anecessaryconditionforoptimalcontrolofforward-backward...
TRANSCRIPT
Research ArticleANecessary Condition for Optimal Control of Forward-BackwardStochastic Control System with Levy Process in NonconvexControl Domain Case
Hong Huang 1 Xiangrong Wang 2 and Ying Li 34
1School of Data and Computer Science Shandong Womenrsquos University Jinan 250300 China2Institute of Financial Engineering Shandong University of Science and Technology Qingdao 266590 China3Office of Academic Research Shandong Womenrsquos University Jinan 250300 China4College of Computer Science and Engineering Shandong University of Science and Technology Qingdao 266590 China
Correspondence should be addressed to Ying Li cherry_jn126com
Received 10 April 2020 Accepted 18 May 2020 Published 3 June 2020
Guest Editor Yi Qi
Copyright copy 2020 Hong Huang et al is 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
is paper analyzes one kind of optimal control problem which is described by forward-backward stochastic differential equations withLevy process (FBSDEL)Wederive a necessary condition for the existence of the optimal control bymeans of spike variational techniquewhile the control domain is not necessarily convex Simultaneously we also get the maximum principle for this control system whenthere are some initial and terminal state constraints Finally a financial example is discussed to illustrate the application of our result
1 Introduction
Stochastic optimal control is an importantmatter that cannot beneglected inmodern control theory in long days As is known toall Pontryaginrsquos [1]maximum principle is one of themain waysto settle the stochastic optimal control problem By introducingtheHamiltonian function a necessary condition for the optimalcontrol of stochastic control systems was given by him whichwas called the maximum condition From that time plenty ofworks on this issue have been done Peng [2] was the first one toprove the general maximum principle of the forward-backwardstochastic control system with diffusion coefficient containingthe control variable by the technique of the second-order Taylorexpansion and the second-order duality He [3] was also the firstone to demonstrate the maximum principle of forward-back-ward stochastic control systems from the view of backwardstochastic differential equations (BSDE) In Pengrsquos paper [3] thecontrol domainwas convex (in local form) Xu [4] extended thisconclusion to the case of the nonconvex control domain (inglobal form) but the control variables were not included in thediffusion coefficient And these results were extended to thefully coupled case in the form of local and global by Shi andWu
[5 6] in 1998 and in 2006 respectively On the basis of theseworks Situ [7] was the first to obtain the maximum principle ofthe forward stochastic control system with global form ofrandom jumps in 1991 Shi andWu [8] and Shi [9] acquired themaximum principle for a kind of forward-backward stochasticcontrol system with Poisson jumps in the form of local andglobal respectively e fully coupled forward-backward sto-chastic control systemwas extended by Liu et al [10] at the baseof Shi andWu [8] and in the meanwhile they also obtained themaximum principle with the control system be constrainedabout initial-terminal state constraints Considering that in reallife the decision makers could only get partial information butnot complete information in most cases many scholars havepaid attention to the partial observable stochastic optimalcontrol problem and have achieved many results (see for ex-ample [11ndash13]) Traditionally when using a stochastic partialdifferential equation called the Zakai equation to transform afull-information optimal control problem to the partially ob-servable case scholars will encounter a difficult problem aninfinite-dimensional optimal control problem Wang and Wu[14] proposed a backward separation approach and replaced theoriginal state and observation equation with the Zakai equation
HindawiMathematical Problems in EngineeringVolume 2020 Article ID 1768507 11 pageshttpsdoiorg10115520201768507
and lots of complicated stochastic calculi in infinite-dimensionalspaces were avoided in this way Based on this approach Xiao[15] studied a partially observed optimal control of forward-backward stochastic systems with random jumps and obtainedthe maximum principle and sufficient conditions of an optimalcontrol under some certain convexity assumptions Wang et al[16] proved the maximum principles for forward-backwardstochastic control systems with correlated state and observationnoises More recent conclusions of the partially observed sto-chastic control problem can be seen from the studies conductedby Wang et al [17] Zhang et al [18] and Xiong et al [19]
In these years through the study of mathematicaleconomics and mathematical finance many scholars turntheir attention to the stochastic control system driven byLevy process In 2000 Nualart and Schoutens [20] built apair of pairwise strongly orthonormal martingales whichwas called the Teugels martingale Meanwhile under someexponential moment conditions they also obtained amartingale representation in that paper Under these twoimportant conclusions for BSDE driven by the Teugelsmartingale they [21] proved the existence and uniquenesstheorem of its solution in the next year From then on anumber of important results were proved Meng and Tang[22] obtained the maximum principle of the forwardstochastic control system driven by Levy process Anecessary and sufficient condition for the existence of theoptimal control of backward stochastic control systemsdriven by Levy process was deduced by Tang and Zhang[23] through convex variation methods and dualitytechniques For the forward-backward stochastic controlsystem driven by Levy process there are also a lot ofachievements based on the existence and uniquenesstheorem of FBSDEL [24] Zhang et al [25] obtained anecessary condition of the optimal control and verifica-tion theorem but in their control system the backwardstate variables yt and zt did not enter the forward partWang and Huang [26] extended this result to the fullycoupled control system and obtained the continuity resultdepending on parameters about FBSDEL and the localform maximum principle Subsequently Huang et al [27]studied this control system with terminal state constraintsand obtained the corresponding necessary maximumprinciple using Ekelandrsquos variational For more recentconclusions about the stochastic control problem drivenby Levy process please refer to [28ndash30]
In this paper we will study the optimal control problemfor forward-backward stochastic control systems driven byLevy process which could be considered as a nonconvexcontrol domain case that is extended from the result of [25]
With the technique of spike variation and Ekelandrsquos vari-ational principle the maximum principle of this type ofcontrol system and the control system with initial and finalstate constraints are obtained
e structure of this paper is as follows Section 2 de-scribes some of the preparations used in this paper emaximum principle and the one with initial and terminalstate constraints as the major results of this paper will beshown in Sections 3 and 4 As an application of the max-imum principle Section 5 gives an optimal consumptionproblem in the financial market Section 6 is the summary ofthis article
2 Preliminary Statement
Let (ΩFt P) be a complete probability space which sat-isfied the usual conditions and the information structure isgiven byFt which is generated by two processes a standardBrownian motion Bt1113864 11138650le tleT valued in Rd and an inde-pendent 1-dimensional Levy process Lt1113864 11138650le tleT of the formLt bt + lt here lt is a pure jump process And assume thatLevy measure ] satisfies the following two conditionsthereby Levy process Lt1113864 11138650le tleT has moments in all orders
(i) 1113938R(1and x2)](dx)ltinfin
(ii) 1113938(minus εε)c eλ |x|](dx)ltinfin forallεgt 0 and some λgt 0
Denote L1t Lt Lt Lt minus Ltminus and Li
t 11139360ltslet(Ls)i
for ige 2 And let Yit Li
t minus E[Lit] (ige 1) be the compensated
power jump process of order i then Teugels martingale isdefined by Hi
t 1113936ij1 cijY
jt the coefficients cij correspond to
orthonormalization of the polynomials 1 x x2 withrespect to the measure μ(dx) υ(dx) + σ2δ0(dx)
en Hit1113864 1113865infini1 are pairwise strongly orthogonal martin-
gales and their predictable quadratic variation processes arelangHi
t Hjt rang δijt δij is an indicator function here And
[Hi Hj]t minus langHit H
jt rang is an Ftminus martingale for more details
of the Teugels martingale see Nualart and Schoutens [20]In the following of this section we shall assume some
notations for a Hilbert space H
l2(H) ϕ |H minus valued 1113936infini1 ϕi2 ltinfin1113966 1113967
L2(ΩH) ξ |
H valuedFT minus measurable E|ξ|2 ltinfinl2(0 TH) ϕi
t | l2(H)minus1113864
valuedFt minus measurable 1113936infini1 E 1113938
T
0 ϕit2dtltinfin
M2(0 TH) ϕ(middot) |1113864
H minus valuedFt minus measurable E 1113938T
0 |ϕt|2dtltinfin
For the following FBSDEL
dxt b t xt yt zt rt( 1113857dt + σ t xt yt zt rt( 1113857dBt + 1113936infin
i1gi t xtminus ytminus zt rt( 1113857dHi
t
minus dyt f t xt yt zt rt( 1113857dt minus ztdBt minus 1113936infin
i1ri
tdHit
x0 a yT Φ xT( 1113857
⎧⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎩
(1)
2 Mathematical Problems in Engineering
where (xt yt zt rt) take the value inΩ times [0 T] times Rn times Rm times
Rmtimesd times l2(Rm) and mappings b σ g and f take the value inRn Rntimesd l2(Rn) and Rm respectively Convenient forwriting set column vector α (x y z)T andA(t α r) (minus Mτf(t α r) Mb(t α r) Mσ(t α r))Twhere M is a m times n full rank matrix
Assumption 1
(i) All mappings in equation (1) are uniformly Lip-schitz continuous in their own argumentsrespectively
(ii) For all (ω t) isin Ω times [0 T] l(ω t 0 0 0 0)
isinM2(0 T Rn+m+mtimesd) times l2(0 T Rm) for l b f σrespectively and g(ω t 0 0 0 0) isin l2(Rn)
(iii) langΦ(x) minus Φ(x) M(x minus x)rangge β|M1113954x|2(iv) langA(tα r) minus A(tα r)α minus αrang + 1113936
infini1langM 1113954gi 1113954rirangle minus μ
1|M1113954x|2 minus μ2(|Mτ1113954y|2 + |Mτ1113954z|2 + 1113936infini1 Mτ 1113954ri2)
where α (x y z) 1113954x x minus x 1113954y y minus y 1113954z z minus z1113954ri ri minus ri 1113954gi gi(tα r) minus gi(tα r) μ1 μ2 and βare nonnegative constants which satisfiedμ1 + μ2 gt 0 μ2 + βgt 0 and μ1 gt 0βgt 0(resp μ2 gt 0)when mgtn (resp ngtm)
en the following existence and uniqueness of thesolution conclusion holds
Lemma 1 9ere exists a unique solution in M2(0 TH)
satisfying FBSDEL (1) under Assumption 19e detailed certification process of this conclusion can be
seen in [24]
3 Stochastic Maximum Principle
In this section for any given admissible control u(middot) weconsider the following stochastic control system
dxt b t xt ut( 1113857dt + σ t xt( 1113857dBt + 1113936infin
i1gi t xtminus( 1113857dHi
t
minus dyt f t xt yt zt rt ut1113872 1113873dt minus ztdBt minus 1113936infin
i1ri
tdHit
x0 a yT Φ xT( 1113857 t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎩
(2)
where a isin Rn is given An admissible controlu(middot) isinM2(0 T Rp) is an Ft-predictable process whichtakes values in a nonempty subset U of Rp Uad is the set ofall admissible controls
And the performance criterion is
J(u) Ec y0( 1113857 (3)
where c Rm⟶ R is a given Frechet differential functionOur optimal control problem amounts to determining
an admissible control ulowast isin Uad such that
J ulowast(middot)( 1113857 inf
u(middot)isinUad
J(u(middot)) (4)
In order to get the necessary conditions for the optimalcontrol we assume ulowastt is the optimal control and thecorresponding solution of (2) is recorded as (xlowastt ylowastt zlowastt rlowastt )
and introduce the ldquospike variational controlrdquo as follows
uεt
vt τ le tle τ + ε
ulowastt otherwise1113896 (5)
and (xεt yε
t zεt rεt) are the state trajectories of uε
t here vt is anarbitrary admissible control and ε is a sufficiently smallconstant
We also need the following assumption and variationalequation (6)
Assumption 2
(i) b f g σ Φ and c are continuously differentiablewith respect to (x y z r u) and the derivatives areall bounded
(ii) ere exists a constant Cgt 0 and it holds that|cy|leC(1 + |y|)
dXt bx(t)Xt + b t uεt( 1113857 minus b t ulowastt( 11138571113858 1113859dt + σx(t)XtdBt + 1113936
infin
i1gi
x(t)XtdHit
minus dYt fx(t)Xt + fy(t)Yt + fz(t)Zt + fr(t)Rt + f t uεt( 1113857 minus f t ulowastt( 11138571113960 1113961dt
minus ZtdBt minus 1113936infin
i1Ri
tdHit
X0 0
YT Φx(t)XT
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(6)
Mathematical Problems in Engineering 3
Here bx(t) bx(txlowastt ulowastt ) σx(t) σx(txlowastt ) gix(t)
gix(txlowastt ) b(tuε
t) b(txεt u
εt) b(tulowastt ) b (txlowastt ulowastt )
fw(t) fw(txlowastt ylowastt zlowastt rlowastt ulowastt ) (w xyz middot r) f(tuεt)
f(xεt y
εt z
εt r
εt u
εt) and f(tulowastt ) f(xlowastt ylowastt zlowastt rlowastt ulowastt )
Lemma 2 Suppose Assumptions 1 and 2 hold for the first-order variation X Y Z R we have the following estimations
sup0letleT
E Xt
111386811138681113868111386811138681113868111386811138682 leCε2 (7)
sup0letleT
E Xt
111386811138681113868111386811138681113868111386811138684 leCε4 (8)
sup0letleT
E Yt
111386811138681113868111386811138681113868111386811138682 leCε2 (9)
sup0letleT
E Yt
111386811138681113868111386811138681113868111386811138684 leCε4 (10)
sup0letleT
E 1113946T
0Zt
111386811138681113868111386811138681113868111386811138682dsleCε2 (11)
sup0letleT
E 1113946T
0Zt
111386811138681113868111386811138681113868111386811138682ds1113888 1113889
2
leCε4 (12)
sup0letleT
E 1113946T
0Rt
2dsleCε2 (13)
sup0letleT
E 1113946T
0Rt
2ds1113888 1113889
2
leCε4 (14)
Proof We first prove inequations (7) and (8) For theforward part of the first-order variation equation we have
E Xt
111386811138681113868111386811138681113868111386811138682
E 1113946t
0bx(s)Xs + b s u
εs( 1113857 minus b s u
lowasts( 11138571113858 1113859ds + σx(s)XsdBs + 1113944
infin
i1g
ix(s)XsdH
is
⎧⎨
⎩
⎫⎬
⎭
2
le 4 E 1113946t
0bx(s)Xsds1113888 1113889
2
+ E 1113946t
0b s u
εs( 1113857 minus b s u
lowasts( 11138571113858 1113859ds1113888 1113889
2
+ E 1113946t
0σx(s)Xs1113858 1113859
2ds + E 1113946t
01113944
infin
i1g
ix(s)Xs
⎡⎣ ⎤⎦2
ds⎧⎨
⎩
⎫⎬
⎭
le 12C2TE 1113946
t
0X
2sds + 4E 1113946
t
0b s u
εs( 1113857 minus b s u
lowasts( 11138571113858 1113859ds1113888 1113889
2
(15)
Applying Gronwallrsquos inequation we have
E Xt
111386811138681113868111386811138681113868111386811138682 leCε2 for all t isin [0 T] (16)
Similarly (8) holdsWe next estimate Yt Zt and Rt the backward part of the
first-order variation equation can be rewritten as
Yt + 1113946T
tZsdBs + 1113946
T
t1113944
infin
i1R
isdH
is Φx(t)XT + 1113946
T
tfx(t)Xt1113858
+ fy(t)Yt + fz(t)Zt
+ fr(t)Rt
+ f t uεt( 1113857 minus f t u
lowastt( 1113857]dt
(17)
Squaring both sides of (17) and using the fact of
EYt 1113946T
tZsdBs 0
EYt 1113946T
t1113944
infin
i1R
isdH
is 0
E 1113946T
tZsdBs 1113946
T
t1113944
infin
i1R
isdH
is 0
(18)
we get
E Yt
111386811138681113868111386811138681113868111386811138682
+ E 1113946T
tZ2sds + E 1113946
T
t1113944
infin
i1R
is
⎛⎝ ⎞⎠
2
dHis
E Φx(t)XT + 1113946T
tfx(t)Xt + fy(t)Yt + fz(t)Zt + fr(t)Rt + f t u
εt( 1113857 minus f t u
lowastt( 11138571113960 1113961dt1113896 1113897
2
le 6C2EX
2T + 6C
2TE 1113946
T
tX
2sds + 6C
2TE 1113946
T
tY2sds + 6C
2(T minus t)E 1113946
T
tZ2sds
+ 6C2(T minus t)E 1113946
T
t1113944
infin
i1R
is
⎛⎝ ⎞⎠
2
ds + 6E 1113946T
tf s u
εs( 1113857 minus f s u
lowasts( 1113857( 1113857ds1113888 1113889
2
(19)
4 Mathematical Problems in Engineering
When t isin [T minus δ T] with δ 112C2 we have
E Yt
111386811138681113868111386811138681113868111386811138682
+12
E 1113946T
tZ2sds +
12
E 1113946T
t1113944
infin
i1R
is
⎛⎝ ⎞⎠
2
dHis
le 6C2EX
2T + 6C
2TE 1113946
T
tX
2sds + 6C
2TE
middot 1113946T
tY2sds + 6E 1113946
T
tf s u
εs( 1113857 minus f s u
lowasts( 1113857( 1113857ds1113888 1113889
2
(20)
Applying Gronwallrsquos inequation we have
sup0letleT
E Yt
111386811138681113868111386811138681113868111386811138682 leCε2 t isin [T minus δ T]
sup0letleT
E 1113946T
0Zt
111386811138681113868111386811138681113868111386811138682dsleCε2 t isin [T minus δ T]
sup0letleT
E 1113946T
0Rt
2dsleCε2 t isin [T minus δ T]
(21)
Consider the BSDE of the first-order variation equationin the interval [t T minus δ]
Yt + 1113946Tminus δ
tZsdBs + 1113946
Tminus δ
t1113944
infin
i1R
isdH
is YTminus δ
+ 1113946Tminus δ
tfx(t)Xt + fy(t)Yt + fz(t)Zt1113960
+ fr(t)Rt + f t uεt( 1113857 minus f t u
lowastt( 11138571113859dt
(22)
us
E Yt
111386811138681113868111386811138681113868111386811138682
+ E 1113946Tminus δ
tZ2sds + E 1113946
Tminus δ
t1113944
infin
i1R
is
⎛⎝ ⎞⎠
2
dHis
E YTminus δ + 1113946Tminus δ
tfx(t)Xt + fy(t)Yt + fz(t)Zt + fr(t)Rt11139601113896
+ f t uεt( 1113857 minus f t u
lowastt( 11138571113859dt1113865
2
le 6C2EX
2T + 6C
2TE 1113946
Tminus δ
tX
2sds + 6C
2TE 1113946
Tminus δ
tY2sds
+ 6C2(T minus t)E 1113946
Tminus δ
tZ2sds + 6C
2(T minus t)E
middot 1113946Tminus δ
t1113944
infin
i1R
is
⎛⎝ ⎞⎠
2
ds + 6E 1113946Tminus δ
tf s u
εs( 1113857 minus f s u
lowasts( 1113857( 1113857ds1113888 1113889
2
(23)
So when t isin [T minus 2δ T] with δ 112C2 we have
sup0letleTE Yt
111386811138681113868111386811138681113868111386811138682 leCε2 t isin [T minus 2δ T]
sup0letleTE 1113946T
0Zt
111386811138681113868111386811138681113868111386811138682dsleCε2 t isin [T minus 2δ T]
sup0letleTE 1113946T
0Rt
2dsleCε2 t isin [T minus 2δ T]
(24)
After a finite number of iterations (9) (11) and (13) areobtained And (10) (12) and (14) can be proved by using a
similar method and the following inequalitiesE(1113938
T
tZsdBs)
4 ge β1E(1113938T
tZ2
sds)2 β1 gt 0E(1113938
T
t1113936infini1 Ri
sdHis)4 ge β2E( 1113938
T
t(1113936infini1 Ri
s)2ds)2 β2 gt 0
Lemma 3 Under hypothesis Assumptions 1 and 2 it holdsthe following four estimations
sup0letleT
E xεt minus xlowastt minus Xt
111386811138681113868111386811138681113868111386811138682 leCεε
2 Cε⟶ 0when ε⟶ 0
(25)
sup0letleT
E yεt minus ylowastt minus Yt
111386811138681113868111386811138681113868111386811138682 leCεε
2 Cε⟶ 0when ε⟶ 0
(26)
E 1113946T
0zεt minus zlowastt minus Zt
111386811138681113868111386811138681113868111386811138682dsleCεε
2 Cε⟶ 0when ε⟶ 0
(27)
E 1113946T
trεt minus rlowastt minus Rt
2dsleCεε
2 Cε⟶ 0when ε⟶ 0
(28)
Proof To prove (25) we observe that
1113946t
0b s x
lowasts + Xs u
εs( 1113857ds + 1113946
t
0σ s x
lowasts + Xs( 1113857dBs
+ 1113946t
01113944
infin
i1g
is xlowastsminus + Xsminus( 1113857dH
is
1113946t
0b s x
lowasts u
εs( 1113857 + 1113946
1
0bx s x
lowasts + λXs u
εs( 1113857dλXs1113890 1113891ds
+ 1113946t
0σ s x
lowasts( 1113857 + 1113946
1
0σx s x
lowasts + λXs( 1113857dλXs1113890 1113891dBs
+ 1113946t
01113944
infin
i1g
is xlowasts( 1113857 + 1113946
1
01113944
infin
i1g
ix s x
lowasts + λXs( 1113857dλXs
⎡⎣ ⎤⎦dHis
1113946t
0b s x
lowasts ulowasts( 1113857ds + 1113946
t
0σ s x
lowasts( 1113857dBs + 1113946
t
01113944
infin
i1g
is xlowasts( 1113857dH
is
+ 1113946t
0bx s x
lowasts ulowasts( 1113857Xsds + 1113946
t
0σx s x
lowasts( 1113857XsdBs
+ 1113946t
01113944
infin
i1g
ix s x
lowastsminus( 1113857dH
is
+ 1113946t
0b s x
lowasts u
εs( 1113857 minus b s x
lowasts ulowasts( 11138571113858 1113859ds + 1113946
t
0Aεds
+ 1113946t
0BεdBs + 1113946
t
0CεdH
is
xlowastt minus x0 + Xt + 1113946
t
0Aεds + 1113946
t
0BεdBs + 1113946
t
0CεdH
is
(29)
Mathematical Problems in Engineering 5
where
Aε
11139461
0bx s x
lowasts + λXs u
εs( 1113857 minus bx s x
lowasts ulowasts( 11138571113858 1113859dλXs
Bε
11139461
0σx s x
lowasts + λXs( 1113857 minus σx s x
lowasts( 11138571113858 1113859dλXs
Cε
11139461
01113944
infin
i1g
ix s x
lowasts + λXs( 1113857 minus g
ix s x
lowasts( 11138571113872 1113873⎡⎣ ⎤⎦dλXs
(30)
It follows easily from Lemma 2 that
sup0letleT
E 1113946t
0A
εds1113888 1113889
2
+ 1113946t
0BεdBs1113888 1113889
2
+ 1113946t
0CεdH
is1113888 1113889
2⎡⎣ ⎤⎦ o ε21113872 1113873
(31)
Since
xεt minus x0 1113946
t
0b s x
εs u
εs( 1113857ds + 1113946
t
0σ s x
εs( 1113857dBs + 1113946
t
01113944
infin
i1g
is x
εsminus( 1113857dH
is
(32)
then
xεt minus xlowastt minus Xt 1113946
t
0b s x
εs u
εs( 1113857 minus b s x
lowasts + Xs u
εs( 11138571113858 1113859ds
+ 1113946t
0σ s x
εs( 1113857 minus σ s x
lowasts + Xs( 11138571113858 1113859dBs
+ 1113946t
01113944
infin
i1g
is x
εsminus( 1113857 minus g
is xlowastsminus + Xsminus( 11138571113872 1113873dH
is
+ 1113946t
0Aεds + 1113946
t
0BεdBs + 1113946
t
0CεdH
is
1113946t
0D
εxεs minus xlowasts minus Xs( 1113857ds + 1113946
t
0Eε
xεs minus xlowasts minus Xs( 1113857dBs
+ 1113946t
0Fε
xεs minus xlowasts minus Xs( 1113857dH
is
(33)
with Dε 111393810 bx(s xlowasts + Xs + λ(xε
s minus xlowasts minus Xs) uεs)dλ
Eε 111393810 σx(s xlowasts + Xs + λ(xε
s minus xlowasts minus Xs))dλ and Fε 111393810
1113936infini1 gi
x(s xlowasts + Xs + λ(xεs minus xlowasts minus Xs))dλ
By Gronwallrsquos inequation we have
sup0letleT
E xεt minus xlowastt minus Xt
111386811138681113868111386811138681113868111386811138682 leCεε
2 Cε⟶ 0when ε⟶ 0
(34)
Next we prove (26) (27) (28) it can be easily checkedthat
minus 1113946T
tf s x
lowasts + Xs y
lowasts + Ys z
lowasts + Zs r
lowasts + Rs u
εs( 1113857ds
+ 1113946T
ts zlowasts + Zs( 1113857dBs + 1113946
T
t1113944
infin
i1g
is xlowastsminus + Xsminus( 1113857dH
is
Φ xlowastT( 1113857 minus y
lowastt +Φx x
lowastT( 1113857XT minus Yt minus 1113946
T
tGεds
(35)
Here
Gε
11139461
0fx s x
lowasts + λXs y
lowasts + λYs z
lowasts((
+ λZs rlowasts + λRs u
εs1113857 minus fx(s)1113857dλXs
+ 11139461
0fy s x
lowasts + λXs y
lowasts + λYs z
lowasts(1113872
+ λZs rlowasts + λRs u
εs1113857 minus fy(s)1113873dλYs
+ 11139461
0fz s x
lowasts + λXs y
lowasts + λYs z
lowasts((
+ λZs rlowasts + λRs u
εs1113857 minus fz(s)1113857dλZs
+ 11139461
0fr s x
lowasts + λXs y
lowasts + λYs z
lowasts((
+ λZs rlowasts + λRs u
εs1113857 minus fr(s)1113857dλRs
(36)
Since
yεt Φ x
εT( 1113857 + 1113946
T
tf s x
εs y
εs z
εs r
εs u
εs( 1113857ds minus 1113946
T
tzsdBs
minus 1113946T
t1113944
infin
i1r
iεs dH
is
(37)
then
yεt minus ylowastt minus Yt Φ x
εT( 1113857 minus Φ x
lowastT( 1113857 minus Φx x
lowastT( 1113857XT + 1113946
T
tf s x
εs y
εs z
εs r
εs u
εs( 1113857 minus f s x
lowasts + Xs y
lowasts + Ys z
lowasts + Zs r
lowasts + Rs u
εs(1113858 1113859ds
minus 1113946T
tzεs minus zlowasts minus Zs( 1113857dBs minus 1113946
T
t1113944
infin
i1r
iεs minus r
ilowasts minus Rs1113872 1113873dH
is + 1113946
T
tGεds
(38)
6 Mathematical Problems in Engineering
Squaring both sides of the equation above we get
E yεt minus ylowastt minus Yt
111386811138681113868111386811138681113868111386811138682
+ E 1113946T
tzεs minus zlowasts minus Zs( 1113857
2ds minus E 1113946T
t1113944
infin
i1r
iεs minus r
ilowasts minus Rs1113872 1113873
2ds
E f s xεs y
εs z
εs r
εs u
εs( 1113857 minus f s x
lowasts + Xs y
lowasts + Ys z
lowasts + Zs r
lowasts + Rs u
εs( 11138571113858 1113859ds +Φ x
εT( 1113857 minus Φ x
lowastT( 1113857 minus Φx x
lowastT( 1113857XT + 1113946
T
tGεds1113896 1113897
2
(39)
From Lemma 2 and equation (25) we have
sup0letleTE 1113946T
tGεds1113888 1113889
2
o ε21113872 1113873
E Φ xεT( 1113857 minus Φ x
lowastT( 1113857 minus Φx x
lowastT( 1113857XT1113858 1113859
2 o ε21113872 1113873
(40)
en we can get (26) (27) and (28) by applying theiterative method to the above relations
Lemma 4 (variational inequality) Under the conditionsthat Assumptions 1 and 2 are established we can get thefollowing variational inequality
Ecy ylowast0( 1113857Y0 ge o(ε) (41)
Proof From the four estimations in Lemma 3 we have thefollowing estimation
E c yε0( 1113857 minus c y
lowast0 + Y0( 11138571113858 1113859 o(ε) (42)
erefore
0leE c ylowast0 + Y0( 1113857 minus c y
lowast0( 11138571113858 1113859 + o(ε) Ecy y
lowast0( 1113857Y0 + o(ε)
(43)
We introduce the following Hamiltonian functionH [0 T] times Rn times Rm times Rmtimesd times l2(Rm) times U times Rn times Rm times
Rntimesdtimes l2(Rn) as
H(t x y z r u p q w k) langp b(t x u)rang +langw σ(t x)rang
+langk g(t x)rang minus langq f(t x y z r u)rang
(44)
and the following adjoint equation
dqt Hy ulowastt( 1113857dt + Hz ulowastt( 1113857dBt + 1113936infin
i1Hi
r ulowastt( 1113857dHit
minus dpt Hx ulowastt( 1113857dt minus wtdBt minus 1113936infin
i1ki
tminus dHit
q0 cy ylowast0( 1113857 pT minus Φx xlowastT( 1113857qT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎩
(45)
where
Hx ulowastt( 1113857 Hx t x
lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857
Hy ulowastt( 1113857 Hy t x
lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857
Hz ulowastt( 1113857 Hz t x
lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857
Hr ulowastt( 1113857 Hr t x
lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857
(46)
It is easily check that adjoint equation (45) has a uniquesolution quartet (pt qt wt kt) isinM2(0 T Rn+m+mtimesd) times
l2(0 T Rm)en we get the main result of this section
Theorem 1 Let hypothesis Assumptions 1 and2 hold ulowastt isan optimal control and the corresponding optimal statetrajectories are (xlowastt ylowastt zlowastt rlowastt ) let (pt qt wt kt) be thesolution of adjoint equation (45) and then for each ad-missible control ut isin Uad[0 T] we have
H txlowastt ylowastt zlowastt rlowastt utpt qtwtkt( 1113857
geH txlowastt ylowastt zlowastt rlowastt ulowastt pt qtwtkt( 1113857asae(47)
Proof Applying It 1113954os formula to langp Xrang and langq Yrang itfollows from (6) (45) and the variational inequality that
E 1113946T
0H t xt yt zt rt u
εt pt qt wt kt( 11138571113858
minus H t xt yt zt rt ulowastt pt qt wt kt( 11138571113859dt Ecy y
lowast0( 1113857Y0 ge o(ε)
(48)
By the definition of uεt we know that for any
u isin Uad[0 T] the following inequation holds
E H t xt yt zt rt uεt pt qt wt kt( 11138571113858
minus H t xt yt zt rt ulowastt pt qt wt kt( 11138571113859ge 0
(49)
en (47) can be easily checked
4 Stochastic Control Problem withState Constraints
In this part we are going to discuss stochastic controlproblems with state constraints in control system (2)Specifically the initial state constraints and final stateconstraints are as follows
EG1 xT( 1113857 0
EG0 y0( 1113857 0(50)
Mathematical Problems in Engineering 7
where G1 Rn⟶ Rn1(n1 lt n) G0 Rm⟶ Rm1(m1 ltm)Our optimal control problem is to find ulowast isin Uad such that
J ulowast(middot)( 1113857 inf
u(middot)isinUad
J(u(middot)) (51)
subject to the state constraints (50) In the following we willapply Ekelandrsquos variational principle to solve this optimalcontrol problem Firstly we need the following assumptions
Assumption 3 Control domain U is assumed to be closedthe mappings in state constraints G1 G0 are continuouslydifferentiable and G1x G0y are bounded
For forallu1(middot) u2(middot) isin Uad let
d u1(middot) u2(middot)( 1113857 E u1(middot) minus u2(middot)1113868111386811138681113868
11138681113868111386811138682 gt 0 t isin [0 T]1113966 1113967
11138681113868111386811138681113868
11138681113868111386811138681113868 (52)
Same as Section 3 we also assume ulowastt be the optimalcontrol and the corresponding optimal state trajectories are(xlowastt ylowastt zlowastt rlowastt ) In order to solve the constraint problemwe need the following penalty cost functional for any ρgt 0
Jρ(u) E G1 xT( 11138571113858 11138591113868111386811138681113868
11138681113868111386811138682
+ E G0 y0( 11138571113858 11138591113868111386811138681113868
11138681113868111386811138682
+ J(u) minus J ulowast( ) + ρ1113868111386811138681113868
11138681113868111386811138682
1113966 111396712
(53)
It can be checked that Jρ(u) Uad⟶ R1 is continuousand for any u(middot) isin Uad
Jρ(u)ge 0 Jρ ulowast
( 1113857 ρ
Jρ ulowast
( 1113857le infu(middot)isinUad
Jρ(u) + ρ(54)
It can be obtained by Ekelandrsquos variational principle thatthere exists u
ρt isin Uad such that
(i) Jρ uρ( )le Jρ ulowast( ) ρ
(ii) d uρ ulowast( )le ρradic
(iii) Jρ(v)ge Jρ uρ( ) minusρradic
d uρ v( ) for forallv(middot) isin Uad
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(55)
For fixed ρ and admissible control uρt we define the spike
variation as follows
uρεt
vt τ le tle τ + ε
uρt otherwise
1113896 (56)
for any εgt 0 and it is easy to check from (iii) of (55) that
Jρ uρε( ) minus Jρ uρ( ) +ρradic
d uρε uρ( )ge 0 (57)
Let (xρt y
ρt z
ρt r
ρt ) be the trajectories corresponding to u
ρt
and (xρεt y
ρεt z
ρεt r
ρεt ) be the trajectories corresponding to
uρεt e variational equation we used in this section is the
same as the one in Section 3 with ulowastt uρt and
(xlowastt ylowastt zlowastt rlowastt ) (xρt y
ρt z
ρt r
ρt ) And we also assume that
the solution of this variational equation is (Xρt Y
ρt Z
ρt R
ρt )
Similar to the approach in Lemmas 2 and 3 it can be shownthat
sup0letleTE xρεt minus x
ρt minus X
ρt
111386811138681113868111386811138681113868111386811138682 leCεε
2 Cε⟶ 0when ε⟶ 0
sup0letleTE yρεt minus y
ρt minus Y
ρt
111386811138681113868111386811138681113868111386811138682 leCεε
2 Cε⟶ 0when ε⟶ 0
E 1113946T
0zρεt minus z
ρt minus Z
ρt
111386811138681113868111386811138681113868111386811138682dsleCεε
2 Cε⟶ 0when ε⟶ 0
E 1113946T
0rρεt minus r
ρt minus R
ρt
2dsleCεε
2 Cε⟶ 0when ε⟶ 0
(58)
en by (57) the following variational inequality holds
Jρ uρε
( 1113857 minus Jρ uρ
( 1113857 +ρ
radicd u
ρε u
ρ( 1113857
J2ρ uρε( ) minus J2ρ uρ( )
Jρ uρε( ) + Jρ uρ( )+ ε
ρ
radic
langhρε1 E G1x x
ρT( 11138571113858 1113859X
ρTrang +langhρε
0 E G0y yρ0( 11138571113960 1113961Y
ρ0rang
+ hρε
E cy yρ0( 1113857Y
ρ01113960 1113961 + ε
ρ
radic+ o(ε)
(59)
where
hρε1
2E G1 xρT( 11138571113858 1113859
Jρ uρε( ) + Jρ uρ( )
hρε0
2E G0 yρ0( 11138571113858 1113859
Jρ uρε( ) + Jρ uρ( )
hρε
2E c y
ρ0( 1113857 minus c y0( 1113857 + ρ1113858 1113859
Jρ uρε( ) + Jρ uρ( )
(60)
Now let (pρεt q
ρεt w
ρεt k
ρεt ) be the solution of
minus dpρεt b
ρx(t)( 1113857
τpρεt minus f
ρx(t)( 1113857
τqρεt + σρx(t)( 1113857
τw
ρεt + 1113936infin
i1gρx(t)( 1113857
τkρεt )i
1113890 1113891dt
minus wρεt dBt minus 1113936
infin
i1kρεt( 1113857
idHit
dqρεt f
ρy(t)( 1113857
τqρεt dt + f
ρz(t)( 1113857
τqρεt dBt + 1113936
infin
i1fρir (t)1113872 1113873
τ)q
ρεt dHi
t
q0 minus G0y yρ0( 1113857h
ρε0 + cy y
ρ0( 1113857hρε pT G1x x
ρT( 1113857h
ρε1 minus Φx x
ρT( 1113857q
ρεT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(61)
8 Mathematical Problems in Engineering
Here bρx(t) bx(t x
ρt u
ρt ) σρx(t) σx(t x
ρt )
fρy(t) fy(t x
ρt y
ρt z
ρt r
ρt u
ρt ) etc Applying It 1113954os formula
to langXρt q
ρεt rang + langY
ρt p
ρεt rang variational inequality (59) can be
rewritten as
H t xρt y
ρt z
ρt r
ρt u
ρεt p
ρεt q
ρεt w
ρεt k
ρεt( 1113857
minus H t xρt y
ρt z
ρt r
ρt u
ρt p
ρεt q
ρεt w
ρεt k
ρεt( 1113857
+ ερ
radic+ o(ε)ge 0 forallvt isin U ae as
(62)
where the Hamiltonian function H is defined as (44) Since
limε⟶0 hρε0
111386811138681113868111386811138681113868111386811138682
+ hρε1
111386811138681113868111386811138681113868111386811138682
+ hρε1113868111386811138681113868
11138681113868111386811138682
1113872 1113873 1 (63)
there exists a convergent subsequence still denoted by(h
ρε0 h
ρε1 hρε) such that (h
ρε0 h
ρε1 hρε)⟶ (h
ρ0 h
ρ1 hρ) when
ε⟶ 0 with |hρ0|2 + |h
ρ1|2 + |hρ|2 1
Let (pρt q
ρt w
ρt k
ρt ) be the solution of
minus dpρt b
ρx(t)( 1113857
τpρt minus f
ρx(t)( 1113857
τqρt + σρx(t)( 1113857
τw
ρt + 1113936infin
i1gρx(t)( 1113857
τkρt )i
1113890 1113891dt
minus wρtdBt minus 1113936
infin
i1kρt( 1113857
idHit
dqρt f
ρy(t)( 1113857
τqρtdt + f
ρz(t)( 1113857
τqρtdBt + 1113936
infin
i1fρir (t)1113872 1113873
τ1113888 1113889q
ρtdHi
t
q0 minus G0y yρ0( 1113857h
ρ0 + cy y
ρ0( 1113857hρ pT G1x x
ρT( 1113857h
ρ1 minus Φx x
ρT( 1113857q
ρT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(64)
From the continuous dependence of the solutions ofFBSDEs with Levy process on the parameters we can provethat the following convergence holds(p
ρεt q
ρεt w
ρεt k
ρεt )⟶ (p
ρt q
ρt w
ρt k
ρt ) then in equation
(62) it implies that
H t xρt y
ρt z
ρt r
ρt vt p
ρt q
ρt w
ρt k
ρt( 1113857
minus H t xρt y
ρt z
ρt r
ρt u
ρt p
ρt q
ρt w
ρt k
ρt( 1113857
+ρ
radic ge 0 forallvt isin U ae as
(65)
Similarly there exists a convergent subsequence(h
ρ0 h
ρ1 h
ρ) such that (h
ρ0 h
ρ1 h
ρ)⟶ (h0 h1 h) when
ρ⟶ 0 with |h0|2 + |h1|
2 + |h|2 1 Since uρt⟶ ulowastt as
ρ⟶ 0 we have (xρt y
ρt z
ρt r
ρt )⟶ (xlowastt ylowastt zlowastt rlowastt ) and
(Xρt Y
ρt Z
ρt R
ρt )⟶ (Xt Yt Zt Rt) which is the solution of
the variational equation as same as (6)e following adjoint equation is introduced
minus dpt bx(t)( 1113857τpt minus fx(t)( 1113857
τqt + σx(t)( 1113857
τwt + 1113936infin
i1gx(t)( 1113857
τkt)
i1113890 1113891dt
minus wtdBt minus 1113936infin
i1kt( 1113857
idHit
dqt fy(t)1113872 1113873τqtdt + fz(t)( 1113857
τqtdBt + 1113936
infin
i1fρir (t)1113872 1113873
τ1113888 1113889qtdHi
t
q0 minus G0y y0( 1113857h0 + cy y0( 1113857h pT G1x xT( 1113857h1 minus Φx xT( 1113857qT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(66)
Similarly it can be proved that(p
ρt q
ρt w
ρt k
ρt )⟶ (pt qt wt kt) then inequation (65)
impliesH t x
lowastt ylowastt zlowastt rlowastt vt pt qt wt kt( 1113857
minus H t xlowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857ge 0forallvt isin U ae as
(67)
en we get the following theorem
Theorem 2 Let Assumptions 1ndash3 hold ulowastt is an optimalcontrol and the corresponding optimal state trajectories are(xlowastt ylowastt zlowastt rlowastt ) and (pt qt wt kt) is the solution of adjoint
equation (66) then there exists nonzero constant(h1 h0 h) isin (Rn1 times Rm1 times R) with |h0|
2 + |h1|2 + |h|2 1
such that for any admissible control vt isin Uad[0 T] themaximum condition (67) holds
9is conclusion can be drawn from the above analysisdirectly
5 A Financial Example
