Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 761306 7 pageshttpdxdoiorg1011552013761306

Research ArticleNonzero-Sum Stochastic Differential Game betweenController and Stopper for Jump Diffusions

Yan Wang12 Aimin Song2 Cheng-De Zheng2 and Enmin Feng1

1 School of Mathematical Sciences Dalian University of Technology Dalian 116023 China2 School of Science Dalian Jiaotong University Dalian 116028 China

Correspondence should be addressed to Yan Wang wymath163com

Received 5 February 2013 Accepted 7 May 2013

Academic Editor Ryan Loxton

Copyright copy 2013 Yan Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We consider a nonzero-sum stochastic differential game which involves two players a controller and a stopper The controllerchooses a control process and the stopper selects the stopping rule which halts the gameThis game is studied in a jump diffusionssetting within Markov control limit By a dynamic programming approach we give a verification theorem in terms of variationalinequality-Hamilton-Jacobi-Bellman (VIHJB) equations for the solutions of the game Furthermore we apply the verificationtheorem to characterize Nash equilibrium of the game in a specific example

1 Introduction

In this paper we study a nonzero-sum stochastic differentialgame with two players a controller and a stopper The state119883(sdot) in this game evolves according to a stochastic differentialequation driven by jump diffusionsThe controller affects thecontrol process 119906(sdot) in the drift and volatility of119883(sdot) at time 119905and the stopper decides the duration of the game in the formof a stopping rule 120591 for the process119883(sdot) The objectives of thetwo players are to maximize their own expected payoff

In order to illustrate the motivation and the backgroundof application for this game we show a model in finance

Example 1 Let (ΩF F119905119905ge0 119875) be a filtered probability

space let 119861(119905) be a 119896-dimensional Brownian Motion and let(119889119905 119889119911) = (

1(119889119905 119889119911)

119896(119889119905 119889119911)) be 119896-independent

compensated Poisson randommeasures independent of 119861(119905)119905 isin [0infin) For 119894 = 1 119896

119894(119889119905 119889119911) = 119873

119894(119889119905 119889119911)minus]

119894(119889119911)119889119905

where ]119894is the Levy measure of a Levy process 120578

119894(119905) with

jump measure 119873119894such that 119864[1205782

119894(119905)] lt infin for all 119905 F

119905119905ge0

is the filtration generated by 119861(119905) and (119889119905 119889119911) (as usualaugmentedwith all the119875-null sets)We refer to [1 2] formoreinformation about Levy processes

We firstly define a financial market model as followsSuppose that there are two investment possibilities

(1) a risk-free asset (eg a bond) with unit price 1198780(119905) at

time 119905 given by1198891198780(119905) = 119903 (119905) 119878

0(119905) 119889119905 119878

0(0) = 1 (1)

(2) a risky asset (eg a stock) with unit price 1198781(119905) at time

119905 given by

1198891198781(119905)

= 1198781(119905minus) [119887 (119905) 119889119905 + 120590 (119905) 119889119861 (119905) + int

R

120574 (119905 119911) (119889119905 119889119911)]

1198781(0) gt 0

(2)

where 119903(119905) is F119905-adapted with int

119879

0|119903(119905)|119889119905 lt infin as 119879 gt 0

is a fixed given constant 119887 120590 120574 is F119905-predictable processes

satisfying 120574(119905 119911) gt minus1 for aa 119905 119911 as and

int

119879

0

|119887 (119905)| + 1205902(119905) + int

R

1205742(119905 119911) ] (119889119911) 119889119905 lt infin forall119879 lt infin

(3)

2 Abstract and Applied Analysis

Assume that an investor hires a portfolio manager tomanage his wealth 119883(119905) from investments The manager(controller) can choose a portfolio 119906(119905) which represents theproportion of the total wealth 119883(119905) invested in the stock attime 119905 And the investor (stopper) can halt the wealth process119883(119905) by selecting a stop-rule 120591 119862[0 119879] rarr [0 119879] Then thedynamics of the corresponding wealth process 119883(119905) = 119883

119906(119905)

is

119889119883119906(119905) = 119883

119906(119905minus) [(1 minus 119906 (119905)) 119903 (119905) + 119906 (119905) 119887 (119905)] 119889119905

+ 119906 (119905) 120590 (119905) 119889119861 (119905)

+119906 (119905minus) intR

120574 (119905 119911) (119889119905 119889119911)

119883119906(0) = 119909 gt 0

(4)

(see eg [3ndash5])We require that 119906(119905minus)120574(119905 119911) gt minus1 for aa 119905 119911as and that

int

119879

0

|(1 minus 119906 (119905)) 119903 (119905)| + |119906 (119905) 119887 (119905)| + 1199062(119905) 1205902(119905)

+1199062(119905) int

R

1205742(119905 119911) ] (119889119911) 119889119905 lt infin as

(5)

At terminal time 120591 the stopper gives the controller apayoff 119862(119883(120591)) where 119862 119862[0 119879] rarr R is a deterministicmapping Therefore the controller aims to maximize hisutility of the following form

J119909

1(119906 120591)=119864

119909[119890minus120575120591

1198801(119862 (119883 (120591)))minusint

120591

0

119890minus120575119905ℎ (119905 119883 (119905) 119906

119905) 119889119905]

(6)

where 120575 gt 0 is the discounting rate ℎ is a cost function and1198801is the controllerrsquos utility We denote 119864119909 the expectation

with respect to119875119909 and119875119909 the probability laws of119883(119905) startingat 119909

Meanwhile it is the stopperrsquos objective to choose thestopping time 120591 such that his own utility

J119909

2(119906 120591) = 119864

119909[119890minus120575120591

1198802119883 (120591) minus 119862 (119883 (120591))] (7)

is maximized where 1198802is the stopperrsquos utility

As this game is typically a nonzero sum we seek a Nashequilibrium namely a pair (119906lowast 120591lowast) such that

J119909

1(119906lowast 120591lowast) ge J

119909

1(119906 120591lowast) forall119906

J119909

2(119906lowast 120591lowast) ge J

119909

2(119906lowast 120591) forall120591

(8)

This means that the choice 119906lowast is optimal for the controllerwhen the stopper uses 120591lowast and vice verse

The game (4) and (8) are a nonzero-sum stochasticdifferential game between a controller and a stopper Theexistence of Nash equilibrium shows that by an appropriatestopping rule design 120591lowast the stopper can induce the controllerto choose the best portfolio he can Similarly by applying asuitable portfolio 119906lowast the controller can force the stopper tostop the employment relationship at a time of the controllerrsquoschoosing

There have been significant advances in the researchof stochastic differential games of control and stoppingFor example in [6ndash9] the authors considered the zero-sumstochastic differential games ofmixed type with both controlsand stopping between two players In these games each of theplayers chooses an optimal strategy which is composed by acontrol 119906(sdot) and a stopping 120591 Under appropriate conditionsthey constructed a saddle pair of optimal strategies For thenonsum case the games of mixed type were discussed in[6 10] The authors presented Nash equilibria rather thansaddle pairs of strategies [10] Moreover the papers [11ndash17]considered a zero-sum stochastic differential game betweencontroller and stopper where one player (controller) choosesa control process 119906(sdot) and the other (stopper) chooses astopping 120591 One player tries to maximize the reward and theother to minimize it They presented a saddle pair for thegame

In this paper we study a nonzero-sum stochastic differen-tial game between a controller and a stopper The controllerand the stopper have different payoffsThe objectives of themare tomaximize their own payoffsThis game is considered ina jumpdiffusion context under theMarkov control conditionWe prove a verification theorem in terms of VIHJB equationsfor the game to characterize Nash equilibrium

Our setup and approach are related to [3 17] Howevertheir games are different fromours In [3] the authors studiedthe stochastic differential games between two controllersThe work in [17] was carried out for a zero-sum stochasticdifferential game between a controller and a stopper

The paper is organized as follows in the next sectionwe formulate the nonzero-sum stochastic differential gamebetween a controller and a stopper and prove a generalverification theorem In Section 3 we apply the generalresults obtained in Section 2 to characterize the solutions of aspecial game Finally we conclude this paper in Section 4

2 A Verification Theorem forNonzero-Sum Stochastic Differential Gamebetween Controller and Stopper

Suppose the state 119884(119905) = 119884119906(119905) at time 119905 is given by the

following stochastic differential equation

119889119884 (119905) = 120572 (119884 (119905) 1199060(119905)) 119889119905 + 120573 (119884 (119905) 119906

