120595 is from119867 times119870 times 1198712(Ξ 119870)
to 119870 however the same arguments apply
Theorem 7 Assume that Hypothesis 1 holds and thatHypothesis 2 holds true in the particular case 119870 = 119877 Thenfor every 119901 gt 2 119902 120575 lt 0 satisfying (39) with 120578 = 120578(119902)and for every 120582 gt 120582
1015840
= minus(120575 + 120583 minus (1198712
1199112)) there exists a
unique solution in H119902
120578(119902)times K
119901
120575of (36) that will be denoted
by (119883(sdot 119905 119909) 119884(sdot 119905 119909) 119885(sdot 119905 119909)) Moreover 119884(sdot 119905 119909) isin
119871119901
P(Ω 119862
120575(119877)) The map (119905 119909) rarr (119884(sdot 119905 119909) 119885(sdot 119905 119909)) is con-
tinuous from [0infin)timesC toK119901
120575 and themap (119905 119909) rarr 119884(sdot 119905 119909)
is continuous from [0infin) timesC to 119871119901P(Ω 119862
120575(119877))
Proof We first notice that the system is decoupled the firstdoes not contain the solution (119884 119885) of the second oneThere-fore under the assumption of Hypothesis 1 by Theorem 2there exists a unique solution 119883(sdot 119905 119909) and 119883
sdot(119905 119909) isin H
119902
120578(119902)
of the first equation Moreover from Theorem 4 it followsthat the map (119905 119909) rarr 119883
sdot(119905 119909) is continuous from [0infin)timesC
toH119902
120578(119902)
Let 119870 = 119877 from Theorem 6 we have that thereexists a unique solution (119884(sdot 119905 119909) 119885(sdot 119905 119909)) isin K
119901
120575of the
second equation and the map 119883sdotrarr (119884(119883
sdot) 119885(119883
sdot)) is
continuous from H119902
120578(119902)to K
119901
120575while X
sdotrarr (Y(X
sdot)) is
continuous fromH119902
120578(119902)to119871119901
P(Ω 119862
120575(119877))We have proved that
(119883(sdot 119905 119909) 119884(sdot 119905 119909) 119885(sdot 119905 119909)) isin H119902
120578(119902)times K
119901
120575is the unique
solution of (36) and the other assertions follow from com-position
Remark 8 From Remark 5 by similar passages we can showthat for fixed 119905 and for 119902 large enough under the assumptionsof Theorem 7 the map 119909 rarr (119884(sdot 119905 119909) 119885(sdot 119905 119909)) is continu-ous from 119871
119902
(ΩCF119905) toK119901
120575(119905)
We also remark that the process 119883(sdot 119905 119909) is F[119905infin)
measurable since C is separable Banach space we have that119883sdot(119905 119909) is F
[119905infin)measurable So that 119884(119905) is measurable
with respect to both F[119905infin)
and F119905 it follows that 119884(119905) is
deterministicFor later use we notice three useful identities for 119905 le 119904 lt
infin the equality 119875-as
119883119897(119904 119883
119904(119905 119909)) = 119883
119897(119905 119909) 119897 isin [119904infin) (41)
is a consequence of the uniqueness of the solution of (13)Since the solution of the backward equation is uniquely
determined on an interval [119904infin) by the values of the process119883sdoton the same interval for 119905 le 119904 lt infin we have 119875-as
119884 (119897 119904 119883119904(119905 119909)) = 119884 (119897 119905 119909) for 119897 isin [119904infin)
119885 (119897 119904 119883119904(119905 119909)) = 119885 (119897 119905 119909) for aa 119897 isin [119904infin)
(42)
Lemma 9 (see [30]) Let 119864 be a metric space with metric 119889and let 119891 Ω rarr 119864 be strongly measurable Then thereexists a sequence 119891
119899 119899 isin 119873 of simple 119864-valued functions
(ie 119891119899isFB(E)measurable and takes only a finite number
of values) such that for arbitrary 120596 isin Ω the sequence119889(119891
119899(120596) 119891(120596)) 119899 isin 119873 is monotonically decreasing to zero
Let now 119891 isin 119871119902
(ΩC) By Lemma 9 we get the existenceof a sequence of simple function 119891
119899 119899 isin 119873 such that
1003816100381610038161003816119891119899 (120596) minus 119891 (120596)1003816100381610038161003816 darr 0 for all 120596 isin Ω as 119899 rarr infin (43)
Hence 119891119899
rarr 119891 in | sdot |119871119902(ΩC) by Lebesguersquos dominated
convergence theoremWe are now in a position of showing the main result in
this section
Theorem 10 Assume that Hypothesis 1 holds true and thatHypothesis 2 holds in the particular case 119870 = 119877 Then thereexist two Borel measurable deterministic functions 120592 [tinfin) times
C rarr 119877 and 120577 [119905infin) times C rarr Ξlowast
= 119871(Ξ 119877) =
1198712(Ξ 119877) such that for 119905 isin [0infin) and x isin C the solution
(119883(119905 119909) 119884(119905 119909) 119885(119905 119909)) of (36) satisfies
119884 (119904 119905 119909) = 120592 (119904 119883119904(119905 119909)) 119885 (119904 119905 119909) = 120577 (119904 119883
119904(119905 119909))
119875-as for aa 119904 isin [119905infin)
(44)
Proof We apply the techniques introduced in [26 Proposi-tion 32] Let 119890
119894 be a basis of Ξlowast and let us define 119885119894119873
=
((119885 119890119894)Ξlowast and119873) or (minus119873) Then for every 0 le 119905
1lt 119905
2lt infin Δ gt
0 and 1199091 119909
2isin C we have that
100381610038161003816100381610038161003816100381610038161003816
119864 int
1199051+Δ
1199051
119885119894119873
(119904 1199051 119909
1) 119889119904 minus 119864int
1199052+Δ
1199052
119885119894119873
(119904 1199052 119909
2) 119889119904
100381610038161003816100381610038161003816100381610038161003816
le 119864int
1199052
1199051
10038161003816100381610038161003816119885119894119873
(119904 1199051 119909
1)10038161003816100381610038161003816119889119904
+ 119864int
1199051+Δ
1199052
10038161003816100381610038161003816119885119894119873
(119904 1199051 119909
1) minus 119885
119894119873
(119904 1199052 119909
2)10038161003816100381610038161003816119889119904
+ 119864int
1199052+Δ
1199051+Δ
10038161003816100381610038161003816119885119894119873
(119904 1199052 119909
2)10038161003816100381610038161003816119889119904
8 Abstract and Applied Analysis
le 211987310038161003816100381610038161199052 minus 1199051
1003816100381610038161003816 + Δ12
119890minus120575(1199051+Δ)
times(119864(int
infin
0
119890212057511990410038161003816100381610038161003816119885119894119873
(119904 1199051 119909
1)minus119885
119894119873
(119904 1199052 119909
2)10038161003816100381610038161003816
2
119889119904)
1199012
)
1119901
le 211987310038161003816100381610038161199052 minus 1199051
1003816100381610038161003816 + Δ12
119890minus120575(1199051+Δ)
times (119864(int
infin
0
11989021205751199041003816100381610038161003816119885 (119904 1199051 1199091)minus119885 (119904 1199052 1199092)
10038161003816100381610038162
119889119904)
1199012
)
1119901
(45)
From Theorem 7 we have that the map (119905 119909) rarr int119905+Δ
119905
119885119894119873
(119904 119905 119909)119889119904 is continuous from [0infin) times C to 119877 ByRemark 8 we also have that for fixed 119905 the map 119909 rarr
119864int119905+Δ
119905
119864119885119894119873
(119904 119905 119909)119889119904 is continuous from 119871119902
(ΩCF119905) to 119877
for 119902 large enough Let us define
120577119894119873
(119905 119909) = lim inf119899rarrinfin
119899119864int
119905+(1119899)
119905
119885119894119873
(119904 119905 119909) 119889119904
119905 isin [0infin) 119909 isin C
(46)
It is clear that 120577119894119873 [0infin) timesC rarr 119877 is a Borel functionWe fix 119909 and 0 le 119905 le 119904 lt infin For 119897 isin [119904infin) we
denote 119864[119885119894119873
(119897 119904 119910)]|119910=119883119904(119905119909)
the random variable obtainedby composing119883
119904(119905 119909) with the map 119910 rarr 119864[119885
119894119873
(119897 119904 119910)]By Lemma 9 there exists a sequence of C-valued F
119904-
measurable simple functions
119891119898 Ω 997888rarr C 119891
119898=
119873119898
sum
119896=1
ℎ(119898)
119896119868119891119898=ℎ(119898)
119896 119873
119898isin 119873 (47)
where ℎ(119898)1 ℎ
(119898)
119898are pairwise distinct andΩ = ⋃
119873119898
119896=1119891
119898=
ℎ(119898)
119896 such that
1003816100381610038161003816119891119898 (120596) minus 119883119904(120596)
1003816100381610038161003816 darr 0 for all 120596 isin Ω as 119899 997888rarr infin (48)
For any 119861 isin F119904 we have
int119861
int
119904+(1119899)
119904
119885119894119873
(119897 119905 119909) 119889119897119889119875
= int119861
int
119904+(1119899)
119904
119885119894119873
(119897 119904 119883119904) 119889119897119889119875
= 119864119868119861int
119904+(1119899)
119904
119885119894119873
(119897 119904 119883119904) 119889119897
= lim119898rarrinfin
119864(119868119861int
119904+(1119899)
119904
119885119894119873
(119897 119904 119891119898) 119889119897)
= lim119898rarrinfin
119873119898
sum
119896=1
119864(119868119861119868119891119898=ℎ(119898)
119896int
119904+(1119899)
119904
119885119894119873
(119897 119904 ℎ(119898)
119896) 119889119897)
= lim119898rarrinfin
119864(119868119861
119873119898
sum
119896=1
119868119891119898=ℎ(119898)
119896)119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 ℎ(119898)
119896) 119889119897
= lim119898rarrinfin
119864119868119861(119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 119910) 119889119897|119910=119891119898
)
= lim119898rarrinfin
int119861
(119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 119910) 119889119897|119910=119891119898
)119889119875
= int119861
(119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 119910) 119889119897|119910=119883119904
)119889119875
(49)
and we get that
120577119894119873
(119904 119883119904(119905 119909)) = lim inf
119899rarrinfin
119899
times [119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 119910) 119889119897|119910=119883119904(119905119909)
]
= lim inf119899rarrinfin
119899119864[int
119904+(1119899)
119904
119885119894119873
(119897 119905 119909) 119889119897
100381610038161003816100381610038161003816100381610038161003816
F119904]
119875-as(50)
Fix 119905 and 119909 Recalling that |119885119894119873
| le 119873 by the Lebesguetheorem on differentiation it follows that 119875-as
lim119899rarrinfin
119899int
119904+(1119899)
119904
119885119894119873
(119897 119905 119909) 119889119897 = 119885119894119873
(119904 119905 119909)
for aa 119904 isin [119905infin)
(51)
By the boundedness of 119885119894119873 applying the dominated conver-gence theorem we get that
120577119894119873
(119904 119883119904(119905 119909)) = 119864 [119885
119894119873
(119904 119905 119909)10038161003816100381610038161003816F
119904] = 119885
119894119873
(119904 119905 119909)
119875-as for aa 119904 isin [119905infin)
(52)
Now we have proved that for every 119905 119909
120577119894119873
(119904 119883119904(119905 119909)) = 119885
119894119873
(119904 119905 119909)
119875-as for aa 119904 isin [119905infin)
(53)
for every 119894 119873 Let 119862 sub [0infin) times C denote the set ofpairs (119905 119909) such that lim
119873rarrinfin120577119894119873
(119905 119909) exists and the seriessuminfin
119894=1(lim
119873rarrinfin120577119894119873
(119905 119909))119890119894converges in Ξlowast We define
120577 (119905 119909) =
infin
sum
119894=1
( lim119873rarrinfin
120577119894119873
(119905 119909)) 119890119894 (119905 119909) isin 119862
120577 (119905 119909) = 0 (119905 119909) notin 119862
(54)
Abstract and Applied Analysis 9
Since 119885 satisfies
119885 (120596 119904 119905 119909) =
infin
sum
119894=1
( lim119873rarrinfin
119885119894119873
(120596 119904 119905 119909)) 119890119894 (55)
for every 120596 119904 119905 119909 From (53) it follows that for every 119905 119909 wehave (119904 119883
119904(120596 119905 119909)) isin 119862 119875-as for almost all 119904 isin [119905infin) and
119885(119904 119905 119909) = 120577(119904 119883119904(119905 119909)) 119875-as for aa 119904 isin [119905infin)
We define 120592(119905 119909) = 119884(119905 119905 119909) since 119884(119905 119905 119909) is deter-ministic so the map (119905 119909) rarr 120592(119905 119909) can be written as acomposition 120592(119905 119909) = Γ
3(Γ2(119905 Γ
1(119905 119909))) with
Γ1 [0infin) timesC 997888rarr 119871
119901
P(Ω 119862
120575(119877))
Γ1(119905 119909) = 119884 (sdot 119905 119909)
Γ2 [0infin) times 119871
119901
P(Ω 119862
120575(119877)) 997888rarr 119871
119901
(Ω 119877)
Γ2(119905 119881) = 119881 (119905)
Γ3 119871
119901
(Ω 119877) 997888rarr 119877 Γ3120585 = 119864120585
(56)
FromTheorem 7 it follows that Γ1is continuous By
|119881(119905) minus 119880(119904)|119871119901(Ω119877)
le |119881 (119905) minus 119881 (119904)|119871119901(Ω119877)
+ 119890minus120575119901119904
|119881 minus 119880||119871
119901
P(Ω119862120575(119877))
(57)
we have that Γ2is continuous It is clear that Γ
3is continuous
Then themap (119905 119909) rarr 120592(119905 119909) is continuous from [0infin)timesCto 119877 therefore 120592(119905 119909)is a Borel measurable function Fromuniqueness of the solution of (36) it follows that 119884(119904 119905 119909) =120592(119904 119883
119904(119905 119909)) 119875-as for aa 119904 isin [119905infin)
4 The Fundamental Relation
Let (ΩF 119875) be a given complete probability space with afiltration F
119905119905ge0
satisfying the usual conditions 119882(119905) 119905 ge 0
is a cylindrical Wiener process in Ξ with respect to F119905119905ge0
We will say that an F
119905ge0-predictable process 119906 with values
in a given measurable space (119880U) is an admissible controlThe function 119877 [0infin) times C times 119880 rarr Ξ is measurableand bounded We consider the following controlled stateequation
119889119883119906
(119904) = 119860119883119906
(119904) 119889119904 + 119865 (119904 119883119906
119904) 119889119904
+ 119866 (119904 119883119906
119904) 119877 (119904 119883
119906
119904 119906 (119904)) 119889119904 + 119866 (119904 119883
119906
119904) 119889119882 (119904)
119904 isin [119905infin)
119883119906
119905= 119909
(58)
Here we assume that there exists a mild solution of (58)which will be denoted by 119883119906
(119904 119905 119909) or simply by 119883119906
(119904) Weconsider a cost function of the form
119869 (119906) = 119864int
infin
119905
119890minus120582119904
119892 (119904 119883119906
119904 119906 (119904)) 119889119904 (59)
Here 119892 is function on [0infin) times C times 119880 with real values Ourpurpose is to minimize the function 119869 over all admissiblecontrols
We define in a classical way the Hamiltonian functionrelative to the previous problem for all 119905 isin [0infin) 119909 isin
C and 119911 isin Ξlowast
120595 (119905 119909 119911) = inf 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) 119906 isin 119880
(60)
and the corresponding possibly empty set of minimizers
Γ (119905 119909 119911) = 119906 isin 119880 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) = 120595 (119905 119909 119911)
(61)
We are now ready to formulate the assumptions we need
Hypothesis 3 (i) 119860 119865 and G verify Hypothesis 1(ii) (119880U) is ameasurable spaceThemap 119892 [0infin)timesCtimes
119880 rarr 119877 is continuous and satisfies |119892(119905 119909 119906)| le 119870119892(1+|119909|
119898119892
119862)
for suitable constants 119870119892gt 0 119898
119892gt 0 and all 119909 isin C119906 isin
119880 The map 119877 [0infin) times C times 119880 rarr Ξ is measurable and|119877(119905 119904 119906)| le 119871
119877for a suitable constant 119870
119877gt 0 and all 119909 isin
C119906 isin 119880 and119911 isin Ξlowast
(iii)TheHamiltonian120595 defined in (60) satisfies the requi-rements of Hypothesis 2 (with119870 = 119877)
(iv) We fix here 119901 gt 2 q and 120575 lt 0 satisfying (39) with120578 = 120578(119902) and such that 119902 gt 119898
119892
We are in a position to prove the main result of thissection
Theorem11 Assume thatHypothesis 3 holds and suppose that120582 verifies
120582 gt (minus120575 minus 120583 +1198712
119911
2) or (minus120575 +
1198712
119877
2 (119901 minus 1))
or (1198712
119877119898119892
2 (119902 minus 119898119892)minus 120578 (119902)119898
119892)
(62)
Let 120592 120577 denote the function in the statement of Theorem 10Then for every admissible control 119906 and for the correspondingtrajectory119883 starting at (119905 119909) one has
119869 (119906) = 120592 (119905 119909) + 119864int
infin
119905
119890minus120582119904
[minus120595 (119904 119883119906
119904 120577 (119904 119883
119906
119904)) + 120577 (119904 119883
119906
119904)
times 119877 (119904 119883119906
119904 119906 (119904)) + 119892 (119904 119883
119906
119904 119906 (119904))] 119889119904
(63)
Proof Consider (58) in the probability space (ΩF 119875) withfiltration F
119905119905ge0
and with an F119905119905ge0
-cylindrical Wienerprocess 119882(119905) 119905 ge 0 Let us define
119882119906
(119904) = 119882 (119904) + int
119904
119905and119904
119877 (120590119883119906
120590 119906 (120590)) 119889120590 119904 isin [0infin)
120588 (119879) = exp(int119879
119905
minus119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882 (119904)
minus1
2int
119879
119905
1003816100381610038161003816119877 (119904 119883119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)
(64)
10 Abstract and Applied Analysis
Let 119875119906 be the unique probability onF[0infin)
such that
119875119906
|F119879
= 120588 (119879) 119875|F119879
(65)
We notice that under 119875119906 the process119882119906 is aWiener processLet us denote by F119906
119905119905ge0
the filtration generated by119882119906 andcompleted in the usual way Relatively to 119882119906 (58) can berewritten as
119889119883119906
(119904) = 119860119883119906
(119904) 119889119904 + 119865 (119904 119883119906
119904) 119889119904
+ 119866 (119904 119883119906
119904) 119889119882
119906
(119904) 119904 isin [119905infin)
119883119906
119905= 119909
(66)
In the space (ΩF[0infin)
F119906
119905119905ge0 119875
119906
) we consider the follow-ing system of forward-backward equations
119883119906
(119904) = 119890(119904minus119905)119860
119909 (0) + int
119904
119905
119890(119904minus120590)119860
119865 (120590119883119906
120590) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883119906
120590) 119889119882
119906
(120590) 119904 isin [119905infin)
119883119906
119905= 119909 isin C
119884119906
(119904) minus 119884119906
(119879) + int
119879
119904
119885119906
(120590) 119889119882119906
(120590) + 120582int
119879
119904
119884119906
(120590) 119889120590
= int
119879
119904
120595 (120590119883119906
120590 119885
119906
(120590)) 119889120590 0 le 119904 le 119879 lt infin
(67)
Applying the Ito formula to 119890minus120582119904119884119906(119904) and writing the back-ward equation in (67) with respect to the process119882 we get
119884119906
(119904) + int
119879
119904
119890minus120582120590
119885119906
(120590) 119889119882 (120590)
= int
119879
119904
119890minus120582120590
[120595 (120590119883119906
120590 119885
119906
(120590))
minus119885119906
(120590) 119877 (120590119883119906
120590 119906 (120590))] 119889120590
+ 119890minus120582119879
119884119906
(119879)
(68)
Recalling that 119877 is bounded we get for all 119903 ge 1 and someconstant 119862
119864119906
[120588(119879)minus119903
] = 119864119906
[exp 119903 (int119879
119905
119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882
119906
(119904)
minus1
2int
119879
119905
1003816100381610038161003816119877 (119904 119883119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)]
= 119864119906
[exp(int119879
119905
119903119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882
119906
(119904)
minus1
2int
119879
119905
11990321003816100381610038161003816119877 (119904 119883
119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)
times exp 119903 (119903 minus 1)2
int
119879
119905
1003816100381610038161003816119877 (119904 119883119906
119904 119906 (119904))
10038161003816100381610038162
119889119904]
le 119890(12)119903(119903minus1)119879119871
2
119877119864119906
times exp(int119879
119905
2119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882
119906
(119904)
minus1
2int
119879
119905
41003816100381610038161003816119877 (119904 119883
119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)
= 119890(12)119903(119903minus1)119879119871
2
119877
(69)
It follows that
119864(int
119879
119905
|119890minus120582119904
119885119906
(119904)|2
119889119904)
12
= 119864119906
[(int
119879
119905
10038161003816100381610038161003816119890minus120582119904
119885119906
(119904)10038161003816100381610038161003816
2
119889119904)
12
120588minus1
]
le (119864119906
int
119879
119905
10038161003816100381610038161003816119890minus120582119904
119885119906
(119904)10038161003816100381610038161003816
2
119889119904)
12
times (119864119906
120588minus2
)12
lt infin
(70)
We conclude that the stochastic integral in (68) has zeroexpectation If we set 119904 = 119905 in (68) and we take expectationwith respect to 119875 we obtain
119890minus120582119879
119864119884119906
(119879) minus 119884119906
(119905)
= 119864int
119879
119905
119890minus120582120590
[minus120595 (120590119883119906
120590 119885
119906
(120590))
+119885119906
(120590) 119877 (120590119883119906
120590 119906 (120590))] 119889120590
(71)
ByTheorem 7 119884119906(sdot 119905 119909) isin 119871119901P(Ω 119862
120575(119877)) so that
119864119906
|119884(119879 119905 119909)|119901
le 119862 exp (minus119901120575119879) (72)
By the Holder inequality we have that for suitable constant119862 gt 0
119864 |119884 (119879 119905 119909)| = 119864119906
(120588minus1
(119879) |119884 (119879 119905 119909)|)
le 119864(120588minus119901(119901minus1)
)(119901minus1)119901
119864(|119884 (119879 119905 119909)|119901
)1119901
le 119862119890((1198712
1198772(119901minus1))minus120575))119879
(73)
From Theorem 2 we obtain 119864119906sup119904ge119905119890120578119902119904
|119883119906
119904|119902
lt infin by thesimilar process we get that
1198641003816100381610038161003816119883
119906
119879
1003816100381610038161003816119898119892
le 119862119890(1198712
119877119898119892(2119902minus2119898
119892)minus1
minus120578(119902)119898119892)119879
(74)
Abstract and Applied Analysis 11
for suitable constant 119862 gt 0 and
119864int
infin
119905
119890minus120582120590 1003816100381610038161003816119892 (120590119883
119906
120590 119906 (120590))
1003816100381610038161003816 119889120590 lt infin (75)
Since 119884119906(119905 119905 119909) = 120592(119905 119909) and 119885119906
(119904 119905 119909) = 120577(119904 119883119906
119904(119905 119909)) 119875-
as for aa 119904 isin [119905infin) we have that
119890minus120582119879
119864119884119906
(119879) minus 119907 (119905 119909)
= 119864int
119879
119905
119890minus120582120590
[minus120595 (120590119883119906
120590 120577 (120590 119883
119906
120590))
+120577 (120590119883119906
120590) 119877 (120590119883
119906
120590 119906 (120590))] 119889120590
(76)
Thus adding and subtracting119864intinfin119905
119890minus120582120590
119892(120590119883119906
120590 119906(120590))119889120590 and
letting 119879 rarr infin we conclude that
119869 (119906) = 120592 (119905 119909)
+ 119864int
infin
119905
119890minus120582119904
[minus120595 (119904 119883119906
119904 120577 (119904 119883
119906
119904)) + 120577 (119904 119883
119906
119904) 119877
times (119904 119883119906
119904 119906 (119904)) + 119892 (119904 119883
119906
119904 119906 (119904))] 119889119904
(77)
The proof is finished
We immediately deduce the following consequences
Theorem 12 Let 119905 isin [0infin) and 119909 isin C be fixed assumethat the set-valued map Γ has nonempty values and it admitsa measurable selection Γ
0 [0infin)timesCtimesΞlowast rarr 119880 and assume
that a control 119906(sdot) satisfies
119906 (119904) = Γ0(119904 119883
119906
119904 120577 (119904 119883
119906
119904))
119875-as for almost every 119904 isin [119905infin)
(78)
Then 119869(119905 119909 119906) = 120592(119905 119909) and the pair (119906(sdot) 119883) is optimal forthe control problem starting from 119909 at time 119905
Such a control can be shown to exist if there exists a solutionfor the so-called closed-loop equation as follows
119889119883 (119904)=119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904+119866 (119904 119883
119904)
times(119877 (119904 119883119904 Γ
0(119904 119883
119904 120577 (119904 119883
119904))) 119889119904 + 119889119882 (119904))
119904 isin [119905infin)
119883119905= 119909
(79)
since in this case we can define an optimal control setting
119906 (119904) = Γ0(119904 119883
119904 120577 (119904 119883
119904)) (80)
However under the present assumptions we cannot guaranteethat the closed-loop equation has a solution in the mildsense To circumvent this difficulty we will revert to a weakformulation of the optimal control problem
5 Existence of Optimal Control
We formulate the optimal control problem in the weak sensefollowing the approach of [31]Themain advantage is that wewill be able to solve the closed-loop equation in a weak senseand hence to find an optimal control even if the feedbacklaw is nonsmooth
We call (ΩF F119905119905ge0 119875119882) an admissible setup if
(ΩF F119905119905ge0 119875) is a filtered probability space satisfying the
usual conditions and 119882 is a cylindrical 119875-Wiener processwith values in Ξ with respect to the filtration F
119905119905ge0
By an admissible control system we mean (ΩF
F119905119905ge0 119875119882 119906119883
119906
) where (ΩF F119905119905ge0 119875 119882) is an
admissible setup 119906 is an F119905-predictable process with values
in119880 and119883119906 is a mild solution of (58) An admissible controlsystem will be briefly denoted by (119882 119906119883119906
) in the followingOur purpose is to minimize the cost functional
119869 (119906) = 119864int
infin
119905
119890minus120582119904
119892 (119904 119883119906
119904 119906 (119904)) 119889119904 (81)
over all the admissible control systemOur main result in this section is based on the solvability
of the closed-loop equation
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904 + 119866 (119904 119883
119904)
times (119877 (119904 119883119904 Γ
0(119904 119883
119904 120577 (119904 119883
119904))) 119889119904 +119889119882 (119904))
119904 isin [119905infin)
119883119905= 119909
(82)
In the following sense we say that 119883 is a weak solution of(82) if there exists an admissible setup (ΩF F
119905119905ge0 119875119882)
and anF119905-adapted continuous process119883(119905)with values in119867
which solves the equation in the mild sense namely 119875-as
119883 (119904) = 119890(119904minus119905)119860
119909 (0) + int
119904
119905
119890(119904minus120590)119860
119865 (120590119883120590) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883120590) 119877
times (120590119883120590 Γ
0(120590119883
120590 120577 (120590 119883
120590))) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883120590) 119889119882
120590 119904 isin [119905infin)
(83)
119883119905= 119909 (84)
Theorem 13 Assume that Hypothesis 3 holds Then thereexists a weak solution of the closed-loop equation (82) whichis unique in law
12 Abstract and Applied Analysis
Proof (uniqueness) Let 119883 be a weak solution of (82) in anadmissible setup (ΩF F
119905119905ge0 119875119882)We define
120588 (119879) = exp(int119879
119905
minus119877lowast
(120590 119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889119882 (120590)
minus1
2int
119879
119905
1003816100381610038161003816119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)))10038161003816100381610038162
119889120590)
(85)
Since 119877 is bounded the Girsanov theorem ensures that thereexists a probability measure 1198750 such that the process
1198820
(119904) = 119882 (119904) + int
119904
119905and119904
119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889120590
119904 isin [0infin)
(86)
is a 1198750-Wiener process and
1198750
|F119879
= 120588 (119879) 119875|F119879
(87)
Let us denote by F0
119905119905ge0
the filtration generated by1198820 andcompleted in the usual way In (ΩF
[0infin) F0
119905119905ge0 119875
0
) 119883 isa mild solution of
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119905 119883119904) 119889119904
+ 119866 (119904 119883119904) 119889119882
0
(119904) 119904 isin [119905infin)
119883119905= 119909
120588 (119879) = exp(int119879
119905
minus119877lowast
(120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)) 119889119882
0
(120590)
+ 12int
119879
119905
1003816100381610038161003816119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)))10038161003816100381610038162
119889120590)
(88)
By Hypothesis 3 the joint law of 119883 and 1198820 is uniquely
determined by 119860 119865 119866 and 119909 Taking into account the lastdisplayed formula we conclude that the joint law of 119883 and120588(119879) under1198750 is also uniquely determined and consequentlyso is the law of 119883 under 119875 This completes the proof of theuniqueness part
Proof (existence) Let (ΩF 119875) be a given complete probabil-ity space 119882(119905) 119905 ge 0 is a cylindrical Wiener process on(ΩF 119875)with values inΞ and F
119905119905ge0
is the natural filtrationof 119882(119905) 119905 ge 0 augmented with the family of119875-null sets Let119883(sdot) be the mild solution of
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904
+ 119866 (119904 119883119904) 119889119882 (119904) 119904 isin [119905infin)
119883119905= 119909
(89)
and by the Girsanov theorem let 1198751 be the probability on Ωunder which
1198821
(119904) = 119882 (119904) minus int
119904
119905and119904
119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889120590
(90)
is aWiener process (notice that 119877 is bounded)Then119883 is theweak solution of (82) relatively to the probability 1198751 and theWiener process1198821
Now we can state the main result of this section
Corollary 14 Assume that Hypothesis 3 holds true and 120582
verifies (62) Also assume that the set-valued map Γ hasnonempty values and it admits a measurable selection Γ
0
[0infin) timesC times Ξlowast
rarr 119880 Then for every 119905 isin [0infin) and x isin Cand for all admissible control system (119882 119906119883
119906
) one has
119869 (119906 119905 119909) ge 120592 (119905 119909) (91)
and the equality holds if
119906 (119904) = Γ0(119904 119883
119906
119904 120577 (119904 119883
119906
119904))
119875 minus 119886119904 119891119900119903 119886119897119898119900119904119905 119890119907119890119903119910 119904 isin [119905infin)
(92)
Moreover from Theorem 13 it follows that the closed-loop equation (82) admits a weak solution (ΩFF
119905119905ge0 119875119882119883) which is unique in law and setting
119906 (119904) = Γ0(119904 119883
119904 120577 (119904 119883
119904)) (93)
we obtain an optimal admissible control system (119882 119906119883)
6 Applications
In this section we present a simple application of the previousresults We consider the stochastic delay partial differentialequation in the bounded domain 119861 sub 119877
119899 with smoothboundary 120597119861 as follows
119889119911119906
(119905 120585) = Δ119911119906
(119905 120585) 119889119905 + 119891 (119905 119911119906
119905(120585)) 119889119905
+
119889
sum
119894=1
119892119894(119905 119911
119906
119905(120585)) [119903
119894
(120585) 119906119894
(119905) 119889119905 + 119889119882119894
(119905)]
119911119906
0(120579 120585) = 119909 (120579 120585) 120585 isin 119861 120579 isin [minus1 0]
119911119906
(119905 120585) = 0 119905 isin [0infin) 120585 isin 120597119861
(94)
Here119882 = (1198821
1198822
119882119889
) is a standardWiener process in119877119889 and the functions 119891 [0 +infin) times 119862([minus1 0] 119877) rarr 119877 and119892119894 [0 +infin) times 119862([minus1 0] 119877) rarr 119877 are Lipschitz continuous
and bounded Setting 119880 as a bounded subset of 119877119889 Ξ = 119877119889
119867 = 1198712
(119861) and 119909 isin 119862([minus1 0]119867) We define 119865 and 119866 asfollowing
119865 (119905 119909) (120585) = 119891 (119905 119909 (120585))
(119866 (119905 119909) 119911) (120585) =
119889
sum
119894=1
119892119894(119905 119909 (120585)) 119911
119894
(120585)
120585 isin 119861 119909 isin 119862 ([minus1 0] 119867) 119911 isin 119871 (Ξ119867)
(95)
and let 119860 denote the Laplace operator Δ in 1198712
(119861) withdomain11988222
(119861)⋂11988212
0(119861) then (94) has the form (58) and
Hypothesis 1 holds
Abstract and Applied Analysis 13
Let us consider the optimal control problem associatedwith the cost
119869 (119906) = 119864int
infin
0
119890minus120582119905
[int119861
120590 (120585 119911119906
119905(120585)) 119889120585 + 119906
2
(119905)] 119889119905 (96)
where 120582 verifies (62) and 120590 119862([minus1 0] 119877) times 119880 rarr [0infin) isa bounded measurable function Define 119892 119862([minus1 0]119867) times
119880 rarr [0infin) and 119877 119862([minus1 0]119867) times 119880 rarr Ξ by119892(119910 119906) = int
119861
120590(119905 119910(120585) 119906)119889120585 + 1199062 and 119877(119910 119906) = (int
119861
1199031
(120585)1199061
119889120585
int119861
1199032
(120585)1199062
119889120585 int119861
119903119889
(120585)119906119889
119889120585) for 119910 isin 119862([minus1 0]119867) 119906 =
(1199061
1199062
119906119889
) isin 119880 respectively It can be easily verifiedthat Hypothesis 3 holds true and the set-valued map Γ hasnonempty values and it admits a measurable selection Γ
0
[0infin) times C times Ξlowast
rarr 119880 Then the closed-loop equation(82) admits a weak solution (ΩF F
119905119905ge0 119875119882 119906 119911
sdot(sdot)) and
setting
119906 (119904) = Γ0(119904 119911
119904(sdot) 120577 (119904 119911
119904(sdot))) (97)
we obtain an optimal admissible control system (119882 119906 119911(sdot))
References
[1] J Bismut ldquoOn optimal control of linear stochastic equationswith a linear-quadratic criterionrdquo SIAM Journal on Control andOptimization vol 15 no 3 pp 1ndash4 1977
[2] N Nagase ldquoOn the existence of optimal control for controlledstochastic partial differential equationsrdquo Nagoya MathematicsJournal vol 115 pp 73ndash85 1989
[3] N El Karoui D Huu Nguyen and M Jeanblanc-Pique ldquoCom-pactification methods in the control of degenerate diffusionsrdquoStochastics vol 20 pp 169ndash219 1987
[4] M Nisio ldquoOptimal control for stochastic partial differentialequations and viscosity solutions of Bellman equationsrdquoNagoyaMathematics Journal vol 123 pp 13ndash37 1991
[5] M Nisio ldquoOn sensitive control for stochastic partial differentialequationsrdquo in Stochastic Analysis on Infinite Dimensional SpacesProceedings of the US Japan Bilateral Seminar H Kunita et alEd vol 310 of Pitman Research Notes Mathematical Series pp231ndash241 Longman Scientific and Technical Baton Rouge LaUSA January 1994
[6] R Buckdahn and A Rascanu ldquoOn the existence of stochasticoptimal control of distributed state systemrdquoNonlinear AnalysisTheory Methods and Applications vol 52 no 4 pp 1153ndash11842003
[7] V Barbu and G Da Prato Equations in Hilbert Spaces vol 86 ofPitman Research Notes in Mathematics Pitman 1983
[8] P Cannarsa and G Da Prato ldquoSecond-order Hamilton-Jacobiequations in infinite dimensionsrdquo SIAM Journal on Control andOptimization vol 29 no 2 pp 474ndash492 1991
[9] P Cannarsa and G Da Prato ldquoDirect solution of a second-orderHamilton-Jacobi equations in Hilbert spacesrdquo in StochasticPartial Differential Equations and Applications G Da Pratoand L Tubaro Eds vol 268 of Pitman Research Notes inMathematics Pitman 1992
[10] F Gozzi ldquoRegularity of solutions of second order Hamilton-Jacobi equations and application to a control problemrdquoCommu-nications in Partial Differential Equations vol 20 pp 775ndash8261995
[11] F Gozzi ldquoGlobal regular solutions of second order Hamilton-Jacobi equations in Hilbert spaces with locally Lipschitz non-linearitiesrdquo Journal of Mathematical Analysis and Applicationsvol 198 no 2 pp 399ndash443 1996
[12] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems and Control Lettersvol 14 no 1 pp 55ndash61 1990
[13] N El Karoui and LMazliak Eds Backward Stochastic Differen-tial Equations vol 364 of Pitman ResearchNotes inMathematicsSeries Longman 1997
[14] E Pardoux and BSDEs ldquoweak convergence and homogeneiza-tion of semilinear PDEsrdquo in Non- Linear Analysis DifferentialEquations and Control F H Clarke and R J Stern Eds pp503ndash549 Kluwer Dordrecht The Netherlands 1999
[15] S Peng ldquoA generalized dynamic programming principle andHamilton-Jacobi-Bellman equationrdquo Stochastics and StochasticsReports vol 38 pp 119ndash134 1992
[16] N E Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997
[17] SHamad120583ene and J P Lepeltier ldquoBackward equations stochas-tic control and zero-sum stochastic differential gamesrdquo Stochas-tics and Stochastics Reports vol 54 pp 221ndash231 1995
[18] N El-Karoui and S Hamadene ldquoBSDEs and risk-sensitive con-trol zero-sum and nonzero-sum game problems of stochasticfunctional differential equationsrdquo Stochastic Processes and theirApplications vol 107 no 1 pp 145ndash169 2003
[19] M Fuhrman and G Tessiture ldquoExistence of optimal stochasticcontrols and global solutions of forward-backward stochasticdifferential equationsrdquo SIAM Journal on Control and Optimiza-tion vol 43 no 3 pp 813ndash830 2005
[20] M Fuhrman Y Hu and G Tessitore ldquoOn a class of stochasticoptimal control problems related to bsdes with quadraticgrowthrdquo SIAM Journal on Control and Optimization vol 45 no4 pp 1279ndash1296 2006
[21] M Fuhrman and G Tessitore ldquoNonlinear kolmogorov equa-tions in infinite dimensional spaces the backward stochasticdifferential equations approach and applications to optimalcontrolrdquoAnnals of Probability vol 30 no 3 pp 1397ndash1465 2002
[22] F Masiero ldquoSemilinear kolmogorov equations and applicationsto stochastic optimal controlrdquo Applied Mathematics and Opti-mization vol 51 no 1 pp 201ndash250 2005
[23] M Fuhrman FMasiero andG Tessitore ldquoStochastic equationswith delay optimal control via BSDEs and regular solutions ofHamilton-jacobi-bellman equationsrdquo SIAM Journal on Controland Optimization vol 48 no 7 pp 4624ndash4651 2010
[24] M Fuhrman and G Tessiture ldquoInfinite horizon backwardstochastic differential equations and elliptic equations in hilbertspacesrdquo Annals of Probability vol 32 no 1 pp 607ndash660 2004
[25] F Masiero ldquoInfinite horizon stochastic optimal control prob-lems with degenerate noise and elliptic equations in Hilbertspacesrdquo Applied Mathematics and Optimization vol 55 no 3pp 285ndash326 2007
[26] M Fuhrman ldquoA class of stochastic optimal control problemsin Hilbert spaces BSDEs and optimal control laws stateconstraints conditioned processesrdquo Stochastic Processes andtheir Applications vol 108 no 2 pp 263ndash298 2003
[27] F Masiero ldquoStochastic optimal control problems and parabolicequations in banach spacesrdquo SIAM Journal on Control andOptimization vol 47 no 1 pp 251ndash300 2008
[28] G Da Prato and J Zabczyk Ergodicity For Infinite-DimensionalSystems Cambridge University Press 1996
14 Abstract and Applied Analysis
[29] J Zabczyk ldquoParabolic equations on Hilbert spacesrdquo in StochaS-tic PDErsquoS and Kolmogorov Equations in Infinite Dimensionsvol 1715 of Lecture Notes in Math pp 117ndash213 Springer BerlinGermany 1999
[30] G Da Prato and J Zabczyk Stochstic Equations in InfiniteDimensions Cambridge University Press 1992
[31] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 of Applications of MathematicsSpringer New York NY USA 1993
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OptimizationJournal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
the corresponding regularity properties of the transitionsemigroup of the associated Ornstein-Uhlenbeck process
The main tools for the control problem are techniquesfrom the theory of backward stochastic differential equations(BSDEs) in the sense of Pardoux and Peng first consideredin the nonlinear case in [12] see [13 14] as general referencesBSDEs have been successfully applied to control problemssee for example [15 16] and we also refer the reader to[17ndash20] Fuhrman and Tessiture [19] considered the optimalcontrol problem for stochastic differential equation in thestrong form assuming Lipschitz conditions and allowingdegeneracy of the diffusion coefficient under some structuralconstraint on the state equation Existence of an optimalcontrol for stochastic systems in infinite dimensional spacesalso has been obtained in [21ndash27] In [21] Fuhrman and Tes-sitore showed the regularity with respect to parameters andthe regularity in the Malliavin spaces for the solution of thebackward-forward system and defined the feedback law byMalliavin calculus Finally the optimal control is obtained bythe feedback Appealing to the Malliavin calculus comparedwith Fuhrman et al [23] the existence of optimal controlfor stochastic differential equations with delay is proved bythe feedback law Fuhrman and Tessiture [24] dealt with aninfinite horizon optimal control problem for the stochas-tic evolution equation in Hilbert space and the optimalcontrol is showed by means of infinite horizon backwardstochastic differential equation in infinite dimensional spacesand Malliavin calculus In Masiero [25] the infinite horizonoptimal control problem for stochastic evolution equationis also studied by means of the Hamilton-Jacobi-Bellmanequation In Fuhrman [26] a class of optimal control prob-lems governed by stochastic evolution equations in Hilbertspaces which includes state constraints is considered andthe optimal control is obtained by the Fleming logarithmictransformation Masiero [27] studied stochastic evolutionequations evolving in a Banach space where 119866 is a constantand characterized the optimal control via a feedback law byavoiding use of Malliavin calculus Since there is a lack ofregularity of 119865 and 119866 Malliavin calculus is not available inthis case the method in [27] also cannot be used as 119866 isnot a constant but we can prove a theorem similar to [26Proposition 32] which will be used to define the feedbacklaw
In the present paper we study the infinite horizon optimalcontrol problem for stochastic delay evolution equations inHilbert spaces and by usingTheorem 10 the optimal controlis obtained Since we do not relate the optimal feedback lawwith the gradient of the value function and do not considerthe associated Hamilton-Jacobi-Bellman equation we candrop the Gateaux differentiability of the drift term and thediffusion term
The plan of the paper is as follows In the next sectionsome notations are fixed and the stochastic delay evolutionequations are considered with an infinite horizon in particu-lar continuous dependence on initial value (119905 119909) is proved InSection 3 we give the proof of Theorem 10 which is the keyof many subsequent results The addressed optimal controlproblem is considered and the fundamental relation betweenthe optimal control problem and BSDEs is established in
Section 4 Section 5 is devoted to proving the existence anduniqueness of optimal control in the weak sense Finally anapplication is given in Section 6
2 Preliminaries
We list some notations that are used in this paper We usethe symbol | sdot | to denote the norm in a Banach space 119865with a subscript if necessary Let Ξ 119867 and 119870 denote realseparable Hilbert spaces with scalar products (sdot sdot)
Ξ (sdot sdot)
119867
and (sdot sdot)119870 respectively For fixed 120591 gt 0 C = 119862([minus120591 0]119867)
denotes the space of continuous functions from [minus120591 0] to119867endowed with the usual norm |119891|
119862= sup
120579isin[minus1205910]|119891(120579)|
119867 Let
Ξlowast denote the dual space of Ξ with scalar product (sdot sdot)
Ξlowast and
let 119871(Ξ119867) denote the space of all bounded linear operatorsfrom Ξ into 119867 the subspace of Hilbert-Schmidt operatorswith the Hilbert-Schmidt norm is denoted by 119871
2(Ξ119867)
Let (ΩF 119875) be a complete space with a filtration F119905119905ge0
which satisfies the usual condition By a cylindrical Wienerprocess with values in aHilbert spaceΞ defined on (ΩF 119875)we mean a family 119882(119905) 119905 ge 0 of linear mappings Ξ rarr
1198712
(Ω) such that for every 120585 120578 isin Ξ 119882(119905)120585 119905 ge 0 is areal Wiener process and 119864(119882(119905)120585 sdot 119882(119905)120578) = (120585 120578)
Ξ In
the following 119882(119905) 119905 ge 0 is a cylindrical Wiener processadapted to the filtration F
119905119905ge0
In this section and the next section F
119905119905ge0
will denotethe natural filtration of 119882 augmented with the family Nof 119875-null of F The filtration F
119905 119905 ge 0 satisfies the usual
conditions For [119886 119887] [119886infin) sub [0infin) we also use thefollowing notations
F[119886119887]
= 120590 (119882 (119904) minus 119882 (119886) 119904 isin [119886 119887]) orN
F[119886infin)
= 120590 (119882 (119904) minus 119882 (119886) 119904 isin [ainfin)) orN
(4)
By P we denote the predictable 120590-algebra and by B(Λ) wedenote the Borel 120590-algebra of any topological space Λ
Similar to [24] we define several classes of stochasticprocesses with values in a Banach space 119865 as follows
(i) 1198712P(Ω times [119905infin) 119865) denotes the space of equivalenceclasses of processes 119884 isin 119871
2
(Ω times [119905infin) 119865) admittinga predictable version 1198712P(Ω times [119905infin) 119865) is endowedwith the norm
|119884|2
= 119864int
infin
119905
|119884(119904)|2
119889119904 (5)
(ii) 119871119901P(Ω 119871
119902
120573([119905infin) 119865)) defined for 120573 isin 119877 and 119901 119902 isin
[1infin) denotes the space of equivalence classes ofprocesses 119884(119904) 119904 ge 119905 with values in 119865 such that thenorm
|119884|119901
= 119864(int
infin
119905
119890119902120573119904
|119884 (119904)|119902
119889119904)
119901119902
(6)
is finite and 119884 admits a predictable version(iii) K119901
120573(119905) denotes the space 119871
119901
P(Ω 119871
2
120573([119905infin) 119865)) times
119871119901
P(Ω 119871
2
120573([119905infin) 119871
2(Ξ 119865))) The norm of an element
Abstract and Applied Analysis 3
(119884 119885) isinK119901
120573is |(119884 119885)| = |119884|+ |119885| Here 119865 is a Hilbert
space(iv) 119871119901
P(Ω 119862([119905 119879] 119865)) defined for 119879 gt 119905 ge 0 and
119901 isin [1infin) denotes the space of predictable processes119884(119904) 119904 isin [119905 119879]with continuous paths in 119865 such thatthe norm
|119884|119901
= 119864 sup119904isin[119905119879]
|119884 (119904)|119901
(7)
is finite Elements of 119871119901P(Ω 119862([119905 119879] 119865)) are identified
up to indistinguishability
(v) 119871119902P(Ω 119862
120578([119905infin) 119865)) defined for 120578 isin 119877 and 119902 isin
[1infin) denotes the space of predictable processes119884(119904) 119904 ge 119905 with continuous paths in 119865 such that thenorm
|119884|119902
= 119864 sup119904ge119905
119890120578119902119904
|119884 (119904)|119902
(8)
is finite Elements of 119871119902P(Ω 119862
120578(119865)) are identified up
to indistinguishability(vi) Finally for 120578 isin 119877 and 119902 isin [1infin) we
defined Hq120578(119905) as the space 119871119902
P(Ω 119871
119902
120578([119905infin) 119865)) cap
119871119902
P(Ω 119862
120578([119905infin) 119865)) endowed with the norm
|119884|H119902
120578
= |119884|119871119902
P(Ω119871119902
120578([119905infin)119865))
+ |119884|119871119902
P(Ω119862120578([119905infin)119865))
(9)
For simplicity we denote 119871119901
P(Ω 119871
119902
120573([0infin) 119865)) 119871119902
P(Ω
119862120578([0infin) 119865)) H119902
120578(0) and K
119901
120573(0) by 119871119901
P(Ω 119871
119902
120573(119865)) 119871119902
P(Ω
119862120578(119865))H119902
120578 andK
119901
120573 respectively
Now for every fixed 119905 ge 0 we consider the followingstochastic delay evolution equation
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119905 119883119904) 119889119904 + 119866 (119904 119883
119904) 119889119882 (119904)
119904 isin [119905infin)
119883119905= 119909 isin C
(10)
We make the following assumptions
Hypothesis 1 (i)The operator119860 is the generator of a stronglycontinuous semigroup 119890
119905119860
119905 ge 0 of bounded linearoperators in the Hilbert space119867 We denote by119872 and 120596 twoconstants such that |119890119905119860| le 119872119890120596119905 for 119905 ge 0
(ii) The mapping 119865 [0infin) timesC rarr 119867 is measurable andsatisfies for some constant 119871 gt 0 and 0 le 120579 lt 1
10038161003816100381610038161003816119890119904119860
119865 (119905 119909)10038161003816100381610038161003816le 119871119890
120596119904
119904minus120579
(1 + |119909|119862)
10038161003816100381610038161003816119890119904119860
119865 (119905 119909) minus 119890119904119860
119865 (119905 119910)10038161003816100381610038161003816le 119871119890
120596119904
119904minus1205791003816100381610038161003816119909 minus 119910
1003816100381610038161003816119862
119904 gt 0 119905 isin [0 +infin) 119909 119910 isin C
(11)
(iii) 119866 is a mapping [0infin) timesC rarr 119871(Ξ119867) such that forevery 119907 isin Ξ the map 119866119907 [0infin) times C rarr 119867 is measurable
119890119904119860
119866(119905 119909) isin 1198712(Ξ119867) for every s gt 0 119905 isin [0infin) and 119909 isin C
and
10038161003816100381610038161003816119890119904119860
119866 (119905 119909)100381610038161003816100381610038161198712(Ξ119867)
le 119871119890120596119904
119904minus120574
(1 + |119909|119862)
10038161003816100381610038161003816119890119904119860
119866 (119905 119909) minus 119890119904119860
119866 (119905 119910)100381610038161003816100381610038161198712(Ξ119867)
le 119871119890120596119904
119904minus120574
(1003816100381610038161003816119909 minus 119910
1003816100381610038161003816119862)
119904 gt 0 119905 isin [0 +infin) 119909 119910 isin C
(12)
for some constants 119871 gt 0 and 120574 isin [0 12)We say that119883 is amild solution of (10) if it is a continuous
F119905119905ge0
-predictable process with values in 119867 and it satisfies119875-as
119883 (119904) = 119890(119904minus119905)119860
119909 (0) + int
119904
119905
119890(119904minus120590)119860
119865 (120590119883120590) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883120590) 119889119882 (120590) 119904 isin [119905infin)
119883119905= 119909 isin C
(13)
To stress dependence on initial data we denote the solutionby 119883(119904 119905 119909) Note that 119883(119904 119905 119909) is F
[119905119904]measurable hence
independent ofF119905
We first recall a well-known result on solvability of (10)on bounded interval
Theorem 1 Assume that Hypothesis 1 holds Then for all119902 isin [2infin) and 119879 gt 0 there exists a unique process 119883 isin
119871119902
P(Ω 119862([119905 119879]119867)) as mild solution of (10) Moreover
119864 sup119904isin[119905119879]
|119883 (119904)|119902
le 119862(1 + |119909|119862)119902
(14)
for some constant C depending only on 119902 120574 120579T 120591 L 120596 andM
By Theorem 1 and the arbitrariness of 119879 in its statementthe solution is defined for every 119904 ge 119905 We have the followingresult
Theorem 2 Assume that Hypothesis 1 holds and the process119883(sdot 119905 119909) is mild solution of (10) with initial value (119905 119909) isin
[0infin) timesC Then for every 119902 isin [1infin) there exists a constant120578(119902) such that the process 119883
sdot(119905 119909) isin H
119902
120578(119902)(119905) Moreover for a
suitable constant 119862 gt 0 one has
119864supsget119890120578(119902)1199021199041003816100381610038161003816119883119904
1003816100381610038161003816119902
119862+ 119864int
infin
119905
119890120578(119902)1199021199041003816100381610038161003816119883119904
1003816100381610038161003816119902
119862119889119904 le 119862(1 + |119909|
119862)119902
(15)
with the constant 120578(119902) depending only on 119902 120574 120579 120591 119871 120596 and119872
4 Abstract and Applied Analysis
Proof We define a mapping Φ from H119902
120578(119905) times [0infin) times C to
H119902
120578(119905) by the formula
Φ(119883sdot 119905 119909)
119904(119897) = 119890
(119904+119897minus119905)119860
119909 (0) + int
119904+119897
119905
119890(119904+119897minus120590)119860
119865 (120590119883120590) 119889120590
+ int
119904+119897
119905
119890(119904+119897minus120590)119860
119866 (120590119883120590) 119889119882 (120590)
119904 isin [119905infin) 119897 isin [minus120591 0] 119904 + 119897 ge 119905
Φ(119883sdot 119905 119909)
119904(119897) = 119909 (119904 + 119897 minus 119905)
119904 isin [119905infin) 119897 isin [minus120591 0] 119904 + 119897 lt 119905
(16)
We are going to show that provided 120578 is suitably chosenΦ(sdot 119905 119909) is well defined and that it is a contraction inH119902
120578(119905)
that is there exists 119888 lt 1 such that10038161003816100381610038161003816Φ (119883
1
sdot 119905 119909) minus Φ (119883
1
sdot 119905 119909)
10038161003816100381610038161003816H119902
120578(119905)
le 119888100381610038161003816100381610038161198831
sdotminus 119883
2
sdot
10038161003816100381610038161003816H119902
120578(119905)
1198831
sdot 119883
2
sdotisinH
119902
120578(119905)
(17)
For simplicity we set 119905 = 0 and we treat only the case 119865 = 0the general case being handled in a similar way We will usethe so called factorization method see [28 Theorem 525]Let us take 119902 gt 1 and 120572 isin (0 1) such that 1119902 lt 120572 lt (12) minus120574 and let 119888minus1
120572= int
s120590
(119904 minus 119903)120572minus1
(119903 minus 120590)minus120572
119889119903By the stochastic Fubini theorem
Φ(119883sdot 0 119909)
119904(119897) = 119890
(119904+119897)119860
119909 (0)
+ 119888120572int
119904+119897
0
int
119904+119897
120590
(119904 + 119897 minus 119903)120572minus1
(119903 minus 120590)minus120572
times 119890(119904+119897minus119903)119860
119890(119903minus120590)119860
119889119903119866 (120590119883120590) 119889119882 (120590)
= 119890(119904+119897)119860
119909 (0) + Φ1015840
(119883119904) (119897)
119904 isin [0infin) 119897 isin [minus120591 0] 119904 + 119897 ge 0
Φ(119883sdot 0 119909)
119904(119897) = 119909 (119904 + 119897)
119904 isin [0infin) 119897 isin [minus120591 0] 119904 + 119897 lt 0
(18)
where
Φ1015840
(119883sdot)119904(119897) = 119888
120572int
119904+119897
0
(119904 + 119897 minus 119903)120572minus1
119890(119904+119897minus119903)119860
119884 (119903) 119889119903
119884 (119903) = int
119903
0
(119903 minus 120590)minus120572
119890(119903minus120590)119860
119866 (120590119883120590) 119889119882 (120590)
(19)
Since supminus120591le119897le0
|119890(119904+119897)119860
119909(0)| le 119872119890120596119904
|119909|119862 the process 119890(119904+sdot)119860
119909(0) 119904 ge 0 belongs to H119902
120578provided 120596 + 120578 lt 0 Next we
estimate Φ1015840
(119883sdot) where
10038161003816100381610038161003816Φ1015840
(119883sdot)119904(119897)10038161003816100381610038161003816le 119888
120572int
119904+119897
0
(119904 + 119897 minus 119903)120572minus1
119872119890(119904+119897minus119903)120596
|119884 (119903)| 119889119903
(20)
setting 1199021015840 = 119902(119902 minus 1) so that
11989011990212057811990410038161003816100381610038161003816Φ1015840
(119883sdot)119904
10038161003816100381610038161003816
119902
le 119888119902
120572119872
119902 supminus120591le119897le0
119890119902120578119904
(int
119904+119897
0
(119904+119897minus 119903)120572minus1
119890120596(119904+119897minus119903)
|119884 (119903)| 119889119903)
119902
le 119888119902
120572119872
119902 supminus120591le119897le0
(int
119904+119897
0
(119904 + 119897 minus 119903)120572minus1
119890((120596+120578)119902
1015840
)(119904+119897minus119903)
times119890((120596+120578)119902)(119904minus119903)
119890120578119903
|119884 (119903)| 119889119903)
119902
le 119888119902
120572119872
119902 supminus120591le119897le0
(int
119904+119897
0
119890(120578+120596)(119904+119897minus119903)
(119904 + 119897 minus 119903)(120572minus1)119902
1015840
119889119903)
1199021199021015840
times int
119904+119897
0
119890(120578+120596)(119904minus119903)
119890119902120578119903
|119884 (119903)|119902
119889119903
le 119888119902
120572119872
119902
(int
119904
0
119890(120578+120596)119903
1199031199021015840
(120572minus1)
119889119903)
1199021199021015840
times int
119904
0
119890(120578+120596)(119904minus119903)
119890119902120578119903
|119884 (119903)|119902
119889119903
(21)
Applying the Young inequality for convolutions we have
int
infin
0
11989011990212057811990410038161003816100381610038161003816Φ1015840
(119883sdot)119904
10038161003816100381610038161003816
119902
119889119904 le 119888119902
120572119872
119902
(int
infin
0
119890(120578+120596)119904
1199041199021015840
(120572minus1)
119889119904)
1199021199021015840
times int
infin
0
119890(120578+120596)119904
119889119904int
infin
0
119890119902120578119904
|119884 (119904)|119902
119889119904
(22)
and we conclude that10038161003816100381610038161003816Φ1015840
(119883sdot)10038161003816100381610038161003816119871119902
P(Ω119871119902
120578(C))
le 119888120572119872|119884|
119871119902
P(Ω119871119902
120578(119867))
times (int
infin
0
119890(120578+120596)119904
1199041199021015840
(120572minus1)
119889119904)
11199021015840
times (int
infin
0
119890(120578+120596)119904
119889119904)
1119902
(23)
If we start again from (20) and apply theHolder inequality weobtain
10038161003816100381610038161003816119890120578(119904+119897)
Φ1015840
(119883sdot)119904(119897)10038161003816100381610038161003816le 119888
120572119872(int
119904+119897
0
119903(120572minus1)119902
1015840
119890(120596+120578)119903119902
1015840
119889119903)
11199021015840
times (int
119904+119897
0
119890120578119903119902
|119884 (119903)|119902
119889119903)
1119902
10038161003816100381610038161003816119890120578119904
Φ1015840
(119883sdot)119904
10038161003816100381610038161003816le 119888
120572119872(int
119904
0
119903(120572minus1)119902
1015840
119890(120596+120578)119903119902
1015840
119889119903)
11199021015840
times (int
119904
0
119890120578119903119902
|119884(119903)|119902
119889119903)
1119902
(24)
Abstract and Applied Analysis 5
So we conclude that10038161003816100381610038161003816Φ1015840
(119883sdot)10038161003816100381610038161003816119871119902
P(Ω119862120578(C))
le 119888120572119872|119884|
119871119902
P(Ω119871119902
120578(119867))
times (int
infin
0
119903(120572minus1)119902
1015840
119890(120596+120578)119903119902
1015840
119889119903)
11199021015840
(25)
On the other hand by the Burkholder-Davis-Gundy inequal-ities for some constant 119888
119902depending only on 119902 we have
119864|119884 (119903)|119902
le 119888119902119864(int
119903
0
(119903 minus 120590)minus212057210038161003816100381610038161003816119890(119903minus120590)119860
119866 (120590119883120590)10038161003816100381610038161003816
2
1198712(Ξ119867)
119889120590)
1199022
le 119871119902
119888119902119864
times (int
119903
0
(119903 minus 120590)minus2120572minus2120574
1198902120596(119903minus120590)
(1 +1003816100381610038161003816119883120590
10038161003816100381610038162
119862) 119889120590)
1199022
(26)
which implies that
[119864|119884 (119903)|119902
]2119902
le 1198712
1198882119902
119902int
119903
0
(119903 minus 120590)minus2120572minus2120574
times 1198902120596(119903minus120590)
[119864(1 +1003816100381610038161003816119883120590
1003816100381610038161003816119862)119902
]2119902
119889120590
(27)
so that
1198902120578119903
[119864|119884 (119903)|119902
]2119902
le 1198621int
119903
0
(119903 minus 120590)minus2120572minus2120574
1198902(120596+120578)(119903minus120590)
1198902120578120590
119889120590
+ 1198622int
119903
0
(119903 minus 120590)minus2120572minus2120574
1198902(120596+120578)(119903minus120590)
times 1198902120578120590
[1198641003816100381610038161003816119883120590
1003816100381610038161003816119902
119862]2119902
119889120590
(28)
for suitable constants 1198621 119862
2 Applying the Young inequality
for convolutions we obtain
int
infin
0
119890119902120578119903
119864|119884 (119903)|119902
119889119904le 1198621(int
infin
0
119904minus2120572minus2120574
1198902(120596+120578)119904
119889119904)
1199022
int
infin
0
119890119902120578119904
119889119904
+ 1198622(int
infin
0
119904minus2120572minus2120574
1198902(120596+120578)119904
119889119904)
1199022
times int
infin
0
119890119902120578119904
1198641003816100381610038161003816119883119904
1003816100381610038161003816119902
119862119889119904
(29)
This shows that |119884|119871119902
P(Ω119871119902
120578(119867))
is finite provided we assumethat 120578 lt 0 and 120596 + 120578 lt 0 so the map is well defined
If 1198831
sdot 119883
2
sdotare processes belonging to H119902
120578and 1198841 1198842 are
defined accordingly the entirely analogous passages showthat100381610038161003816100381610038161198841
minus 119884210038161003816100381610038161003816119871119902
P(Ω119871119902
120578(119867))
le 1198711198881119902
120572
100381610038161003816100381610038161198831
sdotminus 119883
2
sdot
10038161003816100381610038161003816119871119902
P(Ω119871119902
120578(C))
times (int
infin
0
119904minus2120572minus2120574
1198902(120596+120578)119904
119889119904)
12
(30)
Recalling the inequalities (23) and (25) and noting that themap Y rarr Φ
1015840
(Xsdot) is linear we obtain an explicit expression
for the constant 119888 in (17) and it is immediate to verify that119888 lt 1 provided 120578 lt 0 is chosen sufficiently large We fixsuch a value of 120578(119902) The first result is a consequence of thecontraction principle The estimate (15) also follows from thecontraction property ofΦ(sdot 119905 119909)