In this section we will study the problem of optimal con-sumption rate selection in the financial market which willnaturally inspire our research to the forward-backwardstochastic optimal control problem in Section 3
Mathematical Problems in Engineering 9
Assume the investorrsquos asset process xt (tge 0) in the fi-nancial market is described by the following stochasticdifferential equation with the Teugels martingale
dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin
i1gi
txtdHit
x0 Wgt 0 t isin [0 T]
⎧⎪⎪⎨
⎪⎪⎩(68)
where μt and σt ne 0 are the expected return and volatility ofthe value process xt at time t respectively and Ct is theconsumption rate process Assume μt σt gt and Ct are alluniformly bounded Ftminus measurable random processes epurpose of investors is to select the optimal consumptionstrategy Clowastt at time tge 0 minimizing the following recursiveutility
J(C(middot)) E y01113858 1113859 (69)
where yt is the following backward stochastic process
minus dyt Leminus rtC1minus R
t
1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936
infin
i1ri
tdHit
yT minus xT gt 0 t isin [0 T]
⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
(70)
with constant Lgt 0 the discount factor rgt 0 and theArrowndashPratt measure of risk aversion R isin (0 1)
If the consumption process Ct is regarded as a controlvariable then combining (68) with (70) we encounter thefollowing control system
dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin
i1gi
txtdHit
minus dyt Leminus rtC1minus R
t
1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936
infin
i1ri
tdHit
x0 Wgt 0 yT minus xT gt 0 t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(71)
which is obviously a special case of stochastic control system(2) with b(t x u) μtxt minus Ct σ(t x) σtxt g(t x)
gitxt Φx(xT) minus XT and f(t x y z r u) Leminus rt(C1minus R
t (1 minus R)) minus ryt
We can check that both Assumptions 1 and 2 are sat-isfieden we can use ourmaximum principle (eorem 1)to solve the above optimization problem Let Clowastt be anoptimal consumption rate and xlowastt ylowastt be the correspondingwealth process and recursive utility process In this case theHamiltonian function H reduces to
H t xlowastt ylowastt Ct p
lowastt qlowastt wlowastt klowastt( 1113857 langplowastt μtx
lowastt minus Ctrang
+langwlowastt σtxlowastt rang +langklowastt g
itxlowastt rang minus langqlowastt Le
minus rt C1minus Rt
1 minus Rminus rylowastt rang
(72)
and (plowastt qlowastt wlowastt klowastt ) is the solution of the following adjointequation
dqt rqtdt
minus dpt μtpt + σtwt + 1113936infin
i1ki
t( 1113857τgi
t1113890 1113891dt minus wtdBt minus 1113936infin
i1ki
tdHit
q0 1 pT qT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
(73)
According to the maximum principle (47) we have
plowastt minus Le
minus rtClowastt( 1113857
minus Rqlowastt (74)
and the optimal consumption rate
Clowastt minus
ert
L
plowasttqlowastt
1113888 1113889
minus 1R
(75)
By solving FBSDE (73) we get qlowastt ert plowastt erteμ(Tminus t)and wlowastt klowastt 0 en we get the optimal consumptionrate of the investor which is
Clowastt minus
erT
Leμ(Tminus t)
1113888 1113889
minus 1R
t isin [0 T] (76)
6 Conclusions
In this paper a nonconvex control domain case of theforward-backward stochastic control driven by Levy processis considered and we obtain the global stochastic maximumprinciple for this stochastic control problem And then theproblem of stochastic control with initial and final stateconstraints on the state variables is discussed and a nec-essary condition about existence of the optimal control isalso acquired A financial example of optimal consumptionis discussed to illustrate the application of the stochasticmaximum principle
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the NNSF (11701335) Shan-dong NSF (ZR2019BG015) NSSF (17BGL058) ShandongProvincial Higher Education Science and Technology PlanProject (J18KA236) and Shandong Womenrsquos College High-level Research Project Cultivation Foundation(2018GSPGJ07)
References
[1] L S Pontryakin V G Boltyanskti R V Gamkrelidze andE F Mischenko9eMathematical9eory of Optimal ControlProcesses John Wiley New York NY USA 1962
[2] S Peng ldquoA general stochastic maximum principle for optimalcontrol problemsrdquo SIAM Journal on Control and Optimiza-tion vol 28 no 4 pp 966ndash979 1990
10 Mathematical Problems in Engineering
[3] S G Peng ldquoBackward stochastic differrntial equations andapplication to optimal controlrdquo Applied Mathematics ampOptimization vol 27 no 4 pp 125ndash144 1993
[4] W S Xu ldquoStochastic maximum principle for optimal controlproblem of forward and backward systemrdquo Journal of Aus-tralian Mathematical Sciences vol 37 pp 249ndash259 1998B
[5] Z Wu ldquoMaximum principle for optimal control problem offully coupled forward-backward stochastic control systemrdquoSystem Science and Mathematical Sciences vol 11 no 3pp 249ndash259 1998
[6] J T Shi and ZWu ldquoemaximum principle for fully coupledforward-backward stochastic control systemrdquo Acta Auto-matica Sinica vol 32 no 2 pp 161ndash169 2006
[7] R Situ ldquoA maximum principle for optimal controls of sto-chastic systems with random jumpsrdquo in Proceedings of theNational Conference on Control 9eory and its ApplicationsQingdao China 1991
[8] J Shi and Z Wu ldquoMaximum principle for forward-backwardstochastic control system with random jumps and applica-tions to financerdquo Journal of Systems Science and Complexityvol 23 no 2 pp 219ndash231 2010
[9] J T Shi ldquoGlobal maximum principle for forward-backwardstochastic control system with Poisson jumpsrdquo Asian Journalof Control vol 16 no 1 pp 1ndash11 2014
[10] M Liu X Wang and H Huang ldquoMaximum principle forforward-backward control system driven by Ito-Levy pro-cesses under initial-terminal constraintsrdquo MathematicalProblems in Engineering vol 2017 pp 1ndash13 2017
[11] S Tang ldquoe maximum principle for partially observedoptimal control of stochastic differential equationsrdquo SIAMJournal on Control and Optimization vol 36 no 5pp 1596ndash1617 1998
[12] Q Meng ldquoA maximum principle for optimal control problemof fully coupled forward-backward stochastic systems withpartial informationrdquo Science in China Series A Mathematicsvol 52 no 7 pp 1579ndash1588 2009
[13] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward backward stichastic differential equationswith jumpsrdquo SIAM J Control Optimenleadertwodots vol 48pp 2845ndash2976 2009
[14] G Wang and Z Wu ldquoKalman-Bucy filtering equations offorward and backward stochastic systems and applications torecursive optimal control problemsrdquo Journal of MathematicalAnalysis and Applications vol 342 no 2 pp 1280ndash12962008
[15] H Xiao ldquoe maximum principle for partially observedoptimal control of forward-backward stochastic systems withrandom jumpsrdquo Journal of Systems Science and Complexityvol 24 no 6 pp 1083ndash1099 2011
[16] G Wang Z Wu and J Xiong ldquoMaximum principles forforward-backward stochastic control systems with correlatedstate and observation noisesrdquo SIAM Journal on Control andOptimization vol 51 no 1 pp 491ndash524 2013
[17] G Wang C Zhang and W Zhang ldquoStochastic maximumprinciple for mean-field type optimal control under partialinformationrdquo IEEE Transactions on Automatic Controlvol 59 no 2 pp 522ndash528 2014
[18] S Q Zhang J Xiong and X D Liu ldquoStochastic maximumprinciple for partially observed forward backward stochasticdifferential equations with jumps and regime switchingrdquoScience China Information Sciences vol 61 no 7 pp 1ndash132018
[19] J Xiong S Q Zhang S Zhang and Y Zhuang ldquoA partiallyobserved non-zero sum differential game of forward-
backward stochastic differential equations and its applicationin financerdquo Mathematical Control amp Related Fields vol 9no 2 pp 257ndash276 2019
[20] D Nualart and W Schoutens ldquoChaotic and predictablerepresentations for Levy processesrdquo Stochastic Processes and9eir Applications vol 90 no 1 pp 109ndash122 2000
[21] D Nualart and W Schoutens ldquoBackward stochastic differ-ential equations and Feynman-Kac formula for Levy pro-cesses with applications in financerdquo Bernoulli vol 7 no 5pp 761ndash776 2001
[22] Q Meng and M Tang ldquoNecessary and sufficient conditionsfor optimal control of stochastic systems associated with Levyprocessesrdquo Science in China Series F Information Sciencesvol 52 no 11 pp 1982ndash1992 2009
[23] M Tang and Q Zhang ldquoOptimal variational principle forbackward stochastic control systems associated with Levyprocessesrdquo Science China Mathematics vol 55 no 4pp 745ndash761 2012
[24] F Baghery N Khelfallah B Mezerdi and I Turpin ldquoFullycoupled forward backward stochastic differential equationsdriven by Levy processes and application to differentialgamesrdquo De Gruyter vol 22 no 3 pp 151ndash161 2014
[25] F Zhang M N Tang and Q X Meng ldquoSochastic maximumprinciple for forward-backward stochastic control systemsassociated with Levy processesrdquo Chinese Annals of Mathe-matics vol 35A no 1 pp 83ndash100 2014
[26] X R Wang and H Huang ldquoMaximum principle for forward-backward stochastic control system driven by Levy processrdquoMathematical Problems in Engineering vol 2015 Article ID702802 12 pages 2015
[27] H Huang X Wang and M Liu ldquoA maximum principle forfully coupled forward-backward stochastic control systemdriven by Levy process with terminal state constraintsrdquoJournal of Systems Science and Complexity vol 31 no 4pp 859ndash874 2018
[28] P Muthukumar and R Deepa ldquoInfinite horizon optimalcontrol of forward backward stochastic system driven byTeugels martingales with Levy processrdquo Stochastic and Dy-namics vol 3 no 17 pp 1ndash17 2017
[29] J B Wu and Z M Liu ldquoOptimal control of mean-fieldbackward doubly stochastic systems driven by Ito-Levyprocessesrdquo International Journal of Control vol 93 no 4pp 953ndash970 2018
[30] W Wang J Wu and Z Liu ldquoe optimal control of fully-coupled forward-backward doubly stochastic systems drivenby Ito-Levy processesrdquo Journal of Systems Science andComplexity vol 32 no 4 pp 997ndash1018 2019
Mathematical Problems in Engineering 11
and lots of complicated stochastic calculi in infinite-dimensionalspaces were avoided in this way Based on this approach Xiao[15] studied a partially observed optimal control of forward-backward stochastic systems with random jumps and obtainedthe maximum principle and sufficient conditions of an optimalcontrol under some certain convexity assumptions Wang et al[16] proved the maximum principles for forward-backwardstochastic control systems with correlated state and observationnoises More recent conclusions of the partially observed sto-chastic control problem can be seen from the studies conductedby Wang et al [17] Zhang et al [18] and Xiong et al [19]
In these years through the study of mathematicaleconomics and mathematical finance many scholars turntheir attention to the stochastic control system driven byLevy process In 2000 Nualart and Schoutens [20] built apair of pairwise strongly orthonormal martingales whichwas called the Teugels martingale Meanwhile under someexponential moment conditions they also obtained amartingale representation in that paper Under these twoimportant conclusions for BSDE driven by the Teugelsmartingale they [21] proved the existence and uniquenesstheorem of its solution in the next year From then on anumber of important results were proved Meng and Tang[22] obtained the maximum principle of the forwardstochastic control system driven by Levy process Anecessary and sufficient condition for the existence of theoptimal control of backward stochastic control systemsdriven by Levy process was deduced by Tang and Zhang[23] through convex variation methods and dualitytechniques For the forward-backward stochastic controlsystem driven by Levy process there are also a lot ofachievements based on the existence and uniquenesstheorem of FBSDEL [24] Zhang et al [25] obtained anecessary condition of the optimal control and verifica-tion theorem but in their control system the backwardstate variables yt and zt did not enter the forward partWang and Huang [26] extended this result to the fullycoupled control system and obtained the continuity resultdepending on parameters about FBSDEL and the localform maximum principle Subsequently Huang et al [27]studied this control system with terminal state constraintsand obtained the corresponding necessary maximumprinciple using Ekelandrsquos variational For more recentconclusions about the stochastic control problem drivenby Levy process please refer to [28ndash30]
In this paper we will study the optimal control problemfor forward-backward stochastic control systems driven byLevy process which could be considered as a nonconvexcontrol domain case that is extended from the result of [25]
With the technique of spike variation and Ekelandrsquos vari-ational principle the maximum principle of this type ofcontrol system and the control system with initial and finalstate constraints are obtained
e structure of this paper is as follows Section 2 de-scribes some of the preparations used in this paper emaximum principle and the one with initial and terminalstate constraints as the major results of this paper will beshown in Sections 3 and 4 As an application of the max-imum principle Section 5 gives an optimal consumptionproblem in the financial market Section 6 is the summary ofthis article
2 Preliminary Statement
Let (ΩFt P) be a complete probability space which sat-isfied the usual conditions and the information structure isgiven byFt which is generated by two processes a standardBrownian motion Bt1113864 11138650le tleT valued in Rd and an inde-pendent 1-dimensional Levy process Lt1113864 11138650le tleT of the formLt bt + lt here lt is a pure jump process And assume thatLevy measure ] satisfies the following two conditionsthereby Levy process Lt1113864 11138650le tleT has moments in all orders
(i) 1113938R(1and x2)](dx)ltinfin
(ii) 1113938(minus εε)c eλ |x|](dx)ltinfin forallεgt 0 and some λgt 0
Denote L1t Lt Lt Lt minus Ltminus and Li
t 11139360ltslet(Ls)i
for ige 2 And let Yit Li
t minus E[Lit] (ige 1) be the compensated
power jump process of order i then Teugels martingale isdefined by Hi
t 1113936ij1 cijY
jt the coefficients cij correspond to
orthonormalization of the polynomials 1 x x2 withrespect to the measure μ(dx) υ(dx) + σ2δ0(dx)
en Hit1113864 1113865infini1 are pairwise strongly orthogonal martin-
gales and their predictable quadratic variation processes arelangHi
t Hjt rang δijt δij is an indicator function here And
[Hi Hj]t minus langHit H
jt rang is an Ftminus martingale for more details
of the Teugels martingale see Nualart and Schoutens [20]In the following of this section we shall assume some
notations for a Hilbert space H
l2(H) ϕ |H minus valued 1113936infini1 ϕi2 ltinfin1113966 1113967
L2(ΩH) ξ |
H valuedFT minus measurable E|ξ|2 ltinfinl2(0 TH) ϕi
t | l2(H)minus1113864
valuedFt minus measurable 1113936infini1 E 1113938
T
0 ϕit2dtltinfin
M2(0 TH) ϕ(middot) |1113864
H minus valuedFt minus measurable E 1113938T
0 |ϕt|2dtltinfin
For the following FBSDEL
dxt b t xt yt zt rt( 1113857dt + σ t xt yt zt rt( 1113857dBt + 1113936infin
i1gi t xtminus ytminus zt rt( 1113857dHi
t
minus dyt f t xt yt zt rt( 1113857dt minus ztdBt minus 1113936infin
i1ri
tdHit
x0 a yT Φ xT( 1113857
⎧⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎩
(1)
2 Mathematical Problems in Engineering
where (xt yt zt rt) take the value inΩ times [0 T] times Rn times Rm times
Rmtimesd times l2(Rm) and mappings b σ g and f take the value inRn Rntimesd l2(Rn) and Rm respectively Convenient forwriting set column vector α (x y z)T andA(t α r) (minus Mτf(t α r) Mb(t α r) Mσ(t α r))Twhere M is a m times n full rank matrix
Assumption 1
(i) All mappings in equation (1) are uniformly Lip-schitz continuous in their own argumentsrespectively
(ii) For all (ω t) isin Ω times [0 T] l(ω t 0 0 0 0)
isinM2(0 T Rn+m+mtimesd) times l2(0 T Rm) for l b f σrespectively and g(ω t 0 0 0 0) isin l2(Rn)
(iii) langΦ(x) minus Φ(x) M(x minus x)rangge β|M1113954x|2(iv) langA(tα r) minus A(tα r)α minus αrang + 1113936
infini1langM 1113954gi 1113954rirangle minus μ
1|M1113954x|2 minus μ2(|Mτ1113954y|2 + |Mτ1113954z|2 + 1113936infini1 Mτ 1113954ri2)
where α (x y z) 1113954x x minus x 1113954y y minus y 1113954z z minus z1113954ri ri minus ri 1113954gi gi(tα r) minus gi(tα r) μ1 μ2 and βare nonnegative constants which satisfiedμ1 + μ2 gt 0 μ2 + βgt 0 and μ1 gt 0βgt 0(resp μ2 gt 0)when mgtn (resp ngtm)
en the following existence and uniqueness of thesolution conclusion holds
Lemma 1 9ere exists a unique solution in M2(0 TH)
satisfying FBSDEL (1) under Assumption 19e detailed certification process of this conclusion can be
seen in [24]
3 Stochastic Maximum Principle
In this section for any given admissible control u(middot) weconsider the following stochastic control system
dxt b t xt ut( 1113857dt + σ t xt( 1113857dBt + 1113936infin
i1gi t xtminus( 1113857dHi
t
minus dyt f t xt yt zt rt ut1113872 1113873dt minus ztdBt minus 1113936infin
i1ri
tdHit
x0 a yT Φ xT( 1113857 t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎩
(2)
where a isin Rn is given An admissible controlu(middot) isinM2(0 T Rp) is an Ft-predictable process whichtakes values in a nonempty subset U of Rp Uad is the set ofall admissible controls
And the performance criterion is
J(u) Ec y0( 1113857 (3)
where c Rm⟶ R is a given Frechet differential functionOur optimal control problem amounts to determining
an admissible control ulowast isin Uad such that
J ulowast(middot)( 1113857 inf
u(middot)isinUad
J(u(middot)) (4)
In order to get the necessary conditions for the optimalcontrol we assume ulowastt is the optimal control and thecorresponding solution of (2) is recorded as (xlowastt ylowastt zlowastt rlowastt )
and introduce the ldquospike variational controlrdquo as follows
uεt
vt τ le tle τ + ε
ulowastt otherwise1113896 (5)
and (xεt yε
t zεt rεt) are the state trajectories of uε
t here vt is anarbitrary admissible control and ε is a sufficiently smallconstant
We also need the following assumption and variationalequation (6)
Assumption 2
(i) b f g σ Φ and c are continuously differentiablewith respect to (x y z r u) and the derivatives areall bounded
(ii) ere exists a constant Cgt 0 and it holds that|cy|leC(1 + |y|)
dXt bx(t)Xt + b t uεt( 1113857 minus b t ulowastt( 11138571113858 1113859dt + σx(t)XtdBt + 1113936
infin
i1gi
x(t)XtdHit
minus dYt fx(t)Xt + fy(t)Yt + fz(t)Zt + fr(t)Rt + f t uεt( 1113857 minus f t ulowastt( 11138571113960 1113961dt
minus ZtdBt minus 1113936infin
i1Ri
tdHit
X0 0
YT Φx(t)XT
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(6)
Mathematical Problems in Engineering 3
Here bx(t) bx(txlowastt ulowastt ) σx(t) σx(txlowastt ) gix(t)
gix(txlowastt ) b(tuε
t) b(txεt u
εt) b(tulowastt ) b (txlowastt ulowastt )
fw(t) fw(txlowastt ylowastt zlowastt rlowastt ulowastt ) (w xyz middot r) f(tuεt)
f(xεt y
εt z
εt r
εt u
εt) and f(tulowastt ) f(xlowastt ylowastt zlowastt rlowastt ulowastt )
Lemma 2 Suppose Assumptions 1 and 2 hold for the first-order variation X Y Z R we have the following estimations
sup0letleT
E Xt
111386811138681113868111386811138681113868111386811138682 leCε2 (7)
sup0letleT
E Xt
111386811138681113868111386811138681113868111386811138684 leCε4 (8)
sup0letleT
E Yt
111386811138681113868111386811138681113868111386811138682 leCε2 (9)
sup0letleT
E Yt
111386811138681113868111386811138681113868111386811138684 leCε4 (10)
sup0letleT
E 1113946T
0Zt
111386811138681113868111386811138681113868111386811138682dsleCε2 (11)
sup0letleT
E 1113946T
0Zt
111386811138681113868111386811138681113868111386811138682ds1113888 1113889
2
leCε4 (12)
sup0letleT
E 1113946T
0Rt
2dsleCε2 (13)
sup0letleT
E 1113946T
0Rt
2ds1113888 1113889
2
leCε4 (14)
Proof We first prove inequations (7) and (8) For theforward part of the first-order variation equation we have
E Xt
111386811138681113868111386811138681113868111386811138682
E 1113946t
0bx(s)Xs + b s u
εs( 1113857 minus b s u
lowasts( 11138571113858 1113859ds + σx(s)XsdBs + 1113944
infin
i1g
ix(s)XsdH
is
⎧⎨
⎩
⎫⎬
⎭
2
le 4 E 1113946t
0bx(s)Xsds1113888 1113889
2
+ E 1113946t
0b s u
εs( 1113857 minus b s u
lowasts( 11138571113858 1113859ds1113888 1113889
2
+ E 1113946t
0σx(s)Xs1113858 1113859
2ds + E 1113946t
01113944
infin
i1g
ix(s)Xs
⎡⎣ ⎤⎦2
ds⎧⎨
⎩
⎫⎬
⎭
le 12C2TE 1113946
t
0X
2sds + 4E 1113946
t
0b s u
εs( 1113857 minus b s u
lowasts( 11138571113858 1113859ds1113888 1113889
2
(15)
Applying Gronwallrsquos inequation we have
E Xt
111386811138681113868111386811138681113868111386811138682 leCε2 for all t isin [0 T] (16)
Similarly (8) holdsWe next estimate Yt Zt and Rt the backward part of the
first-order variation equation can be rewritten as
Yt + 1113946T
tZsdBs + 1113946
T
t1113944
infin
i1R
isdH
is Φx(t)XT + 1113946
T
tfx(t)Xt1113858
+ fy(t)Yt + fz(t)Zt
+ fr(t)Rt
+ f t uεt( 1113857 minus f t u
lowastt( 1113857]dt
(17)
Squaring both sides of (17) and using the fact of
EYt 1113946T
tZsdBs 0
EYt 1113946T
t1113944
infin
i1R
isdH
is 0
E 1113946T
tZsdBs 1113946
T
t1113944
infin
i1R
isdH
is 0
(18)
we get
E Yt
111386811138681113868111386811138681113868111386811138682
+ E 1113946T
tZ2sds + E 1113946
T
t1113944
infin
i1R
is
⎛⎝ ⎞⎠
2
dHis
E Φx(t)XT + 1113946T
tfx(t)Xt + fy(t)Yt + fz(t)Zt + fr(t)Rt + f t u
εt( 1113857 minus f t u
lowastt( 11138571113960 1113961dt1113896 1113897
2
le 6C2EX
2T + 6C
2TE 1113946
T
tX
2sds + 6C
2TE 1113946
T
tY2sds + 6C
2(T minus t)E 1113946
T
tZ2sds
+ 6C2(T minus t)E 1113946
T
t1113944
infin
i1R
is
⎛⎝ ⎞⎠
2
ds + 6E 1113946T
tf s u
εs( 1113857 minus f s u
lowasts( 1113857( 1113857ds1113888 1113889
2
(19)
4 Mathematical Problems in Engineering
When t isin [T minus δ T] with δ 112C2 we have
E Yt
111386811138681113868111386811138681113868111386811138682
+12
E 1113946T
tZ2sds +
12
E 1113946T
t1113944
infin
i1R
is
⎛⎝ ⎞⎠
2
dHis
le 6C2EX
2T + 6C
2TE 1113946
T
tX
2sds + 6C
2TE
middot 1113946T
tY2sds + 6E 1113946
T
tf s u
εs( 1113857 minus f s u
lowasts( 1113857( 1113857ds1113888 1113889
2
(20)
Applying Gronwallrsquos inequation we have
sup0letleT
E Yt
111386811138681113868111386811138681113868111386811138682 leCε2 t isin [T minus δ T]
sup0letleT
E 1113946T
0Zt
111386811138681113868111386811138681113868111386811138682dsleCε2 t isin [T minus δ T]
sup0letleT
E 1113946T
0Rt
2dsleCε2 t isin [T minus δ T]
(21)
Consider the BSDE of the first-order variation equationin the interval [t T minus δ]
Yt + 1113946Tminus δ
tZsdBs + 1113946
Tminus δ
t1113944
infin
i1R
isdH
is YTminus δ
+ 1113946Tminus δ
tfx(t)Xt + fy(t)Yt + fz(t)Zt1113960
+ fr(t)Rt + f t uεt( 1113857 minus f t u
lowastt( 11138571113859dt
(22)
us
E Yt
111386811138681113868111386811138681113868111386811138682
+ E 1113946Tminus δ
tZ2sds + E 1113946
Tminus δ
t1113944
infin
i1R
is
⎛⎝ ⎞⎠
2
dHis
E YTminus δ + 1113946Tminus δ
tfx(t)Xt + fy(t)Yt + fz(t)Zt + fr(t)Rt11139601113896
+ f t uεt( 1113857 minus f t u
lowastt( 11138571113859dt1113865
2
le 6C2EX
2T + 6C
2TE 1113946
Tminus δ
tX
2sds + 6C
2TE 1113946
Tminus δ
tY2sds
+ 6C2(T minus t)E 1113946
Tminus δ
tZ2sds + 6C
2(T minus t)E
middot 1113946Tminus δ
t1113944
infin
i1R
is
⎛⎝ ⎞⎠
2
ds + 6E 1113946Tminus δ
tf s u
εs( 1113857 minus f s u
lowasts( 1113857( 1113857ds1113888 1113889
2
(23)
So when t isin [T minus 2δ T] with δ 112C2 we have
sup0letleTE Yt
111386811138681113868111386811138681113868111386811138682 leCε2 t isin [T minus 2δ T]
sup0letleTE 1113946T
0Zt
111386811138681113868111386811138681113868111386811138682dsleCε2 t isin [T minus 2δ T]
sup0letleTE 1113946T
0Rt
2dsleCε2 t isin [T minus 2δ T]
(24)
After a finite number of iterations (9) (11) and (13) areobtained And (10) (12) and (14) can be proved by using a
similar method and the following inequalitiesE(1113938
T
tZsdBs)
4 ge β1E(1113938T
tZ2
sds)2 β1 gt 0E(1113938
T
t1113936infini1 Ri
sdHis)4 ge β2E( 1113938
T
t(1113936infini1 Ri
s)2ds)2 β2 gt 0
Lemma 3 Under hypothesis Assumptions 1 and 2 it holdsthe following four estimations
sup0letleT
E xεt minus xlowastt minus Xt
111386811138681113868111386811138681113868111386811138682 leCεε
2 Cε⟶ 0when ε⟶ 0
(25)
sup0letleT
E yεt minus ylowastt minus Yt
111386811138681113868111386811138681113868111386811138682 leCεε
2 Cε⟶ 0when ε⟶ 0
(26)
E 1113946T
0zεt minus zlowastt minus Zt
111386811138681113868111386811138681113868111386811138682dsleCεε
2 Cε⟶ 0when ε⟶ 0
(27)
E 1113946T
trεt minus rlowastt minus Rt
2dsleCεε
2 Cε⟶ 0when ε⟶ 0
(28)
Proof To prove (25) we observe that
1113946t
0b s x
lowasts + Xs u
εs( 1113857ds + 1113946
t
0σ s x
lowasts + Xs( 1113857dBs
+ 1113946t
01113944
infin
i1g
is xlowastsminus + Xsminus( 1113857dH
is
1113946t
0b s x
lowasts u
εs( 1113857 + 1113946
1
0bx s x
lowasts + λXs u
εs( 1113857dλXs1113890 1113891ds
+ 1113946t
0σ s x
lowasts( 1113857 + 1113946
1
0σx s x
lowasts + λXs( 1113857dλXs1113890 1113891dBs
+ 1113946t
01113944
infin
i1g
is xlowasts( 1113857 + 1113946
1
01113944
infin
i1g
ix s x
lowasts + λXs( 1113857dλXs
⎡⎣ ⎤⎦dHis
1113946t
0b s x
lowasts ulowasts( 1113857ds + 1113946
t
0σ s x
lowasts( 1113857dBs + 1113946
t
01113944
infin
i1g
is xlowasts( 1113857dH
is
+ 1113946t
0bx s x
lowasts ulowasts( 1113857Xsds + 1113946
t
0σx s x
lowasts( 1113857XsdBs
+ 1113946t
01113944
infin
i1g
ix s x
lowastsminus( 1113857dH
is
+ 1113946t
0b s x
lowasts u
εs( 1113857 minus b s x
lowasts ulowasts( 11138571113858 1113859ds + 1113946
t
0Aεds
+ 1113946t
0BεdBs + 1113946
t
0CεdH
is
xlowastt minus x0 + Xt + 1113946
t
0Aεds + 1113946
t
0BεdBs + 1113946
t
0CεdH
is
(29)
Mathematical Problems in Engineering 5
where
Aε
11139461
0bx s x
lowasts + λXs u
εs( 1113857 minus bx s x
lowasts ulowasts( 11138571113858 1113859dλXs
Bε
11139461
0σx s x
lowasts + λXs( 1113857 minus σx s x
lowasts( 11138571113858 1113859dλXs
Cε
11139461
01113944
infin
i1g
ix s x
lowasts + λXs( 1113857 minus g
ix s x
lowasts( 11138571113872 1113873⎡⎣ ⎤⎦dλXs
(30)
It follows easily from Lemma 2 that
sup0letleT
E 1113946t
0A
εds1113888 1113889
2
+ 1113946t
0BεdBs1113888 1113889
2
+ 1113946t
0CεdH
is1113888 1113889
2⎡⎣ ⎤⎦ o ε21113872 1113873
(31)
Since
xεt minus x0 1113946
t
0b s x
εs u
εs( 1113857ds + 1113946
t
0σ s x
εs( 1113857dBs + 1113946
t
01113944
infin
i1g
is x
εsminus( 1113857dH
is
(32)
then
xεt minus xlowastt minus Xt 1113946
t
0b s x
εs u
εs( 1113857 minus b s x
lowasts + Xs u
εs( 11138571113858 1113859ds
+ 1113946t
0σ s x
εs( 1113857 minus σ s x
lowasts + Xs( 11138571113858 1113859dBs
+ 1113946t
01113944
infin
i1g
is x
εsminus( 1113857 minus g
is xlowastsminus + Xsminus( 11138571113872 1113873dH
is
+ 1113946t
0Aεds + 1113946
t
0BεdBs + 1113946
t
0CεdH
is
1113946t
0D
εxεs minus xlowasts minus Xs( 1113857ds + 1113946
t
0Eε
xεs minus xlowasts minus Xs( 1113857dBs
+ 1113946t
0Fε
xεs minus xlowasts minus Xs( 1113857dH
is
(33)
with Dε 111393810 bx(s xlowasts + Xs + λ(xε
s minus xlowasts minus Xs) uεs)dλ
Eε 111393810 σx(s xlowasts + Xs + λ(xε
s minus xlowasts minus Xs))dλ and Fε 111393810
1113936infini1 gi
x(s xlowasts + Xs + λ(xεs minus xlowasts minus Xs))dλ
By Gronwallrsquos inequation we have
sup0letleT
E xεt minus xlowastt minus Xt
111386811138681113868111386811138681113868111386811138682 leCεε
2 Cε⟶ 0when ε⟶ 0
(34)
Next we prove (26) (27) (28) it can be easily checkedthat
minus 1113946T
tf s x
lowasts + Xs y
lowasts + Ys z
lowasts + Zs r
lowasts + Rs u
εs( 1113857ds
+ 1113946T
ts zlowasts + Zs( 1113857dBs + 1113946
T
t1113944
infin
i1g
is xlowastsminus + Xsminus( 1113857dH
is
Φ xlowastT( 1113857 minus y
lowastt +Φx x
lowastT( 1113857XT minus Yt minus 1113946
T
tGεds
(35)
Here
Gε
11139461
0fx s x
lowasts + λXs y
lowasts + λYs z
lowasts((
+ λZs rlowasts + λRs u
εs1113857 minus fx(s)1113857dλXs
+ 11139461
0fy s x
lowasts + λXs y
lowasts + λYs z
lowasts(1113872
+ λZs rlowasts + λRs u
εs1113857 minus fy(s)1113873dλYs
+ 11139461
0fz s x
lowasts + λXs y
lowasts + λYs z
lowasts((
+ λZs rlowasts + λRs u
εs1113857 minus fz(s)1113857dλZs
+ 11139461
0fr s x
lowasts + λXs y
lowasts + λYs z
lowasts((
+ λZs rlowasts + λRs u
εs1113857 minus fr(s)1113857dλRs
(36)
Since
yεt Φ x
εT( 1113857 + 1113946
T
tf s x
εs y
εs z
εs r
εs u
εs( 1113857ds minus 1113946
T
tzsdBs
minus 1113946T
t1113944
infin
i1r
iεs dH
is
(37)
then
yεt minus ylowastt minus Yt Φ x
εT( 1113857 minus Φ x
lowastT( 1113857 minus Φx x
lowastT( 1113857XT + 1113946
T
tf s x
εs y
εs z
εs r
εs u
εs( 1113857 minus f s x
lowasts + Xs y
lowasts + Ys z
lowasts + Zs r
lowasts + Rs u
εs(1113858 1113859ds
minus 1113946T
tzεs minus zlowasts minus Zs( 1113857dBs minus 1113946
T
t1113944
infin
i1r
iεs minus r
ilowasts minus Rs1113872 1113873dH
is + 1113946
T
tGεds
(38)
6 Mathematical Problems in Engineering
Squaring both sides of the equation above we get
E yεt minus ylowastt minus Yt
111386811138681113868111386811138681113868111386811138682
+ E 1113946T
tzεs minus zlowasts minus Zs( 1113857
2ds minus E 1113946T
t1113944
infin
i1r
iεs minus r
ilowasts minus Rs1113872 1113873
2ds
E f s xεs y
εs z
εs r
εs u
εs( 1113857 minus f s x
lowasts + Xs y
lowasts + Ys z
lowasts + Zs r
lowasts + Rs u
εs( 11138571113858 1113859ds +Φ x
εT( 1113857 minus Φ x
lowastT( 1113857 minus Φx x
lowastT( 1113857XT + 1113946
T
tGεds1113896 1113897
2
(39)
From Lemma 2 and equation (25) we have
sup0letleTE 1113946T
tGεds1113888 1113889
2
o ε21113872 1113873
E Φ xεT( 1113857 minus Φ x
lowastT( 1113857 minus Φx x
lowastT( 1113857XT1113858 1113859
2 o ε21113872 1113873
(40)
en we can get (26) (27) and (28) by applying theiterative method to the above relations
Lemma 4 (variational inequality) Under the conditionsthat Assumptions 1 and 2 are established we can get thefollowing variational inequality
Ecy ylowast0( 1113857Y0 ge o(ε) (41)
Proof From the four estimations in Lemma 3 we have thefollowing estimation
E c yε0( 1113857 minus c y
lowast0 + Y0( 11138571113858 1113859 o(ε) (42)
erefore
0leE c ylowast0 + Y0( 1113857 minus c y
lowast0( 11138571113858 1113859 + o(ε) Ecy y
lowast0( 1113857Y0 + o(ε)
(43)
We introduce the following Hamiltonian functionH [0 T] times Rn times Rm times Rmtimesd times l2(Rm) times U times Rn times Rm times
Rntimesdtimes l2(Rn) as
H(t x y z r u p q w k) langp b(t x u)rang +langw σ(t x)rang
+langk g(t x)rang minus langq f(t x y z r u)rang
(44)
and the following adjoint equation
dqt Hy ulowastt( 1113857dt + Hz ulowastt( 1113857dBt + 1113936infin
i1Hi
r ulowastt( 1113857dHit
minus dpt Hx ulowastt( 1113857dt minus wtdBt minus 1113936infin
i1ki
tminus dHit
q0 cy ylowast0( 1113857 pT minus Φx xlowastT( 1113857qT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎩
(45)
where
Hx ulowastt( 1113857 Hx t x
lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857
Hy ulowastt( 1113857 Hy t x
lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857
Hz ulowastt( 1113857 Hz t x
lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857
Hr ulowastt( 1113857 Hr t x
lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857
(46)
It is easily check that adjoint equation (45) has a uniquesolution quartet (pt qt wt kt) isinM2(0 T Rn+m+mtimesd) times
l2(0 T Rm)en we get the main result of this section
Theorem 1 Let hypothesis Assumptions 1 and2 hold ulowastt isan optimal control and the corresponding optimal statetrajectories are (xlowastt ylowastt zlowastt rlowastt ) let (pt qt wt kt) be thesolution of adjoint equation (45) and then for each ad-missible control ut isin Uad[0 T] we have
H txlowastt ylowastt zlowastt rlowastt utpt qtwtkt( 1113857
geH txlowastt ylowastt zlowastt rlowastt ulowastt pt qtwtkt( 1113857asae(47)
Proof Applying It 1113954os formula to langp Xrang and langq Yrang itfollows from (6) (45) and the variational inequality that
E 1113946T
0H t xt yt zt rt u
εt pt qt wt kt( 11138571113858
minus H t xt yt zt rt ulowastt pt qt wt kt( 11138571113859dt Ecy y
lowast0( 1113857Y0 ge o(ε)
(48)
By the definition of uεt we know that for any
u isin Uad[0 T] the following inequation holds
E H t xt yt zt rt uεt pt qt wt kt( 11138571113858
minus H t xt yt zt rt ulowastt pt qt wt kt( 11138571113859ge 0
(49)
en (47) can be easily checked
4 Stochastic Control Problem withState Constraints
In this part we are going to discuss stochastic controlproblems with state constraints in control system (2)Specifically the initial state constraints and final stateconstraints are as follows
EG1 xT( 1113857 0
EG0 y0( 1113857 0(50)
Mathematical Problems in Engineering 7
where G1 Rn⟶ Rn1(n1 lt n) G0 Rm⟶ Rm1(m1 ltm)Our optimal control problem is to find ulowast isin Uad such that
J ulowast(middot)( 1113857 inf
u(middot)isinUad
J(u(middot)) (51)
subject to the state constraints (50) In the following we willapply Ekelandrsquos variational principle to solve this optimalcontrol problem Firstly we need the following assumptions
Assumption 3 Control domain U is assumed to be closedthe mappings in state constraints G1 G0 are continuouslydifferentiable and G1x G0y are bounded
For forallu1(middot) u2(middot) isin Uad let
d u1(middot) u2(middot)( 1113857 E u1(middot) minus u2(middot)1113868111386811138681113868
11138681113868111386811138682 gt 0 t isin [0 T]1113966 1113967
11138681113868111386811138681113868
11138681113868111386811138681113868 (52)
Same as Section 3 we also assume ulowastt be the optimalcontrol and the corresponding optimal state trajectories are(xlowastt ylowastt zlowastt rlowastt ) In order to solve the constraint problemwe need the following penalty cost functional for any ρgt 0
Jρ(u) E G1 xT( 11138571113858 11138591113868111386811138681113868
11138681113868111386811138682
+ E G0 y0( 11138571113858 11138591113868111386811138681113868
11138681113868111386811138682
+ J(u) minus J ulowast( ) + ρ1113868111386811138681113868
11138681113868111386811138682
1113966 111396712
(53)
It can be checked that Jρ(u) Uad⟶ R1 is continuousand for any u(middot) isin Uad
Jρ(u)ge 0 Jρ ulowast
( 1113857 ρ
Jρ ulowast