0(119905)) 119889119861 (119905)

+ intR1198960

120579 (119884 (119905minus) 1199061(119905minus 119911) 119911) (119889119905 119889119911)

119884 (0) = 119910 isin R119896

(9)

where 120572 R119896 times U rarr R119896 120573 R119896 times U rarr R119896times119896 120579 R119896 times

U times R119896 rarr R119896times119896 are given functions and U denotes a givensubset of R119901

We regard 1199060(119905) = 119906

0(119905 120596) and 119906

1(119905 119911) = 119906

1(119905 119911 120596)

as the control processes assumed to be cadlag F119905-adapted

and with values in 1199060(119905) isin U 119906

1(119905 119911) isin U for aa 119905 119911 120596

And we put 119906(119905) = (1199060(119905) 1199061(119905 119911)) Then 119884(119905) = 119884

(119906)(119905) is

a controlled jump diffusion (see [18] for more informationabout stochastic control of jump diffusion)

Abstract and Applied Analysis 3

Fix an open solvency set S sub R119896 Let

120591S = inf 119905 gt 0 119884 (119905) notin S (10)

be the bankruptcy time 120591S is the first time at which thestochastic process 119884(119905) exits the solvency set S Similaroptimal control problems in which the terminal time isgoverned by a stopping criterion are considered in [19ndash21] inthe deterministic case

Let 119891119894 R119896 times 119870 rarr R and 119892

119894 R119896 rarr R be given

functions for 119894 = 1 2 Let A be a family of admissiblecontrols contained in the set of 119906(sdot) such that (9) has a uniquestrong solution and

119864119910[int

120591S

0

1003816100381610038161003816119891119894 (119884119905 119906119905)1003816100381610038161003816 119889119905] lt infin 119894 = 1 2 (11)

for all 119910 isin S where 119864119910 denotes expectation given that119884(0) =119910 isin R119896 Denote by Γ the set of all stopping times 120591 le 120591SMoreover we assume that

the family 119892minus

119894(119884120591) 120591 isin Γ is uniformly integrable

for 119894 = 1 2

(12)

Then for 119906 isin A and 120591 isin Γ we define the performancefunctionals as follows

J119910

119894(119906 120591) = 119864

119910[int

120591

0

119891119894(119884 (119905) 119906 (119905)) 119889119905 + 119892

119894(119884 (120591))]

119894 = 1 2

(13)

We interpret 119892119894(119884(120591)) as 0 if 120591 = infin We may regardJ

119910

1(119906 120591)

as the payoff to the controller who controls 119906 andJ1199102(119906 120591) as

the payoff to the stopper who decides 120591

Definition 2 (nash equilibrium) A pair (119906lowast 120591lowast) isin A times Γ iscalled a Nash equilibrium for the stochastic differential game(9) and (13) if the following holds

J119910

1(119906lowast 120591lowast) ge J

119910

1(119906 120591lowast) forall119906 isin U 119910 isin S (14)

J119910

2(119906lowast 120591lowast) ge J

119910

2(119906lowast 120591) forall120591 isin Γ 119910 isin S (15)

Condition (14) states that if the stopper chooses 120591lowast it isoptimal for the controller to use the control 119906lowast Similarlycondition (15) states that if the controller uses 119906lowast it is optimalfor the stopper to decide 120591lowast Thus (119906lowast 120591lowast) is an equilibriumpoint in the sense that there is no reason for each individualplayer to deviate from it as long as the other player does not

We restrict ourselves to Markov controls that is weassume that 119906

0(119905) =

0(119884(119905)) 119906

1(119905 119911) =

1(119884(119905) 119911) As

customary we do not distinguish between 1199060and

0 1199061and

1 Then the controls 119906

0and 119906

1can simply be identified with

functions 0(119910) and

1(119910 119911) where 119910 isin S and 119911 isin R119896

When the control 119906 is Markovian the correspondingprocess 119884(119906)(119905) becomes a Markov process with the generator119860119906 of 120601 isin 1198622(R119896) given by

119860120601 (119910) =

119896

sum

119894=1

120572119894(119910 1199060(119910))

120597120601

120597119910119894

(119910)

+1

2

119896

sum

119894119895=1

(120573120573119879)119894119895(119910 1199060(119910))

1205972120601

120597119910119894120597119910119895

(119910)

+

119896

sum

119895=1

intR

120601 (119910 + 120579119895(119910 1199061(119910 119911) 119911)) minus 120601 (119910)

minusnabla120601 (119910) 120579119895(119910 1199061(119910 119911) 119911) ] (119889119911)

(16)

where nabla120601 = (1205971206011205971199101 120597120601120597119910

119896) is the gradient of 120601 and 120579119895

is the 119895th column of the 119896 times 119896matrix 120579Now we can state the main result of this section

Theorem 3 (verification theorem for game (9) and (13))Suppose there exist two functions 120601

119894 S rarr R 119894 = 1 2 such

that

(i) 120601119894isin 1198621(S0)⋂119862(S) 119894 = 1 2 whereS is the closure of

S and S0 is the interior of S(ii) 120601119894ge 119892119894on S 119894 = 1 2

Define the following continuation regions

119863119894= 119910 isin S 120601

119894(119910) gt 119892

119894(119910) 119894 = 1 2 (17)

Suppose 119884(119905) = 119884119906(119905) spends 0 time on 120597119863

119894as 119894 =

1 2 that is(iii) 119864119910[int120591119878

0120594120597119863119894(119884(119905))119889119905] = 0 for all 119910 isin S 119906 isin U 119894 = 1 2

(iv) 120597119863119894is a Lipschitz surface 119894 = 1 2

(v) 120601119894isin 1198622(S120597119863

119894)The second-order derivatives of 120601

119894are

locally bounded near 120597119863119894 respectively 119894 = 1 2

(vi) 1198631= 1198632= 119863

(vii) there exists isin A such that for 119894 = 1 2

119860120601119894(119910) + 119891

119894(119910 (119910))

= sup119906isinA

119860119906120601119894(119910) + 119891

119894(119910 119906 (119910))

= 0 119910 isin 119863

le 0 119910 isin S 119863

(18)

(viii) 119864119910[|120601119894(119884120591)| + int120591

0|119860119906120601119894(119884119906(119905))|119889119905] lt infin 119894 = 1 2 for all

119906 isin U 120591 isin ΓFor 119906 isin A define

120591119863= 120591119906

119863= inf 119905 gt 0 119884

119906(119905) notin 119863 lt infin (19)

and in particular

120591119863= 120591

119863= inf 119905 gt 0 119884

(119905) notin 119863 lt infin (20)

4 Abstract and Applied Analysis

Suppose that(ix) the family 120601

119894(119884(120591)) 120591 isin Γ 120591 le 120591

119863 is uniformly

integrable for all 119906 isin A and 119910 isin S 119894 = 1 2

Then (120591119863 ) isin Γ timesA is a Nash equilibrium for game (9) (13)

and

1206011(119910) = sup

119906isinA

J119906120591119863

1(119910) = J

120591119863

1(119910) (21)

1206012(119910) = sup

120591isinΓ

J120591

2(119910) = J

120591119863

2(119910) (22)

Proof From (i) (iv) and (v) we may assume by an approx-imation theorem (see Theorem 21 in [18]) that 120601

119894isin 1198622(S)

119894 = 1 2For a given 119906 isin A we define with 119884(119905) = 119884

119906(119905)

120591119863= 120591119906

119863= inf 119905 gt 0 119884

119906(119905) notin 119863 (23)

In particular let be as in (vii) Then

120591119863= 120591

119863= inf 119905 gt 0 119884

(119905) notin 119863 (24)

We first prove that (21) holds Let 120591119863isin Γ be as in (24) For

arbitrary 119906 isin A by (vii) and the Dynkinrsquos formula for jumpdiffusion (see Theorem 124 in [18]) we have

1206011(119910) = 119864

119910[int

120591119863and119898

0

minus1198601199061206011(119884 (119905)) 119889119905 + 120601

1(119884 (120591119863and 119898))]

ge 119864119910[int

120591119863and119898

0

1198911(119884 (119905) 119906 (119905)) 119889119905 + 120601

1(119884 (120591119863and 119898))]