For investigating the dependence of the solution119883(119904 119905 119909)on the initial data 119909 and 119905 we reformulate (13) as an equationon [0infin) We set
119878 (119904) = 119890119904119860
for 119904 ge 0 119878 (119904) = 119868 for 119904 lt 0 (31)
and we consider the equation
119883(119904) = 119878 (119904 minus 119905) 119909 ((0 and (119904 minus 119905)) or (minus120591))
+ int
119904
0
119868[119905infin)
(120590) 119878 (119904 minus 120590) 119865 (120590119883120590) 119889120590
+ int
119904
0
119868[119905infin)
(120590) 119878 (119904 minus 120590)
times 119866 (120590119883120590) 119889119882 (120590) 119904 isin [0infin)
1198830(120579) = 119909 ((minus119905 + 120579) or (minus120591)) 120579 isin [minus120591 0]
(32)
Under the assumptions of Hypothesis 1 by Theorem 2 it iseasy to prove that equation (32) has a unique solution 119883 and119883sdotisin H
119902
120578(119902)for every 119902 isin [2infin) It clearly satisfies 119883(119904) =
119909((119904 minus 119905) or (minus120591)) for 119904 isin [minus120591 119905) and its restriction to the timeinternal [119905infin) is the unique mild solution of (10) From nowon we denote by119883(119904 119905 119909) 119904 isin [0infin) the solution of (32)
We need the following parameter-depending contractionprinciple which is stated in the following lemma and provedin [29 Theorems 101 and 102]
Lemma3 (ParameterDependingContraction Principle) Let119861119863 denote Banach spaces Let ℎ 119861times119863 rarr 119861 be a continuousmapping satisfying
1003816100381610038161003816ℎ (1199091 119910) minus ℎ (1199092 119910)1003816100381610038161003816 le 120572
10038161003816100381610038161199091 minus 11990921003816100381610038161003816 (33)
for some 120572 isin [0 1) and every 1199091 119909
2isin 119861 y isin 119863 Let 120601(119910)
denote the unique fixed point of the mapping ℎ(sdot 119910) 119861 rarr 119861Then 120601 119863 rarr 119861 is continuous
Theorem 4 Assume that Hypothesis 1 holds true Then forevery 119902 isin [1infin) the map (119905 119909) rarr 119883
sdot(119905 119909) is continuous
from [0infin) timesC toHq120578(q)
Proof Clearly it is enough to prove the claim for 119902 large Letus consider the map Φ defined in the proof of Theorem 2 In
6 Abstract and Applied Analysis
our present notationΦ can be seen as a mapping fromH119902
120578times
[0infin) timesC toH119902
120578as follows
Φ(119883sdot 119905 119909)
119904(119897) = 119878 (119904 + 119897 minus 119905) 119909 (0)
+ int
119904+119897
0
119868[119905infin)
(120590) 119878 (119904 + 119897 minus 120590) 119865 (120590119883120590) 119889120590
+ int
119904+119897
0
119868[119905infin)
(120590) 119878 (119904 + 119897 minus 120590)
times 119866 (120590119883120590) 119889119882 (120590)
119904 isin [0infin) 119897 isin [minus120591 0] 119904 + 119897 ge 119905
Φ(119883sdot 119905 119909)
119904(119897) = 119909 ((119904 + 119897 minus 119905) or (minus120591))
119904 isin [0infin) 119897 isin [minus120591 0] 119904 + 119897 le 119905
(34)
By the arguments of the proof of Theorem 2 Φ(sdot 119905 119909)is a contraction in H119902
120578uniformly with respect to 119905 119909
The process 119883sdot(119905 119909) is the unique fixed point of Φ(sdot 119905 119909)
So by the parameter-depending contraction principle(Lemma 3) it suffices to show that Φ is continuous fromH119902
120578times [0infin) times C to H119902
120578 From the contraction property
of Φ(sdot 119905 119909) mentioned earlier we have that Φ(sdot 119905 119909) iscontinuous uniformly in 119905 119909 Moreover for fixed 119883
sdot it is
easy to verify that Φ(119883sdot sdot sdot) is continuous from [0infin) times C
toH119902
120578 The proof is finished
Remark 5 By similar passages we can show that for fixed119905 Theorem 4 still holds true for 119902 large enough if the spaces[0infin) times C and H119902
120578are replaced by the spaces 119871119902(ΩCF
119905)
and H119902
120578(119905) respectively where 119871119902(ΩCF
119905) denotes that the
space of F119905-measurable function with value in C such that
the norm
|119909|119902
= 119864|119909|119902
119862 (35)
is finite
3 The Backward-Forward System
In this section we consider the system of stochastic differen-tial equations 119875-as
119883 (119904) = 119890(119904minus119905)119860
119909 (0) + int
119904
119905
119890(119904minus120590)119860
119865 (120590119883120590) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883120590) 119889119882 (120590) 119904 isin [119905infin)
119883119905= 119909 isin C
119884 (119904) minus 119884 (119879) + int
119879
119904
119885 (120590) 119889119882 (120590) + 120582int
119879
119904
119884 (120590) 119889120590
= int
119879
119904
120595 (120590119883120590 119884 (120590) 119885 (120590)) 119889120590 0 le 119904 le 119879 lt infin
(36)
for 119904 varying on the time interval [119905infin) sub [0infin) Asin Section 2 we extend the domain of the solution setting119883(119904 119905 119909) = 119909((119904 minus 119905) or (minus120591)) for 119904 isin [minus120591 119905)
We make the following assumptions
Hypothesis 2 Themapping 120595 [0infin)timesCtimes119870times1198712(Ξ 119870) rarr
119870 is Borelmeasurable such that for all 119905 isin [0infin)120595(119905 sdot) Ctimes119870 times 119871
2(Ξ 119870) rarr 119870 is continuous and for some 119871
119910 119871
119911gt 0
120583 isin 119877 and119898 ge 11003816100381610038161003816120595 (119904 119909 1199101 1199111) minus 120595 (119904 119909 1199102 1199112)
1003816100381610038161003816
le 119871119910
10038161003816100381610038161199101 minus 11991021003816100381610038161003816 + 119871119911
10038161003816100381610038161199111 minus 11991121003816100381610038161003816
1003816100381610038161003816120595 (119904 119909 119910 119911)1003816100381610038161003816 le 119871 (1 + |119909|
119898
119862+10038161003816100381610038161199101003816100381610038161003816 + |119911|)
⟨120595 (119904 119909 1199101 119911) minus 120595 (119904 119909 119910
2 119911) 119910
1minus 119910
2⟩119870ge 120583
10038161003816100381610038161199101 minus 119910210038161003816100381610038162
(37)
for every 119904 isin [0infin) 119909 isin C 119910 1199101 119910
2isin 119870 119911 119911
1 and 119911
2isin
1198712(Ξ 119870)We note that the third inequality in (37) follows from the
first one taking 120583 = minus119871119910but that the third inequalitymay also
hold for different values of 120583Firstly we consider the backward stochastic differential
equation
119884 (119904) minus 119884 (119879) + int
119879
119904
119885 (120590) 119889119882 (120590) + 120582int
119879
119904
119884 (120590) 119889120590
= int
119879
119904
120595 (120590119883120590 119884 (120590) 119885 (120590)) 119889120590 0 le 119904 le 119879 lt infin
(38)
119870 is a Hilbert space the mapping 120595 [0infin) times C times 119870 times
1198712(Ξ 119870) rarr 119870 is a given measurable function 119883
sdotis a
predictable process with values in another Banach space Cand 120582 is a real number
Theorem 6 Assume that Hypothesis 2 holds Let 119901 gt 2 and120575 lt 0 be given and choose
119902 ge 119898119901 120578 gt120575
119898 (39)
Then the following hold
(i) For 119883sdotisin 119871
119902
P(Ω 119871
119902
120578(C)) and 120582 gt minus(120575 + 120583 minus (119871
2
1199112))
(38) has a unique solution in Kp120575that will be denoted
by (119884(119883sdot)(119904) 119885(119883
sdot)(119904)) 119904 ge 0
(ii) The estimate
119864sup119904ge0
(119884 (119883sdot) (119904))
119901
119890119901120575119904
+ 119864(int
infin
0
11989021205751205901003816100381610038161003816119884(119883sdot
)(120590)10038161003816100381610038162
119889120590)
1199012
+ 119864(int
infin
0
11989021205751205901003816100381610038161003816119885 (119883sdot
) (120590)10038161003816100381610038162
119889120590)
1199012
le 119888(1 +1003816100381610038161003816119883sdot
1003816100381610038161003816119898
119871119902
P(Ω119871119902
120578(C))
)119901
(40)
Abstract and Applied Analysis 7
holds for a suitable constant 119888 In particular 119884(119883sdot) isin
119871119901
P(Ω 119862
120575(119870))
(iii) The map 119883sdotrarr (119884(119883
sdot) 119885(119883
sdot)) is continuous from
119871119902
P(Ω 119871
119902
120578(C)) toK119901
120575 and 119883
sdotrarr 119884(119883
sdot) is continuous
from 119871119902
P(Ω 119871
119902
120578(C)) to 119871119901
P(Ω 119862
120575(119870))
(iv) The statements of points (i) (ii) and (iii) still holdtrue if the space 119871119902
P(Ω 119871
119902
120578(C)) is replaced by the space
119871119902
P(Ω 119862
120578(C))
Proof The theorem is very similar to Proposition 311 in [24]The only minor difference is that the mapping 120595 [0infin) times
Ctimes119870times1198712(Ξ 119870) rarr 119870 is a givenmeasurable function while
in [24] the measurable function 120595 is from119867 times119870 times 1198712(Ξ 119870)
to 119870 however the same arguments apply
Theorem 7 Assume that Hypothesis 1 holds and thatHypothesis 2 holds true in the particular case 119870 = 119877 Thenfor every 119901 gt 2 119902 120575 lt 0 satisfying (39) with 120578 = 120578(119902)and for every 120582 gt 120582
1015840
= minus(120575 + 120583 minus (1198712
1199112)) there exists a
unique solution in H119902
120578(119902)times K
119901
120575of (36) that will be denoted
by (119883(sdot 119905 119909) 119884(sdot 119905 119909) 119885(sdot 119905 119909)) Moreover 119884(sdot 119905 119909) isin
119871119901
P(Ω 119862
120575(119877)) The map (119905 119909) rarr (119884(sdot 119905 119909) 119885(sdot 119905 119909)) is con-
tinuous from [0infin)timesC toK119901
120575 and themap (119905 119909) rarr 119884(sdot 119905 119909)
is continuous from [0infin) timesC to 119871119901P(Ω 119862
120575(119877))
Proof We first notice that the system is decoupled the firstdoes not contain the solution (119884 119885) of the second oneThere-fore under the assumption of Hypothesis 1 by Theorem 2there exists a unique solution 119883(sdot 119905 119909) and 119883
sdot(119905 119909) isin H
119902
120578(119902)
of the first equation Moreover from Theorem 4 it followsthat the map (119905 119909) rarr 119883
sdot(119905 119909) is continuous from [0infin)timesC
toH119902
120578(119902)
Let 119870 = 119877 from Theorem 6 we have that thereexists a unique solution (119884(sdot 119905 119909) 119885(sdot 119905 119909)) isin K
119901
120575of the
second equation and the map 119883sdotrarr (119884(119883
sdot) 119885(119883
sdot)) is
continuous from H119902
120578(119902)to K
119901
120575while X
sdotrarr (Y(X
sdot)) is
continuous fromH119902
120578(119902)to119871119901
P(Ω 119862
120575(119877))We have proved that
(119883(sdot 119905 119909) 119884(sdot 119905 119909) 119885(sdot 119905 119909)) isin H119902
120578(119902)times K
119901
120575is the unique
solution of (36) and the other assertions follow from com-position
Remark 8 From Remark 5 by similar passages we can showthat for fixed 119905 and for 119902 large enough under the assumptionsof Theorem 7 the map 119909 rarr (119884(sdot 119905 119909) 119885(sdot 119905 119909)) is continu-ous from 119871
119902
(ΩCF119905) toK119901
120575(119905)
We also remark that the process 119883(sdot 119905 119909) is F[119905infin)
measurable since C is separable Banach space we have that119883sdot(119905 119909) is F
[119905infin)measurable So that 119884(119905) is measurable
with respect to both F[119905infin)
and F119905 it follows that 119884(119905) is
deterministicFor later use we notice three useful identities for 119905 le 119904 lt
infin the equality 119875-as
119883119897(119904 119883
119904(119905 119909)) = 119883
119897(119905 119909) 119897 isin [119904infin) (41)
is a consequence of the uniqueness of the solution of (13)Since the solution of the backward equation is uniquely
determined on an interval [119904infin) by the values of the process119883sdoton the same interval for 119905 le 119904 lt infin we have 119875-as
119884 (119897 119904 119883119904(119905 119909)) = 119884 (119897 119905 119909) for 119897 isin [119904infin)
119885 (119897 119904 119883119904(119905 119909)) = 119885 (119897 119905 119909) for aa 119897 isin [119904infin)
(42)
Lemma 9 (see [30]) Let 119864 be a metric space with metric 119889and let 119891 Ω rarr 119864 be strongly measurable Then thereexists a sequence 119891
119899 119899 isin 119873 of simple 119864-valued functions
(ie 119891119899isFB(E)measurable and takes only a finite number
of values) such that for arbitrary 120596 isin Ω the sequence119889(119891
119899(120596) 119891(120596)) 119899 isin 119873 is monotonically decreasing to zero
Let now 119891 isin 119871119902
(ΩC) By Lemma 9 we get the existenceof a sequence of simple function 119891
119899 119899 isin 119873 such that
1003816100381610038161003816119891119899 (120596) minus 119891 (120596)1003816100381610038161003816 darr 0 for all 120596 isin Ω as 119899 rarr infin (43)
Hence 119891119899
rarr 119891 in | sdot |119871119902(ΩC) by Lebesguersquos dominated
convergence theoremWe are now in a position of showing the main result in
this section
Theorem 10 Assume that Hypothesis 1 holds true and thatHypothesis 2 holds in the particular case 119870 = 119877 Then thereexist two Borel measurable deterministic functions 120592 [tinfin) times
C rarr 119877 and 120577 [119905infin) times C rarr Ξlowast
= 119871(Ξ 119877) =
1198712(Ξ 119877) such that for 119905 isin [0infin) and x isin C the solution
(119883(119905 119909) 119884(119905 119909) 119885(119905 119909)) of (36) satisfies
119884 (119904 119905 119909) = 120592 (119904 119883119904(119905 119909)) 119885 (119904 119905 119909) = 120577 (119904 119883
119904(119905 119909))
119875-as for aa 119904 isin [119905infin)
(44)
Proof We apply the techniques introduced in [26 Proposi-tion 32] Let 119890
119894 be a basis of Ξlowast and let us define 119885119894119873
=
((119885 119890119894)Ξlowast and119873) or (minus119873) Then for every 0 le 119905
1lt 119905
2lt infin Δ gt
0 and 1199091 119909
2isin C we have that
100381610038161003816100381610038161003816100381610038161003816
119864 int
1199051+Δ
1199051
119885119894119873
(119904 1199051 119909
1) 119889119904 minus 119864int
1199052+Δ
1199052
119885119894119873
(119904 1199052 119909
2) 119889119904
100381610038161003816100381610038161003816100381610038161003816
le 119864int
1199052
1199051
10038161003816100381610038161003816119885119894119873
(119904 1199051 119909
1)10038161003816100381610038161003816119889119904
+ 119864int
1199051+Δ
1199052
10038161003816100381610038161003816119885119894119873
(119904 1199051 119909
1) minus 119885
119894119873
(119904 1199052 119909
2)10038161003816100381610038161003816119889119904
+ 119864int
1199052+Δ
1199051+Δ
10038161003816100381610038161003816119885119894119873
(119904 1199052 119909
2)10038161003816100381610038161003816119889119904
8 Abstract and Applied Analysis
le 211987310038161003816100381610038161199052 minus 1199051
1003816100381610038161003816 + Δ12
119890minus120575(1199051+Δ)
times(119864(int
infin
0
119890212057511990410038161003816100381610038161003816119885119894119873
(119904 1199051 119909
1)minus119885
119894119873
(119904 1199052 119909
2)10038161003816100381610038161003816
2
119889119904)
1199012
)
1119901
le 211987310038161003816100381610038161199052 minus 1199051
1003816100381610038161003816 + Δ12
119890minus120575(1199051+Δ)
times (119864(int
infin
0
11989021205751199041003816100381610038161003816119885 (119904 1199051 1199091)minus119885 (119904 1199052 1199092)
10038161003816100381610038162
119889119904)
1199012
)
1119901
(45)
From Theorem 7 we have that the map (119905 119909) rarr int119905+Δ
119905
119885119894119873
(119904 119905 119909)119889119904 is continuous from [0infin) times C to 119877 ByRemark 8 we also have that for fixed 119905 the map 119909 rarr
119864int119905+Δ
119905
119864119885119894119873
(119904 119905 119909)119889119904 is continuous from 119871119902
(ΩCF119905) to 119877
for 119902 large enough Let us define
120577119894119873
(119905 119909) = lim inf119899rarrinfin
119899119864int
119905+(1119899)
119905
119885119894119873
(119904 119905 119909) 119889119904
119905 isin [0infin) 119909 isin C
(46)
It is clear that 120577119894119873 [0infin) timesC rarr 119877 is a Borel functionWe fix 119909 and 0 le 119905 le 119904 lt infin For 119897 isin [119904infin) we
denote 119864[119885119894119873
(119897 119904 119910)]|119910=119883119904(119905119909)
the random variable obtainedby composing119883
119904(119905 119909) with the map 119910 rarr 119864[119885
119894119873
(119897 119904 119910)]By Lemma 9 there exists a sequence of C-valued F
119904-
measurable simple functions
119891119898 Ω 997888rarr C 119891
119898=
119873119898
sum
119896=1
ℎ(119898)
119896119868119891119898=ℎ(119898)
119896 119873
119898isin 119873 (47)
where ℎ(119898)1 ℎ
(119898)
119898are pairwise distinct andΩ = ⋃
119873119898
119896=1119891
119898=
ℎ(119898)
119896 such that
1003816100381610038161003816119891119898 (120596) minus 119883119904(120596)
1003816100381610038161003816 darr 0 for all 120596 isin Ω as 119899 997888rarr infin (48)
For any 119861 isin F119904 we have
int119861
int
119904+(1119899)
119904
119885119894119873
(119897 119905 119909) 119889119897119889119875
= int119861
int
119904+(1119899)
119904
119885119894119873
(119897 119904 119883119904) 119889119897119889119875
= 119864119868119861int
119904+(1119899)
119904
119885119894119873
(119897 119904 119883119904) 119889119897
= lim119898rarrinfin
119864(119868119861int
119904+(1119899)
119904
119885119894119873
(119897 119904 119891119898) 119889119897)
= lim119898rarrinfin
119873119898
sum
119896=1
119864(119868119861119868119891119898=ℎ(119898)
119896int
119904+(1119899)
119904
119885119894119873
(119897 119904 ℎ(119898)
119896) 119889119897)
= lim119898rarrinfin
119864(119868119861
119873119898
sum
119896=1
119868119891119898=ℎ(119898)
119896)119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 ℎ(119898)
119896) 119889119897
= lim119898rarrinfin
119864119868119861(119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 119910) 119889119897|119910=119891119898
)
= lim119898rarrinfin
int119861
(119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 119910) 119889119897|119910=119891119898
)119889119875
= int119861
(119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 119910) 119889119897|119910=119883119904
)119889119875
(49)
and we get that
120577119894119873
(119904 119883119904(119905 119909)) = lim inf
119899rarrinfin
119899
times [119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 119910) 119889119897|119910=119883119904(119905119909)
]
= lim inf119899rarrinfin
119899119864[int
119904+(1119899)
119904
119885119894119873
(119897 119905 119909) 119889119897
100381610038161003816100381610038161003816100381610038161003816
F119904]
119875-as(50)
Fix 119905 and 119909 Recalling that |119885119894119873
| le 119873 by the Lebesguetheorem on differentiation it follows that 119875-as
lim119899rarrinfin
119899int
119904+(1119899)
119904
119885119894119873
(119897 119905 119909) 119889119897 = 119885119894119873
(119904 119905 119909)
for aa 119904 isin [119905infin)
(51)
By the boundedness of 119885119894119873 applying the dominated conver-gence theorem we get that
120577119894119873
(119904 119883119904(119905 119909)) = 119864 [119885
119894119873
(119904 119905 119909)10038161003816100381610038161003816F
119904] = 119885
119894119873
(119904 119905 119909)
119875-as for aa 119904 isin [119905infin)
(52)
Now we have proved that for every 119905 119909
120577119894119873
(119904 119883119904(119905 119909)) = 119885
119894119873
(119904 119905 119909)
119875-as for aa 119904 isin [119905infin)
(53)
for every 119894 119873 Let 119862 sub [0infin) times C denote the set ofpairs (119905 119909) such that lim
119873rarrinfin120577119894119873
(119905 119909) exists and the seriessuminfin
119894=1(lim
119873rarrinfin120577119894119873
(119905 119909))119890119894converges in Ξlowast We define
120577 (119905 119909) =
infin
sum
119894=1
( lim119873rarrinfin
120577119894119873
(119905 119909)) 119890119894 (119905 119909) isin 119862
120577 (119905 119909) = 0 (119905 119909) notin 119862
(54)
Abstract and Applied Analysis 9
Since 119885 satisfies
119885 (120596 119904 119905 119909) =
infin
sum
119894=1
( lim119873rarrinfin
119885119894119873
(120596 119904 119905 119909)) 119890119894 (55)
for every 120596 119904 119905 119909 From (53) it follows that for every 119905 119909 wehave (119904 119883
119904(120596 119905 119909)) isin 119862 119875-as for almost all 119904 isin [119905infin) and
119885(119904 119905 119909) = 120577(119904 119883119904(119905 119909)) 119875-as for aa 119904 isin [119905infin)
We define 120592(119905 119909) = 119884(119905 119905 119909) since 119884(119905 119905 119909) is deter-ministic so the map (119905 119909) rarr 120592(119905 119909) can be written as acomposition 120592(119905 119909) = Γ
3(Γ2(119905 Γ
1(119905 119909))) with
Γ1 [0infin) timesC 997888rarr 119871
119901
P(Ω 119862
120575(119877))
Γ1(119905 119909) = 119884 (sdot 119905 119909)
Γ2 [0infin) times 119871
119901
P(Ω 119862
120575(119877)) 997888rarr 119871
119901
(Ω 119877)
Γ2(119905 119881) = 119881 (119905)
Γ3 119871
119901
(Ω 119877) 997888rarr 119877 Γ3120585 = 119864120585
(56)
FromTheorem 7 it follows that Γ1is continuous By
|119881(119905) minus 119880(119904)|119871119901(Ω119877)
le |119881 (119905) minus 119881 (119904)|119871119901(Ω119877)
+ 119890minus120575119901119904
|119881 minus 119880||119871
119901
P(Ω119862120575(119877))
(57)
we have that Γ2is continuous It is clear that Γ
3is continuous
Then themap (119905 119909) rarr 120592(119905 119909) is continuous from [0infin)timesCto 119877 therefore 120592(119905 119909)is a Borel measurable function Fromuniqueness of the solution of (36) it follows that 119884(119904 119905 119909) =120592(119904 119883
119904(119905 119909)) 119875-as for aa 119904 isin [119905infin)
4 The Fundamental Relation
Let (ΩF 119875) be a given complete probability space with afiltration F
119905119905ge0
satisfying the usual conditions 119882(119905) 119905 ge 0
is a cylindrical Wiener process in Ξ with respect to F119905119905ge0
We will say that an F
119905ge0-predictable process 119906 with values
in a given measurable space (119880U) is an admissible controlThe function 119877 [0infin) times C times 119880 rarr Ξ is measurableand bounded We consider the following controlled stateequation
119889119883119906
(119904) = 119860119883119906
(119904) 119889119904 + 119865 (119904 119883119906
119904) 119889119904
+ 119866 (119904 119883119906
119904) 119877 (119904 119883
119906
119904 119906 (119904)) 119889119904 + 119866 (119904 119883
119906
119904) 119889119882 (119904)
119904 isin [119905infin)
119883119906
119905= 119909
(58)
Here we assume that there exists a mild solution of (58)which will be denoted by 119883119906
(119904 119905 119909) or simply by 119883119906
(119904) Weconsider a cost function of the form
119869 (119906) = 119864int
infin
119905
119890minus120582119904
119892 (119904 119883119906
119904 119906 (119904)) 119889119904 (59)
Here 119892 is function on [0infin) times C times 119880 with real values Ourpurpose is to minimize the function 119869 over all admissiblecontrols
We define in a classical way the Hamiltonian functionrelative to the previous problem for all 119905 isin [0infin) 119909 isin
C and 119911 isin Ξlowast
120595 (119905 119909 119911) = inf 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) 119906 isin 119880
(60)
and the corresponding possibly empty set of minimizers
Γ (119905 119909 119911) = 119906 isin 119880 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) = 120595 (119905 119909 119911)
(61)
We are now ready to formulate the assumptions we need
Hypothesis 3 (i) 119860 119865 and G verify Hypothesis 1(ii) (119880U) is ameasurable spaceThemap 119892 [0infin)timesCtimes
119880 rarr 119877 is continuous and satisfies |119892(119905 119909 119906)| le 119870119892(1+|119909|
119898119892
119862)
for suitable constants 119870119892gt 0 119898
119892gt 0 and all 119909 isin C119906 isin
119880 The map 119877 [0infin) times C times 119880 rarr Ξ is measurable and|119877(119905 119904 119906)| le 119871
119877for a suitable constant 119870
119877gt 0 and all 119909 isin
C119906 isin 119880 and119911 isin Ξlowast
(iii)TheHamiltonian120595 defined in (60) satisfies the requi-rements of Hypothesis 2 (with119870 = 119877)
(iv) We fix here 119901 gt 2 q and 120575 lt 0 satisfying (39) with120578 = 120578(119902) and such that 119902 gt 119898
119892
We are in a position to prove the main result of thissection
Theorem11 Assume thatHypothesis 3 holds and suppose that120582 verifies
120582 gt (minus120575 minus 120583 +1198712
119911
2) or (minus120575 +
1198712
119877
2 (119901 minus 1))
or (1198712
119877119898119892
2 (119902 minus 119898119892)minus 120578 (119902)119898
119892)
(62)
Let 120592 120577 denote the function in the statement of Theorem 10Then for every admissible control 119906 and for the correspondingtrajectory119883 starting at (119905 119909) one has
119869 (119906) = 120592 (119905 119909) + 119864int
infin
119905
119890minus120582119904
[minus120595 (119904 119883119906
119904 120577 (119904 119883
119906
119904)) + 120577 (119904 119883
119906
119904)
times 119877 (119904 119883119906
119904 119906 (119904)) + 119892 (119904 119883
119906
119904 119906 (119904))] 119889119904
(63)
Proof Consider (58) in the probability space (ΩF 119875) withfiltration F
119905119905ge0
and with an F119905119905ge0
-cylindrical Wienerprocess 119882(119905) 119905 ge 0 Let us define
119882119906
(119904) = 119882 (119904) + int
119904
119905and119904
119877 (120590119883119906
120590 119906 (120590)) 119889120590 119904 isin [0infin)
120588 (119879) = exp(int119879
119905
minus119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882 (119904)
minus1
2int
119879
119905
1003816100381610038161003816119877 (119904 119883119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)
(64)
10 Abstract and Applied Analysis
Let 119875119906 be the unique probability onF[0infin)
such that
119875119906
|F119879
= 120588 (119879) 119875|F119879
(65)
We notice that under 119875119906 the process119882119906 is aWiener processLet us denote by F119906
119905119905ge0
the filtration generated by119882119906 andcompleted in the usual way Relatively to 119882119906 (58) can berewritten as
119889119883119906
(119904) = 119860119883119906
(119904) 119889119904 + 119865 (119904 119883119906
119904) 119889119904
+ 119866 (119904 119883119906
119904) 119889119882
119906
(119904) 119904 isin [119905infin)
119883119906
119905= 119909
(66)
In the space (ΩF[0infin)
F119906
119905119905ge0 119875
119906
) we consider the follow-ing system of forward-backward equations
119883119906
(119904) = 119890(119904minus119905)119860
119909 (0) + int
119904
119905
119890(119904minus120590)119860
119865 (120590119883119906
120590) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883119906
120590) 119889119882
119906
(120590) 119904 isin [119905infin)
119883119906
119905= 119909 isin C
119884119906
(119904) minus 119884119906
(119879) + int
119879
119904
119885119906
(120590) 119889119882119906
(120590) + 120582int
119879
119904
119884119906
(120590) 119889120590
= int
119879
119904
120595 (120590119883119906
120590 119885
119906
(120590)) 119889120590 0 le 119904 le 119879 lt infin
(67)
Applying the Ito formula to 119890minus120582119904119884119906(119904) and writing the back-ward equation in (67) with respect to the process119882 we get
119884119906
(119904) + int
119879
119904
119890minus120582120590
119885119906
(120590) 119889119882 (120590)
= int
119879
119904
119890minus120582120590
[120595 (120590119883119906
120590 119885
119906
(120590))
minus119885119906
(120590) 119877 (120590119883119906
120590 119906 (120590))] 119889120590
+ 119890minus120582119879
119884119906
(119879)
(68)
Recalling that 119877 is bounded we get for all 119903 ge 1 and someconstant 119862
119864119906
[120588(119879)minus119903
] = 119864119906
[exp 119903 (int119879
119905
119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882
119906
(119904)
minus1
2int
119879
119905
1003816100381610038161003816119877 (119904 119883119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)]
= 119864119906
[exp(int119879
119905
119903119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882
119906
(119904)
minus1
2int
119879
119905
11990321003816100381610038161003816119877 (119904 119883
119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)
times exp 119903 (119903 minus 1)2
int
119879
119905
1003816100381610038161003816119877 (119904 119883119906
119904 119906 (119904))
10038161003816100381610038162
119889119904]
le 119890(12)119903(119903minus1)119879119871
2
119877119864119906
times exp(int119879
119905
2119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882
119906
(119904)
minus1
2int
119879
119905
41003816100381610038161003816119877 (119904 119883
119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)
= 119890(12)119903(119903minus1)119879119871
2
119877
(69)
It follows that
119864(int
119879
119905
|119890minus120582119904
119885119906
(119904)|2
119889119904)
12
= 119864119906
[(int
119879
119905
10038161003816100381610038161003816119890minus120582119904
119885119906
(119904)10038161003816100381610038161003816
2
119889119904)
12
120588minus1
]
le (119864119906
int
119879
119905
10038161003816100381610038161003816119890minus120582119904
119885119906
(119904)10038161003816100381610038161003816
2
119889119904)
12
times (119864119906
120588minus2
)12
lt infin
(70)
We conclude that the stochastic integral in (68) has zeroexpectation If we set 119904 = 119905 in (68) and we take expectationwith respect to 119875 we obtain
119890minus120582119879
119864119884119906
(119879) minus 119884119906
(119905)
= 119864int
119879
119905
119890minus120582120590
[minus120595 (120590119883119906
120590 119885
119906
(120590))
+119885119906
(120590) 119877 (120590119883119906
120590 119906 (120590))] 119889120590
(71)
ByTheorem 7 119884119906(sdot 119905 119909) isin 119871119901P(Ω 119862
120575(119877)) so that
119864119906
|119884(119879 119905 119909)|119901
le 119862 exp (minus119901120575119879) (72)
By the Holder inequality we have that for suitable constant119862 gt 0
119864 |119884 (119879 119905 119909)| = 119864119906
(120588minus1
(119879) |119884 (119879 119905 119909)|)
le 119864(120588minus119901(119901minus1)
)(119901minus1)119901
119864(|119884 (119879 119905 119909)|119901
)1119901
le 119862119890((1198712
1198772(119901minus1))minus120575))119879
(73)
From Theorem 2 we obtain 119864119906sup119904ge119905119890120578119902119904
|119883119906
119904|119902
lt infin by thesimilar process we get that
1198641003816100381610038161003816119883
119906
119879
1003816100381610038161003816119898119892
le 119862119890(1198712
119877119898119892(2119902minus2119898
119892)minus1
minus120578(119902)119898119892)119879
(74)
Abstract and Applied Analysis 11
for suitable constant 119862 gt 0 and
119864int
infin
119905
119890minus120582120590 1003816100381610038161003816119892 (120590119883
119906
120590 119906 (120590))
1003816100381610038161003816 119889120590 lt infin (75)
Since 119884119906(119905 119905 119909) = 120592(119905 119909) and 119885119906
(119904 119905 119909) = 120577(119904 119883119906
119904(119905 119909)) 119875-
as for aa 119904 isin [119905infin) we have that
119890minus120582119879
119864119884119906
(119879) minus 119907 (119905 119909)
= 119864int
119879
119905
119890minus120582120590
[minus120595 (120590119883119906
120590 120577 (120590 119883
119906
120590))
+120577 (120590119883119906
120590) 119877 (120590119883
119906
120590 119906 (120590))] 119889120590
(76)
Thus adding and subtracting119864intinfin119905
119890minus120582120590
119892(120590119883119906
120590 119906(120590))119889120590 and
letting 119879 rarr infin we conclude that
119869 (119906) = 120592 (119905 119909)
+ 119864int
infin
119905
119890minus120582119904
[minus120595 (119904 119883119906
119904 120577 (119904 119883
119906
119904)) + 120577 (119904 119883
119906
119904) 119877
times (119904 119883119906
119904 119906 (119904)) + 119892 (119904 119883
119906
119904 119906 (119904))] 119889119904
(77)
The proof is finished
We immediately deduce the following consequences
Theorem 12 Let 119905 isin [0infin) and 119909 isin C be fixed assumethat the set-valued map Γ has nonempty values and it admitsa measurable selection Γ
0 [0infin)timesCtimesΞlowast rarr 119880 and assume
that a control 119906(sdot) satisfies
119906 (119904) = Γ0(119904 119883
119906
119904 120577 (119904 119883
119906
119904))
119875-as for almost every 119904 isin [119905infin)
(78)
Then 119869(119905 119909 119906) = 120592(119905 119909) and the pair (119906(sdot) 119883) is optimal forthe control problem starting from 119909 at time 119905
Such a control can be shown to exist if there exists a solutionfor the so-called closed-loop equation as follows
119889119883 (119904)=119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904+119866 (119904 119883
119904)
times(119877 (119904 119883119904 Γ
0(119904 119883
119904 120577 (119904 119883
119904))) 119889119904 + 119889119882 (119904))
119904 isin [119905infin)
119883119905= 119909
(79)
since in this case we can define an optimal control setting
119906 (119904) = Γ0(119904 119883
119904 120577 (119904 119883
119904)) (80)
However under the present assumptions we cannot guaranteethat the closed-loop equation has a solution in the mildsense To circumvent this difficulty we will revert to a weakformulation of the optimal control problem
5 Existence of Optimal Control
We formulate the optimal control problem in the weak sensefollowing the approach of [31]Themain advantage is that wewill be able to solve the closed-loop equation in a weak senseand hence to find an optimal control even if the feedbacklaw is nonsmooth
We call (ΩF F119905119905ge0 119875119882) an admissible setup if
(ΩF F119905119905ge0 119875) is a filtered probability space satisfying the
usual conditions and 119882 is a cylindrical 119875-Wiener processwith values in Ξ with respect to the filtration F
119905119905ge0
By an admissible control system we mean (ΩF
F119905119905ge0 119875119882 119906119883
119906
) where (ΩF F119905119905ge0 119875 119882) is an
admissible setup 119906 is an F119905-predictable process with values
in119880 and119883119906 is a mild solution of (58) An admissible controlsystem will be briefly denoted by (119882 119906119883119906
) in the followingOur purpose is to minimize the cost functional
119869 (119906) = 119864int
infin
119905
119890minus120582119904
119892 (119904 119883119906
119904 119906 (119904)) 119889119904 (81)
over all the admissible control systemOur main result in this section is based on the solvability
of the closed-loop equation
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904 + 119866 (119904 119883
119904)
times (119877 (119904 119883119904 Γ
0(119904 119883
119904 120577 (119904 119883
119904))) 119889119904 +119889119882 (119904))
119904 isin [119905infin)
119883119905= 119909
(82)
In the following sense we say that 119883 is a weak solution of(82) if there exists an admissible setup (ΩF F
119905119905ge0 119875119882)
and anF119905-adapted continuous process119883(119905)with values in119867
which solves the equation in the mild sense namely 119875-as
119883 (119904) = 119890(119904minus119905)119860
119909 (0) + int
119904
119905
119890(119904minus120590)119860
119865 (120590119883120590) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883120590) 119877
times (120590119883120590 Γ
0(120590119883
120590 120577 (120590 119883
120590))) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883120590) 119889119882
120590 119904 isin [119905infin)
(83)
119883119905= 119909 (84)
Theorem 13 Assume that Hypothesis 3 holds Then thereexists a weak solution of the closed-loop equation (82) whichis unique in law
12 Abstract and Applied Analysis
Proof (uniqueness) Let 119883 be a weak solution of (82) in anadmissible setup (ΩF F
119905119905ge0 119875119882)We define
120588 (119879) = exp(int119879
119905
minus119877lowast
(120590 119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889119882 (120590)
minus1
2int
119879
119905
1003816100381610038161003816119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)))10038161003816100381610038162
119889120590)
(85)
Since 119877 is bounded the Girsanov theorem ensures that thereexists a probability measure 1198750 such that the process
1198820
(119904) = 119882 (119904) + int
119904
119905and119904
119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889120590
119904 isin [0infin)
(86)
is a 1198750-Wiener process and
1198750
|F119879
= 120588 (119879) 119875|F119879
(87)
Let us denote by F0
119905119905ge0
the filtration generated by1198820 andcompleted in the usual way In (ΩF
[0infin) F0
119905119905ge0 119875
0
) 119883 isa mild solution of
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119905 119883119904) 119889119904
+ 119866 (119904 119883119904) 119889119882
0
(119904) 119904 isin [119905infin)
119883119905= 119909
120588 (119879) = exp(int119879
119905
minus119877lowast
(120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)) 119889119882
0
(120590)
+ 12int
119879
119905
1003816100381610038161003816119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)))10038161003816100381610038162
119889120590)
(88)
By Hypothesis 3 the joint law of 119883 and 1198820 is uniquely
determined by 119860 119865 119866 and 119909 Taking into account the lastdisplayed formula we conclude that the joint law of 119883 and120588(119879) under1198750 is also uniquely determined and consequentlyso is the law of 119883 under 119875 This completes the proof of theuniqueness part
Proof (existence) Let (ΩF 119875) be a given complete probabil-ity space 119882(119905) 119905 ge 0 is a cylindrical Wiener process on(ΩF 119875)with values inΞ and F
119905119905ge0
is the natural filtrationof 119882(119905) 119905 ge 0 augmented with the family of119875-null sets Let119883(sdot) be the mild solution of
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904
+ 119866 (119904 119883119904) 119889119882 (119904) 119904 isin [119905infin)
119883119905= 119909
(89)
and by the Girsanov theorem let 1198751 be the probability on Ωunder which
1198821
(119904) = 119882 (119904) minus int
119904
119905and119904
119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889120590
(90)
is aWiener process (notice that 119877 is bounded)Then119883 is theweak solution of (82) relatively to the probability 1198751 and theWiener process1198821
Now we can state the main result of this section
Corollary 14 Assume that Hypothesis 3 holds true and 120582
verifies (62) Also assume that the set-valued map Γ hasnonempty values and it admits a measurable selection Γ
0
[0infin) timesC times Ξlowast
rarr 119880 Then for every 119905 isin [0infin) and x isin Cand for all admissible control system (119882 119906119883
119906
) one has
119869 (119906 119905 119909) ge 120592 (119905 119909) (91)
and the equality holds if
119906 (119904) = Γ0(119904 119883
119906
119904 120577 (119904 119883
119906
119904))
119875 minus 119886119904 119891119900119903 119886119897119898119900119904119905 119890119907119890119903119910 119904 isin [119905infin)
(92)
Moreover from Theorem 13 it follows that the closed-loop equation (82) admits a weak solution (ΩFF
119905119905ge0 119875119882119883) which is unique in law and setting
119906 (119904) = Γ0(119904 119883
119904 120577 (119904 119883
119904)) (93)
we obtain an optimal admissible control system (119882 119906119883)
6 Applications
In this section we present a simple application of the previousresults We consider the stochastic delay partial differentialequation in the bounded domain 119861 sub 119877
119899 with smoothboundary 120597119861 as follows
119889119911119906
(119905 120585) = Δ119911119906
(119905 120585) 119889119905 + 119891 (119905 119911119906
119905(120585)) 119889119905
+
119889
sum
119894=1
119892119894(119905 119911
119906
119905(120585)) [119903
119894
(120585) 119906119894
(119905) 119889119905 + 119889119882119894
(119905)]
119911119906
0(120579 120585) = 119909 (120579 120585) 120585 isin 119861 120579 isin [minus1 0]
119911119906
(119905 120585) = 0 119905 isin [0infin) 120585 isin 120597119861
(94)
Here119882 = (1198821
1198822
119882119889
) is a standardWiener process in119877119889 and the functions 119891 [0 +infin) times 119862([minus1 0] 119877) rarr 119877 and119892119894 [0 +infin) times 119862([minus1 0] 119877) rarr 119877 are Lipschitz continuous
and bounded Setting 119880 as a bounded subset of 119877119889 Ξ = 119877119889
119867 = 1198712
(119861) and 119909 isin 119862([minus1 0]119867) We define 119865 and 119866 asfollowing
119865 (119905 119909) (120585) = 119891 (119905 119909 (120585))
(119866 (119905 119909) 119911) (120585) =
119889
sum
119894=1
119892119894(119905 119909 (120585)) 119911
119894
(120585)
120585 isin 119861 119909 isin 119862 ([minus1 0] 119867) 119911 isin 119871 (Ξ119867)
(95)
and let 119860 denote the Laplace operator Δ in 1198712
(119861) withdomain11988222
(119861)⋂11988212
0(119861) then (94) has the form (58) and
Hypothesis 1 holds
Abstract and Applied Analysis 13
Let us consider the optimal control problem associatedwith the cost
119869 (119906) = 119864int
infin
0
119890minus120582119905
[int119861
120590 (120585 119911119906
119905(120585)) 119889120585 + 119906
2
(119905)] 119889119905 (96)
where 120582 verifies (62) and 120590 119862([minus1 0] 119877) times 119880 rarr [0infin) isa bounded measurable function Define 119892 119862([minus1 0]119867) times
119880 rarr [0infin) and 119877 119862([minus1 0]119867) times 119880 rarr Ξ by119892(119910 119906) = int
119861
120590(119905 119910(120585) 119906)119889120585 + 1199062 and 119877(119910 119906) = (int
119861
1199031
(120585)1199061
119889120585
int119861
1199032
(120585)1199062
119889120585 int119861
119903119889
(120585)119906119889
119889120585) for 119910 isin 119862([minus1 0]119867) 119906 =
(1199061
1199062
119906119889
) isin 119880 respectively It can be easily verifiedthat Hypothesis 3 holds true and the set-valued map Γ hasnonempty values and it admits a measurable selection Γ
0
[0infin) times C times Ξlowast
rarr 119880 Then the closed-loop equation(82) admits a weak solution (ΩF F
119905119905ge0 119875119882 119906 119911
sdot(sdot)) and
setting
119906 (119904) = Γ0(119904 119911
119904(sdot) 120577 (119904 119911
119904(sdot))) (97)
we obtain an optimal admissible control system (119882 119906 119911(sdot))
References
[1] J Bismut ldquoOn optimal control of linear stochastic equationswith a linear-quadratic criterionrdquo SIAM Journal on Control andOptimization vol 15 no 3 pp 1ndash4 1977
[2] N Nagase ldquoOn the existence of optimal control for controlledstochastic partial differential equationsrdquo Nagoya MathematicsJournal vol 115 pp 73ndash85 1989
[3] N El Karoui D Huu Nguyen and M Jeanblanc-Pique ldquoCom-pactification methods in the control of degenerate diffusionsrdquoStochastics vol 20 pp 169ndash219 1987
[4] M Nisio ldquoOptimal control for stochastic partial differentialequations and viscosity solutions of Bellman equationsrdquoNagoyaMathematics Journal vol 123 pp 13ndash37 1991
[5] M Nisio ldquoOn sensitive control for stochastic partial differentialequationsrdquo in Stochastic Analysis on Infinite Dimensional SpacesProceedings of the US Japan Bilateral Seminar H Kunita et alEd vol 310 of Pitman Research Notes Mathematical Series pp231ndash241 Longman Scientific and Technical Baton Rouge LaUSA January 1994
[6] R Buckdahn and A Rascanu ldquoOn the existence of stochasticoptimal control of distributed state systemrdquoNonlinear AnalysisTheory Methods and Applications vol 52 no 4 pp 1153ndash11842003
[7] V Barbu and G Da Prato Equations in Hilbert Spaces vol 86 ofPitman Research Notes in Mathematics Pitman 1983
[8] P Cannarsa and G Da Prato ldquoSecond-order Hamilton-Jacobiequations in infinite dimensionsrdquo SIAM Journal on Control andOptimization vol 29 no 2 pp 474ndash492 1991
[9] P Cannarsa and G Da Prato ldquoDirect solution of a second-orderHamilton-Jacobi equations in Hilbert spacesrdquo in StochasticPartial Differential Equations and Applications G Da Pratoand L Tubaro Eds vol 268 of Pitman Research Notes inMathematics Pitman 1992
[10] F Gozzi ldquoRegularity of solutions of second order Hamilton-Jacobi equations and application to a control problemrdquoCommu-nications in Partial Differential Equations vol 20 pp 775ndash8261995
[11] F Gozzi ldquoGlobal regular solutions of second order Hamilton-Jacobi equations in Hilbert spaces with locally Lipschitz non-linearitiesrdquo Journal of Mathematical Analysis and Applicationsvol 198 no 2 pp 399ndash443 1996
[12] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems and Control Lettersvol 14 no 1 pp 55ndash61 1990
[13] N El Karoui and LMazliak Eds Backward Stochastic Differen-tial Equations vol 364 of Pitman ResearchNotes inMathematicsSeries Longman 1997
[14] E Pardoux and BSDEs ldquoweak convergence and homogeneiza-tion of semilinear PDEsrdquo in Non- Linear Analysis DifferentialEquations and Control F H Clarke and R J Stern Eds pp503ndash549 Kluwer Dordrecht The Netherlands 1999
[15] S Peng ldquoA generalized dynamic programming principle andHamilton-Jacobi-Bellman equationrdquo Stochastics and StochasticsReports vol 38 pp 119ndash134 1992
[16] N E Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997
[17] SHamad120583ene and J P Lepeltier ldquoBackward equations stochas-tic control and zero-sum stochastic differential gamesrdquo Stochas-tics and Stochastics Reports vol 54 pp 221ndash231 1995
[18] N El-Karoui and S Hamadene ldquoBSDEs and risk-sensitive con-trol zero-sum and nonzero-sum game problems of stochasticfunctional differential equationsrdquo Stochastic Processes and theirApplications vol 107 no 1 pp 145ndash169 2003
[19] M Fuhrman and G Tessiture ldquoExistence of optimal stochasticcontrols and global solutions of forward-backward stochasticdifferential equationsrdquo SIAM Journal on Control and Optimiza-tion vol 43 no 3 pp 813ndash830 2005
[20] M Fuhrman Y Hu and G Tessitore ldquoOn a class of stochasticoptimal control problems related to bsdes with quadraticgrowthrdquo SIAM Journal on Control and Optimization vol 45 no4 pp 1279ndash1296 2006
[21] M Fuhrman and G Tessitore ldquoNonlinear kolmogorov equa-tions in infinite dimensional spaces the backward stochasticdifferential equations approach and applications to optimalcontrolrdquoAnnals of Probability vol 30 no 3 pp 1397ndash1465 2002
[22] F Masiero ldquoSemilinear kolmogorov equations and applicationsto stochastic optimal controlrdquo Applied Mathematics and Opti-mization vol 51 no 1 pp 201ndash250 2005
[23] M Fuhrman FMasiero andG Tessitore ldquoStochastic equationswith delay optimal control via BSDEs and regular solutions ofHamilton-jacobi-bellman equationsrdquo SIAM Journal on Controland Optimization vol 48 no 7 pp 4624ndash4651 2010
[24] M Fuhrman and G Tessiture ldquoInfinite horizon backwardstochastic differential equations and elliptic equations in hilbertspacesrdquo Annals of Probability vol 32 no 1 pp 607ndash660 2004
[25] F Masiero ldquoInfinite horizon stochastic optimal control prob-lems with degenerate noise and elliptic equations in Hilbertspacesrdquo Applied Mathematics and Optimization vol 55 no 3pp 285ndash326 2007
[26] M Fuhrman ldquoA class of stochastic optimal control problemsin Hilbert spaces BSDEs and optimal control laws stateconstraints conditioned processesrdquo Stochastic Processes andtheir Applications vol 108 no 2 pp 263ndash298 2003
[27] F Masiero ldquoStochastic optimal control problems and parabolicequations in banach spacesrdquo SIAM Journal on Control andOptimization vol 47 no 1 pp 251ndash300 2008
[28] G Da Prato and J Zabczyk Ergodicity For Infinite-DimensionalSystems Cambridge University Press 1996
14 Abstract and Applied Analysis
[29] J Zabczyk ldquoParabolic equations on Hilbert spacesrdquo in StochaS-tic PDErsquoS and Kolmogorov Equations in Infinite Dimensionsvol 1715 of Lecture Notes in Math pp 117ndash213 Springer BerlinGermany 1999
[30] G Da Prato and J Zabczyk Stochstic Equations in InfiniteDimensions Cambridge University Press 1992
[31] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 of Applications of MathematicsSpringer New York NY USA 1993
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Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
(119884 119885) isinK119901
120573is |(119884 119885)| = |119884|+ |119885| Here 119865 is a Hilbert
space(iv) 119871119901
P(Ω 119862([119905 119879] 119865)) defined for 119879 gt 119905 ge 0 and
119901 isin [1infin) denotes the space of predictable processes119884(119904) 119904 isin [119905 119879]with continuous paths in 119865 such thatthe norm
|119884|119901
= 119864 sup119904isin[119905119879]
|119884 (119904)|119901
(7)
is finite Elements of 119871119901P(Ω 119862([119905 119879] 119865)) are identified
up to indistinguishability
(v) 119871119902P(Ω 119862
120578([119905infin) 119865)) defined for 120578 isin 119877 and 119902 isin
[1infin) denotes the space of predictable processes119884(119904) 119904 ge 119905 with continuous paths in 119865 such that thenorm
|119884|119902
= 119864 sup119904ge119905
119890120578119902119904
|119884 (119904)|119902
(8)
is finite Elements of 119871119902P(Ω 119862
120578(119865)) are identified up
to indistinguishability(vi) Finally for 120578 isin 119877 and 119902 isin [1infin) we
defined Hq120578(119905) as the space 119871119902
P(Ω 119871
119902
120578([119905infin) 119865)) cap
119871119902
P(Ω 119862
120578([119905infin) 119865)) endowed with the norm
|119884|H119902
120578
= |119884|119871119902
P(Ω119871119902
120578([119905infin)119865))
+ |119884|119871119902
P(Ω119862120578([119905infin)119865))
(9)
For simplicity we denote 119871119901
P(Ω 119871
119902
120573([0infin) 119865)) 119871119902
P(Ω
119862120578([0infin) 119865)) H119902
120578(0) and K
119901
120573(0) by 119871119901
P(Ω 119871
119902
120573(119865)) 119871119902
P(Ω
119862120578(119865))H119902
120578 andK
119901
120573 respectively
Now for every fixed 119905 ge 0 we consider the followingstochastic delay evolution equation
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119905 119883119904) 119889119904 + 119866 (119904 119883
119904) 119889119882 (119904)
119904 isin [119905infin)
119883119905= 119909 isin C
(10)
We make the following assumptions
Hypothesis 1 (i)The operator119860 is the generator of a stronglycontinuous semigroup 119890
119905119860
119905 ge 0 of bounded linearoperators in the Hilbert space119867 We denote by119872 and 120596 twoconstants such that |119890119905119860| le 119872119890120596119905 for 119905 ge 0
(ii) The mapping 119865 [0infin) timesC rarr 119867 is measurable andsatisfies for some constant 119871 gt 0 and 0 le 120579 lt 1
10038161003816100381610038161003816119890119904119860
119865 (119905 119909)10038161003816100381610038161003816le 119871119890
120596119904
119904minus120579
(1 + |119909|119862)
10038161003816100381610038161003816119890119904119860
119865 (119905 119909) minus 119890119904119860
119865 (119905 119910)10038161003816100381610038161003816le 119871119890
120596119904
119904minus1205791003816100381610038161003816119909 minus 119910
1003816100381610038161003816119862
119904 gt 0 119905 isin [0 +infin) 119909 119910 isin C
(11)
(iii) 119866 is a mapping [0infin) timesC rarr 119871(Ξ119867) such that forevery 119907 isin Ξ the map 119866119907 [0infin) times C rarr 119867 is measurable
119890119904119860
119866(119905 119909) isin 1198712(Ξ119867) for every s gt 0 119905 isin [0infin) and 119909 isin C
and
10038161003816100381610038161003816119890119904119860
119866 (119905 119909)100381610038161003816100381610038161198712(Ξ119867)
le 119871119890120596119904
119904minus120574
(1 + |119909|119862)
10038161003816100381610038161003816119890119904119860
119866 (119905 119909) minus 119890119904119860
119866 (119905 119910)100381610038161003816100381610038161198712(Ξ119867)
le 119871119890120596119904
119904minus120574
(1003816100381610038161003816119909 minus 119910
1003816100381610038161003816119862)
119904 gt 0 119905 isin [0 +infin) 119909 119910 isin C
(12)
for some constants 119871 gt 0 and 120574 isin [0 12)We say that119883 is amild solution of (10) if it is a continuous
F119905119905ge0
-predictable process with values in 119867 and it satisfies119875-as
119883 (119904) = 119890(119904minus119905)119860
119909 (0) + int
119904
119905
119890(119904minus120590)119860
119865 (120590119883120590) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883120590) 119889119882 (120590) 119904 isin [119905infin)
119883119905= 119909 isin C
(13)
To stress dependence on initial data we denote the solutionby 119883(119904 119905 119909) Note that 119883(119904 119905 119909) is F
[119905119904]measurable hence
independent ofF119905
We first recall a well-known result on solvability of (10)on bounded interval
Theorem 1 Assume that Hypothesis 1 holds Then for all119902 isin [2infin) and 119879 gt 0 there exists a unique process 119883 isin
119871119902
P(Ω 119862([119905 119879]119867)) as mild solution of (10) Moreover
119864 sup119904isin[119905119879]
|119883 (119904)|119902
le 119862(1 + |119909|119862)119902
(14)
for some constant C depending only on 119902 120574 120579T 120591 L 120596 andM
By Theorem 1 and the arbitrariness of 119879 in its statementthe solution is defined for every 119904 ge 119905 We have the followingresult
Theorem 2 Assume that Hypothesis 1 holds and the process119883(sdot 119905 119909) is mild solution of (10) with initial value (119905 119909) isin
[0infin) timesC Then for every 119902 isin [1infin) there exists a constant120578(119902) such that the process 119883
sdot(119905 119909) isin H
119902
120578(119902)(119905) Moreover for a
suitable constant 119862 gt 0 one has
119864supsget119890120578(119902)1199021199041003816100381610038161003816119883119904
1003816100381610038161003816119902
119862+ 119864int
infin
119905
119890120578(119902)1199021199041003816100381610038161003816119883119904
1003816100381610038161003816119902
119862119889119904 le 119862(1 + |119909|
119862)119902
(15)
with the constant 120578(119902) depending only on 119902 120574 120579 120591 119871 120596 and119872
4 Abstract and Applied Analysis
Proof We define a mapping Φ from H119902
120578(119905) times [0infin) times C to
H119902
120578(119905) by the formula
Φ(119883sdot 119905 119909)
119904(119897) = 119890
(119904+119897minus119905)119860
119909 (0) + int
119904+119897
119905
119890(119904+119897minus120590)119860
119865 (120590119883120590) 119889120590
+ int
119904+119897
119905
119890(119904+119897minus120590)119860
119866 (120590119883120590) 119889119882 (120590)
119904 isin [119905infin) 119897 isin [minus120591 0] 119904 + 119897 ge 119905
Φ(119883sdot 119905 119909)
119904(119897) = 119909 (119904 + 119897 minus 119905)
119904 isin [119905infin) 119897 isin [minus120591 0] 119904 + 119897 lt 119905
(16)
We are going to show that provided 120578 is suitably chosenΦ(sdot 119905 119909) is well defined and that it is a contraction inH119902
120578(119905)
that is there exists 119888 lt 1 such that10038161003816100381610038161003816Φ (119883
1
sdot 119905 119909) minus Φ (119883
1
sdot 119905 119909)
10038161003816100381610038161003816H119902
120578(119905)
le 119888100381610038161003816100381610038161198831
sdotminus 119883
2
sdot
10038161003816100381610038161003816H119902
120578(119905)
1198831
sdot 119883
2
sdotisinH
119902
120578(119905)
(17)
For simplicity we set 119905 = 0 and we treat only the case 119865 = 0the general case being handled in a similar way We will usethe so called factorization method see [28 Theorem 525]Let us take 119902 gt 1 and 120572 isin (0 1) such that 1119902 lt 120572 lt (12) minus120574 and let 119888minus1
120572= int
s120590
(119904 minus 119903)120572minus1
(119903 minus 120590)minus120572
119889119903By the stochastic Fubini theorem
Φ(119883sdot 0 119909)
119904(119897) = 119890
(119904+119897)119860
119909 (0)
+ 119888120572int
119904+119897
0
int
119904+119897
120590
(119904 + 119897 minus 119903)120572minus1
(119903 minus 120590)minus120572
times 119890(119904+119897minus119903)119860
119890(119903minus120590)119860
119889119903119866 (120590119883120590) 119889119882 (120590)
= 119890(119904+119897)119860
119909 (0) + Φ1015840
(119883119904) (119897)
119904 isin [0infin) 119897 isin [minus120591 0] 119904 + 119897 ge 0
Φ(119883sdot 0 119909)
119904(119897) = 119909 (119904 + 119897)
119904 isin [0infin) 119897 isin [minus120591 0] 119904 + 119897 lt 0
(18)
where
Φ1015840
(119883sdot)119904(119897) = 119888
120572int
119904+119897
0
(119904 + 119897 minus 119903)120572minus1
119890(119904+119897minus119903)119860
119884 (119903) 119889119903
119884 (119903) = int
119903
0
(119903 minus 120590)minus120572
119890(119903minus120590)119860
119866 (120590119883120590) 119889119882 (120590)
(19)
Since supminus120591le119897le0
|119890(119904+119897)119860
119909(0)| le 119872119890120596119904
|119909|119862 the process 119890(119904+sdot)119860
119909(0) 119904 ge 0 belongs to H119902
120578provided 120596 + 120578 lt 0 Next we
estimate Φ1015840
(119883sdot) where
10038161003816100381610038161003816Φ1015840
(119883sdot)119904(119897)10038161003816100381610038161003816le 119888
120572int
119904+119897
0
(119904 + 119897 minus 119903)120572minus1
119872119890(119904+119897minus119903)120596
|119884 (119903)| 119889119903
(20)
setting 1199021015840 = 119902(119902 minus 1) so that
11989011990212057811990410038161003816100381610038161003816Φ1015840
(119883sdot)119904
10038161003816100381610038161003816
119902
le 119888119902
120572119872
119902 supminus120591le119897le0
119890119902120578119904
(int
119904+119897
0
(119904+119897minus 119903)120572minus1
119890120596(119904+119897minus119903)
|119884 (119903)| 119889119903)
119902
le 119888119902
120572119872
119902 supminus120591le119897le0
(int
119904+119897
0
(119904 + 119897 minus 119903)120572minus1
119890((120596+120578)119902
1015840
)(119904+119897minus119903)
times119890((120596+120578)119902)(119904minus119903)
119890120578119903
|119884 (119903)| 119889119903)
119902
le 119888119902
120572119872
119902 supminus120591le119897le0
(int
119904+119897
0
119890(120578+120596)(119904+119897minus119903)
(119904 + 119897 minus 119903)(120572minus1)119902
1015840
119889119903)
1199021199021015840
times int
119904+119897
0
119890(120578+120596)(119904minus119903)
119890119902120578119903
|119884 (119903)|119902
119889119903
le 119888119902
120572119872
119902
(int
119904
0
119890(120578+120596)119903
1199031199021015840
(120572minus1)
119889119903)
1199021199021015840
times int
119904
0
119890(120578+120596)(119904minus119903)
119890119902120578119903
|119884 (119903)|119902
119889119903
(21)
Applying the Young inequality for convolutions we have
int
infin
0
11989011990212057811990410038161003816100381610038161003816Φ1015840
(119883sdot)119904
10038161003816100381610038161003816
119902
119889119904 le 119888119902
120572119872
119902
(int
infin
0
119890(120578+120596)119904
1199041199021015840
(120572minus1)
119889119904)
1199021199021015840
times int
infin
0
119890(120578+120596)119904
119889119904int
infin
0
119890119902120578119904
|119884 (119904)|119902
119889119904
(22)
and we conclude that10038161003816100381610038161003816Φ1015840
(119883sdot)10038161003816100381610038161003816119871119902
P(Ω119871119902
120578(C))
le 119888120572119872|119884|
119871119902
P(Ω119871119902
120578(119867))
times (int
infin
0
119890(120578+120596)119904
1199041199021015840
(120572minus1)
119889119904)
11199021015840
times (int
infin
0
119890(120578+120596)119904
119889119904)
1119902
(23)
If we start again from (20) and apply theHolder inequality weobtain
10038161003816100381610038161003816119890120578(119904+119897)
Φ1015840
(119883sdot)119904(119897)10038161003816100381610038161003816le 119888
120572119872(int
119904+119897
0
119903(120572minus1)119902
1015840
119890(120596+120578)119903119902
1015840
119889119903)
11199021015840
times (int
119904+119897
0
119890120578119903119902
|119884 (119903)|119902
119889119903)
1119902
10038161003816100381610038161003816119890120578119904
Φ1015840
(119883sdot)119904
10038161003816100381610038161003816le 119888
120572119872(int
119904
0
119903(120572minus1)119902
1015840
119890(120596+120578)119903119902
1015840
119889119903)
11199021015840
times (int
119904
0
119890120578119903119902
|119884(119903)|119902
119889119903)
1119902
(24)
Abstract and Applied Analysis 5
So we conclude that10038161003816100381610038161003816Φ1015840
(119883sdot)10038161003816100381610038161003816119871119902
P(Ω119862120578(C))
le 119888120572119872|119884|
119871119902
P(Ω119871119902
120578(119867))
times (int
infin
0
119903(120572minus1)119902
1015840
119890(120596+120578)119903119902
1015840
119889119903)
11199021015840
(25)
On the other hand by the Burkholder-Davis-Gundy inequal-ities for some constant 119888
119902depending only on 119902 we have
119864|119884 (119903)|119902
le 119888119902119864(int
119903
0
(119903 minus 120590)minus212057210038161003816100381610038161003816119890(119903minus120590)119860
119866 (120590119883120590)10038161003816100381610038161003816
2
1198712(Ξ119867)
119889120590)
1199022
le 119871119902
119888119902119864
times (int
119903
0
(119903 minus 120590)minus2120572minus2120574
1198902120596(119903minus120590)
(1 +1003816100381610038161003816119883120590
10038161003816100381610038162
119862) 119889120590)
1199022
(26)
which implies that
[119864|119884 (119903)|119902
]2119902
le 1198712
1198882119902