( 1113857le infu(middot)isinUad
Jρ(u) + ρ(54)
It can be obtained by Ekelandrsquos variational principle thatthere exists u
ρt isin Uad such that
(i) Jρ uρ( )le Jρ ulowast( ) ρ
(ii) d uρ ulowast( )le ρradic
(iii) Jρ(v)ge Jρ uρ( ) minusρradic
d uρ v( ) for forallv(middot) isin Uad
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(55)
For fixed ρ and admissible control uρt we define the spike
variation as follows
uρεt
vt τ le tle τ + ε
uρt otherwise
1113896 (56)
for any εgt 0 and it is easy to check from (iii) of (55) that
Jρ uρε( ) minus Jρ uρ( ) +ρradic
d uρε uρ( )ge 0 (57)
Let (xρt y
ρt z
ρt r
ρt ) be the trajectories corresponding to u
ρt
and (xρεt y
ρεt z
ρεt r
ρεt ) be the trajectories corresponding to
uρεt e variational equation we used in this section is the
same as the one in Section 3 with ulowastt uρt and
(xlowastt ylowastt zlowastt rlowastt ) (xρt y
ρt z
ρt r
ρt ) And we also assume that
the solution of this variational equation is (Xρt Y
ρt Z
ρt R
ρt )
Similar to the approach in Lemmas 2 and 3 it can be shownthat
sup0letleTE xρεt minus x
ρt minus X
ρt
111386811138681113868111386811138681113868111386811138682 leCεε
2 Cε⟶ 0when ε⟶ 0
sup0letleTE yρεt minus y
ρt minus Y
ρt
111386811138681113868111386811138681113868111386811138682 leCεε
2 Cε⟶ 0when ε⟶ 0
E 1113946T
0zρεt minus z
ρt minus Z
ρt
111386811138681113868111386811138681113868111386811138682dsleCεε
2 Cε⟶ 0when ε⟶ 0
E 1113946T
0rρεt minus r
ρt minus R
ρt
2dsleCεε
2 Cε⟶ 0when ε⟶ 0
(58)
en by (57) the following variational inequality holds
Jρ uρε
( 1113857 minus Jρ uρ
( 1113857 +ρ
radicd u
ρε u
ρ( 1113857
J2ρ uρε( ) minus J2ρ uρ( )
Jρ uρε( ) + Jρ uρ( )+ ε
ρ
radic
langhρε1 E G1x x
ρT( 11138571113858 1113859X
ρTrang +langhρε
0 E G0y yρ0( 11138571113960 1113961Y
ρ0rang
+ hρε
E cy yρ0( 1113857Y
ρ01113960 1113961 + ε
ρ
radic+ o(ε)
(59)
where
hρε1
2E G1 xρT( 11138571113858 1113859
Jρ uρε( ) + Jρ uρ( )
hρε0
2E G0 yρ0( 11138571113858 1113859
Jρ uρε( ) + Jρ uρ( )
hρε
2E c y
ρ0( 1113857 minus c y0( 1113857 + ρ1113858 1113859
Jρ uρε( ) + Jρ uρ( )
(60)
Now let (pρεt q
ρεt w
ρεt k
ρεt ) be the solution of
minus dpρεt b
ρx(t)( 1113857
τpρεt minus f
ρx(t)( 1113857
τqρεt + σρx(t)( 1113857
τw
ρεt + 1113936infin
i1gρx(t)( 1113857
τkρεt )i
1113890 1113891dt
minus wρεt dBt minus 1113936
infin
i1kρεt( 1113857
idHit
dqρεt f
ρy(t)( 1113857
τqρεt dt + f
ρz(t)( 1113857
τqρεt dBt + 1113936
infin
i1fρir (t)1113872 1113873
τ)q
ρεt dHi
t
q0 minus G0y yρ0( 1113857h
ρε0 + cy y
ρ0( 1113857hρε pT G1x x
ρT( 1113857h
ρε1 minus Φx x
ρT( 1113857q
ρεT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(61)
8 Mathematical Problems in Engineering
Here bρx(t) bx(t x
ρt u
ρt ) σρx(t) σx(t x
ρt )
fρy(t) fy(t x
ρt y
ρt z
ρt r
ρt u
ρt ) etc Applying It 1113954os formula
to langXρt q
ρεt rang + langY
ρt p
ρεt rang variational inequality (59) can be
rewritten as
H t xρt y
ρt z
ρt r
ρt u
ρεt p
ρεt q
ρεt w
ρεt k
ρεt( 1113857
minus H t xρt y
ρt z
ρt r
ρt u
ρt p
ρεt q
ρεt w
ρεt k
ρεt( 1113857
+ ερ
radic+ o(ε)ge 0 forallvt isin U ae as
(62)
where the Hamiltonian function H is defined as (44) Since
limε⟶0 hρε0
111386811138681113868111386811138681113868111386811138682
+ hρε1
111386811138681113868111386811138681113868111386811138682
+ hρε1113868111386811138681113868
11138681113868111386811138682
1113872 1113873 1 (63)
there exists a convergent subsequence still denoted by(h
ρε0 h
ρε1 hρε) such that (h
ρε0 h
ρε1 hρε)⟶ (h
ρ0 h
ρ1 hρ) when
ε⟶ 0 with |hρ0|2 + |h
ρ1|2 + |hρ|2 1
Let (pρt q
ρt w
ρt k
ρt ) be the solution of
minus dpρt b
ρx(t)( 1113857
τpρt minus f
ρx(t)( 1113857
τqρt + σρx(t)( 1113857
τw
ρt + 1113936infin
i1gρx(t)( 1113857
τkρt )i
1113890 1113891dt
minus wρtdBt minus 1113936
infin
i1kρt( 1113857
idHit
dqρt f
ρy(t)( 1113857
τqρtdt + f
ρz(t)( 1113857
τqρtdBt + 1113936
infin
i1fρir (t)1113872 1113873
τ1113888 1113889q
ρtdHi
t
q0 minus G0y yρ0( 1113857h
ρ0 + cy y
ρ0( 1113857hρ pT G1x x
ρT( 1113857h
ρ1 minus Φx x
ρT( 1113857q
ρT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(64)
From the continuous dependence of the solutions ofFBSDEs with Levy process on the parameters we can provethat the following convergence holds(p
ρεt q
ρεt w
ρεt k
ρεt )⟶ (p
ρt q
ρt w
ρt k
ρt ) then in equation
(62) it implies that
H t xρt y
ρt z
ρt r
ρt vt p
ρt q
ρt w
ρt k
ρt( 1113857
minus H t xρt y
ρt z
ρt r
ρt u
ρt p
ρt q
ρt w
ρt k
ρt( 1113857
+ρ
radic ge 0 forallvt isin U ae as
(65)
Similarly there exists a convergent subsequence(h
ρ0 h
ρ1 h
ρ) such that (h
ρ0 h
ρ1 h
ρ)⟶ (h0 h1 h) when
ρ⟶ 0 with |h0|2 + |h1|
2 + |h|2 1 Since uρt⟶ ulowastt as
ρ⟶ 0 we have (xρt y
ρt z
ρt r
ρt )⟶ (xlowastt ylowastt zlowastt rlowastt ) and
(Xρt Y
ρt Z
ρt R
ρt )⟶ (Xt Yt Zt Rt) which is the solution of
the variational equation as same as (6)e following adjoint equation is introduced
minus dpt bx(t)( 1113857τpt minus fx(t)( 1113857
τqt + σx(t)( 1113857
τwt + 1113936infin
i1gx(t)( 1113857
τkt)
i1113890 1113891dt
minus wtdBt minus 1113936infin
i1kt( 1113857
idHit
dqt fy(t)1113872 1113873τqtdt + fz(t)( 1113857
τqtdBt + 1113936
infin
i1fρir (t)1113872 1113873
τ1113888 1113889qtdHi
t
q0 minus G0y y0( 1113857h0 + cy y0( 1113857h pT G1x xT( 1113857h1 minus Φx xT( 1113857qT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(66)
Similarly it can be proved that(p
ρt q
ρt w
ρt k
ρt )⟶ (pt qt wt kt) then inequation (65)
impliesH t x
lowastt ylowastt zlowastt rlowastt vt pt qt wt kt( 1113857
minus H t xlowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857ge 0forallvt isin U ae as
(67)
en we get the following theorem
Theorem 2 Let Assumptions 1ndash3 hold ulowastt is an optimalcontrol and the corresponding optimal state trajectories are(xlowastt ylowastt zlowastt rlowastt ) and (pt qt wt kt) is the solution of adjoint
equation (66) then there exists nonzero constant(h1 h0 h) isin (Rn1 times Rm1 times R) with |h0|
2 + |h1|2 + |h|2 1
such that for any admissible control vt isin Uad[0 T] themaximum condition (67) holds
9is conclusion can be drawn from the above analysisdirectly
5 A Financial Example
In this section we will study the problem of optimal con-sumption rate selection in the financial market which willnaturally inspire our research to the forward-backwardstochastic optimal control problem in Section 3
Mathematical Problems in Engineering 9
Assume the investorrsquos asset process xt (tge 0) in the fi-nancial market is described by the following stochasticdifferential equation with the Teugels martingale
dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin
i1gi
txtdHit
x0 Wgt 0 t isin [0 T]
⎧⎪⎪⎨
⎪⎪⎩(68)
where μt and σt ne 0 are the expected return and volatility ofthe value process xt at time t respectively and Ct is theconsumption rate process Assume μt σt gt and Ct are alluniformly bounded Ftminus measurable random processes epurpose of investors is to select the optimal consumptionstrategy Clowastt at time tge 0 minimizing the following recursiveutility
J(C(middot)) E y01113858 1113859 (69)
where yt is the following backward stochastic process
minus dyt Leminus rtC1minus R
t
1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936
infin
i1ri
tdHit
yT minus xT gt 0 t isin [0 T]
⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
(70)
with constant Lgt 0 the discount factor rgt 0 and theArrowndashPratt measure of risk aversion R isin (0 1)
If the consumption process Ct is regarded as a controlvariable then combining (68) with (70) we encounter thefollowing control system
dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin
i1gi
txtdHit
minus dyt Leminus rtC1minus R
t
1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936
infin
i1ri
tdHit
x0 Wgt 0 yT minus xT gt 0 t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(71)
which is obviously a special case of stochastic control system(2) with b(t x u) μtxt minus Ct σ(t x) σtxt g(t x)
gitxt Φx(xT) minus XT and f(t x y z r u) Leminus rt(C1minus R
t (1 minus R)) minus ryt
We can check that both Assumptions 1 and 2 are sat-isfieden we can use ourmaximum principle (eorem 1)to solve the above optimization problem Let Clowastt be anoptimal consumption rate and xlowastt ylowastt be the correspondingwealth process and recursive utility process In this case theHamiltonian function H reduces to
H t xlowastt ylowastt Ct p
lowastt qlowastt wlowastt klowastt( 1113857 langplowastt μtx
lowastt minus Ctrang
+langwlowastt σtxlowastt rang +langklowastt g
itxlowastt rang minus langqlowastt Le
minus rt C1minus Rt
1 minus Rminus rylowastt rang
(72)
and (plowastt qlowastt wlowastt klowastt ) is the solution of the following adjointequation
dqt rqtdt
minus dpt μtpt + σtwt + 1113936infin
i1ki
t( 1113857τgi
t1113890 1113891dt minus wtdBt minus 1113936infin
i1ki
tdHit
q0 1 pT qT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
(73)
According to the maximum principle (47) we have
plowastt minus Le
minus rtClowastt( 1113857
minus Rqlowastt (74)
and the optimal consumption rate
Clowastt minus
ert
L
plowasttqlowastt
1113888 1113889
minus 1R
(75)
By solving FBSDE (73) we get qlowastt ert plowastt erteμ(Tminus t)and wlowastt klowastt 0 en we get the optimal consumptionrate of the investor which is
Clowastt minus
erT
Leμ(Tminus t)
1113888 1113889
minus 1R
t isin [0 T] (76)
6 Conclusions
In this paper a nonconvex control domain case of theforward-backward stochastic control driven by Levy processis considered and we obtain the global stochastic maximumprinciple for this stochastic control problem And then theproblem of stochastic control with initial and final stateconstraints on the state variables is discussed and a nec-essary condition about existence of the optimal control isalso acquired A financial example of optimal consumptionis discussed to illustrate the application of the stochasticmaximum principle
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the NNSF (11701335) Shan-dong NSF (ZR2019BG015) NSSF (17BGL058) ShandongProvincial Higher Education Science and Technology PlanProject (J18KA236) and Shandong Womenrsquos College High-level Research Project Cultivation Foundation(2018GSPGJ07)
References
[1] L S Pontryakin V G Boltyanskti R V Gamkrelidze andE F Mischenko9eMathematical9eory of Optimal ControlProcesses John Wiley New York NY USA 1962
[2] S Peng ldquoA general stochastic maximum principle for optimalcontrol problemsrdquo SIAM Journal on Control and Optimiza-tion vol 28 no 4 pp 966ndash979 1990
10 Mathematical Problems in Engineering
[3] S G Peng ldquoBackward stochastic differrntial equations andapplication to optimal controlrdquo Applied Mathematics ampOptimization vol 27 no 4 pp 125ndash144 1993
[4] W S Xu ldquoStochastic maximum principle for optimal controlproblem of forward and backward systemrdquo Journal of Aus-tralian Mathematical Sciences vol 37 pp 249ndash259 1998B
[5] Z Wu ldquoMaximum principle for optimal control problem offully coupled forward-backward stochastic control systemrdquoSystem Science and Mathematical Sciences vol 11 no 3pp 249ndash259 1998
[6] J T Shi and ZWu ldquoemaximum principle for fully coupledforward-backward stochastic control systemrdquo Acta Auto-matica Sinica vol 32 no 2 pp 161ndash169 2006
[7] R Situ ldquoA maximum principle for optimal controls of sto-chastic systems with random jumpsrdquo in Proceedings of theNational Conference on Control 9eory and its ApplicationsQingdao China 1991
[8] J Shi and Z Wu ldquoMaximum principle for forward-backwardstochastic control system with random jumps and applica-tions to financerdquo Journal of Systems Science and Complexityvol 23 no 2 pp 219ndash231 2010
[9] J T Shi ldquoGlobal maximum principle for forward-backwardstochastic control system with Poisson jumpsrdquo Asian Journalof Control vol 16 no 1 pp 1ndash11 2014
[10] M Liu X Wang and H Huang ldquoMaximum principle forforward-backward control system driven by Ito-Levy pro-cesses under initial-terminal constraintsrdquo MathematicalProblems in Engineering vol 2017 pp 1ndash13 2017
[11] S Tang ldquoe maximum principle for partially observedoptimal control of stochastic differential equationsrdquo SIAMJournal on Control and Optimization vol 36 no 5pp 1596ndash1617 1998
[12] Q Meng ldquoA maximum principle for optimal control problemof fully coupled forward-backward stochastic systems withpartial informationrdquo Science in China Series A Mathematicsvol 52 no 7 pp 1579ndash1588 2009
[13] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward backward stichastic differential equationswith jumpsrdquo SIAM J Control Optimenleadertwodots vol 48pp 2845ndash2976 2009
[14] G Wang and Z Wu ldquoKalman-Bucy filtering equations offorward and backward stochastic systems and applications torecursive optimal control problemsrdquo Journal of MathematicalAnalysis and Applications vol 342 no 2 pp 1280ndash12962008
[15] H Xiao ldquoe maximum principle for partially observedoptimal control of forward-backward stochastic systems withrandom jumpsrdquo Journal of Systems Science and Complexityvol 24 no 6 pp 1083ndash1099 2011
[16] G Wang Z Wu and J Xiong ldquoMaximum principles forforward-backward stochastic control systems with correlatedstate and observation noisesrdquo SIAM Journal on Control andOptimization vol 51 no 1 pp 491ndash524 2013
[17] G Wang C Zhang and W Zhang ldquoStochastic maximumprinciple for mean-field type optimal control under partialinformationrdquo IEEE Transactions on Automatic Controlvol 59 no 2 pp 522ndash528 2014
[18] S Q Zhang J Xiong and X D Liu ldquoStochastic maximumprinciple for partially observed forward backward stochasticdifferential equations with jumps and regime switchingrdquoScience China Information Sciences vol 61 no 7 pp 1ndash132018
[19] J Xiong S Q Zhang S Zhang and Y Zhuang ldquoA partiallyobserved non-zero sum differential game of forward-
backward stochastic differential equations and its applicationin financerdquo Mathematical Control amp Related Fields vol 9no 2 pp 257ndash276 2019
[20] D Nualart and W Schoutens ldquoChaotic and predictablerepresentations for Levy processesrdquo Stochastic Processes and9eir Applications vol 90 no 1 pp 109ndash122 2000
[21] D Nualart and W Schoutens ldquoBackward stochastic differ-ential equations and Feynman-Kac formula for Levy pro-cesses with applications in financerdquo Bernoulli vol 7 no 5pp 761ndash776 2001
[22] Q Meng and M Tang ldquoNecessary and sufficient conditionsfor optimal control of stochastic systems associated with Levyprocessesrdquo Science in China Series F Information Sciencesvol 52 no 11 pp 1982ndash1992 2009
[23] M Tang and Q Zhang ldquoOptimal variational principle forbackward stochastic control systems associated with Levyprocessesrdquo Science China Mathematics vol 55 no 4pp 745ndash761 2012
[24] F Baghery N Khelfallah B Mezerdi and I Turpin ldquoFullycoupled forward backward stochastic differential equationsdriven by Levy processes and application to differentialgamesrdquo De Gruyter vol 22 no 3 pp 151ndash161 2014
[25] F Zhang M N Tang and Q X Meng ldquoSochastic maximumprinciple for forward-backward stochastic control systemsassociated with Levy processesrdquo Chinese Annals of Mathe-matics vol 35A no 1 pp 83ndash100 2014
[26] X R Wang and H Huang ldquoMaximum principle for forward-backward stochastic control system driven by Levy processrdquoMathematical Problems in Engineering vol 2015 Article ID702802 12 pages 2015
[27] H Huang X Wang and M Liu ldquoA maximum principle forfully coupled forward-backward stochastic control systemdriven by Levy process with terminal state constraintsrdquoJournal of Systems Science and Complexity vol 31 no 4pp 859ndash874 2018
[28] P Muthukumar and R Deepa ldquoInfinite horizon optimalcontrol of forward backward stochastic system driven byTeugels martingales with Levy processrdquo Stochastic and Dy-namics vol 3 no 17 pp 1ndash17 2017
[29] J B Wu and Z M Liu ldquoOptimal control of mean-fieldbackward doubly stochastic systems driven by Ito-Levyprocessesrdquo International Journal of Control vol 93 no 4pp 953ndash970 2018
[30] W Wang J Wu and Z Liu ldquoe optimal control of fully-coupled forward-backward doubly stochastic systems drivenby Ito-Levy processesrdquo Journal of Systems Science andComplexity vol 32 no 4 pp 997ndash1018 2019
Mathematical Problems in Engineering 11
where (xt yt zt rt) take the value inΩ times [0 T] times Rn times Rm times
Rmtimesd times l2(Rm) and mappings b σ g and f take the value inRn Rntimesd l2(Rn) and Rm respectively Convenient forwriting set column vector α (x y z)T andA(t α r) (minus Mτf(t α r) Mb(t α r) Mσ(t α r))Twhere M is a m times n full rank matrix
Assumption 1
(i) All mappings in equation (1) are uniformly Lip-schitz continuous in their own argumentsrespectively
(ii) For all (ω t) isin Ω times [0 T] l(ω t 0 0 0 0)
isinM2(0 T Rn+m+mtimesd) times l2(0 T Rm) for l b f σrespectively and g(ω t 0 0 0 0) isin l2(Rn)
(iii) langΦ(x) minus Φ(x) M(x minus x)rangge β|M1113954x|2(iv) langA(tα r) minus A(tα r)α minus αrang + 1113936
infini1langM 1113954gi 1113954rirangle minus μ
1|M1113954x|2 minus μ2(|Mτ1113954y|2 + |Mτ1113954z|2 + 1113936infini1 Mτ 1113954ri2)
where α (x y z) 1113954x x minus x 1113954y y minus y 1113954z z minus z1113954ri ri minus ri 1113954gi gi(tα r) minus gi(tα r) μ1 μ2 and βare nonnegative constants which satisfiedμ1 + μ2 gt 0 μ2 + βgt 0 and μ1 gt 0βgt 0(resp μ2 gt 0)when mgtn (resp ngtm)
en the following existence and uniqueness of thesolution conclusion holds
Lemma 1 9ere exists a unique solution in M2(0 TH)
satisfying FBSDEL (1) under Assumption 19e detailed certification process of this conclusion can be
seen in [24]
3 Stochastic Maximum Principle
In this section for any given admissible control u(middot) weconsider the following stochastic control system
dxt b t xt ut( 1113857dt + σ t xt( 1113857dBt + 1113936infin
i1gi t xtminus( 1113857dHi
t
minus dyt f t xt yt zt rt ut1113872 1113873dt minus ztdBt minus 1113936infin
i1ri
tdHit
x0 a yT Φ xT( 1113857 t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎩
(2)
where a isin Rn is given An admissible controlu(middot) isinM2(0 T Rp) is an Ft-predictable process whichtakes values in a nonempty subset U of Rp Uad is the set ofall admissible controls
And the performance criterion is
J(u) Ec y0( 1113857 (3)
where c Rm⟶ R is a given Frechet differential functionOur optimal control problem amounts to determining
an admissible control ulowast isin Uad such that
J ulowast(middot)( 1113857 inf
u(middot)isinUad
J(u(middot)) (4)
In order to get the necessary conditions for the optimalcontrol we assume ulowastt is the optimal control and thecorresponding solution of (2) is recorded as (xlowastt ylowastt zlowastt rlowastt )
and introduce the ldquospike variational controlrdquo as follows
uεt
vt τ le tle τ + ε
ulowastt otherwise1113896 (5)
and (xεt yε
t zεt rεt) are the state trajectories of uε
t here vt is anarbitrary admissible control and ε is a sufficiently smallconstant
We also need the following assumption and variationalequation (6)
Assumption 2
(i) b f g σ Φ and c are continuously differentiablewith respect to (x y z r u) and the derivatives areall bounded
(ii) ere exists a constant Cgt 0 and it holds that|cy|leC(1 + |y|)
dXt bx(t)Xt + b t uεt( 1113857 minus b t ulowastt( 11138571113858 1113859dt + σx(t)XtdBt + 1113936
infin
i1gi
x(t)XtdHit
minus dYt fx(t)Xt + fy(t)Yt + fz(t)Zt + fr(t)Rt + f t uεt( 1113857 minus f t ulowastt( 11138571113960 1113961dt
minus ZtdBt minus 1113936infin
i1Ri
tdHit
X0 0
YT Φx(t)XT
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(6)
Mathematical Problems in Engineering 3
Here bx(t) bx(txlowastt ulowastt ) σx(t) σx(txlowastt ) gix(t)
gix(txlowastt ) b(tuε
t) b(txεt u
εt) b(tulowastt ) b (txlowastt ulowastt )
fw(t) fw(txlowastt ylowastt zlowastt rlowastt ulowastt ) (w xyz middot r) f(tuεt)
f(xεt y
εt z
εt r
εt u
εt) and f(tulowastt ) f(xlowastt ylowastt zlowastt rlowastt ulowastt )
Lemma 2 Suppose Assumptions 1 and 2 hold for the first-order variation X Y Z R we have the following estimations
sup0letleT
E Xt
111386811138681113868111386811138681113868111386811138682 leCε2 (7)
sup0letleT
E Xt
111386811138681113868111386811138681113868111386811138684 leCε4 (8)
sup0letleT
E Yt
111386811138681113868111386811138681113868111386811138682 leCε2 (9)
sup0letleT
E Yt
111386811138681113868111386811138681113868111386811138684 leCε4 (10)
sup0letleT
E 1113946T
0Zt
111386811138681113868111386811138681113868111386811138682dsleCε2 (11)
sup0letleT
E 1113946T
0Zt
111386811138681113868111386811138681113868111386811138682ds1113888 1113889
2
leCε4 (12)
sup0letleT
E 1113946T
0Rt
2dsleCε2 (13)
sup0letleT
E 1113946T
0Rt
2ds1113888 1113889
2
leCε4 (14)
Proof We first prove inequations (7) and (8) For theforward part of the first-order variation equation we have
E Xt
111386811138681113868111386811138681113868111386811138682
E 1113946t
0bx(s)Xs + b s u
εs( 1113857 minus b s u
lowasts( 11138571113858 1113859ds + σx(s)XsdBs + 1113944
infin
i1g
ix(s)XsdH
is
⎧⎨
⎩
⎫⎬
⎭
2
le 4 E 1113946t
0bx(s)Xsds1113888 1113889
2
+ E 1113946t
0b s u
εs( 1113857 minus b s u
lowasts( 11138571113858 1113859ds1113888 1113889
2
+ E 1113946t
0σx(s)Xs1113858 1113859
2ds + E 1113946t
01113944
infin
i1g
ix(s)Xs
⎡⎣ ⎤⎦2
ds⎧⎨
⎩
⎫⎬
⎭
le 12C2TE 1113946
t
0X
2sds + 4E 1113946
t
0b s u
εs( 1113857 minus b s u
lowasts( 11138571113858 1113859ds1113888 1113889
2
(15)
Applying Gronwallrsquos inequation we have
E Xt
111386811138681113868111386811138681113868111386811138682 leCε2 for all t isin [0 T] (16)
Similarly (8) holdsWe next estimate Yt Zt and Rt the backward part of the
first-order variation equation can be rewritten as
Yt + 1113946T
tZsdBs + 1113946
T
t1113944
infin
i1R
isdH
is Φx(t)XT + 1113946
T
tfx(t)Xt1113858
+ fy(t)Yt + fz(t)Zt
+ fr(t)Rt
+ f t uεt( 1113857 minus f t u
lowastt( 1113857]dt
(17)
Squaring both sides of (17) and using the fact of
EYt 1113946T
tZsdBs 0
EYt 1113946T
t1113944
infin
i1R
isdH
is 0
E 1113946T
tZsdBs 1113946
T
t1113944
infin
i1R
isdH
is 0
(18)
we get
E Yt
111386811138681113868111386811138681113868111386811138682
+ E 1113946T
tZ2sds + E 1113946
T
t1113944
infin
i1R
is
⎛⎝ ⎞⎠
2
dHis
E Φx(t)XT + 1113946T
tfx(t)Xt + fy(t)Yt + fz(t)Zt + fr(t)Rt + f t u
εt( 1113857 minus f t u
lowastt( 11138571113960 1113961dt1113896 1113897
2
le 6C2EX
2T + 6C
2TE 1113946
T
tX
2sds + 6C
2TE 1113946
T
tY2sds + 6C
2(T minus t)E 1113946
T
tZ2sds
+ 6C2(T minus t)E 1113946
T
t1113944
infin
i1R
is
⎛⎝ ⎞⎠
2
ds + 6E 1113946T
tf s u
εs( 1113857 minus f s u
lowasts( 1113857( 1113857ds1113888 1113889
2
(19)
4 Mathematical Problems in Engineering
When t isin [T minus δ T] with δ 112C2 we have
E Yt
111386811138681113868111386811138681113868111386811138682
+12
E 1113946T
tZ2sds +
12
E 1113946T
t1113944
infin
i1R
is
⎛⎝ ⎞⎠
2
dHis
le 6C2EX
2T + 6C
2TE 1113946
T
tX
2sds + 6C
2TE
middot 1113946T
tY2sds + 6E 1113946
T
tf s u
εs( 1113857 minus f s u
lowasts( 1113857( 1113857ds1113888 1113889
2
(20)
Applying Gronwallrsquos inequation we have
sup0letleT
E Yt
111386811138681113868111386811138681113868111386811138682 leCε2 t isin [T minus δ T]
sup0letleT
E 1113946T
0Zt
111386811138681113868111386811138681113868111386811138682dsleCε2 t isin [T minus δ T]
sup0letleT
E 1113946T
0Rt
2dsleCε2 t isin [T minus δ T]
(21)
Consider the BSDE of the first-order variation equationin the interval [t T minus δ]
Yt + 1113946Tminus δ
tZsdBs + 1113946
Tminus δ
t1113944
infin
i1R
isdH
is YTminus δ
+ 1113946Tminus δ
tfx(t)Xt + fy(t)Yt + fz(t)Zt1113960
+ fr(t)Rt + f t uεt( 1113857 minus f t u
lowastt( 11138571113859dt
(22)
us
E Yt
111386811138681113868111386811138681113868111386811138682
+ E 1113946Tminus δ
tZ2sds + E 1113946
Tminus δ
t1113944
infin
i1R
is
⎛⎝ ⎞⎠
2
dHis
E YTminus δ + 1113946Tminus δ
tfx(t)Xt + fy(t)Yt + fz(t)Zt + fr(t)Rt11139601113896
+ f t uεt( 1113857 minus f t u
lowastt( 11138571113859dt1113865
2
le 6C2EX
2T + 6C
2TE 1113946
Tminus δ
tX
2sds + 6C
2TE 1113946
Tminus δ
tY2sds
+ 6C2(T minus t)E 1113946
Tminus δ
tZ2sds + 6C
2(T minus t)E
middot 1113946Tminus δ
t1113944
infin
i1R
is
⎛⎝ ⎞⎠
2
ds + 6E 1113946Tminus δ
tf s u
εs( 1113857 minus f s u
lowasts( 1113857( 1113857ds1113888 1113889
2
(23)
So when t isin [T minus 2δ T] with δ 112C2 we have
sup0letleTE Yt
111386811138681113868111386811138681113868111386811138682 leCε2 t isin [T minus 2δ T]
sup0letleTE 1113946T
0Zt
111386811138681113868111386811138681113868111386811138682dsleCε2 t isin [T minus 2δ T]
sup0letleTE 1113946T
0Rt
2dsleCε2 t isin [T minus 2δ T]
(24)
After a finite number of iterations (9) (11) and (13) areobtained And (10) (12) and (14) can be proved by using a
similar method and the following inequalitiesE(1113938
T
tZsdBs)
4 ge β1E(1113938T
tZ2
sds)2 β1 gt 0E(1113938
T
t1113936infini1 Ri
sdHis)4 ge β2E( 1113938
T
t(1113936infini1 Ri
s)2ds)2 β2 gt 0
Lemma 3 Under hypothesis Assumptions 1 and 2 it holdsthe following four estimations
sup0letleT
E xεt minus xlowastt minus Xt
111386811138681113868111386811138681113868111386811138682 leCεε
2 Cε⟶ 0when ε⟶ 0
(25)
sup0letleT
E yεt minus ylowastt minus Yt
111386811138681113868111386811138681113868111386811138682 leCεε
2 Cε⟶ 0when ε⟶ 0
(26)
E 1113946T
0zεt minus zlowastt minus Zt
111386811138681113868111386811138681113868111386811138682dsleCεε
2 Cε⟶ 0when ε⟶ 0
(27)
E 1113946T
trεt minus rlowastt minus Rt
2dsleCεε
2 Cε⟶ 0when ε⟶ 0
(28)
Proof To prove (25) we observe that
1113946t
0b s x
lowasts + Xs u
εs( 1113857ds + 1113946
t
0σ s x
lowasts + Xs( 1113857dBs
+ 1113946t
01113944
infin
i1g
is xlowastsminus + Xsminus( 1113857dH
is
1113946t
0b s x
lowasts u
εs( 1113857 + 1113946
1
0bx s x
lowasts + λXs u
εs( 1113857dλXs1113890 1113891ds
+ 1113946t
0σ s x
lowasts( 1113857 + 1113946
1
0σx s x
lowasts + λXs( 1113857dλXs1113890 1113891dBs
+ 1113946t
01113944
infin
i1g
is xlowasts( 1113857 + 1113946
1
01113944
infin
i1g
ix s x
lowasts + λXs( 1113857dλXs
⎡⎣ ⎤⎦dHis
1113946t
0b s x
lowasts ulowasts( 1113857ds + 1113946
t
0σ s x
lowasts( 1113857dBs + 1113946
t
01113944
infin
i1g
is xlowasts( 1113857dH
is
+ 1113946t
0bx s x
lowasts ulowasts( 1113857Xsds + 1113946
t
0σx s x
lowasts( 1113857XsdBs
+ 1113946t
01113944
infin
i1g
ix s x
lowastsminus( 1113857dH
is
+ 1113946t
0b s x
lowasts u
εs( 1113857 minus b s x
lowasts ulowasts( 11138571113858 1113859ds + 1113946
t
0Aεds
+ 1113946t
0BεdBs + 1113946
t
0CεdH
is
xlowastt minus x0 + Xt + 1113946
t
0Aεds + 1113946
t
0BεdBs + 1113946
t
0CεdH
is
(29)
Mathematical Problems in Engineering 5
where
Aε
11139461
0bx s x
lowasts + λXs u
εs( 1113857 minus bx s x
lowasts ulowasts( 11138571113858 1113859dλXs
Bε
11139461
0σx s x
lowasts + λXs( 1113857 minus σx s x
lowasts( 11138571113858 1113859dλXs
Cε
11139461
01113944
infin
i1g
ix s x
lowasts + λXs( 1113857 minus g
ix s x
lowasts( 11138571113872 1113873⎡⎣ ⎤⎦dλXs
(30)
It follows easily from Lemma 2 that
sup0letleT
E 1113946t
0A
εds1113888 1113889
2
+ 1113946t
0BεdBs1113888 1113889
2
+ 1113946t
0CεdH
is1113888 1113889
2⎡⎣ ⎤⎦ o ε21113872 1113873
(31)
Since
xεt minus x0 1113946
t
0b s x
εs u
εs( 1113857ds + 1113946
t
0σ s x
εs( 1113857dBs + 1113946
t
01113944
infin
i1g
is x
εsminus( 1113857dH
is
(32)
then
xεt minus xlowastt minus Xt 1113946
t
0b s x
εs u
εs( 1113857 minus b s x
lowasts + Xs u
εs( 11138571113858 1113859ds
+ 1113946t
0σ s x
εs( 1113857 minus σ s x
lowasts + Xs( 11138571113858 1113859dBs
+ 1113946t
01113944
infin
i1g
is x
εsminus( 1113857 minus g
is xlowastsminus + Xsminus( 11138571113872 1113873dH
is
+ 1113946t
0Aεds + 1113946
t
0BεdBs + 1113946
t
0CεdH
is
1113946t
0D
εxεs minus xlowasts minus Xs( 1113857ds + 1113946
t
0Eε
xεs minus xlowasts minus Xs( 1113857dBs
+ 1113946t
0Fε
xεs minus xlowasts minus Xs( 1113857dH
is
(33)
with Dε 111393810 bx(s xlowasts + Xs + λ(xε
s minus xlowasts minus Xs) uεs)dλ
Eε 111393810 σx(s xlowasts + Xs + λ(xε
s minus xlowasts minus Xs))dλ and Fε 111393810
1113936infini1 gi
x(s xlowasts + Xs + λ(xεs minus xlowasts minus Xs))dλ
By Gronwallrsquos inequation we have
sup0letleT
E xεt minus xlowastt minus Xt
111386811138681113868111386811138681113868111386811138682 leCεε
2 Cε⟶ 0when ε⟶ 0
(34)
Next we prove (26) (27) (28) it can be easily checkedthat
minus 1113946T
tf s x
lowasts + Xs y
lowasts + Ys z
lowasts + Zs r
lowasts + Rs u
εs( 1113857ds
+ 1113946T
ts zlowasts + Zs( 1113857dBs + 1113946
T
t1113944
infin
i1g
is xlowastsminus + Xsminus( 1113857dH
is
Φ xlowastT( 1113857 minus y
lowastt +Φx x
lowastT( 1113857XT minus Yt minus 1113946
T
tGεds
(35)
Here
Gε
11139461
0fx s x
lowasts + λXs y
lowasts + λYs z
lowasts((
+ λZs rlowasts + λRs u
εs1113857 minus fx(s)1113857dλXs
+ 11139461
0fy s x
lowasts + λXs y
lowasts + λYs z
lowasts(1113872
+ λZs rlowasts + λRs u
εs1113857 minus fy(s)1113873dλYs
+ 11139461
0fz s x
lowasts + λXs y
lowasts + λYs z
lowasts((
+ λZs rlowasts + λRs u
εs1113857 minus fz(s)1113857dλZs
+ 11139461
0fr s x
lowasts + λXs y
lowasts + λYs z
lowasts((
+ λZs rlowasts + λRs u
εs1113857 minus fr(s)1113857dλRs
(36)
Since
yεt Φ x
εT( 1113857 + 1113946
T
tf s x
εs y
εs z
εs r
εs u
εs( 1113857ds minus 1113946
T
tzsdBs
minus 1113946T
t1113944
infin
i1r
iεs dH
is
(37)
then
yεt minus ylowastt minus Yt Φ x
εT( 1113857 minus Φ x
lowastT( 1113857 minus Φx x
lowastT( 1113857XT + 1113946
T
tf s x
εs y
εs z
εs r
εs u
εs( 1113857 minus f s x
lowasts + Xs y
lowasts + Ys z
lowasts + Zs r
lowasts + Rs u
εs(1113858 1113859ds
minus 1113946T
tzεs minus zlowasts minus Zs( 1113857dBs minus 1113946
T
t1113944
infin
i1r
iεs minus r
ilowasts minus Rs1113872 1113873dH
is + 1113946
T
tGεds
(38)
6 Mathematical Problems in Engineering
Squaring both sides of the equation above we get
E yεt minus ylowastt minus Yt
111386811138681113868111386811138681113868111386811138682
+ E 1113946T
tzεs minus zlowasts minus Zs( 1113857
2ds minus E 1113946T
t1113944
infin
i1r
iεs minus r
ilowasts minus Rs1113872 1113873
2ds
E f s xεs y
εs z
εs r
εs u
εs( 1113857 minus f s x
lowasts + Xs y
lowasts + Ys z
lowasts + Zs r
lowasts + Rs u
εs( 11138571113858 1113859ds +Φ x
εT( 1113857 minus Φ x
lowastT( 1113857 minus Φx x
lowastT( 1113857XT + 1113946
T
tGεds1113896 1113897
2
(39)
From Lemma 2 and equation (25) we have
sup0letleTE 1113946T
tGεds1113888 1113889
2
o ε21113872 1113873
E Φ xεT( 1113857 minus Φ x
lowastT( 1113857 minus Φx x
lowastT( 1113857XT1113858 1113859
2 o ε21113872 1113873
(40)
en we can get (26) (27) and (28) by applying theiterative method to the above relations
Lemma 4 (variational inequality) Under the conditionsthat Assumptions 1 and 2 are established we can get thefollowing variational inequality
Ecy ylowast0( 1113857Y0 ge o(ε) (41)
Proof From the four estimations in Lemma 3 we have thefollowing estimation
E c yε0( 1113857 minus c y
lowast0 + Y0( 11138571113858 1113859 o(ε) (42)
erefore
0leE c ylowast0 + Y0( 1113857 minus c y
lowast0( 11138571113858 1113859 + o(ε) Ecy y
lowast0( 1113857Y0 + o(ε)
(43)
We introduce the following Hamiltonian functionH [0 T] times Rn times Rm times Rmtimesd times l2(Rm) times U times Rn times Rm times
Rntimesdtimes l2(Rn) as
H(t x y z r u p q w k) langp b(t x u)rang +langw σ(t x)rang
+langk g(t x)rang minus langq f(t x y z r u)rang
(44)
and the following adjoint equation
dqt Hy ulowastt( 1113857dt + Hz ulowastt( 1113857dBt + 1113936infin
i1Hi
r ulowastt( 1113857dHit
minus dpt Hx ulowastt( 1113857dt minus wtdBt minus 1113936infin
i1ki
tminus dHit
q0 cy ylowast0( 1113857 pT minus Φx xlowastT( 1113857qT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎩
(45)
where
Hx ulowastt( 1113857 Hx t x
lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857
Hy ulowastt( 1113857 Hy t x
lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857
Hz ulowastt( 1113857 Hz t x
lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857
Hr ulowastt( 1113857 Hr t x
lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857
(46)
It is easily check that adjoint equation (45) has a uniquesolution quartet (pt qt wt kt) isinM2(0 T Rn+m+mtimesd) times
l2(0 T Rm)en we get the main result of this section
Theorem 1 Let hypothesis Assumptions 1 and2 hold ulowastt isan optimal control and the corresponding optimal statetrajectories are (xlowastt ylowastt zlowastt rlowastt ) let (pt qt wt kt) be thesolution of adjoint equation (45) and then for each ad-missible control ut isin Uad[0 T] we have
H txlowastt ylowastt zlowastt rlowastt utpt qtwtkt( 1113857
geH txlowastt ylowastt zlowastt rlowastt ulowastt pt qtwtkt( 1113857asae(47)
Proof Applying It 1113954os formula to langp Xrang and langq Yrang itfollows from (6) (45) and the variational inequality that
E 1113946T
0H t xt yt zt rt u
εt pt qt wt kt( 11138571113858
minus H t xt yt zt rt ulowastt pt qt wt kt( 11138571113859dt Ecy y
lowast0( 1113857Y0 ge o(ε)
(48)
By the definition of uεt we know that for any
u isin Uad[0 T] the following inequation holds
E H t xt yt zt rt uεt pt qt wt kt( 11138571113858
minus H t xt yt zt rt ulowastt pt qt wt kt( 11138571113859ge 0
(49)
en (47) can be easily checked
4 Stochastic Control Problem withState Constraints
In this part we are going to discuss stochastic controlproblems with state constraints in control system (2)Specifically the initial state constraints and final stateconstraints are as follows
EG1 xT( 1113857 0
EG0 y0( 1113857 0(50)
Mathematical Problems in Engineering 7
where G1 Rn⟶ Rn1(n1 lt n) G0 Rm⟶ Rm1(m1 ltm)Our optimal control problem is to find ulowast isin Uad such that
J ulowast(middot)( 1113857 inf
u(middot)isinUad
J(u(middot)) (51)
subject to the state constraints (50) In the following we willapply Ekelandrsquos variational principle to solve this optimalcontrol problem Firstly we need the following assumptions
Assumption 3 Control domain U is assumed to be closedthe mappings in state constraints G1 G0 are continuouslydifferentiable and G1x G0y are bounded
For forallu1(middot) u2(middot) isin Uad let
d u1(middot) u2(middot)( 1113857 E u1(middot) minus u2(middot)1113868111386811138681113868
11138681113868111386811138682 gt 0 t isin [0 T]1113966 1113967
11138681113868111386811138681113868
11138681113868111386811138681113868 (52)