(25)

where119898 = 1 2 Therefore by (11) (12) (i) (ii) (viii) andthe Fatou lemma

1206011(119910)

ge lim inf119898rarrinfin

119864119910[int

120591119863and119898

0

1198911(119884 (119905) 119906 (119905)) 119889119905+120601

1(119884 (120591119863and 119898))]

ge 119864119910[int

120591119863

0

1198911(119884 (119905) 119906 (119905)) 119889119905 + 119892

1(119884 (120591119863))]

= J119910

1(119906 120591119863)

(26)

Since this holds for all 119906 isin A we have

1206011(119910) ge sup

119906isinA

J119910

1(119906 120591119863) (27)

In particular applying the Dynkinrsquos formula to 119906 = we getan equality that is

1206011(119910) = 119864

119910[int

120591119863and119898

0

minus1198601206011( (119905)) 119889119905 + 120601

1( (120591119863and 119898))]

= 119864119910[int

120591119863and119898

0

1198911( (119905) (119905)) 119889119905 + 120601

1( (120591119863and 119898))]

(28)

where (119905) = 119884(119905) and119898 = 1 2 Hence we deduce that

1206011(119910) = J

119910

1( 120591119863) (29)

Since we always have

J119910

1( 120591119863) le sup119906isinA

J119910

1(119906 120591119863) (30)

we conclude by combining (27) (29) and (30) that

1206011(119910) = J

119910

1( 120591119863) = sup119906isinA

J119910

1(119906 120591119863) (31)

which is (21)Next we prove that (22) holds Let isin A be as in (vii)

For 120591 isin Γ by the Dynkinrsquos formula and (vii) we have

119864119910[1206012( (120591119898))] = 120601

2(119910) + 119864

119910[int

120591119898

0

1198601206012( (119905)) 119889119905]

le 1206012(119910) minus 119864

119910[int

120591119898

0

1198912( (119905) (119905)) 119889119905]

(32)

where 120591119898= 120591 and 119898119898 = 1 2

Letting119898 rarr infin gives by (11) (12) (i) (ii) (viii) and theFatou Lemma

1206012(119910) ge lim inf

119898rarrinfin119864119910[int

120591and119898

0

1198912( (119905) (119905)) 119889119905 + 120601

2( (120591119898))]

ge 119864119910[int

120591

0

1198912( (119905) (119905)) 119889119905 + 119892

2( (120591)) 120594

120591ltinfin]

= J119910

2( 120591)

(33)

The inequality (33) holds for all 120591 isin Γ Then we have

1206012(119910) ge sup

120591isinΓ

J119910

2( 120591) (34)

Similarly applying the above argument to the pair ( 120591119863) we

get an equality in (34) that is

1206012(119910) = J

119910

2( 120591119863) (35)

We always have

J119910

2( 120591119863) le sup120591isinΓ

J119910

2( 120591) (36)

Therefore combining (34) (35) and (36) we get

1206012(119910) = J

119910

2( 120591119863) = sup120591isinΓ

J119910

2( 120591) (37)

which is (22) The proof is completed

Abstract and Applied Analysis 5

3 An Example

In this section we come back to Example 1 and useTheorem 3to study the solutions of game (4) and (8) Here and in thefollowing all the processes are assumed to be one-dimensionfor simplicity

To put game (4) and (8) into the framework of Section 2we define the process 119884(119905) = (119884

0(119905) 1198841(119905)) 119884(0) = 119910 = (119904 119910

1)

by

1198891198840(119905) = 119889119905 119884

0(0) = 119904 isin R

1198891198841(119905) = 119889119883 (119905)

= 1198841(119905) [(1 minus 119906 (119905)) 119903 (119905) + 119906 (119905) 119887 (119905)] 119889119905

+ 119906 (119905) 120590 (119905) 119889119861 (119905)

+119906 (119905minus) intR

120574 (119905 119911) (119889119905 119889119911)

1198841(0) = 119909 = 119910

1

(38)

Then the performance functionals to the controller (6) andthe stopper (7) can be formulated as follows

J119910

1(119906 120591) = 119864

1199041199101 [119890minus120575(119904+120591)

1198801(119862 (1198841(120591)))

minusint

120591

0

119890minus120575(119904+119905)

ℎ (1198840(119905) 1198841(119905) 119906 (119905)) 119889119905]

J119910

2(119906 120591) = 119864

1199041199101 [119890minus120575(119904+120591)

1198802(1198841(120591) minus 119862 (119884

1(120591)))]

(39)

In this case the generator 119860119906 in (16) has the form

119860119906120601 (119904 119909)

=120597120601

120597119904+ [(1 minus 119906) 119903119909 + 119906119887119909]

120597120601

120597119909+1

2119909211990621205902 1205972120601

1205971199092

+ intR0

120601 (119904 119909 + 119909119906120574 (119911)) minus 120601 (119904 119909) minus 119909119906120574 (119911)120597120601

120597119909

times ] (119889119911)

(40)

To obtain a possible Nash equilibrium ( 120591) isin A times Γ forgame (4) and (8) according to Theorem 3 it is necessary tofind a subset 119863 of S = R2

+= [0infin)

2 and 120601119894(119904 119909) 119894 = 1 2

such that

(i) 1206011(119904 119909) = 119890

minus1205751199041198801(119862(119909)) and 120601

2(119904 119909) = 119890

minus1205751199041198802(119909 minus

119862(119909)) for all (119904 119909) isin 119863(ii) 1206011(119904 119909) ge 119890

minus1205751199041198801(119862(119909)) and 120601

2(119904 119909) ge 119890

minus1205751199041198802(119909 minus

119862(119909)) for all (119904 119909) isin S(iii) 119860119906120601

1(119904 119909) minus ℎ(119904 119909 119906) le 0 and 119860119906120601

2(119904 119909) le 0 for all

(119904 119909) isin S 119863 and 119906(iv) there exists such that 119860120601

1(119904 119909) minus ℎ(119904 119909 ) =

1198601206012(119904 119909) = 0 for all (119904 119909) isin 119863

Imposing the first-order condition on 1198601199061206011(119904 119909) minus

ℎ(119904 119909 119906) and 1198601199061206012(119904 119909) we get the following equations for

the optimal control processes

(119887119909 minus 119903119909)1205971206011

120597119909(119904 119909) + 119909

2120590212059721206011

1205971199092(119904 119909)

+ intR

119903120574 (119911) [1205971206011

120597119909(119904 119909 + 119903120574 (119911)) minus

1205971206011

120597119909(119904 119909)]

minus120597ℎ

120597119906(119904 119909 ) = 0

(119887119909 minus 119903119909)1205971206012

120597119909(119904 119909) + 119909

2120590212059721206012

1205971199092(119904 119909)

+ intR

119903120574 (119911) [1205971206012

120597119909(119904 119909 + 119903120574 (119911)) minus

1205971206012

120597119909(119904 119909)] = 0

(41)

With as in (41) we put

119860120601119894(119904 119909)

=120597120601119894

120597119904+ [(1 minus ) 119903119909 + 119887119909]

120597120601119894

120597119909+1

2119909221205902 1205972120601119894

1205971199092

+ intR0

120601119894(119904 119909 + 119909120574 (119911)) minus 120601

119894(119904 119909) minus 119909120574 (119911)

120597120601119894

120597119909

times ] (119889119911)

(42)

Thus we may reduce game (4) and (8) to the problem ofsolving a family of nonlinear variational-integro inequalitiesWe summarize as follows

Theorem 4 Suppose there exist satisfying (41) and two 1198621-functions 120601

119894 119894 = 1 2 such that

(1)

119863 = (119904 119909) 1206012(119904 119909) gt 119890

minus1205751199041198802(119909 minus 119862 (119909))

= (119904 119909) 1206011(119904 119909) gt 119890

minus1205751199041198801(119862 (119909))

(43)

(2) 120601119894isin 1198622(119863) 119894 = 1 2

(3) 1206012(119904 119909) = 119890

minus1205751199041198802(119909 minus 119862(119909)) and 120601

1(119904 119909) =

119890minus1205751199041198801(119862(119909)) for all (119904 119909) isin S 119863

(4) 1198601199061206011(119904 119909) minus ℎ(119904 119909 119906) le 0 and 119860119906120601