119902int
119903
0
(119903 minus 120590)minus2120572minus2120574
times 1198902120596(119903minus120590)
[119864(1 +1003816100381610038161003816119883120590
1003816100381610038161003816119862)119902
]2119902
119889120590
(27)
so that
1198902120578119903
[119864|119884 (119903)|119902
]2119902
le 1198621int
119903
0
(119903 minus 120590)minus2120572minus2120574
1198902(120596+120578)(119903minus120590)
1198902120578120590
119889120590
+ 1198622int
119903
0
(119903 minus 120590)minus2120572minus2120574
1198902(120596+120578)(119903minus120590)
times 1198902120578120590
[1198641003816100381610038161003816119883120590
1003816100381610038161003816119902
119862]2119902
119889120590
(28)
for suitable constants 1198621 119862
2 Applying the Young inequality
for convolutions we obtain
int
infin
0
119890119902120578119903
119864|119884 (119903)|119902
119889119904le 1198621(int
infin
0
119904minus2120572minus2120574
1198902(120596+120578)119904
119889119904)
1199022
int
infin
0
119890119902120578119904
119889119904
+ 1198622(int
infin
0
119904minus2120572minus2120574
1198902(120596+120578)119904
119889119904)
1199022
times int
infin
0
119890119902120578119904
1198641003816100381610038161003816119883119904
1003816100381610038161003816119902
119862119889119904
(29)
This shows that |119884|119871119902
P(Ω119871119902
120578(119867))
is finite provided we assumethat 120578 lt 0 and 120596 + 120578 lt 0 so the map is well defined
If 1198831
sdot 119883
2
sdotare processes belonging to H119902
120578and 1198841 1198842 are
defined accordingly the entirely analogous passages showthat100381610038161003816100381610038161198841
minus 119884210038161003816100381610038161003816119871119902
P(Ω119871119902
120578(119867))
le 1198711198881119902
120572
100381610038161003816100381610038161198831
sdotminus 119883
2
sdot
10038161003816100381610038161003816119871119902
P(Ω119871119902
120578(C))
times (int
infin
0
119904minus2120572minus2120574
1198902(120596+120578)119904
119889119904)
12
(30)
Recalling the inequalities (23) and (25) and noting that themap Y rarr Φ
1015840
(Xsdot) is linear we obtain an explicit expression
for the constant 119888 in (17) and it is immediate to verify that119888 lt 1 provided 120578 lt 0 is chosen sufficiently large We fixsuch a value of 120578(119902) The first result is a consequence of thecontraction principle The estimate (15) also follows from thecontraction property ofΦ(sdot 119905 119909)
For investigating the dependence of the solution119883(119904 119905 119909)on the initial data 119909 and 119905 we reformulate (13) as an equationon [0infin) We set
119878 (119904) = 119890119904119860
for 119904 ge 0 119878 (119904) = 119868 for 119904 lt 0 (31)
and we consider the equation
119883(119904) = 119878 (119904 minus 119905) 119909 ((0 and (119904 minus 119905)) or (minus120591))
+ int
119904
0
119868[119905infin)
(120590) 119878 (119904 minus 120590) 119865 (120590119883120590) 119889120590
+ int
119904
0
119868[119905infin)
(120590) 119878 (119904 minus 120590)
times 119866 (120590119883120590) 119889119882 (120590) 119904 isin [0infin)
1198830(120579) = 119909 ((minus119905 + 120579) or (minus120591)) 120579 isin [minus120591 0]
(32)
Under the assumptions of Hypothesis 1 by Theorem 2 it iseasy to prove that equation (32) has a unique solution 119883 and119883sdotisin H
119902
120578(119902)for every 119902 isin [2infin) It clearly satisfies 119883(119904) =
119909((119904 minus 119905) or (minus120591)) for 119904 isin [minus120591 119905) and its restriction to the timeinternal [119905infin) is the unique mild solution of (10) From nowon we denote by119883(119904 119905 119909) 119904 isin [0infin) the solution of (32)
We need the following parameter-depending contractionprinciple which is stated in the following lemma and provedin [29 Theorems 101 and 102]
Lemma3 (ParameterDependingContraction Principle) Let119861119863 denote Banach spaces Let ℎ 119861times119863 rarr 119861 be a continuousmapping satisfying
1003816100381610038161003816ℎ (1199091 119910) minus ℎ (1199092 119910)1003816100381610038161003816 le 120572
10038161003816100381610038161199091 minus 11990921003816100381610038161003816 (33)
for some 120572 isin [0 1) and every 1199091 119909
2isin 119861 y isin 119863 Let 120601(119910)
denote the unique fixed point of the mapping ℎ(sdot 119910) 119861 rarr 119861Then 120601 119863 rarr 119861 is continuous
Theorem 4 Assume that Hypothesis 1 holds true Then forevery 119902 isin [1infin) the map (119905 119909) rarr 119883
sdot(119905 119909) is continuous
from [0infin) timesC toHq120578(q)
Proof Clearly it is enough to prove the claim for 119902 large Letus consider the map Φ defined in the proof of Theorem 2 In
6 Abstract and Applied Analysis
our present notationΦ can be seen as a mapping fromH119902
120578times
[0infin) timesC toH119902
120578as follows
Φ(119883sdot 119905 119909)
119904(119897) = 119878 (119904 + 119897 minus 119905) 119909 (0)
+ int
119904+119897
0
119868[119905infin)
(120590) 119878 (119904 + 119897 minus 120590) 119865 (120590119883120590) 119889120590
+ int
119904+119897
0
119868[119905infin)
(120590) 119878 (119904 + 119897 minus 120590)
times 119866 (120590119883120590) 119889119882 (120590)
119904 isin [0infin) 119897 isin [minus120591 0] 119904 + 119897 ge 119905
Φ(119883sdot 119905 119909)
119904(119897) = 119909 ((119904 + 119897 minus 119905) or (minus120591))
119904 isin [0infin) 119897 isin [minus120591 0] 119904 + 119897 le 119905
(34)
By the arguments of the proof of Theorem 2 Φ(sdot 119905 119909)is a contraction in H119902
120578uniformly with respect to 119905 119909
The process 119883sdot(119905 119909) is the unique fixed point of Φ(sdot 119905 119909)
So by the parameter-depending contraction principle(Lemma 3) it suffices to show that Φ is continuous fromH119902
120578times [0infin) times C to H119902
120578 From the contraction property
of Φ(sdot 119905 119909) mentioned earlier we have that Φ(sdot 119905 119909) iscontinuous uniformly in 119905 119909 Moreover for fixed 119883
sdot it is
easy to verify that Φ(119883sdot sdot sdot) is continuous from [0infin) times C
toH119902
120578 The proof is finished
Remark 5 By similar passages we can show that for fixed119905 Theorem 4 still holds true for 119902 large enough if the spaces[0infin) times C and H119902
120578are replaced by the spaces 119871119902(ΩCF
119905)
and H119902
120578(119905) respectively where 119871119902(ΩCF
119905) denotes that the
space of F119905-measurable function with value in C such that
the norm
|119909|119902
= 119864|119909|119902
119862 (35)
is finite
3 The Backward-Forward System
In this section we consider the system of stochastic differen-tial equations 119875-as
119883 (119904) = 119890(119904minus119905)119860
119909 (0) + int
119904
119905
119890(119904minus120590)119860
119865 (120590119883120590) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883120590) 119889119882 (120590) 119904 isin [119905infin)
119883119905= 119909 isin C
119884 (119904) minus 119884 (119879) + int
119879
119904
119885 (120590) 119889119882 (120590) + 120582int
119879
119904
119884 (120590) 119889120590
= int
119879
119904
120595 (120590119883120590 119884 (120590) 119885 (120590)) 119889120590 0 le 119904 le 119879 lt infin
(36)
for 119904 varying on the time interval [119905infin) sub [0infin) Asin Section 2 we extend the domain of the solution setting119883(119904 119905 119909) = 119909((119904 minus 119905) or (minus120591)) for 119904 isin [minus120591 119905)
We make the following assumptions
Hypothesis 2 Themapping 120595 [0infin)timesCtimes119870times1198712(Ξ 119870) rarr
119870 is Borelmeasurable such that for all 119905 isin [0infin)120595(119905 sdot) Ctimes119870 times 119871
2(Ξ 119870) rarr 119870 is continuous and for some 119871
119910 119871
119911gt 0
120583 isin 119877 and119898 ge 11003816100381610038161003816120595 (119904 119909 1199101 1199111) minus 120595 (119904 119909 1199102 1199112)
1003816100381610038161003816
le 119871119910
10038161003816100381610038161199101 minus 11991021003816100381610038161003816 + 119871119911
10038161003816100381610038161199111 minus 11991121003816100381610038161003816
1003816100381610038161003816120595 (119904 119909 119910 119911)1003816100381610038161003816 le 119871 (1 + |119909|
119898
119862+10038161003816100381610038161199101003816100381610038161003816 + |119911|)
⟨120595 (119904 119909 1199101 119911) minus 120595 (119904 119909 119910
2 119911) 119910
1minus 119910
2⟩119870ge 120583
10038161003816100381610038161199101 minus 119910210038161003816100381610038162
(37)
for every 119904 isin [0infin) 119909 isin C 119910 1199101 119910
2isin 119870 119911 119911
1 and 119911
2isin
1198712(Ξ 119870)We note that the third inequality in (37) follows from the
first one taking 120583 = minus119871119910but that the third inequalitymay also
hold for different values of 120583Firstly we consider the backward stochastic differential
equation
119884 (119904) minus 119884 (119879) + int
119879
119904
119885 (120590) 119889119882 (120590) + 120582int
119879
119904
119884 (120590) 119889120590
= int
119879
119904
120595 (120590119883120590 119884 (120590) 119885 (120590)) 119889120590 0 le 119904 le 119879 lt infin
(38)
119870 is a Hilbert space the mapping 120595 [0infin) times C times 119870 times
1198712(Ξ 119870) rarr 119870 is a given measurable function 119883
sdotis a
predictable process with values in another Banach space Cand 120582 is a real number
Theorem 6 Assume that Hypothesis 2 holds Let 119901 gt 2 and120575 lt 0 be given and choose
119902 ge 119898119901 120578 gt120575
119898 (39)
Then the following hold
(i) For 119883sdotisin 119871
119902
P(Ω 119871
119902
120578(C)) and 120582 gt minus(120575 + 120583 minus (119871
2
1199112))
(38) has a unique solution in Kp120575that will be denoted
by (119884(119883sdot)(119904) 119885(119883
sdot)(119904)) 119904 ge 0
(ii) The estimate
119864sup119904ge0
(119884 (119883sdot) (119904))
119901
119890119901120575119904
+ 119864(int
infin
0
11989021205751205901003816100381610038161003816119884(119883sdot
)(120590)10038161003816100381610038162
119889120590)
1199012
+ 119864(int
infin
0
11989021205751205901003816100381610038161003816119885 (119883sdot
) (120590)10038161003816100381610038162
119889120590)
1199012
le 119888(1 +1003816100381610038161003816119883sdot
1003816100381610038161003816119898
119871119902
P(Ω119871119902
120578(C))
)119901
(40)
Abstract and Applied Analysis 7
holds for a suitable constant 119888 In particular 119884(119883sdot) isin
119871119901
P(Ω 119862
120575(119870))
(iii) The map 119883sdotrarr (119884(119883
sdot) 119885(119883
sdot)) is continuous from
119871119902
P(Ω 119871
119902
120578(C)) toK119901
120575 and 119883
sdotrarr 119884(119883
sdot) is continuous
from 119871119902
P(Ω 119871
119902
120578(C)) to 119871119901
P(Ω 119862
120575(119870))
(iv) The statements of points (i) (ii) and (iii) still holdtrue if the space 119871119902
P(Ω 119871
119902
120578(C)) is replaced by the space
119871119902
P(Ω 119862
120578(C))
Proof The theorem is very similar to Proposition 311 in [24]The only minor difference is that the mapping 120595 [0infin) times
Ctimes119870times1198712(Ξ 119870) rarr 119870 is a givenmeasurable function while
in [24] the measurable function 120595 is from119867 times119870 times 1198712(Ξ 119870)
to 119870 however the same arguments apply
Theorem 7 Assume that Hypothesis 1 holds and thatHypothesis 2 holds true in the particular case 119870 = 119877 Thenfor every 119901 gt 2 119902 120575 lt 0 satisfying (39) with 120578 = 120578(119902)and for every 120582 gt 120582
1015840
= minus(120575 + 120583 minus (1198712
1199112)) there exists a
unique solution in H119902
120578(119902)times K
119901
120575of (36) that will be denoted
by (119883(sdot 119905 119909) 119884(sdot 119905 119909) 119885(sdot 119905 119909)) Moreover 119884(sdot 119905 119909) isin
119871119901
P(Ω 119862
120575(119877)) The map (119905 119909) rarr (119884(sdot 119905 119909) 119885(sdot 119905 119909)) is con-
tinuous from [0infin)timesC toK119901
120575 and themap (119905 119909) rarr 119884(sdot 119905 119909)
is continuous from [0infin) timesC to 119871119901P(Ω 119862
120575(119877))
Proof We first notice that the system is decoupled the firstdoes not contain the solution (119884 119885) of the second oneThere-fore under the assumption of Hypothesis 1 by Theorem 2there exists a unique solution 119883(sdot 119905 119909) and 119883
sdot(119905 119909) isin H
119902
120578(119902)
of the first equation Moreover from Theorem 4 it followsthat the map (119905 119909) rarr 119883
sdot(119905 119909) is continuous from [0infin)timesC
toH119902
120578(119902)
Let 119870 = 119877 from Theorem 6 we have that thereexists a unique solution (119884(sdot 119905 119909) 119885(sdot 119905 119909)) isin K
119901
120575of the
second equation and the map 119883sdotrarr (119884(119883
sdot) 119885(119883
sdot)) is
continuous from H119902
120578(119902)to K
119901
120575while X
sdotrarr (Y(X
sdot)) is
continuous fromH119902
120578(119902)to119871119901
P(Ω 119862
120575(119877))We have proved that
(119883(sdot 119905 119909) 119884(sdot 119905 119909) 119885(sdot 119905 119909)) isin H119902
120578(119902)times K
119901
120575is the unique
solution of (36) and the other assertions follow from com-position
Remark 8 From Remark 5 by similar passages we can showthat for fixed 119905 and for 119902 large enough under the assumptionsof Theorem 7 the map 119909 rarr (119884(sdot 119905 119909) 119885(sdot 119905 119909)) is continu-ous from 119871
119902
(ΩCF119905) toK119901
120575(119905)
We also remark that the process 119883(sdot 119905 119909) is F[119905infin)
measurable since C is separable Banach space we have that119883sdot(119905 119909) is F
[119905infin)measurable So that 119884(119905) is measurable
with respect to both F[119905infin)
and F119905 it follows that 119884(119905) is
deterministicFor later use we notice three useful identities for 119905 le 119904 lt
infin the equality 119875-as
119883119897(119904 119883
119904(119905 119909)) = 119883
119897(119905 119909) 119897 isin [119904infin) (41)
is a consequence of the uniqueness of the solution of (13)Since the solution of the backward equation is uniquely
determined on an interval [119904infin) by the values of the process119883sdoton the same interval for 119905 le 119904 lt infin we have 119875-as
119884 (119897 119904 119883119904(119905 119909)) = 119884 (119897 119905 119909) for 119897 isin [119904infin)
119885 (119897 119904 119883119904(119905 119909)) = 119885 (119897 119905 119909) for aa 119897 isin [119904infin)
(42)
Lemma 9 (see [30]) Let 119864 be a metric space with metric 119889and let 119891 Ω rarr 119864 be strongly measurable Then thereexists a sequence 119891
119899 119899 isin 119873 of simple 119864-valued functions
(ie 119891119899isFB(E)measurable and takes only a finite number
of values) such that for arbitrary 120596 isin Ω the sequence119889(119891
119899(120596) 119891(120596)) 119899 isin 119873 is monotonically decreasing to zero
Let now 119891 isin 119871119902
(ΩC) By Lemma 9 we get the existenceof a sequence of simple function 119891
119899 119899 isin 119873 such that
1003816100381610038161003816119891119899 (120596) minus 119891 (120596)1003816100381610038161003816 darr 0 for all 120596 isin Ω as 119899 rarr infin (43)
Hence 119891119899
rarr 119891 in | sdot |119871119902(ΩC) by Lebesguersquos dominated
convergence theoremWe are now in a position of showing the main result in
this section
Theorem 10 Assume that Hypothesis 1 holds true and thatHypothesis 2 holds in the particular case 119870 = 119877 Then thereexist two Borel measurable deterministic functions 120592 [tinfin) times
C rarr 119877 and 120577 [119905infin) times C rarr Ξlowast
= 119871(Ξ 119877) =
1198712(Ξ 119877) such that for 119905 isin [0infin) and x isin C the solution
(119883(119905 119909) 119884(119905 119909) 119885(119905 119909)) of (36) satisfies
119884 (119904 119905 119909) = 120592 (119904 119883119904(119905 119909)) 119885 (119904 119905 119909) = 120577 (119904 119883
119904(119905 119909))
119875-as for aa 119904 isin [119905infin)
(44)
Proof We apply the techniques introduced in [26 Proposi-tion 32] Let 119890
119894 be a basis of Ξlowast and let us define 119885119894119873
=
((119885 119890119894)Ξlowast and119873) or (minus119873) Then for every 0 le 119905
1lt 119905
2lt infin Δ gt
0 and 1199091 119909
2isin C we have that
100381610038161003816100381610038161003816100381610038161003816
119864 int
1199051+Δ
1199051
119885119894119873
(119904 1199051 119909
1) 119889119904 minus 119864int
1199052+Δ
1199052
119885119894119873
(119904 1199052 119909
2) 119889119904
100381610038161003816100381610038161003816100381610038161003816
le 119864int
1199052
1199051
10038161003816100381610038161003816119885119894119873
(119904 1199051 119909
1)10038161003816100381610038161003816119889119904
+ 119864int
1199051+Δ
1199052
10038161003816100381610038161003816119885119894119873
(119904 1199051 119909
1) minus 119885
119894119873
(119904 1199052 119909
2)10038161003816100381610038161003816119889119904
+ 119864int
1199052+Δ
1199051+Δ
10038161003816100381610038161003816119885119894119873
(119904 1199052 119909
2)10038161003816100381610038161003816119889119904
8 Abstract and Applied Analysis
le 211987310038161003816100381610038161199052 minus 1199051
1003816100381610038161003816 + Δ12
119890minus120575(1199051+Δ)
times(119864(int
infin
0
119890212057511990410038161003816100381610038161003816119885119894119873
(119904 1199051 119909
1)minus119885
119894119873
(119904 1199052 119909
2)10038161003816100381610038161003816
2
119889119904)
1199012
)
1119901
le 211987310038161003816100381610038161199052 minus 1199051
1003816100381610038161003816 + Δ12
119890minus120575(1199051+Δ)
times (119864(int
infin
0
11989021205751199041003816100381610038161003816119885 (119904 1199051 1199091)minus119885 (119904 1199052 1199092)
10038161003816100381610038162
119889119904)
1199012
)
1119901
(45)
From Theorem 7 we have that the map (119905 119909) rarr int119905+Δ
119905
119885119894119873
(119904 119905 119909)119889119904 is continuous from [0infin) times C to 119877 ByRemark 8 we also have that for fixed 119905 the map 119909 rarr
119864int119905+Δ
119905
119864119885119894119873
(119904 119905 119909)119889119904 is continuous from 119871119902
(ΩCF119905) to 119877
for 119902 large enough Let us define
120577119894119873
(119905 119909) = lim inf119899rarrinfin
119899119864int
119905+(1119899)
119905
119885119894119873
(119904 119905 119909) 119889119904
119905 isin [0infin) 119909 isin C
(46)
It is clear that 120577119894119873 [0infin) timesC rarr 119877 is a Borel functionWe fix 119909 and 0 le 119905 le 119904 lt infin For 119897 isin [119904infin) we
denote 119864[119885119894119873
(119897 119904 119910)]|119910=119883119904(119905119909)
the random variable obtainedby composing119883
119904(119905 119909) with the map 119910 rarr 119864[119885
119894119873
(119897 119904 119910)]By Lemma 9 there exists a sequence of C-valued F
119904-
measurable simple functions
119891119898 Ω 997888rarr C 119891
119898=
119873119898
sum
119896=1
ℎ(119898)
119896119868119891119898=ℎ(119898)
119896 119873
119898isin 119873 (47)
where ℎ(119898)1 ℎ
(119898)
119898are pairwise distinct andΩ = ⋃
119873119898
119896=1119891
119898=
ℎ(119898)
119896 such that
1003816100381610038161003816119891119898 (120596) minus 119883119904(120596)
1003816100381610038161003816 darr 0 for all 120596 isin Ω as 119899 997888rarr infin (48)
For any 119861 isin F119904 we have
int119861
int
119904+(1119899)
119904
119885119894119873
(119897 119905 119909) 119889119897119889119875
= int119861
int
119904+(1119899)
119904
119885119894119873
(119897 119904 119883119904) 119889119897119889119875
= 119864119868119861int
119904+(1119899)
119904
119885119894119873
(119897 119904 119883119904) 119889119897
= lim119898rarrinfin
119864(119868119861int
119904+(1119899)
119904
119885119894119873
(119897 119904 119891119898) 119889119897)
= lim119898rarrinfin
119873119898
sum
119896=1
119864(119868119861119868119891119898=ℎ(119898)
119896int
119904+(1119899)
119904
119885119894119873
(119897 119904 ℎ(119898)
119896) 119889119897)
= lim119898rarrinfin
119864(119868119861
119873119898
sum
119896=1
119868119891119898=ℎ(119898)
119896)119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 ℎ(119898)
119896) 119889119897
= lim119898rarrinfin
119864119868119861(119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 119910) 119889119897|119910=119891119898
)
= lim119898rarrinfin
int119861
(119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 119910) 119889119897|119910=119891119898
)119889119875
= int119861
(119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 119910) 119889119897|119910=119883119904
)119889119875
(49)
and we get that
120577119894119873
(119904 119883119904(119905 119909)) = lim inf
119899rarrinfin
119899
times [119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 119910) 119889119897|119910=119883119904(119905119909)
]
= lim inf119899rarrinfin
119899119864[int
119904+(1119899)
119904
119885119894119873
(119897 119905 119909) 119889119897
100381610038161003816100381610038161003816100381610038161003816
F119904]
119875-as(50)
Fix 119905 and 119909 Recalling that |119885119894119873
| le 119873 by the Lebesguetheorem on differentiation it follows that 119875-as
lim119899rarrinfin
119899int
119904+(1119899)
119904
119885119894119873
(119897 119905 119909) 119889119897 = 119885119894119873
(119904 119905 119909)
for aa 119904 isin [119905infin)
(51)
By the boundedness of 119885119894119873 applying the dominated conver-gence theorem we get that
120577119894119873
(119904 119883119904(119905 119909)) = 119864 [119885
119894119873
(119904 119905 119909)10038161003816100381610038161003816F
119904] = 119885
119894119873
(119904 119905 119909)
119875-as for aa 119904 isin [119905infin)
(52)
Now we have proved that for every 119905 119909
120577119894119873
(119904 119883119904(119905 119909)) = 119885
119894119873
(119904 119905 119909)
119875-as for aa 119904 isin [119905infin)
(53)
for every 119894 119873 Let 119862 sub [0infin) times C denote the set ofpairs (119905 119909) such that lim
119873rarrinfin120577119894119873
(119905 119909) exists and the seriessuminfin
119894=1(lim
119873rarrinfin120577119894119873
(119905 119909))119890119894converges in Ξlowast We define
120577 (119905 119909) =
infin
sum
119894=1
( lim119873rarrinfin
120577119894119873
(119905 119909)) 119890119894 (119905 119909) isin 119862
120577 (119905 119909) = 0 (119905 119909) notin 119862
(54)
Abstract and Applied Analysis 9
Since 119885 satisfies
119885 (120596 119904 119905 119909) =
infin
sum
119894=1
( lim119873rarrinfin
119885119894119873
(120596 119904 119905 119909)) 119890119894 (55)
for every 120596 119904 119905 119909 From (53) it follows that for every 119905 119909 wehave (119904 119883
119904(120596 119905 119909)) isin 119862 119875-as for almost all 119904 isin [119905infin) and
119885(119904 119905 119909) = 120577(119904 119883119904(119905 119909)) 119875-as for aa 119904 isin [119905infin)
We define 120592(119905 119909) = 119884(119905 119905 119909) since 119884(119905 119905 119909) is deter-ministic so the map (119905 119909) rarr 120592(119905 119909) can be written as acomposition 120592(119905 119909) = Γ
3(Γ2(119905 Γ
1(119905 119909))) with
Γ1 [0infin) timesC 997888rarr 119871
119901
P(Ω 119862
120575(119877))
Γ1(119905 119909) = 119884 (sdot 119905 119909)
Γ2 [0infin) times 119871
119901
P(Ω 119862
120575(119877)) 997888rarr 119871
119901
(Ω 119877)
Γ2(119905 119881) = 119881 (119905)
Γ3 119871
119901
(Ω 119877) 997888rarr 119877 Γ3120585 = 119864120585
(56)
FromTheorem 7 it follows that Γ1is continuous By
|119881(119905) minus 119880(119904)|119871119901(Ω119877)
le |119881 (119905) minus 119881 (119904)|119871119901(Ω119877)
+ 119890minus120575119901119904
|119881 minus 119880||119871
119901
P(Ω119862120575(119877))
(57)
we have that Γ2is continuous It is clear that Γ
3is continuous
Then themap (119905 119909) rarr 120592(119905 119909) is continuous from [0infin)timesCto 119877 therefore 120592(119905 119909)is a Borel measurable function Fromuniqueness of the solution of (36) it follows that 119884(119904 119905 119909) =120592(119904 119883
119904(119905 119909)) 119875-as for aa 119904 isin [119905infin)
4 The Fundamental Relation
Let (ΩF 119875) be a given complete probability space with afiltration F
119905119905ge0
satisfying the usual conditions 119882(119905) 119905 ge 0
is a cylindrical Wiener process in Ξ with respect to F119905119905ge0
We will say that an F
119905ge0-predictable process 119906 with values
in a given measurable space (119880U) is an admissible controlThe function 119877 [0infin) times C times 119880 rarr Ξ is measurableand bounded We consider the following controlled stateequation
119889119883119906
(119904) = 119860119883119906
(119904) 119889119904 + 119865 (119904 119883119906
119904) 119889119904
+ 119866 (119904 119883119906
119904) 119877 (119904 119883
119906
119904 119906 (119904)) 119889119904 + 119866 (119904 119883
119906
119904) 119889119882 (119904)
119904 isin [119905infin)
119883119906
119905= 119909
(58)
Here we assume that there exists a mild solution of (58)which will be denoted by 119883119906
(119904 119905 119909) or simply by 119883119906
(119904) Weconsider a cost function of the form
119869 (119906) = 119864int
infin
119905
119890minus120582119904
119892 (119904 119883119906
119904 119906 (119904)) 119889119904 (59)
Here 119892 is function on [0infin) times C times 119880 with real values Ourpurpose is to minimize the function 119869 over all admissiblecontrols
We define in a classical way the Hamiltonian functionrelative to the previous problem for all 119905 isin [0infin) 119909 isin
C and 119911 isin Ξlowast
120595 (119905 119909 119911) = inf 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) 119906 isin 119880
(60)
and the corresponding possibly empty set of minimizers
Γ (119905 119909 119911) = 119906 isin 119880 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) = 120595 (119905 119909 119911)
(61)
We are now ready to formulate the assumptions we need
Hypothesis 3 (i) 119860 119865 and G verify Hypothesis 1(ii) (119880U) is ameasurable spaceThemap 119892 [0infin)timesCtimes
119880 rarr 119877 is continuous and satisfies |119892(119905 119909 119906)| le 119870119892(1+|119909|
119898119892
119862)
for suitable constants 119870119892gt 0 119898
119892gt 0 and all 119909 isin C119906 isin
119880 The map 119877 [0infin) times C times 119880 rarr Ξ is measurable and|119877(119905 119904 119906)| le 119871
119877for a suitable constant 119870
119877gt 0 and all 119909 isin
C119906 isin 119880 and119911 isin Ξlowast
(iii)TheHamiltonian120595 defined in (60) satisfies the requi-rements of Hypothesis 2 (with119870 = 119877)
(iv) We fix here 119901 gt 2 q and 120575 lt 0 satisfying (39) with120578 = 120578(119902) and such that 119902 gt 119898
119892
We are in a position to prove the main result of thissection
Theorem11 Assume thatHypothesis 3 holds and suppose that120582 verifies
120582 gt (minus120575 minus 120583 +1198712
119911
2) or (minus120575 +
1198712
119877
2 (119901 minus 1))
or (1198712
119877119898119892
2 (119902 minus 119898119892)minus 120578 (119902)119898
119892)
(62)
Let 120592 120577 denote the function in the statement of Theorem 10Then for every admissible control 119906 and for the correspondingtrajectory119883 starting at (119905 119909) one has
119869 (119906) = 120592 (119905 119909) + 119864int
infin
119905
119890minus120582119904
[minus120595 (119904 119883119906
119904 120577 (119904 119883
119906
119904)) + 120577 (119904 119883
119906
119904)
times 119877 (119904 119883119906
119904 119906 (119904)) + 119892 (119904 119883
119906
119904 119906 (119904))] 119889119904
(63)
Proof Consider (58) in the probability space (ΩF 119875) withfiltration F
119905119905ge0
and with an F119905119905ge0
-cylindrical Wienerprocess 119882(119905) 119905 ge 0 Let us define
119882119906
(119904) = 119882 (119904) + int
119904
119905and119904
119877 (120590119883119906
120590 119906 (120590)) 119889120590 119904 isin [0infin)
120588 (119879) = exp(int119879
119905
minus119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882 (119904)
minus1
2int
119879
119905
1003816100381610038161003816119877 (119904 119883119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)
(64)
10 Abstract and Applied Analysis
Let 119875119906 be the unique probability onF[0infin)
such that
119875119906
|F119879
= 120588 (119879) 119875|F119879
(65)
We notice that under 119875119906 the process119882119906 is aWiener processLet us denote by F119906
119905119905ge0
the filtration generated by119882119906 andcompleted in the usual way Relatively to 119882119906 (58) can berewritten as
119889119883119906
(119904) = 119860119883119906
(119904) 119889119904 + 119865 (119904 119883119906
119904) 119889119904
+ 119866 (119904 119883119906
119904) 119889119882
119906
(119904) 119904 isin [119905infin)
119883119906
119905= 119909
(66)
In the space (ΩF[0infin)
F119906
119905119905ge0 119875
119906
) we consider the follow-ing system of forward-backward equations
119883119906
(119904) = 119890(119904minus119905)119860
119909 (0) + int
119904
119905
119890(119904minus120590)119860
119865 (120590119883119906
120590) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883119906
120590) 119889119882
119906
(120590) 119904 isin [119905infin)
119883119906
119905= 119909 isin C
119884119906
(119904) minus 119884119906
(119879) + int
119879
119904
119885119906
(120590) 119889119882119906
(120590) + 120582int
119879
119904
119884119906
(120590) 119889120590
= int
119879
119904
120595 (120590119883119906
120590 119885
119906
(120590)) 119889120590 0 le 119904 le 119879 lt infin
(67)
Applying the Ito formula to 119890minus120582119904119884119906(119904) and writing the back-ward equation in (67) with respect to the process119882 we get
119884119906
(119904) + int
119879
119904
119890minus120582120590
119885119906
(120590) 119889119882 (120590)
= int
119879
119904
119890minus120582120590
[120595 (120590119883119906
120590 119885
119906
(120590))
minus119885119906
(120590) 119877 (120590119883119906
120590 119906 (120590))] 119889120590
+ 119890minus120582119879
119884119906
(119879)
(68)
Recalling that 119877 is bounded we get for all 119903 ge 1 and someconstant 119862
119864119906
[120588(119879)minus119903
] = 119864119906
[exp 119903 (int119879
119905
119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882
119906
(119904)
minus1
2int
119879
119905
1003816100381610038161003816119877 (119904 119883119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)]
= 119864119906
[exp(int119879
119905
119903119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882
119906
(119904)
minus1
2int
119879
119905
11990321003816100381610038161003816119877 (119904 119883
119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)
times exp 119903 (119903 minus 1)2
int
119879
119905
1003816100381610038161003816119877 (119904 119883119906
119904 119906 (119904))
10038161003816100381610038162
119889119904]
le 119890(12)119903(119903minus1)119879119871
2
119877119864119906
times exp(int119879
119905
2119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882
119906
(119904)
minus1
2int
119879
119905
41003816100381610038161003816119877 (119904 119883
119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)
= 119890(12)119903(119903minus1)119879119871
2
119877
(69)
It follows that
119864(int
119879
119905
|119890minus120582119904
119885119906
(119904)|2
119889119904)
12
= 119864119906
[(int
119879
119905
10038161003816100381610038161003816119890minus120582119904
119885119906
(119904)10038161003816100381610038161003816
2
119889119904)
12
120588minus1
]
le (119864119906
int
119879
119905
10038161003816100381610038161003816119890minus120582119904
119885119906
(119904)10038161003816100381610038161003816
2
119889119904)
12
times (119864119906
120588minus2
)12
lt infin
(70)
We conclude that the stochastic integral in (68) has zeroexpectation If we set 119904 = 119905 in (68) and we take expectationwith respect to 119875 we obtain
119890minus120582119879
119864119884119906
(119879) minus 119884119906
(119905)
= 119864int
119879
119905
119890minus120582120590
[minus120595 (120590119883119906
120590 119885
119906
(120590))
+119885119906
(120590) 119877 (120590119883119906
120590 119906 (120590))] 119889120590
(71)
ByTheorem 7 119884119906(sdot 119905 119909) isin 119871119901P(Ω 119862
120575(119877)) so that
119864119906
|119884(119879 119905 119909)|119901
le 119862 exp (minus119901120575119879) (72)
By the Holder inequality we have that for suitable constant119862 gt 0
119864 |119884 (119879 119905 119909)| = 119864119906
(120588minus1
(119879) |119884 (119879 119905 119909)|)
le 119864(120588minus119901(119901minus1)
)(119901minus1)119901
119864(|119884 (119879 119905 119909)|119901
)1119901
le 119862119890((1198712
1198772(119901minus1))minus120575))119879
(73)
From Theorem 2 we obtain 119864119906sup119904ge119905119890120578119902119904
|119883119906
119904|119902
lt infin by thesimilar process we get that
1198641003816100381610038161003816119883
119906
119879
1003816100381610038161003816119898119892
le 119862119890(1198712
119877119898119892(2119902minus2119898
119892)minus1
minus120578(119902)119898119892)119879
(74)
Abstract and Applied Analysis 11
for suitable constant 119862 gt 0 and
119864int
infin
119905
119890minus120582120590 1003816100381610038161003816119892 (120590119883
119906
120590 119906 (120590))
1003816100381610038161003816 119889120590 lt infin (75)
Since 119884119906(119905 119905 119909) = 120592(119905 119909) and 119885119906
(119904 119905 119909) = 120577(119904 119883119906
119904(119905 119909)) 119875-
as for aa 119904 isin [119905infin) we have that
119890minus120582119879
119864119884119906
(119879) minus 119907 (119905 119909)
= 119864int
119879
119905
119890minus120582120590
[minus120595 (120590119883119906
120590 120577 (120590 119883
119906
120590))
+120577 (120590119883119906
120590) 119877 (120590119883
119906
120590 119906 (120590))] 119889120590
(76)
Thus adding and subtracting119864intinfin119905
119890minus120582120590
119892(120590119883119906
120590 119906(120590))119889120590 and
letting 119879 rarr infin we conclude that
119869 (119906) = 120592 (119905 119909)
+ 119864int
infin
119905
119890minus120582119904
[minus120595 (119904 119883119906
119904 120577 (119904 119883
119906
119904)) + 120577 (119904 119883
119906
119904) 119877
times (119904 119883119906
119904 119906 (119904)) + 119892 (119904 119883
119906
119904 119906 (119904))] 119889119904
(77)
The proof is finished
We immediately deduce the following consequences
Theorem 12 Let 119905 isin [0infin) and 119909 isin C be fixed assumethat the set-valued map Γ has nonempty values and it admitsa measurable selection Γ
0 [0infin)timesCtimesΞlowast rarr 119880 and assume
that a control 119906(sdot) satisfies
119906 (119904) = Γ0(119904 119883
119906
119904 120577 (119904 119883
119906
119904))
119875-as for almost every 119904 isin [119905infin)
(78)
Then 119869(119905 119909 119906) = 120592(119905 119909) and the pair (119906(sdot) 119883) is optimal forthe control problem starting from 119909 at time 119905
Such a control can be shown to exist if there exists a solutionfor the so-called closed-loop equation as follows
119889119883 (119904)=119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904+119866 (119904 119883
119904)
times(119877 (119904 119883119904 Γ
0(119904 119883
119904 120577 (119904 119883
119904))) 119889119904 + 119889119882 (119904))
119904 isin [119905infin)
119883119905= 119909
(79)
since in this case we can define an optimal control setting
119906 (119904) = Γ0(119904 119883
119904 120577 (119904 119883
119904)) (80)
However under the present assumptions we cannot guaranteethat the closed-loop equation has a solution in the mildsense To circumvent this difficulty we will revert to a weakformulation of the optimal control problem
5 Existence of Optimal Control
We formulate the optimal control problem in the weak sensefollowing the approach of [31]Themain advantage is that wewill be able to solve the closed-loop equation in a weak senseand hence to find an optimal control even if the feedbacklaw is nonsmooth
We call (ΩF F119905119905ge0 119875119882) an admissible setup if
(ΩF F119905119905ge0 119875) is a filtered probability space satisfying the
usual conditions and 119882 is a cylindrical 119875-Wiener processwith values in Ξ with respect to the filtration F
119905119905ge0
By an admissible control system we mean (ΩF
F119905119905ge0 119875119882 119906119883
119906
) where (ΩF F119905119905ge0 119875 119882) is an
admissible setup 119906 is an F119905-predictable process with values
in119880 and119883119906 is a mild solution of (58) An admissible controlsystem will be briefly denoted by (119882 119906119883119906
) in the followingOur purpose is to minimize the cost functional
119869 (119906) = 119864int
infin
119905
119890minus120582119904
119892 (119904 119883119906
119904 119906 (119904)) 119889119904 (81)
over all the admissible control systemOur main result in this section is based on the solvability
of the closed-loop equation
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904 + 119866 (119904 119883
119904)
times (119877 (119904 119883119904 Γ
0(119904 119883
119904 120577 (119904 119883
119904))) 119889119904 +119889119882 (119904))
119904 isin [119905infin)
119883119905= 119909
(82)
In the following sense we say that 119883 is a weak solution of(82) if there exists an admissible setup (ΩF F
119905119905ge0 119875119882)
and anF119905-adapted continuous process119883(119905)with values in119867
which solves the equation in the mild sense namely 119875-as
119883 (119904) = 119890(119904minus119905)119860
119909 (0) + int
119904
119905
119890(119904minus120590)119860
119865 (120590119883120590) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883120590) 119877
times (120590119883120590 Γ
0(120590119883
120590 120577 (120590 119883
120590))) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883120590) 119889119882
120590 119904 isin [119905infin)
(83)
119883119905= 119909 (84)
Theorem 13 Assume that Hypothesis 3 holds Then thereexists a weak solution of the closed-loop equation (82) whichis unique in law
12 Abstract and Applied Analysis
Proof (uniqueness) Let 119883 be a weak solution of (82) in anadmissible setup (ΩF F
119905119905ge0 119875119882)We define
120588 (119879) = exp(int119879
119905
minus119877lowast
(120590 119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889119882 (120590)
minus1
2int
119879
119905
1003816100381610038161003816119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)))10038161003816100381610038162
119889120590)
(85)
Since 119877 is bounded the Girsanov theorem ensures that thereexists a probability measure 1198750 such that the process
1198820
(119904) = 119882 (119904) + int
119904
119905and119904
119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889120590
119904 isin [0infin)
(86)
is a 1198750-Wiener process and
1198750
|F119879
= 120588 (119879) 119875|F119879
(87)
Let us denote by F0
119905119905ge0
the filtration generated by1198820 andcompleted in the usual way In (ΩF
[0infin) F0
119905119905ge0 119875
0
) 119883 isa mild solution of
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119905 119883119904) 119889119904
+ 119866 (119904 119883119904) 119889119882
0
(119904) 119904 isin [119905infin)
119883119905= 119909
120588 (119879) = exp(int119879
119905
minus119877lowast
(120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)) 119889119882
0
(120590)
+ 12int
119879
119905
1003816100381610038161003816119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)))10038161003816100381610038162
119889120590)
(88)
By Hypothesis 3 the joint law of 119883 and 1198820 is uniquely
determined by 119860 119865 119866 and 119909 Taking into account the lastdisplayed formula we conclude that the joint law of 119883 and120588(119879) under1198750 is also uniquely determined and consequentlyso is the law of 119883 under 119875 This completes the proof of theuniqueness part
Proof (existence) Let (ΩF 119875) be a given complete probabil-ity space 119882(119905) 119905 ge 0 is a cylindrical Wiener process on(ΩF 119875)with values inΞ and F
119905119905ge0
is the natural filtrationof 119882(119905) 119905 ge 0 augmented with the family of119875-null sets Let119883(sdot) be the mild solution of
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904
+ 119866 (119904 119883119904) 119889119882 (119904) 119904 isin [119905infin)
119883119905= 119909
(89)
and by the Girsanov theorem let 1198751 be the probability on Ωunder which
1198821
(119904) = 119882 (119904) minus int
119904
119905and119904
119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889120590
(90)
is aWiener process (notice that 119877 is bounded)Then119883 is theweak solution of (82) relatively to the probability 1198751 and theWiener process1198821
Now we can state the main result of this section
Corollary 14 Assume that Hypothesis 3 holds true and 120582
verifies (62) Also assume that the set-valued map Γ hasnonempty values and it admits a measurable selection Γ
0
[0infin) timesC times Ξlowast
rarr 119880 Then for every 119905 isin [0infin) and x isin Cand for all admissible control system (119882 119906119883
119906
) one has
119869 (119906 119905 119909) ge 120592 (119905 119909) (91)
and the equality holds if
119906 (119904) = Γ0(119904 119883
119906
119904 120577 (119904 119883
119906
119904))
119875 minus 119886119904 119891119900119903 119886119897119898119900119904119905 119890119907119890119903119910 119904 isin [119905infin)
(92)
Moreover from Theorem 13 it follows that the closed-loop equation (82) admits a weak solution (ΩFF
119905119905ge0 119875119882119883) which is unique in law and setting
119906 (119904) = Γ0(119904 119883
119904 120577 (119904 119883
119904)) (93)
we obtain an optimal admissible control system (119882 119906119883)
6 Applications
In this section we present a simple application of the previousresults We consider the stochastic delay partial differentialequation in the bounded domain 119861 sub 119877
119899 with smoothboundary 120597119861 as follows
119889119911119906
(119905 120585) = Δ119911119906
(119905 120585) 119889119905 + 119891 (119905 119911119906
119905(120585)) 119889119905
+
119889
sum
119894=1
119892119894(119905 119911
119906
119905(120585)) [119903
119894
(120585) 119906119894
(119905) 119889119905 + 119889119882119894
(119905)]
119911119906
0(120579 120585) = 119909 (120579 120585) 120585 isin 119861 120579 isin [minus1 0]
119911119906
(119905 120585) = 0 119905 isin [0infin) 120585 isin 120597119861
(94)
Here119882 = (1198821
1198822
119882119889
) is a standardWiener process in119877119889 and the functions 119891 [0 +infin) times 119862([minus1 0] 119877) rarr 119877 and119892119894 [0 +infin) times 119862([minus1 0] 119877) rarr 119877 are Lipschitz continuous
and bounded Setting 119880 as a bounded subset of 119877119889 Ξ = 119877119889
119867 = 1198712
(119861) and 119909 isin 119862([minus1 0]119867) We define 119865 and 119866 asfollowing
119865 (119905 119909) (120585) = 119891 (119905 119909 (120585))
(119866 (119905 119909) 119911) (120585) =
119889
sum
119894=1
119892119894(119905 119909 (120585)) 119911
119894
(120585)
120585 isin 119861 119909 isin 119862 ([minus1 0] 119867) 119911 isin 119871 (Ξ119867)
(95)
and let 119860 denote the Laplace operator Δ in 1198712
(119861) withdomain11988222
(119861)⋂11988212
0(119861) then (94) has the form (58) and
Hypothesis 1 holds
Abstract and Applied Analysis 13
Let us consider the optimal control problem associatedwith the cost
119869 (119906) = 119864int
infin
0
119890minus120582119905
[int119861
120590 (120585 119911119906
119905(120585)) 119889120585 + 119906
2
(119905)] 119889119905 (96)
where 120582 verifies (62) and 120590 119862([minus1 0] 119877) times 119880 rarr [0infin) isa bounded measurable function Define 119892 119862([minus1 0]119867) times
119880 rarr [0infin) and 119877 119862([minus1 0]119867) times 119880 rarr Ξ by119892(119910 119906) = int
119861
120590(119905 119910(120585) 119906)119889120585 + 1199062 and 119877(119910 119906) = (int
119861
1199031
(120585)1199061
119889120585
int119861
1199032
(120585)1199062
119889120585 int119861
119903119889
(120585)119906119889
119889120585) for 119910 isin 119862([minus1 0]119867) 119906 =
(1199061
1199062
119906119889
) isin 119880 respectively It can be easily verifiedthat Hypothesis 3 holds true and the set-valued map Γ hasnonempty values and it admits a measurable selection Γ
0
[0infin) times C times Ξlowast
rarr 119880 Then the closed-loop equation(82) admits a weak solution (ΩF F
119905119905ge0 119875119882 119906 119911
sdot(sdot)) and
setting
119906 (119904) = Γ0(119904 119911
119904(sdot) 120577 (119904 119911
119904(sdot))) (97)
we obtain an optimal admissible control system (119882 119906 119911(sdot))
References
[1] J Bismut ldquoOn optimal control of linear stochastic equationswith a linear-quadratic criterionrdquo SIAM Journal on Control andOptimization vol 15 no 3 pp 1ndash4 1977
[2] N Nagase ldquoOn the existence of optimal control for controlledstochastic partial differential equationsrdquo Nagoya MathematicsJournal vol 115 pp 73ndash85 1989
[3] N El Karoui D Huu Nguyen and M Jeanblanc-Pique ldquoCom-pactification methods in the control of degenerate diffusionsrdquoStochastics vol 20 pp 169ndash219 1987
[4] M Nisio ldquoOptimal control for stochastic partial differentialequations and viscosity solutions of Bellman equationsrdquoNagoyaMathematics Journal vol 123 pp 13ndash37 1991
[5] M Nisio ldquoOn sensitive control for stochastic partial differentialequationsrdquo in Stochastic Analysis on Infinite Dimensional SpacesProceedings of the US Japan Bilateral Seminar H Kunita et alEd vol 310 of Pitman Research Notes Mathematical Series pp231ndash241 Longman Scientific and Technical Baton Rouge LaUSA January 1994
[6] R Buckdahn and A Rascanu ldquoOn the existence of stochasticoptimal control of distributed state systemrdquoNonlinear AnalysisTheory Methods and Applications vol 52 no 4 pp 1153ndash11842003
[7] V Barbu and G Da Prato Equations in Hilbert Spaces vol 86 ofPitman Research Notes in Mathematics Pitman 1983
[8] P Cannarsa and G Da Prato ldquoSecond-order Hamilton-Jacobiequations in infinite dimensionsrdquo SIAM Journal on Control andOptimization vol 29 no 2 pp 474ndash492 1991
[9] P Cannarsa and G Da Prato ldquoDirect solution of a second-orderHamilton-Jacobi equations in Hilbert spacesrdquo in StochasticPartial Differential Equations and Applications G Da Pratoand L Tubaro Eds vol 268 of Pitman Research Notes inMathematics Pitman 1992
[10] F Gozzi ldquoRegularity of solutions of second order Hamilton-Jacobi equations and application to a control problemrdquoCommu-nications in Partial Differential Equations vol 20 pp 775ndash8261995
[11] F Gozzi ldquoGlobal regular solutions of second order Hamilton-Jacobi equations in Hilbert spaces with locally Lipschitz non-linearitiesrdquo Journal of Mathematical Analysis and Applicationsvol 198 no 2 pp 399ndash443 1996
[12] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems and Control Lettersvol 14 no 1 pp 55ndash61 1990
[13] N El Karoui and LMazliak Eds Backward Stochastic Differen-tial Equations vol 364 of Pitman ResearchNotes inMathematicsSeries Longman 1997
[14] E Pardoux and BSDEs ldquoweak convergence and homogeneiza-tion of semilinear PDEsrdquo in Non- Linear Analysis DifferentialEquations and Control F H Clarke and R J Stern Eds pp503ndash549 Kluwer Dordrecht The Netherlands 1999
[15] S Peng ldquoA generalized dynamic programming principle andHamilton-Jacobi-Bellman equationrdquo Stochastics and StochasticsReports vol 38 pp 119ndash134 1992
[16] N E Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997
[17] SHamad120583ene and J P Lepeltier ldquoBackward equations stochas-tic control and zero-sum stochastic differential gamesrdquo Stochas-tics and Stochastics Reports vol 54 pp 221ndash231 1995
[18] N El-Karoui and S Hamadene ldquoBSDEs and risk-sensitive con-trol zero-sum and nonzero-sum game problems of stochasticfunctional differential equationsrdquo Stochastic Processes and theirApplications vol 107 no 1 pp 145ndash169 2003
[19] M Fuhrman and G Tessiture ldquoExistence of optimal stochasticcontrols and global solutions of forward-backward stochasticdifferential equationsrdquo SIAM Journal on Control and Optimiza-tion vol 43 no 3 pp 813ndash830 2005
[20] M Fuhrman Y Hu and G Tessitore ldquoOn a class of stochasticoptimal control problems related to bsdes with quadraticgrowthrdquo SIAM Journal on Control and Optimization vol 45 no4 pp 1279ndash1296 2006
[21] M Fuhrman and G Tessitore ldquoNonlinear kolmogorov equa-tions in infinite dimensional spaces the backward stochasticdifferential equations approach and applications to optimalcontrolrdquoAnnals of Probability vol 30 no 3 pp 1397ndash1465 2002
[22] F Masiero ldquoSemilinear kolmogorov equations and applicationsto stochastic optimal controlrdquo Applied Mathematics and Opti-mization vol 51 no 1 pp 201ndash250 2005
[23] M Fuhrman FMasiero andG Tessitore ldquoStochastic equationswith delay optimal control via BSDEs and regular solutions ofHamilton-jacobi-bellman equationsrdquo SIAM Journal on Controland Optimization vol 48 no 7 pp 4624ndash4651 2010
[24] M Fuhrman and G Tessiture ldquoInfinite horizon backwardstochastic differential equations and elliptic equations in hilbertspacesrdquo Annals of Probability vol 32 no 1 pp 607ndash660 2004
[25] F Masiero ldquoInfinite horizon stochastic optimal control prob-lems with degenerate noise and elliptic equations in Hilbertspacesrdquo Applied Mathematics and Optimization vol 55 no 3pp 285ndash326 2007
[26] M Fuhrman ldquoA class of stochastic optimal control problemsin Hilbert spaces BSDEs and optimal control laws stateconstraints conditioned processesrdquo Stochastic Processes andtheir Applications vol 108 no 2 pp 263ndash298 2003
[27] F Masiero ldquoStochastic optimal control problems and parabolicequations in banach spacesrdquo SIAM Journal on Control andOptimization vol 47 no 1 pp 251ndash300 2008
[28] G Da Prato and J Zabczyk Ergodicity For Infinite-DimensionalSystems Cambridge University Press 1996
14 Abstract and Applied Analysis
[29] J Zabczyk ldquoParabolic equations on Hilbert spacesrdquo in StochaS-tic PDErsquoS and Kolmogorov Equations in Infinite Dimensionsvol 1715 of Lecture Notes in Math pp 117ndash213 Springer BerlinGermany 1999
[30] G Da Prato and J Zabczyk Stochstic Equations in InfiniteDimensions Cambridge University Press 1992
[31] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 of Applications of MathematicsSpringer New York NY USA 1993
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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Mathematical PhysicsAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
Proof We define a mapping Φ from H119902
120578(119905) times [0infin) times C to
H119902
120578(119905) by the formula
Φ(119883sdot 119905 119909)
119904(119897) = 119890
(119904+119897minus119905)119860
119909 (0) + int
119904+119897
119905
119890(119904+119897minus120590)119860
119865 (120590119883120590) 119889120590
+ int
119904+119897
119905
119890(119904+119897minus120590)119860
119866 (120590119883120590) 119889119882 (120590)
119904 isin [119905infin) 119897 isin [minus120591 0] 119904 + 119897 ge 119905
Φ(119883sdot 119905 119909)
119904(119897) = 119909 (119904 + 119897 minus 119905)
119904 isin [119905infin) 119897 isin [minus120591 0] 119904 + 119897 lt 119905
(16)
We are going to show that provided 120578 is suitably chosenΦ(sdot 119905 119909) is well defined and that it is a contraction inH119902
120578(119905)
that is there exists 119888 lt 1 such that10038161003816100381610038161003816Φ (119883
1
sdot 119905 119909) minus Φ (119883
1
sdot 119905 119909)
10038161003816100381610038161003816H119902
120578(119905)
le 119888100381610038161003816100381610038161198831
sdotminus 119883
2
sdot
10038161003816100381610038161003816H119902
120578(119905)
1198831
sdot 119883
2
sdotisinH
119902
120578(119905)
(17)
For simplicity we set 119905 = 0 and we treat only the case 119865 = 0the general case being handled in a similar way We will usethe so called factorization method see [28 Theorem 525]Let us take 119902 gt 1 and 120572 isin (0 1) such that 1119902 lt 120572 lt (12) minus120574 and let 119888minus1
120572= int
s120590
(119904 minus 119903)120572minus1
(119903 minus 120590)minus120572
119889119903By the stochastic Fubini theorem
Φ(119883sdot 0 119909)
119904(119897) = 119890
(119904+119897)119860
119909 (0)
+ 119888120572int
119904+119897
0
int
119904+119897
120590
(119904 + 119897 minus 119903)120572minus1
(119903 minus 120590)minus120572
times 119890(119904+119897minus119903)119860
119890(119903minus120590)119860
119889119903119866 (120590119883120590) 119889119882 (120590)
= 119890(119904+119897)119860
119909 (0) + Φ1015840
(119883119904) (119897)
119904 isin [0infin) 119897 isin [minus120591 0] 119904 + 119897 ge 0
Φ(119883sdot 0 119909)
119904(119897) = 119909 (119904 + 119897)
119904 isin [0infin) 119897 isin [minus120591 0] 119904 + 119897 lt 0
(18)
where
Φ1015840
(119883sdot)119904(119897) = 119888
120572int
119904+119897
0
(119904 + 119897 minus 119903)120572minus1
119890(119904+119897minus119903)119860
119884 (119903) 119889119903
119884 (119903) = int
119903
0
(119903 minus 120590)minus120572
119890(119903minus120590)119860
119866 (120590119883120590) 119889119882 (120590)
(19)
Since supminus120591le119897le0
|119890(119904+119897)119860
119909(0)| le 119872119890120596119904
|119909|119862 the process 119890(119904+sdot)119860
119909(0) 119904 ge 0 belongs to H119902
120578provided 120596 + 120578 lt 0 Next we
estimate Φ1015840
(119883sdot) where
10038161003816100381610038161003816Φ1015840
(119883sdot)119904(119897)10038161003816100381610038161003816le 119888
120572int
119904+119897
0
(119904 + 119897 minus 119903)120572minus1
119872119890(119904+119897minus119903)120596
|119884 (119903)| 119889119903
(20)
setting 1199021015840 = 119902(119902 minus 1) so that
11989011990212057811990410038161003816100381610038161003816Φ1015840
(119883sdot)119904
10038161003816100381610038161003816
119902
le 119888119902
120572119872
119902 supminus120591le119897le0
119890119902120578119904
(int
119904+119897
0
(119904+119897minus 119903)120572minus1
119890120596(119904+119897minus119903)
|119884 (119903)| 119889119903)
119902
le 119888119902
120572119872
119902 supminus120591le119897le0
(int
119904+119897
0
(119904 + 119897 minus 119903)120572minus1
119890((120596+120578)119902
1015840
)(119904+119897minus119903)
times119890((120596+120578)119902)(119904minus119903)
119890120578119903
|119884 (119903)| 119889119903)
119902
le 119888119902
120572119872
119902 supminus120591le119897le0
(int
119904+119897
0
119890(120578+120596)(119904+119897minus119903)
(119904 + 119897 minus 119903)(120572minus1)119902
1015840
119889119903)
1199021199021015840
times int
119904+119897
0
119890(120578+120596)(119904minus119903)
119890119902120578119903
|119884 (119903)|119902
119889119903
le 119888119902
120572119872
119902
(int
119904
0
119890(120578+120596)119903
1199031199021015840
(120572minus1)
119889119903)
1199021199021015840
times int
119904
0
119890(120578+120596)(119904minus119903)
119890119902120578119903
|119884 (119903)|119902
119889119903
(21)
Applying the Young inequality for convolutions we have
int
infin
0
11989011990212057811990410038161003816100381610038161003816Φ1015840
(119883sdot)119904
10038161003816100381610038161003816
119902
119889119904 le 119888119902
120572119872
119902
(int
infin
0
119890(120578+120596)119904
1199041199021015840
(120572minus1)
119889119904)
1199021199021015840
times int
infin
0
119890(120578+120596)119904
119889119904int
infin
0
119890119902120578119904
|119884 (119904)|119902
119889119904
(22)
and we conclude that10038161003816100381610038161003816Φ1015840
(119883sdot)10038161003816100381610038161003816119871119902
P(Ω119871119902
120578(C))
le 119888120572119872|119884|
119871119902
P(Ω119871119902
120578(119867))
times (int
infin
0
119890(120578+120596)119904
1199041199021015840
(120572minus1)
119889119904)
11199021015840
times (int
infin
0
119890(120578+120596)119904
119889119904)
1119902
(23)
If we start again from (20) and apply theHolder inequality weobtain
10038161003816100381610038161003816119890120578(119904+119897)
Φ1015840
(119883sdot)119904(119897)10038161003816100381610038161003816le 119888
120572119872(int
119904+119897
0
119903(120572minus1)119902
1015840
119890(120596+120578)119903119902
1015840
119889119903)
11199021015840
times (int
119904+119897
0
119890120578119903119902
|119884 (119903)|119902
119889119903)
1119902
10038161003816100381610038161003816119890120578119904
Φ1015840
(119883sdot)119904
10038161003816100381610038161003816le 119888
120572119872(int
119904
0
119903(120572minus1)119902
1015840
119890(120596+120578)119903119902
1015840
119889119903)
11199021015840
times (int
119904
0
119890120578119903119902
|119884(119903)|119902
119889119903)
1119902
(24)
Abstract and Applied Analysis 5
So we conclude that10038161003816100381610038161003816Φ1015840
(119883sdot)10038161003816100381610038161003816119871119902
P(Ω119862120578(C))
le 119888120572119872|119884|
119871119902
P(Ω119871119902
120578(119867))
times (int
infin
0
119903(120572minus1)119902
1015840
119890(120596+120578)119903119902
1015840
119889119903)
11199021015840
(25)
On the other hand by the Burkholder-Davis-Gundy inequal-ities for some constant 119888
119902depending only on 119902 we have
119864|119884 (119903)|119902
le 119888119902119864(int
119903
0
(119903 minus 120590)minus212057210038161003816100381610038161003816119890(119903minus120590)119860
119866 (120590119883120590)10038161003816100381610038161003816
2
1198712(Ξ119867)
119889120590)
1199022
le 119871119902
119888119902119864
times (int
119903
0
(119903 minus 120590)minus2120572minus2120574
1198902120596(119903minus120590)
(1 +1003816100381610038161003816119883120590
10038161003816100381610038162
119862) 119889120590)
1199022
(26)
which implies that
[119864|119884 (119903)|119902
]2119902
le 1198712
1198882119902
119902int
119903
0
(119903 minus 120590)minus2120572minus2120574
times 1198902120596(119903minus120590)
[119864(1 +1003816100381610038161003816119883120590
1003816100381610038161003816119862)119902
]2119902
119889120590
(27)
so that
1198902120578119903
[119864|119884 (119903)|119902
]2119902
le 1198621int
119903
0
(119903 minus 120590)minus2120572minus2120574
1198902(120596+120578)(119903minus120590)
1198902120578120590
119889120590
+ 1198622int
119903
0
(119903 minus 120590)minus2120572minus2120574
1198902(120596+120578)(119903minus120590)
times 1198902120578120590
[1198641003816100381610038161003816119883120590
1003816100381610038161003816119902
119862]2119902
119889120590
(28)
for suitable constants 1198621 119862
2 Applying the Young inequality
for convolutions we obtain
int
infin
0
119890119902120578119903
119864|119884 (119903)|119902
119889119904le 1198621(int
infin
0
119904minus2120572minus2120574
1198902(120596+120578)119904
119889119904)
1199022
int
infin
0
119890119902120578119904
119889119904
+ 1198622(int
infin
0
119904minus2120572minus2120574
1198902(120596+120578)119904
119889119904)
1199022
times int
infin
0
119890119902120578119904
1198641003816100381610038161003816119883119904
1003816100381610038161003816119902
119862119889119904
(29)
This shows that |119884|119871119902
P(Ω119871119902
120578(119867))
is finite provided we assumethat 120578 lt 0 and 120596 + 120578 lt 0 so the map is well defined
If 1198831
sdot 119883
2
sdotare processes belonging to H119902
120578and 1198841 1198842 are
defined accordingly the entirely analogous passages showthat100381610038161003816100381610038161198841
minus 119884210038161003816100381610038161003816119871119902
P(Ω119871119902
120578(119867))
le 1198711198881119902
120572
100381610038161003816100381610038161198831
sdotminus 119883
2
sdot
10038161003816100381610038161003816119871119902
P(Ω119871119902
120578(C))
times (int
infin
0
119904minus2120572minus2120574
1198902(120596+120578)119904
119889119904)
12
(30)
Recalling the inequalities (23) and (25) and noting that themap Y rarr Φ
1015840
(Xsdot) is linear we obtain an explicit expression
for the constant 119888 in (17) and it is immediate to verify that119888 lt 1 provided 120578 lt 0 is chosen sufficiently large We fixsuch a value of 120578(119902) The first result is a consequence of thecontraction principle The estimate (15) also follows from thecontraction property ofΦ(sdot 119905 119909)
For investigating the dependence of the solution119883(119904 119905 119909)on the initial data 119909 and 119905 we reformulate (13) as an equationon [0infin) We set
119878 (119904) = 119890119904119860
for 119904 ge 0 119878 (119904) = 119868 for 119904 lt 0 (31)
and we consider the equation
119883(119904) = 119878 (119904 minus 119905) 119909 ((0 and (119904 minus 119905)) or (minus120591))
+ int
119904
0
119868[119905infin)
(120590) 119878 (119904 minus 120590) 119865 (120590119883120590) 119889120590
+ int
119904
0
119868[119905infin)
(120590) 119878 (119904 minus 120590)
times 119866 (120590119883120590) 119889119882 (120590) 119904 isin [0infin)
1198830(120579) = 119909 ((minus119905 + 120579) or (minus120591)) 120579 isin [minus120591 0]
(32)
Under the assumptions of Hypothesis 1 by Theorem 2 it iseasy to prove that equation (32) has a unique solution 119883 and119883sdotisin H
119902
120578(119902)for every 119902 isin [2infin) It clearly satisfies 119883(119904) =