Same as Section 3 we also assume ulowastt be the optimalcontrol and the corresponding optimal state trajectories are(xlowastt ylowastt zlowastt rlowastt ) In order to solve the constraint problemwe need the following penalty cost functional for any ρgt 0
Jρ(u) E G1 xT( 11138571113858 11138591113868111386811138681113868
11138681113868111386811138682
+ E G0 y0( 11138571113858 11138591113868111386811138681113868
11138681113868111386811138682
+ J(u) minus J ulowast( ) + ρ1113868111386811138681113868
11138681113868111386811138682
1113966 111396712
(53)
It can be checked that Jρ(u) Uad⟶ R1 is continuousand for any u(middot) isin Uad
Jρ(u)ge 0 Jρ ulowast
( 1113857 ρ
Jρ ulowast
( 1113857le infu(middot)isinUad
Jρ(u) + ρ(54)
It can be obtained by Ekelandrsquos variational principle thatthere exists u
ρt isin Uad such that
(i) Jρ uρ( )le Jρ ulowast( ) ρ
(ii) d uρ ulowast( )le ρradic
(iii) Jρ(v)ge Jρ uρ( ) minusρradic
d uρ v( ) for forallv(middot) isin Uad
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(55)
For fixed ρ and admissible control uρt we define the spike
variation as follows
uρεt
vt τ le tle τ + ε
uρt otherwise
1113896 (56)
for any εgt 0 and it is easy to check from (iii) of (55) that
Jρ uρε( ) minus Jρ uρ( ) +ρradic
d uρε uρ( )ge 0 (57)
Let (xρt y
ρt z
ρt r
ρt ) be the trajectories corresponding to u
ρt
and (xρεt y
ρεt z
ρεt r
ρεt ) be the trajectories corresponding to
uρεt e variational equation we used in this section is the
same as the one in Section 3 with ulowastt uρt and
(xlowastt ylowastt zlowastt rlowastt ) (xρt y
ρt z
ρt r
ρt ) And we also assume that
the solution of this variational equation is (Xρt Y
ρt Z
ρt R
ρt )
Similar to the approach in Lemmas 2 and 3 it can be shownthat
sup0letleTE xρεt minus x
ρt minus X
ρt
111386811138681113868111386811138681113868111386811138682 leCεε
2 Cε⟶ 0when ε⟶ 0
sup0letleTE yρεt minus y
ρt minus Y
ρt
111386811138681113868111386811138681113868111386811138682 leCεε
2 Cε⟶ 0when ε⟶ 0
E 1113946T
0zρεt minus z
ρt minus Z
ρt
111386811138681113868111386811138681113868111386811138682dsleCεε
2 Cε⟶ 0when ε⟶ 0
E 1113946T
0rρεt minus r
ρt minus R
ρt
2dsleCεε
2 Cε⟶ 0when ε⟶ 0
(58)
en by (57) the following variational inequality holds
Jρ uρε
( 1113857 minus Jρ uρ
( 1113857 +ρ
radicd u
ρε u
ρ( 1113857
J2ρ uρε( ) minus J2ρ uρ( )
Jρ uρε( ) + Jρ uρ( )+ ε
ρ
radic
langhρε1 E G1x x
ρT( 11138571113858 1113859X
ρTrang +langhρε
0 E G0y yρ0( 11138571113960 1113961Y
ρ0rang
+ hρε
E cy yρ0( 1113857Y
ρ01113960 1113961 + ε
ρ
radic+ o(ε)
(59)
where
hρε1
2E G1 xρT( 11138571113858 1113859
Jρ uρε( ) + Jρ uρ( )
hρε0
2E G0 yρ0( 11138571113858 1113859
Jρ uρε( ) + Jρ uρ( )
hρε
2E c y
ρ0( 1113857 minus c y0( 1113857 + ρ1113858 1113859
Jρ uρε( ) + Jρ uρ( )
(60)
Now let (pρεt q
ρεt w
ρεt k
ρεt ) be the solution of
minus dpρεt b
ρx(t)( 1113857
τpρεt minus f
ρx(t)( 1113857
τqρεt + σρx(t)( 1113857
τw
ρεt + 1113936infin
i1gρx(t)( 1113857
τkρεt )i
1113890 1113891dt
minus wρεt dBt minus 1113936
infin
i1kρεt( 1113857
idHit
dqρεt f
ρy(t)( 1113857
τqρεt dt + f
ρz(t)( 1113857
τqρεt dBt + 1113936
infin
i1fρir (t)1113872 1113873
τ)q
ρεt dHi
t
q0 minus G0y yρ0( 1113857h
ρε0 + cy y
ρ0( 1113857hρε pT G1x x
ρT( 1113857h
ρε1 minus Φx x
ρT( 1113857q
ρεT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(61)
8 Mathematical Problems in Engineering
Here bρx(t) bx(t x
ρt u
ρt ) σρx(t) σx(t x
ρt )
fρy(t) fy(t x
ρt y
ρt z
ρt r
ρt u
ρt ) etc Applying It 1113954os formula
to langXρt q
ρεt rang + langY
ρt p
ρεt rang variational inequality (59) can be
rewritten as
H t xρt y
ρt z
ρt r
ρt u
ρεt p
ρεt q
ρεt w
ρεt k
ρεt( 1113857
minus H t xρt y
ρt z
ρt r
ρt u
ρt p
ρεt q
ρεt w
ρεt k
ρεt( 1113857
+ ερ
radic+ o(ε)ge 0 forallvt isin U ae as
(62)
where the Hamiltonian function H is defined as (44) Since
limε⟶0 hρε0
111386811138681113868111386811138681113868111386811138682
+ hρε1
111386811138681113868111386811138681113868111386811138682
+ hρε1113868111386811138681113868
11138681113868111386811138682
1113872 1113873 1 (63)
there exists a convergent subsequence still denoted by(h
ρε0 h
ρε1 hρε) such that (h
ρε0 h
ρε1 hρε)⟶ (h
ρ0 h
ρ1 hρ) when
ε⟶ 0 with |hρ0|2 + |h
ρ1|2 + |hρ|2 1
Let (pρt q
ρt w
ρt k
ρt ) be the solution of
minus dpρt b
ρx(t)( 1113857
τpρt minus f
ρx(t)( 1113857
τqρt + σρx(t)( 1113857
τw
ρt + 1113936infin
i1gρx(t)( 1113857
τkρt )i
1113890 1113891dt
minus wρtdBt minus 1113936
infin
i1kρt( 1113857
idHit
dqρt f
ρy(t)( 1113857
τqρtdt + f
ρz(t)( 1113857
τqρtdBt + 1113936
infin
i1fρir (t)1113872 1113873
τ1113888 1113889q
ρtdHi
t
q0 minus G0y yρ0( 1113857h
ρ0 + cy y
ρ0( 1113857hρ pT G1x x
ρT( 1113857h
ρ1 minus Φx x
ρT( 1113857q
ρT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(64)
From the continuous dependence of the solutions ofFBSDEs with Levy process on the parameters we can provethat the following convergence holds(p
ρεt q
ρεt w
ρεt k
ρεt )⟶ (p
ρt q
ρt w
ρt k
ρt ) then in equation
(62) it implies that
H t xρt y
ρt z
ρt r
ρt vt p
ρt q
ρt w
ρt k
ρt( 1113857
minus H t xρt y
ρt z
ρt r
ρt u
ρt p
ρt q
ρt w
ρt k
ρt( 1113857
+ρ
radic ge 0 forallvt isin U ae as
(65)
Similarly there exists a convergent subsequence(h
ρ0 h
ρ1 h
ρ) such that (h
ρ0 h
ρ1 h
ρ)⟶ (h0 h1 h) when
ρ⟶ 0 with |h0|2 + |h1|
2 + |h|2 1 Since uρt⟶ ulowastt as
ρ⟶ 0 we have (xρt y
ρt z
ρt r
ρt )⟶ (xlowastt ylowastt zlowastt rlowastt ) and
(Xρt Y
ρt Z
ρt R
ρt )⟶ (Xt Yt Zt Rt) which is the solution of
the variational equation as same as (6)e following adjoint equation is introduced
minus dpt bx(t)( 1113857τpt minus fx(t)( 1113857
τqt + σx(t)( 1113857
τwt + 1113936infin
i1gx(t)( 1113857
τkt)
i1113890 1113891dt
minus wtdBt minus 1113936infin
i1kt( 1113857
idHit
dqt fy(t)1113872 1113873τqtdt + fz(t)( 1113857
τqtdBt + 1113936
infin
i1fρir (t)1113872 1113873
τ1113888 1113889qtdHi
t
q0 minus G0y y0( 1113857h0 + cy y0( 1113857h pT G1x xT( 1113857h1 minus Φx xT( 1113857qT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(66)
Similarly it can be proved that(p
ρt q
ρt w
ρt k
ρt )⟶ (pt qt wt kt) then inequation (65)
impliesH t x
lowastt ylowastt zlowastt rlowastt vt pt qt wt kt( 1113857
minus H t xlowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857ge 0forallvt isin U ae as
(67)
en we get the following theorem
Theorem 2 Let Assumptions 1ndash3 hold ulowastt is an optimalcontrol and the corresponding optimal state trajectories are(xlowastt ylowastt zlowastt rlowastt ) and (pt qt wt kt) is the solution of adjoint
equation (66) then there exists nonzero constant(h1 h0 h) isin (Rn1 times Rm1 times R) with |h0|
2 + |h1|2 + |h|2 1
such that for any admissible control vt isin Uad[0 T] themaximum condition (67) holds
9is conclusion can be drawn from the above analysisdirectly
5 A Financial Example
In this section we will study the problem of optimal con-sumption rate selection in the financial market which willnaturally inspire our research to the forward-backwardstochastic optimal control problem in Section 3
Mathematical Problems in Engineering 9
Assume the investorrsquos asset process xt (tge 0) in the fi-nancial market is described by the following stochasticdifferential equation with the Teugels martingale
dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin
i1gi
txtdHit
x0 Wgt 0 t isin [0 T]
⎧⎪⎪⎨
⎪⎪⎩(68)
where μt and σt ne 0 are the expected return and volatility ofthe value process xt at time t respectively and Ct is theconsumption rate process Assume μt σt gt and Ct are alluniformly bounded Ftminus measurable random processes epurpose of investors is to select the optimal consumptionstrategy Clowastt at time tge 0 minimizing the following recursiveutility
J(C(middot)) E y01113858 1113859 (69)
where yt is the following backward stochastic process
minus dyt Leminus rtC1minus R
t
1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936
infin
i1ri
tdHit
yT minus xT gt 0 t isin [0 T]
⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
(70)
with constant Lgt 0 the discount factor rgt 0 and theArrowndashPratt measure of risk aversion R isin (0 1)
If the consumption process Ct is regarded as a controlvariable then combining (68) with (70) we encounter thefollowing control system
dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin
i1gi
txtdHit
minus dyt Leminus rtC1minus R
t
1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936
infin
i1ri
tdHit
x0 Wgt 0 yT minus xT gt 0 t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(71)
which is obviously a special case of stochastic control system(2) with b(t x u) μtxt minus Ct σ(t x) σtxt g(t x)
gitxt Φx(xT) minus XT and f(t x y z r u) Leminus rt(C1minus R
t (1 minus R)) minus ryt
We can check that both Assumptions 1 and 2 are sat-isfieden we can use ourmaximum principle (eorem 1)to solve the above optimization problem Let Clowastt be anoptimal consumption rate and xlowastt ylowastt be the correspondingwealth process and recursive utility process In this case theHamiltonian function H reduces to
H t xlowastt ylowastt Ct p
lowastt qlowastt wlowastt klowastt( 1113857 langplowastt μtx
lowastt minus Ctrang
+langwlowastt σtxlowastt rang +langklowastt g
itxlowastt rang minus langqlowastt Le
minus rt C1minus Rt
1 minus Rminus rylowastt rang
(72)
and (plowastt qlowastt wlowastt klowastt ) is the solution of the following adjointequation
dqt rqtdt
minus dpt μtpt + σtwt + 1113936infin
i1ki
t( 1113857τgi
t1113890 1113891dt minus wtdBt minus 1113936infin
i1ki
tdHit
q0 1 pT qT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
(73)
According to the maximum principle (47) we have
plowastt minus Le
minus rtClowastt( 1113857
minus Rqlowastt (74)
and the optimal consumption rate
Clowastt minus
ert
L
plowasttqlowastt
1113888 1113889
minus 1R
(75)
By solving FBSDE (73) we get qlowastt ert plowastt erteμ(Tminus t)and wlowastt klowastt 0 en we get the optimal consumptionrate of the investor which is
Clowastt minus
erT
Leμ(Tminus t)
1113888 1113889
minus 1R
t isin [0 T] (76)
6 Conclusions
In this paper a nonconvex control domain case of theforward-backward stochastic control driven by Levy processis considered and we obtain the global stochastic maximumprinciple for this stochastic control problem And then theproblem of stochastic control with initial and final stateconstraints on the state variables is discussed and a nec-essary condition about existence of the optimal control isalso acquired A financial example of optimal consumptionis discussed to illustrate the application of the stochasticmaximum principle
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the NNSF (11701335) Shan-dong NSF (ZR2019BG015) NSSF (17BGL058) ShandongProvincial Higher Education Science and Technology PlanProject (J18KA236) and Shandong Womenrsquos College High-level Research Project Cultivation Foundation(2018GSPGJ07)
References
[1] L S Pontryakin V G Boltyanskti R V Gamkrelidze andE F Mischenko9eMathematical9eory of Optimal ControlProcesses John Wiley New York NY USA 1962
[2] S Peng ldquoA general stochastic maximum principle for optimalcontrol problemsrdquo SIAM Journal on Control and Optimiza-tion vol 28 no 4 pp 966ndash979 1990
10 Mathematical Problems in Engineering
[3] S G Peng ldquoBackward stochastic differrntial equations andapplication to optimal controlrdquo Applied Mathematics ampOptimization vol 27 no 4 pp 125ndash144 1993
[4] W S Xu ldquoStochastic maximum principle for optimal controlproblem of forward and backward systemrdquo Journal of Aus-tralian Mathematical Sciences vol 37 pp 249ndash259 1998B
[5] Z Wu ldquoMaximum principle for optimal control problem offully coupled forward-backward stochastic control systemrdquoSystem Science and Mathematical Sciences vol 11 no 3pp 249ndash259 1998
[6] J T Shi and ZWu ldquoemaximum principle for fully coupledforward-backward stochastic control systemrdquo Acta Auto-matica Sinica vol 32 no 2 pp 161ndash169 2006
[7] R Situ ldquoA maximum principle for optimal controls of sto-chastic systems with random jumpsrdquo in Proceedings of theNational Conference on Control 9eory and its ApplicationsQingdao China 1991
[8] J Shi and Z Wu ldquoMaximum principle for forward-backwardstochastic control system with random jumps and applica-tions to financerdquo Journal of Systems Science and Complexityvol 23 no 2 pp 219ndash231 2010
[9] J T Shi ldquoGlobal maximum principle for forward-backwardstochastic control system with Poisson jumpsrdquo Asian Journalof Control vol 16 no 1 pp 1ndash11 2014
[10] M Liu X Wang and H Huang ldquoMaximum principle forforward-backward control system driven by Ito-Levy pro-cesses under initial-terminal constraintsrdquo MathematicalProblems in Engineering vol 2017 pp 1ndash13 2017
[11] S Tang ldquoe maximum principle for partially observedoptimal control of stochastic differential equationsrdquo SIAMJournal on Control and Optimization vol 36 no 5pp 1596ndash1617 1998
[12] Q Meng ldquoA maximum principle for optimal control problemof fully coupled forward-backward stochastic systems withpartial informationrdquo Science in China Series A Mathematicsvol 52 no 7 pp 1579ndash1588 2009
[13] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward backward stichastic differential equationswith jumpsrdquo SIAM J Control Optimenleadertwodots vol 48pp 2845ndash2976 2009
[14] G Wang and Z Wu ldquoKalman-Bucy filtering equations offorward and backward stochastic systems and applications torecursive optimal control problemsrdquo Journal of MathematicalAnalysis and Applications vol 342 no 2 pp 1280ndash12962008
[15] H Xiao ldquoe maximum principle for partially observedoptimal control of forward-backward stochastic systems withrandom jumpsrdquo Journal of Systems Science and Complexityvol 24 no 6 pp 1083ndash1099 2011
[16] G Wang Z Wu and J Xiong ldquoMaximum principles forforward-backward stochastic control systems with correlatedstate and observation noisesrdquo SIAM Journal on Control andOptimization vol 51 no 1 pp 491ndash524 2013
[17] G Wang C Zhang and W Zhang ldquoStochastic maximumprinciple for mean-field type optimal control under partialinformationrdquo IEEE Transactions on Automatic Controlvol 59 no 2 pp 522ndash528 2014
[18] S Q Zhang J Xiong and X D Liu ldquoStochastic maximumprinciple for partially observed forward backward stochasticdifferential equations with jumps and regime switchingrdquoScience China Information Sciences vol 61 no 7 pp 1ndash132018
[19] J Xiong S Q Zhang S Zhang and Y Zhuang ldquoA partiallyobserved non-zero sum differential game of forward-
backward stochastic differential equations and its applicationin financerdquo Mathematical Control amp Related Fields vol 9no 2 pp 257ndash276 2019
[20] D Nualart and W Schoutens ldquoChaotic and predictablerepresentations for Levy processesrdquo Stochastic Processes and9eir Applications vol 90 no 1 pp 109ndash122 2000
[21] D Nualart and W Schoutens ldquoBackward stochastic differ-ential equations and Feynman-Kac formula for Levy pro-cesses with applications in financerdquo Bernoulli vol 7 no 5pp 761ndash776 2001
[22] Q Meng and M Tang ldquoNecessary and sufficient conditionsfor optimal control of stochastic systems associated with Levyprocessesrdquo Science in China Series F Information Sciencesvol 52 no 11 pp 1982ndash1992 2009
[23] M Tang and Q Zhang ldquoOptimal variational principle forbackward stochastic control systems associated with Levyprocessesrdquo Science China Mathematics vol 55 no 4pp 745ndash761 2012
[24] F Baghery N Khelfallah B Mezerdi and I Turpin ldquoFullycoupled forward backward stochastic differential equationsdriven by Levy processes and application to differentialgamesrdquo De Gruyter vol 22 no 3 pp 151ndash161 2014
[25] F Zhang M N Tang and Q X Meng ldquoSochastic maximumprinciple for forward-backward stochastic control systemsassociated with Levy processesrdquo Chinese Annals of Mathe-matics vol 35A no 1 pp 83ndash100 2014
[26] X R Wang and H Huang ldquoMaximum principle for forward-backward stochastic control system driven by Levy processrdquoMathematical Problems in Engineering vol 2015 Article ID702802 12 pages 2015
[27] H Huang X Wang and M Liu ldquoA maximum principle forfully coupled forward-backward stochastic control systemdriven by Levy process with terminal state constraintsrdquoJournal of Systems Science and Complexity vol 31 no 4pp 859ndash874 2018
[28] P Muthukumar and R Deepa ldquoInfinite horizon optimalcontrol of forward backward stochastic system driven byTeugels martingales with Levy processrdquo Stochastic and Dy-namics vol 3 no 17 pp 1ndash17 2017
[29] J B Wu and Z M Liu ldquoOptimal control of mean-fieldbackward doubly stochastic systems driven by Ito-Levyprocessesrdquo International Journal of Control vol 93 no 4pp 953ndash970 2018
[30] W Wang J Wu and Z Liu ldquoe optimal control of fully-coupled forward-backward doubly stochastic systems drivenby Ito-Levy processesrdquo Journal of Systems Science andComplexity vol 32 no 4 pp 997ndash1018 2019
Mathematical Problems in Engineering 11
Here bx(t) bx(txlowastt ulowastt ) σx(t) σx(txlowastt ) gix(t)
gix(txlowastt ) b(tuε
t) b(txεt u
εt) b(tulowastt ) b (txlowastt ulowastt )
fw(t) fw(txlowastt ylowastt zlowastt rlowastt ulowastt ) (w xyz middot r) f(tuεt)
f(xεt y
εt z
εt r
εt u
εt) and f(tulowastt ) f(xlowastt ylowastt zlowastt rlowastt ulowastt )
Lemma 2 Suppose Assumptions 1 and 2 hold for the first-order variation X Y Z R we have the following estimations
sup0letleT
E Xt
111386811138681113868111386811138681113868111386811138682 leCε2 (7)
sup0letleT
E Xt
111386811138681113868111386811138681113868111386811138684 leCε4 (8)
sup0letleT
E Yt
111386811138681113868111386811138681113868111386811138682 leCε2 (9)
sup0letleT
E Yt
111386811138681113868111386811138681113868111386811138684 leCε4 (10)
sup0letleT
E 1113946T
0Zt
111386811138681113868111386811138681113868111386811138682dsleCε2 (11)
sup0letleT
E 1113946T
0Zt
111386811138681113868111386811138681113868111386811138682ds1113888 1113889
2
leCε4 (12)
sup0letleT
E 1113946T
0Rt
2dsleCε2 (13)
sup0letleT
E 1113946T
0Rt
2ds1113888 1113889
2
leCε4 (14)
Proof We first prove inequations (7) and (8) For theforward part of the first-order variation equation we have
E Xt
111386811138681113868111386811138681113868111386811138682
E 1113946t
0bx(s)Xs + b s u
εs( 1113857 minus b s u
lowasts( 11138571113858 1113859ds + σx(s)XsdBs + 1113944
infin
i1g
ix(s)XsdH
is
⎧⎨
⎩
⎫⎬
⎭
2
le 4 E 1113946t
0bx(s)Xsds1113888 1113889
2
+ E 1113946t
0b s u
εs( 1113857 minus b s u
lowasts( 11138571113858 1113859ds1113888 1113889
2
+ E 1113946t
0σx(s)Xs1113858 1113859
2ds + E 1113946t
01113944
infin
i1g
ix(s)Xs
⎡⎣ ⎤⎦2
ds⎧⎨
⎩
⎫⎬
⎭
le 12C2TE 1113946
t
0X
2sds + 4E 1113946
t
0b s u
εs( 1113857 minus b s u
lowasts( 11138571113858 1113859ds1113888 1113889
2
(15)
Applying Gronwallrsquos inequation we have
E Xt
111386811138681113868111386811138681113868111386811138682 leCε2 for all t isin [0 T] (16)
Similarly (8) holdsWe next estimate Yt Zt and Rt the backward part of the
first-order variation equation can be rewritten as
Yt + 1113946T
tZsdBs + 1113946
T
t1113944
infin
i1R
isdH
is Φx(t)XT + 1113946
T
tfx(t)Xt1113858
+ fy(t)Yt + fz(t)Zt
+ fr(t)Rt
+ f t uεt( 1113857 minus f t u
lowastt( 1113857]dt
(17)
Squaring both sides of (17) and using the fact of
EYt 1113946T
tZsdBs 0
EYt 1113946T
t1113944
infin
i1R
isdH
is 0
E 1113946T
tZsdBs 1113946
T
t1113944
infin
i1R
isdH
is 0
(18)
we get
E Yt
111386811138681113868111386811138681113868111386811138682
+ E 1113946T
tZ2sds + E 1113946
T
t1113944
infin
i1R
is
⎛⎝ ⎞⎠
2
dHis
E Φx(t)XT + 1113946T
tfx(t)Xt + fy(t)Yt + fz(t)Zt + fr(t)Rt + f t u
εt( 1113857 minus f t u
lowastt( 11138571113960 1113961dt1113896 1113897
2
le 6C2EX
2T + 6C
2TE 1113946
T
tX
2sds + 6C
2TE 1113946
T
tY2sds + 6C
2(T minus t)E 1113946
T
tZ2sds
+ 6C2(T minus t)E 1113946
T
t1113944
infin
i1R
is
⎛⎝ ⎞⎠
2
ds + 6E 1113946T
tf s u
εs( 1113857 minus f s u
lowasts( 1113857( 1113857ds1113888 1113889
2
(19)
4 Mathematical Problems in Engineering
When t isin [T minus δ T] with δ 112C2 we have
E Yt
111386811138681113868111386811138681113868111386811138682
+12
E 1113946T
tZ2sds +
12
E 1113946T
t1113944
infin
i1R
is
⎛⎝ ⎞⎠
2
dHis
le 6C2EX
2T + 6C
2TE 1113946
T
tX
2sds + 6C
2TE
middot 1113946T
tY2sds + 6E 1113946
T
tf s u
εs( 1113857 minus f s u
lowasts( 1113857( 1113857ds1113888 1113889
2
(20)
Applying Gronwallrsquos inequation we have
sup0letleT
E Yt
111386811138681113868111386811138681113868111386811138682 leCε2 t isin [T minus δ T]
sup0letleT
E 1113946T
0Zt
111386811138681113868111386811138681113868111386811138682dsleCε2 t isin [T minus δ T]
sup0letleT
E 1113946T
0Rt
2dsleCε2 t isin [T minus δ T]
(21)
Consider the BSDE of the first-order variation equationin the interval [t T minus δ]
Yt + 1113946Tminus δ
tZsdBs + 1113946
Tminus δ
t1113944
infin
i1R
isdH
is YTminus δ
+ 1113946Tminus δ
tfx(t)Xt + fy(t)Yt + fz(t)Zt1113960
+ fr(t)Rt + f t uεt( 1113857 minus f t u
lowastt( 11138571113859dt
(22)
us
E Yt
111386811138681113868111386811138681113868111386811138682
+ E 1113946Tminus δ
tZ2sds + E 1113946
Tminus δ
t1113944
infin
i1R
is
⎛⎝ ⎞⎠
2
dHis
E YTminus δ + 1113946Tminus δ
tfx(t)Xt + fy(t)Yt + fz(t)Zt + fr(t)Rt11139601113896
+ f t uεt( 1113857 minus f t u
lowastt( 11138571113859dt1113865
2
le 6C2EX
2T + 6C
2TE 1113946
Tminus δ
tX
2sds + 6C
2TE 1113946
Tminus δ
tY2sds
+ 6C2(T minus t)E 1113946
Tminus δ
tZ2sds + 6C
2(T minus t)E
middot 1113946Tminus δ
t1113944
infin
i1R
is
⎛⎝ ⎞⎠
2
ds + 6E 1113946Tminus δ
tf s u
εs( 1113857 minus f s u
lowasts( 1113857( 1113857ds1113888 1113889
2
(23)
So when t isin [T minus 2δ T] with δ 112C2 we have
sup0letleTE Yt
111386811138681113868111386811138681113868111386811138682 leCε2 t isin [T minus 2δ T]
sup0letleTE 1113946T
0Zt
111386811138681113868111386811138681113868111386811138682dsleCε2 t isin [T minus 2δ T]
sup0letleTE 1113946T
0Rt
2dsleCε2 t isin [T minus 2δ T]
(24)
After a finite number of iterations (9) (11) and (13) areobtained And (10) (12) and (14) can be proved by using a
similar method and the following inequalitiesE(1113938
T
tZsdBs)
4 ge β1E(1113938T
tZ2
sds)2 β1 gt 0E(1113938
T
t1113936infini1 Ri
sdHis)4 ge β2E( 1113938
T
t(1113936infini1 Ri
s)2ds)2 β2 gt 0
Lemma 3 Under hypothesis Assumptions 1 and 2 it holdsthe following four estimations
sup0letleT
E xεt minus xlowastt minus Xt
111386811138681113868111386811138681113868111386811138682 leCεε
2 Cε⟶ 0when ε⟶ 0
(25)
sup0letleT
E yεt minus ylowastt minus Yt
111386811138681113868111386811138681113868111386811138682 leCεε
2 Cε⟶ 0when ε⟶ 0
(26)
E 1113946T
0zεt minus zlowastt minus Zt
111386811138681113868111386811138681113868111386811138682dsleCεε
2 Cε⟶ 0when ε⟶ 0
(27)
E 1113946T
trεt minus rlowastt minus Rt
2dsleCεε
2 Cε⟶ 0when ε⟶ 0
(28)
Proof To prove (25) we observe that
1113946t
0b s x
lowasts + Xs u
εs( 1113857ds + 1113946
t
0σ s x
lowasts + Xs( 1113857dBs
+ 1113946t
01113944
infin
i1g
is xlowastsminus + Xsminus( 1113857dH
is
1113946t
0b s x
lowasts u
εs( 1113857 + 1113946
1
0bx s x
lowasts + λXs u
εs( 1113857dλXs1113890 1113891ds
+ 1113946t
0σ s x
lowasts( 1113857 + 1113946
1
0σx s x
lowasts + λXs( 1113857dλXs1113890 1113891dBs
+ 1113946t
01113944
infin
i1g
is xlowasts( 1113857 + 1113946
1
01113944
infin
i1g
ix s x
lowasts + λXs( 1113857dλXs
⎡⎣ ⎤⎦dHis
1113946t
0b s x
lowasts ulowasts( 1113857ds + 1113946
t
0σ s x
lowasts( 1113857dBs + 1113946
t
01113944
infin
i1g
is xlowasts( 1113857dH
is
+ 1113946t
0bx s x
lowasts ulowasts( 1113857Xsds + 1113946
t
0σx s x
lowasts( 1113857XsdBs
+ 1113946t
01113944
infin
i1g
ix s x
lowastsminus( 1113857dH
is
+ 1113946t
0b s x
lowasts u
εs( 1113857 minus b s x
lowasts ulowasts( 11138571113858 1113859ds + 1113946
t
0Aεds
+ 1113946t
0BεdBs + 1113946
t
0CεdH
is
xlowastt minus x0 + Xt + 1113946
t
0Aεds + 1113946
t
0BεdBs + 1113946
t
0CεdH
is
(29)
Mathematical Problems in Engineering 5
where
Aε
11139461
0bx s x
lowasts + λXs u
εs( 1113857 minus bx s x
lowasts ulowasts( 11138571113858 1113859dλXs
Bε
11139461
0σx s x
lowasts + λXs( 1113857 minus σx s x
lowasts( 11138571113858 1113859dλXs
Cε
11139461
01113944
infin
i1g
ix s x
lowasts + λXs( 1113857 minus g
ix s x
lowasts( 11138571113872 1113873⎡⎣ ⎤⎦dλXs
(30)
It follows easily from Lemma 2 that
sup0letleT
E 1113946t
0A
εds1113888 1113889
2
+ 1113946t
0BεdBs1113888 1113889
2
+ 1113946t
0CεdH
is1113888 1113889
2⎡⎣ ⎤⎦ o ε21113872 1113873
(31)
Since
xεt minus x0 1113946
t
0b s x
εs u
εs( 1113857ds + 1113946
t
0σ s x
εs( 1113857dBs + 1113946
t
01113944
infin
i1g
is x
εsminus( 1113857dH
is
(32)
then
xεt minus xlowastt minus Xt 1113946
t
0b s x
εs u
εs( 1113857 minus b s x
lowasts + Xs u
εs( 11138571113858 1113859ds
+ 1113946t
0σ s x
εs( 1113857 minus σ s x
lowasts + Xs( 11138571113858 1113859dBs
+ 1113946t
01113944
infin
i1g
is x
εsminus( 1113857 minus g
is xlowastsminus + Xsminus( 11138571113872 1113873dH
is
+ 1113946t
0Aεds + 1113946
t
0BεdBs + 1113946
t
0CεdH
is
1113946t
0D
εxεs minus xlowasts minus Xs( 1113857ds + 1113946
t
0Eε
xεs minus xlowasts minus Xs( 1113857dBs
+ 1113946t
0Fε
xεs minus xlowasts minus Xs( 1113857dH
is
(33)
with Dε 111393810 bx(s xlowasts + Xs + λ(xε
s minus xlowasts minus Xs) uεs)dλ
Eε 111393810 σx(s xlowasts + Xs + λ(xε
s minus xlowasts minus Xs))dλ and Fε 111393810
1113936infini1 gi
x(s xlowasts + Xs + λ(xεs minus xlowasts minus Xs))dλ
By Gronwallrsquos inequation we have
sup0letleT
E xεt minus xlowastt minus Xt
111386811138681113868111386811138681113868111386811138682 leCεε
2 Cε⟶ 0when ε⟶ 0
(34)
Next we prove (26) (27) (28) it can be easily checkedthat
minus 1113946T
tf s x
lowasts + Xs y
lowasts + Ys z
lowasts + Zs r
lowasts + Rs u
εs( 1113857ds
+ 1113946T
ts zlowasts + Zs( 1113857dBs + 1113946
T
t1113944
infin
i1g
is xlowastsminus + Xsminus( 1113857dH
is
Φ xlowastT( 1113857 minus y
lowastt +Φx x
lowastT( 1113857XT minus Yt minus 1113946
T
tGεds
(35)
Here
Gε
11139461
0fx s x
lowasts + λXs y
lowasts + λYs z
lowasts((
+ λZs rlowasts + λRs u
εs1113857 minus fx(s)1113857dλXs
+ 11139461
0fy s x
lowasts + λXs y
lowasts + λYs z
lowasts(1113872
+ λZs rlowasts + λRs u
εs1113857 minus fy(s)1113873dλYs
+ 11139461
0fz s x
lowasts + λXs y
lowasts + λYs z
lowasts((
+ λZs rlowasts + λRs u
εs1113857 minus fz(s)1113857dλZs
+ 11139461
0fr s x
lowasts + λXs y
lowasts + λYs z
lowasts((
+ λZs rlowasts + λRs u
εs1113857 minus fr(s)1113857dλRs
(36)
Since
yεt Φ x
εT( 1113857 + 1113946
T
tf s x
εs y
εs z
εs r
εs u
εs( 1113857ds minus 1113946
T
tzsdBs
minus 1113946T
t1113944
infin
i1r
iεs dH
is
(37)
then
yεt minus ylowastt minus Yt Φ x
εT( 1113857 minus Φ x
lowastT( 1113857 minus Φx x
lowastT( 1113857XT + 1113946
T
tf s x
εs y
εs z
εs r
εs u
εs( 1113857 minus f s x
lowasts + Xs y
lowasts + Ys z
lowasts + Zs r
lowasts + Rs u
εs(1113858 1113859ds
minus 1113946T
tzεs minus zlowasts minus Zs( 1113857dBs minus 1113946
T
t1113944
infin
i1r
iεs minus r
ilowasts minus Rs1113872 1113873dH
is + 1113946
T
tGεds
(38)
6 Mathematical Problems in Engineering
Squaring both sides of the equation above we get
E yεt minus ylowastt minus Yt
111386811138681113868111386811138681113868111386811138682
+ E 1113946T
tzεs minus zlowasts minus Zs( 1113857
2ds minus E 1113946T
t1113944
infin
i1r
iεs minus r
ilowasts minus Rs1113872 1113873
2ds
E f s xεs y
εs z
εs r
εs u
εs( 1113857 minus f s x
lowasts + Xs y
lowasts + Ys z
lowasts + Zs r
lowasts + Rs u
εs( 11138571113858 1113859ds +Φ x
εT( 1113857 minus Φ x
lowastT( 1113857 minus Φx x
lowastT( 1113857XT + 1113946
T
tGεds1113896 1113897
2
(39)
From Lemma 2 and equation (25) we have
sup0letleTE 1113946T
tGεds1113888 1113889
2
o ε21113872 1113873
E Φ xεT( 1113857 minus Φ x
lowastT( 1113857 minus Φx x
lowastT( 1113857XT1113858 1113859
2 o ε21113872 1113873
(40)
en we can get (26) (27) and (28) by applying theiterative method to the above relations
Lemma 4 (variational inequality) Under the conditionsthat Assumptions 1 and 2 are established we can get thefollowing variational inequality
Ecy ylowast0( 1113857Y0 ge o(ε) (41)
Proof From the four estimations in Lemma 3 we have thefollowing estimation
E c yε0( 1113857 minus c y
lowast0 + Y0( 11138571113858 1113859 o(ε) (42)
erefore
0leE c ylowast0 + Y0( 1113857 minus c y
lowast0( 11138571113858 1113859 + o(ε) Ecy y
lowast0( 1113857Y0 + o(ε)
(43)
We introduce the following Hamiltonian functionH [0 T] times Rn times Rm times Rmtimesd times l2(Rm) times U times Rn times Rm times
Rntimesdtimes l2(Rn) as
H(t x y z r u p q w k) langp b(t x u)rang +langw σ(t x)rang
+langk g(t x)rang minus langq f(t x y z r u)rang
(44)
and the following adjoint equation
dqt Hy ulowastt( 1113857dt + Hz ulowastt( 1113857dBt + 1113936infin
i1Hi
r ulowastt( 1113857dHit
minus dpt Hx ulowastt( 1113857dt minus wtdBt minus 1113936infin
i1ki
tminus dHit
q0 cy ylowast0( 1113857 pT minus Φx xlowastT( 1113857qT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎩
(45)
where
Hx ulowastt( 1113857 Hx t x
lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857
Hy ulowastt( 1113857 Hy t x
lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857
Hz ulowastt( 1113857 Hz t x
lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857
Hr ulowastt( 1113857 Hr t x
lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857
(46)
It is easily check that adjoint equation (45) has a uniquesolution quartet (pt qt wt kt) isinM2(0 T Rn+m+mtimesd) times
l2(0 T Rm)en we get the main result of this section
Theorem 1 Let hypothesis Assumptions 1 and2 hold ulowastt isan optimal control and the corresponding optimal statetrajectories are (xlowastt ylowastt zlowastt rlowastt ) let (pt qt wt kt) be thesolution of adjoint equation (45) and then for each ad-missible control ut isin Uad[0 T] we have
H txlowastt ylowastt zlowastt rlowastt utpt qtwtkt( 1113857
geH txlowastt ylowastt zlowastt rlowastt ulowastt pt qtwtkt( 1113857asae(47)
Proof Applying It 1113954os formula to langp Xrang and langq Yrang itfollows from (6) (45) and the variational inequality that
E 1113946T
0H t xt yt zt rt u
εt pt qt wt kt( 11138571113858
minus H t xt yt zt rt ulowastt pt qt wt kt( 11138571113859dt Ecy y
lowast0( 1113857Y0 ge o(ε)
(48)
By the definition of uεt we know that for any
u isin Uad[0 T] the following inequation holds
E H t xt yt zt rt uεt pt qt wt kt( 11138571113858
minus H t xt yt zt rt ulowastt pt qt wt kt( 11138571113859ge 0
(49)
en (47) can be easily checked
4 Stochastic Control Problem withState Constraints
In this part we are going to discuss stochastic controlproblems with state constraints in control system (2)Specifically the initial state constraints and final stateconstraints are as follows
EG1 xT( 1113857 0
EG0 y0( 1113857 0(50)
Mathematical Problems in Engineering 7
where G1 Rn⟶ Rn1(n1 lt n) G0 Rm⟶ Rm1(m1 ltm)Our optimal control problem is to find ulowast isin Uad such that
J ulowast(middot)( 1113857 inf
u(middot)isinUad
J(u(middot)) (51)
subject to the state constraints (50) In the following we willapply Ekelandrsquos variational principle to solve this optimalcontrol problem Firstly we need the following assumptions
Assumption 3 Control domain U is assumed to be closedthe mappings in state constraints G1 G0 are continuouslydifferentiable and G1x G0y are bounded
For forallu1(middot) u2(middot) isin Uad let
d u1(middot) u2(middot)( 1113857 E u1(middot) minus u2(middot)1113868111386811138681113868
11138681113868111386811138682 gt 0 t isin [0 T]1113966 1113967
11138681113868111386811138681113868
11138681113868111386811138681113868 (52)
Same as Section 3 we also assume ulowastt be the optimalcontrol and the corresponding optimal state trajectories are(xlowastt ylowastt zlowastt rlowastt ) In order to solve the constraint problemwe need the following penalty cost functional for any ρgt 0
Jρ(u) E G1 xT( 11138571113858 11138591113868111386811138681113868
11138681113868111386811138682
+ E G0 y0( 11138571113858 11138591113868111386811138681113868
11138681113868111386811138682
+ J(u) minus J ulowast( ) + ρ1113868111386811138681113868
11138681113868111386811138682
1113966 111396712
(53)
It can be checked that Jρ(u) Uad⟶ R1 is continuousand for any u(middot) isin Uad
Jρ(u)ge 0 Jρ ulowast
( 1113857 ρ
Jρ ulowast
( 1113857le infu(middot)isinUad
Jρ(u) + ρ(54)
It can be obtained by Ekelandrsquos variational principle thatthere exists u
ρt isin Uad such that
(i) Jρ uρ( )le Jρ ulowast( ) ρ
(ii) d uρ ulowast( )le ρradic
(iii) Jρ(v)ge Jρ uρ( ) minusρradic
d uρ v( ) for forallv(middot) isin Uad
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(55)
For fixed ρ and admissible control uρt we define the spike
variation as follows
uρεt
vt τ le tle τ + ε
uρt otherwise
1113896 (56)
for any εgt 0 and it is easy to check from (iii) of (55) that
Jρ uρε( ) minus Jρ uρ( ) +ρradic
d uρε uρ( )ge 0 (57)
Let (xρt y
ρt z
ρt r
ρt ) be the trajectories corresponding to u
ρt
and (xρεt y
ρεt z
ρεt r
ρεt ) be the trajectories corresponding to
uρεt e variational equation we used in this section is the
same as the one in Section 3 with ulowastt uρt and
(xlowastt ylowastt zlowastt rlowastt ) (xρt y
ρt z
ρt r
ρt ) And we also assume that
the solution of this variational equation is (Xρt Y
ρt Z
ρt R
ρt )
Similar to the approach in Lemmas 2 and 3 it can be shownthat
sup0letleTE xρεt minus x
ρt minus X
ρt
111386811138681113868111386811138681113868111386811138682 leCεε
2 Cε⟶ 0when ε⟶ 0
sup0letleTE yρεt minus y
ρt minus Y
ρt
111386811138681113868111386811138681113868111386811138682 leCεε
2 Cε⟶ 0when ε⟶ 0
E 1113946T
0zρεt minus z
ρt minus Z
ρt
111386811138681113868111386811138681113868111386811138682dsleCεε
2 Cε⟶ 0when ε⟶ 0
E 1113946T
0rρεt minus r
ρt minus R
ρt
2dsleCεε
2 Cε⟶ 0when ε⟶ 0
(58)
en by (57) the following variational inequality holds
Jρ uρε
( 1113857 minus Jρ uρ
( 1113857 +ρ
radicd u
ρε u
ρ( 1113857
J2ρ uρε( ) minus J2ρ uρ( )
Jρ uρε( ) + Jρ uρ( )+ ε
ρ
radic
langhρε1 E G1x x
ρT( 11138571113858 1113859X
ρTrang +langhρε
0 E G0y yρ0( 11138571113960 1113961Y
ρ0rang
+ hρε
E cy yρ0( 1113857Y
ρ01113960 1113961 + ε
ρ
radic+ o(ε)
(59)
where
hρε1
2E G1 xρT( 11138571113858 1113859
Jρ uρε( ) + Jρ uρ( )
hρε0
2E G0 yρ0( 11138571113858 1113859