2(119904 119909) le 0 for all

(119904 119909) isin S 119863 and for all 119906 isin A(5) 119860120601

1(119904 119909) minus ℎ(119904 119909 ) = 0 for all (119904 119909) isin 119863 where

1198601206011is given by (42)

(6) 1198601206012(119904 119909) = 0 for all (119904 119909) isin 119863 where 119860120601

2is given

by (42)

Then the pair ( 120591) is a Nash equilibrium of the stochasticdifferential game (4) and (8) where

120591 = inf 119905 gt 0 119884(119905) notin 119863 (44)

6 Abstract and Applied Analysis

Moreover the corresponding equilibrium performances are

1206011(119904 119909) = J

(119904119909)

1( 120591)

1206012(119904 119909) = J

(119904119909)

2( 120591)

(45)

In this paper we will not discuss general solutions of thisfamily of nonlinear variational-integro inequalities Insteadwe discuss a solution in special case when

120574 (119905 119911) = 0 ℎ (119904 119909 119906) =1199062

2 (46)

Let us try the functions 120601119894 119894 = 1 2 of the form

120601119894(119904 119909) = 119890

minus120575119904120595119894(119909) (47)

and a continuation region119863 of the form

119863 = (119904 119909) 119909 lt 1199090 for some 119909

0gt 0 (48)

Then we have

119860119906120601119894(119904 119909) = 119890

minus120575119904119860119906120595119894(119909) (49)

where119860119906120595119894(119909) = minus 120575120595

119894(119909) + [(1 minus 119906) 119903119909 + 119906119887119909] 120595

1015840

119894(119909)

+1

211990921199062120590212059510158401015840

119894(119909)

(50)

By conditions (1) and (3) in Theorem 4 we get

1205951(119909) = 119880

1(119862 (119909)) 119909 ge 119909

0

1205951(119909) gt 119880

1(119862 (119909)) 0 lt 119909 lt 119909

0

1205952(119909) = 119880

2(119909 minus 119862 (119909)) 119909 ge 119909

0

1205952(119909) gt 119880

2(119909 minus 119862 (119909)) 0 lt 119909 lt 119909

0

(51)

From conditions (4) (5) and (6) ofTheorem 4 we get thecandidate for the optimal control as follows

= Argmax119906isinA

1198601199061205951(119909) minus

1199062

2

= Argmax119906isinA

minus 1205751205951(119909) + [(1 minus 119906) 119903119909 + 119906119887119909] 120595

1015840

1(119909)

+1

211990921199062120590212059510158401015840

1(119909) minus

1199062

2

=(119887119909 minus 119903119909) 120595

1015840

1(119909)

1 minus 1199092120590212059510158401015840

1(119909)

(52)

= Argmax119906isinA

1198601199061205952(119909)

= Argmax119906isinA

minus 1205751205952(119909) + [(1 minus 119906) 119903119909 + 119906119887119909] 120595

1015840

2(119909)

+1

211990921199062120590212059510158401015840

2(119909)

=minus (119887119909 minus 119903119909) 120595

1015840

2(119909)

1199092120590212059510158401015840

2(119909)

(53)

Let120595119894(119909) =

119894(119909) on 0 lt 119909 lt 119909

0 119894 = 1 2 By condition (5)

in Theorem 4 we have 1198601(119909) minus

22 = 0 for 0 lt 119909 lt 119909

0

Substituting (52) into 1198601199061(119909) minus 119906

22 = 0 we obtain

minus 1205751(119909) + [119903119909 +

(119887119909 minus 119903119909)21015840

1(119909)

1 minus 1199092120590210158401015840

1(119909)

] 1015840

1(119909)

+

[(119887119909 minus 119903119909) 1015840

1(119909)]2

2(1 minus 1199092120590210158401015840

1(119909))2[1199092120590210158401015840

1(119909) minus 1] = 0

(54)

Similarly we obtain1198602(119909) = 0 for 0 lt 119909 lt 119909

0by condition

(6) in Theorem 4 And we substitute (53) in 1198601199062(119909) = 0 to

get

minus 1205752(119909) + 119903119909

1015840

2(119909) +

[(119887119909 minus 119903119909) 1015840

2(119909)]2

1 minus 1199092120590210158401015840

2(119909)

+

[119909120590 (119887119909 minus 119903119909) 1015840

2(119909)]2

10158401015840

2(119909)

2(1 minus 1199092120590210158401015840

2(119909))2

= 0

(55)

Therefore we conclude that

1205951(119909) =

1198801(119862 (119909)) 119909 ge 119909

0

1(119909) 0 lt 119909 lt 119909

0

(56)

1205952(119909) =

1198802(119909 minus 119862 (119909)) 119909 ge 119909

0

2(119909) 0 lt 119909 lt 119909

0

(57)

where 1(119909) and

2(119909) are the solutions of (56) and (57)

respectivelyAccording to Theorem 4 we use the continuity and

differentiability of 120595119894at 119909 = 119909

0to determine 119909

0 119894 = 1 2 that

is

1205951(1199090) = 1198801(119862 (1199090))

1205951015840

1(1199090) = 119880

1015840

1(119862 (1199090)) 1198621015840(1199090)

1205952(1199090) = 1198802(1199090minus 119862 (119909

0))

1205951015840

2(1199090) = 119880

1015840

2(1199090minus 119862 (119909

0)) (1 minus 119862

1015840(1199090))

(58)

At the end of this section we summarize the above resultsin the following theorem

Theorem 5 Let 120595119894 119894 = 1 2 and let 119909

0be the solutions of

equations (56)ndash(58) Then the pair ( 120591) given by

=(119887119909 minus 119903119909) 120595

1015840

1(119909)

1 minus 1199092120590212059510158401015840

1(119909)

=minus (119887119909 minus 119903119909) 120595

1015840

2(119909)

1199092120590212059510158401015840

2(119909)

120591 = inf 119905 gt 0 119909 ge 1199090

(59)

is a Nash equilibrium of game (4) and (8) The correspondingequilibrium performances are

120601119894(119904 119909) = 119890

minus120575119904120595119894(119909) 119894 = 1 2 (60)

Abstract and Applied Analysis 7

4 Conclusion

A verification theorem is obtained for the general stochasticdifferential game between a controller and a stopper Inthe special case of quadratic cost we use this theorem tocharacterize the Nash equilibrium However the question ofthe existence and uniqueness of Nash equilibrium for thegame remains open It will be considered in our subsequentwork

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China under Grant 61273022 and 11171050

References

[1] J Bertoin Levy Processes Cambridge University Press Cam-bridge UK 1996

[2] D Applebaum Levy Processes and Stochastic Calculus Cam-bridge University Press Cambridge UK 2003

[3] S Mataramvura and B Oslashksendal ldquoRisk minimizing portfoliosand HJBI equations for stochastic differential gamesrdquo Stochas-tics vol 80 no 4 pp 317ndash337 2008

[4] R J Elliott and T K Siu ldquoOn risk minimizing portfolios underaMarkovian regime-switching Black-Scholes economyrdquoAnnalsof Operations Research vol 176 pp 271ndash291 2010

[5] G Wang and Z Wu ldquoMean-variance hedging and forward-backward stochastic differential filtering equationsrdquo Abstractand Applied Analysis vol 2011 Article ID 310910 20 pages 2011

[6] I Karatzas and W Sudderth ldquoStochastic games of control andstopping for a linear diffusionrdquo in Random Walk SequentialAnalysis and Related Topics A Festschrift in Honor of YS ChowA Hsiung Z L Ying and C H Zhang Eds pp 100ndash117WorldScientific Publishers 2006

[7] S Hamadene ldquoMixed zero-sum stochastic differential gameand American game optionsrdquo SIAM Journal on Control andOptimization vol 45 no 2 pp 496ndash518 2006

[8] M K Ghosh M K S Rao and D Sheetal ldquoDifferential gamesof mixed type with control and stopping timesrdquo NonlinearDifferential Equations and Applications vol 16 no 2 pp 143ndash158 2009

[9] M K Ghosh and K S Mallikarjuna Rao ldquoExistence of value instochastic differential games of mixed typerdquo Stochastic Analysisand Applications vol 30 no 5 pp 895ndash905 2012

[10] I Karatzas and Q Li ldquoBSDE approach to non-zero-sumstochastic differential games of control and stoppingrdquo inStochastic Processes Finance and Control pp 105ndash153 WorldScientific Publishers 2012