119909((119904 minus 119905) or (minus120591)) for 119904 isin [minus120591 119905) and its restriction to the timeinternal [119905infin) is the unique mild solution of (10) From nowon we denote by119883(119904 119905 119909) 119904 isin [0infin) the solution of (32)
We need the following parameter-depending contractionprinciple which is stated in the following lemma and provedin [29 Theorems 101 and 102]
Lemma3 (ParameterDependingContraction Principle) Let119861119863 denote Banach spaces Let ℎ 119861times119863 rarr 119861 be a continuousmapping satisfying
1003816100381610038161003816ℎ (1199091 119910) minus ℎ (1199092 119910)1003816100381610038161003816 le 120572
10038161003816100381610038161199091 minus 11990921003816100381610038161003816 (33)
for some 120572 isin [0 1) and every 1199091 119909
2isin 119861 y isin 119863 Let 120601(119910)
denote the unique fixed point of the mapping ℎ(sdot 119910) 119861 rarr 119861Then 120601 119863 rarr 119861 is continuous
Theorem 4 Assume that Hypothesis 1 holds true Then forevery 119902 isin [1infin) the map (119905 119909) rarr 119883
sdot(119905 119909) is continuous
from [0infin) timesC toHq120578(q)
Proof Clearly it is enough to prove the claim for 119902 large Letus consider the map Φ defined in the proof of Theorem 2 In
6 Abstract and Applied Analysis
our present notationΦ can be seen as a mapping fromH119902
120578times
[0infin) timesC toH119902
120578as follows
Φ(119883sdot 119905 119909)
119904(119897) = 119878 (119904 + 119897 minus 119905) 119909 (0)
+ int
119904+119897
0
119868[119905infin)
(120590) 119878 (119904 + 119897 minus 120590) 119865 (120590119883120590) 119889120590
+ int
119904+119897
0
119868[119905infin)
(120590) 119878 (119904 + 119897 minus 120590)
times 119866 (120590119883120590) 119889119882 (120590)
119904 isin [0infin) 119897 isin [minus120591 0] 119904 + 119897 ge 119905
Φ(119883sdot 119905 119909)
119904(119897) = 119909 ((119904 + 119897 minus 119905) or (minus120591))
119904 isin [0infin) 119897 isin [minus120591 0] 119904 + 119897 le 119905
(34)
By the arguments of the proof of Theorem 2 Φ(sdot 119905 119909)is a contraction in H119902
120578uniformly with respect to 119905 119909
The process 119883sdot(119905 119909) is the unique fixed point of Φ(sdot 119905 119909)
So by the parameter-depending contraction principle(Lemma 3) it suffices to show that Φ is continuous fromH119902
120578times [0infin) times C to H119902
120578 From the contraction property
of Φ(sdot 119905 119909) mentioned earlier we have that Φ(sdot 119905 119909) iscontinuous uniformly in 119905 119909 Moreover for fixed 119883
sdot it is
easy to verify that Φ(119883sdot sdot sdot) is continuous from [0infin) times C
toH119902
120578 The proof is finished
Remark 5 By similar passages we can show that for fixed119905 Theorem 4 still holds true for 119902 large enough if the spaces[0infin) times C and H119902
120578are replaced by the spaces 119871119902(ΩCF
119905)
and H119902
120578(119905) respectively where 119871119902(ΩCF
119905) denotes that the
space of F119905-measurable function with value in C such that
the norm
|119909|119902
= 119864|119909|119902
119862 (35)
is finite
3 The Backward-Forward System
In this section we consider the system of stochastic differen-tial equations 119875-as
119883 (119904) = 119890(119904minus119905)119860
119909 (0) + int
119904
119905
119890(119904minus120590)119860
119865 (120590119883120590) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883120590) 119889119882 (120590) 119904 isin [119905infin)
119883119905= 119909 isin C
119884 (119904) minus 119884 (119879) + int
119879
119904
119885 (120590) 119889119882 (120590) + 120582int
119879
119904
119884 (120590) 119889120590
= int
119879
119904
120595 (120590119883120590 119884 (120590) 119885 (120590)) 119889120590 0 le 119904 le 119879 lt infin
(36)
for 119904 varying on the time interval [119905infin) sub [0infin) Asin Section 2 we extend the domain of the solution setting119883(119904 119905 119909) = 119909((119904 minus 119905) or (minus120591)) for 119904 isin [minus120591 119905)
We make the following assumptions
Hypothesis 2 Themapping 120595 [0infin)timesCtimes119870times1198712(Ξ 119870) rarr
119870 is Borelmeasurable such that for all 119905 isin [0infin)120595(119905 sdot) Ctimes119870 times 119871
2(Ξ 119870) rarr 119870 is continuous and for some 119871
119910 119871
119911gt 0
120583 isin 119877 and119898 ge 11003816100381610038161003816120595 (119904 119909 1199101 1199111) minus 120595 (119904 119909 1199102 1199112)
1003816100381610038161003816
le 119871119910
10038161003816100381610038161199101 minus 11991021003816100381610038161003816 + 119871119911
10038161003816100381610038161199111 minus 11991121003816100381610038161003816
1003816100381610038161003816120595 (119904 119909 119910 119911)1003816100381610038161003816 le 119871 (1 + |119909|
119898
119862+10038161003816100381610038161199101003816100381610038161003816 + |119911|)
⟨120595 (119904 119909 1199101 119911) minus 120595 (119904 119909 119910
2 119911) 119910
1minus 119910
2⟩119870ge 120583
10038161003816100381610038161199101 minus 119910210038161003816100381610038162
(37)
for every 119904 isin [0infin) 119909 isin C 119910 1199101 119910
2isin 119870 119911 119911
1 and 119911
2isin
1198712(Ξ 119870)We note that the third inequality in (37) follows from the
first one taking 120583 = minus119871119910but that the third inequalitymay also
hold for different values of 120583Firstly we consider the backward stochastic differential
equation
119884 (119904) minus 119884 (119879) + int
119879
119904
119885 (120590) 119889119882 (120590) + 120582int
119879
119904
119884 (120590) 119889120590
= int
119879
119904
120595 (120590119883120590 119884 (120590) 119885 (120590)) 119889120590 0 le 119904 le 119879 lt infin
(38)
119870 is a Hilbert space the mapping 120595 [0infin) times C times 119870 times
1198712(Ξ 119870) rarr 119870 is a given measurable function 119883
sdotis a
predictable process with values in another Banach space Cand 120582 is a real number
Theorem 6 Assume that Hypothesis 2 holds Let 119901 gt 2 and120575 lt 0 be given and choose
119902 ge 119898119901 120578 gt120575
119898 (39)
Then the following hold
(i) For 119883sdotisin 119871
119902
P(Ω 119871
119902
120578(C)) and 120582 gt minus(120575 + 120583 minus (119871
2
1199112))
(38) has a unique solution in Kp120575that will be denoted
by (119884(119883sdot)(119904) 119885(119883
sdot)(119904)) 119904 ge 0
(ii) The estimate
119864sup119904ge0
(119884 (119883sdot) (119904))
119901
119890119901120575119904
+ 119864(int
infin
0
11989021205751205901003816100381610038161003816119884(119883sdot
)(120590)10038161003816100381610038162
119889120590)
1199012
+ 119864(int
infin
0
11989021205751205901003816100381610038161003816119885 (119883sdot
) (120590)10038161003816100381610038162
119889120590)
1199012
le 119888(1 +1003816100381610038161003816119883sdot
1003816100381610038161003816119898
119871119902
P(Ω119871119902
120578(C))
)119901
(40)
Abstract and Applied Analysis 7
holds for a suitable constant 119888 In particular 119884(119883sdot) isin
119871119901
P(Ω 119862
120575(119870))
(iii) The map 119883sdotrarr (119884(119883
sdot) 119885(119883
sdot)) is continuous from
119871119902
P(Ω 119871
119902
120578(C)) toK119901
120575 and 119883
sdotrarr 119884(119883
sdot) is continuous
from 119871119902
P(Ω 119871
119902
120578(C)) to 119871119901
P(Ω 119862
120575(119870))
(iv) The statements of points (i) (ii) and (iii) still holdtrue if the space 119871119902
P(Ω 119871
119902
120578(C)) is replaced by the space
119871119902
P(Ω 119862
120578(C))
Proof The theorem is very similar to Proposition 311 in [24]The only minor difference is that the mapping 120595 [0infin) times
Ctimes119870times1198712(Ξ 119870) rarr 119870 is a givenmeasurable function while
in [24] the measurable function 120595 is from119867 times119870 times 1198712(Ξ 119870)
to 119870 however the same arguments apply
Theorem 7 Assume that Hypothesis 1 holds and thatHypothesis 2 holds true in the particular case 119870 = 119877 Thenfor every 119901 gt 2 119902 120575 lt 0 satisfying (39) with 120578 = 120578(119902)and for every 120582 gt 120582
1015840
= minus(120575 + 120583 minus (1198712
1199112)) there exists a
unique solution in H119902
120578(119902)times K
119901
120575of (36) that will be denoted
by (119883(sdot 119905 119909) 119884(sdot 119905 119909) 119885(sdot 119905 119909)) Moreover 119884(sdot 119905 119909) isin
119871119901
P(Ω 119862
120575(119877)) The map (119905 119909) rarr (119884(sdot 119905 119909) 119885(sdot 119905 119909)) is con-
tinuous from [0infin)timesC toK119901
120575 and themap (119905 119909) rarr 119884(sdot 119905 119909)
is continuous from [0infin) timesC to 119871119901P(Ω 119862
120575(119877))
Proof We first notice that the system is decoupled the firstdoes not contain the solution (119884 119885) of the second oneThere-fore under the assumption of Hypothesis 1 by Theorem 2there exists a unique solution 119883(sdot 119905 119909) and 119883
sdot(119905 119909) isin H
119902
120578(119902)
of the first equation Moreover from Theorem 4 it followsthat the map (119905 119909) rarr 119883
sdot(119905 119909) is continuous from [0infin)timesC
toH119902
120578(119902)
Let 119870 = 119877 from Theorem 6 we have that thereexists a unique solution (119884(sdot 119905 119909) 119885(sdot 119905 119909)) isin K
119901
120575of the
second equation and the map 119883sdotrarr (119884(119883
sdot) 119885(119883
sdot)) is
continuous from H119902
120578(119902)to K
119901
120575while X
sdotrarr (Y(X
sdot)) is
continuous fromH119902
120578(119902)to119871119901
P(Ω 119862
120575(119877))We have proved that
(119883(sdot 119905 119909) 119884(sdot 119905 119909) 119885(sdot 119905 119909)) isin H119902
120578(119902)times K
119901
120575is the unique
solution of (36) and the other assertions follow from com-position
Remark 8 From Remark 5 by similar passages we can showthat for fixed 119905 and for 119902 large enough under the assumptionsof Theorem 7 the map 119909 rarr (119884(sdot 119905 119909) 119885(sdot 119905 119909)) is continu-ous from 119871
119902
(ΩCF119905) toK119901
120575(119905)
We also remark that the process 119883(sdot 119905 119909) is F[119905infin)
measurable since C is separable Banach space we have that119883sdot(119905 119909) is F
[119905infin)measurable So that 119884(119905) is measurable
with respect to both F[119905infin)
and F119905 it follows that 119884(119905) is
deterministicFor later use we notice three useful identities for 119905 le 119904 lt
infin the equality 119875-as
119883119897(119904 119883
119904(119905 119909)) = 119883
119897(119905 119909) 119897 isin [119904infin) (41)
is a consequence of the uniqueness of the solution of (13)Since the solution of the backward equation is uniquely
determined on an interval [119904infin) by the values of the process119883sdoton the same interval for 119905 le 119904 lt infin we have 119875-as
119884 (119897 119904 119883119904(119905 119909)) = 119884 (119897 119905 119909) for 119897 isin [119904infin)
119885 (119897 119904 119883119904(119905 119909)) = 119885 (119897 119905 119909) for aa 119897 isin [119904infin)
(42)
Lemma 9 (see [30]) Let 119864 be a metric space with metric 119889and let 119891 Ω rarr 119864 be strongly measurable Then thereexists a sequence 119891
119899 119899 isin 119873 of simple 119864-valued functions
(ie 119891119899isFB(E)measurable and takes only a finite number
of values) such that for arbitrary 120596 isin Ω the sequence119889(119891
119899(120596) 119891(120596)) 119899 isin 119873 is monotonically decreasing to zero
Let now 119891 isin 119871119902
(ΩC) By Lemma 9 we get the existenceof a sequence of simple function 119891
119899 119899 isin 119873 such that
1003816100381610038161003816119891119899 (120596) minus 119891 (120596)1003816100381610038161003816 darr 0 for all 120596 isin Ω as 119899 rarr infin (43)
Hence 119891119899
rarr 119891 in | sdot |119871119902(ΩC) by Lebesguersquos dominated
convergence theoremWe are now in a position of showing the main result in
this section
Theorem 10 Assume that Hypothesis 1 holds true and thatHypothesis 2 holds in the particular case 119870 = 119877 Then thereexist two Borel measurable deterministic functions 120592 [tinfin) times
C rarr 119877 and 120577 [119905infin) times C rarr Ξlowast
= 119871(Ξ 119877) =
1198712(Ξ 119877) such that for 119905 isin [0infin) and x isin C the solution
(119883(119905 119909) 119884(119905 119909) 119885(119905 119909)) of (36) satisfies
119884 (119904 119905 119909) = 120592 (119904 119883119904(119905 119909)) 119885 (119904 119905 119909) = 120577 (119904 119883
119904(119905 119909))
119875-as for aa 119904 isin [119905infin)
(44)
Proof We apply the techniques introduced in [26 Proposi-tion 32] Let 119890
119894 be a basis of Ξlowast and let us define 119885119894119873
=
((119885 119890119894)Ξlowast and119873) or (minus119873) Then for every 0 le 119905
1lt 119905
2lt infin Δ gt
0 and 1199091 119909
2isin C we have that
100381610038161003816100381610038161003816100381610038161003816
119864 int
1199051+Δ
1199051
119885119894119873
(119904 1199051 119909
1) 119889119904 minus 119864int
1199052+Δ
1199052
119885119894119873
(119904 1199052 119909
2) 119889119904
100381610038161003816100381610038161003816100381610038161003816
le 119864int
1199052
1199051
10038161003816100381610038161003816119885119894119873
(119904 1199051 119909
1)10038161003816100381610038161003816119889119904
+ 119864int
1199051+Δ
1199052
10038161003816100381610038161003816119885119894119873
(119904 1199051 119909
1) minus 119885
119894119873
(119904 1199052 119909
2)10038161003816100381610038161003816119889119904
+ 119864int
1199052+Δ
1199051+Δ
10038161003816100381610038161003816119885119894119873
(119904 1199052 119909
2)10038161003816100381610038161003816119889119904
8 Abstract and Applied Analysis
le 211987310038161003816100381610038161199052 minus 1199051
1003816100381610038161003816 + Δ12
119890minus120575(1199051+Δ)
times(119864(int
infin
0
119890212057511990410038161003816100381610038161003816119885119894119873
(119904 1199051 119909
1)minus119885
119894119873
(119904 1199052 119909
2)10038161003816100381610038161003816
2
119889119904)
1199012
)
1119901
le 211987310038161003816100381610038161199052 minus 1199051
1003816100381610038161003816 + Δ12
119890minus120575(1199051+Δ)
times (119864(int
infin
0
11989021205751199041003816100381610038161003816119885 (119904 1199051 1199091)minus119885 (119904 1199052 1199092)
10038161003816100381610038162
119889119904)
1199012
)
1119901
(45)
From Theorem 7 we have that the map (119905 119909) rarr int119905+Δ
119905
119885119894119873
(119904 119905 119909)119889119904 is continuous from [0infin) times C to 119877 ByRemark 8 we also have that for fixed 119905 the map 119909 rarr
119864int119905+Δ
119905
119864119885119894119873
(119904 119905 119909)119889119904 is continuous from 119871119902
(ΩCF119905) to 119877
for 119902 large enough Let us define
120577119894119873
(119905 119909) = lim inf119899rarrinfin
119899119864int
119905+(1119899)
119905
119885119894119873
(119904 119905 119909) 119889119904
119905 isin [0infin) 119909 isin C
(46)
It is clear that 120577119894119873 [0infin) timesC rarr 119877 is a Borel functionWe fix 119909 and 0 le 119905 le 119904 lt infin For 119897 isin [119904infin) we
denote 119864[119885119894119873
(119897 119904 119910)]|119910=119883119904(119905119909)
the random variable obtainedby composing119883
119904(119905 119909) with the map 119910 rarr 119864[119885
119894119873
(119897 119904 119910)]By Lemma 9 there exists a sequence of C-valued F
119904-
measurable simple functions
119891119898 Ω 997888rarr C 119891
119898=
119873119898
sum
119896=1
ℎ(119898)
119896119868119891119898=ℎ(119898)
119896 119873
119898isin 119873 (47)
where ℎ(119898)1 ℎ
(119898)
119898are pairwise distinct andΩ = ⋃
119873119898
119896=1119891
119898=
ℎ(119898)
119896 such that
1003816100381610038161003816119891119898 (120596) minus 119883119904(120596)
1003816100381610038161003816 darr 0 for all 120596 isin Ω as 119899 997888rarr infin (48)
For any 119861 isin F119904 we have
int119861
int
119904+(1119899)
119904
119885119894119873
(119897 119905 119909) 119889119897119889119875
= int119861
int
119904+(1119899)
119904
119885119894119873
(119897 119904 119883119904) 119889119897119889119875
= 119864119868119861int
119904+(1119899)
119904
119885119894119873
(119897 119904 119883119904) 119889119897
= lim119898rarrinfin
119864(119868119861int
119904+(1119899)
119904
119885119894119873
(119897 119904 119891119898) 119889119897)
= lim119898rarrinfin
119873119898
sum
119896=1
119864(119868119861119868119891119898=ℎ(119898)
119896int
119904+(1119899)
119904
119885119894119873
(119897 119904 ℎ(119898)
119896) 119889119897)
= lim119898rarrinfin
119864(119868119861
119873119898
sum
119896=1
119868119891119898=ℎ(119898)
119896)119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 ℎ(119898)
119896) 119889119897
= lim119898rarrinfin
119864119868119861(119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 119910) 119889119897|119910=119891119898
)
= lim119898rarrinfin
int119861
(119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 119910) 119889119897|119910=119891119898
)119889119875
= int119861
(119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 119910) 119889119897|119910=119883119904
)119889119875
(49)
and we get that
120577119894119873
(119904 119883119904(119905 119909)) = lim inf
119899rarrinfin
119899
times [119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 119910) 119889119897|119910=119883119904(119905119909)
]
= lim inf119899rarrinfin
119899119864[int
119904+(1119899)
119904
119885119894119873
(119897 119905 119909) 119889119897
100381610038161003816100381610038161003816100381610038161003816
F119904]
119875-as(50)
Fix 119905 and 119909 Recalling that |119885119894119873
| le 119873 by the Lebesguetheorem on differentiation it follows that 119875-as
lim119899rarrinfin
119899int
119904+(1119899)
119904
119885119894119873
(119897 119905 119909) 119889119897 = 119885119894119873
(119904 119905 119909)
for aa 119904 isin [119905infin)
(51)
By the boundedness of 119885119894119873 applying the dominated conver-gence theorem we get that
120577119894119873
(119904 119883119904(119905 119909)) = 119864 [119885
119894119873
(119904 119905 119909)10038161003816100381610038161003816F
119904] = 119885
119894119873
(119904 119905 119909)
119875-as for aa 119904 isin [119905infin)
(52)
Now we have proved that for every 119905 119909
120577119894119873
(119904 119883119904(119905 119909)) = 119885
119894119873
(119904 119905 119909)
119875-as for aa 119904 isin [119905infin)
(53)
for every 119894 119873 Let 119862 sub [0infin) times C denote the set ofpairs (119905 119909) such that lim
119873rarrinfin120577119894119873
(119905 119909) exists and the seriessuminfin
119894=1(lim
119873rarrinfin120577119894119873
(119905 119909))119890119894converges in Ξlowast We define
120577 (119905 119909) =
infin
sum
119894=1
( lim119873rarrinfin
120577119894119873
(119905 119909)) 119890119894 (119905 119909) isin 119862
120577 (119905 119909) = 0 (119905 119909) notin 119862
(54)
Abstract and Applied Analysis 9
Since 119885 satisfies
119885 (120596 119904 119905 119909) =
infin
sum
119894=1
( lim119873rarrinfin
119885119894119873
(120596 119904 119905 119909)) 119890119894 (55)
for every 120596 119904 119905 119909 From (53) it follows that for every 119905 119909 wehave (119904 119883
119904(120596 119905 119909)) isin 119862 119875-as for almost all 119904 isin [119905infin) and
119885(119904 119905 119909) = 120577(119904 119883119904(119905 119909)) 119875-as for aa 119904 isin [119905infin)
We define 120592(119905 119909) = 119884(119905 119905 119909) since 119884(119905 119905 119909) is deter-ministic so the map (119905 119909) rarr 120592(119905 119909) can be written as acomposition 120592(119905 119909) = Γ
3(Γ2(119905 Γ
1(119905 119909))) with
Γ1 [0infin) timesC 997888rarr 119871
119901
P(Ω 119862
120575(119877))
Γ1(119905 119909) = 119884 (sdot 119905 119909)
Γ2 [0infin) times 119871
119901
P(Ω 119862
120575(119877)) 997888rarr 119871
119901
(Ω 119877)
Γ2(119905 119881) = 119881 (119905)
Γ3 119871
119901
(Ω 119877) 997888rarr 119877 Γ3120585 = 119864120585
(56)
FromTheorem 7 it follows that Γ1is continuous By
|119881(119905) minus 119880(119904)|119871119901(Ω119877)
le |119881 (119905) minus 119881 (119904)|119871119901(Ω119877)
+ 119890minus120575119901119904
|119881 minus 119880||119871
119901
P(Ω119862120575(119877))
(57)
we have that Γ2is continuous It is clear that Γ
3is continuous
Then themap (119905 119909) rarr 120592(119905 119909) is continuous from [0infin)timesCto 119877 therefore 120592(119905 119909)is a Borel measurable function Fromuniqueness of the solution of (36) it follows that 119884(119904 119905 119909) =120592(119904 119883
119904(119905 119909)) 119875-as for aa 119904 isin [119905infin)
4 The Fundamental Relation
Let (ΩF 119875) be a given complete probability space with afiltration F
119905119905ge0
satisfying the usual conditions 119882(119905) 119905 ge 0
is a cylindrical Wiener process in Ξ with respect to F119905119905ge0
We will say that an F
119905ge0-predictable process 119906 with values
in a given measurable space (119880U) is an admissible controlThe function 119877 [0infin) times C times 119880 rarr Ξ is measurableand bounded We consider the following controlled stateequation
119889119883119906
(119904) = 119860119883119906
(119904) 119889119904 + 119865 (119904 119883119906
119904) 119889119904
+ 119866 (119904 119883119906
119904) 119877 (119904 119883
119906
119904 119906 (119904)) 119889119904 + 119866 (119904 119883
119906
119904) 119889119882 (119904)
119904 isin [119905infin)
119883119906
119905= 119909
(58)
Here we assume that there exists a mild solution of (58)which will be denoted by 119883119906
(119904 119905 119909) or simply by 119883119906
(119904) Weconsider a cost function of the form
119869 (119906) = 119864int
infin
119905
119890minus120582119904
119892 (119904 119883119906
119904 119906 (119904)) 119889119904 (59)
Here 119892 is function on [0infin) times C times 119880 with real values Ourpurpose is to minimize the function 119869 over all admissiblecontrols
We define in a classical way the Hamiltonian functionrelative to the previous problem for all 119905 isin [0infin) 119909 isin
C and 119911 isin Ξlowast
120595 (119905 119909 119911) = inf 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) 119906 isin 119880
(60)
and the corresponding possibly empty set of minimizers
Γ (119905 119909 119911) = 119906 isin 119880 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) = 120595 (119905 119909 119911)
(61)
We are now ready to formulate the assumptions we need
Hypothesis 3 (i) 119860 119865 and G verify Hypothesis 1(ii) (119880U) is ameasurable spaceThemap 119892 [0infin)timesCtimes
119880 rarr 119877 is continuous and satisfies |119892(119905 119909 119906)| le 119870119892(1+|119909|
119898119892
119862)
for suitable constants 119870119892gt 0 119898
119892gt 0 and all 119909 isin C119906 isin
119880 The map 119877 [0infin) times C times 119880 rarr Ξ is measurable and|119877(119905 119904 119906)| le 119871
119877for a suitable constant 119870
119877gt 0 and all 119909 isin
C119906 isin 119880 and119911 isin Ξlowast
(iii)TheHamiltonian120595 defined in (60) satisfies the requi-rements of Hypothesis 2 (with119870 = 119877)
(iv) We fix here 119901 gt 2 q and 120575 lt 0 satisfying (39) with120578 = 120578(119902) and such that 119902 gt 119898
119892
We are in a position to prove the main result of thissection
Theorem11 Assume thatHypothesis 3 holds and suppose that120582 verifies
120582 gt (minus120575 minus 120583 +1198712
119911
2) or (minus120575 +
1198712
119877
2 (119901 minus 1))
or (1198712
119877119898119892
2 (119902 minus 119898119892)minus 120578 (119902)119898
119892)
(62)
Let 120592 120577 denote the function in the statement of Theorem 10Then for every admissible control 119906 and for the correspondingtrajectory119883 starting at (119905 119909) one has
119869 (119906) = 120592 (119905 119909) + 119864int
infin
119905
119890minus120582119904
[minus120595 (119904 119883119906
119904 120577 (119904 119883
119906
119904)) + 120577 (119904 119883
119906
119904)
times 119877 (119904 119883119906
119904 119906 (119904)) + 119892 (119904 119883
119906
119904 119906 (119904))] 119889119904
(63)
Proof Consider (58) in the probability space (ΩF 119875) withfiltration F
119905119905ge0
and with an F119905119905ge0
-cylindrical Wienerprocess 119882(119905) 119905 ge 0 Let us define
119882119906
(119904) = 119882 (119904) + int
119904
119905and119904
119877 (120590119883119906
120590 119906 (120590)) 119889120590 119904 isin [0infin)
120588 (119879) = exp(int119879
119905
minus119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882 (119904)
minus1
2int
119879
119905
1003816100381610038161003816119877 (119904 119883119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)
(64)
10 Abstract and Applied Analysis
Let 119875119906 be the unique probability onF[0infin)
such that
119875119906
|F119879
= 120588 (119879) 119875|F119879
(65)
We notice that under 119875119906 the process119882119906 is aWiener processLet us denote by F119906
119905119905ge0
the filtration generated by119882119906 andcompleted in the usual way Relatively to 119882119906 (58) can berewritten as
119889119883119906
(119904) = 119860119883119906
(119904) 119889119904 + 119865 (119904 119883119906
119904) 119889119904
+ 119866 (119904 119883119906
119904) 119889119882
119906
(119904) 119904 isin [119905infin)
119883119906
119905= 119909
(66)
In the space (ΩF[0infin)
F119906
119905119905ge0 119875
119906
) we consider the follow-ing system of forward-backward equations
119883119906
(119904) = 119890(119904minus119905)119860
119909 (0) + int
119904
119905
119890(119904minus120590)119860
119865 (120590119883119906
120590) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883119906
120590) 119889119882
119906
(120590) 119904 isin [119905infin)
119883119906
119905= 119909 isin C
119884119906
(119904) minus 119884119906
(119879) + int
119879
119904
119885119906
(120590) 119889119882119906
(120590) + 120582int
119879
119904
119884119906
(120590) 119889120590
= int
119879
119904
120595 (120590119883119906
120590 119885
119906
(120590)) 119889120590 0 le 119904 le 119879 lt infin
(67)
Applying the Ito formula to 119890minus120582119904119884119906(119904) and writing the back-ward equation in (67) with respect to the process119882 we get
119884119906
(119904) + int
119879
119904
119890minus120582120590
119885119906
(120590) 119889119882 (120590)
= int
119879
119904
119890minus120582120590
[120595 (120590119883119906
120590 119885
119906
(120590))
minus119885119906
(120590) 119877 (120590119883119906
120590 119906 (120590))] 119889120590
+ 119890minus120582119879
119884119906
(119879)
(68)
Recalling that 119877 is bounded we get for all 119903 ge 1 and someconstant 119862
119864119906
[120588(119879)minus119903
] = 119864119906
[exp 119903 (int119879
119905
119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882
119906
(119904)
minus1
2int
119879
119905
1003816100381610038161003816119877 (119904 119883119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)]
= 119864119906
[exp(int119879
119905
119903119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882
119906
(119904)
minus1
2int
119879
119905
11990321003816100381610038161003816119877 (119904 119883
119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)
times exp 119903 (119903 minus 1)2
int
119879
119905
1003816100381610038161003816119877 (119904 119883119906
119904 119906 (119904))
10038161003816100381610038162
119889119904]
le 119890(12)119903(119903minus1)119879119871
2
119877119864119906
times exp(int119879
119905
2119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882
119906
(119904)
minus1
2int
119879
119905
41003816100381610038161003816119877 (119904 119883
119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)
= 119890(12)119903(119903minus1)119879119871
2
119877
(69)
It follows that
119864(int
119879
119905
|119890minus120582119904
119885119906
(119904)|2
119889119904)
12
= 119864119906
[(int
119879
119905
10038161003816100381610038161003816119890minus120582119904
119885119906
(119904)10038161003816100381610038161003816
2
119889119904)
12
120588minus1
]
le (119864119906
int
119879
119905
10038161003816100381610038161003816119890minus120582119904
119885119906
(119904)10038161003816100381610038161003816
2
119889119904)
12
times (119864119906
120588minus2
)12
lt infin
(70)
We conclude that the stochastic integral in (68) has zeroexpectation If we set 119904 = 119905 in (68) and we take expectationwith respect to 119875 we obtain
119890minus120582119879
119864119884119906
(119879) minus 119884119906
(119905)
= 119864int
119879
119905
119890minus120582120590
[minus120595 (120590119883119906
120590 119885
119906
(120590))
+119885119906
(120590) 119877 (120590119883119906
120590 119906 (120590))] 119889120590
(71)
ByTheorem 7 119884119906(sdot 119905 119909) isin 119871119901P(Ω 119862
120575(119877)) so that
119864119906
|119884(119879 119905 119909)|119901
le 119862 exp (minus119901120575119879) (72)
By the Holder inequality we have that for suitable constant119862 gt 0
119864 |119884 (119879 119905 119909)| = 119864119906
(120588minus1
(119879) |119884 (119879 119905 119909)|)
le 119864(120588minus119901(119901minus1)
)(119901minus1)119901
119864(|119884 (119879 119905 119909)|119901
)1119901
le 119862119890((1198712
1198772(119901minus1))minus120575))119879
(73)
From Theorem 2 we obtain 119864119906sup119904ge119905119890120578119902119904
|119883119906
119904|119902
lt infin by thesimilar process we get that
1198641003816100381610038161003816119883
119906
119879
1003816100381610038161003816119898119892
le 119862119890(1198712
119877119898119892(2119902minus2119898
119892)minus1
minus120578(119902)119898119892)119879
(74)
Abstract and Applied Analysis 11
for suitable constant 119862 gt 0 and
119864int
infin
119905
119890minus120582120590 1003816100381610038161003816119892 (120590119883
119906
120590 119906 (120590))
1003816100381610038161003816 119889120590 lt infin (75)
Since 119884119906(119905 119905 119909) = 120592(119905 119909) and 119885119906
(119904 119905 119909) = 120577(119904 119883119906
119904(119905 119909)) 119875-
as for aa 119904 isin [119905infin) we have that
119890minus120582119879
119864119884119906
(119879) minus 119907 (119905 119909)
= 119864int
119879
119905
119890minus120582120590
[minus120595 (120590119883119906
120590 120577 (120590 119883
119906
120590))
+120577 (120590119883119906
120590) 119877 (120590119883
119906
120590 119906 (120590))] 119889120590
(76)
Thus adding and subtracting119864intinfin119905
119890minus120582120590
119892(120590119883119906
120590 119906(120590))119889120590 and
letting 119879 rarr infin we conclude that
119869 (119906) = 120592 (119905 119909)
+ 119864int
infin
119905
119890minus120582119904
[minus120595 (119904 119883119906
119904 120577 (119904 119883
119906
119904)) + 120577 (119904 119883
119906
119904) 119877
times (119904 119883119906
119904 119906 (119904)) + 119892 (119904 119883
119906
119904 119906 (119904))] 119889119904
(77)
The proof is finished
We immediately deduce the following consequences
Theorem 12 Let 119905 isin [0infin) and 119909 isin C be fixed assumethat the set-valued map Γ has nonempty values and it admitsa measurable selection Γ
0 [0infin)timesCtimesΞlowast rarr 119880 and assume
that a control 119906(sdot) satisfies
119906 (119904) = Γ0(119904 119883
119906
119904 120577 (119904 119883
119906
119904))
119875-as for almost every 119904 isin [119905infin)
(78)
Then 119869(119905 119909 119906) = 120592(119905 119909) and the pair (119906(sdot) 119883) is optimal forthe control problem starting from 119909 at time 119905
Such a control can be shown to exist if there exists a solutionfor the so-called closed-loop equation as follows
119889119883 (119904)=119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904+119866 (119904 119883
119904)
times(119877 (119904 119883119904 Γ
0(119904 119883
119904 120577 (119904 119883
119904))) 119889119904 + 119889119882 (119904))
119904 isin [119905infin)
119883119905= 119909
(79)
since in this case we can define an optimal control setting
119906 (119904) = Γ0(119904 119883
119904 120577 (119904 119883
119904)) (80)
However under the present assumptions we cannot guaranteethat the closed-loop equation has a solution in the mildsense To circumvent this difficulty we will revert to a weakformulation of the optimal control problem
5 Existence of Optimal Control
We formulate the optimal control problem in the weak sensefollowing the approach of [31]Themain advantage is that wewill be able to solve the closed-loop equation in a weak senseand hence to find an optimal control even if the feedbacklaw is nonsmooth
We call (ΩF F119905119905ge0 119875119882) an admissible setup if
(ΩF F119905119905ge0 119875) is a filtered probability space satisfying the
usual conditions and 119882 is a cylindrical 119875-Wiener processwith values in Ξ with respect to the filtration F
119905119905ge0
By an admissible control system we mean (ΩF
F119905119905ge0 119875119882 119906119883
119906
) where (ΩF F119905119905ge0 119875 119882) is an
admissible setup 119906 is an F119905-predictable process with values
in119880 and119883119906 is a mild solution of (58) An admissible controlsystem will be briefly denoted by (119882 119906119883119906
) in the followingOur purpose is to minimize the cost functional
119869 (119906) = 119864int
infin
119905
119890minus120582119904
119892 (119904 119883119906
119904 119906 (119904)) 119889119904 (81)
over all the admissible control systemOur main result in this section is based on the solvability
of the closed-loop equation
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904 + 119866 (119904 119883
119904)
times (119877 (119904 119883119904 Γ
0(119904 119883
119904 120577 (119904 119883
119904))) 119889119904 +119889119882 (119904))
119904 isin [119905infin)
119883119905= 119909
(82)
In the following sense we say that 119883 is a weak solution of(82) if there exists an admissible setup (ΩF F
119905119905ge0 119875119882)
and anF119905-adapted continuous process119883(119905)with values in119867
which solves the equation in the mild sense namely 119875-as
119883 (119904) = 119890(119904minus119905)119860
119909 (0) + int
119904
119905
119890(119904minus120590)119860
119865 (120590119883120590) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883120590) 119877
times (120590119883120590 Γ
0(120590119883
120590 120577 (120590 119883
120590))) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883120590) 119889119882
120590 119904 isin [119905infin)
(83)
119883119905= 119909 (84)
Theorem 13 Assume that Hypothesis 3 holds Then thereexists a weak solution of the closed-loop equation (82) whichis unique in law
12 Abstract and Applied Analysis
Proof (uniqueness) Let 119883 be a weak solution of (82) in anadmissible setup (ΩF F
119905119905ge0 119875119882)We define
120588 (119879) = exp(int119879
119905
minus119877lowast
(120590 119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889119882 (120590)
minus1
2int
119879
119905
1003816100381610038161003816119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)))10038161003816100381610038162
119889120590)
(85)
Since 119877 is bounded the Girsanov theorem ensures that thereexists a probability measure 1198750 such that the process
1198820
(119904) = 119882 (119904) + int
119904
119905and119904
119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889120590
119904 isin [0infin)
(86)
is a 1198750-Wiener process and
1198750
|F119879
= 120588 (119879) 119875|F119879
(87)
Let us denote by F0
119905119905ge0
the filtration generated by1198820 andcompleted in the usual way In (ΩF
[0infin) F0
119905119905ge0 119875
0
) 119883 isa mild solution of
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119905 119883119904) 119889119904
+ 119866 (119904 119883119904) 119889119882
0
(119904) 119904 isin [119905infin)
119883119905= 119909
120588 (119879) = exp(int119879
119905
minus119877lowast
(120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)) 119889119882
0
(120590)
+ 12int
119879
119905
1003816100381610038161003816119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)))10038161003816100381610038162
119889120590)
(88)
By Hypothesis 3 the joint law of 119883 and 1198820 is uniquely
determined by 119860 119865 119866 and 119909 Taking into account the lastdisplayed formula we conclude that the joint law of 119883 and120588(119879) under1198750 is also uniquely determined and consequentlyso is the law of 119883 under 119875 This completes the proof of theuniqueness part
Proof (existence) Let (ΩF 119875) be a given complete probabil-ity space 119882(119905) 119905 ge 0 is a cylindrical Wiener process on(ΩF 119875)with values inΞ and F
119905119905ge0
is the natural filtrationof 119882(119905) 119905 ge 0 augmented with the family of119875-null sets Let119883(sdot) be the mild solution of
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904
+ 119866 (119904 119883119904) 119889119882 (119904) 119904 isin [119905infin)
119883119905= 119909
(89)
and by the Girsanov theorem let 1198751 be the probability on Ωunder which
1198821
(119904) = 119882 (119904) minus int
119904
119905and119904
119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889120590
(90)
is aWiener process (notice that 119877 is bounded)Then119883 is theweak solution of (82) relatively to the probability 1198751 and theWiener process1198821
Now we can state the main result of this section
Corollary 14 Assume that Hypothesis 3 holds true and 120582
verifies (62) Also assume that the set-valued map Γ hasnonempty values and it admits a measurable selection Γ
0
[0infin) timesC times Ξlowast
rarr 119880 Then for every 119905 isin [0infin) and x isin Cand for all admissible control system (119882 119906119883
119906
) one has
119869 (119906 119905 119909) ge 120592 (119905 119909) (91)
and the equality holds if
119906 (119904) = Γ0(119904 119883
119906
119904 120577 (119904 119883
119906
119904))
119875 minus 119886119904 119891119900119903 119886119897119898119900119904119905 119890119907119890119903119910 119904 isin [119905infin)
(92)
Moreover from Theorem 13 it follows that the closed-loop equation (82) admits a weak solution (ΩFF
119905119905ge0 119875119882119883) which is unique in law and setting
119906 (119904) = Γ0(119904 119883
119904 120577 (119904 119883
119904)) (93)
we obtain an optimal admissible control system (119882 119906119883)
6 Applications
In this section we present a simple application of the previousresults We consider the stochastic delay partial differentialequation in the bounded domain 119861 sub 119877
119899 with smoothboundary 120597119861 as follows
119889119911119906
(119905 120585) = Δ119911119906
(119905 120585) 119889119905 + 119891 (119905 119911119906
119905(120585)) 119889119905
+
119889
sum
119894=1
119892119894(119905 119911
119906
119905(120585)) [119903
119894
(120585) 119906119894
(119905) 119889119905 + 119889119882119894
(119905)]
119911119906
0(120579 120585) = 119909 (120579 120585) 120585 isin 119861 120579 isin [minus1 0]
119911119906
(119905 120585) = 0 119905 isin [0infin) 120585 isin 120597119861
(94)
Here119882 = (1198821
1198822
119882119889
) is a standardWiener process in119877119889 and the functions 119891 [0 +infin) times 119862([minus1 0] 119877) rarr 119877 and119892119894 [0 +infin) times 119862([minus1 0] 119877) rarr 119877 are Lipschitz continuous
and bounded Setting 119880 as a bounded subset of 119877119889 Ξ = 119877119889
119867 = 1198712
(119861) and 119909 isin 119862([minus1 0]119867) We define 119865 and 119866 asfollowing
119865 (119905 119909) (120585) = 119891 (119905 119909 (120585))
(119866 (119905 119909) 119911) (120585) =
119889
sum
119894=1
119892119894(119905 119909 (120585)) 119911
119894
(120585)
120585 isin 119861 119909 isin 119862 ([minus1 0] 119867) 119911 isin 119871 (Ξ119867)
(95)
and let 119860 denote the Laplace operator Δ in 1198712
(119861) withdomain11988222
(119861)⋂11988212
0(119861) then (94) has the form (58) and
Hypothesis 1 holds
Abstract and Applied Analysis 13
Let us consider the optimal control problem associatedwith the cost
119869 (119906) = 119864int
infin
0
119890minus120582119905
[int119861
120590 (120585 119911119906
119905(120585)) 119889120585 + 119906
2
(119905)] 119889119905 (96)
where 120582 verifies (62) and 120590 119862([minus1 0] 119877) times 119880 rarr [0infin) isa bounded measurable function Define 119892 119862([minus1 0]119867) times
119880 rarr [0infin) and 119877 119862([minus1 0]119867) times 119880 rarr Ξ by119892(119910 119906) = int
119861
120590(119905 119910(120585) 119906)119889120585 + 1199062 and 119877(119910 119906) = (int
119861
1199031
(120585)1199061
119889120585
int119861
1199032
(120585)1199062
119889120585 int119861
119903119889
(120585)119906119889
119889120585) for 119910 isin 119862([minus1 0]119867) 119906 =
(1199061
1199062
119906119889
) isin 119880 respectively It can be easily verifiedthat Hypothesis 3 holds true and the set-valued map Γ hasnonempty values and it admits a measurable selection Γ
0
[0infin) times C times Ξlowast
rarr 119880 Then the closed-loop equation(82) admits a weak solution (ΩF F
119905119905ge0 119875119882 119906 119911
sdot(sdot)) and
setting
119906 (119904) = Γ0(119904 119911
119904(sdot) 120577 (119904 119911
119904(sdot))) (97)
we obtain an optimal admissible control system (119882 119906 119911(sdot))
References
[1] J Bismut ldquoOn optimal control of linear stochastic equationswith a linear-quadratic criterionrdquo SIAM Journal on Control andOptimization vol 15 no 3 pp 1ndash4 1977
[2] N Nagase ldquoOn the existence of optimal control for controlledstochastic partial differential equationsrdquo Nagoya MathematicsJournal vol 115 pp 73ndash85 1989
[3] N El Karoui D Huu Nguyen and M Jeanblanc-Pique ldquoCom-pactification methods in the control of degenerate diffusionsrdquoStochastics vol 20 pp 169ndash219 1987
[4] M Nisio ldquoOptimal control for stochastic partial differentialequations and viscosity solutions of Bellman equationsrdquoNagoyaMathematics Journal vol 123 pp 13ndash37 1991
[5] M Nisio ldquoOn sensitive control for stochastic partial differentialequationsrdquo in Stochastic Analysis on Infinite Dimensional SpacesProceedings of the US Japan Bilateral Seminar H Kunita et alEd vol 310 of Pitman Research Notes Mathematical Series pp231ndash241 Longman Scientific and Technical Baton Rouge LaUSA January 1994
[6] R Buckdahn and A Rascanu ldquoOn the existence of stochasticoptimal control of distributed state systemrdquoNonlinear AnalysisTheory Methods and Applications vol 52 no 4 pp 1153ndash11842003
[7] V Barbu and G Da Prato Equations in Hilbert Spaces vol 86 ofPitman Research Notes in Mathematics Pitman 1983
[8] P Cannarsa and G Da Prato ldquoSecond-order Hamilton-Jacobiequations in infinite dimensionsrdquo SIAM Journal on Control andOptimization vol 29 no 2 pp 474ndash492 1991
[9] P Cannarsa and G Da Prato ldquoDirect solution of a second-orderHamilton-Jacobi equations in Hilbert spacesrdquo in StochasticPartial Differential Equations and Applications G Da Pratoand L Tubaro Eds vol 268 of Pitman Research Notes inMathematics Pitman 1992
[10] F Gozzi ldquoRegularity of solutions of second order Hamilton-Jacobi equations and application to a control problemrdquoCommu-nications in Partial Differential Equations vol 20 pp 775ndash8261995
[11] F Gozzi ldquoGlobal regular solutions of second order Hamilton-Jacobi equations in Hilbert spaces with locally Lipschitz non-linearitiesrdquo Journal of Mathematical Analysis and Applicationsvol 198 no 2 pp 399ndash443 1996
[12] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems and Control Lettersvol 14 no 1 pp 55ndash61 1990
[13] N El Karoui and LMazliak Eds Backward Stochastic Differen-tial Equations vol 364 of Pitman ResearchNotes inMathematicsSeries Longman 1997
[14] E Pardoux and BSDEs ldquoweak convergence and homogeneiza-tion of semilinear PDEsrdquo in Non- Linear Analysis DifferentialEquations and Control F H Clarke and R J Stern Eds pp503ndash549 Kluwer Dordrecht The Netherlands 1999
[15] S Peng ldquoA generalized dynamic programming principle andHamilton-Jacobi-Bellman equationrdquo Stochastics and StochasticsReports vol 38 pp 119ndash134 1992
[16] N E Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997
[17] SHamad120583ene and J P Lepeltier ldquoBackward equations stochas-tic control and zero-sum stochastic differential gamesrdquo Stochas-tics and Stochastics Reports vol 54 pp 221ndash231 1995
[18] N El-Karoui and S Hamadene ldquoBSDEs and risk-sensitive con-trol zero-sum and nonzero-sum game problems of stochasticfunctional differential equationsrdquo Stochastic Processes and theirApplications vol 107 no 1 pp 145ndash169 2003
[19] M Fuhrman and G Tessiture ldquoExistence of optimal stochasticcontrols and global solutions of forward-backward stochasticdifferential equationsrdquo SIAM Journal on Control and Optimiza-tion vol 43 no 3 pp 813ndash830 2005
[20] M Fuhrman Y Hu and G Tessitore ldquoOn a class of stochasticoptimal control problems related to bsdes with quadraticgrowthrdquo SIAM Journal on Control and Optimization vol 45 no4 pp 1279ndash1296 2006
[21] M Fuhrman and G Tessitore ldquoNonlinear kolmogorov equa-tions in infinite dimensional spaces the backward stochasticdifferential equations approach and applications to optimalcontrolrdquoAnnals of Probability vol 30 no 3 pp 1397ndash1465 2002
[22] F Masiero ldquoSemilinear kolmogorov equations and applicationsto stochastic optimal controlrdquo Applied Mathematics and Opti-mization vol 51 no 1 pp 201ndash250 2005
[23] M Fuhrman FMasiero andG Tessitore ldquoStochastic equationswith delay optimal control via BSDEs and regular solutions ofHamilton-jacobi-bellman equationsrdquo SIAM Journal on Controland Optimization vol 48 no 7 pp 4624ndash4651 2010
[24] M Fuhrman and G Tessiture ldquoInfinite horizon backwardstochastic differential equations and elliptic equations in hilbertspacesrdquo Annals of Probability vol 32 no 1 pp 607ndash660 2004
[25] F Masiero ldquoInfinite horizon stochastic optimal control prob-lems with degenerate noise and elliptic equations in Hilbertspacesrdquo Applied Mathematics and Optimization vol 55 no 3pp 285ndash326 2007
[26] M Fuhrman ldquoA class of stochastic optimal control problemsin Hilbert spaces BSDEs and optimal control laws stateconstraints conditioned processesrdquo Stochastic Processes andtheir Applications vol 108 no 2 pp 263ndash298 2003
[27] F Masiero ldquoStochastic optimal control problems and parabolicequations in banach spacesrdquo SIAM Journal on Control andOptimization vol 47 no 1 pp 251ndash300 2008
[28] G Da Prato and J Zabczyk Ergodicity For Infinite-DimensionalSystems Cambridge University Press 1996
14 Abstract and Applied Analysis
[29] J Zabczyk ldquoParabolic equations on Hilbert spacesrdquo in StochaS-tic PDErsquoS and Kolmogorov Equations in Infinite Dimensionsvol 1715 of Lecture Notes in Math pp 117ndash213 Springer BerlinGermany 1999
[30] G Da Prato and J Zabczyk Stochstic Equations in InfiniteDimensions Cambridge University Press 1992
[31] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 of Applications of MathematicsSpringer New York NY USA 1993
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
So we conclude that10038161003816100381610038161003816Φ1015840
(119883sdot)10038161003816100381610038161003816119871119902
P(Ω119862120578(C))
le 119888120572119872|119884|
119871119902
P(Ω119871119902
120578(119867))
times (int
infin
0
119903(120572minus1)119902
1015840
119890(120596+120578)119903119902
1015840
119889119903)
11199021015840
(25)
On the other hand by the Burkholder-Davis-Gundy inequal-ities for some constant 119888
119902depending only on 119902 we have
119864|119884 (119903)|119902
le 119888119902119864(int
119903
0
(119903 minus 120590)minus212057210038161003816100381610038161003816119890(119903minus120590)119860
119866 (120590119883120590)10038161003816100381610038161003816
2
1198712(Ξ119867)
119889120590)
1199022
le 119871119902
119888119902119864
times (int
119903
0
(119903 minus 120590)minus2120572minus2120574
1198902120596(119903minus120590)
(1 +1003816100381610038161003816119883120590
10038161003816100381610038162
119862) 119889120590)
1199022
(26)
which implies that
[119864|119884 (119903)|119902
]2119902
le 1198712
1198882119902
119902int
119903
0
(119903 minus 120590)minus2120572minus2120574
times 1198902120596(119903minus120590)
[119864(1 +1003816100381610038161003816119883120590
1003816100381610038161003816119862)119902
]2119902
119889120590
(27)
so that
1198902120578119903
[119864|119884 (119903)|119902
]2119902
le 1198621int
119903
0
(119903 minus 120590)minus2120572minus2120574
1198902(120596+120578)(119903minus120590)
1198902120578120590
119889120590
+ 1198622int
119903
0
(119903 minus 120590)minus2120572minus2120574
1198902(120596+120578)(119903minus120590)
times 1198902120578120590
[1198641003816100381610038161003816119883120590
1003816100381610038161003816119902
119862]2119902
119889120590
(28)
for suitable constants 1198621 119862
2 Applying the Young inequality
for convolutions we obtain
int
infin
0
119890119902120578119903
119864|119884 (119903)|119902
119889119904le 1198621(int
infin
0
119904minus2120572minus2120574
1198902(120596+120578)119904
119889119904)
1199022
int
infin
0
119890119902120578119904
119889119904
+ 1198622(int
infin
0
119904minus2120572minus2120574
1198902(120596+120578)119904
119889119904)
1199022
times int
infin
0
119890119902120578119904
1198641003816100381610038161003816119883119904
1003816100381610038161003816119902
119862119889119904
(29)
This shows that |119884|119871119902
P(Ω119871119902
120578(119867))
is finite provided we assumethat 120578 lt 0 and 120596 + 120578 lt 0 so the map is well defined
If 1198831
sdot 119883
2
sdotare processes belonging to H119902
120578and 1198841 1198842 are
defined accordingly the entirely analogous passages showthat100381610038161003816100381610038161198841
minus 119884210038161003816100381610038161003816119871119902
P(Ω119871119902
120578(119867))
le 1198711198881119902
120572
100381610038161003816100381610038161198831
sdotminus 119883
2
sdot
10038161003816100381610038161003816119871119902
P(Ω119871119902
120578(C))
times (int
infin
0
119904minus2120572minus2120574
1198902(120596+120578)119904
119889119904)
12
(30)
Recalling the inequalities (23) and (25) and noting that themap Y rarr Φ
1015840
(Xsdot) is linear we obtain an explicit expression
for the constant 119888 in (17) and it is immediate to verify that119888 lt 1 provided 120578 lt 0 is chosen sufficiently large We fixsuch a value of 120578(119902) The first result is a consequence of thecontraction principle The estimate (15) also follows from thecontraction property ofΦ(sdot 119905 119909)
For investigating the dependence of the solution119883(119904 119905 119909)on the initial data 119909 and 119905 we reformulate (13) as an equationon [0infin) We set
119878 (119904) = 119890119904119860
for 119904 ge 0 119878 (119904) = 119868 for 119904 lt 0 (31)
and we consider the equation
119883(119904) = 119878 (119904 minus 119905) 119909 ((0 and (119904 minus 119905)) or (minus120591))
+ int
119904
0
119868[119905infin)
(120590) 119878 (119904 minus 120590) 119865 (120590119883120590) 119889120590
+ int
119904
0
119868[119905infin)
(120590) 119878 (119904 minus 120590)
times 119866 (120590119883120590) 119889119882 (120590) 119904 isin [0infin)
1198830(120579) = 119909 ((minus119905 + 120579) or (minus120591)) 120579 isin [minus120591 0]
(32)
Under the assumptions of Hypothesis 1 by Theorem 2 it iseasy to prove that equation (32) has a unique solution 119883 and119883sdotisin H
119902
120578(119902)for every 119902 isin [2infin) It clearly satisfies 119883(119904) =
119909((119904 minus 119905) or (minus120591)) for 119904 isin [minus120591 119905) and its restriction to the timeinternal [119905infin) is the unique mild solution of (10) From nowon we denote by119883(119904 119905 119909) 119904 isin [0infin) the solution of (32)
We need the following parameter-depending contractionprinciple which is stated in the following lemma and provedin [29 Theorems 101 and 102]
Lemma3 (ParameterDependingContraction Principle) Let119861119863 denote Banach spaces Let ℎ 119861times119863 rarr 119861 be a continuousmapping satisfying
1003816100381610038161003816ℎ (1199091 119910) minus ℎ (1199092 119910)1003816100381610038161003816 le 120572
10038161003816100381610038161199091 minus 11990921003816100381610038161003816 (33)
for some 120572 isin [0 1) and every 1199091 119909
2isin 119861 y isin 119863 Let 120601(119910)
denote the unique fixed point of the mapping ℎ(sdot 119910) 119861 rarr 119861Then 120601 119863 rarr 119861 is continuous
Theorem 4 Assume that Hypothesis 1 holds true Then forevery 119902 isin [1infin) the map (119905 119909) rarr 119883
sdot(119905 119909) is continuous
from [0infin) timesC toHq120578(q)
Proof Clearly it is enough to prove the claim for 119902 large Letus consider the map Φ defined in the proof of Theorem 2 In
6 Abstract and Applied Analysis
our present notationΦ can be seen as a mapping fromH119902
120578times
[0infin) timesC toH119902
120578as follows
Φ(119883sdot 119905 119909)
119904(119897) = 119878 (119904 + 119897 minus 119905) 119909 (0)
+ int
119904+119897
0
119868[119905infin)
(120590) 119878 (119904 + 119897 minus 120590) 119865 (120590119883120590) 119889120590
+ int
119904+119897
0
119868[119905infin)
(120590) 119878 (119904 + 119897 minus 120590)
times 119866 (120590119883120590) 119889119882 (120590)
119904 isin [0infin) 119897 isin [minus120591 0] 119904 + 119897 ge 119905
Φ(119883sdot 119905 119909)
119904(119897) = 119909 ((119904 + 119897 minus 119905) or (minus120591))
119904 isin [0infin) 119897 isin [minus120591 0] 119904 + 119897 le 119905
(34)
By the arguments of the proof of Theorem 2 Φ(sdot 119905 119909)is a contraction in H119902
120578uniformly with respect to 119905 119909
The process 119883sdot(119905 119909) is the unique fixed point of Φ(sdot 119905 119909)
So by the parameter-depending contraction principle(Lemma 3) it suffices to show that Φ is continuous fromH119902
120578times [0infin) times C to H119902
120578 From the contraction property
of Φ(sdot 119905 119909) mentioned earlier we have that Φ(sdot 119905 119909) iscontinuous uniformly in 119905 119909 Moreover for fixed 119883
sdot it is
easy to verify that Φ(119883sdot sdot sdot) is continuous from [0infin) times C
toH119902
120578 The proof is finished
Remark 5 By similar passages we can show that for fixed119905 Theorem 4 still holds true for 119902 large enough if the spaces[0infin) times C and H119902
120578are replaced by the spaces 119871119902(ΩCF
119905)
and H119902
120578(119905) respectively where 119871119902(ΩCF
119905) denotes that the
space of F119905-measurable function with value in C such that
the norm
|119909|119902
= 119864|119909|119902
119862 (35)
is finite
3 The Backward-Forward System
In this section we consider the system of stochastic differen-tial equations 119875-as
119883 (119904) = 119890(119904minus119905)119860
119909 (0) + int
119904
119905
119890(119904minus120590)119860
119865 (120590119883120590) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883120590) 119889119882 (120590) 119904 isin [119905infin)
119883119905= 119909 isin C
119884 (119904) minus 119884 (119879) + int
119879
119904
119885 (120590) 119889119882 (120590) + 120582int
119879
119904
119884 (120590) 119889120590
= int
119879
119904
120595 (120590119883120590 119884 (120590) 119885 (120590)) 119889120590 0 le 119904 le 119879 lt infin
(36)
for 119904 varying on the time interval [119905infin) sub [0infin) Asin Section 2 we extend the domain of the solution setting119883(119904 119905 119909) = 119909((119904 minus 119905) or (minus120591)) for 119904 isin [minus120591 119905)
We make the following assumptions
Hypothesis 2 Themapping 120595 [0infin)timesCtimes119870times1198712(Ξ 119870) rarr
119870 is Borelmeasurable such that for all 119905 isin [0infin)120595(119905 sdot) Ctimes119870 times 119871
2(Ξ 119870) rarr 119870 is continuous and for some 119871
119910 119871
119911gt 0
120583 isin 119877 and119898 ge 11003816100381610038161003816120595 (119904 119909 1199101 1199111) minus 120595 (119904 119909 1199102 1199112)
1003816100381610038161003816
le 119871119910
10038161003816100381610038161199101 minus 11991021003816100381610038161003816 + 119871119911
10038161003816100381610038161199111 minus 11991121003816100381610038161003816
1003816100381610038161003816120595 (119904 119909 119910 119911)1003816100381610038161003816 le 119871 (1 + |119909|
119898
119862+10038161003816100381610038161199101003816100381610038161003816 + |119911|)
⟨120595 (119904 119909 1199101 119911) minus 120595 (119904 119909 119910
2 119911) 119910
1minus 119910
2⟩119870ge 120583
10038161003816100381610038161199101 minus 119910210038161003816100381610038162
(37)
for every 119904 isin [0infin) 119909 isin C 119910 1199101 119910
2isin 119870 119911 119911
1 and 119911
2isin
1198712(Ξ 119870)We note that the third inequality in (37) follows from the
first one taking 120583 = minus119871119910but that the third inequalitymay also
hold for different values of 120583Firstly we consider the backward stochastic differential
equation
119884 (119904) minus 119884 (119879) + int
119879
119904
119885 (120590) 119889119882 (120590) + 120582int
119879
119904
119884 (120590) 119889120590
= int
119879
119904
120595 (120590119883120590 119884 (120590) 119885 (120590)) 119889120590 0 le 119904 le 119879 lt infin
(38)
119870 is a Hilbert space the mapping 120595 [0infin) times C times 119870 times
1198712(Ξ 119870) rarr 119870 is a given measurable function 119883
sdotis a
predictable process with values in another Banach space Cand 120582 is a real number
Theorem 6 Assume that Hypothesis 2 holds Let 119901 gt 2 and120575 lt 0 be given and choose
119902 ge 119898119901 120578 gt120575
119898 (39)
Then the following hold
(i) For 119883sdotisin 119871
119902
P(Ω 119871
119902
120578(C)) and 120582 gt minus(120575 + 120583 minus (119871
2
1199112))
(38) has a unique solution in Kp120575that will be denoted
by (119884(119883sdot)(119904) 119885(119883
sdot)(119904)) 119904 ge 0
(ii) The estimate
119864sup119904ge0
(119884 (119883sdot) (119904))
119901
119890119901120575119904
+ 119864(int
infin
0
11989021205751205901003816100381610038161003816119884(119883sdot
)(120590)10038161003816100381610038162
119889120590)
1199012
+ 119864(int
infin
0
11989021205751205901003816100381610038161003816119885 (119883sdot
) (120590)10038161003816100381610038162
119889120590)
1199012
le 119888(1 +1003816100381610038161003816119883sdot
1003816100381610038161003816119898
119871119902
P(Ω119871119902
120578(C))
)119901
(40)
Abstract and Applied Analysis 7
holds for a suitable constant 119888 In particular 119884(119883sdot) isin
119871119901
P(Ω 119862
120575(119870))
(iii) The map 119883sdotrarr (119884(119883
sdot) 119885(119883
sdot)) is continuous from
119871119902
P(Ω 119871
119902
120578(C)) toK119901
120575 and 119883
sdotrarr 119884(119883
sdot) is continuous
from 119871119902
P(Ω 119871
119902
120578(C)) to 119871119901
P(Ω 119862
120575(119870))
(iv) The statements of points (i) (ii) and (iii) still holdtrue if the space 119871119902
P(Ω 119871
119902
120578(C)) is replaced by the space
119871119902
P(Ω 119862
120578(C))
Proof The theorem is very similar to Proposition 311 in [24]The only minor difference is that the mapping 120595 [0infin) times
Ctimes119870times1198712(Ξ 119870) rarr 119870 is a givenmeasurable function while
in [24] the measurable function 120595 is from119867 times119870 times 1198712(Ξ 119870)
to 119870 however the same arguments apply
Theorem 7 Assume that Hypothesis 1 holds and thatHypothesis 2 holds true in the particular case 119870 = 119877 Thenfor every 119901 gt 2 119902 120575 lt 0 satisfying (39) with 120578 = 120578(119902)and for every 120582 gt 120582
1015840
= minus(120575 + 120583 minus (1198712
1199112)) there exists a
unique solution in H119902
120578(119902)times K
119901
120575of (36) that will be denoted
by (119883(sdot 119905 119909) 119884(sdot 119905 119909) 119885(sdot 119905 119909)) Moreover 119884(sdot 119905 119909) isin
119871119901
P(Ω 119862
120575(119877)) The map (119905 119909) rarr (119884(sdot 119905 119909) 119885(sdot 119905 119909)) is con-
tinuous from [0infin)timesC toK119901
120575 and themap (119905 119909) rarr 119884(sdot 119905 119909)
is continuous from [0infin) timesC to 119871119901P(Ω 119862
120575(119877))
Proof We first notice that the system is decoupled the firstdoes not contain the solution (119884 119885) of the second oneThere-fore under the assumption of Hypothesis 1 by Theorem 2there exists a unique solution 119883(sdot 119905 119909) and 119883
sdot(119905 119909) isin H
119902
120578(119902)
of the first equation Moreover from Theorem 4 it followsthat the map (119905 119909) rarr 119883
sdot(119905 119909) is continuous from [0infin)timesC
toH119902
120578(119902)
Let 119870 = 119877 from Theorem 6 we have that thereexists a unique solution (119884(sdot 119905 119909) 119885(sdot 119905 119909)) isin K
119901
120575of the
second equation and the map 119883sdotrarr (119884(119883
sdot) 119885(119883
sdot)) is
continuous from H119902
120578(119902)to K
119901
120575while X
sdotrarr (Y(X
sdot)) is
continuous fromH119902
120578(119902)to119871119901
P(Ω 119862
120575(119877))We have proved that
(119883(sdot 119905 119909) 119884(sdot 119905 119909) 119885(sdot 119905 119909)) isin H119902
120578(119902)times K
119901
120575is the unique