Jρ uρε( ) + Jρ uρ( )
hρε
2E c y
ρ0( 1113857 minus c y0( 1113857 + ρ1113858 1113859
Jρ uρε( ) + Jρ uρ( )
(60)
Now let (pρεt q
ρεt w
ρεt k
ρεt ) be the solution of
minus dpρεt b
ρx(t)( 1113857
τpρεt minus f
ρx(t)( 1113857
τqρεt + σρx(t)( 1113857
τw
ρεt + 1113936infin
i1gρx(t)( 1113857
τkρεt )i
1113890 1113891dt
minus wρεt dBt minus 1113936
infin
i1kρεt( 1113857
idHit
dqρεt f
ρy(t)( 1113857
τqρεt dt + f
ρz(t)( 1113857
τqρεt dBt + 1113936
infin
i1fρir (t)1113872 1113873
τ)q
ρεt dHi
t
q0 minus G0y yρ0( 1113857h
ρε0 + cy y
ρ0( 1113857hρε pT G1x x
ρT( 1113857h
ρε1 minus Φx x
ρT( 1113857q
ρεT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(61)
8 Mathematical Problems in Engineering
Here bρx(t) bx(t x
ρt u
ρt ) σρx(t) σx(t x
ρt )
fρy(t) fy(t x
ρt y
ρt z
ρt r
ρt u
ρt ) etc Applying It 1113954os formula
to langXρt q
ρεt rang + langY
ρt p
ρεt rang variational inequality (59) can be
rewritten as
H t xρt y
ρt z
ρt r
ρt u
ρεt p
ρεt q
ρεt w
ρεt k
ρεt( 1113857
minus H t xρt y
ρt z
ρt r
ρt u
ρt p
ρεt q
ρεt w
ρεt k
ρεt( 1113857
+ ερ
radic+ o(ε)ge 0 forallvt isin U ae as
(62)
where the Hamiltonian function H is defined as (44) Since
limε⟶0 hρε0
111386811138681113868111386811138681113868111386811138682
+ hρε1
111386811138681113868111386811138681113868111386811138682
+ hρε1113868111386811138681113868
11138681113868111386811138682
1113872 1113873 1 (63)
there exists a convergent subsequence still denoted by(h
ρε0 h
ρε1 hρε) such that (h
ρε0 h
ρε1 hρε)⟶ (h
ρ0 h
ρ1 hρ) when
ε⟶ 0 with |hρ0|2 + |h
ρ1|2 + |hρ|2 1
Let (pρt q
ρt w
ρt k
ρt ) be the solution of
minus dpρt b
ρx(t)( 1113857
τpρt minus f
ρx(t)( 1113857
τqρt + σρx(t)( 1113857
τw
ρt + 1113936infin
i1gρx(t)( 1113857
τkρt )i
1113890 1113891dt
minus wρtdBt minus 1113936
infin
i1kρt( 1113857
idHit
dqρt f
ρy(t)( 1113857
τqρtdt + f
ρz(t)( 1113857
τqρtdBt + 1113936
infin
i1fρir (t)1113872 1113873
τ1113888 1113889q
ρtdHi
t
q0 minus G0y yρ0( 1113857h
ρ0 + cy y
ρ0( 1113857hρ pT G1x x
ρT( 1113857h
ρ1 minus Φx x
ρT( 1113857q
ρT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(64)
From the continuous dependence of the solutions ofFBSDEs with Levy process on the parameters we can provethat the following convergence holds(p
ρεt q
ρεt w
ρεt k
ρεt )⟶ (p
ρt q
ρt w
ρt k
ρt ) then in equation
(62) it implies that
H t xρt y
ρt z
ρt r
ρt vt p
ρt q
ρt w
ρt k
ρt( 1113857
minus H t xρt y
ρt z
ρt r
ρt u
ρt p
ρt q
ρt w
ρt k
ρt( 1113857
+ρ
radic ge 0 forallvt isin U ae as
(65)
Similarly there exists a convergent subsequence(h
ρ0 h
ρ1 h
ρ) such that (h
ρ0 h
ρ1 h
ρ)⟶ (h0 h1 h) when
ρ⟶ 0 with |h0|2 + |h1|
2 + |h|2 1 Since uρt⟶ ulowastt as
ρ⟶ 0 we have (xρt y
ρt z
ρt r
ρt )⟶ (xlowastt ylowastt zlowastt rlowastt ) and
(Xρt Y
ρt Z
ρt R
ρt )⟶ (Xt Yt Zt Rt) which is the solution of
the variational equation as same as (6)e following adjoint equation is introduced
minus dpt bx(t)( 1113857τpt minus fx(t)( 1113857
τqt + σx(t)( 1113857
τwt + 1113936infin
i1gx(t)( 1113857
τkt)
i1113890 1113891dt
minus wtdBt minus 1113936infin
i1kt( 1113857
idHit
dqt fy(t)1113872 1113873τqtdt + fz(t)( 1113857
τqtdBt + 1113936
infin
i1fρir (t)1113872 1113873
τ1113888 1113889qtdHi
t
q0 minus G0y y0( 1113857h0 + cy y0( 1113857h pT G1x xT( 1113857h1 minus Φx xT( 1113857qT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(66)
Similarly it can be proved that(p
ρt q
ρt w
ρt k
ρt )⟶ (pt qt wt kt) then inequation (65)
impliesH t x
lowastt ylowastt zlowastt rlowastt vt pt qt wt kt( 1113857
minus H t xlowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857ge 0forallvt isin U ae as
(67)
en we get the following theorem
Theorem 2 Let Assumptions 1ndash3 hold ulowastt is an optimalcontrol and the corresponding optimal state trajectories are(xlowastt ylowastt zlowastt rlowastt ) and (pt qt wt kt) is the solution of adjoint
equation (66) then there exists nonzero constant(h1 h0 h) isin (Rn1 times Rm1 times R) with |h0|
2 + |h1|2 + |h|2 1
such that for any admissible control vt isin Uad[0 T] themaximum condition (67) holds
9is conclusion can be drawn from the above analysisdirectly
5 A Financial Example
In this section we will study the problem of optimal con-sumption rate selection in the financial market which willnaturally inspire our research to the forward-backwardstochastic optimal control problem in Section 3
Mathematical Problems in Engineering 9
Assume the investorrsquos asset process xt (tge 0) in the fi-nancial market is described by the following stochasticdifferential equation with the Teugels martingale
dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin
i1gi
txtdHit
x0 Wgt 0 t isin [0 T]
⎧⎪⎪⎨
⎪⎪⎩(68)
where μt and σt ne 0 are the expected return and volatility ofthe value process xt at time t respectively and Ct is theconsumption rate process Assume μt σt gt and Ct are alluniformly bounded Ftminus measurable random processes epurpose of investors is to select the optimal consumptionstrategy Clowastt at time tge 0 minimizing the following recursiveutility
J(C(middot)) E y01113858 1113859 (69)
where yt is the following backward stochastic process
minus dyt Leminus rtC1minus R
t
1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936
infin
i1ri
tdHit
yT minus xT gt 0 t isin [0 T]
⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
(70)
with constant Lgt 0 the discount factor rgt 0 and theArrowndashPratt measure of risk aversion R isin (0 1)
If the consumption process Ct is regarded as a controlvariable then combining (68) with (70) we encounter thefollowing control system
dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin
i1gi
txtdHit
minus dyt Leminus rtC1minus R
t
1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936
infin
i1ri
tdHit
x0 Wgt 0 yT minus xT gt 0 t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(71)
which is obviously a special case of stochastic control system(2) with b(t x u) μtxt minus Ct σ(t x) σtxt g(t x)
gitxt Φx(xT) minus XT and f(t x y z r u) Leminus rt(C1minus R
t (1 minus R)) minus ryt
We can check that both Assumptions 1 and 2 are sat-isfieden we can use ourmaximum principle (eorem 1)to solve the above optimization problem Let Clowastt be anoptimal consumption rate and xlowastt ylowastt be the correspondingwealth process and recursive utility process In this case theHamiltonian function H reduces to
H t xlowastt ylowastt Ct p
lowastt qlowastt wlowastt klowastt( 1113857 langplowastt μtx
lowastt minus Ctrang
+langwlowastt σtxlowastt rang +langklowastt g
itxlowastt rang minus langqlowastt Le
minus rt C1minus Rt
1 minus Rminus rylowastt rang
(72)
and (plowastt qlowastt wlowastt klowastt ) is the solution of the following adjointequation
dqt rqtdt
minus dpt μtpt + σtwt + 1113936infin
i1ki
t( 1113857τgi
t1113890 1113891dt minus wtdBt minus 1113936infin
i1ki
tdHit
q0 1 pT qT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
(73)
According to the maximum principle (47) we have
plowastt minus Le
minus rtClowastt( 1113857
minus Rqlowastt (74)
and the optimal consumption rate
Clowastt minus
ert
L
plowasttqlowastt
1113888 1113889
minus 1R
(75)
By solving FBSDE (73) we get qlowastt ert plowastt erteμ(Tminus t)and wlowastt klowastt 0 en we get the optimal consumptionrate of the investor which is
Clowastt minus
erT
Leμ(Tminus t)
1113888 1113889
minus 1R
t isin [0 T] (76)
6 Conclusions
In this paper a nonconvex control domain case of theforward-backward stochastic control driven by Levy processis considered and we obtain the global stochastic maximumprinciple for this stochastic control problem And then theproblem of stochastic control with initial and final stateconstraints on the state variables is discussed and a nec-essary condition about existence of the optimal control isalso acquired A financial example of optimal consumptionis discussed to illustrate the application of the stochasticmaximum principle
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the NNSF (11701335) Shan-dong NSF (ZR2019BG015) NSSF (17BGL058) ShandongProvincial Higher Education Science and Technology PlanProject (J18KA236) and Shandong Womenrsquos College High-level Research Project Cultivation Foundation(2018GSPGJ07)
References
[1] L S Pontryakin V G Boltyanskti R V Gamkrelidze andE F Mischenko9eMathematical9eory of Optimal ControlProcesses John Wiley New York NY USA 1962
[2] S Peng ldquoA general stochastic maximum principle for optimalcontrol problemsrdquo SIAM Journal on Control and Optimiza-tion vol 28 no 4 pp 966ndash979 1990
10 Mathematical Problems in Engineering
[3] S G Peng ldquoBackward stochastic differrntial equations andapplication to optimal controlrdquo Applied Mathematics ampOptimization vol 27 no 4 pp 125ndash144 1993
[4] W S Xu ldquoStochastic maximum principle for optimal controlproblem of forward and backward systemrdquo Journal of Aus-tralian Mathematical Sciences vol 37 pp 249ndash259 1998B
[5] Z Wu ldquoMaximum principle for optimal control problem offully coupled forward-backward stochastic control systemrdquoSystem Science and Mathematical Sciences vol 11 no 3pp 249ndash259 1998
[6] J T Shi and ZWu ldquoemaximum principle for fully coupledforward-backward stochastic control systemrdquo Acta Auto-matica Sinica vol 32 no 2 pp 161ndash169 2006
[7] R Situ ldquoA maximum principle for optimal controls of sto-chastic systems with random jumpsrdquo in Proceedings of theNational Conference on Control 9eory and its ApplicationsQingdao China 1991
[8] J Shi and Z Wu ldquoMaximum principle for forward-backwardstochastic control system with random jumps and applica-tions to financerdquo Journal of Systems Science and Complexityvol 23 no 2 pp 219ndash231 2010
[9] J T Shi ldquoGlobal maximum principle for forward-backwardstochastic control system with Poisson jumpsrdquo Asian Journalof Control vol 16 no 1 pp 1ndash11 2014
[10] M Liu X Wang and H Huang ldquoMaximum principle forforward-backward control system driven by Ito-Levy pro-cesses under initial-terminal constraintsrdquo MathematicalProblems in Engineering vol 2017 pp 1ndash13 2017
[11] S Tang ldquoe maximum principle for partially observedoptimal control of stochastic differential equationsrdquo SIAMJournal on Control and Optimization vol 36 no 5pp 1596ndash1617 1998
[12] Q Meng ldquoA maximum principle for optimal control problemof fully coupled forward-backward stochastic systems withpartial informationrdquo Science in China Series A Mathematicsvol 52 no 7 pp 1579ndash1588 2009
[13] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward backward stichastic differential equationswith jumpsrdquo SIAM J Control Optimenleadertwodots vol 48pp 2845ndash2976 2009
[14] G Wang and Z Wu ldquoKalman-Bucy filtering equations offorward and backward stochastic systems and applications torecursive optimal control problemsrdquo Journal of MathematicalAnalysis and Applications vol 342 no 2 pp 1280ndash12962008
[15] H Xiao ldquoe maximum principle for partially observedoptimal control of forward-backward stochastic systems withrandom jumpsrdquo Journal of Systems Science and Complexityvol 24 no 6 pp 1083ndash1099 2011
[16] G Wang Z Wu and J Xiong ldquoMaximum principles forforward-backward stochastic control systems with correlatedstate and observation noisesrdquo SIAM Journal on Control andOptimization vol 51 no 1 pp 491ndash524 2013
[17] G Wang C Zhang and W Zhang ldquoStochastic maximumprinciple for mean-field type optimal control under partialinformationrdquo IEEE Transactions on Automatic Controlvol 59 no 2 pp 522ndash528 2014
[18] S Q Zhang J Xiong and X D Liu ldquoStochastic maximumprinciple for partially observed forward backward stochasticdifferential equations with jumps and regime switchingrdquoScience China Information Sciences vol 61 no 7 pp 1ndash132018
[19] J Xiong S Q Zhang S Zhang and Y Zhuang ldquoA partiallyobserved non-zero sum differential game of forward-
backward stochastic differential equations and its applicationin financerdquo Mathematical Control amp Related Fields vol 9no 2 pp 257ndash276 2019
[20] D Nualart and W Schoutens ldquoChaotic and predictablerepresentations for Levy processesrdquo Stochastic Processes and9eir Applications vol 90 no 1 pp 109ndash122 2000
[21] D Nualart and W Schoutens ldquoBackward stochastic differ-ential equations and Feynman-Kac formula for Levy pro-cesses with applications in financerdquo Bernoulli vol 7 no 5pp 761ndash776 2001
[22] Q Meng and M Tang ldquoNecessary and sufficient conditionsfor optimal control of stochastic systems associated with Levyprocessesrdquo Science in China Series F Information Sciencesvol 52 no 11 pp 1982ndash1992 2009
[23] M Tang and Q Zhang ldquoOptimal variational principle forbackward stochastic control systems associated with Levyprocessesrdquo Science China Mathematics vol 55 no 4pp 745ndash761 2012
[24] F Baghery N Khelfallah B Mezerdi and I Turpin ldquoFullycoupled forward backward stochastic differential equationsdriven by Levy processes and application to differentialgamesrdquo De Gruyter vol 22 no 3 pp 151ndash161 2014
[25] F Zhang M N Tang and Q X Meng ldquoSochastic maximumprinciple for forward-backward stochastic control systemsassociated with Levy processesrdquo Chinese Annals of Mathe-matics vol 35A no 1 pp 83ndash100 2014
[26] X R Wang and H Huang ldquoMaximum principle for forward-backward stochastic control system driven by Levy processrdquoMathematical Problems in Engineering vol 2015 Article ID702802 12 pages 2015
[27] H Huang X Wang and M Liu ldquoA maximum principle forfully coupled forward-backward stochastic control systemdriven by Levy process with terminal state constraintsrdquoJournal of Systems Science and Complexity vol 31 no 4pp 859ndash874 2018
[28] P Muthukumar and R Deepa ldquoInfinite horizon optimalcontrol of forward backward stochastic system driven byTeugels martingales with Levy processrdquo Stochastic and Dy-namics vol 3 no 17 pp 1ndash17 2017
[29] J B Wu and Z M Liu ldquoOptimal control of mean-fieldbackward doubly stochastic systems driven by Ito-Levyprocessesrdquo International Journal of Control vol 93 no 4pp 953ndash970 2018
[30] W Wang J Wu and Z Liu ldquoe optimal control of fully-coupled forward-backward doubly stochastic systems drivenby Ito-Levy processesrdquo Journal of Systems Science andComplexity vol 32 no 4 pp 997ndash1018 2019
Mathematical Problems in Engineering 11
When t isin [T minus δ T] with δ 112C2 we have
E Yt
111386811138681113868111386811138681113868111386811138682
+12
E 1113946T
tZ2sds +
12
E 1113946T
t1113944
infin
i1R
is
⎛⎝ ⎞⎠
2
dHis
le 6C2EX
2T + 6C
2TE 1113946
T
tX
2sds + 6C
2TE
middot 1113946T
tY2sds + 6E 1113946
T
tf s u
εs( 1113857 minus f s u
lowasts( 1113857( 1113857ds1113888 1113889
2
(20)
Applying Gronwallrsquos inequation we have
sup0letleT
E Yt
111386811138681113868111386811138681113868111386811138682 leCε2 t isin [T minus δ T]
sup0letleT
E 1113946T
0Zt
111386811138681113868111386811138681113868111386811138682dsleCε2 t isin [T minus δ T]
sup0letleT
E 1113946T
0Rt
2dsleCε2 t isin [T minus δ T]
(21)
Consider the BSDE of the first-order variation equationin the interval [t T minus δ]
Yt + 1113946Tminus δ
tZsdBs + 1113946
Tminus δ
t1113944
infin
i1R
isdH
is YTminus δ
+ 1113946Tminus δ
tfx(t)Xt + fy(t)Yt + fz(t)Zt1113960
+ fr(t)Rt + f t uεt( 1113857 minus f t u
lowastt( 11138571113859dt
(22)
us
E Yt
111386811138681113868111386811138681113868111386811138682
+ E 1113946Tminus δ
tZ2sds + E 1113946
Tminus δ
t1113944
infin
i1R
is
⎛⎝ ⎞⎠
2
dHis
E YTminus δ + 1113946Tminus δ
tfx(t)Xt + fy(t)Yt + fz(t)Zt + fr(t)Rt11139601113896
+ f t uεt( 1113857 minus f t u
lowastt( 11138571113859dt1113865
2
le 6C2EX
2T + 6C
2TE 1113946
Tminus δ
tX
2sds + 6C
2TE 1113946
Tminus δ
tY2sds
+ 6C2(T minus t)E 1113946
Tminus δ
tZ2sds + 6C
2(T minus t)E
middot 1113946Tminus δ
t1113944
infin
i1R
is
⎛⎝ ⎞⎠
2
ds + 6E 1113946Tminus δ
tf s u
εs( 1113857 minus f s u
lowasts( 1113857( 1113857ds1113888 1113889
2
(23)
So when t isin [T minus 2δ T] with δ 112C2 we have
sup0letleTE Yt
111386811138681113868111386811138681113868111386811138682 leCε2 t isin [T minus 2δ T]
sup0letleTE 1113946T
0Zt
111386811138681113868111386811138681113868111386811138682dsleCε2 t isin [T minus 2δ T]
sup0letleTE 1113946T
0Rt
2dsleCε2 t isin [T minus 2δ T]
(24)
After a finite number of iterations (9) (11) and (13) areobtained And (10) (12) and (14) can be proved by using a
similar method and the following inequalitiesE(1113938
T
tZsdBs)
4 ge β1E(1113938T
tZ2
sds)2 β1 gt 0E(1113938
T
t1113936infini1 Ri
sdHis)4 ge β2E( 1113938
T
t(1113936infini1 Ri
s)2ds)2 β2 gt 0
Lemma 3 Under hypothesis Assumptions 1 and 2 it holdsthe following four estimations
sup0letleT
E xεt minus xlowastt minus Xt
111386811138681113868111386811138681113868111386811138682 leCεε
2 Cε⟶ 0when ε⟶ 0
(25)
sup0letleT
E yεt minus ylowastt minus Yt
111386811138681113868111386811138681113868111386811138682 leCεε
2 Cε⟶ 0when ε⟶ 0
(26)
E 1113946T
0zεt minus zlowastt minus Zt
111386811138681113868111386811138681113868111386811138682dsleCεε
2 Cε⟶ 0when ε⟶ 0
(27)
E 1113946T
trεt minus rlowastt minus Rt
2dsleCεε
2 Cε⟶ 0when ε⟶ 0
(28)
Proof To prove (25) we observe that
1113946t
0b s x
lowasts + Xs u
εs( 1113857ds + 1113946
t
0σ s x
lowasts + Xs( 1113857dBs
+ 1113946t
01113944
infin
i1g
is xlowastsminus + Xsminus( 1113857dH
is
1113946t
0b s x
lowasts u
εs( 1113857 + 1113946
1
0bx s x
lowasts + λXs u
εs( 1113857dλXs1113890 1113891ds
+ 1113946t
0σ s x
lowasts( 1113857 + 1113946
1
0σx s x
lowasts + λXs( 1113857dλXs1113890 1113891dBs
+ 1113946t
01113944
infin
i1g
is xlowasts( 1113857 + 1113946
1
01113944
infin
i1g
ix s x
lowasts + λXs( 1113857dλXs
⎡⎣ ⎤⎦dHis
1113946t
0b s x
lowasts ulowasts( 1113857ds + 1113946
t
0σ s x
lowasts( 1113857dBs + 1113946
t
01113944
infin
i1g
is xlowasts( 1113857dH
is
+ 1113946t
0bx s x
lowasts ulowasts( 1113857Xsds + 1113946
t
0σx s x
lowasts( 1113857XsdBs
+ 1113946t
01113944
infin
i1g
ix s x
lowastsminus( 1113857dH
is
+ 1113946t
0b s x
lowasts u
εs( 1113857 minus b s x
lowasts ulowasts( 11138571113858 1113859ds + 1113946
t
0Aεds
+ 1113946t
0BεdBs + 1113946
t
0CεdH
is
xlowastt minus x0 + Xt + 1113946
t
0Aεds + 1113946
t
0BεdBs + 1113946
t
0CεdH
is
(29)
Mathematical Problems in Engineering 5
where
Aε
11139461
0bx s x
lowasts + λXs u
εs( 1113857 minus bx s x
lowasts ulowasts( 11138571113858 1113859dλXs
Bε
11139461
0σx s x
lowasts + λXs( 1113857 minus σx s x
lowasts( 11138571113858 1113859dλXs
Cε
11139461
01113944
infin
i1g
ix s x
lowasts + λXs( 1113857 minus g
ix s x
lowasts( 11138571113872 1113873⎡⎣ ⎤⎦dλXs
(30)
It follows easily from Lemma 2 that
sup0letleT
E 1113946t
0A
εds1113888 1113889
2
+ 1113946t
0BεdBs1113888 1113889
2
+ 1113946t
0CεdH
is1113888 1113889
2⎡⎣ ⎤⎦ o ε21113872 1113873
(31)
Since
xεt minus x0 1113946
t
0b s x
εs u
εs( 1113857ds + 1113946
t
0σ s x
εs( 1113857dBs + 1113946
t
01113944
infin
i1g
is x
εsminus( 1113857dH
is
(32)
then
xεt minus xlowastt minus Xt 1113946
t
0b s x
εs u
εs( 1113857 minus b s x
lowasts + Xs u
εs( 11138571113858 1113859ds
+ 1113946t
0σ s x
εs( 1113857 minus σ s x
lowasts + Xs( 11138571113858 1113859dBs
+ 1113946t
01113944
infin
i1g
is x
εsminus( 1113857 minus g
is xlowastsminus + Xsminus( 11138571113872 1113873dH
is
+ 1113946t
0Aεds + 1113946
t
0BεdBs + 1113946
t
0CεdH
is
1113946t
0D
εxεs minus xlowasts minus Xs( 1113857ds + 1113946
t
0Eε
xεs minus xlowasts minus Xs( 1113857dBs
+ 1113946t
0Fε
xεs minus xlowasts minus Xs( 1113857dH
is
(33)
with Dε 111393810 bx(s xlowasts + Xs + λ(xε
s minus xlowasts minus Xs) uεs)dλ
Eε 111393810 σx(s xlowasts + Xs + λ(xε
s minus xlowasts minus Xs))dλ and Fε 111393810
1113936infini1 gi
x(s xlowasts + Xs + λ(xεs minus xlowasts minus Xs))dλ
By Gronwallrsquos inequation we have
sup0letleT
E xεt minus xlowastt minus Xt
111386811138681113868111386811138681113868111386811138682 leCεε
2 Cε⟶ 0when ε⟶ 0
(34)
Next we prove (26) (27) (28) it can be easily checkedthat
minus 1113946T
tf s x
lowasts + Xs y
lowasts + Ys z
lowasts + Zs r
lowasts + Rs u
εs( 1113857ds
+ 1113946T
ts zlowasts + Zs( 1113857dBs + 1113946
T
t1113944
infin
i1g
is xlowastsminus + Xsminus( 1113857dH
is
Φ xlowastT( 1113857 minus y
lowastt +Φx x
lowastT( 1113857XT minus Yt minus 1113946
T
tGεds
(35)
Here
Gε
11139461
0fx s x
lowasts + λXs y
lowasts + λYs z
lowasts((
+ λZs rlowasts + λRs u
εs1113857 minus fx(s)1113857dλXs
+ 11139461
0fy s x
lowasts + λXs y
lowasts + λYs z
lowasts(1113872
+ λZs rlowasts + λRs u
εs1113857 minus fy(s)1113873dλYs
+ 11139461
0fz s x
lowasts + λXs y
lowasts + λYs z
lowasts((
+ λZs rlowasts + λRs u
εs1113857 minus fz(s)1113857dλZs
+ 11139461
0fr s x
lowasts + λXs y
lowasts + λYs z
lowasts((
+ λZs rlowasts + λRs u
εs1113857 minus fr(s)1113857dλRs
(36)
Since
yεt Φ x
εT( 1113857 + 1113946
T
tf s x
εs y
εs z
εs r
εs u
εs( 1113857ds minus 1113946
T
tzsdBs
minus 1113946T
t1113944
infin
i1r
iεs dH
is
(37)
then
yεt minus ylowastt minus Yt Φ x
εT( 1113857 minus Φ x
lowastT( 1113857 minus Φx x
lowastT( 1113857XT + 1113946
T
tf s x
εs y
εs z
εs r
εs u
εs( 1113857 minus f s x
lowasts + Xs y
lowasts + Ys z
lowasts + Zs r
lowasts + Rs u
εs(1113858 1113859ds
minus 1113946T
tzεs minus zlowasts minus Zs( 1113857dBs minus 1113946
T
t1113944
infin
i1r
iεs minus r
ilowasts minus Rs1113872 1113873dH
is + 1113946
T
tGεds
(38)
6 Mathematical Problems in Engineering
Squaring both sides of the equation above we get
E yεt minus ylowastt minus Yt
111386811138681113868111386811138681113868111386811138682
+ E 1113946T
tzεs minus zlowasts minus Zs( 1113857
2ds minus E 1113946T
t1113944
infin
i1r
iεs minus r
ilowasts minus Rs1113872 1113873
2ds
E f s xεs y
εs z
εs r
εs u
εs( 1113857 minus f s x
lowasts + Xs y
lowasts + Ys z
lowasts + Zs r
lowasts + Rs u
εs( 11138571113858 1113859ds +Φ x
εT( 1113857 minus Φ x
lowastT( 1113857 minus Φx x
lowastT( 1113857XT + 1113946
T
tGεds1113896 1113897
2
(39)
From Lemma 2 and equation (25) we have
sup0letleTE 1113946T
tGεds1113888 1113889
2
o ε21113872 1113873
E Φ xεT( 1113857 minus Φ x
lowastT( 1113857 minus Φx x
lowastT( 1113857XT1113858 1113859
2 o ε21113872 1113873
(40)
en we can get (26) (27) and (28) by applying theiterative method to the above relations
Lemma 4 (variational inequality) Under the conditionsthat Assumptions 1 and 2 are established we can get thefollowing variational inequality
Ecy ylowast0( 1113857Y0 ge o(ε) (41)
Proof From the four estimations in Lemma 3 we have thefollowing estimation
E c yε0( 1113857 minus c y
lowast0 + Y0( 11138571113858 1113859 o(ε) (42)
erefore
0leE c ylowast0 + Y0( 1113857 minus c y
lowast0( 11138571113858 1113859 + o(ε) Ecy y
lowast0( 1113857Y0 + o(ε)
(43)
We introduce the following Hamiltonian functionH [0 T] times Rn times Rm times Rmtimesd times l2(Rm) times U times Rn times Rm times
Rntimesdtimes l2(Rn) as
H(t x y z r u p q w k) langp b(t x u)rang +langw σ(t x)rang
+langk g(t x)rang minus langq f(t x y z r u)rang
(44)
and the following adjoint equation
dqt Hy ulowastt( 1113857dt + Hz ulowastt( 1113857dBt + 1113936infin
i1Hi
r ulowastt( 1113857dHit
minus dpt Hx ulowastt( 1113857dt minus wtdBt minus 1113936infin
i1ki
tminus dHit
q0 cy ylowast0( 1113857 pT minus Φx xlowastT( 1113857qT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎩
(45)
where
Hx ulowastt( 1113857 Hx t x
lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857
Hy ulowastt( 1113857 Hy t x
lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857
Hz ulowastt( 1113857 Hz t x
lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857
Hr ulowastt( 1113857 Hr t x
lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857
(46)
It is easily check that adjoint equation (45) has a uniquesolution quartet (pt qt wt kt) isinM2(0 T Rn+m+mtimesd) times
l2(0 T Rm)en we get the main result of this section
Theorem 1 Let hypothesis Assumptions 1 and2 hold ulowastt isan optimal control and the corresponding optimal statetrajectories are (xlowastt ylowastt zlowastt rlowastt ) let (pt qt wt kt) be thesolution of adjoint equation (45) and then for each ad-missible control ut isin Uad[0 T] we have
H txlowastt ylowastt zlowastt rlowastt utpt qtwtkt( 1113857
geH txlowastt ylowastt zlowastt rlowastt ulowastt pt qtwtkt( 1113857asae(47)
Proof Applying It 1113954os formula to langp Xrang and langq Yrang itfollows from (6) (45) and the variational inequality that
E 1113946T
0H t xt yt zt rt u
εt pt qt wt kt( 11138571113858
minus H t xt yt zt rt ulowastt pt qt wt kt( 11138571113859dt Ecy y
lowast0( 1113857Y0 ge o(ε)
(48)
By the definition of uεt we know that for any
u isin Uad[0 T] the following inequation holds
E H t xt yt zt rt uεt pt qt wt kt( 11138571113858
minus H t xt yt zt rt ulowastt pt qt wt kt( 11138571113859ge 0
(49)
en (47) can be easily checked
4 Stochastic Control Problem withState Constraints
In this part we are going to discuss stochastic controlproblems with state constraints in control system (2)Specifically the initial state constraints and final stateconstraints are as follows
EG1 xT( 1113857 0
EG0 y0( 1113857 0(50)
Mathematical Problems in Engineering 7
where G1 Rn⟶ Rn1(n1 lt n) G0 Rm⟶ Rm1(m1 ltm)Our optimal control problem is to find ulowast isin Uad such that
J ulowast(middot)( 1113857 inf
u(middot)isinUad
J(u(middot)) (51)
subject to the state constraints (50) In the following we willapply Ekelandrsquos variational principle to solve this optimalcontrol problem Firstly we need the following assumptions
Assumption 3 Control domain U is assumed to be closedthe mappings in state constraints G1 G0 are continuouslydifferentiable and G1x G0y are bounded
For forallu1(middot) u2(middot) isin Uad let
d u1(middot) u2(middot)( 1113857 E u1(middot) minus u2(middot)1113868111386811138681113868
11138681113868111386811138682 gt 0 t isin [0 T]1113966 1113967
11138681113868111386811138681113868
11138681113868111386811138681113868 (52)
Same as Section 3 we also assume ulowastt be the optimalcontrol and the corresponding optimal state trajectories are(xlowastt ylowastt zlowastt rlowastt ) In order to solve the constraint problemwe need the following penalty cost functional for any ρgt 0
Jρ(u) E G1 xT( 11138571113858 11138591113868111386811138681113868
11138681113868111386811138682
+ E G0 y0( 11138571113858 11138591113868111386811138681113868
11138681113868111386811138682
+ J(u) minus J ulowast( ) + ρ1113868111386811138681113868
11138681113868111386811138682
1113966 111396712
(53)
It can be checked that Jρ(u) Uad⟶ R1 is continuousand for any u(middot) isin Uad
Jρ(u)ge 0 Jρ ulowast
( 1113857 ρ
Jρ ulowast
( 1113857le infu(middot)isinUad
Jρ(u) + ρ(54)
It can be obtained by Ekelandrsquos variational principle thatthere exists u
ρt isin Uad such that
(i) Jρ uρ( )le Jρ ulowast( ) ρ
(ii) d uρ ulowast( )le ρradic
(iii) Jρ(v)ge Jρ uρ( ) minusρradic
d uρ v( ) for forallv(middot) isin Uad
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(55)
For fixed ρ and admissible control uρt we define the spike
variation as follows
uρεt
vt τ le tle τ + ε
uρt otherwise
1113896 (56)
for any εgt 0 and it is easy to check from (iii) of (55) that
Jρ uρε( ) minus Jρ uρ( ) +ρradic
d uρε uρ( )ge 0 (57)
Let (xρt y
ρt z
ρt r
ρt ) be the trajectories corresponding to u
ρt
and (xρεt y
ρεt z
ρεt r
ρεt ) be the trajectories corresponding to
uρεt e variational equation we used in this section is the
same as the one in Section 3 with ulowastt uρt and
(xlowastt ylowastt zlowastt rlowastt ) (xρt y
ρt z
ρt r
ρt ) And we also assume that
the solution of this variational equation is (Xρt Y
ρt Z
ρt R
ρt )
Similar to the approach in Lemmas 2 and 3 it can be shownthat
sup0letleTE xρεt minus x
ρt minus X
ρt
111386811138681113868111386811138681113868111386811138682 leCεε
2 Cε⟶ 0when ε⟶ 0
sup0letleTE yρεt minus y
ρt minus Y
ρt
111386811138681113868111386811138681113868111386811138682 leCεε
2 Cε⟶ 0when ε⟶ 0
E 1113946T
0zρεt minus z
ρt minus Z
ρt
111386811138681113868111386811138681113868111386811138682dsleCεε
2 Cε⟶ 0when ε⟶ 0
E 1113946T
0rρεt minus r
ρt minus R
ρt
2dsleCεε
2 Cε⟶ 0when ε⟶ 0
(58)
en by (57) the following variational inequality holds
Jρ uρε
( 1113857 minus Jρ uρ
( 1113857 +ρ
radicd u
ρε u
ρ( 1113857
J2ρ uρε( ) minus J2ρ uρ( )
Jρ uρε( ) + Jρ uρ( )+ ε
ρ
radic
langhρε1 E G1x x
ρT( 11138571113858 1113859X
ρTrang +langhρε
0 E G0y yρ0( 11138571113960 1113961Y
ρ0rang
+ hρε
E cy yρ0( 1113857Y
ρ01113960 1113961 + ε
ρ
radic+ o(ε)
(59)
where
hρε1
2E G1 xρT( 11138571113858 1113859
Jρ uρε( ) + Jρ uρ( )
hρε0
2E G0 yρ0( 11138571113858 1113859
Jρ uρε( ) + Jρ uρ( )
hρε
2E c y
ρ0( 1113857 minus c y0( 1113857 + ρ1113858 1113859
Jρ uρε( ) + Jρ uρ( )
(60)
Now let (pρεt q
ρεt w
ρεt k
ρεt ) be the solution of
minus dpρεt b
ρx(t)( 1113857
τpρεt minus f
ρx(t)( 1113857
τqρεt + σρx(t)( 1113857
τw
ρεt + 1113936infin
i1gρx(t)( 1113857
τkρεt )i
1113890 1113891dt
minus wρεt dBt minus 1113936
infin
i1kρεt( 1113857
idHit
dqρεt f
ρy(t)( 1113857
τqρεt dt + f
ρz(t)( 1113857
τqρεt dBt + 1113936
infin
i1fρir (t)1113872 1113873
τ)q
ρεt dHi
t
q0 minus G0y yρ0( 1113857h
ρε0 + cy y
ρ0( 1113857hρε pT G1x x
ρT( 1113857h
ρε1 minus Φx x
ρT( 1113857q
ρεT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(61)
8 Mathematical Problems in Engineering
Here bρx(t) bx(t x
ρt u
ρt ) σρx(t) σx(t x
ρt )
fρy(t) fy(t x
ρt y
ρt z
ρt r
ρt u
ρt ) etc Applying It 1113954os formula
to langXρt q
ρεt rang + langY
ρt p
ρεt rang variational inequality (59) can be
rewritten as
H t xρt y
ρt z
ρt r
ρt u
ρεt p
ρεt q
ρεt w
ρεt k
ρεt( 1113857
minus H t xρt y
ρt z
ρt r
ρt u
ρt p
ρεt q
ρεt w
ρεt k
ρεt( 1113857
+ ερ
radic+ o(ε)ge 0 forallvt isin U ae as
(62)
where the Hamiltonian function H is defined as (44) Since
limε⟶0 hρε0
111386811138681113868111386811138681113868111386811138682
+ hρε1
111386811138681113868111386811138681113868111386811138682
+ hρε1113868111386811138681113868
11138681113868111386811138682
1113872 1113873 1 (63)
there exists a convergent subsequence still denoted by(h
ρε0 h
ρε1 hρε) such that (h
ρε0 h
ρε1 hρε)⟶ (h
ρ0 h
ρ1 hρ) when
ε⟶ 0 with |hρ0|2 + |h
ρ1|2 + |hρ|2 1
Let (pρt q
ρt w
ρt k
ρt ) be the solution of
minus dpρt b
ρx(t)( 1113857
τpρt minus f
ρx(t)( 1113857
τqρt + σρx(t)( 1113857
τw
ρt + 1113936infin
i1gρx(t)( 1113857
τkρt )i
1113890 1113891dt
minus wρtdBt minus 1113936
infin
i1kρt( 1113857
idHit
dqρt f
ρy(t)( 1113857
τqρtdt + f
ρz(t)( 1113857
τqρtdBt + 1113936
infin
i1fρir (t)1113872 1113873
τ1113888 1113889q
ρtdHi
t
q0 minus G0y yρ0( 1113857h
ρ0 + cy y
ρ0( 1113857hρ pT G1x x
ρT( 1113857h
ρ1 minus Φx x
ρT( 1113857q
ρT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(64)
From the continuous dependence of the solutions ofFBSDEs with Levy process on the parameters we can provethat the following convergence holds(p
ρεt q
ρεt w
ρεt k
ρεt )⟶ (p
ρt q
ρt w
ρt k
ρt ) then in equation
(62) it implies that
H t xρt y
ρt z
ρt r
ρt vt p
ρt q
ρt w
ρt k
ρt( 1113857
minus H t xρt y
ρt z
ρt r
ρt u
ρt p
ρt q
ρt w
ρt k
ρt( 1113857
+ρ
radic ge 0 forallvt isin U ae as
(65)
Similarly there exists a convergent subsequence(h
ρ0 h
ρ1 h
ρ) such that (h
ρ0 h
ρ1 h
ρ)⟶ (h0 h1 h) when
ρ⟶ 0 with |h0|2 + |h1|
2 + |h|2 1 Since uρt⟶ ulowastt as
ρ⟶ 0 we have (xρt y
ρt z
ρt r
ρt )⟶ (xlowastt ylowastt zlowastt rlowastt ) and
(Xρt Y
ρt Z
ρt R
ρt )⟶ (Xt Yt Zt Rt) which is the solution of
the variational equation as same as (6)e following adjoint equation is introduced
minus dpt bx(t)( 1113857τpt minus fx(t)( 1113857
τqt + σx(t)( 1113857
τwt + 1113936infin
i1gx(t)( 1113857
τkt)
i1113890 1113891dt
minus wtdBt minus 1113936infin
i1kt( 1113857
idHit
dqt fy(t)1113872 1113873τqtdt + fz(t)( 1113857
τqtdBt + 1113936
infin
i1fρir (t)1113872 1113873
τ1113888 1113889qtdHi
t
q0 minus G0y y0( 1113857h0 + cy y0( 1113857h pT G1x xT( 1113857h1 minus Φx xT( 1113857qT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(66)
Similarly it can be proved that(p
ρt q
ρt w
ρt k
ρt )⟶ (pt qt wt kt) then inequation (65)
impliesH t x
lowastt ylowastt zlowastt rlowastt vt pt qt wt kt( 1113857
minus H t xlowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857ge 0forallvt isin U ae as
(67)
en we get the following theorem
Theorem 2 Let Assumptions 1ndash3 hold ulowastt is an optimalcontrol and the corresponding optimal state trajectories are(xlowastt ylowastt zlowastt rlowastt ) and (pt qt wt kt) is the solution of adjoint
equation (66) then there exists nonzero constant(h1 h0 h) isin (Rn1 times Rm1 times R) with |h0|
2 + |h1|2 + |h|2 1
such that for any admissible control vt isin Uad[0 T] themaximum condition (67) holds
9is conclusion can be drawn from the above analysisdirectly
5 A Financial Example
In this section we will study the problem of optimal con-sumption rate selection in the financial market which willnaturally inspire our research to the forward-backwardstochastic optimal control problem in Section 3
Mathematical Problems in Engineering 9
Assume the investorrsquos asset process xt (tge 0) in the fi-nancial market is described by the following stochasticdifferential equation with the Teugels martingale
dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin
i1gi
txtdHit
x0 Wgt 0 t isin [0 T]
⎧⎪⎪⎨
⎪⎪⎩(68)
where μt and σt ne 0 are the expected return and volatility ofthe value process xt at time t respectively and Ct is theconsumption rate process Assume μt σt gt and Ct are alluniformly bounded Ftminus measurable random processes epurpose of investors is to select the optimal consumptionstrategy Clowastt at time tge 0 minimizing the following recursiveutility
J(C(middot)) E y01113858 1113859 (69)
where yt is the following backward stochastic process
minus dyt Leminus rtC1minus R
t
1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936
infin
i1ri
tdHit
yT minus xT gt 0 t isin [0 T]
⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
(70)
with constant Lgt 0 the discount factor rgt 0 and theArrowndashPratt measure of risk aversion R isin (0 1)
If the consumption process Ct is regarded as a controlvariable then combining (68) with (70) we encounter thefollowing control system
dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin
i1gi
txtdHit