[11] J-P Lepeltier ldquoOn a general zero-sum stochastic game withstopping strategy for one player and continuous strategy for theotherrdquo Probability and Mathematical Statistics vol 6 no 1 pp43ndash50 1985

[12] I Karatzas and W D Sudderth ldquoThe controller-and-stoppergame for a linear diffusionrdquo The Annals of Probability vol 29no 3 pp 1111ndash1127 2001

[13] AWeerasinghe ldquoA controller and a stopper gamewith degener-ate variance controlrdquo Electronic Communications in Probabilityvol 11 pp 89ndash99 2006

[14] I Karatzas and I-M Zamfirescu ldquoMartingale approach tostochastic differential games of control and stoppingrdquo TheAnnals of Probability vol 36 no 4 pp 1495ndash1527 2008

[15] E Bayraktar I Karatzas and S Yao ldquoOptimal stopping fordynamic convex risk measuresrdquo Illinois Journal of Mathematicsvol 54 no 3 pp 1025ndash1067 2010

[16] E Bayraktar and V R Young ldquoProving regularity of theminimal probability of ruin via a game of stopping and controlrdquoFinance and Stochastics vol 15 no 4 pp 785ndash818 2011

[17] F Bagherya S Haademb B Oslashksendal and I Turpina ldquoOptimalstopping and stochastic control differential games for jumpdiffusionsrdquo Stochastics vol 85 no 1 pp 85ndash97 2013

[18] B Oslashksendal and A Sulem Applied Stochastic Control of JumpDiffusions Springer Berlin Germany 2nd edition 2007

[19] Q Lin R Loxton K L Teo and Y H Wu ldquoA new computa-tional method for a class of free terminal time optimal controlproblemsrdquo Pacific Journal of Optimization vol 7 no 1 pp 63ndash81 2011

[20] Q Lin R Loxton K L Teo and Y H Wu ldquoOptimal controlcomputation for nonlinear systems with state-dependent stop-ping criteriardquo Automatica vol 48 no 9 pp 2116ndash2129 2012

[21] C Jiang Q Lin C Yu K L Teo and G-R Duan ldquoAn exactpenalty method for free terminal time optimal control problemwith continuous inequality constraintsrdquo Journal of OptimizationTheory and Applications vol 154 no 1 pp 30ndash53 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

Assume that an investor hires a portfolio manager tomanage his wealth 119883(119905) from investments The manager(controller) can choose a portfolio 119906(119905) which represents theproportion of the total wealth 119883(119905) invested in the stock attime 119905 And the investor (stopper) can halt the wealth process119883(119905) by selecting a stop-rule 120591 119862[0 119879] rarr [0 119879] Then thedynamics of the corresponding wealth process 119883(119905) = 119883

119906(119905)

is

119889119883119906(119905) = 119883

119906(119905minus) [(1 minus 119906 (119905)) 119903 (119905) + 119906 (119905) 119887 (119905)] 119889119905

+ 119906 (119905) 120590 (119905) 119889119861 (119905)

+119906 (119905minus) intR

120574 (119905 119911) (119889119905 119889119911)

119883119906(0) = 119909 gt 0

(4)

(see eg [3ndash5])We require that 119906(119905minus)120574(119905 119911) gt minus1 for aa 119905 119911as and that

int

119879

0

|(1 minus 119906 (119905)) 119903 (119905)| + |119906 (119905) 119887 (119905)| + 1199062(119905) 1205902(119905)

+1199062(119905) int

R

1205742(119905 119911) ] (119889119911) 119889119905 lt infin as

(5)

At terminal time 120591 the stopper gives the controller apayoff 119862(119883(120591)) where 119862 119862[0 119879] rarr R is a deterministicmapping Therefore the controller aims to maximize hisutility of the following form

J119909

1(119906 120591)=119864

119909[119890minus120575120591

1198801(119862 (119883 (120591)))minusint

120591

0

119890minus120575119905ℎ (119905 119883 (119905) 119906

119905) 119889119905]

(6)

where 120575 gt 0 is the discounting rate ℎ is a cost function and1198801is the controllerrsquos utility We denote 119864119909 the expectation

with respect to119875119909 and119875119909 the probability laws of119883(119905) startingat 119909

Meanwhile it is the stopperrsquos objective to choose thestopping time 120591 such that his own utility

J119909

2(119906 120591) = 119864

119909[119890minus120575120591

1198802119883 (120591) minus 119862 (119883 (120591))] (7)

is maximized where 1198802is the stopperrsquos utility

As this game is typically a nonzero sum we seek a Nashequilibrium namely a pair (119906lowast 120591lowast) such that

J119909

1(119906lowast 120591lowast) ge J

119909

1(119906 120591lowast) forall119906

J119909

2(119906lowast 120591lowast) ge J

119909

2(119906lowast 120591) forall120591

(8)

This means that the choice 119906lowast is optimal for the controllerwhen the stopper uses 120591lowast and vice verse

The game (4) and (8) are a nonzero-sum stochasticdifferential game between a controller and a stopper Theexistence of Nash equilibrium shows that by an appropriatestopping rule design 120591lowast the stopper can induce the controllerto choose the best portfolio he can Similarly by applying asuitable portfolio 119906lowast the controller can force the stopper tostop the employment relationship at a time of the controllerrsquoschoosing

There have been significant advances in the researchof stochastic differential games of control and stoppingFor example in [6ndash9] the authors considered the zero-sumstochastic differential games ofmixed type with both controlsand stopping between two players In these games each of theplayers chooses an optimal strategy which is composed by acontrol 119906(sdot) and a stopping 120591 Under appropriate conditionsthey constructed a saddle pair of optimal strategies For thenonsum case the games of mixed type were discussed in[6 10] The authors presented Nash equilibria rather thansaddle pairs of strategies [10] Moreover the papers [11ndash17]considered a zero-sum stochastic differential game betweencontroller and stopper where one player (controller) choosesa control process 119906(sdot) and the other (stopper) chooses astopping 120591 One player tries to maximize the reward and theother to minimize it They presented a saddle pair for thegame

In this paper we study a nonzero-sum stochastic differen-tial game between a controller and a stopper The controllerand the stopper have different payoffsThe objectives of themare tomaximize their own payoffsThis game is considered ina jumpdiffusion context under theMarkov control conditionWe prove a verification theorem in terms of VIHJB equationsfor the game to characterize Nash equilibrium

Our setup and approach are related to [3 17] Howevertheir games are different fromours In [3] the authors studiedthe stochastic differential games between two controllersThe work in [17] was carried out for a zero-sum stochasticdifferential game between a controller and a stopper

The paper is organized as follows in the next sectionwe formulate the nonzero-sum stochastic differential gamebetween a controller and a stopper and prove a generalverification theorem In Section 3 we apply the generalresults obtained in Section 2 to characterize the solutions of aspecial game Finally we conclude this paper in Section 4

2 A Verification Theorem forNonzero-Sum Stochastic Differential Gamebetween Controller and Stopper

Suppose the state 119884(119905) = 119884119906(119905) at time 119905 is given by the

following stochastic differential equation

119889119884 (119905) = 120572 (119884 (119905) 1199060(119905)) 119889119905 + 120573 (119884 (119905) 119906

0(119905)) 119889119861 (119905)

+ intR1198960

120579 (119884 (119905minus) 1199061(119905minus 119911) 119911) (119889119905 119889119911)

119884 (0) = 119910 isin R119896

(9)

where 120572 R119896 times U rarr R119896 120573 R119896 times U rarr R119896times119896 120579 R119896 times

U times R119896 rarr R119896times119896 are given functions and U denotes a givensubset of R119901

We regard 1199060(119905) = 119906

0(119905 120596) and 119906

1(119905 119911) = 119906

1(119905 119911 120596)

as the control processes assumed to be cadlag F119905-adapted

and with values in 1199060(119905) isin U 119906

1(119905 119911) isin U for aa 119905 119911 120596

And we put 119906(119905) = (1199060(119905) 1199061(119905 119911)) Then 119884(119905) = 119884

(119906)(119905) is

a controlled jump diffusion (see [18] for more informationabout stochastic control of jump diffusion)