solution of (36) and the other assertions follow from com-position
Remark 8 From Remark 5 by similar passages we can showthat for fixed 119905 and for 119902 large enough under the assumptionsof Theorem 7 the map 119909 rarr (119884(sdot 119905 119909) 119885(sdot 119905 119909)) is continu-ous from 119871
119902
(ΩCF119905) toK119901
120575(119905)
We also remark that the process 119883(sdot 119905 119909) is F[119905infin)
measurable since C is separable Banach space we have that119883sdot(119905 119909) is F
[119905infin)measurable So that 119884(119905) is measurable
with respect to both F[119905infin)
and F119905 it follows that 119884(119905) is
deterministicFor later use we notice three useful identities for 119905 le 119904 lt
infin the equality 119875-as
119883119897(119904 119883
119904(119905 119909)) = 119883
119897(119905 119909) 119897 isin [119904infin) (41)
is a consequence of the uniqueness of the solution of (13)Since the solution of the backward equation is uniquely
determined on an interval [119904infin) by the values of the process119883sdoton the same interval for 119905 le 119904 lt infin we have 119875-as
119884 (119897 119904 119883119904(119905 119909)) = 119884 (119897 119905 119909) for 119897 isin [119904infin)
119885 (119897 119904 119883119904(119905 119909)) = 119885 (119897 119905 119909) for aa 119897 isin [119904infin)
(42)
Lemma 9 (see [30]) Let 119864 be a metric space with metric 119889and let 119891 Ω rarr 119864 be strongly measurable Then thereexists a sequence 119891
119899 119899 isin 119873 of simple 119864-valued functions
(ie 119891119899isFB(E)measurable and takes only a finite number
of values) such that for arbitrary 120596 isin Ω the sequence119889(119891
119899(120596) 119891(120596)) 119899 isin 119873 is monotonically decreasing to zero
Let now 119891 isin 119871119902
(ΩC) By Lemma 9 we get the existenceof a sequence of simple function 119891
119899 119899 isin 119873 such that
1003816100381610038161003816119891119899 (120596) minus 119891 (120596)1003816100381610038161003816 darr 0 for all 120596 isin Ω as 119899 rarr infin (43)
Hence 119891119899
rarr 119891 in | sdot |119871119902(ΩC) by Lebesguersquos dominated
convergence theoremWe are now in a position of showing the main result in
this section
Theorem 10 Assume that Hypothesis 1 holds true and thatHypothesis 2 holds in the particular case 119870 = 119877 Then thereexist two Borel measurable deterministic functions 120592 [tinfin) times
C rarr 119877 and 120577 [119905infin) times C rarr Ξlowast
= 119871(Ξ 119877) =
1198712(Ξ 119877) such that for 119905 isin [0infin) and x isin C the solution
(119883(119905 119909) 119884(119905 119909) 119885(119905 119909)) of (36) satisfies
119884 (119904 119905 119909) = 120592 (119904 119883119904(119905 119909)) 119885 (119904 119905 119909) = 120577 (119904 119883
119904(119905 119909))
119875-as for aa 119904 isin [119905infin)
(44)
Proof We apply the techniques introduced in [26 Proposi-tion 32] Let 119890
119894 be a basis of Ξlowast and let us define 119885119894119873
=
((119885 119890119894)Ξlowast and119873) or (minus119873) Then for every 0 le 119905
1lt 119905
2lt infin Δ gt
0 and 1199091 119909
2isin C we have that
100381610038161003816100381610038161003816100381610038161003816
119864 int
1199051+Δ
1199051
119885119894119873
(119904 1199051 119909
1) 119889119904 minus 119864int
1199052+Δ
1199052
119885119894119873
(119904 1199052 119909
2) 119889119904
100381610038161003816100381610038161003816100381610038161003816
le 119864int
1199052
1199051
10038161003816100381610038161003816119885119894119873
(119904 1199051 119909
1)10038161003816100381610038161003816119889119904
+ 119864int
1199051+Δ
1199052
10038161003816100381610038161003816119885119894119873
(119904 1199051 119909
1) minus 119885
119894119873
(119904 1199052 119909
2)10038161003816100381610038161003816119889119904
+ 119864int
1199052+Δ
1199051+Δ
10038161003816100381610038161003816119885119894119873
(119904 1199052 119909
2)10038161003816100381610038161003816119889119904
8 Abstract and Applied Analysis
le 211987310038161003816100381610038161199052 minus 1199051
1003816100381610038161003816 + Δ12
119890minus120575(1199051+Δ)
times(119864(int
infin
0
119890212057511990410038161003816100381610038161003816119885119894119873
(119904 1199051 119909
1)minus119885
119894119873
(119904 1199052 119909
2)10038161003816100381610038161003816
2
119889119904)
1199012
)
1119901
le 211987310038161003816100381610038161199052 minus 1199051
1003816100381610038161003816 + Δ12
119890minus120575(1199051+Δ)
times (119864(int
infin
0
11989021205751199041003816100381610038161003816119885 (119904 1199051 1199091)minus119885 (119904 1199052 1199092)
10038161003816100381610038162
119889119904)
1199012
)
1119901
(45)
From Theorem 7 we have that the map (119905 119909) rarr int119905+Δ
119905
119885119894119873
(119904 119905 119909)119889119904 is continuous from [0infin) times C to 119877 ByRemark 8 we also have that for fixed 119905 the map 119909 rarr
119864int119905+Δ
119905
119864119885119894119873
(119904 119905 119909)119889119904 is continuous from 119871119902
(ΩCF119905) to 119877
for 119902 large enough Let us define
120577119894119873
(119905 119909) = lim inf119899rarrinfin
119899119864int
119905+(1119899)
119905
119885119894119873
(119904 119905 119909) 119889119904
119905 isin [0infin) 119909 isin C
(46)
It is clear that 120577119894119873 [0infin) timesC rarr 119877 is a Borel functionWe fix 119909 and 0 le 119905 le 119904 lt infin For 119897 isin [119904infin) we
denote 119864[119885119894119873
(119897 119904 119910)]|119910=119883119904(119905119909)
the random variable obtainedby composing119883
119904(119905 119909) with the map 119910 rarr 119864[119885
119894119873
(119897 119904 119910)]By Lemma 9 there exists a sequence of C-valued F
119904-
measurable simple functions
119891119898 Ω 997888rarr C 119891
119898=
119873119898
sum
119896=1
ℎ(119898)
119896119868119891119898=ℎ(119898)
119896 119873
119898isin 119873 (47)
where ℎ(119898)1 ℎ
(119898)
119898are pairwise distinct andΩ = ⋃
119873119898
119896=1119891
119898=
ℎ(119898)
119896 such that
1003816100381610038161003816119891119898 (120596) minus 119883119904(120596)
1003816100381610038161003816 darr 0 for all 120596 isin Ω as 119899 997888rarr infin (48)
For any 119861 isin F119904 we have
int119861
int
119904+(1119899)
119904
119885119894119873
(119897 119905 119909) 119889119897119889119875
= int119861
int
119904+(1119899)
119904
119885119894119873
(119897 119904 119883119904) 119889119897119889119875
= 119864119868119861int
119904+(1119899)
119904
119885119894119873
(119897 119904 119883119904) 119889119897
= lim119898rarrinfin
119864(119868119861int
119904+(1119899)
119904
119885119894119873
(119897 119904 119891119898) 119889119897)
= lim119898rarrinfin
119873119898
sum
119896=1
119864(119868119861119868119891119898=ℎ(119898)
119896int
119904+(1119899)
119904
119885119894119873
(119897 119904 ℎ(119898)
119896) 119889119897)
= lim119898rarrinfin
119864(119868119861
119873119898
sum
119896=1
119868119891119898=ℎ(119898)
119896)119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 ℎ(119898)
119896) 119889119897
= lim119898rarrinfin
119864119868119861(119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 119910) 119889119897|119910=119891119898
)
= lim119898rarrinfin
int119861
(119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 119910) 119889119897|119910=119891119898
)119889119875
= int119861
(119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 119910) 119889119897|119910=119883119904
)119889119875
(49)
and we get that
120577119894119873
(119904 119883119904(119905 119909)) = lim inf
119899rarrinfin
119899
times [119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 119910) 119889119897|119910=119883119904(119905119909)
]
= lim inf119899rarrinfin
119899119864[int
119904+(1119899)
119904
119885119894119873
(119897 119905 119909) 119889119897
100381610038161003816100381610038161003816100381610038161003816
F119904]
119875-as(50)
Fix 119905 and 119909 Recalling that |119885119894119873
| le 119873 by the Lebesguetheorem on differentiation it follows that 119875-as
lim119899rarrinfin
119899int
119904+(1119899)
119904
119885119894119873
(119897 119905 119909) 119889119897 = 119885119894119873
(119904 119905 119909)
for aa 119904 isin [119905infin)
(51)
By the boundedness of 119885119894119873 applying the dominated conver-gence theorem we get that
120577119894119873
(119904 119883119904(119905 119909)) = 119864 [119885
119894119873
(119904 119905 119909)10038161003816100381610038161003816F
119904] = 119885
119894119873
(119904 119905 119909)
119875-as for aa 119904 isin [119905infin)
(52)
Now we have proved that for every 119905 119909
120577119894119873
(119904 119883119904(119905 119909)) = 119885
119894119873
(119904 119905 119909)
119875-as for aa 119904 isin [119905infin)
(53)
for every 119894 119873 Let 119862 sub [0infin) times C denote the set ofpairs (119905 119909) such that lim
119873rarrinfin120577119894119873
(119905 119909) exists and the seriessuminfin
119894=1(lim
119873rarrinfin120577119894119873
(119905 119909))119890119894converges in Ξlowast We define
120577 (119905 119909) =
infin
sum
119894=1
( lim119873rarrinfin
120577119894119873
(119905 119909)) 119890119894 (119905 119909) isin 119862
120577 (119905 119909) = 0 (119905 119909) notin 119862
(54)
Abstract and Applied Analysis 9
Since 119885 satisfies
119885 (120596 119904 119905 119909) =
infin
sum
119894=1
( lim119873rarrinfin
119885119894119873
(120596 119904 119905 119909)) 119890119894 (55)
for every 120596 119904 119905 119909 From (53) it follows that for every 119905 119909 wehave (119904 119883
119904(120596 119905 119909)) isin 119862 119875-as for almost all 119904 isin [119905infin) and
119885(119904 119905 119909) = 120577(119904 119883119904(119905 119909)) 119875-as for aa 119904 isin [119905infin)
We define 120592(119905 119909) = 119884(119905 119905 119909) since 119884(119905 119905 119909) is deter-ministic so the map (119905 119909) rarr 120592(119905 119909) can be written as acomposition 120592(119905 119909) = Γ
3(Γ2(119905 Γ
1(119905 119909))) with
Γ1 [0infin) timesC 997888rarr 119871
119901
P(Ω 119862
120575(119877))
Γ1(119905 119909) = 119884 (sdot 119905 119909)
Γ2 [0infin) times 119871
119901
P(Ω 119862
120575(119877)) 997888rarr 119871
119901
(Ω 119877)
Γ2(119905 119881) = 119881 (119905)
Γ3 119871
119901
(Ω 119877) 997888rarr 119877 Γ3120585 = 119864120585
(56)
FromTheorem 7 it follows that Γ1is continuous By
|119881(119905) minus 119880(119904)|119871119901(Ω119877)
le |119881 (119905) minus 119881 (119904)|119871119901(Ω119877)
+ 119890minus120575119901119904
|119881 minus 119880||119871
119901
P(Ω119862120575(119877))
(57)
we have that Γ2is continuous It is clear that Γ
3is continuous
Then themap (119905 119909) rarr 120592(119905 119909) is continuous from [0infin)timesCto 119877 therefore 120592(119905 119909)is a Borel measurable function Fromuniqueness of the solution of (36) it follows that 119884(119904 119905 119909) =120592(119904 119883
119904(119905 119909)) 119875-as for aa 119904 isin [119905infin)
4 The Fundamental Relation
Let (ΩF 119875) be a given complete probability space with afiltration F
119905119905ge0
satisfying the usual conditions 119882(119905) 119905 ge 0
is a cylindrical Wiener process in Ξ with respect to F119905119905ge0
We will say that an F
119905ge0-predictable process 119906 with values
in a given measurable space (119880U) is an admissible controlThe function 119877 [0infin) times C times 119880 rarr Ξ is measurableand bounded We consider the following controlled stateequation
119889119883119906
(119904) = 119860119883119906
(119904) 119889119904 + 119865 (119904 119883119906
119904) 119889119904
+ 119866 (119904 119883119906
119904) 119877 (119904 119883
119906
119904 119906 (119904)) 119889119904 + 119866 (119904 119883
119906
119904) 119889119882 (119904)
119904 isin [119905infin)
119883119906
119905= 119909
(58)
Here we assume that there exists a mild solution of (58)which will be denoted by 119883119906
(119904 119905 119909) or simply by 119883119906
(119904) Weconsider a cost function of the form
119869 (119906) = 119864int
infin
119905
119890minus120582119904
119892 (119904 119883119906
119904 119906 (119904)) 119889119904 (59)
Here 119892 is function on [0infin) times C times 119880 with real values Ourpurpose is to minimize the function 119869 over all admissiblecontrols
We define in a classical way the Hamiltonian functionrelative to the previous problem for all 119905 isin [0infin) 119909 isin
C and 119911 isin Ξlowast
120595 (119905 119909 119911) = inf 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) 119906 isin 119880
(60)
and the corresponding possibly empty set of minimizers
Γ (119905 119909 119911) = 119906 isin 119880 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) = 120595 (119905 119909 119911)
(61)
We are now ready to formulate the assumptions we need
Hypothesis 3 (i) 119860 119865 and G verify Hypothesis 1(ii) (119880U) is ameasurable spaceThemap 119892 [0infin)timesCtimes
119880 rarr 119877 is continuous and satisfies |119892(119905 119909 119906)| le 119870119892(1+|119909|
119898119892
119862)
for suitable constants 119870119892gt 0 119898
119892gt 0 and all 119909 isin C119906 isin
119880 The map 119877 [0infin) times C times 119880 rarr Ξ is measurable and|119877(119905 119904 119906)| le 119871
119877for a suitable constant 119870
119877gt 0 and all 119909 isin
C119906 isin 119880 and119911 isin Ξlowast
(iii)TheHamiltonian120595 defined in (60) satisfies the requi-rements of Hypothesis 2 (with119870 = 119877)
(iv) We fix here 119901 gt 2 q and 120575 lt 0 satisfying (39) with120578 = 120578(119902) and such that 119902 gt 119898
119892
We are in a position to prove the main result of thissection
Theorem11 Assume thatHypothesis 3 holds and suppose that120582 verifies
120582 gt (minus120575 minus 120583 +1198712
119911
2) or (minus120575 +
1198712
119877
2 (119901 minus 1))
or (1198712
119877119898119892
2 (119902 minus 119898119892)minus 120578 (119902)119898
119892)
(62)
Let 120592 120577 denote the function in the statement of Theorem 10Then for every admissible control 119906 and for the correspondingtrajectory119883 starting at (119905 119909) one has
119869 (119906) = 120592 (119905 119909) + 119864int
infin
119905
119890minus120582119904
[minus120595 (119904 119883119906
119904 120577 (119904 119883
119906
119904)) + 120577 (119904 119883
119906
119904)
times 119877 (119904 119883119906
119904 119906 (119904)) + 119892 (119904 119883
119906
119904 119906 (119904))] 119889119904
(63)
Proof Consider (58) in the probability space (ΩF 119875) withfiltration F
119905119905ge0
and with an F119905119905ge0
-cylindrical Wienerprocess 119882(119905) 119905 ge 0 Let us define
119882119906
(119904) = 119882 (119904) + int
119904
119905and119904
119877 (120590119883119906
120590 119906 (120590)) 119889120590 119904 isin [0infin)
120588 (119879) = exp(int119879
119905
minus119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882 (119904)
minus1
2int
119879
119905
1003816100381610038161003816119877 (119904 119883119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)
(64)
10 Abstract and Applied Analysis
Let 119875119906 be the unique probability onF[0infin)
such that
119875119906
|F119879
= 120588 (119879) 119875|F119879
(65)
We notice that under 119875119906 the process119882119906 is aWiener processLet us denote by F119906
119905119905ge0
the filtration generated by119882119906 andcompleted in the usual way Relatively to 119882119906 (58) can berewritten as
119889119883119906
(119904) = 119860119883119906
(119904) 119889119904 + 119865 (119904 119883119906
119904) 119889119904
+ 119866 (119904 119883119906
119904) 119889119882
119906
(119904) 119904 isin [119905infin)
119883119906
119905= 119909
(66)
In the space (ΩF[0infin)
F119906
119905119905ge0 119875
119906
) we consider the follow-ing system of forward-backward equations
119883119906
(119904) = 119890(119904minus119905)119860
119909 (0) + int
119904
119905
119890(119904minus120590)119860
119865 (120590119883119906
120590) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883119906
120590) 119889119882
119906
(120590) 119904 isin [119905infin)
119883119906
119905= 119909 isin C
119884119906
(119904) minus 119884119906
(119879) + int
119879
119904
119885119906
(120590) 119889119882119906
(120590) + 120582int
119879
119904
119884119906
(120590) 119889120590
= int
119879
119904
120595 (120590119883119906
120590 119885
119906
(120590)) 119889120590 0 le 119904 le 119879 lt infin
(67)
Applying the Ito formula to 119890minus120582119904119884119906(119904) and writing the back-ward equation in (67) with respect to the process119882 we get
119884119906
(119904) + int
119879
119904
119890minus120582120590
119885119906
(120590) 119889119882 (120590)
= int
119879
119904
119890minus120582120590
[120595 (120590119883119906
120590 119885
119906
(120590))
minus119885119906
(120590) 119877 (120590119883119906
120590 119906 (120590))] 119889120590
+ 119890minus120582119879
119884119906
(119879)
(68)
Recalling that 119877 is bounded we get for all 119903 ge 1 and someconstant 119862
119864119906
[120588(119879)minus119903
] = 119864119906
[exp 119903 (int119879
119905
119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882
119906
(119904)
minus1
2int
119879
119905
1003816100381610038161003816119877 (119904 119883119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)]
= 119864119906
[exp(int119879
119905
119903119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882
119906
(119904)
minus1
2int
119879
119905
11990321003816100381610038161003816119877 (119904 119883
119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)
times exp 119903 (119903 minus 1)2
int
119879
119905
1003816100381610038161003816119877 (119904 119883119906
119904 119906 (119904))
10038161003816100381610038162
119889119904]
le 119890(12)119903(119903minus1)119879119871
2
119877119864119906
times exp(int119879
119905
2119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882
119906
(119904)
minus1
2int
119879
119905
41003816100381610038161003816119877 (119904 119883
119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)
= 119890(12)119903(119903minus1)119879119871
2
119877
(69)
It follows that
119864(int
119879
119905
|119890minus120582119904
119885119906
(119904)|2
119889119904)
12
= 119864119906
[(int
119879
119905
10038161003816100381610038161003816119890minus120582119904
119885119906
(119904)10038161003816100381610038161003816
2
119889119904)
12
120588minus1
]
le (119864119906
int
119879
119905
10038161003816100381610038161003816119890minus120582119904
119885119906
(119904)10038161003816100381610038161003816
2
119889119904)
12
times (119864119906
120588minus2
)12
lt infin
(70)
We conclude that the stochastic integral in (68) has zeroexpectation If we set 119904 = 119905 in (68) and we take expectationwith respect to 119875 we obtain
119890minus120582119879
119864119884119906
(119879) minus 119884119906
(119905)
= 119864int
119879
119905
119890minus120582120590
[minus120595 (120590119883119906
120590 119885
119906
(120590))
+119885119906
(120590) 119877 (120590119883119906
120590 119906 (120590))] 119889120590
(71)
ByTheorem 7 119884119906(sdot 119905 119909) isin 119871119901P(Ω 119862
120575(119877)) so that
119864119906
|119884(119879 119905 119909)|119901
le 119862 exp (minus119901120575119879) (72)
By the Holder inequality we have that for suitable constant119862 gt 0
119864 |119884 (119879 119905 119909)| = 119864119906
(120588minus1
(119879) |119884 (119879 119905 119909)|)
le 119864(120588minus119901(119901minus1)
)(119901minus1)119901
119864(|119884 (119879 119905 119909)|119901
)1119901
le 119862119890((1198712
1198772(119901minus1))minus120575))119879
(73)
From Theorem 2 we obtain 119864119906sup119904ge119905119890120578119902119904
|119883119906
119904|119902
lt infin by thesimilar process we get that
1198641003816100381610038161003816119883
119906
119879
1003816100381610038161003816119898119892
le 119862119890(1198712
119877119898119892(2119902minus2119898
119892)minus1
minus120578(119902)119898119892)119879
(74)
Abstract and Applied Analysis 11
for suitable constant 119862 gt 0 and
119864int
infin
119905
119890minus120582120590 1003816100381610038161003816119892 (120590119883
119906
120590 119906 (120590))
1003816100381610038161003816 119889120590 lt infin (75)
Since 119884119906(119905 119905 119909) = 120592(119905 119909) and 119885119906
(119904 119905 119909) = 120577(119904 119883119906
119904(119905 119909)) 119875-
as for aa 119904 isin [119905infin) we have that
119890minus120582119879
119864119884119906
(119879) minus 119907 (119905 119909)
= 119864int
119879
119905
119890minus120582120590
[minus120595 (120590119883119906
120590 120577 (120590 119883
119906
120590))
+120577 (120590119883119906
120590) 119877 (120590119883
119906
120590 119906 (120590))] 119889120590
(76)
Thus adding and subtracting119864intinfin119905
119890minus120582120590
119892(120590119883119906
120590 119906(120590))119889120590 and
letting 119879 rarr infin we conclude that
119869 (119906) = 120592 (119905 119909)
+ 119864int
infin
119905
119890minus120582119904
[minus120595 (119904 119883119906
119904 120577 (119904 119883
119906
119904)) + 120577 (119904 119883
119906
119904) 119877
times (119904 119883119906
119904 119906 (119904)) + 119892 (119904 119883
119906
119904 119906 (119904))] 119889119904
(77)
The proof is finished
We immediately deduce the following consequences
Theorem 12 Let 119905 isin [0infin) and 119909 isin C be fixed assumethat the set-valued map Γ has nonempty values and it admitsa measurable selection Γ
0 [0infin)timesCtimesΞlowast rarr 119880 and assume
that a control 119906(sdot) satisfies
119906 (119904) = Γ0(119904 119883
119906
119904 120577 (119904 119883
119906
119904))
119875-as for almost every 119904 isin [119905infin)
(78)
Then 119869(119905 119909 119906) = 120592(119905 119909) and the pair (119906(sdot) 119883) is optimal forthe control problem starting from 119909 at time 119905
Such a control can be shown to exist if there exists a solutionfor the so-called closed-loop equation as follows
119889119883 (119904)=119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904+119866 (119904 119883
119904)
times(119877 (119904 119883119904 Γ
0(119904 119883
119904 120577 (119904 119883
119904))) 119889119904 + 119889119882 (119904))
119904 isin [119905infin)
119883119905= 119909
(79)
since in this case we can define an optimal control setting
119906 (119904) = Γ0(119904 119883
119904 120577 (119904 119883
119904)) (80)
However under the present assumptions we cannot guaranteethat the closed-loop equation has a solution in the mildsense To circumvent this difficulty we will revert to a weakformulation of the optimal control problem
5 Existence of Optimal Control
We formulate the optimal control problem in the weak sensefollowing the approach of [31]Themain advantage is that wewill be able to solve the closed-loop equation in a weak senseand hence to find an optimal control even if the feedbacklaw is nonsmooth
We call (ΩF F119905119905ge0 119875119882) an admissible setup if
(ΩF F119905119905ge0 119875) is a filtered probability space satisfying the
usual conditions and 119882 is a cylindrical 119875-Wiener processwith values in Ξ with respect to the filtration F
119905119905ge0
By an admissible control system we mean (ΩF
F119905119905ge0 119875119882 119906119883
119906
) where (ΩF F119905119905ge0 119875 119882) is an
admissible setup 119906 is an F119905-predictable process with values
in119880 and119883119906 is a mild solution of (58) An admissible controlsystem will be briefly denoted by (119882 119906119883119906
) in the followingOur purpose is to minimize the cost functional
119869 (119906) = 119864int
infin
119905
119890minus120582119904
119892 (119904 119883119906
119904 119906 (119904)) 119889119904 (81)
over all the admissible control systemOur main result in this section is based on the solvability
of the closed-loop equation
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904 + 119866 (119904 119883
119904)
times (119877 (119904 119883119904 Γ
0(119904 119883
119904 120577 (119904 119883
119904))) 119889119904 +119889119882 (119904))
119904 isin [119905infin)
119883119905= 119909
(82)
In the following sense we say that 119883 is a weak solution of(82) if there exists an admissible setup (ΩF F
119905119905ge0 119875119882)
and anF119905-adapted continuous process119883(119905)with values in119867
which solves the equation in the mild sense namely 119875-as
119883 (119904) = 119890(119904minus119905)119860
119909 (0) + int
119904
119905
119890(119904minus120590)119860
119865 (120590119883120590) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883120590) 119877
times (120590119883120590 Γ
0(120590119883
120590 120577 (120590 119883
120590))) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883120590) 119889119882
120590 119904 isin [119905infin)
(83)
119883119905= 119909 (84)
Theorem 13 Assume that Hypothesis 3 holds Then thereexists a weak solution of the closed-loop equation (82) whichis unique in law
12 Abstract and Applied Analysis
Proof (uniqueness) Let 119883 be a weak solution of (82) in anadmissible setup (ΩF F
119905119905ge0 119875119882)We define
120588 (119879) = exp(int119879
119905
minus119877lowast
(120590 119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889119882 (120590)
minus1
2int
119879
119905
1003816100381610038161003816119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)))10038161003816100381610038162
119889120590)
(85)
Since 119877 is bounded the Girsanov theorem ensures that thereexists a probability measure 1198750 such that the process
1198820
(119904) = 119882 (119904) + int
119904
119905and119904
119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889120590
119904 isin [0infin)
(86)
is a 1198750-Wiener process and
1198750
|F119879
= 120588 (119879) 119875|F119879
(87)
Let us denote by F0
119905119905ge0
the filtration generated by1198820 andcompleted in the usual way In (ΩF
[0infin) F0
119905119905ge0 119875
0
) 119883 isa mild solution of
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119905 119883119904) 119889119904
+ 119866 (119904 119883119904) 119889119882
0
(119904) 119904 isin [119905infin)
119883119905= 119909
120588 (119879) = exp(int119879
119905
minus119877lowast
(120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)) 119889119882
0
(120590)
+ 12int
119879
119905
1003816100381610038161003816119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)))10038161003816100381610038162
119889120590)
(88)
By Hypothesis 3 the joint law of 119883 and 1198820 is uniquely
determined by 119860 119865 119866 and 119909 Taking into account the lastdisplayed formula we conclude that the joint law of 119883 and120588(119879) under1198750 is also uniquely determined and consequentlyso is the law of 119883 under 119875 This completes the proof of theuniqueness part
Proof (existence) Let (ΩF 119875) be a given complete probabil-ity space 119882(119905) 119905 ge 0 is a cylindrical Wiener process on(ΩF 119875)with values inΞ and F
119905119905ge0
is the natural filtrationof 119882(119905) 119905 ge 0 augmented with the family of119875-null sets Let119883(sdot) be the mild solution of
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904
+ 119866 (119904 119883119904) 119889119882 (119904) 119904 isin [119905infin)
119883119905= 119909
(89)
and by the Girsanov theorem let 1198751 be the probability on Ωunder which
1198821
(119904) = 119882 (119904) minus int
119904
119905and119904
119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889120590
(90)
is aWiener process (notice that 119877 is bounded)Then119883 is theweak solution of (82) relatively to the probability 1198751 and theWiener process1198821
Now we can state the main result of this section
Corollary 14 Assume that Hypothesis 3 holds true and 120582
verifies (62) Also assume that the set-valued map Γ hasnonempty values and it admits a measurable selection Γ
0
[0infin) timesC times Ξlowast
rarr 119880 Then for every 119905 isin [0infin) and x isin Cand for all admissible control system (119882 119906119883
119906
) one has
119869 (119906 119905 119909) ge 120592 (119905 119909) (91)
and the equality holds if
119906 (119904) = Γ0(119904 119883
119906
119904 120577 (119904 119883
119906
119904))
119875 minus 119886119904 119891119900119903 119886119897119898119900119904119905 119890119907119890119903119910 119904 isin [119905infin)
(92)
Moreover from Theorem 13 it follows that the closed-loop equation (82) admits a weak solution (ΩFF
119905119905ge0 119875119882119883) which is unique in law and setting
119906 (119904) = Γ0(119904 119883
119904 120577 (119904 119883
119904)) (93)
we obtain an optimal admissible control system (119882 119906119883)
6 Applications
In this section we present a simple application of the previousresults We consider the stochastic delay partial differentialequation in the bounded domain 119861 sub 119877
119899 with smoothboundary 120597119861 as follows
119889119911119906
(119905 120585) = Δ119911119906
(119905 120585) 119889119905 + 119891 (119905 119911119906
119905(120585)) 119889119905
+
119889
sum
119894=1
119892119894(119905 119911
119906
119905(120585)) [119903
119894
(120585) 119906119894
(119905) 119889119905 + 119889119882119894
(119905)]
119911119906
0(120579 120585) = 119909 (120579 120585) 120585 isin 119861 120579 isin [minus1 0]
119911119906
(119905 120585) = 0 119905 isin [0infin) 120585 isin 120597119861
(94)
Here119882 = (1198821
1198822
119882119889
) is a standardWiener process in119877119889 and the functions 119891 [0 +infin) times 119862([minus1 0] 119877) rarr 119877 and119892119894 [0 +infin) times 119862([minus1 0] 119877) rarr 119877 are Lipschitz continuous
and bounded Setting 119880 as a bounded subset of 119877119889 Ξ = 119877119889
119867 = 1198712
(119861) and 119909 isin 119862([minus1 0]119867) We define 119865 and 119866 asfollowing
119865 (119905 119909) (120585) = 119891 (119905 119909 (120585))
(119866 (119905 119909) 119911) (120585) =
119889
sum
119894=1
119892119894(119905 119909 (120585)) 119911
119894
(120585)
120585 isin 119861 119909 isin 119862 ([minus1 0] 119867) 119911 isin 119871 (Ξ119867)
(95)
and let 119860 denote the Laplace operator Δ in 1198712
(119861) withdomain11988222
(119861)⋂11988212
0(119861) then (94) has the form (58) and
Hypothesis 1 holds
Abstract and Applied Analysis 13
Let us consider the optimal control problem associatedwith the cost
119869 (119906) = 119864int
infin
0
119890minus120582119905
[int119861
120590 (120585 119911119906
119905(120585)) 119889120585 + 119906
2
(119905)] 119889119905 (96)
where 120582 verifies (62) and 120590 119862([minus1 0] 119877) times 119880 rarr [0infin) isa bounded measurable function Define 119892 119862([minus1 0]119867) times
119880 rarr [0infin) and 119877 119862([minus1 0]119867) times 119880 rarr Ξ by119892(119910 119906) = int
119861
120590(119905 119910(120585) 119906)119889120585 + 1199062 and 119877(119910 119906) = (int
119861
1199031
(120585)1199061
119889120585
int119861
1199032
(120585)1199062
119889120585 int119861
119903119889
(120585)119906119889
119889120585) for 119910 isin 119862([minus1 0]119867) 119906 =
(1199061
1199062
119906119889
) isin 119880 respectively It can be easily verifiedthat Hypothesis 3 holds true and the set-valued map Γ hasnonempty values and it admits a measurable selection Γ
0
[0infin) times C times Ξlowast
rarr 119880 Then the closed-loop equation(82) admits a weak solution (ΩF F
119905119905ge0 119875119882 119906 119911
sdot(sdot)) and
setting
119906 (119904) = Γ0(119904 119911
119904(sdot) 120577 (119904 119911
119904(sdot))) (97)
we obtain an optimal admissible control system (119882 119906 119911(sdot))
References
[1] J Bismut ldquoOn optimal control of linear stochastic equationswith a linear-quadratic criterionrdquo SIAM Journal on Control andOptimization vol 15 no 3 pp 1ndash4 1977
[2] N Nagase ldquoOn the existence of optimal control for controlledstochastic partial differential equationsrdquo Nagoya MathematicsJournal vol 115 pp 73ndash85 1989
[3] N El Karoui D Huu Nguyen and M Jeanblanc-Pique ldquoCom-pactification methods in the control of degenerate diffusionsrdquoStochastics vol 20 pp 169ndash219 1987
[4] M Nisio ldquoOptimal control for stochastic partial differentialequations and viscosity solutions of Bellman equationsrdquoNagoyaMathematics Journal vol 123 pp 13ndash37 1991
[5] M Nisio ldquoOn sensitive control for stochastic partial differentialequationsrdquo in Stochastic Analysis on Infinite Dimensional SpacesProceedings of the US Japan Bilateral Seminar H Kunita et alEd vol 310 of Pitman Research Notes Mathematical Series pp231ndash241 Longman Scientific and Technical Baton Rouge LaUSA January 1994
[6] R Buckdahn and A Rascanu ldquoOn the existence of stochasticoptimal control of distributed state systemrdquoNonlinear AnalysisTheory Methods and Applications vol 52 no 4 pp 1153ndash11842003
[7] V Barbu and G Da Prato Equations in Hilbert Spaces vol 86 ofPitman Research Notes in Mathematics Pitman 1983
[8] P Cannarsa and G Da Prato ldquoSecond-order Hamilton-Jacobiequations in infinite dimensionsrdquo SIAM Journal on Control andOptimization vol 29 no 2 pp 474ndash492 1991
[9] P Cannarsa and G Da Prato ldquoDirect solution of a second-orderHamilton-Jacobi equations in Hilbert spacesrdquo in StochasticPartial Differential Equations and Applications G Da Pratoand L Tubaro Eds vol 268 of Pitman Research Notes inMathematics Pitman 1992
[10] F Gozzi ldquoRegularity of solutions of second order Hamilton-Jacobi equations and application to a control problemrdquoCommu-nications in Partial Differential Equations vol 20 pp 775ndash8261995
[11] F Gozzi ldquoGlobal regular solutions of second order Hamilton-Jacobi equations in Hilbert spaces with locally Lipschitz non-linearitiesrdquo Journal of Mathematical Analysis and Applicationsvol 198 no 2 pp 399ndash443 1996
[12] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems and Control Lettersvol 14 no 1 pp 55ndash61 1990
[13] N El Karoui and LMazliak Eds Backward Stochastic Differen-tial Equations vol 364 of Pitman ResearchNotes inMathematicsSeries Longman 1997
[14] E Pardoux and BSDEs ldquoweak convergence and homogeneiza-tion of semilinear PDEsrdquo in Non- Linear Analysis DifferentialEquations and Control F H Clarke and R J Stern Eds pp503ndash549 Kluwer Dordrecht The Netherlands 1999
[15] S Peng ldquoA generalized dynamic programming principle andHamilton-Jacobi-Bellman equationrdquo Stochastics and StochasticsReports vol 38 pp 119ndash134 1992
[16] N E Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997
[17] SHamad120583ene and J P Lepeltier ldquoBackward equations stochas-tic control and zero-sum stochastic differential gamesrdquo Stochas-tics and Stochastics Reports vol 54 pp 221ndash231 1995
[18] N El-Karoui and S Hamadene ldquoBSDEs and risk-sensitive con-trol zero-sum and nonzero-sum game problems of stochasticfunctional differential equationsrdquo Stochastic Processes and theirApplications vol 107 no 1 pp 145ndash169 2003
[19] M Fuhrman and G Tessiture ldquoExistence of optimal stochasticcontrols and global solutions of forward-backward stochasticdifferential equationsrdquo SIAM Journal on Control and Optimiza-tion vol 43 no 3 pp 813ndash830 2005
[20] M Fuhrman Y Hu and G Tessitore ldquoOn a class of stochasticoptimal control problems related to bsdes with quadraticgrowthrdquo SIAM Journal on Control and Optimization vol 45 no4 pp 1279ndash1296 2006
[21] M Fuhrman and G Tessitore ldquoNonlinear kolmogorov equa-tions in infinite dimensional spaces the backward stochasticdifferential equations approach and applications to optimalcontrolrdquoAnnals of Probability vol 30 no 3 pp 1397ndash1465 2002
[22] F Masiero ldquoSemilinear kolmogorov equations and applicationsto stochastic optimal controlrdquo Applied Mathematics and Opti-mization vol 51 no 1 pp 201ndash250 2005
[23] M Fuhrman FMasiero andG Tessitore ldquoStochastic equationswith delay optimal control via BSDEs and regular solutions ofHamilton-jacobi-bellman equationsrdquo SIAM Journal on Controland Optimization vol 48 no 7 pp 4624ndash4651 2010
[24] M Fuhrman and G Tessiture ldquoInfinite horizon backwardstochastic differential equations and elliptic equations in hilbertspacesrdquo Annals of Probability vol 32 no 1 pp 607ndash660 2004
[25] F Masiero ldquoInfinite horizon stochastic optimal control prob-lems with degenerate noise and elliptic equations in Hilbertspacesrdquo Applied Mathematics and Optimization vol 55 no 3pp 285ndash326 2007
[26] M Fuhrman ldquoA class of stochastic optimal control problemsin Hilbert spaces BSDEs and optimal control laws stateconstraints conditioned processesrdquo Stochastic Processes andtheir Applications vol 108 no 2 pp 263ndash298 2003
[27] F Masiero ldquoStochastic optimal control problems and parabolicequations in banach spacesrdquo SIAM Journal on Control andOptimization vol 47 no 1 pp 251ndash300 2008
[28] G Da Prato and J Zabczyk Ergodicity For Infinite-DimensionalSystems Cambridge University Press 1996
14 Abstract and Applied Analysis
[29] J Zabczyk ldquoParabolic equations on Hilbert spacesrdquo in StochaS-tic PDErsquoS and Kolmogorov Equations in Infinite Dimensionsvol 1715 of Lecture Notes in Math pp 117ndash213 Springer BerlinGermany 1999
[30] G Da Prato and J Zabczyk Stochstic Equations in InfiniteDimensions Cambridge University Press 1992
[31] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 of Applications of MathematicsSpringer New York NY USA 1993
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6 Abstract and Applied Analysis
our present notationΦ can be seen as a mapping fromH119902
120578times
[0infin) timesC toH119902
120578as follows
Φ(119883sdot 119905 119909)
119904(119897) = 119878 (119904 + 119897 minus 119905) 119909 (0)
+ int
119904+119897
0
119868[119905infin)
(120590) 119878 (119904 + 119897 minus 120590) 119865 (120590119883120590) 119889120590
+ int
119904+119897
0
119868[119905infin)
(120590) 119878 (119904 + 119897 minus 120590)
times 119866 (120590119883120590) 119889119882 (120590)
119904 isin [0infin) 119897 isin [minus120591 0] 119904 + 119897 ge 119905
Φ(119883sdot 119905 119909)
119904(119897) = 119909 ((119904 + 119897 minus 119905) or (minus120591))
119904 isin [0infin) 119897 isin [minus120591 0] 119904 + 119897 le 119905
(34)
By the arguments of the proof of Theorem 2 Φ(sdot 119905 119909)is a contraction in H119902
120578uniformly with respect to 119905 119909
The process 119883sdot(119905 119909) is the unique fixed point of Φ(sdot 119905 119909)
So by the parameter-depending contraction principle(Lemma 3) it suffices to show that Φ is continuous fromH119902
120578times [0infin) times C to H119902
120578 From the contraction property
of Φ(sdot 119905 119909) mentioned earlier we have that Φ(sdot 119905 119909) iscontinuous uniformly in 119905 119909 Moreover for fixed 119883
sdot it is
easy to verify that Φ(119883sdot sdot sdot) is continuous from [0infin) times C
toH119902
120578 The proof is finished
Remark 5 By similar passages we can show that for fixed119905 Theorem 4 still holds true for 119902 large enough if the spaces[0infin) times C and H119902
120578are replaced by the spaces 119871119902(ΩCF
119905)
and H119902
120578(119905) respectively where 119871119902(ΩCF
119905) denotes that the
space of F119905-measurable function with value in C such that
the norm
|119909|119902
= 119864|119909|119902
119862 (35)
is finite
3 The Backward-Forward System
In this section we consider the system of stochastic differen-tial equations 119875-as
119883 (119904) = 119890(119904minus119905)119860
119909 (0) + int
119904
119905
119890(119904minus120590)119860
119865 (120590119883120590) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883120590) 119889119882 (120590) 119904 isin [119905infin)
119883119905= 119909 isin C
119884 (119904) minus 119884 (119879) + int
119879
119904
119885 (120590) 119889119882 (120590) + 120582int
119879
119904
119884 (120590) 119889120590
= int
119879
119904
120595 (120590119883120590 119884 (120590) 119885 (120590)) 119889120590 0 le 119904 le 119879 lt infin
(36)
for 119904 varying on the time interval [119905infin) sub [0infin) Asin Section 2 we extend the domain of the solution setting119883(119904 119905 119909) = 119909((119904 minus 119905) or (minus120591)) for 119904 isin [minus120591 119905)
We make the following assumptions
Hypothesis 2 Themapping 120595 [0infin)timesCtimes119870times1198712(Ξ 119870) rarr
119870 is Borelmeasurable such that for all 119905 isin [0infin)120595(119905 sdot) Ctimes119870 times 119871
2(Ξ 119870) rarr 119870 is continuous and for some 119871
119910 119871
119911gt 0
120583 isin 119877 and119898 ge 11003816100381610038161003816120595 (119904 119909 1199101 1199111) minus 120595 (119904 119909 1199102 1199112)
1003816100381610038161003816
le 119871119910
10038161003816100381610038161199101 minus 11991021003816100381610038161003816 + 119871119911
10038161003816100381610038161199111 minus 11991121003816100381610038161003816
1003816100381610038161003816120595 (119904 119909 119910 119911)1003816100381610038161003816 le 119871 (1 + |119909|
119898
119862+10038161003816100381610038161199101003816100381610038161003816 + |119911|)
⟨120595 (119904 119909 1199101 119911) minus 120595 (119904 119909 119910
2 119911) 119910
1minus 119910
2⟩119870ge 120583
10038161003816100381610038161199101 minus 119910210038161003816100381610038162
(37)
for every 119904 isin [0infin) 119909 isin C 119910 1199101 119910
2isin 119870 119911 119911
1 and 119911
2isin
1198712(Ξ 119870)We note that the third inequality in (37) follows from the
first one taking 120583 = minus119871119910but that the third inequalitymay also
hold for different values of 120583Firstly we consider the backward stochastic differential
equation
119884 (119904) minus 119884 (119879) + int
119879
119904
119885 (120590) 119889119882 (120590) + 120582int
119879
119904
119884 (120590) 119889120590
= int
119879
119904
120595 (120590119883120590 119884 (120590) 119885 (120590)) 119889120590 0 le 119904 le 119879 lt infin
(38)
119870 is a Hilbert space the mapping 120595 [0infin) times C times 119870 times
1198712(Ξ 119870) rarr 119870 is a given measurable function 119883
sdotis a
predictable process with values in another Banach space Cand 120582 is a real number
Theorem 6 Assume that Hypothesis 2 holds Let 119901 gt 2 and120575 lt 0 be given and choose
119902 ge 119898119901 120578 gt120575
119898 (39)
Then the following hold
(i) For 119883sdotisin 119871
119902
P(Ω 119871
119902
120578(C)) and 120582 gt minus(120575 + 120583 minus (119871
2
1199112))
(38) has a unique solution in Kp120575that will be denoted
by (119884(119883sdot)(119904) 119885(119883
sdot)(119904)) 119904 ge 0
(ii) The estimate
119864sup119904ge0
(119884 (119883sdot) (119904))
119901
119890119901120575119904
+ 119864(int
infin
0
11989021205751205901003816100381610038161003816119884(119883sdot
)(120590)10038161003816100381610038162
119889120590)
1199012
+ 119864(int
infin
0
11989021205751205901003816100381610038161003816119885 (119883sdot
) (120590)10038161003816100381610038162
119889120590)
1199012
le 119888(1 +1003816100381610038161003816119883sdot
1003816100381610038161003816119898
119871119902
P(Ω119871119902
120578(C))
)119901
(40)
Abstract and Applied Analysis 7
holds for a suitable constant 119888 In particular 119884(119883sdot) isin
119871119901
P(Ω 119862
120575(119870))
(iii) The map 119883sdotrarr (119884(119883
sdot) 119885(119883
sdot)) is continuous from
119871119902
P(Ω 119871
119902
120578(C)) toK119901
120575 and 119883
sdotrarr 119884(119883
sdot) is continuous
from 119871119902
P(Ω 119871
119902
120578(C)) to 119871119901
P(Ω 119862
120575(119870))
(iv) The statements of points (i) (ii) and (iii) still holdtrue if the space 119871119902
P(Ω 119871
119902
120578(C)) is replaced by the space
119871119902
P(Ω 119862
120578(C))
Proof The theorem is very similar to Proposition 311 in [24]The only minor difference is that the mapping 120595 [0infin) times
Ctimes119870times1198712(Ξ 119870) rarr 119870 is a givenmeasurable function while
in [24] the measurable function 120595 is from119867 times119870 times 1198712(Ξ 119870)
to 119870 however the same arguments apply
Theorem 7 Assume that Hypothesis 1 holds and thatHypothesis 2 holds true in the particular case 119870 = 119877 Thenfor every 119901 gt 2 119902 120575 lt 0 satisfying (39) with 120578 = 120578(119902)and for every 120582 gt 120582
1015840
= minus(120575 + 120583 minus (1198712
1199112)) there exists a
unique solution in H119902
120578(119902)times K
119901
120575of (36) that will be denoted
by (119883(sdot 119905 119909) 119884(sdot 119905 119909) 119885(sdot 119905 119909)) Moreover 119884(sdot 119905 119909) isin
119871119901
P(Ω 119862
120575(119877)) The map (119905 119909) rarr (119884(sdot 119905 119909) 119885(sdot 119905 119909)) is con-
tinuous from [0infin)timesC toK119901
120575 and themap (119905 119909) rarr 119884(sdot 119905 119909)
is continuous from [0infin) timesC to 119871119901P(Ω 119862
120575(119877))
Proof We first notice that the system is decoupled the firstdoes not contain the solution (119884 119885) of the second oneThere-fore under the assumption of Hypothesis 1 by Theorem 2there exists a unique solution 119883(sdot 119905 119909) and 119883
sdot(119905 119909) isin H
119902
120578(119902)
of the first equation Moreover from Theorem 4 it followsthat the map (119905 119909) rarr 119883
sdot(119905 119909) is continuous from [0infin)timesC
toH119902
120578(119902)
Let 119870 = 119877 from Theorem 6 we have that thereexists a unique solution (119884(sdot 119905 119909) 119885(sdot 119905 119909)) isin K
119901
120575of the
second equation and the map 119883sdotrarr (119884(119883
sdot) 119885(119883
sdot)) is
continuous from H119902
120578(119902)to K
119901
120575while X
sdotrarr (Y(X
sdot)) is
continuous fromH119902
120578(119902)to119871119901
P(Ω 119862
120575(119877))We have proved that
(119883(sdot 119905 119909) 119884(sdot 119905 119909) 119885(sdot 119905 119909)) isin H119902
120578(119902)times K
119901
120575is the unique
solution of (36) and the other assertions follow from com-position
Remark 8 From Remark 5 by similar passages we can showthat for fixed 119905 and for 119902 large enough under the assumptionsof Theorem 7 the map 119909 rarr (119884(sdot 119905 119909) 119885(sdot 119905 119909)) is continu-ous from 119871
119902
(ΩCF119905) toK119901
120575(119905)
We also remark that the process 119883(sdot 119905 119909) is F[119905infin)
measurable since C is separable Banach space we have that119883sdot(119905 119909) is F
[119905infin)measurable So that 119884(119905) is measurable
with respect to both F[119905infin)
and F119905 it follows that 119884(119905) is
deterministicFor later use we notice three useful identities for 119905 le 119904 lt
infin the equality 119875-as
119883119897(119904 119883
119904(119905 119909)) = 119883
119897(119905 119909) 119897 isin [119904infin) (41)
is a consequence of the uniqueness of the solution of (13)Since the solution of the backward equation is uniquely
determined on an interval [119904infin) by the values of the process119883sdoton the same interval for 119905 le 119904 lt infin we have 119875-as
119884 (119897 119904 119883119904(119905 119909)) = 119884 (119897 119905 119909) for 119897 isin [119904infin)
119885 (119897 119904 119883119904(119905 119909)) = 119885 (119897 119905 119909) for aa 119897 isin [119904infin)
(42)
Lemma 9 (see [30]) Let 119864 be a metric space with metric 119889and let 119891 Ω rarr 119864 be strongly measurable Then thereexists a sequence 119891
119899 119899 isin 119873 of simple 119864-valued functions
(ie 119891119899isFB(E)measurable and takes only a finite number
of values) such that for arbitrary 120596 isin Ω the sequence119889(119891
119899(120596) 119891(120596)) 119899 isin 119873 is monotonically decreasing to zero
Let now 119891 isin 119871119902
(ΩC) By Lemma 9 we get the existenceof a sequence of simple function 119891
119899 119899 isin 119873 such that
1003816100381610038161003816119891119899 (120596) minus 119891 (120596)1003816100381610038161003816 darr 0 for all 120596 isin Ω as 119899 rarr infin (43)
Hence 119891119899
rarr 119891 in | sdot |119871119902(ΩC) by Lebesguersquos dominated
convergence theoremWe are now in a position of showing the main result in
this section
Theorem 10 Assume that Hypothesis 1 holds true and thatHypothesis 2 holds in the particular case 119870 = 119877 Then thereexist two Borel measurable deterministic functions 120592 [tinfin) times
C rarr 119877 and 120577 [119905infin) times C rarr Ξlowast
= 119871(Ξ 119877) =
1198712(Ξ 119877) such that for 119905 isin [0infin) and x isin C the solution
(119883(119905 119909) 119884(119905 119909) 119885(119905 119909)) of (36) satisfies
119884 (119904 119905 119909) = 120592 (119904 119883119904(119905 119909)) 119885 (119904 119905 119909) = 120577 (119904 119883
119904(119905 119909))
119875-as for aa 119904 isin [119905infin)
(44)
Proof We apply the techniques introduced in [26 Proposi-tion 32] Let 119890
119894 be a basis of Ξlowast and let us define 119885119894119873
=
((119885 119890119894)Ξlowast and119873) or (minus119873) Then for every 0 le 119905
1lt 119905
2lt infin Δ gt
0 and 1199091 119909
2isin C we have that
100381610038161003816100381610038161003816100381610038161003816
119864 int
1199051+Δ
1199051
119885119894119873
(119904 1199051 119909
1) 119889119904 minus 119864int
1199052+Δ
1199052
119885119894119873
(119904 1199052 119909
2) 119889119904
100381610038161003816100381610038161003816100381610038161003816
le 119864int
1199052
1199051
10038161003816100381610038161003816119885119894119873
(119904 1199051 119909
1)10038161003816100381610038161003816119889119904
+ 119864int
1199051+Δ
1199052
10038161003816100381610038161003816119885119894119873
(119904 1199051 119909
1) minus 119885
119894119873
(119904 1199052 119909
2)10038161003816100381610038161003816119889119904
+ 119864int
1199052+Δ
1199051+Δ
10038161003816100381610038161003816119885119894119873
(119904 1199052 119909
2)10038161003816100381610038161003816119889119904
8 Abstract and Applied Analysis
le 211987310038161003816100381610038161199052 minus 1199051
1003816100381610038161003816 + Δ12
119890minus120575(1199051+Δ)
times(119864(int
infin
0
119890212057511990410038161003816100381610038161003816119885119894119873
(119904 1199051 119909
1)minus119885
119894119873
(119904 1199052 119909
2)10038161003816100381610038161003816
2
119889119904)
1199012
)
1119901
le 211987310038161003816100381610038161199052 minus 1199051
1003816100381610038161003816 + Δ12
119890minus120575(1199051+Δ)
times (119864(int
infin
0
11989021205751199041003816100381610038161003816119885 (119904 1199051 1199091)minus119885 (119904 1199052 1199092)
10038161003816100381610038162
119889119904)
1199012
)
1119901
(45)
From Theorem 7 we have that the map (119905 119909) rarr int119905+Δ
119905
119885119894119873
(119904 119905 119909)119889119904 is continuous from [0infin) times C to 119877 ByRemark 8 we also have that for fixed 119905 the map 119909 rarr
119864int119905+Δ
119905
119864119885119894119873
(119904 119905 119909)119889119904 is continuous from 119871119902
(ΩCF119905) to 119877
for 119902 large enough Let us define
120577119894119873
(119905 119909) = lim inf119899rarrinfin
119899119864int
119905+(1119899)
119905
119885119894119873
(119904 119905 119909) 119889119904
119905 isin [0infin) 119909 isin C
(46)
It is clear that 120577119894119873 [0infin) timesC rarr 119877 is a Borel functionWe fix 119909 and 0 le 119905 le 119904 lt infin For 119897 isin [119904infin) we
denote 119864[119885119894119873
(119897 119904 119910)]|119910=119883119904(119905119909)
the random variable obtainedby composing119883
119904(119905 119909) with the map 119910 rarr 119864[119885
119894119873
(119897 119904 119910)]By Lemma 9 there exists a sequence of C-valued F
119904-
measurable simple functions
119891119898 Ω 997888rarr C 119891
119898=
119873119898
sum
119896=1
ℎ(119898)
119896119868119891119898=ℎ(119898)
119896 119873
119898isin 119873 (47)
where ℎ(119898)1 ℎ
(119898)
119898are pairwise distinct andΩ = ⋃
119873119898
119896=1119891
119898=
ℎ(119898)
119896 such that
1003816100381610038161003816119891119898 (120596) minus 119883119904(120596)
1003816100381610038161003816 darr 0 for all 120596 isin Ω as 119899 997888rarr infin (48)
For any 119861 isin F119904 we have
int119861
int
119904+(1119899)
119904
119885119894119873
(119897 119905 119909) 119889119897119889119875
= int119861
int
119904+(1119899)
119904
119885119894119873
(119897 119904 119883119904) 119889119897119889119875
= 119864119868119861int
119904+(1119899)
119904
119885119894119873
(119897 119904 119883119904) 119889119897
= lim119898rarrinfin
119864(119868119861int
119904+(1119899)
119904
119885119894119873
(119897 119904 119891119898) 119889119897)
= lim119898rarrinfin
119873119898
sum
119896=1
119864(119868119861119868119891119898=ℎ(119898)
119896int
119904+(1119899)
119904
119885119894119873
(119897 119904 ℎ(119898)
119896) 119889119897)
= lim119898rarrinfin
119864(119868119861
119873119898
sum
119896=1
119868119891119898=ℎ(119898)
119896)119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 ℎ(119898)
119896) 119889119897
= lim119898rarrinfin
119864119868119861(119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 119910) 119889119897|119910=119891119898
)
= lim119898rarrinfin
int119861
(119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 119910) 119889119897|119910=119891119898
)119889119875
= int119861
(119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 119910) 119889119897|119910=119883119904
)119889119875
(49)
and we get that
120577119894119873
(119904 119883119904(119905 119909)) = lim inf
119899rarrinfin
119899
times [119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 119910) 119889119897|119910=119883119904(119905119909)
]
= lim inf119899rarrinfin
119899119864[int
119904+(1119899)
119904
119885119894119873
(119897 119905 119909) 119889119897
100381610038161003816100381610038161003816100381610038161003816
F119904]
119875-as(50)
Fix 119905 and 119909 Recalling that |119885119894119873
| le 119873 by the Lebesguetheorem on differentiation it follows that 119875-as
lim119899rarrinfin
119899int
119904+(1119899)
119904
119885119894119873
(119897 119905 119909) 119889119897 = 119885119894119873
(119904 119905 119909)
for aa 119904 isin [119905infin)
(51)
By the boundedness of 119885119894119873 applying the dominated conver-gence theorem we get that
120577119894119873
(119904 119883119904(119905 119909)) = 119864 [119885
119894119873
(119904 119905 119909)10038161003816100381610038161003816F
119904] = 119885
119894119873
(119904 119905 119909)
119875-as for aa 119904 isin [119905infin)
(52)
Now we have proved that for every 119905 119909
120577119894119873
(119904 119883119904(119905 119909)) = 119885
119894119873
(119904 119905 119909)
119875-as for aa 119904 isin [119905infin)
(53)
for every 119894 119873 Let 119862 sub [0infin) times C denote the set ofpairs (119905 119909) such that lim
119873rarrinfin120577119894119873
(119905 119909) exists and the seriessuminfin
119894=1(lim
119873rarrinfin120577119894119873
(119905 119909))119890119894converges in Ξlowast We define
120577 (119905 119909) =
infin
sum
119894=1
( lim119873rarrinfin
120577119894119873
(119905 119909)) 119890119894 (119905 119909) isin 119862
120577 (119905 119909) = 0 (119905 119909) notin 119862
(54)
Abstract and Applied Analysis 9
Since 119885 satisfies
119885 (120596 119904 119905 119909) =
infin
sum
119894=1
( lim119873rarrinfin
119885119894119873
(120596 119904 119905 119909)) 119890119894 (55)
for every 120596 119904 119905 119909 From (53) it follows that for every 119905 119909 wehave (119904 119883
119904(120596 119905 119909)) isin 119862 119875-as for almost all 119904 isin [119905infin) and
119885(119904 119905 119909) = 120577(119904 119883119904(119905 119909)) 119875-as for aa 119904 isin [119905infin)
We define 120592(119905 119909) = 119884(119905 119905 119909) since 119884(119905 119905 119909) is deter-ministic so the map (119905 119909) rarr 120592(119905 119909) can be written as acomposition 120592(119905 119909) = Γ
3(Γ2(119905 Γ
1(119905 119909))) with
Γ1 [0infin) timesC 997888rarr 119871
119901
P(Ω 119862
120575(119877))
Γ1(119905 119909) = 119884 (sdot 119905 119909)
Γ2 [0infin) times 119871
119901
P(Ω 119862
120575(119877)) 997888rarr 119871
119901
(Ω 119877)
Γ2(119905 119881) = 119881 (119905)
Γ3 119871
119901
(Ω 119877) 997888rarr 119877 Γ3120585 = 119864120585
(56)
FromTheorem 7 it follows that Γ1is continuous By
|119881(119905) minus 119880(119904)|119871119901(Ω119877)
le |119881 (119905) minus 119881 (119904)|119871119901(Ω119877)
+ 119890minus120575119901119904
|119881 minus 119880||119871
119901
P(Ω119862120575(119877))
(57)
we have that Γ2is continuous It is clear that Γ
3is continuous
Then themap (119905 119909) rarr 120592(119905 119909) is continuous from [0infin)timesCto 119877 therefore 120592(119905 119909)is a Borel measurable function Fromuniqueness of the solution of (36) it follows that 119884(119904 119905 119909) =120592(119904 119883
119904(119905 119909)) 119875-as for aa 119904 isin [119905infin)
4 The Fundamental Relation
Let (ΩF 119875) be a given complete probability space with afiltration F
119905119905ge0
satisfying the usual conditions 119882(119905) 119905 ge 0
is a cylindrical Wiener process in Ξ with respect to F119905119905ge0
We will say that an F
119905ge0-predictable process 119906 with values
in a given measurable space (119880U) is an admissible controlThe function 119877 [0infin) times C times 119880 rarr Ξ is measurableand bounded We consider the following controlled stateequation
119889119883119906
(119904) = 119860119883119906
(119904) 119889119904 + 119865 (119904 119883119906
119904) 119889119904
+ 119866 (119904 119883119906
119904) 119877 (119904 119883
119906
119904 119906 (119904)) 119889119904 + 119866 (119904 119883
119906
119904) 119889119882 (119904)
119904 isin [119905infin)
119883119906
119905= 119909
(58)
Here we assume that there exists a mild solution of (58)which will be denoted by 119883119906
(119904 119905 119909) or simply by 119883119906
(119904) Weconsider a cost function of the form
119869 (119906) = 119864int
infin
119905
119890minus120582119904
119892 (119904 119883119906
119904 119906 (119904)) 119889119904 (59)
Here 119892 is function on [0infin) times C times 119880 with real values Ourpurpose is to minimize the function 119869 over all admissiblecontrols
We define in a classical way the Hamiltonian functionrelative to the previous problem for all 119905 isin [0infin) 119909 isin
C and 119911 isin Ξlowast
120595 (119905 119909 119911) = inf 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) 119906 isin 119880
(60)
and the corresponding possibly empty set of minimizers
Γ (119905 119909 119911) = 119906 isin 119880 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) = 120595 (119905 119909 119911)
(61)
We are now ready to formulate the assumptions we need
Hypothesis 3 (i) 119860 119865 and G verify Hypothesis 1(ii) (119880U) is ameasurable spaceThemap 119892 [0infin)timesCtimes
119880 rarr 119877 is continuous and satisfies |119892(119905 119909 119906)| le 119870119892(1+|119909|
119898119892
119862)
for suitable constants 119870119892gt 0 119898
119892gt 0 and all 119909 isin C119906 isin
119880 The map 119877 [0infin) times C times 119880 rarr Ξ is measurable and|119877(119905 119904 119906)| le 119871
119877for a suitable constant 119870
119877gt 0 and all 119909 isin
C119906 isin 119880 and119911 isin Ξlowast
(iii)TheHamiltonian120595 defined in (60) satisfies the requi-rements of Hypothesis 2 (with119870 = 119877)
(iv) We fix here 119901 gt 2 q and 120575 lt 0 satisfying (39) with120578 = 120578(119902) and such that 119902 gt 119898
119892
We are in a position to prove the main result of thissection
Theorem11 Assume thatHypothesis 3 holds and suppose that120582 verifies
120582 gt (minus120575 minus 120583 +1198712
119911
2) or (minus120575 +
1198712
119877
2 (119901 minus 1))
or (1198712
119877119898119892
2 (119902 minus 119898119892)minus 120578 (119902)119898
119892)
(62)
Let 120592 120577 denote the function in the statement of Theorem 10Then for every admissible control 119906 and for the correspondingtrajectory119883 starting at (119905 119909) one has
119869 (119906) = 120592 (119905 119909) + 119864int
infin
119905
119890minus120582119904
[minus120595 (119904 119883119906
119904 120577 (119904 119883
119906
119904)) + 120577 (119904 119883
119906
119904)
times 119877 (119904 119883119906
119904 119906 (119904)) + 119892 (119904 119883
119906
119904 119906 (119904))] 119889119904
(63)
Proof Consider (58) in the probability space (ΩF 119875) withfiltration F
119905119905ge0
and with an F119905119905ge0
-cylindrical Wienerprocess 119882(119905) 119905 ge 0 Let us define
119882119906
(119904) = 119882 (119904) + int
119904
119905and119904
119877 (120590119883119906
120590 119906 (120590)) 119889120590 119904 isin [0infin)
120588 (119879) = exp(int119879
119905
minus119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882 (119904)
minus1
2int
119879
119905
1003816100381610038161003816119877 (119904 119883119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)
(64)
10 Abstract and Applied Analysis
Let 119875119906 be the unique probability onF[0infin)
such that
119875119906
|F119879
= 120588 (119879) 119875|F119879
(65)
We notice that under 119875119906 the process119882119906 is aWiener processLet us denote by F119906
119905119905ge0
the filtration generated by119882119906 andcompleted in the usual way Relatively to 119882119906 (58) can berewritten as