minus dyt Leminus rtC1minus R
t
1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936
infin
i1ri
tdHit
x0 Wgt 0 yT minus xT gt 0 t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(71)
which is obviously a special case of stochastic control system(2) with b(t x u) μtxt minus Ct σ(t x) σtxt g(t x)
gitxt Φx(xT) minus XT and f(t x y z r u) Leminus rt(C1minus R
t (1 minus R)) minus ryt
We can check that both Assumptions 1 and 2 are sat-isfieden we can use ourmaximum principle (eorem 1)to solve the above optimization problem Let Clowastt be anoptimal consumption rate and xlowastt ylowastt be the correspondingwealth process and recursive utility process In this case theHamiltonian function H reduces to
H t xlowastt ylowastt Ct p
lowastt qlowastt wlowastt klowastt( 1113857 langplowastt μtx
lowastt minus Ctrang
+langwlowastt σtxlowastt rang +langklowastt g
itxlowastt rang minus langqlowastt Le
minus rt C1minus Rt
1 minus Rminus rylowastt rang
(72)
and (plowastt qlowastt wlowastt klowastt ) is the solution of the following adjointequation
dqt rqtdt
minus dpt μtpt + σtwt + 1113936infin
i1ki
t( 1113857τgi
t1113890 1113891dt minus wtdBt minus 1113936infin
i1ki
tdHit
q0 1 pT qT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
(73)
According to the maximum principle (47) we have
plowastt minus Le
minus rtClowastt( 1113857
minus Rqlowastt (74)
and the optimal consumption rate
Clowastt minus
ert
L
plowasttqlowastt
1113888 1113889
minus 1R
(75)
By solving FBSDE (73) we get qlowastt ert plowastt erteμ(Tminus t)and wlowastt klowastt 0 en we get the optimal consumptionrate of the investor which is
Clowastt minus
erT
Leμ(Tminus t)
1113888 1113889
minus 1R
t isin [0 T] (76)
6 Conclusions
In this paper a nonconvex control domain case of theforward-backward stochastic control driven by Levy processis considered and we obtain the global stochastic maximumprinciple for this stochastic control problem And then theproblem of stochastic control with initial and final stateconstraints on the state variables is discussed and a nec-essary condition about existence of the optimal control isalso acquired A financial example of optimal consumptionis discussed to illustrate the application of the stochasticmaximum principle
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the NNSF (11701335) Shan-dong NSF (ZR2019BG015) NSSF (17BGL058) ShandongProvincial Higher Education Science and Technology PlanProject (J18KA236) and Shandong Womenrsquos College High-level Research Project Cultivation Foundation(2018GSPGJ07)
References
[1] L S Pontryakin V G Boltyanskti R V Gamkrelidze andE F Mischenko9eMathematical9eory of Optimal ControlProcesses John Wiley New York NY USA 1962
[2] S Peng ldquoA general stochastic maximum principle for optimalcontrol problemsrdquo SIAM Journal on Control and Optimiza-tion vol 28 no 4 pp 966ndash979 1990
10 Mathematical Problems in Engineering
[3] S G Peng ldquoBackward stochastic differrntial equations andapplication to optimal controlrdquo Applied Mathematics ampOptimization vol 27 no 4 pp 125ndash144 1993
[4] W S Xu ldquoStochastic maximum principle for optimal controlproblem of forward and backward systemrdquo Journal of Aus-tralian Mathematical Sciences vol 37 pp 249ndash259 1998B
[5] Z Wu ldquoMaximum principle for optimal control problem offully coupled forward-backward stochastic control systemrdquoSystem Science and Mathematical Sciences vol 11 no 3pp 249ndash259 1998
[6] J T Shi and ZWu ldquoemaximum principle for fully coupledforward-backward stochastic control systemrdquo Acta Auto-matica Sinica vol 32 no 2 pp 161ndash169 2006
[7] R Situ ldquoA maximum principle for optimal controls of sto-chastic systems with random jumpsrdquo in Proceedings of theNational Conference on Control 9eory and its ApplicationsQingdao China 1991
[8] J Shi and Z Wu ldquoMaximum principle for forward-backwardstochastic control system with random jumps and applica-tions to financerdquo Journal of Systems Science and Complexityvol 23 no 2 pp 219ndash231 2010
[9] J T Shi ldquoGlobal maximum principle for forward-backwardstochastic control system with Poisson jumpsrdquo Asian Journalof Control vol 16 no 1 pp 1ndash11 2014
[10] M Liu X Wang and H Huang ldquoMaximum principle forforward-backward control system driven by Ito-Levy pro-cesses under initial-terminal constraintsrdquo MathematicalProblems in Engineering vol 2017 pp 1ndash13 2017
[11] S Tang ldquoe maximum principle for partially observedoptimal control of stochastic differential equationsrdquo SIAMJournal on Control and Optimization vol 36 no 5pp 1596ndash1617 1998
[12] Q Meng ldquoA maximum principle for optimal control problemof fully coupled forward-backward stochastic systems withpartial informationrdquo Science in China Series A Mathematicsvol 52 no 7 pp 1579ndash1588 2009
[13] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward backward stichastic differential equationswith jumpsrdquo SIAM J Control Optimenleadertwodots vol 48pp 2845ndash2976 2009
[14] G Wang and Z Wu ldquoKalman-Bucy filtering equations offorward and backward stochastic systems and applications torecursive optimal control problemsrdquo Journal of MathematicalAnalysis and Applications vol 342 no 2 pp 1280ndash12962008
[15] H Xiao ldquoe maximum principle for partially observedoptimal control of forward-backward stochastic systems withrandom jumpsrdquo Journal of Systems Science and Complexityvol 24 no 6 pp 1083ndash1099 2011
[16] G Wang Z Wu and J Xiong ldquoMaximum principles forforward-backward stochastic control systems with correlatedstate and observation noisesrdquo SIAM Journal on Control andOptimization vol 51 no 1 pp 491ndash524 2013
[17] G Wang C Zhang and W Zhang ldquoStochastic maximumprinciple for mean-field type optimal control under partialinformationrdquo IEEE Transactions on Automatic Controlvol 59 no 2 pp 522ndash528 2014
[18] S Q Zhang J Xiong and X D Liu ldquoStochastic maximumprinciple for partially observed forward backward stochasticdifferential equations with jumps and regime switchingrdquoScience China Information Sciences vol 61 no 7 pp 1ndash132018
[19] J Xiong S Q Zhang S Zhang and Y Zhuang ldquoA partiallyobserved non-zero sum differential game of forward-
backward stochastic differential equations and its applicationin financerdquo Mathematical Control amp Related Fields vol 9no 2 pp 257ndash276 2019
[20] D Nualart and W Schoutens ldquoChaotic and predictablerepresentations for Levy processesrdquo Stochastic Processes and9eir Applications vol 90 no 1 pp 109ndash122 2000
[21] D Nualart and W Schoutens ldquoBackward stochastic differ-ential equations and Feynman-Kac formula for Levy pro-cesses with applications in financerdquo Bernoulli vol 7 no 5pp 761ndash776 2001
[22] Q Meng and M Tang ldquoNecessary and sufficient conditionsfor optimal control of stochastic systems associated with Levyprocessesrdquo Science in China Series F Information Sciencesvol 52 no 11 pp 1982ndash1992 2009
[23] M Tang and Q Zhang ldquoOptimal variational principle forbackward stochastic control systems associated with Levyprocessesrdquo Science China Mathematics vol 55 no 4pp 745ndash761 2012
[24] F Baghery N Khelfallah B Mezerdi and I Turpin ldquoFullycoupled forward backward stochastic differential equationsdriven by Levy processes and application to differentialgamesrdquo De Gruyter vol 22 no 3 pp 151ndash161 2014
[25] F Zhang M N Tang and Q X Meng ldquoSochastic maximumprinciple for forward-backward stochastic control systemsassociated with Levy processesrdquo Chinese Annals of Mathe-matics vol 35A no 1 pp 83ndash100 2014
[26] X R Wang and H Huang ldquoMaximum principle for forward-backward stochastic control system driven by Levy processrdquoMathematical Problems in Engineering vol 2015 Article ID702802 12 pages 2015
[27] H Huang X Wang and M Liu ldquoA maximum principle forfully coupled forward-backward stochastic control systemdriven by Levy process with terminal state constraintsrdquoJournal of Systems Science and Complexity vol 31 no 4pp 859ndash874 2018
[28] P Muthukumar and R Deepa ldquoInfinite horizon optimalcontrol of forward backward stochastic system driven byTeugels martingales with Levy processrdquo Stochastic and Dy-namics vol 3 no 17 pp 1ndash17 2017
[29] J B Wu and Z M Liu ldquoOptimal control of mean-fieldbackward doubly stochastic systems driven by Ito-Levyprocessesrdquo International Journal of Control vol 93 no 4pp 953ndash970 2018
[30] W Wang J Wu and Z Liu ldquoe optimal control of fully-coupled forward-backward doubly stochastic systems drivenby Ito-Levy processesrdquo Journal of Systems Science andComplexity vol 32 no 4 pp 997ndash1018 2019
Mathematical Problems in Engineering 11
where
Aε
11139461
0bx s x
lowasts + λXs u
εs( 1113857 minus bx s x
lowasts ulowasts( 11138571113858 1113859dλXs
Bε
11139461
0σx s x
lowasts + λXs( 1113857 minus σx s x
lowasts( 11138571113858 1113859dλXs
Cε
11139461
01113944
infin
i1g
ix s x
lowasts + λXs( 1113857 minus g
ix s x
lowasts( 11138571113872 1113873⎡⎣ ⎤⎦dλXs
(30)
It follows easily from Lemma 2 that
sup0letleT
E 1113946t
0A
εds1113888 1113889
2
+ 1113946t
0BεdBs1113888 1113889
2
+ 1113946t
0CεdH
is1113888 1113889
2⎡⎣ ⎤⎦ o ε21113872 1113873
(31)
Since
xεt minus x0 1113946
t
0b s x
εs u
εs( 1113857ds + 1113946
t
0σ s x
εs( 1113857dBs + 1113946
t
01113944
infin
i1g
is x
εsminus( 1113857dH
is
(32)
then
xεt minus xlowastt minus Xt 1113946
t
0b s x
εs u
εs( 1113857 minus b s x
lowasts + Xs u
εs( 11138571113858 1113859ds
+ 1113946t
0σ s x
εs( 1113857 minus σ s x
lowasts + Xs( 11138571113858 1113859dBs
+ 1113946t
01113944
infin
i1g
is x
εsminus( 1113857 minus g
is xlowastsminus + Xsminus( 11138571113872 1113873dH
is
+ 1113946t
0Aεds + 1113946
t
0BεdBs + 1113946
t
0CεdH
is
1113946t
0D
εxεs minus xlowasts minus Xs( 1113857ds + 1113946
t
0Eε
xεs minus xlowasts minus Xs( 1113857dBs
+ 1113946t
0Fε
xεs minus xlowasts minus Xs( 1113857dH
is
(33)
with Dε 111393810 bx(s xlowasts + Xs + λ(xε
s minus xlowasts minus Xs) uεs)dλ
Eε 111393810 σx(s xlowasts + Xs + λ(xε
s minus xlowasts minus Xs))dλ and Fε 111393810
1113936infini1 gi
x(s xlowasts + Xs + λ(xεs minus xlowasts minus Xs))dλ
By Gronwallrsquos inequation we have
sup0letleT
E xεt minus xlowastt minus Xt
111386811138681113868111386811138681113868111386811138682 leCεε
2 Cε⟶ 0when ε⟶ 0
(34)
Next we prove (26) (27) (28) it can be easily checkedthat
minus 1113946T
tf s x
lowasts + Xs y
lowasts + Ys z
lowasts + Zs r
lowasts + Rs u
εs( 1113857ds
+ 1113946T
ts zlowasts + Zs( 1113857dBs + 1113946
T
t1113944
infin
i1g
is xlowastsminus + Xsminus( 1113857dH
is
Φ xlowastT( 1113857 minus y
lowastt +Φx x
lowastT( 1113857XT minus Yt minus 1113946
T
tGεds
(35)
Here
Gε
11139461
0fx s x
lowasts + λXs y
lowasts + λYs z
lowasts((
+ λZs rlowasts + λRs u
εs1113857 minus fx(s)1113857dλXs
+ 11139461
0fy s x
lowasts + λXs y
lowasts + λYs z
lowasts(1113872
+ λZs rlowasts + λRs u
εs1113857 minus fy(s)1113873dλYs
+ 11139461
0fz s x
lowasts + λXs y
lowasts + λYs z
lowasts((
+ λZs rlowasts + λRs u
εs1113857 minus fz(s)1113857dλZs
+ 11139461
0fr s x
lowasts + λXs y
lowasts + λYs z
lowasts((
+ λZs rlowasts + λRs u
εs1113857 minus fr(s)1113857dλRs
(36)
Since
yεt Φ x
εT( 1113857 + 1113946
T
tf s x
εs y
εs z
εs r
εs u
εs( 1113857ds minus 1113946
T
tzsdBs
minus 1113946T
t1113944
infin
i1r
iεs dH
is
(37)
then
yεt minus ylowastt minus Yt Φ x
εT( 1113857 minus Φ x
lowastT( 1113857 minus Φx x
lowastT( 1113857XT + 1113946
T
tf s x
εs y
εs z
εs r
εs u
εs( 1113857 minus f s x
lowasts + Xs y
lowasts + Ys z
lowasts + Zs r
lowasts + Rs u
εs(1113858 1113859ds
minus 1113946T
tzεs minus zlowasts minus Zs( 1113857dBs minus 1113946
T
t1113944
infin
i1r
iεs minus r
ilowasts minus Rs1113872 1113873dH
is + 1113946
T
tGεds
(38)
6 Mathematical Problems in Engineering
Squaring both sides of the equation above we get
E yεt minus ylowastt minus Yt
111386811138681113868111386811138681113868111386811138682
+ E 1113946T
tzεs minus zlowasts minus Zs( 1113857
2ds minus E 1113946T
t1113944
infin
i1r
iεs minus r
ilowasts minus Rs1113872 1113873
2ds
E f s xεs y
εs z
εs r
εs u
εs( 1113857 minus f s x
lowasts + Xs y
lowasts + Ys z
lowasts + Zs r
lowasts + Rs u
εs( 11138571113858 1113859ds +Φ x
εT( 1113857 minus Φ x
lowastT( 1113857 minus Φx x
lowastT( 1113857XT + 1113946
T
tGεds1113896 1113897
2
(39)
From Lemma 2 and equation (25) we have
sup0letleTE 1113946T
tGεds1113888 1113889
2
o ε21113872 1113873
E Φ xεT( 1113857 minus Φ x
lowastT( 1113857 minus Φx x
lowastT( 1113857XT1113858 1113859
2 o ε21113872 1113873
(40)
en we can get (26) (27) and (28) by applying theiterative method to the above relations
Lemma 4 (variational inequality) Under the conditionsthat Assumptions 1 and 2 are established we can get thefollowing variational inequality
Ecy ylowast0( 1113857Y0 ge o(ε) (41)
Proof From the four estimations in Lemma 3 we have thefollowing estimation
E c yε0( 1113857 minus c y
lowast0 + Y0( 11138571113858 1113859 o(ε) (42)
erefore
0leE c ylowast0 + Y0( 1113857 minus c y
lowast0( 11138571113858 1113859 + o(ε) Ecy y
lowast0( 1113857Y0 + o(ε)
(43)
We introduce the following Hamiltonian functionH [0 T] times Rn times Rm times Rmtimesd times l2(Rm) times U times Rn times Rm times
Rntimesdtimes l2(Rn) as
H(t x y z r u p q w k) langp b(t x u)rang +langw σ(t x)rang
+langk g(t x)rang minus langq f(t x y z r u)rang
(44)
and the following adjoint equation
dqt Hy ulowastt( 1113857dt + Hz ulowastt( 1113857dBt + 1113936infin
i1Hi
r ulowastt( 1113857dHit
minus dpt Hx ulowastt( 1113857dt minus wtdBt minus 1113936infin
i1ki
tminus dHit
q0 cy ylowast0( 1113857 pT minus Φx xlowastT( 1113857qT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎩
(45)
where
Hx ulowastt( 1113857 Hx t x
lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857
Hy ulowastt( 1113857 Hy t x
lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857
Hz ulowastt( 1113857 Hz t x
lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857
Hr ulowastt( 1113857 Hr t x
lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857
(46)
It is easily check that adjoint equation (45) has a uniquesolution quartet (pt qt wt kt) isinM2(0 T Rn+m+mtimesd) times
l2(0 T Rm)en we get the main result of this section
Theorem 1 Let hypothesis Assumptions 1 and2 hold ulowastt isan optimal control and the corresponding optimal statetrajectories are (xlowastt ylowastt zlowastt rlowastt ) let (pt qt wt kt) be thesolution of adjoint equation (45) and then for each ad-missible control ut isin Uad[0 T] we have
H txlowastt ylowastt zlowastt rlowastt utpt qtwtkt( 1113857
geH txlowastt ylowastt zlowastt rlowastt ulowastt pt qtwtkt( 1113857asae(47)
Proof Applying It 1113954os formula to langp Xrang and langq Yrang itfollows from (6) (45) and the variational inequality that
E 1113946T
0H t xt yt zt rt u
εt pt qt wt kt( 11138571113858
minus H t xt yt zt rt ulowastt pt qt wt kt( 11138571113859dt Ecy y
lowast0( 1113857Y0 ge o(ε)
(48)
By the definition of uεt we know that for any
u isin Uad[0 T] the following inequation holds
E H t xt yt zt rt uεt pt qt wt kt( 11138571113858
minus H t xt yt zt rt ulowastt pt qt wt kt( 11138571113859ge 0
(49)
en (47) can be easily checked
4 Stochastic Control Problem withState Constraints
In this part we are going to discuss stochastic controlproblems with state constraints in control system (2)Specifically the initial state constraints and final stateconstraints are as follows
EG1 xT( 1113857 0
EG0 y0( 1113857 0(50)
Mathematical Problems in Engineering 7
where G1 Rn⟶ Rn1(n1 lt n) G0 Rm⟶ Rm1(m1 ltm)Our optimal control problem is to find ulowast isin Uad such that
J ulowast(middot)( 1113857 inf
u(middot)isinUad
J(u(middot)) (51)
subject to the state constraints (50) In the following we willapply Ekelandrsquos variational principle to solve this optimalcontrol problem Firstly we need the following assumptions
Assumption 3 Control domain U is assumed to be closedthe mappings in state constraints G1 G0 are continuouslydifferentiable and G1x G0y are bounded
For forallu1(middot) u2(middot) isin Uad let
d u1(middot) u2(middot)( 1113857 E u1(middot) minus u2(middot)1113868111386811138681113868
11138681113868111386811138682 gt 0 t isin [0 T]1113966 1113967
11138681113868111386811138681113868
11138681113868111386811138681113868 (52)
Same as Section 3 we also assume ulowastt be the optimalcontrol and the corresponding optimal state trajectories are(xlowastt ylowastt zlowastt rlowastt ) In order to solve the constraint problemwe need the following penalty cost functional for any ρgt 0
Jρ(u) E G1 xT( 11138571113858 11138591113868111386811138681113868
11138681113868111386811138682
+ E G0 y0( 11138571113858 11138591113868111386811138681113868
11138681113868111386811138682
+ J(u) minus J ulowast( ) + ρ1113868111386811138681113868
11138681113868111386811138682
1113966 111396712
(53)
It can be checked that Jρ(u) Uad⟶ R1 is continuousand for any u(middot) isin Uad
Jρ(u)ge 0 Jρ ulowast
( 1113857 ρ
Jρ ulowast
( 1113857le infu(middot)isinUad
Jρ(u) + ρ(54)
It can be obtained by Ekelandrsquos variational principle thatthere exists u
ρt isin Uad such that
(i) Jρ uρ( )le Jρ ulowast( ) ρ
(ii) d uρ ulowast( )le ρradic
(iii) Jρ(v)ge Jρ uρ( ) minusρradic
d uρ v( ) for forallv(middot) isin Uad
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(55)
For fixed ρ and admissible control uρt we define the spike
variation as follows
uρεt
vt τ le tle τ + ε
uρt otherwise
1113896 (56)
for any εgt 0 and it is easy to check from (iii) of (55) that
Jρ uρε( ) minus Jρ uρ( ) +ρradic
d uρε uρ( )ge 0 (57)
Let (xρt y
ρt z
ρt r
ρt ) be the trajectories corresponding to u
ρt
and (xρεt y
ρεt z
ρεt r
ρεt ) be the trajectories corresponding to
uρεt e variational equation we used in this section is the
same as the one in Section 3 with ulowastt uρt and
(xlowastt ylowastt zlowastt rlowastt ) (xρt y
ρt z
ρt r
ρt ) And we also assume that
the solution of this variational equation is (Xρt Y
ρt Z
ρt R
ρt )
Similar to the approach in Lemmas 2 and 3 it can be shownthat
sup0letleTE xρεt minus x
ρt minus X
ρt
111386811138681113868111386811138681113868111386811138682 leCεε
2 Cε⟶ 0when ε⟶ 0
sup0letleTE yρεt minus y
ρt minus Y
ρt
111386811138681113868111386811138681113868111386811138682 leCεε
2 Cε⟶ 0when ε⟶ 0
E 1113946T
0zρεt minus z
ρt minus Z
ρt
111386811138681113868111386811138681113868111386811138682dsleCεε
2 Cε⟶ 0when ε⟶ 0
E 1113946T
0rρεt minus r
ρt minus R
ρt
2dsleCεε
2 Cε⟶ 0when ε⟶ 0
(58)
en by (57) the following variational inequality holds
Jρ uρε
( 1113857 minus Jρ uρ
( 1113857 +ρ
radicd u
ρε u
ρ( 1113857
J2ρ uρε( ) minus J2ρ uρ( )
Jρ uρε( ) + Jρ uρ( )+ ε
ρ
radic
langhρε1 E G1x x
ρT( 11138571113858 1113859X
ρTrang +langhρε
0 E G0y yρ0( 11138571113960 1113961Y
ρ0rang
+ hρε
E cy yρ0( 1113857Y
ρ01113960 1113961 + ε
ρ
radic+ o(ε)
(59)
where
hρε1
2E G1 xρT( 11138571113858 1113859
Jρ uρε( ) + Jρ uρ( )
hρε0
2E G0 yρ0( 11138571113858 1113859
Jρ uρε( ) + Jρ uρ( )
hρε
2E c y
ρ0( 1113857 minus c y0( 1113857 + ρ1113858 1113859
Jρ uρε( ) + Jρ uρ( )
(60)
Now let (pρεt q
ρεt w
ρεt k
ρεt ) be the solution of
minus dpρεt b
ρx(t)( 1113857
τpρεt minus f
ρx(t)( 1113857
τqρεt + σρx(t)( 1113857
τw
ρεt + 1113936infin
i1gρx(t)( 1113857
τkρεt )i
1113890 1113891dt
minus wρεt dBt minus 1113936
infin
i1kρεt( 1113857
idHit
dqρεt f
ρy(t)( 1113857
τqρεt dt + f
ρz(t)( 1113857
τqρεt dBt + 1113936
infin
i1fρir (t)1113872 1113873
τ)q
ρεt dHi
t
q0 minus G0y yρ0( 1113857h
ρε0 + cy y
ρ0( 1113857hρε pT G1x x
ρT( 1113857h
ρε1 minus Φx x
ρT( 1113857q
ρεT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(61)
8 Mathematical Problems in Engineering
Here bρx(t) bx(t x
ρt u
ρt ) σρx(t) σx(t x
ρt )
fρy(t) fy(t x
ρt y
ρt z
ρt r
ρt u
ρt ) etc Applying It 1113954os formula
to langXρt q
ρεt rang + langY
ρt p
ρεt rang variational inequality (59) can be
rewritten as
H t xρt y
ρt z
ρt r
ρt u
ρεt p
ρεt q
ρεt w
ρεt k
ρεt( 1113857
minus H t xρt y
ρt z
ρt r
ρt u
ρt p
ρεt q
ρεt w
ρεt k
ρεt( 1113857
+ ερ
radic+ o(ε)ge 0 forallvt isin U ae as
(62)
where the Hamiltonian function H is defined as (44) Since
limε⟶0 hρε0
111386811138681113868111386811138681113868111386811138682
+ hρε1
111386811138681113868111386811138681113868111386811138682
+ hρε1113868111386811138681113868
11138681113868111386811138682
1113872 1113873 1 (63)
there exists a convergent subsequence still denoted by(h
ρε0 h
ρε1 hρε) such that (h
ρε0 h
ρε1 hρε)⟶ (h
ρ0 h
ρ1 hρ) when
ε⟶ 0 with |hρ0|2 + |h
ρ1|2 + |hρ|2 1
Let (pρt q
ρt w
ρt k
ρt ) be the solution of
minus dpρt b
ρx(t)( 1113857
τpρt minus f
ρx(t)( 1113857
τqρt + σρx(t)( 1113857
τw
ρt + 1113936infin
i1gρx(t)( 1113857
τkρt )i
1113890 1113891dt
minus wρtdBt minus 1113936
infin
i1kρt( 1113857
idHit
dqρt f
ρy(t)( 1113857
τqρtdt + f
ρz(t)( 1113857
τqρtdBt + 1113936
infin
i1fρir (t)1113872 1113873
τ1113888 1113889q
ρtdHi
t
q0 minus G0y yρ0( 1113857h
ρ0 + cy y
ρ0( 1113857hρ pT G1x x
ρT( 1113857h
ρ1 minus Φx x
ρT( 1113857q
ρT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(64)
From the continuous dependence of the solutions ofFBSDEs with Levy process on the parameters we can provethat the following convergence holds(p
ρεt q
ρεt w
ρεt k
ρεt )⟶ (p
ρt q
ρt w
ρt k
ρt ) then in equation
(62) it implies that
H t xρt y
ρt z
ρt r
ρt vt p
ρt q
ρt w
ρt k
ρt( 1113857
minus H t xρt y
ρt z
ρt r
ρt u
ρt p
ρt q
ρt w
ρt k
ρt( 1113857
+ρ
radic ge 0 forallvt isin U ae as
(65)
Similarly there exists a convergent subsequence(h
ρ0 h
ρ1 h
ρ) such that (h
ρ0 h
ρ1 h
ρ)⟶ (h0 h1 h) when
ρ⟶ 0 with |h0|2 + |h1|
2 + |h|2 1 Since uρt⟶ ulowastt as
ρ⟶ 0 we have (xρt y
ρt z
ρt r
ρt )⟶ (xlowastt ylowastt zlowastt rlowastt ) and
(Xρt Y
ρt Z
ρt R
ρt )⟶ (Xt Yt Zt Rt) which is the solution of
the variational equation as same as (6)e following adjoint equation is introduced
minus dpt bx(t)( 1113857τpt minus fx(t)( 1113857
τqt + σx(t)( 1113857
τwt + 1113936infin
i1gx(t)( 1113857
τkt)
i1113890 1113891dt
minus wtdBt minus 1113936infin
i1kt( 1113857
idHit
dqt fy(t)1113872 1113873τqtdt + fz(t)( 1113857
τqtdBt + 1113936
infin
i1fρir (t)1113872 1113873
τ1113888 1113889qtdHi
t
q0 minus G0y y0( 1113857h0 + cy y0( 1113857h pT G1x xT( 1113857h1 minus Φx xT( 1113857qT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(66)
Similarly it can be proved that(p
ρt q
ρt w
ρt k
ρt )⟶ (pt qt wt kt) then inequation (65)
impliesH t x
lowastt ylowastt zlowastt rlowastt vt pt qt wt kt( 1113857
minus H t xlowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857ge 0forallvt isin U ae as
(67)
en we get the following theorem
Theorem 2 Let Assumptions 1ndash3 hold ulowastt is an optimalcontrol and the corresponding optimal state trajectories are(xlowastt ylowastt zlowastt rlowastt ) and (pt qt wt kt) is the solution of adjoint
equation (66) then there exists nonzero constant(h1 h0 h) isin (Rn1 times Rm1 times R) with |h0|
2 + |h1|2 + |h|2 1
such that for any admissible control vt isin Uad[0 T] themaximum condition (67) holds
9is conclusion can be drawn from the above analysisdirectly
5 A Financial Example
In this section we will study the problem of optimal con-sumption rate selection in the financial market which willnaturally inspire our research to the forward-backwardstochastic optimal control problem in Section 3
Mathematical Problems in Engineering 9
Assume the investorrsquos asset process xt (tge 0) in the fi-nancial market is described by the following stochasticdifferential equation with the Teugels martingale
dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin
i1gi
txtdHit
x0 Wgt 0 t isin [0 T]
⎧⎪⎪⎨
⎪⎪⎩(68)
where μt and σt ne 0 are the expected return and volatility ofthe value process xt at time t respectively and Ct is theconsumption rate process Assume μt σt gt and Ct are alluniformly bounded Ftminus measurable random processes epurpose of investors is to select the optimal consumptionstrategy Clowastt at time tge 0 minimizing the following recursiveutility
J(C(middot)) E y01113858 1113859 (69)
where yt is the following backward stochastic process
minus dyt Leminus rtC1minus R
t
1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936
infin
i1ri
tdHit
yT minus xT gt 0 t isin [0 T]
⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
(70)
with constant Lgt 0 the discount factor rgt 0 and theArrowndashPratt measure of risk aversion R isin (0 1)
If the consumption process Ct is regarded as a controlvariable then combining (68) with (70) we encounter thefollowing control system
dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin
i1gi
txtdHit
minus dyt Leminus rtC1minus R
t
1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936
infin
i1ri
tdHit
x0 Wgt 0 yT minus xT gt 0 t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(71)
which is obviously a special case of stochastic control system(2) with b(t x u) μtxt minus Ct σ(t x) σtxt g(t x)
gitxt Φx(xT) minus XT and f(t x y z r u) Leminus rt(C1minus R
t (1 minus R)) minus ryt
We can check that both Assumptions 1 and 2 are sat-isfieden we can use ourmaximum principle (eorem 1)to solve the above optimization problem Let Clowastt be anoptimal consumption rate and xlowastt ylowastt be the correspondingwealth process and recursive utility process In this case theHamiltonian function H reduces to
H t xlowastt ylowastt Ct p
lowastt qlowastt wlowastt klowastt( 1113857 langplowastt μtx
lowastt minus Ctrang
+langwlowastt σtxlowastt rang +langklowastt g
itxlowastt rang minus langqlowastt Le
minus rt C1minus Rt
1 minus Rminus rylowastt rang
(72)
and (plowastt qlowastt wlowastt klowastt ) is the solution of the following adjointequation
dqt rqtdt
minus dpt μtpt + σtwt + 1113936infin
i1ki
t( 1113857τgi
t1113890 1113891dt minus wtdBt minus 1113936infin
i1ki
tdHit
q0 1 pT qT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
(73)
According to the maximum principle (47) we have
plowastt minus Le
minus rtClowastt( 1113857
minus Rqlowastt (74)
and the optimal consumption rate
Clowastt minus
ert
L
plowasttqlowastt
1113888 1113889
minus 1R
(75)
By solving FBSDE (73) we get qlowastt ert plowastt erteμ(Tminus t)and wlowastt klowastt 0 en we get the optimal consumptionrate of the investor which is
Clowastt minus
erT
Leμ(Tminus t)
1113888 1113889
minus 1R
t isin [0 T] (76)
6 Conclusions
In this paper a nonconvex control domain case of theforward-backward stochastic control driven by Levy processis considered and we obtain the global stochastic maximumprinciple for this stochastic control problem And then theproblem of stochastic control with initial and final stateconstraints on the state variables is discussed and a nec-essary condition about existence of the optimal control isalso acquired A financial example of optimal consumptionis discussed to illustrate the application of the stochasticmaximum principle
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the NNSF (11701335) Shan-dong NSF (ZR2019BG015) NSSF (17BGL058) ShandongProvincial Higher Education Science and Technology PlanProject (J18KA236) and Shandong Womenrsquos College High-level Research Project Cultivation Foundation(2018GSPGJ07)
References
[1] L S Pontryakin V G Boltyanskti R V Gamkrelidze andE F Mischenko9eMathematical9eory of Optimal ControlProcesses John Wiley New York NY USA 1962
[2] S Peng ldquoA general stochastic maximum principle for optimalcontrol problemsrdquo SIAM Journal on Control and Optimiza-tion vol 28 no 4 pp 966ndash979 1990
10 Mathematical Problems in Engineering
[3] S G Peng ldquoBackward stochastic differrntial equations andapplication to optimal controlrdquo Applied Mathematics ampOptimization vol 27 no 4 pp 125ndash144 1993
[4] W S Xu ldquoStochastic maximum principle for optimal controlproblem of forward and backward systemrdquo Journal of Aus-tralian Mathematical Sciences vol 37 pp 249ndash259 1998B
[5] Z Wu ldquoMaximum principle for optimal control problem offully coupled forward-backward stochastic control systemrdquoSystem Science and Mathematical Sciences vol 11 no 3pp 249ndash259 1998
[6] J T Shi and ZWu ldquoemaximum principle for fully coupledforward-backward stochastic control systemrdquo Acta Auto-matica Sinica vol 32 no 2 pp 161ndash169 2006
[7] R Situ ldquoA maximum principle for optimal controls of sto-chastic systems with random jumpsrdquo in Proceedings of theNational Conference on Control 9eory and its ApplicationsQingdao China 1991
[8] J Shi and Z Wu ldquoMaximum principle for forward-backwardstochastic control system with random jumps and applica-tions to financerdquo Journal of Systems Science and Complexityvol 23 no 2 pp 219ndash231 2010
[9] J T Shi ldquoGlobal maximum principle for forward-backwardstochastic control system with Poisson jumpsrdquo Asian Journalof Control vol 16 no 1 pp 1ndash11 2014
[10] M Liu X Wang and H Huang ldquoMaximum principle forforward-backward control system driven by Ito-Levy pro-cesses under initial-terminal constraintsrdquo MathematicalProblems in Engineering vol 2017 pp 1ndash13 2017
[11] S Tang ldquoe maximum principle for partially observedoptimal control of stochastic differential equationsrdquo SIAMJournal on Control and Optimization vol 36 no 5pp 1596ndash1617 1998
[12] Q Meng ldquoA maximum principle for optimal control problemof fully coupled forward-backward stochastic systems withpartial informationrdquo Science in China Series A Mathematicsvol 52 no 7 pp 1579ndash1588 2009
[13] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward backward stichastic differential equationswith jumpsrdquo SIAM J Control Optimenleadertwodots vol 48pp 2845ndash2976 2009
[14] G Wang and Z Wu ldquoKalman-Bucy filtering equations offorward and backward stochastic systems and applications torecursive optimal control problemsrdquo Journal of MathematicalAnalysis and Applications vol 342 no 2 pp 1280ndash12962008
[15] H Xiao ldquoe maximum principle for partially observedoptimal control of forward-backward stochastic systems withrandom jumpsrdquo Journal of Systems Science and Complexityvol 24 no 6 pp 1083ndash1099 2011
[16] G Wang Z Wu and J Xiong ldquoMaximum principles forforward-backward stochastic control systems with correlatedstate and observation noisesrdquo SIAM Journal on Control andOptimization vol 51 no 1 pp 491ndash524 2013
[17] G Wang C Zhang and W Zhang ldquoStochastic maximumprinciple for mean-field type optimal control under partialinformationrdquo IEEE Transactions on Automatic Controlvol 59 no 2 pp 522ndash528 2014
[18] S Q Zhang J Xiong and X D Liu ldquoStochastic maximumprinciple for partially observed forward backward stochasticdifferential equations with jumps and regime switchingrdquoScience China Information Sciences vol 61 no 7 pp 1ndash132018
[19] J Xiong S Q Zhang S Zhang and Y Zhuang ldquoA partiallyobserved non-zero sum differential game of forward-
backward stochastic differential equations and its applicationin financerdquo Mathematical Control amp Related Fields vol 9no 2 pp 257ndash276 2019
[20] D Nualart and W Schoutens ldquoChaotic and predictablerepresentations for Levy processesrdquo Stochastic Processes and9eir Applications vol 90 no 1 pp 109ndash122 2000
[21] D Nualart and W Schoutens ldquoBackward stochastic differ-ential equations and Feynman-Kac formula for Levy pro-cesses with applications in financerdquo Bernoulli vol 7 no 5pp 761ndash776 2001
[22] Q Meng and M Tang ldquoNecessary and sufficient conditionsfor optimal control of stochastic systems associated with Levyprocessesrdquo Science in China Series F Information Sciencesvol 52 no 11 pp 1982ndash1992 2009
[23] M Tang and Q Zhang ldquoOptimal variational principle forbackward stochastic control systems associated with Levyprocessesrdquo Science China Mathematics vol 55 no 4pp 745ndash761 2012
[24] F Baghery N Khelfallah B Mezerdi and I Turpin ldquoFullycoupled forward backward stochastic differential equationsdriven by Levy processes and application to differentialgamesrdquo De Gruyter vol 22 no 3 pp 151ndash161 2014
[25] F Zhang M N Tang and Q X Meng ldquoSochastic maximumprinciple for forward-backward stochastic control systemsassociated with Levy processesrdquo Chinese Annals of Mathe-matics vol 35A no 1 pp 83ndash100 2014
[26] X R Wang and H Huang ldquoMaximum principle for forward-backward stochastic control system driven by Levy processrdquoMathematical Problems in Engineering vol 2015 Article ID702802 12 pages 2015
[27] H Huang X Wang and M Liu ldquoA maximum principle forfully coupled forward-backward stochastic control systemdriven by Levy process with terminal state constraintsrdquoJournal of Systems Science and Complexity vol 31 no 4pp 859ndash874 2018
[28] P Muthukumar and R Deepa ldquoInfinite horizon optimalcontrol of forward backward stochastic system driven byTeugels martingales with Levy processrdquo Stochastic and Dy-namics vol 3 no 17 pp 1ndash17 2017
[29] J B Wu and Z M Liu ldquoOptimal control of mean-fieldbackward doubly stochastic systems driven by Ito-Levyprocessesrdquo International Journal of Control vol 93 no 4pp 953ndash970 2018
[30] W Wang J Wu and Z Liu ldquoe optimal control of fully-coupled forward-backward doubly stochastic systems drivenby Ito-Levy processesrdquo Journal of Systems Science andComplexity vol 32 no 4 pp 997ndash1018 2019
Mathematical Problems in Engineering 11
Squaring both sides of the equation above we get
E yεt minus ylowastt minus Yt
111386811138681113868111386811138681113868111386811138682
+ E 1113946T
tzεs minus zlowasts minus Zs( 1113857
2ds minus E 1113946T
t1113944
infin
i1r
iεs minus r
ilowasts minus Rs1113872 1113873
2ds
E f s xεs y
εs z
εs r
εs u
εs( 1113857 minus f s x
lowasts + Xs y
lowasts + Ys z
lowasts + Zs r
lowasts + Rs u
εs( 11138571113858 1113859ds +Φ x
εT( 1113857 minus Φ x
lowastT( 1113857 minus Φx x
lowastT( 1113857XT + 1113946
T
tGεds1113896 1113897
2
(39)
From Lemma 2 and equation (25) we have
sup0letleTE 1113946T
tGεds1113888 1113889
2
o ε21113872 1113873
E Φ xεT( 1113857 minus Φ x
lowastT( 1113857 minus Φx x