Abstract and Applied Analysis 3

Fix an open solvency set S sub R119896 Let

120591S = inf 119905 gt 0 119884 (119905) notin S (10)

be the bankruptcy time 120591S is the first time at which thestochastic process 119884(119905) exits the solvency set S Similaroptimal control problems in which the terminal time isgoverned by a stopping criterion are considered in [19ndash21] inthe deterministic case

Let 119891119894 R119896 times 119870 rarr R and 119892

119894 R119896 rarr R be given

functions for 119894 = 1 2 Let A be a family of admissiblecontrols contained in the set of 119906(sdot) such that (9) has a uniquestrong solution and

119864119910[int

120591S

0

1003816100381610038161003816119891119894 (119884119905 119906119905)1003816100381610038161003816 119889119905] lt infin 119894 = 1 2 (11)

for all 119910 isin S where 119864119910 denotes expectation given that119884(0) =119910 isin R119896 Denote by Γ the set of all stopping times 120591 le 120591SMoreover we assume that

the family 119892minus

119894(119884120591) 120591 isin Γ is uniformly integrable

for 119894 = 1 2

(12)

Then for 119906 isin A and 120591 isin Γ we define the performancefunctionals as follows

J119910

119894(119906 120591) = 119864

119910[int

120591

0

119891119894(119884 (119905) 119906 (119905)) 119889119905 + 119892

119894(119884 (120591))]

119894 = 1 2

(13)

We interpret 119892119894(119884(120591)) as 0 if 120591 = infin We may regardJ

119910

1(119906 120591)

as the payoff to the controller who controls 119906 andJ1199102(119906 120591) as

the payoff to the stopper who decides 120591

Definition 2 (nash equilibrium) A pair (119906lowast 120591lowast) isin A times Γ iscalled a Nash equilibrium for the stochastic differential game(9) and (13) if the following holds

J119910

1(119906lowast 120591lowast) ge J

119910

1(119906 120591lowast) forall119906 isin U 119910 isin S (14)

J119910

2(119906lowast 120591lowast) ge J

119910

2(119906lowast 120591) forall120591 isin Γ 119910 isin S (15)

Condition (14) states that if the stopper chooses 120591lowast it isoptimal for the controller to use the control 119906lowast Similarlycondition (15) states that if the controller uses 119906lowast it is optimalfor the stopper to decide 120591lowast Thus (119906lowast 120591lowast) is an equilibriumpoint in the sense that there is no reason for each individualplayer to deviate from it as long as the other player does not

We restrict ourselves to Markov controls that is weassume that 119906

0(119905) =

0(119884(119905)) 119906

1(119905 119911) =

1(119884(119905) 119911) As

customary we do not distinguish between 1199060and

0 1199061and

1 Then the controls 119906

0and 119906

1can simply be identified with

functions 0(119910) and

1(119910 119911) where 119910 isin S and 119911 isin R119896

When the control 119906 is Markovian the correspondingprocess 119884(119906)(119905) becomes a Markov process with the generator119860119906 of 120601 isin 1198622(R119896) given by

119860120601 (119910) =

119896

sum

119894=1

120572119894(119910 1199060(119910))

120597120601

120597119910119894

(119910)

+1

2

119896

sum

119894119895=1

(120573120573119879)119894119895(119910 1199060(119910))

1205972120601

120597119910119894120597119910119895

(119910)

+

119896

sum

119895=1

intR

120601 (119910 + 120579119895(119910 1199061(119910 119911) 119911)) minus 120601 (119910)

minusnabla120601 (119910) 120579119895(119910 1199061(119910 119911) 119911) ] (119889119911)

(16)

where nabla120601 = (1205971206011205971199101 120597120601120597119910

119896) is the gradient of 120601 and 120579119895

is the 119895th column of the 119896 times 119896matrix 120579Now we can state the main result of this section

Theorem 3 (verification theorem for game (9) and (13))Suppose there exist two functions 120601

119894 S rarr R 119894 = 1 2 such

that

(i) 120601119894isin 1198621(S0)⋂119862(S) 119894 = 1 2 whereS is the closure of

S and S0 is the interior of S(ii) 120601119894ge 119892119894on S 119894 = 1 2

Define the following continuation regions

119863119894= 119910 isin S 120601

119894(119910) gt 119892

119894(119910) 119894 = 1 2 (17)

Suppose 119884(119905) = 119884119906(119905) spends 0 time on 120597119863

119894as 119894 =

1 2 that is(iii) 119864119910[int120591119878

0120594120597119863119894(119884(119905))119889119905] = 0 for all 119910 isin S 119906 isin U 119894 = 1 2

(iv) 120597119863119894is a Lipschitz surface 119894 = 1 2

(v) 120601119894isin 1198622(S120597119863

119894)The second-order derivatives of 120601

119894are

locally bounded near 120597119863119894 respectively 119894 = 1 2

(vi) 1198631= 1198632= 119863

(vii) there exists isin A such that for 119894 = 1 2

119860120601119894(119910) + 119891

119894(119910 (119910))

= sup119906isinA

119860119906120601119894(119910) + 119891

119894(119910 119906 (119910))

= 0 119910 isin 119863

le 0 119910 isin S 119863

(18)

(viii) 119864119910[|120601119894(119884120591)| + int120591

0|119860119906120601119894(119884119906(119905))|119889119905] lt infin 119894 = 1 2 for all

119906 isin U 120591 isin ΓFor 119906 isin A define

120591119863= 120591119906

119863= inf 119905 gt 0 119884

119906(119905) notin 119863 lt infin (19)

and in particular

120591119863= 120591

119863= inf 119905 gt 0 119884

(119905) notin 119863 lt infin (20)

4 Abstract and Applied Analysis

Suppose that(ix) the family 120601

119894(119884(120591)) 120591 isin Γ 120591 le 120591

119863 is uniformly

integrable for all 119906 isin A and 119910 isin S 119894 = 1 2

Then (120591119863 ) isin Γ timesA is a Nash equilibrium for game (9) (13)

and

1206011(119910) = sup

119906isinA

J119906120591119863

1(119910) = J

120591119863

1(119910) (21)

1206012(119910) = sup

120591isinΓ

J120591

2(119910) = J

120591119863

2(119910) (22)

Proof From (i) (iv) and (v) we may assume by an approx-imation theorem (see Theorem 21 in [18]) that 120601

119894isin 1198622(S)

119894 = 1 2For a given 119906 isin A we define with 119884(119905) = 119884

119906(119905)

120591119863= 120591119906

119863= inf 119905 gt 0 119884

119906(119905) notin 119863 (23)

In particular let be as in (vii) Then

120591119863= 120591

119863= inf 119905 gt 0 119884

(119905) notin 119863 (24)

We first prove that (21) holds Let 120591119863isin Γ be as in (24) For

arbitrary 119906 isin A by (vii) and the Dynkinrsquos formula for jumpdiffusion (see Theorem 124 in [18]) we have

1206011(119910) = 119864

119910[int

120591119863and119898

0

minus1198601199061206011(119884 (119905)) 119889119905 + 120601

1(119884 (120591119863and 119898))]

ge 119864119910[int

120591119863and119898

0

1198911(119884 (119905) 119906 (119905)) 119889119905 + 120601

1(119884 (120591119863and 119898))]

(25)

where119898 = 1 2 Therefore by (11) (12) (i) (ii) (viii) andthe Fatou lemma

1206011(119910)

ge lim inf119898rarrinfin

119864119910[int

120591119863and119898

0

1198911(119884 (119905) 119906 (119905)) 119889119905+120601

1(119884 (120591119863and 119898))]

ge 119864119910[int

120591119863

0

1198911(119884 (119905) 119906 (119905)) 119889119905 + 119892

1(119884 (120591119863))]

= J119910

1(119906 120591119863)

(26)

Since this holds for all 119906 isin A we have

1206011(119910) ge sup

119906isinA

J119910

1(119906 120591119863) (27)

In particular applying the Dynkinrsquos formula to 119906 = we getan equality that is

1206011(119910) = 119864

119910[int

120591119863and119898

0

minus1198601206011( (119905)) 119889119905 + 120601

1( (120591119863and 119898))]

= 119864119910[int

120591119863and119898

0

1198911( (119905) (119905)) 119889119905 + 120601

1( (120591119863and 119898))]

(28)

where (119905) = 119884(119905) and119898 = 1 2 Hence we deduce that

1206011(119910) = J

119910

1( 120591119863) (29)