119889119883119906
(119904) = 119860119883119906
(119904) 119889119904 + 119865 (119904 119883119906
119904) 119889119904
+ 119866 (119904 119883119906
119904) 119889119882
119906
(119904) 119904 isin [119905infin)
119883119906
119905= 119909
(66)
In the space (ΩF[0infin)
F119906
119905119905ge0 119875
119906
) we consider the follow-ing system of forward-backward equations
119883119906
(119904) = 119890(119904minus119905)119860
119909 (0) + int
119904
119905
119890(119904minus120590)119860
119865 (120590119883119906
120590) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883119906
120590) 119889119882
119906
(120590) 119904 isin [119905infin)
119883119906
119905= 119909 isin C
119884119906
(119904) minus 119884119906
(119879) + int
119879
119904
119885119906
(120590) 119889119882119906
(120590) + 120582int
119879
119904
119884119906
(120590) 119889120590
= int
119879
119904
120595 (120590119883119906
120590 119885
119906
(120590)) 119889120590 0 le 119904 le 119879 lt infin
(67)
Applying the Ito formula to 119890minus120582119904119884119906(119904) and writing the back-ward equation in (67) with respect to the process119882 we get
119884119906
(119904) + int
119879
119904
119890minus120582120590
119885119906
(120590) 119889119882 (120590)
= int
119879
119904
119890minus120582120590
[120595 (120590119883119906
120590 119885
119906
(120590))
minus119885119906
(120590) 119877 (120590119883119906
120590 119906 (120590))] 119889120590
+ 119890minus120582119879
119884119906
(119879)
(68)
Recalling that 119877 is bounded we get for all 119903 ge 1 and someconstant 119862
119864119906
[120588(119879)minus119903
] = 119864119906
[exp 119903 (int119879
119905
119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882
119906
(119904)
minus1
2int
119879
119905
1003816100381610038161003816119877 (119904 119883119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)]
= 119864119906
[exp(int119879
119905
119903119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882
119906
(119904)
minus1
2int
119879
119905
11990321003816100381610038161003816119877 (119904 119883
119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)
times exp 119903 (119903 minus 1)2
int
119879
119905
1003816100381610038161003816119877 (119904 119883119906
119904 119906 (119904))
10038161003816100381610038162
119889119904]
le 119890(12)119903(119903minus1)119879119871
2
119877119864119906
times exp(int119879
119905
2119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882
119906
(119904)
minus1
2int
119879
119905
41003816100381610038161003816119877 (119904 119883
119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)
= 119890(12)119903(119903minus1)119879119871
2
119877
(69)
It follows that
119864(int
119879
119905
|119890minus120582119904
119885119906
(119904)|2
119889119904)
12
= 119864119906
[(int
119879
119905
10038161003816100381610038161003816119890minus120582119904
119885119906
(119904)10038161003816100381610038161003816
2
119889119904)
12
120588minus1
]
le (119864119906
int
119879
119905
10038161003816100381610038161003816119890minus120582119904
119885119906
(119904)10038161003816100381610038161003816
2
119889119904)
12
times (119864119906
120588minus2
)12
lt infin
(70)
We conclude that the stochastic integral in (68) has zeroexpectation If we set 119904 = 119905 in (68) and we take expectationwith respect to 119875 we obtain
119890minus120582119879
119864119884119906
(119879) minus 119884119906
(119905)
= 119864int
119879
119905
119890minus120582120590
[minus120595 (120590119883119906
120590 119885
119906
(120590))
+119885119906
(120590) 119877 (120590119883119906
120590 119906 (120590))] 119889120590
(71)
ByTheorem 7 119884119906(sdot 119905 119909) isin 119871119901P(Ω 119862
120575(119877)) so that
119864119906
|119884(119879 119905 119909)|119901
le 119862 exp (minus119901120575119879) (72)
By the Holder inequality we have that for suitable constant119862 gt 0
119864 |119884 (119879 119905 119909)| = 119864119906
(120588minus1
(119879) |119884 (119879 119905 119909)|)
le 119864(120588minus119901(119901minus1)
)(119901minus1)119901
119864(|119884 (119879 119905 119909)|119901
)1119901
le 119862119890((1198712
1198772(119901minus1))minus120575))119879
(73)
From Theorem 2 we obtain 119864119906sup119904ge119905119890120578119902119904
|119883119906
119904|119902
lt infin by thesimilar process we get that
1198641003816100381610038161003816119883
119906
119879
1003816100381610038161003816119898119892
le 119862119890(1198712
119877119898119892(2119902minus2119898
119892)minus1
minus120578(119902)119898119892)119879
(74)
Abstract and Applied Analysis 11
for suitable constant 119862 gt 0 and
119864int
infin
119905
119890minus120582120590 1003816100381610038161003816119892 (120590119883
119906
120590 119906 (120590))
1003816100381610038161003816 119889120590 lt infin (75)
Since 119884119906(119905 119905 119909) = 120592(119905 119909) and 119885119906
(119904 119905 119909) = 120577(119904 119883119906
119904(119905 119909)) 119875-
as for aa 119904 isin [119905infin) we have that
119890minus120582119879
119864119884119906
(119879) minus 119907 (119905 119909)
= 119864int
119879
119905
119890minus120582120590
[minus120595 (120590119883119906
120590 120577 (120590 119883
119906
120590))
+120577 (120590119883119906
120590) 119877 (120590119883
119906
120590 119906 (120590))] 119889120590
(76)
Thus adding and subtracting119864intinfin119905
119890minus120582120590
119892(120590119883119906
120590 119906(120590))119889120590 and
letting 119879 rarr infin we conclude that
119869 (119906) = 120592 (119905 119909)
+ 119864int
infin
119905
119890minus120582119904
[minus120595 (119904 119883119906
119904 120577 (119904 119883
119906
119904)) + 120577 (119904 119883
119906
119904) 119877
times (119904 119883119906
119904 119906 (119904)) + 119892 (119904 119883
119906
119904 119906 (119904))] 119889119904
(77)
The proof is finished
We immediately deduce the following consequences
Theorem 12 Let 119905 isin [0infin) and 119909 isin C be fixed assumethat the set-valued map Γ has nonempty values and it admitsa measurable selection Γ
0 [0infin)timesCtimesΞlowast rarr 119880 and assume
that a control 119906(sdot) satisfies
119906 (119904) = Γ0(119904 119883
119906
119904 120577 (119904 119883
119906
119904))
119875-as for almost every 119904 isin [119905infin)
(78)
Then 119869(119905 119909 119906) = 120592(119905 119909) and the pair (119906(sdot) 119883) is optimal forthe control problem starting from 119909 at time 119905
Such a control can be shown to exist if there exists a solutionfor the so-called closed-loop equation as follows
119889119883 (119904)=119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904+119866 (119904 119883
119904)
times(119877 (119904 119883119904 Γ
0(119904 119883
119904 120577 (119904 119883
119904))) 119889119904 + 119889119882 (119904))
119904 isin [119905infin)
119883119905= 119909
(79)
since in this case we can define an optimal control setting
119906 (119904) = Γ0(119904 119883
119904 120577 (119904 119883
119904)) (80)
However under the present assumptions we cannot guaranteethat the closed-loop equation has a solution in the mildsense To circumvent this difficulty we will revert to a weakformulation of the optimal control problem
5 Existence of Optimal Control
We formulate the optimal control problem in the weak sensefollowing the approach of [31]Themain advantage is that wewill be able to solve the closed-loop equation in a weak senseand hence to find an optimal control even if the feedbacklaw is nonsmooth
We call (ΩF F119905119905ge0 119875119882) an admissible setup if
(ΩF F119905119905ge0 119875) is a filtered probability space satisfying the
usual conditions and 119882 is a cylindrical 119875-Wiener processwith values in Ξ with respect to the filtration F
119905119905ge0
By an admissible control system we mean (ΩF
F119905119905ge0 119875119882 119906119883
119906
) where (ΩF F119905119905ge0 119875 119882) is an
admissible setup 119906 is an F119905-predictable process with values
in119880 and119883119906 is a mild solution of (58) An admissible controlsystem will be briefly denoted by (119882 119906119883119906
) in the followingOur purpose is to minimize the cost functional
119869 (119906) = 119864int
infin
119905
119890minus120582119904
119892 (119904 119883119906
119904 119906 (119904)) 119889119904 (81)
over all the admissible control systemOur main result in this section is based on the solvability
of the closed-loop equation
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904 + 119866 (119904 119883
119904)
times (119877 (119904 119883119904 Γ
0(119904 119883
119904 120577 (119904 119883
119904))) 119889119904 +119889119882 (119904))
119904 isin [119905infin)
119883119905= 119909
(82)
In the following sense we say that 119883 is a weak solution of(82) if there exists an admissible setup (ΩF F
119905119905ge0 119875119882)
and anF119905-adapted continuous process119883(119905)with values in119867
which solves the equation in the mild sense namely 119875-as
119883 (119904) = 119890(119904minus119905)119860
119909 (0) + int
119904
119905
119890(119904minus120590)119860
119865 (120590119883120590) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883120590) 119877
times (120590119883120590 Γ
0(120590119883
120590 120577 (120590 119883
120590))) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883120590) 119889119882
120590 119904 isin [119905infin)
(83)
119883119905= 119909 (84)
Theorem 13 Assume that Hypothesis 3 holds Then thereexists a weak solution of the closed-loop equation (82) whichis unique in law
12 Abstract and Applied Analysis
Proof (uniqueness) Let 119883 be a weak solution of (82) in anadmissible setup (ΩF F
119905119905ge0 119875119882)We define
120588 (119879) = exp(int119879
119905
minus119877lowast
(120590 119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889119882 (120590)
minus1
2int
119879
119905
1003816100381610038161003816119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)))10038161003816100381610038162
119889120590)
(85)
Since 119877 is bounded the Girsanov theorem ensures that thereexists a probability measure 1198750 such that the process
1198820
(119904) = 119882 (119904) + int
119904
119905and119904
119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889120590
119904 isin [0infin)
(86)
is a 1198750-Wiener process and
1198750
|F119879
= 120588 (119879) 119875|F119879
(87)
Let us denote by F0
119905119905ge0
the filtration generated by1198820 andcompleted in the usual way In (ΩF
[0infin) F0
119905119905ge0 119875
0
) 119883 isa mild solution of
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119905 119883119904) 119889119904
+ 119866 (119904 119883119904) 119889119882
0
(119904) 119904 isin [119905infin)
119883119905= 119909
120588 (119879) = exp(int119879
119905
minus119877lowast
(120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)) 119889119882
0
(120590)
+ 12int
119879
119905
1003816100381610038161003816119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)))10038161003816100381610038162
119889120590)
(88)
By Hypothesis 3 the joint law of 119883 and 1198820 is uniquely
determined by 119860 119865 119866 and 119909 Taking into account the lastdisplayed formula we conclude that the joint law of 119883 and120588(119879) under1198750 is also uniquely determined and consequentlyso is the law of 119883 under 119875 This completes the proof of theuniqueness part
Proof (existence) Let (ΩF 119875) be a given complete probabil-ity space 119882(119905) 119905 ge 0 is a cylindrical Wiener process on(ΩF 119875)with values inΞ and F
119905119905ge0
is the natural filtrationof 119882(119905) 119905 ge 0 augmented with the family of119875-null sets Let119883(sdot) be the mild solution of
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904
+ 119866 (119904 119883119904) 119889119882 (119904) 119904 isin [119905infin)
119883119905= 119909
(89)
and by the Girsanov theorem let 1198751 be the probability on Ωunder which
1198821
(119904) = 119882 (119904) minus int
119904
119905and119904
119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889120590
(90)
is aWiener process (notice that 119877 is bounded)Then119883 is theweak solution of (82) relatively to the probability 1198751 and theWiener process1198821
Now we can state the main result of this section
Corollary 14 Assume that Hypothesis 3 holds true and 120582
verifies (62) Also assume that the set-valued map Γ hasnonempty values and it admits a measurable selection Γ
0
[0infin) timesC times Ξlowast
rarr 119880 Then for every 119905 isin [0infin) and x isin Cand for all admissible control system (119882 119906119883
119906
) one has
119869 (119906 119905 119909) ge 120592 (119905 119909) (91)
and the equality holds if
119906 (119904) = Γ0(119904 119883
119906
119904 120577 (119904 119883
119906
119904))
119875 minus 119886119904 119891119900119903 119886119897119898119900119904119905 119890119907119890119903119910 119904 isin [119905infin)
(92)
Moreover from Theorem 13 it follows that the closed-loop equation (82) admits a weak solution (ΩFF
119905119905ge0 119875119882119883) which is unique in law and setting
119906 (119904) = Γ0(119904 119883
119904 120577 (119904 119883
119904)) (93)
we obtain an optimal admissible control system (119882 119906119883)
6 Applications
In this section we present a simple application of the previousresults We consider the stochastic delay partial differentialequation in the bounded domain 119861 sub 119877
119899 with smoothboundary 120597119861 as follows
119889119911119906
(119905 120585) = Δ119911119906
(119905 120585) 119889119905 + 119891 (119905 119911119906
119905(120585)) 119889119905
+
119889
sum
119894=1
119892119894(119905 119911
119906
119905(120585)) [119903
119894
(120585) 119906119894
(119905) 119889119905 + 119889119882119894
(119905)]
119911119906
0(120579 120585) = 119909 (120579 120585) 120585 isin 119861 120579 isin [minus1 0]
119911119906
(119905 120585) = 0 119905 isin [0infin) 120585 isin 120597119861
(94)
Here119882 = (1198821
1198822
119882119889
) is a standardWiener process in119877119889 and the functions 119891 [0 +infin) times 119862([minus1 0] 119877) rarr 119877 and119892119894 [0 +infin) times 119862([minus1 0] 119877) rarr 119877 are Lipschitz continuous
and bounded Setting 119880 as a bounded subset of 119877119889 Ξ = 119877119889
119867 = 1198712
(119861) and 119909 isin 119862([minus1 0]119867) We define 119865 and 119866 asfollowing
119865 (119905 119909) (120585) = 119891 (119905 119909 (120585))
(119866 (119905 119909) 119911) (120585) =
119889
sum
119894=1
119892119894(119905 119909 (120585)) 119911
119894
(120585)
120585 isin 119861 119909 isin 119862 ([minus1 0] 119867) 119911 isin 119871 (Ξ119867)
(95)
and let 119860 denote the Laplace operator Δ in 1198712
(119861) withdomain11988222
(119861)⋂11988212
0(119861) then (94) has the form (58) and
Hypothesis 1 holds
Abstract and Applied Analysis 13
Let us consider the optimal control problem associatedwith the cost
119869 (119906) = 119864int
infin
0
119890minus120582119905
[int119861
120590 (120585 119911119906
119905(120585)) 119889120585 + 119906
2
(119905)] 119889119905 (96)
where 120582 verifies (62) and 120590 119862([minus1 0] 119877) times 119880 rarr [0infin) isa bounded measurable function Define 119892 119862([minus1 0]119867) times
119880 rarr [0infin) and 119877 119862([minus1 0]119867) times 119880 rarr Ξ by119892(119910 119906) = int
119861
120590(119905 119910(120585) 119906)119889120585 + 1199062 and 119877(119910 119906) = (int
119861
1199031
(120585)1199061
119889120585
int119861
1199032
(120585)1199062
119889120585 int119861
119903119889
(120585)119906119889
119889120585) for 119910 isin 119862([minus1 0]119867) 119906 =
(1199061
1199062
119906119889
) isin 119880 respectively It can be easily verifiedthat Hypothesis 3 holds true and the set-valued map Γ hasnonempty values and it admits a measurable selection Γ
0
[0infin) times C times Ξlowast
rarr 119880 Then the closed-loop equation(82) admits a weak solution (ΩF F
119905119905ge0 119875119882 119906 119911
sdot(sdot)) and
setting
119906 (119904) = Γ0(119904 119911
119904(sdot) 120577 (119904 119911
119904(sdot))) (97)
we obtain an optimal admissible control system (119882 119906 119911(sdot))
References
[1] J Bismut ldquoOn optimal control of linear stochastic equationswith a linear-quadratic criterionrdquo SIAM Journal on Control andOptimization vol 15 no 3 pp 1ndash4 1977
[2] N Nagase ldquoOn the existence of optimal control for controlledstochastic partial differential equationsrdquo Nagoya MathematicsJournal vol 115 pp 73ndash85 1989
[3] N El Karoui D Huu Nguyen and M Jeanblanc-Pique ldquoCom-pactification methods in the control of degenerate diffusionsrdquoStochastics vol 20 pp 169ndash219 1987
[4] M Nisio ldquoOptimal control for stochastic partial differentialequations and viscosity solutions of Bellman equationsrdquoNagoyaMathematics Journal vol 123 pp 13ndash37 1991
[5] M Nisio ldquoOn sensitive control for stochastic partial differentialequationsrdquo in Stochastic Analysis on Infinite Dimensional SpacesProceedings of the US Japan Bilateral Seminar H Kunita et alEd vol 310 of Pitman Research Notes Mathematical Series pp231ndash241 Longman Scientific and Technical Baton Rouge LaUSA January 1994
[6] R Buckdahn and A Rascanu ldquoOn the existence of stochasticoptimal control of distributed state systemrdquoNonlinear AnalysisTheory Methods and Applications vol 52 no 4 pp 1153ndash11842003
[7] V Barbu and G Da Prato Equations in Hilbert Spaces vol 86 ofPitman Research Notes in Mathematics Pitman 1983
[8] P Cannarsa and G Da Prato ldquoSecond-order Hamilton-Jacobiequations in infinite dimensionsrdquo SIAM Journal on Control andOptimization vol 29 no 2 pp 474ndash492 1991
[9] P Cannarsa and G Da Prato ldquoDirect solution of a second-orderHamilton-Jacobi equations in Hilbert spacesrdquo in StochasticPartial Differential Equations and Applications G Da Pratoand L Tubaro Eds vol 268 of Pitman Research Notes inMathematics Pitman 1992
[10] F Gozzi ldquoRegularity of solutions of second order Hamilton-Jacobi equations and application to a control problemrdquoCommu-nications in Partial Differential Equations vol 20 pp 775ndash8261995
[11] F Gozzi ldquoGlobal regular solutions of second order Hamilton-Jacobi equations in Hilbert spaces with locally Lipschitz non-linearitiesrdquo Journal of Mathematical Analysis and Applicationsvol 198 no 2 pp 399ndash443 1996
[12] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems and Control Lettersvol 14 no 1 pp 55ndash61 1990
[13] N El Karoui and LMazliak Eds Backward Stochastic Differen-tial Equations vol 364 of Pitman ResearchNotes inMathematicsSeries Longman 1997
[14] E Pardoux and BSDEs ldquoweak convergence and homogeneiza-tion of semilinear PDEsrdquo in Non- Linear Analysis DifferentialEquations and Control F H Clarke and R J Stern Eds pp503ndash549 Kluwer Dordrecht The Netherlands 1999
[15] S Peng ldquoA generalized dynamic programming principle andHamilton-Jacobi-Bellman equationrdquo Stochastics and StochasticsReports vol 38 pp 119ndash134 1992
[16] N E Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997
[17] SHamad120583ene and J P Lepeltier ldquoBackward equations stochas-tic control and zero-sum stochastic differential gamesrdquo Stochas-tics and Stochastics Reports vol 54 pp 221ndash231 1995
[18] N El-Karoui and S Hamadene ldquoBSDEs and risk-sensitive con-trol zero-sum and nonzero-sum game problems of stochasticfunctional differential equationsrdquo Stochastic Processes and theirApplications vol 107 no 1 pp 145ndash169 2003
[19] M Fuhrman and G Tessiture ldquoExistence of optimal stochasticcontrols and global solutions of forward-backward stochasticdifferential equationsrdquo SIAM Journal on Control and Optimiza-tion vol 43 no 3 pp 813ndash830 2005
[20] M Fuhrman Y Hu and G Tessitore ldquoOn a class of stochasticoptimal control problems related to bsdes with quadraticgrowthrdquo SIAM Journal on Control and Optimization vol 45 no4 pp 1279ndash1296 2006
[21] M Fuhrman and G Tessitore ldquoNonlinear kolmogorov equa-tions in infinite dimensional spaces the backward stochasticdifferential equations approach and applications to optimalcontrolrdquoAnnals of Probability vol 30 no 3 pp 1397ndash1465 2002
[22] F Masiero ldquoSemilinear kolmogorov equations and applicationsto stochastic optimal controlrdquo Applied Mathematics and Opti-mization vol 51 no 1 pp 201ndash250 2005
[23] M Fuhrman FMasiero andG Tessitore ldquoStochastic equationswith delay optimal control via BSDEs and regular solutions ofHamilton-jacobi-bellman equationsrdquo SIAM Journal on Controland Optimization vol 48 no 7 pp 4624ndash4651 2010
[24] M Fuhrman and G Tessiture ldquoInfinite horizon backwardstochastic differential equations and elliptic equations in hilbertspacesrdquo Annals of Probability vol 32 no 1 pp 607ndash660 2004
[25] F Masiero ldquoInfinite horizon stochastic optimal control prob-lems with degenerate noise and elliptic equations in Hilbertspacesrdquo Applied Mathematics and Optimization vol 55 no 3pp 285ndash326 2007
[26] M Fuhrman ldquoA class of stochastic optimal control problemsin Hilbert spaces BSDEs and optimal control laws stateconstraints conditioned processesrdquo Stochastic Processes andtheir Applications vol 108 no 2 pp 263ndash298 2003
[27] F Masiero ldquoStochastic optimal control problems and parabolicequations in banach spacesrdquo SIAM Journal on Control andOptimization vol 47 no 1 pp 251ndash300 2008
[28] G Da Prato and J Zabczyk Ergodicity For Infinite-DimensionalSystems Cambridge University Press 1996
14 Abstract and Applied Analysis
[29] J Zabczyk ldquoParabolic equations on Hilbert spacesrdquo in StochaS-tic PDErsquoS and Kolmogorov Equations in Infinite Dimensionsvol 1715 of Lecture Notes in Math pp 117ndash213 Springer BerlinGermany 1999
[30] G Da Prato and J Zabczyk Stochstic Equations in InfiniteDimensions Cambridge University Press 1992
[31] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 of Applications of MathematicsSpringer New York NY USA 1993
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014
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Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 7
holds for a suitable constant 119888 In particular 119884(119883sdot) isin
119871119901
P(Ω 119862
120575(119870))
(iii) The map 119883sdotrarr (119884(119883
sdot) 119885(119883
sdot)) is continuous from
119871119902
P(Ω 119871
119902
120578(C)) toK119901
120575 and 119883
sdotrarr 119884(119883
sdot) is continuous
from 119871119902
P(Ω 119871
119902
120578(C)) to 119871119901
P(Ω 119862
120575(119870))
(iv) The statements of points (i) (ii) and (iii) still holdtrue if the space 119871119902
P(Ω 119871
119902
120578(C)) is replaced by the space
119871119902
P(Ω 119862
120578(C))
Proof The theorem is very similar to Proposition 311 in [24]The only minor difference is that the mapping 120595 [0infin) times
Ctimes119870times1198712(Ξ 119870) rarr 119870 is a givenmeasurable function while
in [24] the measurable function 120595 is from119867 times119870 times 1198712(Ξ 119870)
to 119870 however the same arguments apply
Theorem 7 Assume that Hypothesis 1 holds and thatHypothesis 2 holds true in the particular case 119870 = 119877 Thenfor every 119901 gt 2 119902 120575 lt 0 satisfying (39) with 120578 = 120578(119902)and for every 120582 gt 120582
1015840
= minus(120575 + 120583 minus (1198712
1199112)) there exists a
unique solution in H119902
120578(119902)times K
119901
120575of (36) that will be denoted
by (119883(sdot 119905 119909) 119884(sdot 119905 119909) 119885(sdot 119905 119909)) Moreover 119884(sdot 119905 119909) isin
119871119901
P(Ω 119862
120575(119877)) The map (119905 119909) rarr (119884(sdot 119905 119909) 119885(sdot 119905 119909)) is con-
tinuous from [0infin)timesC toK119901
120575 and themap (119905 119909) rarr 119884(sdot 119905 119909)
is continuous from [0infin) timesC to 119871119901P(Ω 119862
120575(119877))
Proof We first notice that the system is decoupled the firstdoes not contain the solution (119884 119885) of the second oneThere-fore under the assumption of Hypothesis 1 by Theorem 2there exists a unique solution 119883(sdot 119905 119909) and 119883
sdot(119905 119909) isin H
119902
120578(119902)
of the first equation Moreover from Theorem 4 it followsthat the map (119905 119909) rarr 119883
sdot(119905 119909) is continuous from [0infin)timesC
toH119902
120578(119902)
Let 119870 = 119877 from Theorem 6 we have that thereexists a unique solution (119884(sdot 119905 119909) 119885(sdot 119905 119909)) isin K
119901
120575of the
second equation and the map 119883sdotrarr (119884(119883
sdot) 119885(119883
sdot)) is
continuous from H119902
120578(119902)to K
119901
120575while X
sdotrarr (Y(X
sdot)) is
continuous fromH119902
120578(119902)to119871119901
P(Ω 119862
120575(119877))We have proved that
(119883(sdot 119905 119909) 119884(sdot 119905 119909) 119885(sdot 119905 119909)) isin H119902
120578(119902)times K
119901
120575is the unique
solution of (36) and the other assertions follow from com-position
Remark 8 From Remark 5 by similar passages we can showthat for fixed 119905 and for 119902 large enough under the assumptionsof Theorem 7 the map 119909 rarr (119884(sdot 119905 119909) 119885(sdot 119905 119909)) is continu-ous from 119871
119902
(ΩCF119905) toK119901
120575(119905)
We also remark that the process 119883(sdot 119905 119909) is F[119905infin)
measurable since C is separable Banach space we have that119883sdot(119905 119909) is F
[119905infin)measurable So that 119884(119905) is measurable
with respect to both F[119905infin)
and F119905 it follows that 119884(119905) is
deterministicFor later use we notice three useful identities for 119905 le 119904 lt
infin the equality 119875-as
119883119897(119904 119883
119904(119905 119909)) = 119883
119897(119905 119909) 119897 isin [119904infin) (41)
is a consequence of the uniqueness of the solution of (13)Since the solution of the backward equation is uniquely
determined on an interval [119904infin) by the values of the process119883sdoton the same interval for 119905 le 119904 lt infin we have 119875-as
119884 (119897 119904 119883119904(119905 119909)) = 119884 (119897 119905 119909) for 119897 isin [119904infin)
119885 (119897 119904 119883119904(119905 119909)) = 119885 (119897 119905 119909) for aa 119897 isin [119904infin)
(42)
Lemma 9 (see [30]) Let 119864 be a metric space with metric 119889and let 119891 Ω rarr 119864 be strongly measurable Then thereexists a sequence 119891
119899 119899 isin 119873 of simple 119864-valued functions
(ie 119891119899isFB(E)measurable and takes only a finite number
of values) such that for arbitrary 120596 isin Ω the sequence119889(119891
119899(120596) 119891(120596)) 119899 isin 119873 is monotonically decreasing to zero
Let now 119891 isin 119871119902
(ΩC) By Lemma 9 we get the existenceof a sequence of simple function 119891
119899 119899 isin 119873 such that
1003816100381610038161003816119891119899 (120596) minus 119891 (120596)1003816100381610038161003816 darr 0 for all 120596 isin Ω as 119899 rarr infin (43)
Hence 119891119899
rarr 119891 in | sdot |119871119902(ΩC) by Lebesguersquos dominated
convergence theoremWe are now in a position of showing the main result in
this section
Theorem 10 Assume that Hypothesis 1 holds true and thatHypothesis 2 holds in the particular case 119870 = 119877 Then thereexist two Borel measurable deterministic functions 120592 [tinfin) times
C rarr 119877 and 120577 [119905infin) times C rarr Ξlowast
= 119871(Ξ 119877) =
1198712(Ξ 119877) such that for 119905 isin [0infin) and x isin C the solution
(119883(119905 119909) 119884(119905 119909) 119885(119905 119909)) of (36) satisfies
119884 (119904 119905 119909) = 120592 (119904 119883119904(119905 119909)) 119885 (119904 119905 119909) = 120577 (119904 119883
119904(119905 119909))
119875-as for aa 119904 isin [119905infin)
(44)
Proof We apply the techniques introduced in [26 Proposi-tion 32] Let 119890
119894 be a basis of Ξlowast and let us define 119885119894119873
=
((119885 119890119894)Ξlowast and119873) or (minus119873) Then for every 0 le 119905
1lt 119905
2lt infin Δ gt
0 and 1199091 119909
2isin C we have that
100381610038161003816100381610038161003816100381610038161003816
119864 int
1199051+Δ
1199051
119885119894119873
(119904 1199051 119909
1) 119889119904 minus 119864int
1199052+Δ
1199052
119885119894119873
(119904 1199052 119909
2) 119889119904
100381610038161003816100381610038161003816100381610038161003816
le 119864int
1199052
1199051
10038161003816100381610038161003816119885119894119873
(119904 1199051 119909
1)10038161003816100381610038161003816119889119904
+ 119864int
1199051+Δ
1199052
10038161003816100381610038161003816119885119894119873
(119904 1199051 119909
1) minus 119885
119894119873
(119904 1199052 119909
2)10038161003816100381610038161003816119889119904
+ 119864int
1199052+Δ
1199051+Δ
10038161003816100381610038161003816119885119894119873
(119904 1199052 119909
2)10038161003816100381610038161003816119889119904
8 Abstract and Applied Analysis
le 211987310038161003816100381610038161199052 minus 1199051
1003816100381610038161003816 + Δ12
119890minus120575(1199051+Δ)
times(119864(int
infin
0
119890212057511990410038161003816100381610038161003816119885119894119873
(119904 1199051 119909
1)minus119885
119894119873
(119904 1199052 119909
2)10038161003816100381610038161003816
2
119889119904)
1199012
)
1119901
le 211987310038161003816100381610038161199052 minus 1199051
1003816100381610038161003816 + Δ12
119890minus120575(1199051+Δ)
times (119864(int
infin
0
11989021205751199041003816100381610038161003816119885 (119904 1199051 1199091)minus119885 (119904 1199052 1199092)
10038161003816100381610038162
119889119904)
1199012
)
1119901
(45)
From Theorem 7 we have that the map (119905 119909) rarr int119905+Δ
119905
119885119894119873
(119904 119905 119909)119889119904 is continuous from [0infin) times C to 119877 ByRemark 8 we also have that for fixed 119905 the map 119909 rarr
119864int119905+Δ
119905
119864119885119894119873
(119904 119905 119909)119889119904 is continuous from 119871119902
(ΩCF119905) to 119877
for 119902 large enough Let us define
120577119894119873
(119905 119909) = lim inf119899rarrinfin
119899119864int
119905+(1119899)
119905
119885119894119873
(119904 119905 119909) 119889119904
119905 isin [0infin) 119909 isin C
(46)
It is clear that 120577119894119873 [0infin) timesC rarr 119877 is a Borel functionWe fix 119909 and 0 le 119905 le 119904 lt infin For 119897 isin [119904infin) we
denote 119864[119885119894119873
(119897 119904 119910)]|119910=119883119904(119905119909)
the random variable obtainedby composing119883
119904(119905 119909) with the map 119910 rarr 119864[119885
119894119873
(119897 119904 119910)]By Lemma 9 there exists a sequence of C-valued F
119904-
measurable simple functions
119891119898 Ω 997888rarr C 119891
119898=
119873119898
sum
119896=1
ℎ(119898)
119896119868119891119898=ℎ(119898)
119896 119873
119898isin 119873 (47)
where ℎ(119898)1 ℎ
(119898)
119898are pairwise distinct andΩ = ⋃
119873119898
119896=1119891
119898=
ℎ(119898)
119896 such that
1003816100381610038161003816119891119898 (120596) minus 119883119904(120596)
1003816100381610038161003816 darr 0 for all 120596 isin Ω as 119899 997888rarr infin (48)
For any 119861 isin F119904 we have
int119861
int
119904+(1119899)
119904
119885119894119873
(119897 119905 119909) 119889119897119889119875
= int119861
int
119904+(1119899)
119904
119885119894119873
(119897 119904 119883119904) 119889119897119889119875
= 119864119868119861int
119904+(1119899)
119904
119885119894119873
(119897 119904 119883119904) 119889119897
= lim119898rarrinfin
119864(119868119861int
119904+(1119899)
119904
119885119894119873
(119897 119904 119891119898) 119889119897)
= lim119898rarrinfin
119873119898
sum
119896=1
119864(119868119861119868119891119898=ℎ(119898)
119896int
119904+(1119899)
119904
119885119894119873
(119897 119904 ℎ(119898)
119896) 119889119897)
= lim119898rarrinfin
119864(119868119861
119873119898
sum
119896=1
119868119891119898=ℎ(119898)
119896)119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 ℎ(119898)
119896) 119889119897
= lim119898rarrinfin
119864119868119861(119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 119910) 119889119897|119910=119891119898
)
= lim119898rarrinfin
int119861
(119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 119910) 119889119897|119910=119891119898
)119889119875
= int119861
(119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 119910) 119889119897|119910=119883119904
)119889119875
(49)
and we get that
120577119894119873
(119904 119883119904(119905 119909)) = lim inf
119899rarrinfin
119899
times [119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 119910) 119889119897|119910=119883119904(119905119909)
]
= lim inf119899rarrinfin
119899119864[int
119904+(1119899)
119904
119885119894119873
(119897 119905 119909) 119889119897
100381610038161003816100381610038161003816100381610038161003816
F119904]
119875-as(50)
Fix 119905 and 119909 Recalling that |119885119894119873
| le 119873 by the Lebesguetheorem on differentiation it follows that 119875-as
lim119899rarrinfin
119899int
119904+(1119899)
119904
119885119894119873
(119897 119905 119909) 119889119897 = 119885119894119873
(119904 119905 119909)
for aa 119904 isin [119905infin)
(51)
By the boundedness of 119885119894119873 applying the dominated conver-gence theorem we get that
120577119894119873
(119904 119883119904(119905 119909)) = 119864 [119885
119894119873
(119904 119905 119909)10038161003816100381610038161003816F
119904] = 119885
119894119873
(119904 119905 119909)
119875-as for aa 119904 isin [119905infin)
(52)
Now we have proved that for every 119905 119909
120577119894119873
(119904 119883119904(119905 119909)) = 119885
119894119873
(119904 119905 119909)
119875-as for aa 119904 isin [119905infin)
(53)
for every 119894 119873 Let 119862 sub [0infin) times C denote the set ofpairs (119905 119909) such that lim
119873rarrinfin120577119894119873
(119905 119909) exists and the seriessuminfin
119894=1(lim
119873rarrinfin120577119894119873
(119905 119909))119890119894converges in Ξlowast We define
120577 (119905 119909) =
infin
sum
119894=1
( lim119873rarrinfin
120577119894119873
(119905 119909)) 119890119894 (119905 119909) isin 119862
120577 (119905 119909) = 0 (119905 119909) notin 119862
(54)
Abstract and Applied Analysis 9
Since 119885 satisfies
119885 (120596 119904 119905 119909) =
infin
sum
119894=1
( lim119873rarrinfin
119885119894119873
(120596 119904 119905 119909)) 119890119894 (55)
for every 120596 119904 119905 119909 From (53) it follows that for every 119905 119909 wehave (119904 119883
119904(120596 119905 119909)) isin 119862 119875-as for almost all 119904 isin [119905infin) and
119885(119904 119905 119909) = 120577(119904 119883119904(119905 119909)) 119875-as for aa 119904 isin [119905infin)
We define 120592(119905 119909) = 119884(119905 119905 119909) since 119884(119905 119905 119909) is deter-ministic so the map (119905 119909) rarr 120592(119905 119909) can be written as acomposition 120592(119905 119909) = Γ
3(Γ2(119905 Γ
1(119905 119909))) with
Γ1 [0infin) timesC 997888rarr 119871
119901
P(Ω 119862
120575(119877))
Γ1(119905 119909) = 119884 (sdot 119905 119909)
Γ2 [0infin) times 119871
119901
P(Ω 119862
120575(119877)) 997888rarr 119871
119901
(Ω 119877)
Γ2(119905 119881) = 119881 (119905)
Γ3 119871
119901
(Ω 119877) 997888rarr 119877 Γ3120585 = 119864120585
(56)
FromTheorem 7 it follows that Γ1is continuous By
|119881(119905) minus 119880(119904)|119871119901(Ω119877)
le |119881 (119905) minus 119881 (119904)|119871119901(Ω119877)
+ 119890minus120575119901119904
|119881 minus 119880||119871
119901
P(Ω119862120575(119877))
(57)
we have that Γ2is continuous It is clear that Γ
3is continuous
Then themap (119905 119909) rarr 120592(119905 119909) is continuous from [0infin)timesCto 119877 therefore 120592(119905 119909)is a Borel measurable function Fromuniqueness of the solution of (36) it follows that 119884(119904 119905 119909) =120592(119904 119883
119904(119905 119909)) 119875-as for aa 119904 isin [119905infin)
4 The Fundamental Relation
Let (ΩF 119875) be a given complete probability space with afiltration F
119905119905ge0
satisfying the usual conditions 119882(119905) 119905 ge 0
is a cylindrical Wiener process in Ξ with respect to F119905119905ge0
We will say that an F
119905ge0-predictable process 119906 with values
in a given measurable space (119880U) is an admissible controlThe function 119877 [0infin) times C times 119880 rarr Ξ is measurableand bounded We consider the following controlled stateequation
119889119883119906
(119904) = 119860119883119906
(119904) 119889119904 + 119865 (119904 119883119906
119904) 119889119904
+ 119866 (119904 119883119906
119904) 119877 (119904 119883
119906
119904 119906 (119904)) 119889119904 + 119866 (119904 119883
119906
119904) 119889119882 (119904)
119904 isin [119905infin)
119883119906
119905= 119909
(58)
Here we assume that there exists a mild solution of (58)which will be denoted by 119883119906
(119904 119905 119909) or simply by 119883119906
(119904) Weconsider a cost function of the form
119869 (119906) = 119864int
infin
119905
119890minus120582119904
119892 (119904 119883119906
119904 119906 (119904)) 119889119904 (59)
Here 119892 is function on [0infin) times C times 119880 with real values Ourpurpose is to minimize the function 119869 over all admissiblecontrols
We define in a classical way the Hamiltonian functionrelative to the previous problem for all 119905 isin [0infin) 119909 isin
C and 119911 isin Ξlowast
120595 (119905 119909 119911) = inf 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) 119906 isin 119880
(60)
and the corresponding possibly empty set of minimizers
Γ (119905 119909 119911) = 119906 isin 119880 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) = 120595 (119905 119909 119911)
(61)
We are now ready to formulate the assumptions we need
Hypothesis 3 (i) 119860 119865 and G verify Hypothesis 1(ii) (119880U) is ameasurable spaceThemap 119892 [0infin)timesCtimes
119880 rarr 119877 is continuous and satisfies |119892(119905 119909 119906)| le 119870119892(1+|119909|
119898119892
119862)
for suitable constants 119870119892gt 0 119898
119892gt 0 and all 119909 isin C119906 isin
119880 The map 119877 [0infin) times C times 119880 rarr Ξ is measurable and|119877(119905 119904 119906)| le 119871
119877for a suitable constant 119870
119877gt 0 and all 119909 isin
C119906 isin 119880 and119911 isin Ξlowast
(iii)TheHamiltonian120595 defined in (60) satisfies the requi-rements of Hypothesis 2 (with119870 = 119877)
(iv) We fix here 119901 gt 2 q and 120575 lt 0 satisfying (39) with120578 = 120578(119902) and such that 119902 gt 119898
119892
We are in a position to prove the main result of thissection
Theorem11 Assume thatHypothesis 3 holds and suppose that120582 verifies
120582 gt (minus120575 minus 120583 +1198712
119911
2) or (minus120575 +
1198712
119877
2 (119901 minus 1))
or (1198712
119877119898119892
2 (119902 minus 119898119892)minus 120578 (119902)119898
119892)
(62)
Let 120592 120577 denote the function in the statement of Theorem 10Then for every admissible control 119906 and for the correspondingtrajectory119883 starting at (119905 119909) one has
119869 (119906) = 120592 (119905 119909) + 119864int
infin
119905
119890minus120582119904
[minus120595 (119904 119883119906
119904 120577 (119904 119883
119906
119904)) + 120577 (119904 119883
119906
119904)
times 119877 (119904 119883119906
119904 119906 (119904)) + 119892 (119904 119883
119906
119904 119906 (119904))] 119889119904
(63)
Proof Consider (58) in the probability space (ΩF 119875) withfiltration F
119905119905ge0
and with an F119905119905ge0
-cylindrical Wienerprocess 119882(119905) 119905 ge 0 Let us define
119882119906
(119904) = 119882 (119904) + int
119904
119905and119904
119877 (120590119883119906
120590 119906 (120590)) 119889120590 119904 isin [0infin)
120588 (119879) = exp(int119879
119905
minus119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882 (119904)
minus1
2int
119879
119905
1003816100381610038161003816119877 (119904 119883119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)
(64)
10 Abstract and Applied Analysis
Let 119875119906 be the unique probability onF[0infin)
such that
119875119906
|F119879
= 120588 (119879) 119875|F119879
(65)
We notice that under 119875119906 the process119882119906 is aWiener processLet us denote by F119906
119905119905ge0
the filtration generated by119882119906 andcompleted in the usual way Relatively to 119882119906 (58) can berewritten as
119889119883119906
(119904) = 119860119883119906
(119904) 119889119904 + 119865 (119904 119883119906
119904) 119889119904
+ 119866 (119904 119883119906
119904) 119889119882
119906
(119904) 119904 isin [119905infin)
119883119906
119905= 119909
(66)
In the space (ΩF[0infin)
F119906
119905119905ge0 119875
119906
) we consider the follow-ing system of forward-backward equations
119883119906
(119904) = 119890(119904minus119905)119860
119909 (0) + int
119904
119905
119890(119904minus120590)119860
119865 (120590119883119906
120590) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883119906
120590) 119889119882
119906
(120590) 119904 isin [119905infin)
119883119906
119905= 119909 isin C
119884119906
(119904) minus 119884119906
(119879) + int
119879
119904
119885119906
(120590) 119889119882119906
(120590) + 120582int
119879
119904
119884119906
(120590) 119889120590
= int
119879
119904
120595 (120590119883119906
120590 119885
119906
(120590)) 119889120590 0 le 119904 le 119879 lt infin
(67)
Applying the Ito formula to 119890minus120582119904119884119906(119904) and writing the back-ward equation in (67) with respect to the process119882 we get
119884119906
(119904) + int
119879
119904
119890minus120582120590
119885119906
(120590) 119889119882 (120590)
= int
119879
119904
119890minus120582120590
[120595 (120590119883119906
120590 119885
119906
(120590))
minus119885119906
(120590) 119877 (120590119883119906
120590 119906 (120590))] 119889120590
+ 119890minus120582119879
119884119906
(119879)
(68)
Recalling that 119877 is bounded we get for all 119903 ge 1 and someconstant 119862
119864119906
[120588(119879)minus119903
] = 119864119906
[exp 119903 (int119879
119905
119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882
119906
(119904)
minus1
2int
119879
119905
1003816100381610038161003816119877 (119904 119883119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)]
= 119864119906
[exp(int119879
119905
119903119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882
119906
(119904)
minus1
2int
119879
119905
11990321003816100381610038161003816119877 (119904 119883
119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)
times exp 119903 (119903 minus 1)2
int
119879
119905
1003816100381610038161003816119877 (119904 119883119906
119904 119906 (119904))
10038161003816100381610038162
119889119904]
le 119890(12)119903(119903minus1)119879119871
2
119877119864119906
times exp(int119879
119905
2119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882
119906
(119904)
minus1
2int
119879
119905
41003816100381610038161003816119877 (119904 119883
119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)
= 119890(12)119903(119903minus1)119879119871
2
119877
(69)
It follows that
119864(int
119879
119905
|119890minus120582119904
119885119906
(119904)|2
119889119904)
12
= 119864119906
[(int
119879
119905
10038161003816100381610038161003816119890minus120582119904
119885119906
(119904)10038161003816100381610038161003816
2
119889119904)
12
120588minus1
]
le (119864119906
int
119879
119905
10038161003816100381610038161003816119890minus120582119904
119885119906
(119904)10038161003816100381610038161003816
2
119889119904)
12
times (119864119906
120588minus2
)12
lt infin
(70)
We conclude that the stochastic integral in (68) has zeroexpectation If we set 119904 = 119905 in (68) and we take expectationwith respect to 119875 we obtain
119890minus120582119879
119864119884119906
(119879) minus 119884119906
(119905)
= 119864int
119879
119905
119890minus120582120590
[minus120595 (120590119883119906
120590 119885
119906
(120590))
+119885119906
(120590) 119877 (120590119883119906
120590 119906 (120590))] 119889120590
(71)
ByTheorem 7 119884119906(sdot 119905 119909) isin 119871119901P(Ω 119862
120575(119877)) so that
119864119906
|119884(119879 119905 119909)|119901
le 119862 exp (minus119901120575119879) (72)
By the Holder inequality we have that for suitable constant119862 gt 0
119864 |119884 (119879 119905 119909)| = 119864119906
(120588minus1
(119879) |119884 (119879 119905 119909)|)
le 119864(120588minus119901(119901minus1)
)(119901minus1)119901
119864(|119884 (119879 119905 119909)|119901
)1119901
le 119862119890((1198712
1198772(119901minus1))minus120575))119879
(73)
From Theorem 2 we obtain 119864119906sup119904ge119905119890120578119902119904
|119883119906
119904|119902
lt infin by thesimilar process we get that
1198641003816100381610038161003816119883
119906
119879
1003816100381610038161003816119898119892
le 119862119890(1198712
119877119898119892(2119902minus2119898
119892)minus1
minus120578(119902)119898119892)119879
(74)
Abstract and Applied Analysis 11
for suitable constant 119862 gt 0 and
119864int
infin
119905
119890minus120582120590 1003816100381610038161003816119892 (120590119883
119906
120590 119906 (120590))
1003816100381610038161003816 119889120590 lt infin (75)
Since 119884119906(119905 119905 119909) = 120592(119905 119909) and 119885119906
(119904 119905 119909) = 120577(119904 119883119906
119904(119905 119909)) 119875-
as for aa 119904 isin [119905infin) we have that
119890minus120582119879
119864119884119906
(119879) minus 119907 (119905 119909)
= 119864int
119879
119905
119890minus120582120590
[minus120595 (120590119883119906
120590 120577 (120590 119883
119906
120590))
+120577 (120590119883119906
120590) 119877 (120590119883
119906
120590 119906 (120590))] 119889120590
(76)
Thus adding and subtracting119864intinfin119905
119890minus120582120590
119892(120590119883119906
120590 119906(120590))119889120590 and
letting 119879 rarr infin we conclude that
119869 (119906) = 120592 (119905 119909)
+ 119864int
infin
119905
119890minus120582119904
[minus120595 (119904 119883119906
119904 120577 (119904 119883
119906
119904)) + 120577 (119904 119883
119906
119904) 119877
times (119904 119883119906
119904 119906 (119904)) + 119892 (119904 119883
119906
119904 119906 (119904))] 119889119904
(77)
The proof is finished
We immediately deduce the following consequences
Theorem 12 Let 119905 isin [0infin) and 119909 isin C be fixed assumethat the set-valued map Γ has nonempty values and it admitsa measurable selection Γ
0 [0infin)timesCtimesΞlowast rarr 119880 and assume
that a control 119906(sdot) satisfies
119906 (119904) = Γ0(119904 119883
119906
119904 120577 (119904 119883
119906
119904))
119875-as for almost every 119904 isin [119905infin)
(78)
Then 119869(119905 119909 119906) = 120592(119905 119909) and the pair (119906(sdot) 119883) is optimal forthe control problem starting from 119909 at time 119905
Such a control can be shown to exist if there exists a solutionfor the so-called closed-loop equation as follows
119889119883 (119904)=119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904+119866 (119904 119883
119904)
times(119877 (119904 119883119904 Γ
0(119904 119883
119904 120577 (119904 119883
119904))) 119889119904 + 119889119882 (119904))
119904 isin [119905infin)
119883119905= 119909
(79)
since in this case we can define an optimal control setting
119906 (119904) = Γ0(119904 119883
119904 120577 (119904 119883
119904)) (80)
However under the present assumptions we cannot guaranteethat the closed-loop equation has a solution in the mildsense To circumvent this difficulty we will revert to a weakformulation of the optimal control problem
5 Existence of Optimal Control
We formulate the optimal control problem in the weak sensefollowing the approach of [31]Themain advantage is that wewill be able to solve the closed-loop equation in a weak senseand hence to find an optimal control even if the feedbacklaw is nonsmooth
We call (ΩF F119905119905ge0 119875119882) an admissible setup if
(ΩF F119905119905ge0 119875) is a filtered probability space satisfying the
usual conditions and 119882 is a cylindrical 119875-Wiener processwith values in Ξ with respect to the filtration F
119905119905ge0
By an admissible control system we mean (ΩF
F119905119905ge0 119875119882 119906119883
119906
) where (ΩF F119905119905ge0 119875 119882) is an
admissible setup 119906 is an F119905-predictable process with values
in119880 and119883119906 is a mild solution of (58) An admissible controlsystem will be briefly denoted by (119882 119906119883119906
) in the followingOur purpose is to minimize the cost functional
119869 (119906) = 119864int
infin
119905
119890minus120582119904
119892 (119904 119883119906
119904 119906 (119904)) 119889119904 (81)
over all the admissible control systemOur main result in this section is based on the solvability
of the closed-loop equation
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904 + 119866 (119904 119883
119904)
times (119877 (119904 119883119904 Γ
0(119904 119883
119904 120577 (119904 119883
119904))) 119889119904 +119889119882 (119904))
119904 isin [119905infin)
119883119905= 119909
(82)
In the following sense we say that 119883 is a weak solution of(82) if there exists an admissible setup (ΩF F
119905119905ge0 119875119882)
and anF119905-adapted continuous process119883(119905)with values in119867
which solves the equation in the mild sense namely 119875-as
119883 (119904) = 119890(119904minus119905)119860
119909 (0) + int
119904
119905
119890(119904minus120590)119860
119865 (120590119883120590) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883120590) 119877
times (120590119883120590 Γ
0(120590119883
120590 120577 (120590 119883
120590))) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883120590) 119889119882
120590 119904 isin [119905infin)
(83)
119883119905= 119909 (84)
Theorem 13 Assume that Hypothesis 3 holds Then thereexists a weak solution of the closed-loop equation (82) whichis unique in law
12 Abstract and Applied Analysis
Proof (uniqueness) Let 119883 be a weak solution of (82) in anadmissible setup (ΩF F
119905119905ge0 119875119882)We define
120588 (119879) = exp(int119879
119905
minus119877lowast
(120590 119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889119882 (120590)
minus1
2int
119879
119905
1003816100381610038161003816119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)))10038161003816100381610038162
119889120590)
(85)
Since 119877 is bounded the Girsanov theorem ensures that thereexists a probability measure 1198750 such that the process
1198820
(119904) = 119882 (119904) + int
119904
119905and119904
119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889120590
119904 isin [0infin)
(86)
is a 1198750-Wiener process and
1198750
|F119879
= 120588 (119879) 119875|F119879
(87)
Let us denote by F0
119905119905ge0
the filtration generated by1198820 andcompleted in the usual way In (ΩF
[0infin) F0
119905119905ge0 119875
0
) 119883 isa mild solution of
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119905 119883119904) 119889119904
+ 119866 (119904 119883119904) 119889119882
0
(119904) 119904 isin [119905infin)
119883119905= 119909
120588 (119879) = exp(int119879
119905
minus119877lowast
(120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)) 119889119882
0
(120590)
+ 12int
119879
119905
1003816100381610038161003816119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)))10038161003816100381610038162
119889120590)
(88)
By Hypothesis 3 the joint law of 119883 and 1198820 is uniquely
determined by 119860 119865 119866 and 119909 Taking into account the lastdisplayed formula we conclude that the joint law of 119883 and120588(119879) under1198750 is also uniquely determined and consequentlyso is the law of 119883 under 119875 This completes the proof of theuniqueness part
Proof (existence) Let (ΩF 119875) be a given complete probabil-ity space 119882(119905) 119905 ge 0 is a cylindrical Wiener process on(ΩF 119875)with values inΞ and F
119905119905ge0
is the natural filtrationof 119882(119905) 119905 ge 0 augmented with the family of119875-null sets Let119883(sdot) be the mild solution of
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904
+ 119866 (119904 119883119904) 119889119882 (119904) 119904 isin [119905infin)
119883119905= 119909
(89)
and by the Girsanov theorem let 1198751 be the probability on Ωunder which
1198821
(119904) = 119882 (119904) minus int
119904
119905and119904
119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889120590
(90)
is aWiener process (notice that 119877 is bounded)Then119883 is theweak solution of (82) relatively to the probability 1198751 and theWiener process1198821
Now we can state the main result of this section
Corollary 14 Assume that Hypothesis 3 holds true and 120582
verifies (62) Also assume that the set-valued map Γ hasnonempty values and it admits a measurable selection Γ
0
[0infin) timesC times Ξlowast
rarr 119880 Then for every 119905 isin [0infin) and x isin Cand for all admissible control system (119882 119906119883
119906
) one has
119869 (119906 119905 119909) ge 120592 (119905 119909) (91)
and the equality holds if
119906 (119904) = Γ0(119904 119883
119906
119904 120577 (119904 119883
119906
119904))
119875 minus 119886119904 119891119900119903 119886119897119898119900119904119905 119890119907119890119903119910 119904 isin [119905infin)
(92)
Moreover from Theorem 13 it follows that the closed-loop equation (82) admits a weak solution (ΩFF
119905119905ge0 119875119882119883) which is unique in law and setting
119906 (119904) = Γ0(119904 119883
119904 120577 (119904 119883
119904)) (93)
we obtain an optimal admissible control system (119882 119906119883)
6 Applications
In this section we present a simple application of the previousresults We consider the stochastic delay partial differentialequation in the bounded domain 119861 sub 119877
119899 with smoothboundary 120597119861 as follows
119889119911119906
(119905 120585) = Δ119911119906
(119905 120585) 119889119905 + 119891 (119905 119911119906
119905(120585)) 119889119905
+
119889
sum
119894=1
119892119894(119905 119911
119906
119905(120585)) [119903
119894
(120585) 119906119894
(119905) 119889119905 + 119889119882119894
(119905)]
119911119906
0(120579 120585) = 119909 (120579 120585) 120585 isin 119861 120579 isin [minus1 0]
119911119906
(119905 120585) = 0 119905 isin [0infin) 120585 isin 120597119861
(94)
Here119882 = (1198821
1198822
119882119889
) is a standardWiener process in119877119889 and the functions 119891 [0 +infin) times 119862([minus1 0] 119877) rarr 119877 and119892119894 [0 +infin) times 119862([minus1 0] 119877) rarr 119877 are Lipschitz continuous
and bounded Setting 119880 as a bounded subset of 119877119889 Ξ = 119877119889
119867 = 1198712
(119861) and 119909 isin 119862([minus1 0]119867) We define 119865 and 119866 asfollowing
119865 (119905 119909) (120585) = 119891 (119905 119909 (120585))
(119866 (119905 119909) 119911) (120585) =
119889
sum
119894=1
119892119894(119905 119909 (120585)) 119911
119894
(120585)
120585 isin 119861 119909 isin 119862 ([minus1 0] 119867) 119911 isin 119871 (Ξ119867)
(95)
and let 119860 denote the Laplace operator Δ in 1198712
(119861) withdomain11988222
(119861)⋂11988212
0(119861) then (94) has the form (58) and
Hypothesis 1 holds
Abstract and Applied Analysis 13
Let us consider the optimal control problem associatedwith the cost
119869 (119906) = 119864int
infin
0
119890minus120582119905
[int119861
120590 (120585 119911119906
119905(120585)) 119889120585 + 119906
2
(119905)] 119889119905 (96)
where 120582 verifies (62) and 120590 119862([minus1 0] 119877) times 119880 rarr [0infin) isa bounded measurable function Define 119892 119862([minus1 0]119867) times
119880 rarr [0infin) and 119877 119862([minus1 0]119867) times 119880 rarr Ξ by119892(119910 119906) = int
119861
120590(119905 119910(120585) 119906)119889120585 + 1199062 and 119877(119910 119906) = (int
119861
1199031
(120585)1199061
119889120585
int119861
1199032
(120585)1199062
119889120585 int119861
119903119889
(120585)119906119889
119889120585) for 119910 isin 119862([minus1 0]119867) 119906 =
(1199061
1199062
119906119889
) isin 119880 respectively It can be easily verifiedthat Hypothesis 3 holds true and the set-valued map Γ hasnonempty values and it admits a measurable selection Γ
0
[0infin) times C times Ξlowast
rarr 119880 Then the closed-loop equation(82) admits a weak solution (ΩF F
119905119905ge0 119875119882 119906 119911
sdot(sdot)) and
setting
119906 (119904) = Γ0(119904 119911
119904(sdot) 120577 (119904 119911
119904(sdot))) (97)
we obtain an optimal admissible control system (119882 119906 119911(sdot))
References
[1] J Bismut ldquoOn optimal control of linear stochastic equationswith a linear-quadratic criterionrdquo SIAM Journal on Control andOptimization vol 15 no 3 pp 1ndash4 1977
[2] N Nagase ldquoOn the existence of optimal control for controlledstochastic partial differential equationsrdquo Nagoya MathematicsJournal vol 115 pp 73ndash85 1989
[3] N El Karoui D Huu Nguyen and M Jeanblanc-Pique ldquoCom-pactification methods in the control of degenerate diffusionsrdquoStochastics vol 20 pp 169ndash219 1987
[4] M Nisio ldquoOptimal control for stochastic partial differentialequations and viscosity solutions of Bellman equationsrdquoNagoyaMathematics Journal vol 123 pp 13ndash37 1991
[5] M Nisio ldquoOn sensitive control for stochastic partial differentialequationsrdquo in Stochastic Analysis on Infinite Dimensional SpacesProceedings of the US Japan Bilateral Seminar H Kunita et alEd vol 310 of Pitman Research Notes Mathematical Series pp231ndash241 Longman Scientific and Technical Baton Rouge LaUSA January 1994
[6] R Buckdahn and A Rascanu ldquoOn the existence of stochasticoptimal control of distributed state systemrdquoNonlinear AnalysisTheory Methods and Applications vol 52 no 4 pp 1153ndash11842003
[7] V Barbu and G Da Prato Equations in Hilbert Spaces vol 86 ofPitman Research Notes in Mathematics Pitman 1983
[8] P Cannarsa and G Da Prato ldquoSecond-order Hamilton-Jacobiequations in infinite dimensionsrdquo SIAM Journal on Control andOptimization vol 29 no 2 pp 474ndash492 1991
[9] P Cannarsa and G Da Prato ldquoDirect solution of a second-orderHamilton-Jacobi equations in Hilbert spacesrdquo in StochasticPartial Differential Equations and Applications G Da Pratoand L Tubaro Eds vol 268 of Pitman Research Notes inMathematics Pitman 1992
[10] F Gozzi ldquoRegularity of solutions of second order Hamilton-Jacobi equations and application to a control problemrdquoCommu-nications in Partial Differential Equations vol 20 pp 775ndash8261995
[11] F Gozzi ldquoGlobal regular solutions of second order Hamilton-Jacobi equations in Hilbert spaces with locally Lipschitz non-linearitiesrdquo Journal of Mathematical Analysis and Applicationsvol 198 no 2 pp 399ndash443 1996
[12] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems and Control Lettersvol 14 no 1 pp 55ndash61 1990
[13] N El Karoui and LMazliak Eds Backward Stochastic Differen-tial Equations vol 364 of Pitman ResearchNotes inMathematicsSeries Longman 1997
[14] E Pardoux and BSDEs ldquoweak convergence and homogeneiza-tion of semilinear PDEsrdquo in Non- Linear Analysis DifferentialEquations and Control F H Clarke and R J Stern Eds pp503ndash549 Kluwer Dordrecht The Netherlands 1999
[15] S Peng ldquoA generalized dynamic programming principle andHamilton-Jacobi-Bellman equationrdquo Stochastics and StochasticsReports vol 38 pp 119ndash134 1992
[16] N E Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997
[17] SHamad120583ene and J P Lepeltier ldquoBackward equations stochas-tic control and zero-sum stochastic differential gamesrdquo Stochas-tics and Stochastics Reports vol 54 pp 221ndash231 1995
[18] N El-Karoui and S Hamadene ldquoBSDEs and risk-sensitive con-trol zero-sum and nonzero-sum game problems of stochasticfunctional differential equationsrdquo Stochastic Processes and theirApplications vol 107 no 1 pp 145ndash169 2003
[19] M Fuhrman and G Tessiture ldquoExistence of optimal stochasticcontrols and global solutions of forward-backward stochasticdifferential equationsrdquo SIAM Journal on Control and Optimiza-tion vol 43 no 3 pp 813ndash830 2005
[20] M Fuhrman Y Hu and G Tessitore ldquoOn a class of stochasticoptimal control problems related to bsdes with quadraticgrowthrdquo SIAM Journal on Control and Optimization vol 45 no4 pp 1279ndash1296 2006
[21] M Fuhrman and G Tessitore ldquoNonlinear kolmogorov equa-tions in infinite dimensional spaces the backward stochasticdifferential equations approach and applications to optimalcontrolrdquoAnnals of Probability vol 30 no 3 pp 1397ndash1465 2002
[22] F Masiero ldquoSemilinear kolmogorov equations and applicationsto stochastic optimal controlrdquo Applied Mathematics and Opti-mization vol 51 no 1 pp 201ndash250 2005
[23] M Fuhrman FMasiero andG Tessitore ldquoStochastic equationswith delay optimal control via BSDEs and regular solutions ofHamilton-jacobi-bellman equationsrdquo SIAM Journal on Controland Optimization vol 48 no 7 pp 4624ndash4651 2010
[24] M Fuhrman and G Tessiture ldquoInfinite horizon backwardstochastic differential equations and elliptic equations in hilbertspacesrdquo Annals of Probability vol 32 no 1 pp 607ndash660 2004
[25] F Masiero ldquoInfinite horizon stochastic optimal control prob-lems with degenerate noise and elliptic equations in Hilbertspacesrdquo Applied Mathematics and Optimization vol 55 no 3pp 285ndash326 2007