lowastT( 1113857XT1113858 1113859
2 o ε21113872 1113873
(40)
en we can get (26) (27) and (28) by applying theiterative method to the above relations
Lemma 4 (variational inequality) Under the conditionsthat Assumptions 1 and 2 are established we can get thefollowing variational inequality
Ecy ylowast0( 1113857Y0 ge o(ε) (41)
Proof From the four estimations in Lemma 3 we have thefollowing estimation
E c yε0( 1113857 minus c y
lowast0 + Y0( 11138571113858 1113859 o(ε) (42)
erefore
0leE c ylowast0 + Y0( 1113857 minus c y
lowast0( 11138571113858 1113859 + o(ε) Ecy y
lowast0( 1113857Y0 + o(ε)
(43)
We introduce the following Hamiltonian functionH [0 T] times Rn times Rm times Rmtimesd times l2(Rm) times U times Rn times Rm times
Rntimesdtimes l2(Rn) as
H(t x y z r u p q w k) langp b(t x u)rang +langw σ(t x)rang
+langk g(t x)rang minus langq f(t x y z r u)rang
(44)
and the following adjoint equation
dqt Hy ulowastt( 1113857dt + Hz ulowastt( 1113857dBt + 1113936infin
i1Hi
r ulowastt( 1113857dHit
minus dpt Hx ulowastt( 1113857dt minus wtdBt minus 1113936infin
i1ki
tminus dHit
q0 cy ylowast0( 1113857 pT minus Φx xlowastT( 1113857qT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎩
(45)
where
Hx ulowastt( 1113857 Hx t x
lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857
Hy ulowastt( 1113857 Hy t x
lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857
Hz ulowastt( 1113857 Hz t x
lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857
Hr ulowastt( 1113857 Hr t x
lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857
(46)
It is easily check that adjoint equation (45) has a uniquesolution quartet (pt qt wt kt) isinM2(0 T Rn+m+mtimesd) times
l2(0 T Rm)en we get the main result of this section
Theorem 1 Let hypothesis Assumptions 1 and2 hold ulowastt isan optimal control and the corresponding optimal statetrajectories are (xlowastt ylowastt zlowastt rlowastt ) let (pt qt wt kt) be thesolution of adjoint equation (45) and then for each ad-missible control ut isin Uad[0 T] we have
H txlowastt ylowastt zlowastt rlowastt utpt qtwtkt( 1113857
geH txlowastt ylowastt zlowastt rlowastt ulowastt pt qtwtkt( 1113857asae(47)
Proof Applying It 1113954os formula to langp Xrang and langq Yrang itfollows from (6) (45) and the variational inequality that
E 1113946T
0H t xt yt zt rt u
εt pt qt wt kt( 11138571113858
minus H t xt yt zt rt ulowastt pt qt wt kt( 11138571113859dt Ecy y
lowast0( 1113857Y0 ge o(ε)
(48)
By the definition of uεt we know that for any
u isin Uad[0 T] the following inequation holds
E H t xt yt zt rt uεt pt qt wt kt( 11138571113858
minus H t xt yt zt rt ulowastt pt qt wt kt( 11138571113859ge 0
(49)
en (47) can be easily checked
4 Stochastic Control Problem withState Constraints
In this part we are going to discuss stochastic controlproblems with state constraints in control system (2)Specifically the initial state constraints and final stateconstraints are as follows
EG1 xT( 1113857 0
EG0 y0( 1113857 0(50)
Mathematical Problems in Engineering 7
where G1 Rn⟶ Rn1(n1 lt n) G0 Rm⟶ Rm1(m1 ltm)Our optimal control problem is to find ulowast isin Uad such that
J ulowast(middot)( 1113857 inf
u(middot)isinUad
J(u(middot)) (51)
subject to the state constraints (50) In the following we willapply Ekelandrsquos variational principle to solve this optimalcontrol problem Firstly we need the following assumptions
Assumption 3 Control domain U is assumed to be closedthe mappings in state constraints G1 G0 are continuouslydifferentiable and G1x G0y are bounded
For forallu1(middot) u2(middot) isin Uad let
d u1(middot) u2(middot)( 1113857 E u1(middot) minus u2(middot)1113868111386811138681113868
11138681113868111386811138682 gt 0 t isin [0 T]1113966 1113967
11138681113868111386811138681113868
11138681113868111386811138681113868 (52)
Same as Section 3 we also assume ulowastt be the optimalcontrol and the corresponding optimal state trajectories are(xlowastt ylowastt zlowastt rlowastt ) In order to solve the constraint problemwe need the following penalty cost functional for any ρgt 0
Jρ(u) E G1 xT( 11138571113858 11138591113868111386811138681113868
11138681113868111386811138682
+ E G0 y0( 11138571113858 11138591113868111386811138681113868
11138681113868111386811138682
+ J(u) minus J ulowast( ) + ρ1113868111386811138681113868
11138681113868111386811138682
1113966 111396712
(53)
It can be checked that Jρ(u) Uad⟶ R1 is continuousand for any u(middot) isin Uad
Jρ(u)ge 0 Jρ ulowast
( 1113857 ρ
Jρ ulowast
( 1113857le infu(middot)isinUad
Jρ(u) + ρ(54)
It can be obtained by Ekelandrsquos variational principle thatthere exists u
ρt isin Uad such that
(i) Jρ uρ( )le Jρ ulowast( ) ρ
(ii) d uρ ulowast( )le ρradic
(iii) Jρ(v)ge Jρ uρ( ) minusρradic
d uρ v( ) for forallv(middot) isin Uad
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(55)
For fixed ρ and admissible control uρt we define the spike
variation as follows
uρεt
vt τ le tle τ + ε
uρt otherwise
1113896 (56)
for any εgt 0 and it is easy to check from (iii) of (55) that
Jρ uρε( ) minus Jρ uρ( ) +ρradic
d uρε uρ( )ge 0 (57)
Let (xρt y
ρt z
ρt r
ρt ) be the trajectories corresponding to u
ρt
and (xρεt y
ρεt z
ρεt r
ρεt ) be the trajectories corresponding to
uρεt e variational equation we used in this section is the
same as the one in Section 3 with ulowastt uρt and
(xlowastt ylowastt zlowastt rlowastt ) (xρt y
ρt z
ρt r
ρt ) And we also assume that
the solution of this variational equation is (Xρt Y
ρt Z
ρt R
ρt )
Similar to the approach in Lemmas 2 and 3 it can be shownthat
sup0letleTE xρεt minus x
ρt minus X
ρt
111386811138681113868111386811138681113868111386811138682 leCεε
2 Cε⟶ 0when ε⟶ 0
sup0letleTE yρεt minus y
ρt minus Y
ρt
111386811138681113868111386811138681113868111386811138682 leCεε
2 Cε⟶ 0when ε⟶ 0
E 1113946T
0zρεt minus z
ρt minus Z
ρt
111386811138681113868111386811138681113868111386811138682dsleCεε
2 Cε⟶ 0when ε⟶ 0
E 1113946T
0rρεt minus r
ρt minus R
ρt
2dsleCεε
2 Cε⟶ 0when ε⟶ 0
(58)
en by (57) the following variational inequality holds
Jρ uρε
( 1113857 minus Jρ uρ
( 1113857 +ρ
radicd u
ρε u
ρ( 1113857
J2ρ uρε( ) minus J2ρ uρ( )
Jρ uρε( ) + Jρ uρ( )+ ε
ρ
radic
langhρε1 E G1x x
ρT( 11138571113858 1113859X
ρTrang +langhρε
0 E G0y yρ0( 11138571113960 1113961Y
ρ0rang
+ hρε
E cy yρ0( 1113857Y
ρ01113960 1113961 + ε
ρ
radic+ o(ε)
(59)
where
hρε1
2E G1 xρT( 11138571113858 1113859
Jρ uρε( ) + Jρ uρ( )
hρε0
2E G0 yρ0( 11138571113858 1113859
Jρ uρε( ) + Jρ uρ( )
hρε
2E c y
ρ0( 1113857 minus c y0( 1113857 + ρ1113858 1113859
Jρ uρε( ) + Jρ uρ( )
(60)
Now let (pρεt q
ρεt w
ρεt k
ρεt ) be the solution of
minus dpρεt b
ρx(t)( 1113857
τpρεt minus f
ρx(t)( 1113857
τqρεt + σρx(t)( 1113857
τw
ρεt + 1113936infin
i1gρx(t)( 1113857
τkρεt )i
1113890 1113891dt
minus wρεt dBt minus 1113936
infin
i1kρεt( 1113857
idHit
dqρεt f
ρy(t)( 1113857
τqρεt dt + f
ρz(t)( 1113857
τqρεt dBt + 1113936
infin
i1fρir (t)1113872 1113873
τ)q
ρεt dHi
t
q0 minus G0y yρ0( 1113857h
ρε0 + cy y
ρ0( 1113857hρε pT G1x x
ρT( 1113857h
ρε1 minus Φx x
ρT( 1113857q
ρεT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(61)
8 Mathematical Problems in Engineering
Here bρx(t) bx(t x
ρt u
ρt ) σρx(t) σx(t x
ρt )
fρy(t) fy(t x
ρt y
ρt z
ρt r
ρt u
ρt ) etc Applying It 1113954os formula
to langXρt q
ρεt rang + langY
ρt p
ρεt rang variational inequality (59) can be
rewritten as
H t xρt y
ρt z
ρt r
ρt u
ρεt p
ρεt q
ρεt w
ρεt k
ρεt( 1113857
minus H t xρt y
ρt z
ρt r
ρt u
ρt p
ρεt q
ρεt w
ρεt k
ρεt( 1113857
+ ερ
radic+ o(ε)ge 0 forallvt isin U ae as
(62)
where the Hamiltonian function H is defined as (44) Since
limε⟶0 hρε0
111386811138681113868111386811138681113868111386811138682
+ hρε1
111386811138681113868111386811138681113868111386811138682
+ hρε1113868111386811138681113868
11138681113868111386811138682
1113872 1113873 1 (63)
there exists a convergent subsequence still denoted by(h
ρε0 h
ρε1 hρε) such that (h
ρε0 h
ρε1 hρε)⟶ (h
ρ0 h
ρ1 hρ) when
ε⟶ 0 with |hρ0|2 + |h
ρ1|2 + |hρ|2 1
Let (pρt q
ρt w
ρt k
ρt ) be the solution of
minus dpρt b
ρx(t)( 1113857
τpρt minus f
ρx(t)( 1113857
τqρt + σρx(t)( 1113857
τw
ρt + 1113936infin
i1gρx(t)( 1113857
τkρt )i
1113890 1113891dt
minus wρtdBt minus 1113936
infin
i1kρt( 1113857
idHit
dqρt f
ρy(t)( 1113857
τqρtdt + f
ρz(t)( 1113857
τqρtdBt + 1113936
infin
i1fρir (t)1113872 1113873
τ1113888 1113889q
ρtdHi
t
q0 minus G0y yρ0( 1113857h
ρ0 + cy y
ρ0( 1113857hρ pT G1x x
ρT( 1113857h
ρ1 minus Φx x
ρT( 1113857q
ρT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(64)
From the continuous dependence of the solutions ofFBSDEs with Levy process on the parameters we can provethat the following convergence holds(p
ρεt q
ρεt w
ρεt k
ρεt )⟶ (p
ρt q
ρt w
ρt k
ρt ) then in equation
(62) it implies that
H t xρt y
ρt z
ρt r
ρt vt p
ρt q
ρt w
ρt k
ρt( 1113857
minus H t xρt y
ρt z
ρt r
ρt u
ρt p
ρt q
ρt w
ρt k
ρt( 1113857
+ρ
radic ge 0 forallvt isin U ae as
(65)
Similarly there exists a convergent subsequence(h
ρ0 h
ρ1 h
ρ) such that (h
ρ0 h
ρ1 h
ρ)⟶ (h0 h1 h) when
ρ⟶ 0 with |h0|2 + |h1|
2 + |h|2 1 Since uρt⟶ ulowastt as
ρ⟶ 0 we have (xρt y
ρt z
ρt r
ρt )⟶ (xlowastt ylowastt zlowastt rlowastt ) and
(Xρt Y
ρt Z
ρt R
ρt )⟶ (Xt Yt Zt Rt) which is the solution of
the variational equation as same as (6)e following adjoint equation is introduced
minus dpt bx(t)( 1113857τpt minus fx(t)( 1113857
τqt + σx(t)( 1113857
τwt + 1113936infin
i1gx(t)( 1113857
τkt)
i1113890 1113891dt
minus wtdBt minus 1113936infin
i1kt( 1113857
idHit
dqt fy(t)1113872 1113873τqtdt + fz(t)( 1113857
τqtdBt + 1113936
infin
i1fρir (t)1113872 1113873
τ1113888 1113889qtdHi
t
q0 minus G0y y0( 1113857h0 + cy y0( 1113857h pT G1x xT( 1113857h1 minus Φx xT( 1113857qT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(66)
Similarly it can be proved that(p
ρt q
ρt w
ρt k
ρt )⟶ (pt qt wt kt) then inequation (65)
impliesH t x
lowastt ylowastt zlowastt rlowastt vt pt qt wt kt( 1113857
minus H t xlowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857ge 0forallvt isin U ae as
(67)
en we get the following theorem
Theorem 2 Let Assumptions 1ndash3 hold ulowastt is an optimalcontrol and the corresponding optimal state trajectories are(xlowastt ylowastt zlowastt rlowastt ) and (pt qt wt kt) is the solution of adjoint
equation (66) then there exists nonzero constant(h1 h0 h) isin (Rn1 times Rm1 times R) with |h0|
2 + |h1|2 + |h|2 1
such that for any admissible control vt isin Uad[0 T] themaximum condition (67) holds
9is conclusion can be drawn from the above analysisdirectly
5 A Financial Example
In this section we will study the problem of optimal con-sumption rate selection in the financial market which willnaturally inspire our research to the forward-backwardstochastic optimal control problem in Section 3
Mathematical Problems in Engineering 9
Assume the investorrsquos asset process xt (tge 0) in the fi-nancial market is described by the following stochasticdifferential equation with the Teugels martingale
dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin
i1gi
txtdHit
x0 Wgt 0 t isin [0 T]
⎧⎪⎪⎨
⎪⎪⎩(68)
where μt and σt ne 0 are the expected return and volatility ofthe value process xt at time t respectively and Ct is theconsumption rate process Assume μt σt gt and Ct are alluniformly bounded Ftminus measurable random processes epurpose of investors is to select the optimal consumptionstrategy Clowastt at time tge 0 minimizing the following recursiveutility
J(C(middot)) E y01113858 1113859 (69)
where yt is the following backward stochastic process
minus dyt Leminus rtC1minus R
t
1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936
infin
i1ri
tdHit
yT minus xT gt 0 t isin [0 T]
⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
(70)
with constant Lgt 0 the discount factor rgt 0 and theArrowndashPratt measure of risk aversion R isin (0 1)
If the consumption process Ct is regarded as a controlvariable then combining (68) with (70) we encounter thefollowing control system
dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin
i1gi
txtdHit
minus dyt Leminus rtC1minus R
t
1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936
infin
i1ri
tdHit
x0 Wgt 0 yT minus xT gt 0 t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(71)
which is obviously a special case of stochastic control system(2) with b(t x u) μtxt minus Ct σ(t x) σtxt g(t x)
gitxt Φx(xT) minus XT and f(t x y z r u) Leminus rt(C1minus R
t (1 minus R)) minus ryt
We can check that both Assumptions 1 and 2 are sat-isfieden we can use ourmaximum principle (eorem 1)to solve the above optimization problem Let Clowastt be anoptimal consumption rate and xlowastt ylowastt be the correspondingwealth process and recursive utility process In this case theHamiltonian function H reduces to
H t xlowastt ylowastt Ct p
lowastt qlowastt wlowastt klowastt( 1113857 langplowastt μtx
lowastt minus Ctrang
+langwlowastt σtxlowastt rang +langklowastt g
itxlowastt rang minus langqlowastt Le
minus rt C1minus Rt
1 minus Rminus rylowastt rang
(72)
and (plowastt qlowastt wlowastt klowastt ) is the solution of the following adjointequation
dqt rqtdt
minus dpt μtpt + σtwt + 1113936infin
i1ki
t( 1113857τgi
t1113890 1113891dt minus wtdBt minus 1113936infin
i1ki
tdHit
q0 1 pT qT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
(73)
According to the maximum principle (47) we have
plowastt minus Le
minus rtClowastt( 1113857
minus Rqlowastt (74)
and the optimal consumption rate
Clowastt minus
ert
L
plowasttqlowastt
1113888 1113889
minus 1R
(75)
By solving FBSDE (73) we get qlowastt ert plowastt erteμ(Tminus t)and wlowastt klowastt 0 en we get the optimal consumptionrate of the investor which is
Clowastt minus
erT
Leμ(Tminus t)
1113888 1113889
minus 1R
t isin [0 T] (76)
6 Conclusions
In this paper a nonconvex control domain case of theforward-backward stochastic control driven by Levy processis considered and we obtain the global stochastic maximumprinciple for this stochastic control problem And then theproblem of stochastic control with initial and final stateconstraints on the state variables is discussed and a nec-essary condition about existence of the optimal control isalso acquired A financial example of optimal consumptionis discussed to illustrate the application of the stochasticmaximum principle
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the NNSF (11701335) Shan-dong NSF (ZR2019BG015) NSSF (17BGL058) ShandongProvincial Higher Education Science and Technology PlanProject (J18KA236) and Shandong Womenrsquos College High-level Research Project Cultivation Foundation(2018GSPGJ07)
References
[1] L S Pontryakin V G Boltyanskti R V Gamkrelidze andE F Mischenko9eMathematical9eory of Optimal ControlProcesses John Wiley New York NY USA 1962
[2] S Peng ldquoA general stochastic maximum principle for optimalcontrol problemsrdquo SIAM Journal on Control and Optimiza-tion vol 28 no 4 pp 966ndash979 1990
10 Mathematical Problems in Engineering
[3] S G Peng ldquoBackward stochastic differrntial equations andapplication to optimal controlrdquo Applied Mathematics ampOptimization vol 27 no 4 pp 125ndash144 1993
[4] W S Xu ldquoStochastic maximum principle for optimal controlproblem of forward and backward systemrdquo Journal of Aus-tralian Mathematical Sciences vol 37 pp 249ndash259 1998B
[5] Z Wu ldquoMaximum principle for optimal control problem offully coupled forward-backward stochastic control systemrdquoSystem Science and Mathematical Sciences vol 11 no 3pp 249ndash259 1998
[6] J T Shi and ZWu ldquoemaximum principle for fully coupledforward-backward stochastic control systemrdquo Acta Auto-matica Sinica vol 32 no 2 pp 161ndash169 2006
[7] R Situ ldquoA maximum principle for optimal controls of sto-chastic systems with random jumpsrdquo in Proceedings of theNational Conference on Control 9eory and its ApplicationsQingdao China 1991
[8] J Shi and Z Wu ldquoMaximum principle for forward-backwardstochastic control system with random jumps and applica-tions to financerdquo Journal of Systems Science and Complexityvol 23 no 2 pp 219ndash231 2010
[9] J T Shi ldquoGlobal maximum principle for forward-backwardstochastic control system with Poisson jumpsrdquo Asian Journalof Control vol 16 no 1 pp 1ndash11 2014
[10] M Liu X Wang and H Huang ldquoMaximum principle forforward-backward control system driven by Ito-Levy pro-cesses under initial-terminal constraintsrdquo MathematicalProblems in Engineering vol 2017 pp 1ndash13 2017
[11] S Tang ldquoe maximum principle for partially observedoptimal control of stochastic differential equationsrdquo SIAMJournal on Control and Optimization vol 36 no 5pp 1596ndash1617 1998
[12] Q Meng ldquoA maximum principle for optimal control problemof fully coupled forward-backward stochastic systems withpartial informationrdquo Science in China Series A Mathematicsvol 52 no 7 pp 1579ndash1588 2009
[13] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward backward stichastic differential equationswith jumpsrdquo SIAM J Control Optimenleadertwodots vol 48pp 2845ndash2976 2009
[14] G Wang and Z Wu ldquoKalman-Bucy filtering equations offorward and backward stochastic systems and applications torecursive optimal control problemsrdquo Journal of MathematicalAnalysis and Applications vol 342 no 2 pp 1280ndash12962008
[15] H Xiao ldquoe maximum principle for partially observedoptimal control of forward-backward stochastic systems withrandom jumpsrdquo Journal of Systems Science and Complexityvol 24 no 6 pp 1083ndash1099 2011
[16] G Wang Z Wu and J Xiong ldquoMaximum principles forforward-backward stochastic control systems with correlatedstate and observation noisesrdquo SIAM Journal on Control andOptimization vol 51 no 1 pp 491ndash524 2013
[17] G Wang C Zhang and W Zhang ldquoStochastic maximumprinciple for mean-field type optimal control under partialinformationrdquo IEEE Transactions on Automatic Controlvol 59 no 2 pp 522ndash528 2014
[18] S Q Zhang J Xiong and X D Liu ldquoStochastic maximumprinciple for partially observed forward backward stochasticdifferential equations with jumps and regime switchingrdquoScience China Information Sciences vol 61 no 7 pp 1ndash132018
[19] J Xiong S Q Zhang S Zhang and Y Zhuang ldquoA partiallyobserved non-zero sum differential game of forward-
backward stochastic differential equations and its applicationin financerdquo Mathematical Control amp Related Fields vol 9no 2 pp 257ndash276 2019
[20] D Nualart and W Schoutens ldquoChaotic and predictablerepresentations for Levy processesrdquo Stochastic Processes and9eir Applications vol 90 no 1 pp 109ndash122 2000
[21] D Nualart and W Schoutens ldquoBackward stochastic differ-ential equations and Feynman-Kac formula for Levy pro-cesses with applications in financerdquo Bernoulli vol 7 no 5pp 761ndash776 2001
[22] Q Meng and M Tang ldquoNecessary and sufficient conditionsfor optimal control of stochastic systems associated with Levyprocessesrdquo Science in China Series F Information Sciencesvol 52 no 11 pp 1982ndash1992 2009
[23] M Tang and Q Zhang ldquoOptimal variational principle forbackward stochastic control systems associated with Levyprocessesrdquo Science China Mathematics vol 55 no 4pp 745ndash761 2012
[24] F Baghery N Khelfallah B Mezerdi and I Turpin ldquoFullycoupled forward backward stochastic differential equationsdriven by Levy processes and application to differentialgamesrdquo De Gruyter vol 22 no 3 pp 151ndash161 2014
[25] F Zhang M N Tang and Q X Meng ldquoSochastic maximumprinciple for forward-backward stochastic control systemsassociated with Levy processesrdquo Chinese Annals of Mathe-matics vol 35A no 1 pp 83ndash100 2014
[26] X R Wang and H Huang ldquoMaximum principle for forward-backward stochastic control system driven by Levy processrdquoMathematical Problems in Engineering vol 2015 Article ID702802 12 pages 2015
[27] H Huang X Wang and M Liu ldquoA maximum principle forfully coupled forward-backward stochastic control systemdriven by Levy process with terminal state constraintsrdquoJournal of Systems Science and Complexity vol 31 no 4pp 859ndash874 2018
[28] P Muthukumar and R Deepa ldquoInfinite horizon optimalcontrol of forward backward stochastic system driven byTeugels martingales with Levy processrdquo Stochastic and Dy-namics vol 3 no 17 pp 1ndash17 2017
[29] J B Wu and Z M Liu ldquoOptimal control of mean-fieldbackward doubly stochastic systems driven by Ito-Levyprocessesrdquo International Journal of Control vol 93 no 4pp 953ndash970 2018
[30] W Wang J Wu and Z Liu ldquoe optimal control of fully-coupled forward-backward doubly stochastic systems drivenby Ito-Levy processesrdquo Journal of Systems Science andComplexity vol 32 no 4 pp 997ndash1018 2019
Mathematical Problems in Engineering 11
where G1 Rn⟶ Rn1(n1 lt n) G0 Rm⟶ Rm1(m1 ltm)Our optimal control problem is to find ulowast isin Uad such that
J ulowast(middot)( 1113857 inf
u(middot)isinUad
J(u(middot)) (51)
subject to the state constraints (50) In the following we willapply Ekelandrsquos variational principle to solve this optimalcontrol problem Firstly we need the following assumptions
Assumption 3 Control domain U is assumed to be closedthe mappings in state constraints G1 G0 are continuouslydifferentiable and G1x G0y are bounded
For forallu1(middot) u2(middot) isin Uad let
d u1(middot) u2(middot)( 1113857 E u1(middot) minus u2(middot)1113868111386811138681113868
11138681113868111386811138682 gt 0 t isin [0 T]1113966 1113967
11138681113868111386811138681113868
11138681113868111386811138681113868 (52)
Same as Section 3 we also assume ulowastt be the optimalcontrol and the corresponding optimal state trajectories are(xlowastt ylowastt zlowastt rlowastt ) In order to solve the constraint problemwe need the following penalty cost functional for any ρgt 0
Jρ(u) E G1 xT( 11138571113858 11138591113868111386811138681113868
11138681113868111386811138682
+ E G0 y0( 11138571113858 11138591113868111386811138681113868
11138681113868111386811138682
+ J(u) minus J ulowast( ) + ρ1113868111386811138681113868
11138681113868111386811138682
1113966 111396712
(53)
It can be checked that Jρ(u) Uad⟶ R1 is continuousand for any u(middot) isin Uad
Jρ(u)ge 0 Jρ ulowast
( 1113857 ρ
Jρ ulowast
( 1113857le infu(middot)isinUad
Jρ(u) + ρ(54)
It can be obtained by Ekelandrsquos variational principle thatthere exists u
ρt isin Uad such that
(i) Jρ uρ( )le Jρ ulowast( ) ρ
(ii) d uρ ulowast( )le ρradic
(iii) Jρ(v)ge Jρ uρ( ) minusρradic
d uρ v( ) for forallv(middot) isin Uad
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(55)
For fixed ρ and admissible control uρt we define the spike
variation as follows
uρεt
vt τ le tle τ + ε
uρt otherwise
1113896 (56)
for any εgt 0 and it is easy to check from (iii) of (55) that
Jρ uρε( ) minus Jρ uρ( ) +ρradic
d uρε uρ( )ge 0 (57)
Let (xρt y
ρt z
ρt r
ρt ) be the trajectories corresponding to u
ρt
and (xρεt y
ρεt z
ρεt r
ρεt ) be the trajectories corresponding to
uρεt e variational equation we used in this section is the
same as the one in Section 3 with ulowastt uρt and
(xlowastt ylowastt zlowastt rlowastt ) (xρt y
ρt z
ρt r
ρt ) And we also assume that
the solution of this variational equation is (Xρt Y
ρt Z
ρt R
ρt )
Similar to the approach in Lemmas 2 and 3 it can be shownthat
sup0letleTE xρεt minus x
ρt minus X
ρt
111386811138681113868111386811138681113868111386811138682 leCεε
2 Cε⟶ 0when ε⟶ 0
sup0letleTE yρεt minus y
ρt minus Y
ρt
111386811138681113868111386811138681113868111386811138682 leCεε
2 Cε⟶ 0when ε⟶ 0
E 1113946T
0zρεt minus z
ρt minus Z
ρt
111386811138681113868111386811138681113868111386811138682dsleCεε
2 Cε⟶ 0when ε⟶ 0
E 1113946T
0rρεt minus r
ρt minus R
ρt
2dsleCεε
2 Cε⟶ 0when ε⟶ 0
(58)
en by (57) the following variational inequality holds
Jρ uρε
( 1113857 minus Jρ uρ
( 1113857 +ρ
radicd u
ρε u
ρ( 1113857
J2ρ uρε( ) minus J2ρ uρ( )
Jρ uρε( ) + Jρ uρ( )+ ε
ρ
radic
langhρε1 E G1x x
ρT( 11138571113858 1113859X
ρTrang +langhρε
0 E G0y yρ0( 11138571113960 1113961Y
ρ0rang
+ hρε
E cy yρ0( 1113857Y
ρ01113960 1113961 + ε
ρ
radic+ o(ε)
(59)
where
hρε1
2E G1 xρT( 11138571113858 1113859
Jρ uρε( ) + Jρ uρ( )
hρε0
2E G0 yρ0( 11138571113858 1113859
Jρ uρε( ) + Jρ uρ( )
hρε
2E c y
ρ0( 1113857 minus c y0( 1113857 + ρ1113858 1113859
Jρ uρε( ) + Jρ uρ( )
(60)
Now let (pρεt q
ρεt w
ρεt k
ρεt ) be the solution of
minus dpρεt b
ρx(t)( 1113857
τpρεt minus f
ρx(t)( 1113857
τqρεt + σρx(t)( 1113857
τw
ρεt + 1113936infin
i1gρx(t)( 1113857
τkρεt )i
1113890 1113891dt
minus wρεt dBt minus 1113936
infin
i1kρεt( 1113857
idHit
dqρεt f
ρy(t)( 1113857
τqρεt dt + f
ρz(t)( 1113857
τqρεt dBt + 1113936
infin
i1fρir (t)1113872 1113873
τ)q
ρεt dHi
t
q0 minus G0y yρ0( 1113857h
ρε0 + cy y
ρ0( 1113857hρε pT G1x x
ρT( 1113857h
ρε1 minus Φx x
ρT( 1113857q
ρεT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(61)
8 Mathematical Problems in Engineering
Here bρx(t) bx(t x
ρt u
ρt ) σρx(t) σx(t x
ρt )
fρy(t) fy(t x
ρt y
ρt z
ρt r
ρt u
ρt ) etc Applying It 1113954os formula
to langXρt q
ρεt rang + langY
ρt p
ρεt rang variational inequality (59) can be
rewritten as
H t xρt y
ρt z
ρt r
ρt u
ρεt p
ρεt q
ρεt w
ρεt k
ρεt( 1113857
minus H t xρt y
ρt z
ρt r
ρt u
ρt p
ρεt q
ρεt w
ρεt k
ρεt( 1113857
+ ερ
radic+ o(ε)ge 0 forallvt isin U ae as
(62)
where the Hamiltonian function H is defined as (44) Since
limε⟶0 hρε0
111386811138681113868111386811138681113868111386811138682
+ hρε1
111386811138681113868111386811138681113868111386811138682
+ hρε1113868111386811138681113868
11138681113868111386811138682
1113872 1113873 1 (63)
there exists a convergent subsequence still denoted by(h
ρε0 h
ρε1 hρε) such that (h
ρε0 h
ρε1 hρε)⟶ (h
ρ0 h
ρ1 hρ) when
ε⟶ 0 with |hρ0|2 + |h
ρ1|2 + |hρ|2 1
Let (pρt q
ρt w
ρt k
ρt ) be the solution of
minus dpρt b
ρx(t)( 1113857
τpρt minus f
ρx(t)( 1113857
τqρt + σρx(t)( 1113857
τw
ρt + 1113936infin
i1gρx(t)( 1113857
τkρt )i
1113890 1113891dt
minus wρtdBt minus 1113936
infin
i1kρt( 1113857
idHit
dqρt f
ρy(t)( 1113857
τqρtdt + f
ρz(t)( 1113857
τqρtdBt + 1113936
infin
i1fρir (t)1113872 1113873
τ1113888 1113889q
ρtdHi
t
q0 minus G0y yρ0( 1113857h
ρ0 + cy y
ρ0( 1113857hρ pT G1x x
ρT( 1113857h
ρ1 minus Φx x
ρT( 1113857q
ρT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(64)
From the continuous dependence of the solutions ofFBSDEs with Levy process on the parameters we can provethat the following convergence holds(p
ρεt q
ρεt w
ρεt k
ρεt )⟶ (p
ρt q
ρt w
ρt k
ρt ) then in equation
(62) it implies that
H t xρt y
ρt z
ρt r
ρt vt p
ρt q
ρt w
ρt k
ρt( 1113857
minus H t xρt y
ρt z
ρt r
ρt u
ρt p
ρt q
ρt w
ρt k
ρt( 1113857
+ρ
radic ge 0 forallvt isin U ae as
(65)
Similarly there exists a convergent subsequence(h
ρ0 h
ρ1 h
ρ) such that (h
ρ0 h
ρ1 h
ρ)⟶ (h0 h1 h) when
ρ⟶ 0 with |h0|2 + |h1|
2 + |h|2 1 Since uρt⟶ ulowastt as
ρ⟶ 0 we have (xρt y
ρt z
ρt r
ρt )⟶ (xlowastt ylowastt zlowastt rlowastt ) and
(Xρt Y
ρt Z
ρt R
ρt )⟶ (Xt Yt Zt Rt) which is the solution of
the variational equation as same as (6)e following adjoint equation is introduced
minus dpt bx(t)( 1113857τpt minus fx(t)( 1113857
τqt + σx(t)( 1113857
τwt + 1113936infin
i1gx(t)( 1113857
τkt)
i1113890 1113891dt
minus wtdBt minus 1113936infin
i1kt( 1113857
idHit
dqt fy(t)1113872 1113873τqtdt + fz(t)( 1113857
τqtdBt + 1113936
infin
i1fρir (t)1113872 1113873
τ1113888 1113889qtdHi
t
q0 minus G0y y0( 1113857h0 + cy y0( 1113857h pT G1x xT( 1113857h1 minus Φx xT( 1113857qT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(66)
Similarly it can be proved that(p
ρt q
ρt w
ρt k
ρt )⟶ (pt qt wt kt) then inequation (65)
impliesH t x
lowastt ylowastt zlowastt rlowastt vt pt qt wt kt( 1113857
minus H t xlowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857ge 0forallvt isin U ae as
(67)
en we get the following theorem
Theorem 2 Let Assumptions 1ndash3 hold ulowastt is an optimalcontrol and the corresponding optimal state trajectories are(xlowastt ylowastt zlowastt rlowastt ) and (pt qt wt kt) is the solution of adjoint
equation (66) then there exists nonzero constant(h1 h0 h) isin (Rn1 times Rm1 times R) with |h0|
2 + |h1|2 + |h|2 1
such that for any admissible control vt isin Uad[0 T] themaximum condition (67) holds
9is conclusion can be drawn from the above analysisdirectly
5 A Financial Example
In this section we will study the problem of optimal con-sumption rate selection in the financial market which willnaturally inspire our research to the forward-backwardstochastic optimal control problem in Section 3
Mathematical Problems in Engineering 9
Assume the investorrsquos asset process xt (tge 0) in the fi-nancial market is described by the following stochasticdifferential equation with the Teugels martingale
dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin
i1gi
txtdHit
x0 Wgt 0 t isin [0 T]
⎧⎪⎪⎨
⎪⎪⎩(68)
where μt and σt ne 0 are the expected return and volatility ofthe value process xt at time t respectively and Ct is theconsumption rate process Assume μt σt gt and Ct are alluniformly bounded Ftminus measurable random processes epurpose of investors is to select the optimal consumptionstrategy Clowastt at time tge 0 minimizing the following recursiveutility
J(C(middot)) E y01113858 1113859 (69)
where yt is the following backward stochastic process
minus dyt Leminus rtC1minus R
t
1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936
infin
i1ri
tdHit
yT minus xT gt 0 t isin [0 T]
⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
(70)
with constant Lgt 0 the discount factor rgt 0 and theArrowndashPratt measure of risk aversion R isin (0 1)
If the consumption process Ct is regarded as a controlvariable then combining (68) with (70) we encounter thefollowing control system
dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin
i1gi
txtdHit
minus dyt Leminus rtC1minus R
t
1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936
infin
i1ri
tdHit
x0 Wgt 0 yT minus xT gt 0 t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(71)
which is obviously a special case of stochastic control system(2) with b(t x u) μtxt minus Ct σ(t x) σtxt g(t x)
gitxt Φx(xT) minus XT and f(t x y z r u) Leminus rt(C1minus R
t (1 minus R)) minus ryt
We can check that both Assumptions 1 and 2 are sat-isfieden we can use ourmaximum principle (eorem 1)to solve the above optimization problem Let Clowastt be anoptimal consumption rate and xlowastt ylowastt be the correspondingwealth process and recursive utility process In this case theHamiltonian function H reduces to
H t xlowastt ylowastt Ct p
lowastt qlowastt wlowastt klowastt( 1113857 langplowastt μtx
lowastt minus Ctrang
+langwlowastt σtxlowastt rang +langklowastt g
itxlowastt rang minus langqlowastt Le
minus rt C1minus Rt
1 minus Rminus rylowastt rang
(72)
and (plowastt qlowastt wlowastt klowastt ) is the solution of the following adjointequation
dqt rqtdt
minus dpt μtpt + σtwt + 1113936infin
i1ki
t( 1113857τgi
t1113890 1113891dt minus wtdBt minus 1113936infin
i1ki
tdHit
q0 1 pT qT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
(73)
According to the maximum principle (47) we have
plowastt minus Le
minus rtClowastt( 1113857
minus Rqlowastt (74)
and the optimal consumption rate
Clowastt minus
ert
L
plowasttqlowastt
1113888 1113889
minus 1R
(75)
By solving FBSDE (73) we get qlowastt ert plowastt erteμ(Tminus t)and wlowastt klowastt 0 en we get the optimal consumptionrate of the investor which is
Clowastt minus
erT
Leμ(Tminus t)
1113888 1113889
minus 1R
t isin [0 T] (76)
6 Conclusions
In this paper a nonconvex control domain case of theforward-backward stochastic control driven by Levy processis considered and we obtain the global stochastic maximumprinciple for this stochastic control problem And then theproblem of stochastic control with initial and final stateconstraints on the state variables is discussed and a nec-essary condition about existence of the optimal control isalso acquired A financial example of optimal consumptionis discussed to illustrate the application of the stochasticmaximum principle
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the NNSF (11701335) Shan-dong NSF (ZR2019BG015) NSSF (17BGL058) ShandongProvincial Higher Education Science and Technology PlanProject (J18KA236) and Shandong Womenrsquos College High-level Research Project Cultivation Foundation(2018GSPGJ07)
References
[1] L S Pontryakin V G Boltyanskti R V Gamkrelidze andE F Mischenko9eMathematical9eory of Optimal ControlProcesses John Wiley New York NY USA 1962
[2] S Peng ldquoA general stochastic maximum principle for optimalcontrol problemsrdquo SIAM Journal on Control and Optimiza-tion vol 28 no 4 pp 966ndash979 1990
10 Mathematical Problems in Engineering
[3] S G Peng ldquoBackward stochastic differrntial equations andapplication to optimal controlrdquo Applied Mathematics ampOptimization vol 27 no 4 pp 125ndash144 1993
[4] W S Xu ldquoStochastic maximum principle for optimal controlproblem of forward and backward systemrdquo Journal of Aus-tralian Mathematical Sciences vol 37 pp 249ndash259 1998B
[5] Z Wu ldquoMaximum principle for optimal control problem offully coupled forward-backward stochastic control systemrdquoSystem Science and Mathematical Sciences vol 11 no 3pp 249ndash259 1998
[6] J T Shi and ZWu ldquoemaximum principle for fully coupledforward-backward stochastic control systemrdquo Acta Auto-matica Sinica vol 32 no 2 pp 161ndash169 2006
[7] R Situ ldquoA maximum principle for optimal controls of sto-chastic systems with random jumpsrdquo in Proceedings of theNational Conference on Control 9eory and its ApplicationsQingdao China 1991
[8] J Shi and Z Wu ldquoMaximum principle for forward-backwardstochastic control system with random jumps and applica-tions to financerdquo Journal of Systems Science and Complexityvol 23 no 2 pp 219ndash231 2010
[9] J T Shi ldquoGlobal maximum principle for forward-backwardstochastic control system with Poisson jumpsrdquo Asian Journalof Control vol 16 no 1 pp 1ndash11 2014