Since we always have

J119910

1( 120591119863) le sup119906isinA

J119910

1(119906 120591119863) (30)

we conclude by combining (27) (29) and (30) that

1206011(119910) = J

119910

1( 120591119863) = sup119906isinA

J119910

1(119906 120591119863) (31)

which is (21)Next we prove that (22) holds Let isin A be as in (vii)

For 120591 isin Γ by the Dynkinrsquos formula and (vii) we have

119864119910[1206012( (120591119898))] = 120601

2(119910) + 119864

119910[int

120591119898

0

1198601206012( (119905)) 119889119905]

le 1206012(119910) minus 119864

119910[int

120591119898

0

1198912( (119905) (119905)) 119889119905]

(32)

where 120591119898= 120591 and 119898119898 = 1 2

Letting119898 rarr infin gives by (11) (12) (i) (ii) (viii) and theFatou Lemma

1206012(119910) ge lim inf

119898rarrinfin119864119910[int

120591and119898

0

1198912( (119905) (119905)) 119889119905 + 120601

2( (120591119898))]

ge 119864119910[int

120591

0

1198912( (119905) (119905)) 119889119905 + 119892

2( (120591)) 120594

120591ltinfin]

= J119910

2( 120591)

(33)

The inequality (33) holds for all 120591 isin Γ Then we have

1206012(119910) ge sup

120591isinΓ

J119910

2( 120591) (34)

Similarly applying the above argument to the pair ( 120591119863) we

get an equality in (34) that is

1206012(119910) = J

119910

2( 120591119863) (35)

We always have

J119910

2( 120591119863) le sup120591isinΓ

J119910

2( 120591) (36)

Therefore combining (34) (35) and (36) we get

1206012(119910) = J

119910

2( 120591119863) = sup120591isinΓ

J119910

2( 120591) (37)

which is (22) The proof is completed

Abstract and Applied Analysis 5

3 An Example

In this section we come back to Example 1 and useTheorem 3to study the solutions of game (4) and (8) Here and in thefollowing all the processes are assumed to be one-dimensionfor simplicity

To put game (4) and (8) into the framework of Section 2we define the process 119884(119905) = (119884

0(119905) 1198841(119905)) 119884(0) = 119910 = (119904 119910

1)

by

1198891198840(119905) = 119889119905 119884

0(0) = 119904 isin R

1198891198841(119905) = 119889119883 (119905)

= 1198841(119905) [(1 minus 119906 (119905)) 119903 (119905) + 119906 (119905) 119887 (119905)] 119889119905

+ 119906 (119905) 120590 (119905) 119889119861 (119905)

+119906 (119905minus) intR

120574 (119905 119911) (119889119905 119889119911)

1198841(0) = 119909 = 119910

1

(38)

Then the performance functionals to the controller (6) andthe stopper (7) can be formulated as follows

J119910

1(119906 120591) = 119864

1199041199101 [119890minus120575(119904+120591)

1198801(119862 (1198841(120591)))

minusint

120591

0

119890minus120575(119904+119905)

ℎ (1198840(119905) 1198841(119905) 119906 (119905)) 119889119905]

J119910

2(119906 120591) = 119864

1199041199101 [119890minus120575(119904+120591)

1198802(1198841(120591) minus 119862 (119884

1(120591)))]

(39)

In this case the generator 119860119906 in (16) has the form

119860119906120601 (119904 119909)

=120597120601

120597119904+ [(1 minus 119906) 119903119909 + 119906119887119909]

120597120601

120597119909+1

2119909211990621205902 1205972120601

1205971199092

+ intR0

120601 (119904 119909 + 119909119906120574 (119911)) minus 120601 (119904 119909) minus 119909119906120574 (119911)120597120601

120597119909

times ] (119889119911)

(40)

To obtain a possible Nash equilibrium ( 120591) isin A times Γ forgame (4) and (8) according to Theorem 3 it is necessary tofind a subset 119863 of S = R2

+= [0infin)

2 and 120601119894(119904 119909) 119894 = 1 2

such that

(i) 1206011(119904 119909) = 119890

minus1205751199041198801(119862(119909)) and 120601

2(119904 119909) = 119890

minus1205751199041198802(119909 minus

119862(119909)) for all (119904 119909) isin 119863(ii) 1206011(119904 119909) ge 119890

minus1205751199041198801(119862(119909)) and 120601

2(119904 119909) ge 119890

minus1205751199041198802(119909 minus

119862(119909)) for all (119904 119909) isin S(iii) 119860119906120601

1(119904 119909) minus ℎ(119904 119909 119906) le 0 and 119860119906120601

2(119904 119909) le 0 for all

(119904 119909) isin S 119863 and 119906(iv) there exists such that 119860120601

1(119904 119909) minus ℎ(119904 119909 ) =

1198601206012(119904 119909) = 0 for all (119904 119909) isin 119863

Imposing the first-order condition on 1198601199061206011(119904 119909) minus

ℎ(119904 119909 119906) and 1198601199061206012(119904 119909) we get the following equations for

the optimal control processes

(119887119909 minus 119903119909)1205971206011

120597119909(119904 119909) + 119909

2120590212059721206011

1205971199092(119904 119909)

+ intR

119903120574 (119911) [1205971206011

120597119909(119904 119909 + 119903120574 (119911)) minus

1205971206011

120597119909(119904 119909)]

minus120597ℎ

120597119906(119904 119909 ) = 0

(119887119909 minus 119903119909)1205971206012

120597119909(119904 119909) + 119909

2120590212059721206012

1205971199092(119904 119909)

+ intR

119903120574 (119911) [1205971206012

120597119909(119904 119909 + 119903120574 (119911)) minus

1205971206012

120597119909(119904 119909)] = 0

(41)

With as in (41) we put

119860120601119894(119904 119909)

=120597120601119894

120597119904+ [(1 minus ) 119903119909 + 119887119909]

120597120601119894

120597119909+1

2119909221205902 1205972120601119894

1205971199092

+ intR0

120601119894(119904 119909 + 119909120574 (119911)) minus 120601

119894(119904 119909) minus 119909120574 (119911)

120597120601119894

120597119909

times ] (119889119911)

(42)

Thus we may reduce game (4) and (8) to the problem ofsolving a family of nonlinear variational-integro inequalitiesWe summarize as follows

Theorem 4 Suppose there exist satisfying (41) and two 1198621-functions 120601

119894 119894 = 1 2 such that

(1)

119863 = (119904 119909) 1206012(119904 119909) gt 119890

minus1205751199041198802(119909 minus 119862 (119909))

= (119904 119909) 1206011(119904 119909) gt 119890

minus1205751199041198801(119862 (119909))

(43)

(2) 120601119894isin 1198622(119863) 119894 = 1 2

(3) 1206012(119904 119909) = 119890

minus1205751199041198802(119909 minus 119862(119909)) and 120601

1(119904 119909) =

119890minus1205751199041198801(119862(119909)) for all (119904 119909) isin S 119863

(4) 1198601199061206011(119904 119909) minus ℎ(119904 119909 119906) le 0 and 119860119906120601

2(119904 119909) le 0 for all

(119904 119909) isin S 119863 and for all 119906 isin A(5) 119860120601

1(119904 119909) minus ℎ(119904 119909 ) = 0 for all (119904 119909) isin 119863 where

1198601206011is given by (42)

(6) 1198601206012(119904 119909) = 0 for all (119904 119909) isin 119863 where 119860120601

2is given

by (42)

Then the pair ( 120591) is a Nash equilibrium of the stochasticdifferential game (4) and (8) where

120591 = inf 119905 gt 0 119884(119905) notin 119863 (44)

6 Abstract and Applied Analysis

Moreover the corresponding equilibrium performances are

1206011(119904 119909) = J

(119904119909)

1( 120591)

1206012(119904 119909) = J

(119904119909)

2( 120591)

(45)

In this paper we will not discuss general solutions of thisfamily of nonlinear variational-integro inequalities Insteadwe discuss a solution in special case when

120574 (119905 119911) = 0 ℎ (119904 119909 119906) =1199062

2 (46)

Let us try the functions 120601119894 119894 = 1 2 of the form

120601119894(119904 119909) = 119890

minus120575119904120595119894(119909) (47)

and a continuation region119863 of the form

119863 = (119904 119909) 119909 lt 1199090 for some 119909

0gt 0 (48)