[26] M Fuhrman ldquoA class of stochastic optimal control problemsin Hilbert spaces BSDEs and optimal control laws stateconstraints conditioned processesrdquo Stochastic Processes andtheir Applications vol 108 no 2 pp 263ndash298 2003
[27] F Masiero ldquoStochastic optimal control problems and parabolicequations in banach spacesrdquo SIAM Journal on Control andOptimization vol 47 no 1 pp 251ndash300 2008
[28] G Da Prato and J Zabczyk Ergodicity For Infinite-DimensionalSystems Cambridge University Press 1996
14 Abstract and Applied Analysis
[29] J Zabczyk ldquoParabolic equations on Hilbert spacesrdquo in StochaS-tic PDErsquoS and Kolmogorov Equations in Infinite Dimensionsvol 1715 of Lecture Notes in Math pp 117ndash213 Springer BerlinGermany 1999
[30] G Da Prato and J Zabczyk Stochstic Equations in InfiniteDimensions Cambridge University Press 1992
[31] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 of Applications of MathematicsSpringer New York NY USA 1993
Submit your manuscripts athttpwwwhindawicom
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Abstract and Applied Analysis
le 211987310038161003816100381610038161199052 minus 1199051
1003816100381610038161003816 + Δ12
119890minus120575(1199051+Δ)
times(119864(int
infin
0
119890212057511990410038161003816100381610038161003816119885119894119873
(119904 1199051 119909
1)minus119885
119894119873
(119904 1199052 119909
2)10038161003816100381610038161003816
2
119889119904)
1199012
)
1119901
le 211987310038161003816100381610038161199052 minus 1199051
1003816100381610038161003816 + Δ12
119890minus120575(1199051+Δ)
times (119864(int
infin
0
11989021205751199041003816100381610038161003816119885 (119904 1199051 1199091)minus119885 (119904 1199052 1199092)
10038161003816100381610038162
119889119904)
1199012
)
1119901
(45)
From Theorem 7 we have that the map (119905 119909) rarr int119905+Δ
119905
119885119894119873
(119904 119905 119909)119889119904 is continuous from [0infin) times C to 119877 ByRemark 8 we also have that for fixed 119905 the map 119909 rarr
119864int119905+Δ
119905
119864119885119894119873
(119904 119905 119909)119889119904 is continuous from 119871119902
(ΩCF119905) to 119877
for 119902 large enough Let us define
120577119894119873
(119905 119909) = lim inf119899rarrinfin
119899119864int
119905+(1119899)
119905
119885119894119873
(119904 119905 119909) 119889119904
119905 isin [0infin) 119909 isin C
(46)
It is clear that 120577119894119873 [0infin) timesC rarr 119877 is a Borel functionWe fix 119909 and 0 le 119905 le 119904 lt infin For 119897 isin [119904infin) we
denote 119864[119885119894119873
(119897 119904 119910)]|119910=119883119904(119905119909)
the random variable obtainedby composing119883
119904(119905 119909) with the map 119910 rarr 119864[119885
119894119873
(119897 119904 119910)]By Lemma 9 there exists a sequence of C-valued F
119904-
measurable simple functions
119891119898 Ω 997888rarr C 119891
119898=
119873119898
sum
119896=1
ℎ(119898)
119896119868119891119898=ℎ(119898)
119896 119873
119898isin 119873 (47)
where ℎ(119898)1 ℎ
(119898)
119898are pairwise distinct andΩ = ⋃
119873119898
119896=1119891
119898=
ℎ(119898)
119896 such that
1003816100381610038161003816119891119898 (120596) minus 119883119904(120596)
1003816100381610038161003816 darr 0 for all 120596 isin Ω as 119899 997888rarr infin (48)
For any 119861 isin F119904 we have
int119861
int
119904+(1119899)
119904
119885119894119873
(119897 119905 119909) 119889119897119889119875
= int119861
int
119904+(1119899)
119904
119885119894119873
(119897 119904 119883119904) 119889119897119889119875
= 119864119868119861int
119904+(1119899)
119904
119885119894119873
(119897 119904 119883119904) 119889119897
= lim119898rarrinfin
119864(119868119861int
119904+(1119899)
119904
119885119894119873
(119897 119904 119891119898) 119889119897)
= lim119898rarrinfin
119873119898
sum
119896=1
119864(119868119861119868119891119898=ℎ(119898)
119896int
119904+(1119899)
119904
119885119894119873
(119897 119904 ℎ(119898)
119896) 119889119897)
= lim119898rarrinfin
119864(119868119861
119873119898
sum
119896=1
119868119891119898=ℎ(119898)
119896)119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 ℎ(119898)
119896) 119889119897
= lim119898rarrinfin
119864119868119861(119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 119910) 119889119897|119910=119891119898
)
= lim119898rarrinfin
int119861
(119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 119910) 119889119897|119910=119891119898
)119889119875
= int119861
(119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 119910) 119889119897|119910=119883119904
)119889119875
(49)
and we get that
120577119894119873
(119904 119883119904(119905 119909)) = lim inf
119899rarrinfin
119899
times [119864int
119904+(1119899)
119904
119885119894119873
(119897 119904 119910) 119889119897|119910=119883119904(119905119909)
]
= lim inf119899rarrinfin
119899119864[int
119904+(1119899)
119904
119885119894119873
(119897 119905 119909) 119889119897
100381610038161003816100381610038161003816100381610038161003816
F119904]
119875-as(50)
Fix 119905 and 119909 Recalling that |119885119894119873
| le 119873 by the Lebesguetheorem on differentiation it follows that 119875-as
lim119899rarrinfin
119899int
119904+(1119899)
119904
119885119894119873
(119897 119905 119909) 119889119897 = 119885119894119873
(119904 119905 119909)
for aa 119904 isin [119905infin)
(51)
By the boundedness of 119885119894119873 applying the dominated conver-gence theorem we get that
120577119894119873
(119904 119883119904(119905 119909)) = 119864 [119885
119894119873
(119904 119905 119909)10038161003816100381610038161003816F
119904] = 119885
119894119873
(119904 119905 119909)
119875-as for aa 119904 isin [119905infin)
(52)
Now we have proved that for every 119905 119909
120577119894119873
(119904 119883119904(119905 119909)) = 119885
119894119873
(119904 119905 119909)
119875-as for aa 119904 isin [119905infin)
(53)
for every 119894 119873 Let 119862 sub [0infin) times C denote the set ofpairs (119905 119909) such that lim
119873rarrinfin120577119894119873
(119905 119909) exists and the seriessuminfin
119894=1(lim
119873rarrinfin120577119894119873
(119905 119909))119890119894converges in Ξlowast We define
120577 (119905 119909) =
infin
sum
119894=1
( lim119873rarrinfin
120577119894119873
(119905 119909)) 119890119894 (119905 119909) isin 119862
120577 (119905 119909) = 0 (119905 119909) notin 119862
(54)
Abstract and Applied Analysis 9
Since 119885 satisfies
119885 (120596 119904 119905 119909) =
infin
sum
119894=1
( lim119873rarrinfin
119885119894119873
(120596 119904 119905 119909)) 119890119894 (55)
for every 120596 119904 119905 119909 From (53) it follows that for every 119905 119909 wehave (119904 119883
119904(120596 119905 119909)) isin 119862 119875-as for almost all 119904 isin [119905infin) and
119885(119904 119905 119909) = 120577(119904 119883119904(119905 119909)) 119875-as for aa 119904 isin [119905infin)
We define 120592(119905 119909) = 119884(119905 119905 119909) since 119884(119905 119905 119909) is deter-ministic so the map (119905 119909) rarr 120592(119905 119909) can be written as acomposition 120592(119905 119909) = Γ
3(Γ2(119905 Γ
1(119905 119909))) with
Γ1 [0infin) timesC 997888rarr 119871
119901
P(Ω 119862
120575(119877))
Γ1(119905 119909) = 119884 (sdot 119905 119909)
Γ2 [0infin) times 119871
119901
P(Ω 119862
120575(119877)) 997888rarr 119871
119901
(Ω 119877)
Γ2(119905 119881) = 119881 (119905)
Γ3 119871
119901
(Ω 119877) 997888rarr 119877 Γ3120585 = 119864120585
(56)
FromTheorem 7 it follows that Γ1is continuous By
|119881(119905) minus 119880(119904)|119871119901(Ω119877)
le |119881 (119905) minus 119881 (119904)|119871119901(Ω119877)
+ 119890minus120575119901119904
|119881 minus 119880||119871
119901
P(Ω119862120575(119877))
(57)
we have that Γ2is continuous It is clear that Γ
3is continuous
Then themap (119905 119909) rarr 120592(119905 119909) is continuous from [0infin)timesCto 119877 therefore 120592(119905 119909)is a Borel measurable function Fromuniqueness of the solution of (36) it follows that 119884(119904 119905 119909) =120592(119904 119883
119904(119905 119909)) 119875-as for aa 119904 isin [119905infin)
4 The Fundamental Relation
Let (ΩF 119875) be a given complete probability space with afiltration F
119905119905ge0
satisfying the usual conditions 119882(119905) 119905 ge 0
is a cylindrical Wiener process in Ξ with respect to F119905119905ge0
We will say that an F
119905ge0-predictable process 119906 with values
in a given measurable space (119880U) is an admissible controlThe function 119877 [0infin) times C times 119880 rarr Ξ is measurableand bounded We consider the following controlled stateequation
119889119883119906
(119904) = 119860119883119906
(119904) 119889119904 + 119865 (119904 119883119906
119904) 119889119904
+ 119866 (119904 119883119906
119904) 119877 (119904 119883
119906
119904 119906 (119904)) 119889119904 + 119866 (119904 119883
119906
119904) 119889119882 (119904)
119904 isin [119905infin)
119883119906
119905= 119909
(58)
Here we assume that there exists a mild solution of (58)which will be denoted by 119883119906
(119904 119905 119909) or simply by 119883119906
(119904) Weconsider a cost function of the form
119869 (119906) = 119864int
infin
119905
119890minus120582119904
119892 (119904 119883119906
119904 119906 (119904)) 119889119904 (59)
Here 119892 is function on [0infin) times C times 119880 with real values Ourpurpose is to minimize the function 119869 over all admissiblecontrols
We define in a classical way the Hamiltonian functionrelative to the previous problem for all 119905 isin [0infin) 119909 isin
C and 119911 isin Ξlowast
120595 (119905 119909 119911) = inf 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) 119906 isin 119880
(60)
and the corresponding possibly empty set of minimizers
Γ (119905 119909 119911) = 119906 isin 119880 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) = 120595 (119905 119909 119911)
(61)
We are now ready to formulate the assumptions we need
Hypothesis 3 (i) 119860 119865 and G verify Hypothesis 1(ii) (119880U) is ameasurable spaceThemap 119892 [0infin)timesCtimes
119880 rarr 119877 is continuous and satisfies |119892(119905 119909 119906)| le 119870119892(1+|119909|
119898119892
119862)
for suitable constants 119870119892gt 0 119898
119892gt 0 and all 119909 isin C119906 isin
119880 The map 119877 [0infin) times C times 119880 rarr Ξ is measurable and|119877(119905 119904 119906)| le 119871
119877for a suitable constant 119870
119877gt 0 and all 119909 isin
C119906 isin 119880 and119911 isin Ξlowast
(iii)TheHamiltonian120595 defined in (60) satisfies the requi-rements of Hypothesis 2 (with119870 = 119877)
(iv) We fix here 119901 gt 2 q and 120575 lt 0 satisfying (39) with120578 = 120578(119902) and such that 119902 gt 119898
119892
We are in a position to prove the main result of thissection
Theorem11 Assume thatHypothesis 3 holds and suppose that120582 verifies
120582 gt (minus120575 minus 120583 +1198712
119911
2) or (minus120575 +
1198712
119877
2 (119901 minus 1))
or (1198712
119877119898119892
2 (119902 minus 119898119892)minus 120578 (119902)119898
119892)
(62)
Let 120592 120577 denote the function in the statement of Theorem 10Then for every admissible control 119906 and for the correspondingtrajectory119883 starting at (119905 119909) one has
119869 (119906) = 120592 (119905 119909) + 119864int
infin
119905
119890minus120582119904
[minus120595 (119904 119883119906
119904 120577 (119904 119883
119906
119904)) + 120577 (119904 119883
119906
119904)
times 119877 (119904 119883119906
119904 119906 (119904)) + 119892 (119904 119883
119906
119904 119906 (119904))] 119889119904
(63)
Proof Consider (58) in the probability space (ΩF 119875) withfiltration F
119905119905ge0
and with an F119905119905ge0
-cylindrical Wienerprocess 119882(119905) 119905 ge 0 Let us define
119882119906
(119904) = 119882 (119904) + int
119904
119905and119904
119877 (120590119883119906
120590 119906 (120590)) 119889120590 119904 isin [0infin)
120588 (119879) = exp(int119879
119905
minus119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882 (119904)
minus1
2int
119879
119905
1003816100381610038161003816119877 (119904 119883119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)
(64)
10 Abstract and Applied Analysis
Let 119875119906 be the unique probability onF[0infin)
such that
119875119906
|F119879
= 120588 (119879) 119875|F119879
(65)
We notice that under 119875119906 the process119882119906 is aWiener processLet us denote by F119906
119905119905ge0
the filtration generated by119882119906 andcompleted in the usual way Relatively to 119882119906 (58) can berewritten as
119889119883119906
(119904) = 119860119883119906
(119904) 119889119904 + 119865 (119904 119883119906
119904) 119889119904
+ 119866 (119904 119883119906
119904) 119889119882
119906
(119904) 119904 isin [119905infin)
119883119906
119905= 119909
(66)
In the space (ΩF[0infin)
F119906
119905119905ge0 119875
119906
) we consider the follow-ing system of forward-backward equations
119883119906
(119904) = 119890(119904minus119905)119860
119909 (0) + int
119904
119905
119890(119904minus120590)119860
119865 (120590119883119906
120590) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883119906
120590) 119889119882
119906
(120590) 119904 isin [119905infin)
119883119906
119905= 119909 isin C
119884119906
(119904) minus 119884119906
(119879) + int
119879
119904
119885119906
(120590) 119889119882119906
(120590) + 120582int
119879
119904
119884119906
(120590) 119889120590
= int
119879
119904
120595 (120590119883119906
120590 119885
119906
(120590)) 119889120590 0 le 119904 le 119879 lt infin
(67)
Applying the Ito formula to 119890minus120582119904119884119906(119904) and writing the back-ward equation in (67) with respect to the process119882 we get
119884119906
(119904) + int
119879
119904
119890minus120582120590
119885119906
(120590) 119889119882 (120590)
= int
119879
119904
119890minus120582120590
[120595 (120590119883119906
120590 119885
119906
(120590))
minus119885119906
(120590) 119877 (120590119883119906
120590 119906 (120590))] 119889120590
+ 119890minus120582119879
119884119906
(119879)
(68)
Recalling that 119877 is bounded we get for all 119903 ge 1 and someconstant 119862
119864119906
[120588(119879)minus119903
] = 119864119906
[exp 119903 (int119879
119905
119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882
119906
(119904)
minus1
2int
119879
119905
1003816100381610038161003816119877 (119904 119883119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)]
= 119864119906
[exp(int119879
119905
119903119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882
119906
(119904)
minus1
2int
119879
119905
11990321003816100381610038161003816119877 (119904 119883
119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)
times exp 119903 (119903 minus 1)2
int
119879
119905
1003816100381610038161003816119877 (119904 119883119906
119904 119906 (119904))
10038161003816100381610038162
119889119904]
le 119890(12)119903(119903minus1)119879119871
2
119877119864119906
times exp(int119879
119905
2119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882
119906
(119904)
minus1
2int
119879
119905
41003816100381610038161003816119877 (119904 119883
119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)
= 119890(12)119903(119903minus1)119879119871
2
119877
(69)
It follows that
119864(int
119879
119905
|119890minus120582119904
119885119906
(119904)|2
119889119904)
12
= 119864119906
[(int
119879
119905
10038161003816100381610038161003816119890minus120582119904
119885119906
(119904)10038161003816100381610038161003816
2
119889119904)
12
120588minus1
]
le (119864119906
int
119879
119905
10038161003816100381610038161003816119890minus120582119904
119885119906
(119904)10038161003816100381610038161003816
2
119889119904)
12
times (119864119906
120588minus2
)12
lt infin
(70)
We conclude that the stochastic integral in (68) has zeroexpectation If we set 119904 = 119905 in (68) and we take expectationwith respect to 119875 we obtain
119890minus120582119879
119864119884119906
(119879) minus 119884119906
(119905)
= 119864int
119879
119905
119890minus120582120590
[minus120595 (120590119883119906
120590 119885
119906
(120590))
+119885119906
(120590) 119877 (120590119883119906
120590 119906 (120590))] 119889120590
(71)
ByTheorem 7 119884119906(sdot 119905 119909) isin 119871119901P(Ω 119862
120575(119877)) so that
119864119906
|119884(119879 119905 119909)|119901
le 119862 exp (minus119901120575119879) (72)
By the Holder inequality we have that for suitable constant119862 gt 0
119864 |119884 (119879 119905 119909)| = 119864119906
(120588minus1
(119879) |119884 (119879 119905 119909)|)
le 119864(120588minus119901(119901minus1)
)(119901minus1)119901
119864(|119884 (119879 119905 119909)|119901
)1119901
le 119862119890((1198712
1198772(119901minus1))minus120575))119879
(73)
From Theorem 2 we obtain 119864119906sup119904ge119905119890120578119902119904
|119883119906
119904|119902
lt infin by thesimilar process we get that
1198641003816100381610038161003816119883
119906
119879
1003816100381610038161003816119898119892
le 119862119890(1198712
119877119898119892(2119902minus2119898
119892)minus1
minus120578(119902)119898119892)119879
(74)
Abstract and Applied Analysis 11
for suitable constant 119862 gt 0 and
119864int
infin
119905
119890minus120582120590 1003816100381610038161003816119892 (120590119883
119906
120590 119906 (120590))
1003816100381610038161003816 119889120590 lt infin (75)
Since 119884119906(119905 119905 119909) = 120592(119905 119909) and 119885119906
(119904 119905 119909) = 120577(119904 119883119906
119904(119905 119909)) 119875-
as for aa 119904 isin [119905infin) we have that
119890minus120582119879
119864119884119906
(119879) minus 119907 (119905 119909)
= 119864int
119879
119905
119890minus120582120590
[minus120595 (120590119883119906
120590 120577 (120590 119883
119906
120590))
+120577 (120590119883119906
120590) 119877 (120590119883
119906
120590 119906 (120590))] 119889120590
(76)
Thus adding and subtracting119864intinfin119905
119890minus120582120590
119892(120590119883119906
120590 119906(120590))119889120590 and
letting 119879 rarr infin we conclude that
119869 (119906) = 120592 (119905 119909)
+ 119864int
infin
119905
119890minus120582119904
[minus120595 (119904 119883119906
119904 120577 (119904 119883
119906
119904)) + 120577 (119904 119883
119906
119904) 119877
times (119904 119883119906
119904 119906 (119904)) + 119892 (119904 119883
119906
119904 119906 (119904))] 119889119904
(77)
The proof is finished
We immediately deduce the following consequences
Theorem 12 Let 119905 isin [0infin) and 119909 isin C be fixed assumethat the set-valued map Γ has nonempty values and it admitsa measurable selection Γ
0 [0infin)timesCtimesΞlowast rarr 119880 and assume
that a control 119906(sdot) satisfies
119906 (119904) = Γ0(119904 119883
119906
119904 120577 (119904 119883
119906
119904))
119875-as for almost every 119904 isin [119905infin)
(78)
Then 119869(119905 119909 119906) = 120592(119905 119909) and the pair (119906(sdot) 119883) is optimal forthe control problem starting from 119909 at time 119905
Such a control can be shown to exist if there exists a solutionfor the so-called closed-loop equation as follows
119889119883 (119904)=119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904+119866 (119904 119883
119904)
times(119877 (119904 119883119904 Γ
0(119904 119883
119904 120577 (119904 119883
119904))) 119889119904 + 119889119882 (119904))
119904 isin [119905infin)
119883119905= 119909
(79)
since in this case we can define an optimal control setting
119906 (119904) = Γ0(119904 119883
119904 120577 (119904 119883
119904)) (80)
However under the present assumptions we cannot guaranteethat the closed-loop equation has a solution in the mildsense To circumvent this difficulty we will revert to a weakformulation of the optimal control problem
5 Existence of Optimal Control
We formulate the optimal control problem in the weak sensefollowing the approach of [31]Themain advantage is that wewill be able to solve the closed-loop equation in a weak senseand hence to find an optimal control even if the feedbacklaw is nonsmooth
We call (ΩF F119905119905ge0 119875119882) an admissible setup if
(ΩF F119905119905ge0 119875) is a filtered probability space satisfying the
usual conditions and 119882 is a cylindrical 119875-Wiener processwith values in Ξ with respect to the filtration F
119905119905ge0
By an admissible control system we mean (ΩF
F119905119905ge0 119875119882 119906119883
119906
) where (ΩF F119905119905ge0 119875 119882) is an
admissible setup 119906 is an F119905-predictable process with values
in119880 and119883119906 is a mild solution of (58) An admissible controlsystem will be briefly denoted by (119882 119906119883119906
) in the followingOur purpose is to minimize the cost functional
119869 (119906) = 119864int
infin
119905
119890minus120582119904
119892 (119904 119883119906
119904 119906 (119904)) 119889119904 (81)
over all the admissible control systemOur main result in this section is based on the solvability
of the closed-loop equation
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904 + 119866 (119904 119883
119904)
times (119877 (119904 119883119904 Γ
0(119904 119883
119904 120577 (119904 119883
119904))) 119889119904 +119889119882 (119904))
119904 isin [119905infin)
119883119905= 119909
(82)
In the following sense we say that 119883 is a weak solution of(82) if there exists an admissible setup (ΩF F
119905119905ge0 119875119882)
and anF119905-adapted continuous process119883(119905)with values in119867
which solves the equation in the mild sense namely 119875-as
119883 (119904) = 119890(119904minus119905)119860
119909 (0) + int
119904
119905
119890(119904minus120590)119860
119865 (120590119883120590) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883120590) 119877
times (120590119883120590 Γ
0(120590119883
120590 120577 (120590 119883
120590))) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883120590) 119889119882
120590 119904 isin [119905infin)
(83)
119883119905= 119909 (84)
Theorem 13 Assume that Hypothesis 3 holds Then thereexists a weak solution of the closed-loop equation (82) whichis unique in law
12 Abstract and Applied Analysis
Proof (uniqueness) Let 119883 be a weak solution of (82) in anadmissible setup (ΩF F
119905119905ge0 119875119882)We define
120588 (119879) = exp(int119879
119905
minus119877lowast
(120590 119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889119882 (120590)
minus1
2int
119879
119905
1003816100381610038161003816119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)))10038161003816100381610038162
119889120590)
(85)
Since 119877 is bounded the Girsanov theorem ensures that thereexists a probability measure 1198750 such that the process
1198820
(119904) = 119882 (119904) + int
119904
119905and119904
119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889120590
119904 isin [0infin)
(86)
is a 1198750-Wiener process and
1198750
|F119879
= 120588 (119879) 119875|F119879
(87)
Let us denote by F0
119905119905ge0
the filtration generated by1198820 andcompleted in the usual way In (ΩF
[0infin) F0
119905119905ge0 119875
0
) 119883 isa mild solution of
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119905 119883119904) 119889119904
+ 119866 (119904 119883119904) 119889119882
0
(119904) 119904 isin [119905infin)
119883119905= 119909
120588 (119879) = exp(int119879
119905
minus119877lowast
(120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)) 119889119882
0
(120590)
+ 12int
119879
119905
1003816100381610038161003816119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)))10038161003816100381610038162
119889120590)
(88)
By Hypothesis 3 the joint law of 119883 and 1198820 is uniquely
determined by 119860 119865 119866 and 119909 Taking into account the lastdisplayed formula we conclude that the joint law of 119883 and120588(119879) under1198750 is also uniquely determined and consequentlyso is the law of 119883 under 119875 This completes the proof of theuniqueness part
Proof (existence) Let (ΩF 119875) be a given complete probabil-ity space 119882(119905) 119905 ge 0 is a cylindrical Wiener process on(ΩF 119875)with values inΞ and F
119905119905ge0
is the natural filtrationof 119882(119905) 119905 ge 0 augmented with the family of119875-null sets Let119883(sdot) be the mild solution of
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904
+ 119866 (119904 119883119904) 119889119882 (119904) 119904 isin [119905infin)
119883119905= 119909
(89)
and by the Girsanov theorem let 1198751 be the probability on Ωunder which
1198821
(119904) = 119882 (119904) minus int
119904
119905and119904
119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889120590
(90)
is aWiener process (notice that 119877 is bounded)Then119883 is theweak solution of (82) relatively to the probability 1198751 and theWiener process1198821
Now we can state the main result of this section
Corollary 14 Assume that Hypothesis 3 holds true and 120582
verifies (62) Also assume that the set-valued map Γ hasnonempty values and it admits a measurable selection Γ
0
[0infin) timesC times Ξlowast
rarr 119880 Then for every 119905 isin [0infin) and x isin Cand for all admissible control system (119882 119906119883
119906
) one has
119869 (119906 119905 119909) ge 120592 (119905 119909) (91)
and the equality holds if
119906 (119904) = Γ0(119904 119883
119906
119904 120577 (119904 119883
119906
119904))
119875 minus 119886119904 119891119900119903 119886119897119898119900119904119905 119890119907119890119903119910 119904 isin [119905infin)
(92)
Moreover from Theorem 13 it follows that the closed-loop equation (82) admits a weak solution (ΩFF
119905119905ge0 119875119882119883) which is unique in law and setting
119906 (119904) = Γ0(119904 119883
119904 120577 (119904 119883
119904)) (93)
we obtain an optimal admissible control system (119882 119906119883)
6 Applications
In this section we present a simple application of the previousresults We consider the stochastic delay partial differentialequation in the bounded domain 119861 sub 119877
119899 with smoothboundary 120597119861 as follows
119889119911119906
(119905 120585) = Δ119911119906
(119905 120585) 119889119905 + 119891 (119905 119911119906
119905(120585)) 119889119905
+
119889
sum
119894=1
119892119894(119905 119911
119906
119905(120585)) [119903
119894
(120585) 119906119894
(119905) 119889119905 + 119889119882119894
(119905)]
119911119906
0(120579 120585) = 119909 (120579 120585) 120585 isin 119861 120579 isin [minus1 0]
119911119906
(119905 120585) = 0 119905 isin [0infin) 120585 isin 120597119861
(94)
Here119882 = (1198821
1198822
119882119889
) is a standardWiener process in119877119889 and the functions 119891 [0 +infin) times 119862([minus1 0] 119877) rarr 119877 and119892119894 [0 +infin) times 119862([minus1 0] 119877) rarr 119877 are Lipschitz continuous
and bounded Setting 119880 as a bounded subset of 119877119889 Ξ = 119877119889
119867 = 1198712
(119861) and 119909 isin 119862([minus1 0]119867) We define 119865 and 119866 asfollowing
119865 (119905 119909) (120585) = 119891 (119905 119909 (120585))
(119866 (119905 119909) 119911) (120585) =
119889
sum
119894=1
119892119894(119905 119909 (120585)) 119911
119894
(120585)
120585 isin 119861 119909 isin 119862 ([minus1 0] 119867) 119911 isin 119871 (Ξ119867)
(95)
and let 119860 denote the Laplace operator Δ in 1198712
(119861) withdomain11988222
(119861)⋂11988212
0(119861) then (94) has the form (58) and
Hypothesis 1 holds
Abstract and Applied Analysis 13
Let us consider the optimal control problem associatedwith the cost
119869 (119906) = 119864int
infin
0
119890minus120582119905
[int119861
120590 (120585 119911119906
119905(120585)) 119889120585 + 119906
2
(119905)] 119889119905 (96)
where 120582 verifies (62) and 120590 119862([minus1 0] 119877) times 119880 rarr [0infin) isa bounded measurable function Define 119892 119862([minus1 0]119867) times
119880 rarr [0infin) and 119877 119862([minus1 0]119867) times 119880 rarr Ξ by119892(119910 119906) = int
119861
120590(119905 119910(120585) 119906)119889120585 + 1199062 and 119877(119910 119906) = (int
119861
1199031
(120585)1199061
119889120585
int119861
1199032
(120585)1199062
119889120585 int119861
119903119889
(120585)119906119889
119889120585) for 119910 isin 119862([minus1 0]119867) 119906 =
(1199061
1199062
119906119889
) isin 119880 respectively It can be easily verifiedthat Hypothesis 3 holds true and the set-valued map Γ hasnonempty values and it admits a measurable selection Γ
0
[0infin) times C times Ξlowast
rarr 119880 Then the closed-loop equation(82) admits a weak solution (ΩF F
119905119905ge0 119875119882 119906 119911
sdot(sdot)) and
setting
119906 (119904) = Γ0(119904 119911
119904(sdot) 120577 (119904 119911
119904(sdot))) (97)
we obtain an optimal admissible control system (119882 119906 119911(sdot))
References
[1] J Bismut ldquoOn optimal control of linear stochastic equationswith a linear-quadratic criterionrdquo SIAM Journal on Control andOptimization vol 15 no 3 pp 1ndash4 1977
[2] N Nagase ldquoOn the existence of optimal control for controlledstochastic partial differential equationsrdquo Nagoya MathematicsJournal vol 115 pp 73ndash85 1989
[3] N El Karoui D Huu Nguyen and M Jeanblanc-Pique ldquoCom-pactification methods in the control of degenerate diffusionsrdquoStochastics vol 20 pp 169ndash219 1987
[4] M Nisio ldquoOptimal control for stochastic partial differentialequations and viscosity solutions of Bellman equationsrdquoNagoyaMathematics Journal vol 123 pp 13ndash37 1991
[5] M Nisio ldquoOn sensitive control for stochastic partial differentialequationsrdquo in Stochastic Analysis on Infinite Dimensional SpacesProceedings of the US Japan Bilateral Seminar H Kunita et alEd vol 310 of Pitman Research Notes Mathematical Series pp231ndash241 Longman Scientific and Technical Baton Rouge LaUSA January 1994
[6] R Buckdahn and A Rascanu ldquoOn the existence of stochasticoptimal control of distributed state systemrdquoNonlinear AnalysisTheory Methods and Applications vol 52 no 4 pp 1153ndash11842003
[7] V Barbu and G Da Prato Equations in Hilbert Spaces vol 86 ofPitman Research Notes in Mathematics Pitman 1983
[8] P Cannarsa and G Da Prato ldquoSecond-order Hamilton-Jacobiequations in infinite dimensionsrdquo SIAM Journal on Control andOptimization vol 29 no 2 pp 474ndash492 1991
[9] P Cannarsa and G Da Prato ldquoDirect solution of a second-orderHamilton-Jacobi equations in Hilbert spacesrdquo in StochasticPartial Differential Equations and Applications G Da Pratoand L Tubaro Eds vol 268 of Pitman Research Notes inMathematics Pitman 1992
[10] F Gozzi ldquoRegularity of solutions of second order Hamilton-Jacobi equations and application to a control problemrdquoCommu-nications in Partial Differential Equations vol 20 pp 775ndash8261995
[11] F Gozzi ldquoGlobal regular solutions of second order Hamilton-Jacobi equations in Hilbert spaces with locally Lipschitz non-linearitiesrdquo Journal of Mathematical Analysis and Applicationsvol 198 no 2 pp 399ndash443 1996
[12] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems and Control Lettersvol 14 no 1 pp 55ndash61 1990
[13] N El Karoui and LMazliak Eds Backward Stochastic Differen-tial Equations vol 364 of Pitman ResearchNotes inMathematicsSeries Longman 1997
[14] E Pardoux and BSDEs ldquoweak convergence and homogeneiza-tion of semilinear PDEsrdquo in Non- Linear Analysis DifferentialEquations and Control F H Clarke and R J Stern Eds pp503ndash549 Kluwer Dordrecht The Netherlands 1999
[15] S Peng ldquoA generalized dynamic programming principle andHamilton-Jacobi-Bellman equationrdquo Stochastics and StochasticsReports vol 38 pp 119ndash134 1992
[16] N E Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997
[17] SHamad120583ene and J P Lepeltier ldquoBackward equations stochas-tic control and zero-sum stochastic differential gamesrdquo Stochas-tics and Stochastics Reports vol 54 pp 221ndash231 1995
[18] N El-Karoui and S Hamadene ldquoBSDEs and risk-sensitive con-trol zero-sum and nonzero-sum game problems of stochasticfunctional differential equationsrdquo Stochastic Processes and theirApplications vol 107 no 1 pp 145ndash169 2003
[19] M Fuhrman and G Tessiture ldquoExistence of optimal stochasticcontrols and global solutions of forward-backward stochasticdifferential equationsrdquo SIAM Journal on Control and Optimiza-tion vol 43 no 3 pp 813ndash830 2005
[20] M Fuhrman Y Hu and G Tessitore ldquoOn a class of stochasticoptimal control problems related to bsdes with quadraticgrowthrdquo SIAM Journal on Control and Optimization vol 45 no4 pp 1279ndash1296 2006
[21] M Fuhrman and G Tessitore ldquoNonlinear kolmogorov equa-tions in infinite dimensional spaces the backward stochasticdifferential equations approach and applications to optimalcontrolrdquoAnnals of Probability vol 30 no 3 pp 1397ndash1465 2002
[22] F Masiero ldquoSemilinear kolmogorov equations and applicationsto stochastic optimal controlrdquo Applied Mathematics and Opti-mization vol 51 no 1 pp 201ndash250 2005
[23] M Fuhrman FMasiero andG Tessitore ldquoStochastic equationswith delay optimal control via BSDEs and regular solutions ofHamilton-jacobi-bellman equationsrdquo SIAM Journal on Controland Optimization vol 48 no 7 pp 4624ndash4651 2010
[24] M Fuhrman and G Tessiture ldquoInfinite horizon backwardstochastic differential equations and elliptic equations in hilbertspacesrdquo Annals of Probability vol 32 no 1 pp 607ndash660 2004
[25] F Masiero ldquoInfinite horizon stochastic optimal control prob-lems with degenerate noise and elliptic equations in Hilbertspacesrdquo Applied Mathematics and Optimization vol 55 no 3pp 285ndash326 2007
[26] M Fuhrman ldquoA class of stochastic optimal control problemsin Hilbert spaces BSDEs and optimal control laws stateconstraints conditioned processesrdquo Stochastic Processes andtheir Applications vol 108 no 2 pp 263ndash298 2003
[27] F Masiero ldquoStochastic optimal control problems and parabolicequations in banach spacesrdquo SIAM Journal on Control andOptimization vol 47 no 1 pp 251ndash300 2008
[28] G Da Prato and J Zabczyk Ergodicity For Infinite-DimensionalSystems Cambridge University Press 1996
14 Abstract and Applied Analysis
[29] J Zabczyk ldquoParabolic equations on Hilbert spacesrdquo in StochaS-tic PDErsquoS and Kolmogorov Equations in Infinite Dimensionsvol 1715 of Lecture Notes in Math pp 117ndash213 Springer BerlinGermany 1999
[30] G Da Prato and J Zabczyk Stochstic Equations in InfiniteDimensions Cambridge University Press 1992
[31] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 of Applications of MathematicsSpringer New York NY USA 1993
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Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 9
Since 119885 satisfies
119885 (120596 119904 119905 119909) =
infin
sum
119894=1
( lim119873rarrinfin
119885119894119873
(120596 119904 119905 119909)) 119890119894 (55)
for every 120596 119904 119905 119909 From (53) it follows that for every 119905 119909 wehave (119904 119883
119904(120596 119905 119909)) isin 119862 119875-as for almost all 119904 isin [119905infin) and
119885(119904 119905 119909) = 120577(119904 119883119904(119905 119909)) 119875-as for aa 119904 isin [119905infin)
We define 120592(119905 119909) = 119884(119905 119905 119909) since 119884(119905 119905 119909) is deter-ministic so the map (119905 119909) rarr 120592(119905 119909) can be written as acomposition 120592(119905 119909) = Γ
3(Γ2(119905 Γ
1(119905 119909))) with
Γ1 [0infin) timesC 997888rarr 119871
119901
P(Ω 119862
120575(119877))
Γ1(119905 119909) = 119884 (sdot 119905 119909)
Γ2 [0infin) times 119871
119901
P(Ω 119862
120575(119877)) 997888rarr 119871
119901
(Ω 119877)
Γ2(119905 119881) = 119881 (119905)
Γ3 119871
119901
(Ω 119877) 997888rarr 119877 Γ3120585 = 119864120585
(56)
FromTheorem 7 it follows that Γ1is continuous By
|119881(119905) minus 119880(119904)|119871119901(Ω119877)
le |119881 (119905) minus 119881 (119904)|119871119901(Ω119877)
+ 119890minus120575119901119904
|119881 minus 119880||119871
119901
P(Ω119862120575(119877))
(57)
we have that Γ2is continuous It is clear that Γ
3is continuous
Then themap (119905 119909) rarr 120592(119905 119909) is continuous from [0infin)timesCto 119877 therefore 120592(119905 119909)is a Borel measurable function Fromuniqueness of the solution of (36) it follows that 119884(119904 119905 119909) =120592(119904 119883
119904(119905 119909)) 119875-as for aa 119904 isin [119905infin)
4 The Fundamental Relation
Let (ΩF 119875) be a given complete probability space with afiltration F
119905119905ge0
satisfying the usual conditions 119882(119905) 119905 ge 0
is a cylindrical Wiener process in Ξ with respect to F119905119905ge0
We will say that an F
119905ge0-predictable process 119906 with values
in a given measurable space (119880U) is an admissible controlThe function 119877 [0infin) times C times 119880 rarr Ξ is measurableand bounded We consider the following controlled stateequation
119889119883119906
(119904) = 119860119883119906
(119904) 119889119904 + 119865 (119904 119883119906
119904) 119889119904
+ 119866 (119904 119883119906
119904) 119877 (119904 119883
119906
119904 119906 (119904)) 119889119904 + 119866 (119904 119883
119906
119904) 119889119882 (119904)
119904 isin [119905infin)
119883119906
119905= 119909
(58)
Here we assume that there exists a mild solution of (58)which will be denoted by 119883119906
(119904 119905 119909) or simply by 119883119906
(119904) Weconsider a cost function of the form
119869 (119906) = 119864int
infin
119905
119890minus120582119904
119892 (119904 119883119906
119904 119906 (119904)) 119889119904 (59)
Here 119892 is function on [0infin) times C times 119880 with real values Ourpurpose is to minimize the function 119869 over all admissiblecontrols
We define in a classical way the Hamiltonian functionrelative to the previous problem for all 119905 isin [0infin) 119909 isin
C and 119911 isin Ξlowast
120595 (119905 119909 119911) = inf 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) 119906 isin 119880
(60)
and the corresponding possibly empty set of minimizers
Γ (119905 119909 119911) = 119906 isin 119880 119892 (119905 119909 119906) + 119911119877 (119905 119909 119906) = 120595 (119905 119909 119911)
(61)
We are now ready to formulate the assumptions we need
Hypothesis 3 (i) 119860 119865 and G verify Hypothesis 1(ii) (119880U) is ameasurable spaceThemap 119892 [0infin)timesCtimes
119880 rarr 119877 is continuous and satisfies |119892(119905 119909 119906)| le 119870119892(1+|119909|
119898119892
119862)
for suitable constants 119870119892gt 0 119898
119892gt 0 and all 119909 isin C119906 isin
119880 The map 119877 [0infin) times C times 119880 rarr Ξ is measurable and|119877(119905 119904 119906)| le 119871
119877for a suitable constant 119870
119877gt 0 and all 119909 isin
C119906 isin 119880 and119911 isin Ξlowast
(iii)TheHamiltonian120595 defined in (60) satisfies the requi-rements of Hypothesis 2 (with119870 = 119877)
(iv) We fix here 119901 gt 2 q and 120575 lt 0 satisfying (39) with120578 = 120578(119902) and such that 119902 gt 119898
119892
We are in a position to prove the main result of thissection
Theorem11 Assume thatHypothesis 3 holds and suppose that120582 verifies
120582 gt (minus120575 minus 120583 +1198712
119911
2) or (minus120575 +
1198712
119877
2 (119901 minus 1))
or (1198712
119877119898119892
2 (119902 minus 119898119892)minus 120578 (119902)119898
119892)
(62)
Let 120592 120577 denote the function in the statement of Theorem 10Then for every admissible control 119906 and for the correspondingtrajectory119883 starting at (119905 119909) one has
119869 (119906) = 120592 (119905 119909) + 119864int
infin
119905
119890minus120582119904
[minus120595 (119904 119883119906
119904 120577 (119904 119883
119906
119904)) + 120577 (119904 119883
119906
119904)
times 119877 (119904 119883119906
119904 119906 (119904)) + 119892 (119904 119883
119906
119904 119906 (119904))] 119889119904
(63)
Proof Consider (58) in the probability space (ΩF 119875) withfiltration F
119905119905ge0
and with an F119905119905ge0
-cylindrical Wienerprocess 119882(119905) 119905 ge 0 Let us define
119882119906
(119904) = 119882 (119904) + int
119904
119905and119904
119877 (120590119883119906
120590 119906 (120590)) 119889120590 119904 isin [0infin)
120588 (119879) = exp(int119879
119905
minus119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882 (119904)
minus1
2int
119879
119905
1003816100381610038161003816119877 (119904 119883119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)
(64)
10 Abstract and Applied Analysis
Let 119875119906 be the unique probability onF[0infin)
such that
119875119906
|F119879
= 120588 (119879) 119875|F119879
(65)
We notice that under 119875119906 the process119882119906 is aWiener processLet us denote by F119906
119905119905ge0
the filtration generated by119882119906 andcompleted in the usual way Relatively to 119882119906 (58) can berewritten as
119889119883119906
(119904) = 119860119883119906
(119904) 119889119904 + 119865 (119904 119883119906
119904) 119889119904
+ 119866 (119904 119883119906
119904) 119889119882
119906
(119904) 119904 isin [119905infin)
119883119906
119905= 119909
(66)
In the space (ΩF[0infin)
F119906
119905119905ge0 119875
119906
) we consider the follow-ing system of forward-backward equations
119883119906
(119904) = 119890(119904minus119905)119860
119909 (0) + int
119904
119905
119890(119904minus120590)119860
119865 (120590119883119906
120590) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883119906
120590) 119889119882
119906
(120590) 119904 isin [119905infin)
119883119906
119905= 119909 isin C
119884119906
(119904) minus 119884119906
(119879) + int
119879
119904
119885119906
(120590) 119889119882119906
(120590) + 120582int
119879
119904
119884119906
(120590) 119889120590
= int
119879
119904
120595 (120590119883119906
120590 119885
119906
(120590)) 119889120590 0 le 119904 le 119879 lt infin
(67)
Applying the Ito formula to 119890minus120582119904119884119906(119904) and writing the back-ward equation in (67) with respect to the process119882 we get
119884119906
(119904) + int
119879
119904
119890minus120582120590
119885119906
(120590) 119889119882 (120590)
= int
119879
119904
119890minus120582120590
[120595 (120590119883119906
120590 119885
119906
(120590))
minus119885119906
(120590) 119877 (120590119883119906
120590 119906 (120590))] 119889120590
+ 119890minus120582119879
119884119906
(119879)
(68)
Recalling that 119877 is bounded we get for all 119903 ge 1 and someconstant 119862
119864119906
[120588(119879)minus119903
] = 119864119906
[exp 119903 (int119879
119905
119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882
119906
(119904)
minus1
2int
119879
119905
1003816100381610038161003816119877 (119904 119883119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)]
= 119864119906
[exp(int119879
119905
119903119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882
119906
(119904)
minus1
2int
119879
119905
11990321003816100381610038161003816119877 (119904 119883
119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)
times exp 119903 (119903 minus 1)2
int
119879
119905
1003816100381610038161003816119877 (119904 119883119906
119904 119906 (119904))
10038161003816100381610038162
119889119904]
le 119890(12)119903(119903minus1)119879119871
2
119877119864119906
times exp(int119879
119905
2119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882
119906
(119904)
minus1
2int
119879
119905
41003816100381610038161003816119877 (119904 119883
119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)
= 119890(12)119903(119903minus1)119879119871
2
119877
(69)
It follows that
119864(int
119879
119905
|119890minus120582119904
119885119906
(119904)|2
119889119904)
12
= 119864119906
[(int
119879
119905
10038161003816100381610038161003816119890minus120582119904
119885119906
(119904)10038161003816100381610038161003816
2
119889119904)
12
120588minus1
]
le (119864119906
int
119879
119905
10038161003816100381610038161003816119890minus120582119904
119885119906
(119904)10038161003816100381610038161003816
2
119889119904)
12
times (119864119906
120588minus2
)12
lt infin
(70)
We conclude that the stochastic integral in (68) has zeroexpectation If we set 119904 = 119905 in (68) and we take expectationwith respect to 119875 we obtain
119890minus120582119879
119864119884119906
(119879) minus 119884119906
(119905)
= 119864int
119879
119905
119890minus120582120590
[minus120595 (120590119883119906
120590 119885
119906
(120590))
+119885119906
(120590) 119877 (120590119883119906
120590 119906 (120590))] 119889120590
(71)
ByTheorem 7 119884119906(sdot 119905 119909) isin 119871119901P(Ω 119862
120575(119877)) so that
119864119906
|119884(119879 119905 119909)|119901
le 119862 exp (minus119901120575119879) (72)
By the Holder inequality we have that for suitable constant119862 gt 0
119864 |119884 (119879 119905 119909)| = 119864119906
(120588minus1
(119879) |119884 (119879 119905 119909)|)
le 119864(120588minus119901(119901minus1)
)(119901minus1)119901
119864(|119884 (119879 119905 119909)|119901
)1119901
le 119862119890((1198712
1198772(119901minus1))minus120575))119879
(73)
From Theorem 2 we obtain 119864119906sup119904ge119905119890120578119902119904
|119883119906
119904|119902
lt infin by thesimilar process we get that
1198641003816100381610038161003816119883
119906
119879
1003816100381610038161003816119898119892
le 119862119890(1198712
119877119898119892(2119902minus2119898
119892)minus1
minus120578(119902)119898119892)119879
(74)
Abstract and Applied Analysis 11
for suitable constant 119862 gt 0 and
119864int
infin
119905
119890minus120582120590 1003816100381610038161003816119892 (120590119883
119906
120590 119906 (120590))
1003816100381610038161003816 119889120590 lt infin (75)
Since 119884119906(119905 119905 119909) = 120592(119905 119909) and 119885119906
(119904 119905 119909) = 120577(119904 119883119906
119904(119905 119909)) 119875-
as for aa 119904 isin [119905infin) we have that
119890minus120582119879
119864119884119906
(119879) minus 119907 (119905 119909)
= 119864int
119879
119905
119890minus120582120590
[minus120595 (120590119883119906
120590 120577 (120590 119883
119906
120590))
+120577 (120590119883119906
120590) 119877 (120590119883
119906
120590 119906 (120590))] 119889120590
(76)
Thus adding and subtracting119864intinfin119905
119890minus120582120590
119892(120590119883119906
120590 119906(120590))119889120590 and
letting 119879 rarr infin we conclude that
119869 (119906) = 120592 (119905 119909)
+ 119864int
infin
119905
119890minus120582119904
[minus120595 (119904 119883119906
119904 120577 (119904 119883
119906
119904)) + 120577 (119904 119883
119906
119904) 119877
times (119904 119883119906
119904 119906 (119904)) + 119892 (119904 119883
119906
119904 119906 (119904))] 119889119904
(77)
The proof is finished
We immediately deduce the following consequences
Theorem 12 Let 119905 isin [0infin) and 119909 isin C be fixed assumethat the set-valued map Γ has nonempty values and it admitsa measurable selection Γ
0 [0infin)timesCtimesΞlowast rarr 119880 and assume
that a control 119906(sdot) satisfies
119906 (119904) = Γ0(119904 119883
119906
119904 120577 (119904 119883
119906
119904))
119875-as for almost every 119904 isin [119905infin)
(78)
Then 119869(119905 119909 119906) = 120592(119905 119909) and the pair (119906(sdot) 119883) is optimal forthe control problem starting from 119909 at time 119905
Such a control can be shown to exist if there exists a solutionfor the so-called closed-loop equation as follows
119889119883 (119904)=119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904+119866 (119904 119883
119904)
times(119877 (119904 119883119904 Γ
0(119904 119883
119904 120577 (119904 119883
119904))) 119889119904 + 119889119882 (119904))
119904 isin [119905infin)
119883119905= 119909
(79)
since in this case we can define an optimal control setting
119906 (119904) = Γ0(119904 119883
119904 120577 (119904 119883
119904)) (80)
However under the present assumptions we cannot guaranteethat the closed-loop equation has a solution in the mildsense To circumvent this difficulty we will revert to a weakformulation of the optimal control problem
5 Existence of Optimal Control
We formulate the optimal control problem in the weak sensefollowing the approach of [31]Themain advantage is that wewill be able to solve the closed-loop equation in a weak senseand hence to find an optimal control even if the feedbacklaw is nonsmooth
We call (ΩF F119905119905ge0 119875119882) an admissible setup if
(ΩF F119905119905ge0 119875) is a filtered probability space satisfying the
usual conditions and 119882 is a cylindrical 119875-Wiener processwith values in Ξ with respect to the filtration F
119905119905ge0
By an admissible control system we mean (ΩF
F119905119905ge0 119875119882 119906119883
119906
) where (ΩF F119905119905ge0 119875 119882) is an
admissible setup 119906 is an F119905-predictable process with values
in119880 and119883119906 is a mild solution of (58) An admissible controlsystem will be briefly denoted by (119882 119906119883119906
) in the followingOur purpose is to minimize the cost functional
119869 (119906) = 119864int
infin
119905
119890minus120582119904
119892 (119904 119883119906
119904 119906 (119904)) 119889119904 (81)
over all the admissible control systemOur main result in this section is based on the solvability
of the closed-loop equation
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904 + 119866 (119904 119883
119904)
times (119877 (119904 119883119904 Γ
0(119904 119883
119904 120577 (119904 119883
119904))) 119889119904 +119889119882 (119904))
119904 isin [119905infin)
119883119905= 119909
(82)
In the following sense we say that 119883 is a weak solution of(82) if there exists an admissible setup (ΩF F
119905119905ge0 119875119882)
and anF119905-adapted continuous process119883(119905)with values in119867
which solves the equation in the mild sense namely 119875-as
119883 (119904) = 119890(119904minus119905)119860
119909 (0) + int
119904
119905
119890(119904minus120590)119860
119865 (120590119883120590) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883120590) 119877
times (120590119883120590 Γ
0(120590119883
120590 120577 (120590 119883
120590))) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883120590) 119889119882
120590 119904 isin [119905infin)
(83)
119883119905= 119909 (84)
Theorem 13 Assume that Hypothesis 3 holds Then thereexists a weak solution of the closed-loop equation (82) whichis unique in law
12 Abstract and Applied Analysis
Proof (uniqueness) Let 119883 be a weak solution of (82) in anadmissible setup (ΩF F
119905119905ge0 119875119882)We define
120588 (119879) = exp(int119879
119905
minus119877lowast
(120590 119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889119882 (120590)
minus1
2int
119879
119905
1003816100381610038161003816119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)))10038161003816100381610038162
119889120590)
(85)
Since 119877 is bounded the Girsanov theorem ensures that thereexists a probability measure 1198750 such that the process
1198820
(119904) = 119882 (119904) + int
119904
119905and119904
119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889120590
119904 isin [0infin)
(86)
is a 1198750-Wiener process and
1198750
|F119879
= 120588 (119879) 119875|F119879
(87)
Let us denote by F0
119905119905ge0
the filtration generated by1198820 andcompleted in the usual way In (ΩF
[0infin) F0
119905119905ge0 119875
0
) 119883 isa mild solution of
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119905 119883119904) 119889119904
+ 119866 (119904 119883119904) 119889119882
0
(119904) 119904 isin [119905infin)
119883119905= 119909
120588 (119879) = exp(int119879
119905
minus119877lowast
(120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)) 119889119882
0
(120590)
+ 12int
119879
119905
1003816100381610038161003816119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)))10038161003816100381610038162
119889120590)
(88)
By Hypothesis 3 the joint law of 119883 and 1198820 is uniquely
determined by 119860 119865 119866 and 119909 Taking into account the lastdisplayed formula we conclude that the joint law of 119883 and120588(119879) under1198750 is also uniquely determined and consequentlyso is the law of 119883 under 119875 This completes the proof of theuniqueness part
Proof (existence) Let (ΩF 119875) be a given complete probabil-ity space 119882(119905) 119905 ge 0 is a cylindrical Wiener process on(ΩF 119875)with values inΞ and F
119905119905ge0
is the natural filtrationof 119882(119905) 119905 ge 0 augmented with the family of119875-null sets Let119883(sdot) be the mild solution of
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904
+ 119866 (119904 119883119904) 119889119882 (119904) 119904 isin [119905infin)
119883119905= 119909
(89)
and by the Girsanov theorem let 1198751 be the probability on Ωunder which
1198821
(119904) = 119882 (119904) minus int
119904
119905and119904
119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889120590
(90)
is aWiener process (notice that 119877 is bounded)Then119883 is theweak solution of (82) relatively to the probability 1198751 and theWiener process1198821
Now we can state the main result of this section
Corollary 14 Assume that Hypothesis 3 holds true and 120582
verifies (62) Also assume that the set-valued map Γ hasnonempty values and it admits a measurable selection Γ
0
[0infin) timesC times Ξlowast
rarr 119880 Then for every 119905 isin [0infin) and x isin Cand for all admissible control system (119882 119906119883
119906
) one has
119869 (119906 119905 119909) ge 120592 (119905 119909) (91)
and the equality holds if
119906 (119904) = Γ0(119904 119883
119906
119904 120577 (119904 119883
119906
119904))
119875 minus 119886119904 119891119900119903 119886119897119898119900119904119905 119890119907119890119903119910 119904 isin [119905infin)
(92)
Moreover from Theorem 13 it follows that the closed-loop equation (82) admits a weak solution (ΩFF
119905119905ge0 119875119882119883) which is unique in law and setting
119906 (119904) = Γ0(119904 119883
119904 120577 (119904 119883
119904)) (93)
we obtain an optimal admissible control system (119882 119906119883)
6 Applications
In this section we present a simple application of the previousresults We consider the stochastic delay partial differentialequation in the bounded domain 119861 sub 119877
119899 with smoothboundary 120597119861 as follows
119889119911119906
(119905 120585) = Δ119911119906
(119905 120585) 119889119905 + 119891 (119905 119911119906
119905(120585)) 119889119905
+
119889
sum
119894=1
119892119894(119905 119911
119906
119905(120585)) [119903
119894
(120585) 119906119894
(119905) 119889119905 + 119889119882119894
(119905)]
119911119906
0(120579 120585) = 119909 (120579 120585) 120585 isin 119861 120579 isin [minus1 0]
119911119906
(119905 120585) = 0 119905 isin [0infin) 120585 isin 120597119861
(94)
Here119882 = (1198821
1198822
119882119889
) is a standardWiener process in119877119889 and the functions 119891 [0 +infin) times 119862([minus1 0] 119877) rarr 119877 and119892119894 [0 +infin) times 119862([minus1 0] 119877) rarr 119877 are Lipschitz continuous
and bounded Setting 119880 as a bounded subset of 119877119889 Ξ = 119877119889
119867 = 1198712
(119861) and 119909 isin 119862([minus1 0]119867) We define 119865 and 119866 asfollowing
119865 (119905 119909) (120585) = 119891 (119905 119909 (120585))
(119866 (119905 119909) 119911) (120585) =
119889
sum
119894=1
119892119894(119905 119909 (120585)) 119911
119894
(120585)
120585 isin 119861 119909 isin 119862 ([minus1 0] 119867) 119911 isin 119871 (Ξ119867)
(95)
and let 119860 denote the Laplace operator Δ in 1198712
(119861) withdomain11988222
(119861)⋂11988212
0(119861) then (94) has the form (58) and
Hypothesis 1 holds
Abstract and Applied Analysis 13
Let us consider the optimal control problem associatedwith the cost
119869 (119906) = 119864int
infin
0
119890minus120582119905
[int119861
120590 (120585 119911119906
119905(120585)) 119889120585 + 119906
2
(119905)] 119889119905 (96)
where 120582 verifies (62) and 120590 119862([minus1 0] 119877) times 119880 rarr [0infin) isa bounded measurable function Define 119892 119862([minus1 0]119867) times
119880 rarr [0infin) and 119877 119862([minus1 0]119867) times 119880 rarr Ξ by119892(119910 119906) = int
119861
120590(119905 119910(120585) 119906)119889120585 + 1199062 and 119877(119910 119906) = (int
119861
1199031
(120585)1199061
119889120585
int119861
1199032
(120585)1199062
119889120585 int119861
119903119889
(120585)119906119889
119889120585) for 119910 isin 119862([minus1 0]119867) 119906 =
(1199061
1199062
119906119889
) isin 119880 respectively It can be easily verifiedthat Hypothesis 3 holds true and the set-valued map Γ hasnonempty values and it admits a measurable selection Γ
0
[0infin) times C times Ξlowast
rarr 119880 Then the closed-loop equation(82) admits a weak solution (ΩF F
119905119905ge0 119875119882 119906 119911
sdot(sdot)) and
setting
119906 (119904) = Γ0(119904 119911
119904(sdot) 120577 (119904 119911
119904(sdot))) (97)
we obtain an optimal admissible control system (119882 119906 119911(sdot))
References
[1] J Bismut ldquoOn optimal control of linear stochastic equationswith a linear-quadratic criterionrdquo SIAM Journal on Control andOptimization vol 15 no 3 pp 1ndash4 1977
[2] N Nagase ldquoOn the existence of optimal control for controlledstochastic partial differential equationsrdquo Nagoya MathematicsJournal vol 115 pp 73ndash85 1989
[3] N El Karoui D Huu Nguyen and M Jeanblanc-Pique ldquoCom-pactification methods in the control of degenerate diffusionsrdquoStochastics vol 20 pp 169ndash219 1987
[4] M Nisio ldquoOptimal control for stochastic partial differentialequations and viscosity solutions of Bellman equationsrdquoNagoyaMathematics Journal vol 123 pp 13ndash37 1991
[5] M Nisio ldquoOn sensitive control for stochastic partial differentialequationsrdquo in Stochastic Analysis on Infinite Dimensional SpacesProceedings of the US Japan Bilateral Seminar H Kunita et alEd vol 310 of Pitman Research Notes Mathematical Series pp231ndash241 Longman Scientific and Technical Baton Rouge LaUSA January 1994
[6] R Buckdahn and A Rascanu ldquoOn the existence of stochasticoptimal control of distributed state systemrdquoNonlinear AnalysisTheory Methods and Applications vol 52 no 4 pp 1153ndash11842003
[7] V Barbu and G Da Prato Equations in Hilbert Spaces vol 86 ofPitman Research Notes in Mathematics Pitman 1983
[8] P Cannarsa and G Da Prato ldquoSecond-order Hamilton-Jacobiequations in infinite dimensionsrdquo SIAM Journal on Control andOptimization vol 29 no 2 pp 474ndash492 1991
[9] P Cannarsa and G Da Prato ldquoDirect solution of a second-orderHamilton-Jacobi equations in Hilbert spacesrdquo in StochasticPartial Differential Equations and Applications G Da Pratoand L Tubaro Eds vol 268 of Pitman Research Notes inMathematics Pitman 1992
[10] F Gozzi ldquoRegularity of solutions of second order Hamilton-Jacobi equations and application to a control problemrdquoCommu-nications in Partial Differential Equations vol 20 pp 775ndash8261995
[11] F Gozzi ldquoGlobal regular solutions of second order Hamilton-Jacobi equations in Hilbert spaces with locally Lipschitz non-linearitiesrdquo Journal of Mathematical Analysis and Applicationsvol 198 no 2 pp 399ndash443 1996
[12] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems and Control Lettersvol 14 no 1 pp 55ndash61 1990
[13] N El Karoui and LMazliak Eds Backward Stochastic Differen-tial Equations vol 364 of Pitman ResearchNotes inMathematicsSeries Longman 1997
[14] E Pardoux and BSDEs ldquoweak convergence and homogeneiza-tion of semilinear PDEsrdquo in Non- Linear Analysis DifferentialEquations and Control F H Clarke and R J Stern Eds pp503ndash549 Kluwer Dordrecht The Netherlands 1999
[15] S Peng ldquoA generalized dynamic programming principle andHamilton-Jacobi-Bellman equationrdquo Stochastics and StochasticsReports vol 38 pp 119ndash134 1992
[16] N E Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997
[17] SHamad120583ene and J P Lepeltier ldquoBackward equations stochas-tic control and zero-sum stochastic differential gamesrdquo Stochas-tics and Stochastics Reports vol 54 pp 221ndash231 1995
[18] N El-Karoui and S Hamadene ldquoBSDEs and risk-sensitive con-trol zero-sum and nonzero-sum game problems of stochasticfunctional differential equationsrdquo Stochastic Processes and theirApplications vol 107 no 1 pp 145ndash169 2003
[19] M Fuhrman and G Tessiture ldquoExistence of optimal stochasticcontrols and global solutions of forward-backward stochasticdifferential equationsrdquo SIAM Journal on Control and Optimiza-tion vol 43 no 3 pp 813ndash830 2005
[20] M Fuhrman Y Hu and G Tessitore ldquoOn a class of stochasticoptimal control problems related to bsdes with quadraticgrowthrdquo SIAM Journal on Control and Optimization vol 45 no4 pp 1279ndash1296 2006
[21] M Fuhrman and G Tessitore ldquoNonlinear kolmogorov equa-tions in infinite dimensional spaces the backward stochasticdifferential equations approach and applications to optimalcontrolrdquoAnnals of Probability vol 30 no 3 pp 1397ndash1465 2002
[22] F Masiero ldquoSemilinear kolmogorov equations and applicationsto stochastic optimal controlrdquo Applied Mathematics and Opti-mization vol 51 no 1 pp 201ndash250 2005
[23] M Fuhrman FMasiero andG Tessitore ldquoStochastic equationswith delay optimal control via BSDEs and regular solutions ofHamilton-jacobi-bellman equationsrdquo SIAM Journal on Controland Optimization vol 48 no 7 pp 4624ndash4651 2010
[24] M Fuhrman and G Tessiture ldquoInfinite horizon backwardstochastic differential equations and elliptic equations in hilbertspacesrdquo Annals of Probability vol 32 no 1 pp 607ndash660 2004