[10] M Liu X Wang and H Huang ldquoMaximum principle forforward-backward control system driven by Ito-Levy pro-cesses under initial-terminal constraintsrdquo MathematicalProblems in Engineering vol 2017 pp 1ndash13 2017
[11] S Tang ldquoe maximum principle for partially observedoptimal control of stochastic differential equationsrdquo SIAMJournal on Control and Optimization vol 36 no 5pp 1596ndash1617 1998
[12] Q Meng ldquoA maximum principle for optimal control problemof fully coupled forward-backward stochastic systems withpartial informationrdquo Science in China Series A Mathematicsvol 52 no 7 pp 1579ndash1588 2009
[13] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward backward stichastic differential equationswith jumpsrdquo SIAM J Control Optimenleadertwodots vol 48pp 2845ndash2976 2009
[14] G Wang and Z Wu ldquoKalman-Bucy filtering equations offorward and backward stochastic systems and applications torecursive optimal control problemsrdquo Journal of MathematicalAnalysis and Applications vol 342 no 2 pp 1280ndash12962008
[15] H Xiao ldquoe maximum principle for partially observedoptimal control of forward-backward stochastic systems withrandom jumpsrdquo Journal of Systems Science and Complexityvol 24 no 6 pp 1083ndash1099 2011
[16] G Wang Z Wu and J Xiong ldquoMaximum principles forforward-backward stochastic control systems with correlatedstate and observation noisesrdquo SIAM Journal on Control andOptimization vol 51 no 1 pp 491ndash524 2013
[17] G Wang C Zhang and W Zhang ldquoStochastic maximumprinciple for mean-field type optimal control under partialinformationrdquo IEEE Transactions on Automatic Controlvol 59 no 2 pp 522ndash528 2014
[18] S Q Zhang J Xiong and X D Liu ldquoStochastic maximumprinciple for partially observed forward backward stochasticdifferential equations with jumps and regime switchingrdquoScience China Information Sciences vol 61 no 7 pp 1ndash132018
[19] J Xiong S Q Zhang S Zhang and Y Zhuang ldquoA partiallyobserved non-zero sum differential game of forward-
backward stochastic differential equations and its applicationin financerdquo Mathematical Control amp Related Fields vol 9no 2 pp 257ndash276 2019
[20] D Nualart and W Schoutens ldquoChaotic and predictablerepresentations for Levy processesrdquo Stochastic Processes and9eir Applications vol 90 no 1 pp 109ndash122 2000
[21] D Nualart and W Schoutens ldquoBackward stochastic differ-ential equations and Feynman-Kac formula for Levy pro-cesses with applications in financerdquo Bernoulli vol 7 no 5pp 761ndash776 2001
[22] Q Meng and M Tang ldquoNecessary and sufficient conditionsfor optimal control of stochastic systems associated with Levyprocessesrdquo Science in China Series F Information Sciencesvol 52 no 11 pp 1982ndash1992 2009
[23] M Tang and Q Zhang ldquoOptimal variational principle forbackward stochastic control systems associated with Levyprocessesrdquo Science China Mathematics vol 55 no 4pp 745ndash761 2012
[24] F Baghery N Khelfallah B Mezerdi and I Turpin ldquoFullycoupled forward backward stochastic differential equationsdriven by Levy processes and application to differentialgamesrdquo De Gruyter vol 22 no 3 pp 151ndash161 2014
[25] F Zhang M N Tang and Q X Meng ldquoSochastic maximumprinciple for forward-backward stochastic control systemsassociated with Levy processesrdquo Chinese Annals of Mathe-matics vol 35A no 1 pp 83ndash100 2014
[26] X R Wang and H Huang ldquoMaximum principle for forward-backward stochastic control system driven by Levy processrdquoMathematical Problems in Engineering vol 2015 Article ID702802 12 pages 2015
[27] H Huang X Wang and M Liu ldquoA maximum principle forfully coupled forward-backward stochastic control systemdriven by Levy process with terminal state constraintsrdquoJournal of Systems Science and Complexity vol 31 no 4pp 859ndash874 2018
[28] P Muthukumar and R Deepa ldquoInfinite horizon optimalcontrol of forward backward stochastic system driven byTeugels martingales with Levy processrdquo Stochastic and Dy-namics vol 3 no 17 pp 1ndash17 2017
[29] J B Wu and Z M Liu ldquoOptimal control of mean-fieldbackward doubly stochastic systems driven by Ito-Levyprocessesrdquo International Journal of Control vol 93 no 4pp 953ndash970 2018
[30] W Wang J Wu and Z Liu ldquoe optimal control of fully-coupled forward-backward doubly stochastic systems drivenby Ito-Levy processesrdquo Journal of Systems Science andComplexity vol 32 no 4 pp 997ndash1018 2019
Mathematical Problems in Engineering 11
Here bρx(t) bx(t x
ρt u
ρt ) σρx(t) σx(t x
ρt )
fρy(t) fy(t x
ρt y
ρt z
ρt r
ρt u
ρt ) etc Applying It 1113954os formula
to langXρt q
ρεt rang + langY
ρt p
ρεt rang variational inequality (59) can be
rewritten as
H t xρt y
ρt z
ρt r
ρt u
ρεt p
ρεt q
ρεt w
ρεt k
ρεt( 1113857
minus H t xρt y
ρt z
ρt r
ρt u
ρt p
ρεt q
ρεt w
ρεt k
ρεt( 1113857
+ ερ
radic+ o(ε)ge 0 forallvt isin U ae as
(62)
where the Hamiltonian function H is defined as (44) Since
limε⟶0 hρε0
111386811138681113868111386811138681113868111386811138682
+ hρε1
111386811138681113868111386811138681113868111386811138682
+ hρε1113868111386811138681113868
11138681113868111386811138682
1113872 1113873 1 (63)
there exists a convergent subsequence still denoted by(h
ρε0 h
ρε1 hρε) such that (h
ρε0 h
ρε1 hρε)⟶ (h
ρ0 h
ρ1 hρ) when
ε⟶ 0 with |hρ0|2 + |h
ρ1|2 + |hρ|2 1
Let (pρt q
ρt w
ρt k
ρt ) be the solution of
minus dpρt b
ρx(t)( 1113857
τpρt minus f
ρx(t)( 1113857
τqρt + σρx(t)( 1113857
τw
ρt + 1113936infin
i1gρx(t)( 1113857
τkρt )i
1113890 1113891dt
minus wρtdBt minus 1113936
infin
i1kρt( 1113857
idHit
dqρt f
ρy(t)( 1113857
τqρtdt + f
ρz(t)( 1113857
τqρtdBt + 1113936
infin
i1fρir (t)1113872 1113873
τ1113888 1113889q
ρtdHi
t
q0 minus G0y yρ0( 1113857h
ρ0 + cy y
ρ0( 1113857hρ pT G1x x
ρT( 1113857h
ρ1 minus Φx x
ρT( 1113857q
ρT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(64)
From the continuous dependence of the solutions ofFBSDEs with Levy process on the parameters we can provethat the following convergence holds(p
ρεt q
ρεt w
ρεt k
ρεt )⟶ (p
ρt q
ρt w
ρt k
ρt ) then in equation
(62) it implies that
H t xρt y
ρt z
ρt r
ρt vt p
ρt q
ρt w
ρt k
ρt( 1113857
minus H t xρt y
ρt z
ρt r
ρt u
ρt p
ρt q
ρt w
ρt k
ρt( 1113857
+ρ
radic ge 0 forallvt isin U ae as
(65)
Similarly there exists a convergent subsequence(h
ρ0 h
ρ1 h
ρ) such that (h
ρ0 h
ρ1 h
ρ)⟶ (h0 h1 h) when
ρ⟶ 0 with |h0|2 + |h1|
2 + |h|2 1 Since uρt⟶ ulowastt as
ρ⟶ 0 we have (xρt y
ρt z
ρt r
ρt )⟶ (xlowastt ylowastt zlowastt rlowastt ) and
(Xρt Y
ρt Z
ρt R
ρt )⟶ (Xt Yt Zt Rt) which is the solution of
the variational equation as same as (6)e following adjoint equation is introduced
minus dpt bx(t)( 1113857τpt minus fx(t)( 1113857
τqt + σx(t)( 1113857
τwt + 1113936infin
i1gx(t)( 1113857
τkt)
i1113890 1113891dt
minus wtdBt minus 1113936infin
i1kt( 1113857
idHit
dqt fy(t)1113872 1113873τqtdt + fz(t)( 1113857
τqtdBt + 1113936
infin
i1fρir (t)1113872 1113873
τ1113888 1113889qtdHi
t
q0 minus G0y y0( 1113857h0 + cy y0( 1113857h pT G1x xT( 1113857h1 minus Φx xT( 1113857qT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(66)
Similarly it can be proved that(p
ρt q
ρt w
ρt k
ρt )⟶ (pt qt wt kt) then inequation (65)
impliesH t x
lowastt ylowastt zlowastt rlowastt vt pt qt wt kt( 1113857
minus H t xlowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857ge 0forallvt isin U ae as
(67)
en we get the following theorem
Theorem 2 Let Assumptions 1ndash3 hold ulowastt is an optimalcontrol and the corresponding optimal state trajectories are(xlowastt ylowastt zlowastt rlowastt ) and (pt qt wt kt) is the solution of adjoint
equation (66) then there exists nonzero constant(h1 h0 h) isin (Rn1 times Rm1 times R) with |h0|
2 + |h1|2 + |h|2 1
such that for any admissible control vt isin Uad[0 T] themaximum condition (67) holds
9is conclusion can be drawn from the above analysisdirectly
5 A Financial Example
In this section we will study the problem of optimal con-sumption rate selection in the financial market which willnaturally inspire our research to the forward-backwardstochastic optimal control problem in Section 3
Mathematical Problems in Engineering 9
Assume the investorrsquos asset process xt (tge 0) in the fi-nancial market is described by the following stochasticdifferential equation with the Teugels martingale
dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin
i1gi
txtdHit
x0 Wgt 0 t isin [0 T]
⎧⎪⎪⎨
⎪⎪⎩(68)
where μt and σt ne 0 are the expected return and volatility ofthe value process xt at time t respectively and Ct is theconsumption rate process Assume μt σt gt and Ct are alluniformly bounded Ftminus measurable random processes epurpose of investors is to select the optimal consumptionstrategy Clowastt at time tge 0 minimizing the following recursiveutility
J(C(middot)) E y01113858 1113859 (69)
where yt is the following backward stochastic process
minus dyt Leminus rtC1minus R
t
1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936
infin
i1ri
tdHit
yT minus xT gt 0 t isin [0 T]
⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
(70)
with constant Lgt 0 the discount factor rgt 0 and theArrowndashPratt measure of risk aversion R isin (0 1)
If the consumption process Ct is regarded as a controlvariable then combining (68) with (70) we encounter thefollowing control system
dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin
i1gi
txtdHit
minus dyt Leminus rtC1minus R
t
1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936
infin
i1ri
tdHit
x0 Wgt 0 yT minus xT gt 0 t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(71)
which is obviously a special case of stochastic control system(2) with b(t x u) μtxt minus Ct σ(t x) σtxt g(t x)
gitxt Φx(xT) minus XT and f(t x y z r u) Leminus rt(C1minus R
t (1 minus R)) minus ryt
We can check that both Assumptions 1 and 2 are sat-isfieden we can use ourmaximum principle (eorem 1)to solve the above optimization problem Let Clowastt be anoptimal consumption rate and xlowastt ylowastt be the correspondingwealth process and recursive utility process In this case theHamiltonian function H reduces to
H t xlowastt ylowastt Ct p
lowastt qlowastt wlowastt klowastt( 1113857 langplowastt μtx
lowastt minus Ctrang
+langwlowastt σtxlowastt rang +langklowastt g
itxlowastt rang minus langqlowastt Le
minus rt C1minus Rt
1 minus Rminus rylowastt rang
(72)
and (plowastt qlowastt wlowastt klowastt ) is the solution of the following adjointequation
dqt rqtdt
minus dpt μtpt + σtwt + 1113936infin
i1ki
t( 1113857τgi
t1113890 1113891dt minus wtdBt minus 1113936infin
i1ki
tdHit
q0 1 pT qT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
(73)
According to the maximum principle (47) we have
plowastt minus Le
minus rtClowastt( 1113857
minus Rqlowastt (74)
and the optimal consumption rate
Clowastt minus
ert
L
plowasttqlowastt
1113888 1113889
minus 1R
(75)
By solving FBSDE (73) we get qlowastt ert plowastt erteμ(Tminus t)and wlowastt klowastt 0 en we get the optimal consumptionrate of the investor which is
Clowastt minus
erT
Leμ(Tminus t)
1113888 1113889
minus 1R
t isin [0 T] (76)
6 Conclusions
In this paper a nonconvex control domain case of theforward-backward stochastic control driven by Levy processis considered and we obtain the global stochastic maximumprinciple for this stochastic control problem And then theproblem of stochastic control with initial and final stateconstraints on the state variables is discussed and a nec-essary condition about existence of the optimal control isalso acquired A financial example of optimal consumptionis discussed to illustrate the application of the stochasticmaximum principle
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the NNSF (11701335) Shan-dong NSF (ZR2019BG015) NSSF (17BGL058) ShandongProvincial Higher Education Science and Technology PlanProject (J18KA236) and Shandong Womenrsquos College High-level Research Project Cultivation Foundation(2018GSPGJ07)
References
[1] L S Pontryakin V G Boltyanskti R V Gamkrelidze andE F Mischenko9eMathematical9eory of Optimal ControlProcesses John Wiley New York NY USA 1962
[2] S Peng ldquoA general stochastic maximum principle for optimalcontrol problemsrdquo SIAM Journal on Control and Optimiza-tion vol 28 no 4 pp 966ndash979 1990
10 Mathematical Problems in Engineering
[3] S G Peng ldquoBackward stochastic differrntial equations andapplication to optimal controlrdquo Applied Mathematics ampOptimization vol 27 no 4 pp 125ndash144 1993
[4] W S Xu ldquoStochastic maximum principle for optimal controlproblem of forward and backward systemrdquo Journal of Aus-tralian Mathematical Sciences vol 37 pp 249ndash259 1998B
[5] Z Wu ldquoMaximum principle for optimal control problem offully coupled forward-backward stochastic control systemrdquoSystem Science and Mathematical Sciences vol 11 no 3pp 249ndash259 1998
[6] J T Shi and ZWu ldquoemaximum principle for fully coupledforward-backward stochastic control systemrdquo Acta Auto-matica Sinica vol 32 no 2 pp 161ndash169 2006
[7] R Situ ldquoA maximum principle for optimal controls of sto-chastic systems with random jumpsrdquo in Proceedings of theNational Conference on Control 9eory and its ApplicationsQingdao China 1991
[8] J Shi and Z Wu ldquoMaximum principle for forward-backwardstochastic control system with random jumps and applica-tions to financerdquo Journal of Systems Science and Complexityvol 23 no 2 pp 219ndash231 2010
[9] J T Shi ldquoGlobal maximum principle for forward-backwardstochastic control system with Poisson jumpsrdquo Asian Journalof Control vol 16 no 1 pp 1ndash11 2014
[10] M Liu X Wang and H Huang ldquoMaximum principle forforward-backward control system driven by Ito-Levy pro-cesses under initial-terminal constraintsrdquo MathematicalProblems in Engineering vol 2017 pp 1ndash13 2017
[11] S Tang ldquoe maximum principle for partially observedoptimal control of stochastic differential equationsrdquo SIAMJournal on Control and Optimization vol 36 no 5pp 1596ndash1617 1998
[12] Q Meng ldquoA maximum principle for optimal control problemof fully coupled forward-backward stochastic systems withpartial informationrdquo Science in China Series A Mathematicsvol 52 no 7 pp 1579ndash1588 2009
[13] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward backward stichastic differential equationswith jumpsrdquo SIAM J Control Optimenleadertwodots vol 48pp 2845ndash2976 2009
[14] G Wang and Z Wu ldquoKalman-Bucy filtering equations offorward and backward stochastic systems and applications torecursive optimal control problemsrdquo Journal of MathematicalAnalysis and Applications vol 342 no 2 pp 1280ndash12962008
[15] H Xiao ldquoe maximum principle for partially observedoptimal control of forward-backward stochastic systems withrandom jumpsrdquo Journal of Systems Science and Complexityvol 24 no 6 pp 1083ndash1099 2011
[16] G Wang Z Wu and J Xiong ldquoMaximum principles forforward-backward stochastic control systems with correlatedstate and observation noisesrdquo SIAM Journal on Control andOptimization vol 51 no 1 pp 491ndash524 2013
[17] G Wang C Zhang and W Zhang ldquoStochastic maximumprinciple for mean-field type optimal control under partialinformationrdquo IEEE Transactions on Automatic Controlvol 59 no 2 pp 522ndash528 2014
[18] S Q Zhang J Xiong and X D Liu ldquoStochastic maximumprinciple for partially observed forward backward stochasticdifferential equations with jumps and regime switchingrdquoScience China Information Sciences vol 61 no 7 pp 1ndash132018
[19] J Xiong S Q Zhang S Zhang and Y Zhuang ldquoA partiallyobserved non-zero sum differential game of forward-
backward stochastic differential equations and its applicationin financerdquo Mathematical Control amp Related Fields vol 9no 2 pp 257ndash276 2019
[20] D Nualart and W Schoutens ldquoChaotic and predictablerepresentations for Levy processesrdquo Stochastic Processes and9eir Applications vol 90 no 1 pp 109ndash122 2000
[21] D Nualart and W Schoutens ldquoBackward stochastic differ-ential equations and Feynman-Kac formula for Levy pro-cesses with applications in financerdquo Bernoulli vol 7 no 5pp 761ndash776 2001
[22] Q Meng and M Tang ldquoNecessary and sufficient conditionsfor optimal control of stochastic systems associated with Levyprocessesrdquo Science in China Series F Information Sciencesvol 52 no 11 pp 1982ndash1992 2009
[23] M Tang and Q Zhang ldquoOptimal variational principle forbackward stochastic control systems associated with Levyprocessesrdquo Science China Mathematics vol 55 no 4pp 745ndash761 2012
[24] F Baghery N Khelfallah B Mezerdi and I Turpin ldquoFullycoupled forward backward stochastic differential equationsdriven by Levy processes and application to differentialgamesrdquo De Gruyter vol 22 no 3 pp 151ndash161 2014
[25] F Zhang M N Tang and Q X Meng ldquoSochastic maximumprinciple for forward-backward stochastic control systemsassociated with Levy processesrdquo Chinese Annals of Mathe-matics vol 35A no 1 pp 83ndash100 2014
[26] X R Wang and H Huang ldquoMaximum principle for forward-backward stochastic control system driven by Levy processrdquoMathematical Problems in Engineering vol 2015 Article ID702802 12 pages 2015
[27] H Huang X Wang and M Liu ldquoA maximum principle forfully coupled forward-backward stochastic control systemdriven by Levy process with terminal state constraintsrdquoJournal of Systems Science and Complexity vol 31 no 4pp 859ndash874 2018
[28] P Muthukumar and R Deepa ldquoInfinite horizon optimalcontrol of forward backward stochastic system driven byTeugels martingales with Levy processrdquo Stochastic and Dy-namics vol 3 no 17 pp 1ndash17 2017
[29] J B Wu and Z M Liu ldquoOptimal control of mean-fieldbackward doubly stochastic systems driven by Ito-Levyprocessesrdquo International Journal of Control vol 93 no 4pp 953ndash970 2018
[30] W Wang J Wu and Z Liu ldquoe optimal control of fully-coupled forward-backward doubly stochastic systems drivenby Ito-Levy processesrdquo Journal of Systems Science andComplexity vol 32 no 4 pp 997ndash1018 2019
Mathematical Problems in Engineering 11
Assume the investorrsquos asset process xt (tge 0) in the fi-nancial market is described by the following stochasticdifferential equation with the Teugels martingale
dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin
i1gi
txtdHit
x0 Wgt 0 t isin [0 T]
⎧⎪⎪⎨
⎪⎪⎩(68)
where μt and σt ne 0 are the expected return and volatility ofthe value process xt at time t respectively and Ct is theconsumption rate process Assume μt σt gt and Ct are alluniformly bounded Ftminus measurable random processes epurpose of investors is to select the optimal consumptionstrategy Clowastt at time tge 0 minimizing the following recursiveutility
J(C(middot)) E y01113858 1113859 (69)
where yt is the following backward stochastic process
minus dyt Leminus rtC1minus R
t
1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936
infin
i1ri
tdHit
yT minus xT gt 0 t isin [0 T]
⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
(70)
with constant Lgt 0 the discount factor rgt 0 and theArrowndashPratt measure of risk aversion R isin (0 1)
If the consumption process Ct is regarded as a controlvariable then combining (68) with (70) we encounter thefollowing control system
dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin
i1gi
txtdHit
minus dyt Leminus rtC1minus R
t
1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936
infin
i1ri
tdHit
x0 Wgt 0 yT minus xT gt 0 t isin [0 T]
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(71)
which is obviously a special case of stochastic control system(2) with b(t x u) μtxt minus Ct σ(t x) σtxt g(t x)
gitxt Φx(xT) minus XT and f(t x y z r u) Leminus rt(C1minus R
t (1 minus R)) minus ryt
We can check that both Assumptions 1 and 2 are sat-isfieden we can use ourmaximum principle (eorem 1)to solve the above optimization problem Let Clowastt be anoptimal consumption rate and xlowastt ylowastt be the correspondingwealth process and recursive utility process In this case theHamiltonian function H reduces to
H t xlowastt ylowastt Ct p
lowastt qlowastt wlowastt klowastt( 1113857 langplowastt μtx
lowastt minus Ctrang
+langwlowastt σtxlowastt rang +langklowastt g
itxlowastt rang minus langqlowastt Le
minus rt C1minus Rt
1 minus Rminus rylowastt rang
(72)
and (plowastt qlowastt wlowastt klowastt ) is the solution of the following adjointequation
dqt rqtdt
minus dpt μtpt + σtwt + 1113936infin
i1ki
t( 1113857τgi
t1113890 1113891dt minus wtdBt minus 1113936infin
i1ki
tdHit
q0 1 pT qT t isin [0 T]
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
(73)
According to the maximum principle (47) we have
plowastt minus Le
minus rtClowastt( 1113857
minus Rqlowastt (74)
and the optimal consumption rate
Clowastt minus
ert
L
plowasttqlowastt
1113888 1113889
minus 1R
(75)
By solving FBSDE (73) we get qlowastt ert plowastt erteμ(Tminus t)and wlowastt klowastt 0 en we get the optimal consumptionrate of the investor which is
Clowastt minus
erT
Leμ(Tminus t)
1113888 1113889
minus 1R
t isin [0 T] (76)
6 Conclusions
In this paper a nonconvex control domain case of theforward-backward stochastic control driven by Levy processis considered and we obtain the global stochastic maximumprinciple for this stochastic control problem And then theproblem of stochastic control with initial and final stateconstraints on the state variables is discussed and a nec-essary condition about existence of the optimal control isalso acquired A financial example of optimal consumptionis discussed to illustrate the application of the stochasticmaximum principle
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the NNSF (11701335) Shan-dong NSF (ZR2019BG015) NSSF (17BGL058) ShandongProvincial Higher Education Science and Technology PlanProject (J18KA236) and Shandong Womenrsquos College High-level Research Project Cultivation Foundation(2018GSPGJ07)
References
[1] L S Pontryakin V G Boltyanskti R V Gamkrelidze andE F Mischenko9eMathematical9eory of Optimal ControlProcesses John Wiley New York NY USA 1962
[2] S Peng ldquoA general stochastic maximum principle for optimalcontrol problemsrdquo SIAM Journal on Control and Optimiza-tion vol 28 no 4 pp 966ndash979 1990
10 Mathematical Problems in Engineering
[3] S G Peng ldquoBackward stochastic differrntial equations andapplication to optimal controlrdquo Applied Mathematics ampOptimization vol 27 no 4 pp 125ndash144 1993
[4] W S Xu ldquoStochastic maximum principle for optimal controlproblem of forward and backward systemrdquo Journal of Aus-tralian Mathematical Sciences vol 37 pp 249ndash259 1998B
[5] Z Wu ldquoMaximum principle for optimal control problem offully coupled forward-backward stochastic control systemrdquoSystem Science and Mathematical Sciences vol 11 no 3pp 249ndash259 1998
[6] J T Shi and ZWu ldquoemaximum principle for fully coupledforward-backward stochastic control systemrdquo Acta Auto-matica Sinica vol 32 no 2 pp 161ndash169 2006
[7] R Situ ldquoA maximum principle for optimal controls of sto-chastic systems with random jumpsrdquo in Proceedings of theNational Conference on Control 9eory and its ApplicationsQingdao China 1991
[8] J Shi and Z Wu ldquoMaximum principle for forward-backwardstochastic control system with random jumps and applica-tions to financerdquo Journal of Systems Science and Complexityvol 23 no 2 pp 219ndash231 2010
[9] J T Shi ldquoGlobal maximum principle for forward-backwardstochastic control system with Poisson jumpsrdquo Asian Journalof Control vol 16 no 1 pp 1ndash11 2014
[10] M Liu X Wang and H Huang ldquoMaximum principle forforward-backward control system driven by Ito-Levy pro-cesses under initial-terminal constraintsrdquo MathematicalProblems in Engineering vol 2017 pp 1ndash13 2017
[11] S Tang ldquoe maximum principle for partially observedoptimal control of stochastic differential equationsrdquo SIAMJournal on Control and Optimization vol 36 no 5pp 1596ndash1617 1998
[12] Q Meng ldquoA maximum principle for optimal control problemof fully coupled forward-backward stochastic systems withpartial informationrdquo Science in China Series A Mathematicsvol 52 no 7 pp 1579ndash1588 2009
[13] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward backward stichastic differential equationswith jumpsrdquo SIAM J Control Optimenleadertwodots vol 48pp 2845ndash2976 2009
[14] G Wang and Z Wu ldquoKalman-Bucy filtering equations offorward and backward stochastic systems and applications torecursive optimal control problemsrdquo Journal of MathematicalAnalysis and Applications vol 342 no 2 pp 1280ndash12962008
[15] H Xiao ldquoe maximum principle for partially observedoptimal control of forward-backward stochastic systems withrandom jumpsrdquo Journal of Systems Science and Complexityvol 24 no 6 pp 1083ndash1099 2011
[16] G Wang Z Wu and J Xiong ldquoMaximum principles forforward-backward stochastic control systems with correlatedstate and observation noisesrdquo SIAM Journal on Control andOptimization vol 51 no 1 pp 491ndash524 2013
[17] G Wang C Zhang and W Zhang ldquoStochastic maximumprinciple for mean-field type optimal control under partialinformationrdquo IEEE Transactions on Automatic Controlvol 59 no 2 pp 522ndash528 2014
[18] S Q Zhang J Xiong and X D Liu ldquoStochastic maximumprinciple for partially observed forward backward stochasticdifferential equations with jumps and regime switchingrdquoScience China Information Sciences vol 61 no 7 pp 1ndash132018
[19] J Xiong S Q Zhang S Zhang and Y Zhuang ldquoA partiallyobserved non-zero sum differential game of forward-
backward stochastic differential equations and its applicationin financerdquo Mathematical Control amp Related Fields vol 9no 2 pp 257ndash276 2019
[20] D Nualart and W Schoutens ldquoChaotic and predictablerepresentations for Levy processesrdquo Stochastic Processes and9eir Applications vol 90 no 1 pp 109ndash122 2000
[21] D Nualart and W Schoutens ldquoBackward stochastic differ-ential equations and Feynman-Kac formula for Levy pro-cesses with applications in financerdquo Bernoulli vol 7 no 5pp 761ndash776 2001
[22] Q Meng and M Tang ldquoNecessary and sufficient conditionsfor optimal control of stochastic systems associated with Levyprocessesrdquo Science in China Series F Information Sciencesvol 52 no 11 pp 1982ndash1992 2009
[23] M Tang and Q Zhang ldquoOptimal variational principle forbackward stochastic control systems associated with Levyprocessesrdquo Science China Mathematics vol 55 no 4pp 745ndash761 2012
[24] F Baghery N Khelfallah B Mezerdi and I Turpin ldquoFullycoupled forward backward stochastic differential equationsdriven by Levy processes and application to differentialgamesrdquo De Gruyter vol 22 no 3 pp 151ndash161 2014
[25] F Zhang M N Tang and Q X Meng ldquoSochastic maximumprinciple for forward-backward stochastic control systemsassociated with Levy processesrdquo Chinese Annals of Mathe-matics vol 35A no 1 pp 83ndash100 2014
[26] X R Wang and H Huang ldquoMaximum principle for forward-backward stochastic control system driven by Levy processrdquoMathematical Problems in Engineering vol 2015 Article ID702802 12 pages 2015
[27] H Huang X Wang and M Liu ldquoA maximum principle forfully coupled forward-backward stochastic control systemdriven by Levy process with terminal state constraintsrdquoJournal of Systems Science and Complexity vol 31 no 4pp 859ndash874 2018
[28] P Muthukumar and R Deepa ldquoInfinite horizon optimalcontrol of forward backward stochastic system driven byTeugels martingales with Levy processrdquo Stochastic and Dy-namics vol 3 no 17 pp 1ndash17 2017
[29] J B Wu and Z M Liu ldquoOptimal control of mean-fieldbackward doubly stochastic systems driven by Ito-Levyprocessesrdquo International Journal of Control vol 93 no 4pp 953ndash970 2018
[30] W Wang J Wu and Z Liu ldquoe optimal control of fully-coupled forward-backward doubly stochastic systems drivenby Ito-Levy processesrdquo Journal of Systems Science andComplexity vol 32 no 4 pp 997ndash1018 2019
Mathematical Problems in Engineering 11
[3] S G Peng ldquoBackward stochastic differrntial equations andapplication to optimal controlrdquo Applied Mathematics ampOptimization vol 27 no 4 pp 125ndash144 1993
[4] W S Xu ldquoStochastic maximum principle for optimal controlproblem of forward and backward systemrdquo Journal of Aus-tralian Mathematical Sciences vol 37 pp 249ndash259 1998B
[5] Z Wu ldquoMaximum principle for optimal control problem offully coupled forward-backward stochastic control systemrdquoSystem Science and Mathematical Sciences vol 11 no 3pp 249ndash259 1998
[6] J T Shi and ZWu ldquoemaximum principle for fully coupledforward-backward stochastic control systemrdquo Acta Auto-matica Sinica vol 32 no 2 pp 161ndash169 2006
[7] R Situ ldquoA maximum principle for optimal controls of sto-chastic systems with random jumpsrdquo in Proceedings of theNational Conference on Control 9eory and its ApplicationsQingdao China 1991
[8] J Shi and Z Wu ldquoMaximum principle for forward-backwardstochastic control system with random jumps and applica-tions to financerdquo Journal of Systems Science and Complexityvol 23 no 2 pp 219ndash231 2010
[9] J T Shi ldquoGlobal maximum principle for forward-backwardstochastic control system with Poisson jumpsrdquo Asian Journalof Control vol 16 no 1 pp 1ndash11 2014
[10] M Liu X Wang and H Huang ldquoMaximum principle forforward-backward control system driven by Ito-Levy pro-cesses under initial-terminal constraintsrdquo MathematicalProblems in Engineering vol 2017 pp 1ndash13 2017
[11] S Tang ldquoe maximum principle for partially observedoptimal control of stochastic differential equationsrdquo SIAMJournal on Control and Optimization vol 36 no 5pp 1596ndash1617 1998
[12] Q Meng ldquoA maximum principle for optimal control problemof fully coupled forward-backward stochastic systems withpartial informationrdquo Science in China Series A Mathematicsvol 52 no 7 pp 1579ndash1588 2009
[13] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward backward stichastic differential equationswith jumpsrdquo SIAM J Control Optimenleadertwodots vol 48pp 2845ndash2976 2009
[14] G Wang and Z Wu ldquoKalman-Bucy filtering equations offorward and backward stochastic systems and applications torecursive optimal control problemsrdquo Journal of MathematicalAnalysis and Applications vol 342 no 2 pp 1280ndash12962008
[15] H Xiao ldquoe maximum principle for partially observedoptimal control of forward-backward stochastic systems withrandom jumpsrdquo Journal of Systems Science and Complexityvol 24 no 6 pp 1083ndash1099 2011
[16] G Wang Z Wu and J Xiong ldquoMaximum principles forforward-backward stochastic control systems with correlatedstate and observation noisesrdquo SIAM Journal on Control andOptimization vol 51 no 1 pp 491ndash524 2013
[17] G Wang C Zhang and W Zhang ldquoStochastic maximumprinciple for mean-field type optimal control under partialinformationrdquo IEEE Transactions on Automatic Controlvol 59 no 2 pp 522ndash528 2014
[18] S Q Zhang J Xiong and X D Liu ldquoStochastic maximumprinciple for partially observed forward backward stochasticdifferential equations with jumps and regime switchingrdquoScience China Information Sciences vol 61 no 7 pp 1ndash132018
[19] J Xiong S Q Zhang S Zhang and Y Zhuang ldquoA partiallyobserved non-zero sum differential game of forward-
backward stochastic differential equations and its applicationin financerdquo Mathematical Control amp Related Fields vol 9no 2 pp 257ndash276 2019
[20] D Nualart and W Schoutens ldquoChaotic and predictablerepresentations for Levy processesrdquo Stochastic Processes and9eir Applications vol 90 no 1 pp 109ndash122 2000
[21] D Nualart and W Schoutens ldquoBackward stochastic differ-ential equations and Feynman-Kac formula for Levy pro-cesses with applications in financerdquo Bernoulli vol 7 no 5pp 761ndash776 2001
[22] Q Meng and M Tang ldquoNecessary and sufficient conditionsfor optimal control of stochastic systems associated with Levyprocessesrdquo Science in China Series F Information Sciencesvol 52 no 11 pp 1982ndash1992 2009
[23] M Tang and Q Zhang ldquoOptimal variational principle forbackward stochastic control systems associated with Levyprocessesrdquo Science China Mathematics vol 55 no 4pp 745ndash761 2012
[24] F Baghery N Khelfallah B Mezerdi and I Turpin ldquoFullycoupled forward backward stochastic differential equationsdriven by Levy processes and application to differentialgamesrdquo De Gruyter vol 22 no 3 pp 151ndash161 2014
[25] F Zhang M N Tang and Q X Meng ldquoSochastic maximumprinciple for forward-backward stochastic control systemsassociated with Levy processesrdquo Chinese Annals of Mathe-matics vol 35A no 1 pp 83ndash100 2014
[26] X R Wang and H Huang ldquoMaximum principle for forward-backward stochastic control system driven by Levy processrdquoMathematical Problems in Engineering vol 2015 Article ID702802 12 pages 2015
[27] H Huang X Wang and M Liu ldquoA maximum principle forfully coupled forward-backward stochastic control systemdriven by Levy process with terminal state constraintsrdquoJournal of Systems Science and Complexity vol 31 no 4pp 859ndash874 2018
[28] P Muthukumar and R Deepa ldquoInfinite horizon optimalcontrol of forward backward stochastic system driven byTeugels martingales with Levy processrdquo Stochastic and Dy-namics vol 3 no 17 pp 1ndash17 2017
[29] J B Wu and Z M Liu ldquoOptimal control of mean-fieldbackward doubly stochastic systems driven by Ito-Levyprocessesrdquo International Journal of Control vol 93 no 4pp 953ndash970 2018
[30] W Wang J Wu and Z Liu ldquoe optimal control of fully-coupled forward-backward doubly stochastic systems drivenby Ito-Levy processesrdquo Journal of Systems Science andComplexity vol 32 no 4 pp 997ndash1018 2019
Mathematical Problems in Engineering 11