[25] F Masiero ldquoInfinite horizon stochastic optimal control prob-lems with degenerate noise and elliptic equations in Hilbertspacesrdquo Applied Mathematics and Optimization vol 55 no 3pp 285ndash326 2007
[26] M Fuhrman ldquoA class of stochastic optimal control problemsin Hilbert spaces BSDEs and optimal control laws stateconstraints conditioned processesrdquo Stochastic Processes andtheir Applications vol 108 no 2 pp 263ndash298 2003
[27] F Masiero ldquoStochastic optimal control problems and parabolicequations in banach spacesrdquo SIAM Journal on Control andOptimization vol 47 no 1 pp 251ndash300 2008
[28] G Da Prato and J Zabczyk Ergodicity For Infinite-DimensionalSystems Cambridge University Press 1996
14 Abstract and Applied Analysis
[29] J Zabczyk ldquoParabolic equations on Hilbert spacesrdquo in StochaS-tic PDErsquoS and Kolmogorov Equations in Infinite Dimensionsvol 1715 of Lecture Notes in Math pp 117ndash213 Springer BerlinGermany 1999
[30] G Da Prato and J Zabczyk Stochstic Equations in InfiniteDimensions Cambridge University Press 1992
[31] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 of Applications of MathematicsSpringer New York NY USA 1993
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
Volume 2014
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OptimizationJournal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Abstract and Applied Analysis
Let 119875119906 be the unique probability onF[0infin)
such that
119875119906
|F119879
= 120588 (119879) 119875|F119879
(65)
We notice that under 119875119906 the process119882119906 is aWiener processLet us denote by F119906
119905119905ge0
the filtration generated by119882119906 andcompleted in the usual way Relatively to 119882119906 (58) can berewritten as
119889119883119906
(119904) = 119860119883119906
(119904) 119889119904 + 119865 (119904 119883119906
119904) 119889119904
+ 119866 (119904 119883119906
119904) 119889119882
119906
(119904) 119904 isin [119905infin)
119883119906
119905= 119909
(66)
In the space (ΩF[0infin)
F119906
119905119905ge0 119875
119906
) we consider the follow-ing system of forward-backward equations
119883119906
(119904) = 119890(119904minus119905)119860
119909 (0) + int
119904
119905
119890(119904minus120590)119860
119865 (120590119883119906
120590) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883119906
120590) 119889119882
119906
(120590) 119904 isin [119905infin)
119883119906
119905= 119909 isin C
119884119906
(119904) minus 119884119906
(119879) + int
119879
119904
119885119906
(120590) 119889119882119906
(120590) + 120582int
119879
119904
119884119906
(120590) 119889120590
= int
119879
119904
120595 (120590119883119906
120590 119885
119906
(120590)) 119889120590 0 le 119904 le 119879 lt infin
(67)
Applying the Ito formula to 119890minus120582119904119884119906(119904) and writing the back-ward equation in (67) with respect to the process119882 we get
119884119906
(119904) + int
119879
119904
119890minus120582120590
119885119906
(120590) 119889119882 (120590)
= int
119879
119904
119890minus120582120590
[120595 (120590119883119906
120590 119885
119906
(120590))
minus119885119906
(120590) 119877 (120590119883119906
120590 119906 (120590))] 119889120590
+ 119890minus120582119879
119884119906
(119879)
(68)
Recalling that 119877 is bounded we get for all 119903 ge 1 and someconstant 119862
119864119906
[120588(119879)minus119903
] = 119864119906
[exp 119903 (int119879
119905
119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882
119906
(119904)
minus1
2int
119879
119905
1003816100381610038161003816119877 (119904 119883119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)]
= 119864119906
[exp(int119879
119905
119903119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882
119906
(119904)
minus1
2int
119879
119905
11990321003816100381610038161003816119877 (119904 119883
119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)
times exp 119903 (119903 minus 1)2
int
119879
119905
1003816100381610038161003816119877 (119904 119883119906
119904 119906 (119904))
10038161003816100381610038162
119889119904]
le 119890(12)119903(119903minus1)119879119871
2
119877119864119906
times exp(int119879
119905
2119877lowast
(119904 119883119906
119904 119906 (119904)) 119889119882
119906
(119904)
minus1
2int
119879
119905
41003816100381610038161003816119877 (119904 119883
119906
119904 119906 (119904))
10038161003816100381610038162
119889119904)
= 119890(12)119903(119903minus1)119879119871
2
119877
(69)
It follows that
119864(int
119879
119905
|119890minus120582119904
119885119906
(119904)|2
119889119904)
12
= 119864119906
[(int
119879
119905
10038161003816100381610038161003816119890minus120582119904
119885119906
(119904)10038161003816100381610038161003816
2
119889119904)
12
120588minus1
]
le (119864119906
int
119879
119905
10038161003816100381610038161003816119890minus120582119904
119885119906
(119904)10038161003816100381610038161003816
2
119889119904)
12
times (119864119906
120588minus2
)12
lt infin
(70)
We conclude that the stochastic integral in (68) has zeroexpectation If we set 119904 = 119905 in (68) and we take expectationwith respect to 119875 we obtain
119890minus120582119879
119864119884119906
(119879) minus 119884119906
(119905)
= 119864int
119879
119905
119890minus120582120590
[minus120595 (120590119883119906
120590 119885
119906
(120590))
+119885119906
(120590) 119877 (120590119883119906
120590 119906 (120590))] 119889120590
(71)
ByTheorem 7 119884119906(sdot 119905 119909) isin 119871119901P(Ω 119862
120575(119877)) so that
119864119906
|119884(119879 119905 119909)|119901
le 119862 exp (minus119901120575119879) (72)
By the Holder inequality we have that for suitable constant119862 gt 0
119864 |119884 (119879 119905 119909)| = 119864119906
(120588minus1
(119879) |119884 (119879 119905 119909)|)
le 119864(120588minus119901(119901minus1)
)(119901minus1)119901
119864(|119884 (119879 119905 119909)|119901
)1119901
le 119862119890((1198712
1198772(119901minus1))minus120575))119879
(73)
From Theorem 2 we obtain 119864119906sup119904ge119905119890120578119902119904
|119883119906
119904|119902
lt infin by thesimilar process we get that
1198641003816100381610038161003816119883
119906
119879
1003816100381610038161003816119898119892
le 119862119890(1198712
119877119898119892(2119902minus2119898
119892)minus1
minus120578(119902)119898119892)119879
(74)
Abstract and Applied Analysis 11
for suitable constant 119862 gt 0 and
119864int
infin
119905
119890minus120582120590 1003816100381610038161003816119892 (120590119883
119906
120590 119906 (120590))
1003816100381610038161003816 119889120590 lt infin (75)
Since 119884119906(119905 119905 119909) = 120592(119905 119909) and 119885119906
(119904 119905 119909) = 120577(119904 119883119906
119904(119905 119909)) 119875-
as for aa 119904 isin [119905infin) we have that
119890minus120582119879
119864119884119906
(119879) minus 119907 (119905 119909)
= 119864int
119879
119905
119890minus120582120590
[minus120595 (120590119883119906
120590 120577 (120590 119883
119906
120590))
+120577 (120590119883119906
120590) 119877 (120590119883
119906
120590 119906 (120590))] 119889120590
(76)
Thus adding and subtracting119864intinfin119905
119890minus120582120590
119892(120590119883119906
120590 119906(120590))119889120590 and
letting 119879 rarr infin we conclude that
119869 (119906) = 120592 (119905 119909)
+ 119864int
infin
119905
119890minus120582119904
[minus120595 (119904 119883119906
119904 120577 (119904 119883
119906
119904)) + 120577 (119904 119883
119906
119904) 119877
times (119904 119883119906
119904 119906 (119904)) + 119892 (119904 119883
119906
119904 119906 (119904))] 119889119904
(77)
The proof is finished
We immediately deduce the following consequences
Theorem 12 Let 119905 isin [0infin) and 119909 isin C be fixed assumethat the set-valued map Γ has nonempty values and it admitsa measurable selection Γ
0 [0infin)timesCtimesΞlowast rarr 119880 and assume
that a control 119906(sdot) satisfies
119906 (119904) = Γ0(119904 119883
119906
119904 120577 (119904 119883
119906
119904))
119875-as for almost every 119904 isin [119905infin)
(78)
Then 119869(119905 119909 119906) = 120592(119905 119909) and the pair (119906(sdot) 119883) is optimal forthe control problem starting from 119909 at time 119905
Such a control can be shown to exist if there exists a solutionfor the so-called closed-loop equation as follows
119889119883 (119904)=119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904+119866 (119904 119883
119904)
times(119877 (119904 119883119904 Γ
0(119904 119883
119904 120577 (119904 119883
119904))) 119889119904 + 119889119882 (119904))
119904 isin [119905infin)
119883119905= 119909
(79)
since in this case we can define an optimal control setting
119906 (119904) = Γ0(119904 119883
119904 120577 (119904 119883
119904)) (80)
However under the present assumptions we cannot guaranteethat the closed-loop equation has a solution in the mildsense To circumvent this difficulty we will revert to a weakformulation of the optimal control problem
5 Existence of Optimal Control
We formulate the optimal control problem in the weak sensefollowing the approach of [31]Themain advantage is that wewill be able to solve the closed-loop equation in a weak senseand hence to find an optimal control even if the feedbacklaw is nonsmooth
We call (ΩF F119905119905ge0 119875119882) an admissible setup if
(ΩF F119905119905ge0 119875) is a filtered probability space satisfying the
usual conditions and 119882 is a cylindrical 119875-Wiener processwith values in Ξ with respect to the filtration F
119905119905ge0
By an admissible control system we mean (ΩF
F119905119905ge0 119875119882 119906119883
119906
) where (ΩF F119905119905ge0 119875 119882) is an
admissible setup 119906 is an F119905-predictable process with values
in119880 and119883119906 is a mild solution of (58) An admissible controlsystem will be briefly denoted by (119882 119906119883119906
) in the followingOur purpose is to minimize the cost functional
119869 (119906) = 119864int
infin
119905
119890minus120582119904
119892 (119904 119883119906
119904 119906 (119904)) 119889119904 (81)
over all the admissible control systemOur main result in this section is based on the solvability
of the closed-loop equation
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904 + 119866 (119904 119883
119904)
times (119877 (119904 119883119904 Γ
0(119904 119883
119904 120577 (119904 119883
119904))) 119889119904 +119889119882 (119904))
119904 isin [119905infin)
119883119905= 119909
(82)
In the following sense we say that 119883 is a weak solution of(82) if there exists an admissible setup (ΩF F
119905119905ge0 119875119882)
and anF119905-adapted continuous process119883(119905)with values in119867
which solves the equation in the mild sense namely 119875-as
119883 (119904) = 119890(119904minus119905)119860
119909 (0) + int
119904
119905
119890(119904minus120590)119860
119865 (120590119883120590) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883120590) 119877
times (120590119883120590 Γ
0(120590119883
120590 120577 (120590 119883
120590))) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883120590) 119889119882
120590 119904 isin [119905infin)
(83)
119883119905= 119909 (84)
Theorem 13 Assume that Hypothesis 3 holds Then thereexists a weak solution of the closed-loop equation (82) whichis unique in law
12 Abstract and Applied Analysis
Proof (uniqueness) Let 119883 be a weak solution of (82) in anadmissible setup (ΩF F
119905119905ge0 119875119882)We define
120588 (119879) = exp(int119879
119905
minus119877lowast
(120590 119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889119882 (120590)
minus1
2int
119879
119905
1003816100381610038161003816119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)))10038161003816100381610038162
119889120590)
(85)
Since 119877 is bounded the Girsanov theorem ensures that thereexists a probability measure 1198750 such that the process
1198820
(119904) = 119882 (119904) + int
119904
119905and119904
119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889120590
119904 isin [0infin)
(86)
is a 1198750-Wiener process and
1198750
|F119879
= 120588 (119879) 119875|F119879
(87)
Let us denote by F0
119905119905ge0
the filtration generated by1198820 andcompleted in the usual way In (ΩF
[0infin) F0
119905119905ge0 119875
0
) 119883 isa mild solution of
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119905 119883119904) 119889119904
+ 119866 (119904 119883119904) 119889119882
0
(119904) 119904 isin [119905infin)
119883119905= 119909
120588 (119879) = exp(int119879
119905
minus119877lowast
(120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)) 119889119882
0
(120590)
+ 12int
119879
119905
1003816100381610038161003816119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)))10038161003816100381610038162
119889120590)
(88)
By Hypothesis 3 the joint law of 119883 and 1198820 is uniquely
determined by 119860 119865 119866 and 119909 Taking into account the lastdisplayed formula we conclude that the joint law of 119883 and120588(119879) under1198750 is also uniquely determined and consequentlyso is the law of 119883 under 119875 This completes the proof of theuniqueness part
Proof (existence) Let (ΩF 119875) be a given complete probabil-ity space 119882(119905) 119905 ge 0 is a cylindrical Wiener process on(ΩF 119875)with values inΞ and F
119905119905ge0
is the natural filtrationof 119882(119905) 119905 ge 0 augmented with the family of119875-null sets Let119883(sdot) be the mild solution of
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904
+ 119866 (119904 119883119904) 119889119882 (119904) 119904 isin [119905infin)
119883119905= 119909
(89)
and by the Girsanov theorem let 1198751 be the probability on Ωunder which
1198821
(119904) = 119882 (119904) minus int
119904
119905and119904
119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889120590
(90)
is aWiener process (notice that 119877 is bounded)Then119883 is theweak solution of (82) relatively to the probability 1198751 and theWiener process1198821
Now we can state the main result of this section
Corollary 14 Assume that Hypothesis 3 holds true and 120582
verifies (62) Also assume that the set-valued map Γ hasnonempty values and it admits a measurable selection Γ
0
[0infin) timesC times Ξlowast
rarr 119880 Then for every 119905 isin [0infin) and x isin Cand for all admissible control system (119882 119906119883
119906
) one has
119869 (119906 119905 119909) ge 120592 (119905 119909) (91)
and the equality holds if
119906 (119904) = Γ0(119904 119883
119906
119904 120577 (119904 119883
119906
119904))
119875 minus 119886119904 119891119900119903 119886119897119898119900119904119905 119890119907119890119903119910 119904 isin [119905infin)
(92)
Moreover from Theorem 13 it follows that the closed-loop equation (82) admits a weak solution (ΩFF
119905119905ge0 119875119882119883) which is unique in law and setting
119906 (119904) = Γ0(119904 119883
119904 120577 (119904 119883
119904)) (93)
we obtain an optimal admissible control system (119882 119906119883)
6 Applications
In this section we present a simple application of the previousresults We consider the stochastic delay partial differentialequation in the bounded domain 119861 sub 119877
119899 with smoothboundary 120597119861 as follows
119889119911119906
(119905 120585) = Δ119911119906
(119905 120585) 119889119905 + 119891 (119905 119911119906
119905(120585)) 119889119905
+
119889
sum
119894=1
119892119894(119905 119911
119906
119905(120585)) [119903
119894
(120585) 119906119894
(119905) 119889119905 + 119889119882119894
(119905)]
119911119906
0(120579 120585) = 119909 (120579 120585) 120585 isin 119861 120579 isin [minus1 0]
119911119906
(119905 120585) = 0 119905 isin [0infin) 120585 isin 120597119861
(94)
Here119882 = (1198821
1198822
119882119889
) is a standardWiener process in119877119889 and the functions 119891 [0 +infin) times 119862([minus1 0] 119877) rarr 119877 and119892119894 [0 +infin) times 119862([minus1 0] 119877) rarr 119877 are Lipschitz continuous
and bounded Setting 119880 as a bounded subset of 119877119889 Ξ = 119877119889
119867 = 1198712
(119861) and 119909 isin 119862([minus1 0]119867) We define 119865 and 119866 asfollowing
119865 (119905 119909) (120585) = 119891 (119905 119909 (120585))
(119866 (119905 119909) 119911) (120585) =
119889
sum
119894=1
119892119894(119905 119909 (120585)) 119911
119894
(120585)
120585 isin 119861 119909 isin 119862 ([minus1 0] 119867) 119911 isin 119871 (Ξ119867)
(95)
and let 119860 denote the Laplace operator Δ in 1198712
(119861) withdomain11988222
(119861)⋂11988212
0(119861) then (94) has the form (58) and
Hypothesis 1 holds
Abstract and Applied Analysis 13
Let us consider the optimal control problem associatedwith the cost
119869 (119906) = 119864int
infin
0
119890minus120582119905
[int119861
120590 (120585 119911119906
119905(120585)) 119889120585 + 119906
2
(119905)] 119889119905 (96)
where 120582 verifies (62) and 120590 119862([minus1 0] 119877) times 119880 rarr [0infin) isa bounded measurable function Define 119892 119862([minus1 0]119867) times
119880 rarr [0infin) and 119877 119862([minus1 0]119867) times 119880 rarr Ξ by119892(119910 119906) = int
119861
120590(119905 119910(120585) 119906)119889120585 + 1199062 and 119877(119910 119906) = (int
119861
1199031
(120585)1199061
119889120585
int119861
1199032
(120585)1199062
119889120585 int119861
119903119889
(120585)119906119889
119889120585) for 119910 isin 119862([minus1 0]119867) 119906 =
(1199061
1199062
119906119889
) isin 119880 respectively It can be easily verifiedthat Hypothesis 3 holds true and the set-valued map Γ hasnonempty values and it admits a measurable selection Γ
0
[0infin) times C times Ξlowast
rarr 119880 Then the closed-loop equation(82) admits a weak solution (ΩF F
119905119905ge0 119875119882 119906 119911
sdot(sdot)) and
setting
119906 (119904) = Γ0(119904 119911
119904(sdot) 120577 (119904 119911
119904(sdot))) (97)
we obtain an optimal admissible control system (119882 119906 119911(sdot))
References
[1] J Bismut ldquoOn optimal control of linear stochastic equationswith a linear-quadratic criterionrdquo SIAM Journal on Control andOptimization vol 15 no 3 pp 1ndash4 1977
[2] N Nagase ldquoOn the existence of optimal control for controlledstochastic partial differential equationsrdquo Nagoya MathematicsJournal vol 115 pp 73ndash85 1989
[3] N El Karoui D Huu Nguyen and M Jeanblanc-Pique ldquoCom-pactification methods in the control of degenerate diffusionsrdquoStochastics vol 20 pp 169ndash219 1987
[4] M Nisio ldquoOptimal control for stochastic partial differentialequations and viscosity solutions of Bellman equationsrdquoNagoyaMathematics Journal vol 123 pp 13ndash37 1991
[5] M Nisio ldquoOn sensitive control for stochastic partial differentialequationsrdquo in Stochastic Analysis on Infinite Dimensional SpacesProceedings of the US Japan Bilateral Seminar H Kunita et alEd vol 310 of Pitman Research Notes Mathematical Series pp231ndash241 Longman Scientific and Technical Baton Rouge LaUSA January 1994
[6] R Buckdahn and A Rascanu ldquoOn the existence of stochasticoptimal control of distributed state systemrdquoNonlinear AnalysisTheory Methods and Applications vol 52 no 4 pp 1153ndash11842003
[7] V Barbu and G Da Prato Equations in Hilbert Spaces vol 86 ofPitman Research Notes in Mathematics Pitman 1983
[8] P Cannarsa and G Da Prato ldquoSecond-order Hamilton-Jacobiequations in infinite dimensionsrdquo SIAM Journal on Control andOptimization vol 29 no 2 pp 474ndash492 1991
[9] P Cannarsa and G Da Prato ldquoDirect solution of a second-orderHamilton-Jacobi equations in Hilbert spacesrdquo in StochasticPartial Differential Equations and Applications G Da Pratoand L Tubaro Eds vol 268 of Pitman Research Notes inMathematics Pitman 1992
[10] F Gozzi ldquoRegularity of solutions of second order Hamilton-Jacobi equations and application to a control problemrdquoCommu-nications in Partial Differential Equations vol 20 pp 775ndash8261995
[11] F Gozzi ldquoGlobal regular solutions of second order Hamilton-Jacobi equations in Hilbert spaces with locally Lipschitz non-linearitiesrdquo Journal of Mathematical Analysis and Applicationsvol 198 no 2 pp 399ndash443 1996
[12] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems and Control Lettersvol 14 no 1 pp 55ndash61 1990
[13] N El Karoui and LMazliak Eds Backward Stochastic Differen-tial Equations vol 364 of Pitman ResearchNotes inMathematicsSeries Longman 1997
[14] E Pardoux and BSDEs ldquoweak convergence and homogeneiza-tion of semilinear PDEsrdquo in Non- Linear Analysis DifferentialEquations and Control F H Clarke and R J Stern Eds pp503ndash549 Kluwer Dordrecht The Netherlands 1999
[15] S Peng ldquoA generalized dynamic programming principle andHamilton-Jacobi-Bellman equationrdquo Stochastics and StochasticsReports vol 38 pp 119ndash134 1992
[16] N E Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997
[17] SHamad120583ene and J P Lepeltier ldquoBackward equations stochas-tic control and zero-sum stochastic differential gamesrdquo Stochas-tics and Stochastics Reports vol 54 pp 221ndash231 1995
[18] N El-Karoui and S Hamadene ldquoBSDEs and risk-sensitive con-trol zero-sum and nonzero-sum game problems of stochasticfunctional differential equationsrdquo Stochastic Processes and theirApplications vol 107 no 1 pp 145ndash169 2003
[19] M Fuhrman and G Tessiture ldquoExistence of optimal stochasticcontrols and global solutions of forward-backward stochasticdifferential equationsrdquo SIAM Journal on Control and Optimiza-tion vol 43 no 3 pp 813ndash830 2005
[20] M Fuhrman Y Hu and G Tessitore ldquoOn a class of stochasticoptimal control problems related to bsdes with quadraticgrowthrdquo SIAM Journal on Control and Optimization vol 45 no4 pp 1279ndash1296 2006
[21] M Fuhrman and G Tessitore ldquoNonlinear kolmogorov equa-tions in infinite dimensional spaces the backward stochasticdifferential equations approach and applications to optimalcontrolrdquoAnnals of Probability vol 30 no 3 pp 1397ndash1465 2002
[22] F Masiero ldquoSemilinear kolmogorov equations and applicationsto stochastic optimal controlrdquo Applied Mathematics and Opti-mization vol 51 no 1 pp 201ndash250 2005
[23] M Fuhrman FMasiero andG Tessitore ldquoStochastic equationswith delay optimal control via BSDEs and regular solutions ofHamilton-jacobi-bellman equationsrdquo SIAM Journal on Controland Optimization vol 48 no 7 pp 4624ndash4651 2010
[24] M Fuhrman and G Tessiture ldquoInfinite horizon backwardstochastic differential equations and elliptic equations in hilbertspacesrdquo Annals of Probability vol 32 no 1 pp 607ndash660 2004
[25] F Masiero ldquoInfinite horizon stochastic optimal control prob-lems with degenerate noise and elliptic equations in Hilbertspacesrdquo Applied Mathematics and Optimization vol 55 no 3pp 285ndash326 2007
[26] M Fuhrman ldquoA class of stochastic optimal control problemsin Hilbert spaces BSDEs and optimal control laws stateconstraints conditioned processesrdquo Stochastic Processes andtheir Applications vol 108 no 2 pp 263ndash298 2003
[27] F Masiero ldquoStochastic optimal control problems and parabolicequations in banach spacesrdquo SIAM Journal on Control andOptimization vol 47 no 1 pp 251ndash300 2008
[28] G Da Prato and J Zabczyk Ergodicity For Infinite-DimensionalSystems Cambridge University Press 1996
14 Abstract and Applied Analysis
[29] J Zabczyk ldquoParabolic equations on Hilbert spacesrdquo in StochaS-tic PDErsquoS and Kolmogorov Equations in Infinite Dimensionsvol 1715 of Lecture Notes in Math pp 117ndash213 Springer BerlinGermany 1999
[30] G Da Prato and J Zabczyk Stochstic Equations in InfiniteDimensions Cambridge University Press 1992
[31] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 of Applications of MathematicsSpringer New York NY USA 1993
Submit your manuscripts athttpwwwhindawicom
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OptimizationJournal of
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014
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Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 11
for suitable constant 119862 gt 0 and
119864int
infin
119905
119890minus120582120590 1003816100381610038161003816119892 (120590119883
119906
120590 119906 (120590))
1003816100381610038161003816 119889120590 lt infin (75)
Since 119884119906(119905 119905 119909) = 120592(119905 119909) and 119885119906
(119904 119905 119909) = 120577(119904 119883119906
119904(119905 119909)) 119875-
as for aa 119904 isin [119905infin) we have that
119890minus120582119879
119864119884119906
(119879) minus 119907 (119905 119909)
= 119864int
119879
119905
119890minus120582120590
[minus120595 (120590119883119906
120590 120577 (120590 119883
119906
120590))
+120577 (120590119883119906
120590) 119877 (120590119883
119906
120590 119906 (120590))] 119889120590
(76)
Thus adding and subtracting119864intinfin119905
119890minus120582120590
119892(120590119883119906
120590 119906(120590))119889120590 and
letting 119879 rarr infin we conclude that
119869 (119906) = 120592 (119905 119909)
+ 119864int
infin
119905
119890minus120582119904
[minus120595 (119904 119883119906
119904 120577 (119904 119883
119906
119904)) + 120577 (119904 119883
119906
119904) 119877
times (119904 119883119906
119904 119906 (119904)) + 119892 (119904 119883
119906
119904 119906 (119904))] 119889119904
(77)
The proof is finished
We immediately deduce the following consequences
Theorem 12 Let 119905 isin [0infin) and 119909 isin C be fixed assumethat the set-valued map Γ has nonempty values and it admitsa measurable selection Γ
0 [0infin)timesCtimesΞlowast rarr 119880 and assume
that a control 119906(sdot) satisfies
119906 (119904) = Γ0(119904 119883
119906
119904 120577 (119904 119883
119906
119904))
119875-as for almost every 119904 isin [119905infin)
(78)
Then 119869(119905 119909 119906) = 120592(119905 119909) and the pair (119906(sdot) 119883) is optimal forthe control problem starting from 119909 at time 119905
Such a control can be shown to exist if there exists a solutionfor the so-called closed-loop equation as follows
119889119883 (119904)=119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904+119866 (119904 119883
119904)
times(119877 (119904 119883119904 Γ
0(119904 119883
119904 120577 (119904 119883
119904))) 119889119904 + 119889119882 (119904))
119904 isin [119905infin)
119883119905= 119909
(79)
since in this case we can define an optimal control setting
119906 (119904) = Γ0(119904 119883
119904 120577 (119904 119883
119904)) (80)
However under the present assumptions we cannot guaranteethat the closed-loop equation has a solution in the mildsense To circumvent this difficulty we will revert to a weakformulation of the optimal control problem
5 Existence of Optimal Control
We formulate the optimal control problem in the weak sensefollowing the approach of [31]Themain advantage is that wewill be able to solve the closed-loop equation in a weak senseand hence to find an optimal control even if the feedbacklaw is nonsmooth
We call (ΩF F119905119905ge0 119875119882) an admissible setup if
(ΩF F119905119905ge0 119875) is a filtered probability space satisfying the
usual conditions and 119882 is a cylindrical 119875-Wiener processwith values in Ξ with respect to the filtration F
119905119905ge0
By an admissible control system we mean (ΩF
F119905119905ge0 119875119882 119906119883
119906
) where (ΩF F119905119905ge0 119875 119882) is an
admissible setup 119906 is an F119905-predictable process with values
in119880 and119883119906 is a mild solution of (58) An admissible controlsystem will be briefly denoted by (119882 119906119883119906
) in the followingOur purpose is to minimize the cost functional
119869 (119906) = 119864int
infin
119905
119890minus120582119904
119892 (119904 119883119906
119904 119906 (119904)) 119889119904 (81)
over all the admissible control systemOur main result in this section is based on the solvability
of the closed-loop equation
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904 + 119866 (119904 119883
119904)
times (119877 (119904 119883119904 Γ
0(119904 119883
119904 120577 (119904 119883
119904))) 119889119904 +119889119882 (119904))
119904 isin [119905infin)
119883119905= 119909
(82)
In the following sense we say that 119883 is a weak solution of(82) if there exists an admissible setup (ΩF F
119905119905ge0 119875119882)
and anF119905-adapted continuous process119883(119905)with values in119867
which solves the equation in the mild sense namely 119875-as
119883 (119904) = 119890(119904minus119905)119860
119909 (0) + int
119904
119905
119890(119904minus120590)119860
119865 (120590119883120590) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883120590) 119877
times (120590119883120590 Γ
0(120590119883
120590 120577 (120590 119883
120590))) 119889120590
+ int
119904
119905
119890(119904minus120590)119860
119866 (120590119883120590) 119889119882
120590 119904 isin [119905infin)
(83)
119883119905= 119909 (84)
Theorem 13 Assume that Hypothesis 3 holds Then thereexists a weak solution of the closed-loop equation (82) whichis unique in law
12 Abstract and Applied Analysis
Proof (uniqueness) Let 119883 be a weak solution of (82) in anadmissible setup (ΩF F
119905119905ge0 119875119882)We define
120588 (119879) = exp(int119879
119905
minus119877lowast
(120590 119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889119882 (120590)
minus1
2int
119879
119905
1003816100381610038161003816119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)))10038161003816100381610038162
119889120590)
(85)
Since 119877 is bounded the Girsanov theorem ensures that thereexists a probability measure 1198750 such that the process
1198820
(119904) = 119882 (119904) + int
119904
119905and119904
119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889120590
119904 isin [0infin)
(86)
is a 1198750-Wiener process and
1198750
|F119879
= 120588 (119879) 119875|F119879
(87)
Let us denote by F0
119905119905ge0
the filtration generated by1198820 andcompleted in the usual way In (ΩF
[0infin) F0
119905119905ge0 119875
0
) 119883 isa mild solution of
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119905 119883119904) 119889119904
+ 119866 (119904 119883119904) 119889119882
0
(119904) 119904 isin [119905infin)
119883119905= 119909
120588 (119879) = exp(int119879
119905
minus119877lowast
(120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)) 119889119882
0
(120590)
+ 12int
119879
119905
1003816100381610038161003816119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)))10038161003816100381610038162
119889120590)
(88)
By Hypothesis 3 the joint law of 119883 and 1198820 is uniquely
determined by 119860 119865 119866 and 119909 Taking into account the lastdisplayed formula we conclude that the joint law of 119883 and120588(119879) under1198750 is also uniquely determined and consequentlyso is the law of 119883 under 119875 This completes the proof of theuniqueness part
Proof (existence) Let (ΩF 119875) be a given complete probabil-ity space 119882(119905) 119905 ge 0 is a cylindrical Wiener process on(ΩF 119875)with values inΞ and F
119905119905ge0
is the natural filtrationof 119882(119905) 119905 ge 0 augmented with the family of119875-null sets Let119883(sdot) be the mild solution of
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904
+ 119866 (119904 119883119904) 119889119882 (119904) 119904 isin [119905infin)
119883119905= 119909
(89)
and by the Girsanov theorem let 1198751 be the probability on Ωunder which
1198821
(119904) = 119882 (119904) minus int
119904
119905and119904
119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889120590
(90)
is aWiener process (notice that 119877 is bounded)Then119883 is theweak solution of (82) relatively to the probability 1198751 and theWiener process1198821
Now we can state the main result of this section
Corollary 14 Assume that Hypothesis 3 holds true and 120582
verifies (62) Also assume that the set-valued map Γ hasnonempty values and it admits a measurable selection Γ
0
[0infin) timesC times Ξlowast
rarr 119880 Then for every 119905 isin [0infin) and x isin Cand for all admissible control system (119882 119906119883
119906
) one has
119869 (119906 119905 119909) ge 120592 (119905 119909) (91)
and the equality holds if
119906 (119904) = Γ0(119904 119883
119906
119904 120577 (119904 119883
119906
119904))
119875 minus 119886119904 119891119900119903 119886119897119898119900119904119905 119890119907119890119903119910 119904 isin [119905infin)
(92)
Moreover from Theorem 13 it follows that the closed-loop equation (82) admits a weak solution (ΩFF
119905119905ge0 119875119882119883) which is unique in law and setting
119906 (119904) = Γ0(119904 119883
119904 120577 (119904 119883
119904)) (93)
we obtain an optimal admissible control system (119882 119906119883)
6 Applications
In this section we present a simple application of the previousresults We consider the stochastic delay partial differentialequation in the bounded domain 119861 sub 119877
119899 with smoothboundary 120597119861 as follows
119889119911119906
(119905 120585) = Δ119911119906
(119905 120585) 119889119905 + 119891 (119905 119911119906
119905(120585)) 119889119905
+
119889
sum
119894=1
119892119894(119905 119911
119906
119905(120585)) [119903
119894
(120585) 119906119894
(119905) 119889119905 + 119889119882119894
(119905)]
119911119906
0(120579 120585) = 119909 (120579 120585) 120585 isin 119861 120579 isin [minus1 0]
119911119906
(119905 120585) = 0 119905 isin [0infin) 120585 isin 120597119861
(94)
Here119882 = (1198821
1198822
119882119889
) is a standardWiener process in119877119889 and the functions 119891 [0 +infin) times 119862([minus1 0] 119877) rarr 119877 and119892119894 [0 +infin) times 119862([minus1 0] 119877) rarr 119877 are Lipschitz continuous
and bounded Setting 119880 as a bounded subset of 119877119889 Ξ = 119877119889
119867 = 1198712
(119861) and 119909 isin 119862([minus1 0]119867) We define 119865 and 119866 asfollowing
119865 (119905 119909) (120585) = 119891 (119905 119909 (120585))
(119866 (119905 119909) 119911) (120585) =
119889
sum
119894=1
119892119894(119905 119909 (120585)) 119911
119894
(120585)
120585 isin 119861 119909 isin 119862 ([minus1 0] 119867) 119911 isin 119871 (Ξ119867)
(95)
and let 119860 denote the Laplace operator Δ in 1198712
(119861) withdomain11988222
(119861)⋂11988212
0(119861) then (94) has the form (58) and
Hypothesis 1 holds
Abstract and Applied Analysis 13
Let us consider the optimal control problem associatedwith the cost
119869 (119906) = 119864int
infin
0
119890minus120582119905
[int119861
120590 (120585 119911119906
119905(120585)) 119889120585 + 119906
2
(119905)] 119889119905 (96)
where 120582 verifies (62) and 120590 119862([minus1 0] 119877) times 119880 rarr [0infin) isa bounded measurable function Define 119892 119862([minus1 0]119867) times
119880 rarr [0infin) and 119877 119862([minus1 0]119867) times 119880 rarr Ξ by119892(119910 119906) = int
119861
120590(119905 119910(120585) 119906)119889120585 + 1199062 and 119877(119910 119906) = (int
119861
1199031
(120585)1199061
119889120585
int119861
1199032
(120585)1199062
119889120585 int119861
119903119889
(120585)119906119889
119889120585) for 119910 isin 119862([minus1 0]119867) 119906 =
(1199061
1199062
119906119889
) isin 119880 respectively It can be easily verifiedthat Hypothesis 3 holds true and the set-valued map Γ hasnonempty values and it admits a measurable selection Γ
0
[0infin) times C times Ξlowast
rarr 119880 Then the closed-loop equation(82) admits a weak solution (ΩF F
119905119905ge0 119875119882 119906 119911
sdot(sdot)) and
setting
119906 (119904) = Γ0(119904 119911
119904(sdot) 120577 (119904 119911
119904(sdot))) (97)
we obtain an optimal admissible control system (119882 119906 119911(sdot))
References
[1] J Bismut ldquoOn optimal control of linear stochastic equationswith a linear-quadratic criterionrdquo SIAM Journal on Control andOptimization vol 15 no 3 pp 1ndash4 1977
[2] N Nagase ldquoOn the existence of optimal control for controlledstochastic partial differential equationsrdquo Nagoya MathematicsJournal vol 115 pp 73ndash85 1989
[3] N El Karoui D Huu Nguyen and M Jeanblanc-Pique ldquoCom-pactification methods in the control of degenerate diffusionsrdquoStochastics vol 20 pp 169ndash219 1987
[4] M Nisio ldquoOptimal control for stochastic partial differentialequations and viscosity solutions of Bellman equationsrdquoNagoyaMathematics Journal vol 123 pp 13ndash37 1991
[5] M Nisio ldquoOn sensitive control for stochastic partial differentialequationsrdquo in Stochastic Analysis on Infinite Dimensional SpacesProceedings of the US Japan Bilateral Seminar H Kunita et alEd vol 310 of Pitman Research Notes Mathematical Series pp231ndash241 Longman Scientific and Technical Baton Rouge LaUSA January 1994
[6] R Buckdahn and A Rascanu ldquoOn the existence of stochasticoptimal control of distributed state systemrdquoNonlinear AnalysisTheory Methods and Applications vol 52 no 4 pp 1153ndash11842003
[7] V Barbu and G Da Prato Equations in Hilbert Spaces vol 86 ofPitman Research Notes in Mathematics Pitman 1983
[8] P Cannarsa and G Da Prato ldquoSecond-order Hamilton-Jacobiequations in infinite dimensionsrdquo SIAM Journal on Control andOptimization vol 29 no 2 pp 474ndash492 1991
[9] P Cannarsa and G Da Prato ldquoDirect solution of a second-orderHamilton-Jacobi equations in Hilbert spacesrdquo in StochasticPartial Differential Equations and Applications G Da Pratoand L Tubaro Eds vol 268 of Pitman Research Notes inMathematics Pitman 1992
[10] F Gozzi ldquoRegularity of solutions of second order Hamilton-Jacobi equations and application to a control problemrdquoCommu-nications in Partial Differential Equations vol 20 pp 775ndash8261995
[11] F Gozzi ldquoGlobal regular solutions of second order Hamilton-Jacobi equations in Hilbert spaces with locally Lipschitz non-linearitiesrdquo Journal of Mathematical Analysis and Applicationsvol 198 no 2 pp 399ndash443 1996
[12] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems and Control Lettersvol 14 no 1 pp 55ndash61 1990
[13] N El Karoui and LMazliak Eds Backward Stochastic Differen-tial Equations vol 364 of Pitman ResearchNotes inMathematicsSeries Longman 1997
[14] E Pardoux and BSDEs ldquoweak convergence and homogeneiza-tion of semilinear PDEsrdquo in Non- Linear Analysis DifferentialEquations and Control F H Clarke and R J Stern Eds pp503ndash549 Kluwer Dordrecht The Netherlands 1999
[15] S Peng ldquoA generalized dynamic programming principle andHamilton-Jacobi-Bellman equationrdquo Stochastics and StochasticsReports vol 38 pp 119ndash134 1992
[16] N E Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997
[17] SHamad120583ene and J P Lepeltier ldquoBackward equations stochas-tic control and zero-sum stochastic differential gamesrdquo Stochas-tics and Stochastics Reports vol 54 pp 221ndash231 1995
[18] N El-Karoui and S Hamadene ldquoBSDEs and risk-sensitive con-trol zero-sum and nonzero-sum game problems of stochasticfunctional differential equationsrdquo Stochastic Processes and theirApplications vol 107 no 1 pp 145ndash169 2003
[19] M Fuhrman and G Tessiture ldquoExistence of optimal stochasticcontrols and global solutions of forward-backward stochasticdifferential equationsrdquo SIAM Journal on Control and Optimiza-tion vol 43 no 3 pp 813ndash830 2005
[20] M Fuhrman Y Hu and G Tessitore ldquoOn a class of stochasticoptimal control problems related to bsdes with quadraticgrowthrdquo SIAM Journal on Control and Optimization vol 45 no4 pp 1279ndash1296 2006
[21] M Fuhrman and G Tessitore ldquoNonlinear kolmogorov equa-tions in infinite dimensional spaces the backward stochasticdifferential equations approach and applications to optimalcontrolrdquoAnnals of Probability vol 30 no 3 pp 1397ndash1465 2002
[22] F Masiero ldquoSemilinear kolmogorov equations and applicationsto stochastic optimal controlrdquo Applied Mathematics and Opti-mization vol 51 no 1 pp 201ndash250 2005
[23] M Fuhrman FMasiero andG Tessitore ldquoStochastic equationswith delay optimal control via BSDEs and regular solutions ofHamilton-jacobi-bellman equationsrdquo SIAM Journal on Controland Optimization vol 48 no 7 pp 4624ndash4651 2010
[24] M Fuhrman and G Tessiture ldquoInfinite horizon backwardstochastic differential equations and elliptic equations in hilbertspacesrdquo Annals of Probability vol 32 no 1 pp 607ndash660 2004
[25] F Masiero ldquoInfinite horizon stochastic optimal control prob-lems with degenerate noise and elliptic equations in Hilbertspacesrdquo Applied Mathematics and Optimization vol 55 no 3pp 285ndash326 2007
[26] M Fuhrman ldquoA class of stochastic optimal control problemsin Hilbert spaces BSDEs and optimal control laws stateconstraints conditioned processesrdquo Stochastic Processes andtheir Applications vol 108 no 2 pp 263ndash298 2003
[27] F Masiero ldquoStochastic optimal control problems and parabolicequations in banach spacesrdquo SIAM Journal on Control andOptimization vol 47 no 1 pp 251ndash300 2008
[28] G Da Prato and J Zabczyk Ergodicity For Infinite-DimensionalSystems Cambridge University Press 1996
14 Abstract and Applied Analysis
[29] J Zabczyk ldquoParabolic equations on Hilbert spacesrdquo in StochaS-tic PDErsquoS and Kolmogorov Equations in Infinite Dimensionsvol 1715 of Lecture Notes in Math pp 117ndash213 Springer BerlinGermany 1999
[30] G Da Prato and J Zabczyk Stochstic Equations in InfiniteDimensions Cambridge University Press 1992
[31] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 of Applications of MathematicsSpringer New York NY USA 1993
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Abstract and Applied Analysis
Proof (uniqueness) Let 119883 be a weak solution of (82) in anadmissible setup (ΩF F
119905119905ge0 119875119882)We define
120588 (119879) = exp(int119879
119905
minus119877lowast
(120590 119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889119882 (120590)
minus1
2int
119879
119905
1003816100381610038161003816119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)))10038161003816100381610038162
119889120590)
(85)
Since 119877 is bounded the Girsanov theorem ensures that thereexists a probability measure 1198750 such that the process
1198820
(119904) = 119882 (119904) + int
119904
119905and119904
119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889120590
119904 isin [0infin)
(86)
is a 1198750-Wiener process and
1198750
|F119879
= 120588 (119879) 119875|F119879
(87)
Let us denote by F0
119905119905ge0
the filtration generated by1198820 andcompleted in the usual way In (ΩF
[0infin) F0
119905119905ge0 119875
0
) 119883 isa mild solution of
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119905 119883119904) 119889119904
+ 119866 (119904 119883119904) 119889119882
0
(119904) 119904 isin [119905infin)
119883119905= 119909
120588 (119879) = exp(int119879
119905
minus119877lowast
(120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)) 119889119882
0
(120590)
+ 12int
119879
119905
1003816100381610038161003816119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590)))10038161003816100381610038162
119889120590)
(88)
By Hypothesis 3 the joint law of 119883 and 1198820 is uniquely
determined by 119860 119865 119866 and 119909 Taking into account the lastdisplayed formula we conclude that the joint law of 119883 and120588(119879) under1198750 is also uniquely determined and consequentlyso is the law of 119883 under 119875 This completes the proof of theuniqueness part
Proof (existence) Let (ΩF 119875) be a given complete probabil-ity space 119882(119905) 119905 ge 0 is a cylindrical Wiener process on(ΩF 119875)with values inΞ and F
119905119905ge0
is the natural filtrationof 119882(119905) 119905 ge 0 augmented with the family of119875-null sets Let119883(sdot) be the mild solution of
119889119883 (119904) = 119860119883 (119904) 119889119904 + 119865 (119904 119883119904) 119889119904
+ 119866 (119904 119883119904) 119889119882 (119904) 119904 isin [119905infin)
119883119905= 119909
(89)
and by the Girsanov theorem let 1198751 be the probability on Ωunder which
1198821
(119904) = 119882 (119904) minus int
119904
119905and119904
119877 (120590119883120590 Γ
0(120590 119883
120590 120577 (120590 119883
120590))) 119889120590
(90)
is aWiener process (notice that 119877 is bounded)Then119883 is theweak solution of (82) relatively to the probability 1198751 and theWiener process1198821
Now we can state the main result of this section
Corollary 14 Assume that Hypothesis 3 holds true and 120582
verifies (62) Also assume that the set-valued map Γ hasnonempty values and it admits a measurable selection Γ
0
[0infin) timesC times Ξlowast
rarr 119880 Then for every 119905 isin [0infin) and x isin Cand for all admissible control system (119882 119906119883
119906
) one has
119869 (119906 119905 119909) ge 120592 (119905 119909) (91)
and the equality holds if
119906 (119904) = Γ0(119904 119883
119906
119904 120577 (119904 119883
119906
119904))
119875 minus 119886119904 119891119900119903 119886119897119898119900119904119905 119890119907119890119903119910 119904 isin [119905infin)
(92)
Moreover from Theorem 13 it follows that the closed-loop equation (82) admits a weak solution (ΩFF
119905119905ge0 119875119882119883) which is unique in law and setting
119906 (119904) = Γ0(119904 119883
119904 120577 (119904 119883
119904)) (93)
we obtain an optimal admissible control system (119882 119906119883)
6 Applications
In this section we present a simple application of the previousresults We consider the stochastic delay partial differentialequation in the bounded domain 119861 sub 119877
119899 with smoothboundary 120597119861 as follows
119889119911119906
(119905 120585) = Δ119911119906
(119905 120585) 119889119905 + 119891 (119905 119911119906
119905(120585)) 119889119905
+
119889
sum
119894=1
119892119894(119905 119911
119906
119905(120585)) [119903
119894
(120585) 119906119894
(119905) 119889119905 + 119889119882119894
(119905)]
119911119906
0(120579 120585) = 119909 (120579 120585) 120585 isin 119861 120579 isin [minus1 0]
119911119906
(119905 120585) = 0 119905 isin [0infin) 120585 isin 120597119861
(94)
Here119882 = (1198821
1198822
119882119889
) is a standardWiener process in119877119889 and the functions 119891 [0 +infin) times 119862([minus1 0] 119877) rarr 119877 and119892119894 [0 +infin) times 119862([minus1 0] 119877) rarr 119877 are Lipschitz continuous
and bounded Setting 119880 as a bounded subset of 119877119889 Ξ = 119877119889
119867 = 1198712
(119861) and 119909 isin 119862([minus1 0]119867) We define 119865 and 119866 asfollowing
119865 (119905 119909) (120585) = 119891 (119905 119909 (120585))
(119866 (119905 119909) 119911) (120585) =
119889
sum
119894=1
119892119894(119905 119909 (120585)) 119911
119894
(120585)
120585 isin 119861 119909 isin 119862 ([minus1 0] 119867) 119911 isin 119871 (Ξ119867)
(95)
and let 119860 denote the Laplace operator Δ in 1198712
(119861) withdomain11988222
(119861)⋂11988212
0(119861) then (94) has the form (58) and
Hypothesis 1 holds
Abstract and Applied Analysis 13
Let us consider the optimal control problem associatedwith the cost
119869 (119906) = 119864int
infin
0
119890minus120582119905
[int119861
120590 (120585 119911119906
119905(120585)) 119889120585 + 119906
2
(119905)] 119889119905 (96)
where 120582 verifies (62) and 120590 119862([minus1 0] 119877) times 119880 rarr [0infin) isa bounded measurable function Define 119892 119862([minus1 0]119867) times
119880 rarr [0infin) and 119877 119862([minus1 0]119867) times 119880 rarr Ξ by119892(119910 119906) = int
119861
120590(119905 119910(120585) 119906)119889120585 + 1199062 and 119877(119910 119906) = (int
119861
1199031
(120585)1199061
119889120585
int119861
1199032
(120585)1199062
119889120585 int119861
119903119889
(120585)119906119889
119889120585) for 119910 isin 119862([minus1 0]119867) 119906 =
(1199061
1199062
119906119889
) isin 119880 respectively It can be easily verifiedthat Hypothesis 3 holds true and the set-valued map Γ hasnonempty values and it admits a measurable selection Γ
0
[0infin) times C times Ξlowast
rarr 119880 Then the closed-loop equation(82) admits a weak solution (ΩF F
119905119905ge0 119875119882 119906 119911
sdot(sdot)) and
setting
119906 (119904) = Γ0(119904 119911
119904(sdot) 120577 (119904 119911
119904(sdot))) (97)
we obtain an optimal admissible control system (119882 119906 119911(sdot))
References
[1] J Bismut ldquoOn optimal control of linear stochastic equationswith a linear-quadratic criterionrdquo SIAM Journal on Control andOptimization vol 15 no 3 pp 1ndash4 1977
[2] N Nagase ldquoOn the existence of optimal control for controlledstochastic partial differential equationsrdquo Nagoya MathematicsJournal vol 115 pp 73ndash85 1989
[3] N El Karoui D Huu Nguyen and M Jeanblanc-Pique ldquoCom-pactification methods in the control of degenerate diffusionsrdquoStochastics vol 20 pp 169ndash219 1987
[4] M Nisio ldquoOptimal control for stochastic partial differentialequations and viscosity solutions of Bellman equationsrdquoNagoyaMathematics Journal vol 123 pp 13ndash37 1991
[5] M Nisio ldquoOn sensitive control for stochastic partial differentialequationsrdquo in Stochastic Analysis on Infinite Dimensional SpacesProceedings of the US Japan Bilateral Seminar H Kunita et alEd vol 310 of Pitman Research Notes Mathematical Series pp231ndash241 Longman Scientific and Technical Baton Rouge LaUSA January 1994
[6] R Buckdahn and A Rascanu ldquoOn the existence of stochasticoptimal control of distributed state systemrdquoNonlinear AnalysisTheory Methods and Applications vol 52 no 4 pp 1153ndash11842003
[7] V Barbu and G Da Prato Equations in Hilbert Spaces vol 86 ofPitman Research Notes in Mathematics Pitman 1983
[8] P Cannarsa and G Da Prato ldquoSecond-order Hamilton-Jacobiequations in infinite dimensionsrdquo SIAM Journal on Control andOptimization vol 29 no 2 pp 474ndash492 1991
[9] P Cannarsa and G Da Prato ldquoDirect solution of a second-orderHamilton-Jacobi equations in Hilbert spacesrdquo in StochasticPartial Differential Equations and Applications G Da Pratoand L Tubaro Eds vol 268 of Pitman Research Notes inMathematics Pitman 1992
[10] F Gozzi ldquoRegularity of solutions of second order Hamilton-Jacobi equations and application to a control problemrdquoCommu-nications in Partial Differential Equations vol 20 pp 775ndash8261995
[11] F Gozzi ldquoGlobal regular solutions of second order Hamilton-Jacobi equations in Hilbert spaces with locally Lipschitz non-linearitiesrdquo Journal of Mathematical Analysis and Applicationsvol 198 no 2 pp 399ndash443 1996
[12] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems and Control Lettersvol 14 no 1 pp 55ndash61 1990
[13] N El Karoui and LMazliak Eds Backward Stochastic Differen-tial Equations vol 364 of Pitman ResearchNotes inMathematicsSeries Longman 1997
[14] E Pardoux and BSDEs ldquoweak convergence and homogeneiza-tion of semilinear PDEsrdquo in Non- Linear Analysis DifferentialEquations and Control F H Clarke and R J Stern Eds pp503ndash549 Kluwer Dordrecht The Netherlands 1999
[15] S Peng ldquoA generalized dynamic programming principle andHamilton-Jacobi-Bellman equationrdquo Stochastics and StochasticsReports vol 38 pp 119ndash134 1992
[16] N E Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997
[17] SHamad120583ene and J P Lepeltier ldquoBackward equations stochas-tic control and zero-sum stochastic differential gamesrdquo Stochas-tics and Stochastics Reports vol 54 pp 221ndash231 1995
[18] N El-Karoui and S Hamadene ldquoBSDEs and risk-sensitive con-trol zero-sum and nonzero-sum game problems of stochasticfunctional differential equationsrdquo Stochastic Processes and theirApplications vol 107 no 1 pp 145ndash169 2003
[19] M Fuhrman and G Tessiture ldquoExistence of optimal stochasticcontrols and global solutions of forward-backward stochasticdifferential equationsrdquo SIAM Journal on Control and Optimiza-tion vol 43 no 3 pp 813ndash830 2005
[20] M Fuhrman Y Hu and G Tessitore ldquoOn a class of stochasticoptimal control problems related to bsdes with quadraticgrowthrdquo SIAM Journal on Control and Optimization vol 45 no4 pp 1279ndash1296 2006
[21] M Fuhrman and G Tessitore ldquoNonlinear kolmogorov equa-tions in infinite dimensional spaces the backward stochasticdifferential equations approach and applications to optimalcontrolrdquoAnnals of Probability vol 30 no 3 pp 1397ndash1465 2002
[22] F Masiero ldquoSemilinear kolmogorov equations and applicationsto stochastic optimal controlrdquo Applied Mathematics and Opti-mization vol 51 no 1 pp 201ndash250 2005
[23] M Fuhrman FMasiero andG Tessitore ldquoStochastic equationswith delay optimal control via BSDEs and regular solutions ofHamilton-jacobi-bellman equationsrdquo SIAM Journal on Controland Optimization vol 48 no 7 pp 4624ndash4651 2010
[24] M Fuhrman and G Tessiture ldquoInfinite horizon backwardstochastic differential equations and elliptic equations in hilbertspacesrdquo Annals of Probability vol 32 no 1 pp 607ndash660 2004
[25] F Masiero ldquoInfinite horizon stochastic optimal control prob-lems with degenerate noise and elliptic equations in Hilbertspacesrdquo Applied Mathematics and Optimization vol 55 no 3pp 285ndash326 2007
[26] M Fuhrman ldquoA class of stochastic optimal control problemsin Hilbert spaces BSDEs and optimal control laws stateconstraints conditioned processesrdquo Stochastic Processes andtheir Applications vol 108 no 2 pp 263ndash298 2003
[27] F Masiero ldquoStochastic optimal control problems and parabolicequations in banach spacesrdquo SIAM Journal on Control andOptimization vol 47 no 1 pp 251ndash300 2008
[28] G Da Prato and J Zabczyk Ergodicity For Infinite-DimensionalSystems Cambridge University Press 1996
14 Abstract and Applied Analysis
[29] J Zabczyk ldquoParabolic equations on Hilbert spacesrdquo in StochaS-tic PDErsquoS and Kolmogorov Equations in Infinite Dimensionsvol 1715 of Lecture Notes in Math pp 117ndash213 Springer BerlinGermany 1999
[30] G Da Prato and J Zabczyk Stochstic Equations in InfiniteDimensions Cambridge University Press 1992
[31] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 of Applications of MathematicsSpringer New York NY USA 1993
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 13
Let us consider the optimal control problem associatedwith the cost
119869 (119906) = 119864int
infin
0
119890minus120582119905
[int119861
120590 (120585 119911119906
119905(120585)) 119889120585 + 119906
2
(119905)] 119889119905 (96)
where 120582 verifies (62) and 120590 119862([minus1 0] 119877) times 119880 rarr [0infin) isa bounded measurable function Define 119892 119862([minus1 0]119867) times
119880 rarr [0infin) and 119877 119862([minus1 0]119867) times 119880 rarr Ξ by119892(119910 119906) = int
119861
120590(119905 119910(120585) 119906)119889120585 + 1199062 and 119877(119910 119906) = (int
119861
1199031
(120585)1199061
119889120585
int119861
1199032
(120585)1199062
119889120585 int119861
119903119889
(120585)119906119889
119889120585) for 119910 isin 119862([minus1 0]119867) 119906 =
(1199061
1199062
119906119889
) isin 119880 respectively It can be easily verifiedthat Hypothesis 3 holds true and the set-valued map Γ hasnonempty values and it admits a measurable selection Γ
0
[0infin) times C times Ξlowast
rarr 119880 Then the closed-loop equation(82) admits a weak solution (ΩF F
119905119905ge0 119875119882 119906 119911
sdot(sdot)) and
setting
119906 (119904) = Γ0(119904 119911
119904(sdot) 120577 (119904 119911
119904(sdot))) (97)
we obtain an optimal admissible control system (119882 119906 119911(sdot))
References
[1] J Bismut ldquoOn optimal control of linear stochastic equationswith a linear-quadratic criterionrdquo SIAM Journal on Control andOptimization vol 15 no 3 pp 1ndash4 1977
[2] N Nagase ldquoOn the existence of optimal control for controlledstochastic partial differential equationsrdquo Nagoya MathematicsJournal vol 115 pp 73ndash85 1989
[3] N El Karoui D Huu Nguyen and M Jeanblanc-Pique ldquoCom-pactification methods in the control of degenerate diffusionsrdquoStochastics vol 20 pp 169ndash219 1987
[4] M Nisio ldquoOptimal control for stochastic partial differentialequations and viscosity solutions of Bellman equationsrdquoNagoyaMathematics Journal vol 123 pp 13ndash37 1991
[5] M Nisio ldquoOn sensitive control for stochastic partial differentialequationsrdquo in Stochastic Analysis on Infinite Dimensional SpacesProceedings of the US Japan Bilateral Seminar H Kunita et alEd vol 310 of Pitman Research Notes Mathematical Series pp231ndash241 Longman Scientific and Technical Baton Rouge LaUSA January 1994
[6] R Buckdahn and A Rascanu ldquoOn the existence of stochasticoptimal control of distributed state systemrdquoNonlinear AnalysisTheory Methods and Applications vol 52 no 4 pp 1153ndash11842003
[7] V Barbu and G Da Prato Equations in Hilbert Spaces vol 86 ofPitman Research Notes in Mathematics Pitman 1983
[8] P Cannarsa and G Da Prato ldquoSecond-order Hamilton-Jacobiequations in infinite dimensionsrdquo SIAM Journal on Control andOptimization vol 29 no 2 pp 474ndash492 1991
[9] P Cannarsa and G Da Prato ldquoDirect solution of a second-orderHamilton-Jacobi equations in Hilbert spacesrdquo in StochasticPartial Differential Equations and Applications G Da Pratoand L Tubaro Eds vol 268 of Pitman Research Notes inMathematics Pitman 1992
[10] F Gozzi ldquoRegularity of solutions of second order Hamilton-Jacobi equations and application to a control problemrdquoCommu-nications in Partial Differential Equations vol 20 pp 775ndash8261995
[11] F Gozzi ldquoGlobal regular solutions of second order Hamilton-Jacobi equations in Hilbert spaces with locally Lipschitz non-linearitiesrdquo Journal of Mathematical Analysis and Applicationsvol 198 no 2 pp 399ndash443 1996
[12] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems and Control Lettersvol 14 no 1 pp 55ndash61 1990
[13] N El Karoui and LMazliak Eds Backward Stochastic Differen-tial Equations vol 364 of Pitman ResearchNotes inMathematicsSeries Longman 1997
[14] E Pardoux and BSDEs ldquoweak convergence and homogeneiza-tion of semilinear PDEsrdquo in Non- Linear Analysis DifferentialEquations and Control F H Clarke and R J Stern Eds pp503ndash549 Kluwer Dordrecht The Netherlands 1999
[15] S Peng ldquoA generalized dynamic programming principle andHamilton-Jacobi-Bellman equationrdquo Stochastics and StochasticsReports vol 38 pp 119ndash134 1992
[16] N E Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997
[17] SHamad120583ene and J P Lepeltier ldquoBackward equations stochas-tic control and zero-sum stochastic differential gamesrdquo Stochas-tics and Stochastics Reports vol 54 pp 221ndash231 1995
[18] N El-Karoui and S Hamadene ldquoBSDEs and risk-sensitive con-trol zero-sum and nonzero-sum game problems of stochasticfunctional differential equationsrdquo Stochastic Processes and theirApplications vol 107 no 1 pp 145ndash169 2003
[19] M Fuhrman and G Tessiture ldquoExistence of optimal stochasticcontrols and global solutions of forward-backward stochasticdifferential equationsrdquo SIAM Journal on Control and Optimiza-tion vol 43 no 3 pp 813ndash830 2005
[20] M Fuhrman Y Hu and G Tessitore ldquoOn a class of stochasticoptimal control problems related to bsdes with quadraticgrowthrdquo SIAM Journal on Control and Optimization vol 45 no4 pp 1279ndash1296 2006
[21] M Fuhrman and G Tessitore ldquoNonlinear kolmogorov equa-tions in infinite dimensional spaces the backward stochasticdifferential equations approach and applications to optimalcontrolrdquoAnnals of Probability vol 30 no 3 pp 1397ndash1465 2002
[22] F Masiero ldquoSemilinear kolmogorov equations and applicationsto stochastic optimal controlrdquo Applied Mathematics and Opti-mization vol 51 no 1 pp 201ndash250 2005
[23] M Fuhrman FMasiero andG Tessitore ldquoStochastic equationswith delay optimal control via BSDEs and regular solutions ofHamilton-jacobi-bellman equationsrdquo SIAM Journal on Controland Optimization vol 48 no 7 pp 4624ndash4651 2010
[24] M Fuhrman and G Tessiture ldquoInfinite horizon backwardstochastic differential equations and elliptic equations in hilbertspacesrdquo Annals of Probability vol 32 no 1 pp 607ndash660 2004
[25] F Masiero ldquoInfinite horizon stochastic optimal control prob-lems with degenerate noise and elliptic equations in Hilbertspacesrdquo Applied Mathematics and Optimization vol 55 no 3pp 285ndash326 2007
[26] M Fuhrman ldquoA class of stochastic optimal control problemsin Hilbert spaces BSDEs and optimal control laws stateconstraints conditioned processesrdquo Stochastic Processes andtheir Applications vol 108 no 2 pp 263ndash298 2003
[27] F Masiero ldquoStochastic optimal control problems and parabolicequations in banach spacesrdquo SIAM Journal on Control andOptimization vol 47 no 1 pp 251ndash300 2008
[28] G Da Prato and J Zabczyk Ergodicity For Infinite-DimensionalSystems Cambridge University Press 1996
14 Abstract and Applied Analysis
[29] J Zabczyk ldquoParabolic equations on Hilbert spacesrdquo in StochaS-tic PDErsquoS and Kolmogorov Equations in Infinite Dimensionsvol 1715 of Lecture Notes in Math pp 117ndash213 Springer BerlinGermany 1999
[30] G Da Prato and J Zabczyk Stochstic Equations in InfiniteDimensions Cambridge University Press 1992
[31] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 of Applications of MathematicsSpringer New York NY USA 1993
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Abstract and Applied Analysis
[29] J Zabczyk ldquoParabolic equations on Hilbert spacesrdquo in StochaS-tic PDErsquoS and Kolmogorov Equations in Infinite Dimensionsvol 1715 of Lecture Notes in Math pp 117ndash213 Springer BerlinGermany 1999
[30] G Da Prato and J Zabczyk Stochstic Equations in InfiniteDimensions Cambridge University Press 1992
[31] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 of Applications of MathematicsSpringer New York NY USA 1993
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of