Then we have

119860119906120601119894(119904 119909) = 119890

minus120575119904119860119906120595119894(119909) (49)

where119860119906120595119894(119909) = minus 120575120595

119894(119909) + [(1 minus 119906) 119903119909 + 119906119887119909] 120595

1015840

119894(119909)

+1

211990921199062120590212059510158401015840

119894(119909)

(50)

By conditions (1) and (3) in Theorem 4 we get

1205951(119909) = 119880

1(119862 (119909)) 119909 ge 119909

0

1205951(119909) gt 119880

1(119862 (119909)) 0 lt 119909 lt 119909

0

1205952(119909) = 119880

2(119909 minus 119862 (119909)) 119909 ge 119909

0

1205952(119909) gt 119880

2(119909 minus 119862 (119909)) 0 lt 119909 lt 119909

0

(51)

From conditions (4) (5) and (6) ofTheorem 4 we get thecandidate for the optimal control as follows

= Argmax119906isinA

1198601199061205951(119909) minus

1199062

2

= Argmax119906isinA

minus 1205751205951(119909) + [(1 minus 119906) 119903119909 + 119906119887119909] 120595

1015840

1(119909)

+1

211990921199062120590212059510158401015840

1(119909) minus

1199062

2

=(119887119909 minus 119903119909) 120595

1015840

1(119909)

1 minus 1199092120590212059510158401015840

1(119909)

(52)

= Argmax119906isinA

1198601199061205952(119909)

= Argmax119906isinA

minus 1205751205952(119909) + [(1 minus 119906) 119903119909 + 119906119887119909] 120595

1015840

2(119909)

+1

211990921199062120590212059510158401015840

2(119909)

=minus (119887119909 minus 119903119909) 120595

1015840

2(119909)

1199092120590212059510158401015840

2(119909)

(53)

Let120595119894(119909) =

119894(119909) on 0 lt 119909 lt 119909

0 119894 = 1 2 By condition (5)

in Theorem 4 we have 1198601(119909) minus

22 = 0 for 0 lt 119909 lt 119909

0

Substituting (52) into 1198601199061(119909) minus 119906

22 = 0 we obtain

minus 1205751(119909) + [119903119909 +

(119887119909 minus 119903119909)21015840

1(119909)

1 minus 1199092120590210158401015840

1(119909)

] 1015840

1(119909)

+

[(119887119909 minus 119903119909) 1015840

1(119909)]2

2(1 minus 1199092120590210158401015840

1(119909))2[1199092120590210158401015840

1(119909) minus 1] = 0

(54)

Similarly we obtain1198602(119909) = 0 for 0 lt 119909 lt 119909

0by condition

(6) in Theorem 4 And we substitute (53) in 1198601199062(119909) = 0 to

get

minus 1205752(119909) + 119903119909

1015840

2(119909) +

[(119887119909 minus 119903119909) 1015840

2(119909)]2

1 minus 1199092120590210158401015840

2(119909)

+

[119909120590 (119887119909 minus 119903119909) 1015840

2(119909)]2

10158401015840

2(119909)

2(1 minus 1199092120590210158401015840

2(119909))2

= 0

(55)

Therefore we conclude that

1205951(119909) =

1198801(119862 (119909)) 119909 ge 119909

0

1(119909) 0 lt 119909 lt 119909

0

(56)

1205952(119909) =

1198802(119909 minus 119862 (119909)) 119909 ge 119909

0

2(119909) 0 lt 119909 lt 119909

0

(57)

where 1(119909) and

2(119909) are the solutions of (56) and (57)

respectivelyAccording to Theorem 4 we use the continuity and

differentiability of 120595119894at 119909 = 119909

0to determine 119909

0 119894 = 1 2 that

is

1205951(1199090) = 1198801(119862 (1199090))

1205951015840

1(1199090) = 119880

1015840

1(119862 (1199090)) 1198621015840(1199090)

1205952(1199090) = 1198802(1199090minus 119862 (119909

0))

1205951015840

2(1199090) = 119880

1015840

2(1199090minus 119862 (119909

0)) (1 minus 119862

1015840(1199090))

(58)

At the end of this section we summarize the above resultsin the following theorem

Theorem 5 Let 120595119894 119894 = 1 2 and let 119909

0be the solutions of

equations (56)ndash(58) Then the pair ( 120591) given by

=(119887119909 minus 119903119909) 120595

1015840

1(119909)

1 minus 1199092120590212059510158401015840

1(119909)

=minus (119887119909 minus 119903119909) 120595

1015840

2(119909)

1199092120590212059510158401015840

2(119909)

120591 = inf 119905 gt 0 119909 ge 1199090

(59)

is a Nash equilibrium of game (4) and (8) The correspondingequilibrium performances are

120601119894(119904 119909) = 119890

minus120575119904120595119894(119909) 119894 = 1 2 (60)

Abstract and Applied Analysis 7

4 Conclusion

A verification theorem is obtained for the general stochasticdifferential game between a controller and a stopper Inthe special case of quadratic cost we use this theorem tocharacterize the Nash equilibrium However the question ofthe existence and uniqueness of Nash equilibrium for thegame remains open It will be considered in our subsequentwork

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China under Grant 61273022 and 11171050

References

[1] J Bertoin Levy Processes Cambridge University Press Cam-bridge UK 1996

[2] D Applebaum Levy Processes and Stochastic Calculus Cam-bridge University Press Cambridge UK 2003

[3] S Mataramvura and B Oslashksendal ldquoRisk minimizing portfoliosand HJBI equations for stochastic differential gamesrdquo Stochas-tics vol 80 no 4 pp 317ndash337 2008

[4] R J Elliott and T K Siu ldquoOn risk minimizing portfolios underaMarkovian regime-switching Black-Scholes economyrdquoAnnalsof Operations Research vol 176 pp 271ndash291 2010

[5] G Wang and Z Wu ldquoMean-variance hedging and forward-backward stochastic differential filtering equationsrdquo Abstractand Applied Analysis vol 2011 Article ID 310910 20 pages 2011

[6] I Karatzas and W Sudderth ldquoStochastic games of control andstopping for a linear diffusionrdquo in Random Walk SequentialAnalysis and Related Topics A Festschrift in Honor of YS ChowA Hsiung Z L Ying and C H Zhang Eds pp 100ndash117WorldScientific Publishers 2006

[7] S Hamadene ldquoMixed zero-sum stochastic differential gameand American game optionsrdquo SIAM Journal on Control andOptimization vol 45 no 2 pp 496ndash518 2006

[8] M K Ghosh M K S Rao and D Sheetal ldquoDifferential gamesof mixed type with control and stopping timesrdquo NonlinearDifferential Equations and Applications vol 16 no 2 pp 143ndash158 2009

[9] M K Ghosh and K S Mallikarjuna Rao ldquoExistence of value instochastic differential games of mixed typerdquo Stochastic Analysisand Applications vol 30 no 5 pp 895ndash905 2012

[10] I Karatzas and Q Li ldquoBSDE approach to non-zero-sumstochastic differential games of control and stoppingrdquo inStochastic Processes Finance and Control pp 105ndash153 WorldScientific Publishers 2012

[11] J-P Lepeltier ldquoOn a general zero-sum stochastic game withstopping strategy for one player and continuous strategy for theotherrdquo Probability and Mathematical Statistics vol 6 no 1 pp43ndash50 1985

[12] I Karatzas and W D Sudderth ldquoThe controller-and-stoppergame for a linear diffusionrdquo The Annals of Probability vol 29no 3 pp 1111ndash1127 2001

[13] AWeerasinghe ldquoA controller and a stopper gamewith degener-ate variance controlrdquo Electronic Communications in Probabilityvol 11 pp 89ndash99 2006

[14] I Karatzas and I-M Zamfirescu ldquoMartingale approach tostochastic differential games of control and stoppingrdquo TheAnnals of Probability vol 36 no 4 pp 1495ndash1527 2008

[15] E Bayraktar I Karatzas and S Yao ldquoOptimal stopping fordynamic convex risk measuresrdquo Illinois Journal of Mathematicsvol 54 no 3 pp 1025ndash1067 2010

[16] E Bayraktar and V R Young ldquoProving regularity of theminimal probability of ruin via a game of stopping and controlrdquoFinance and Stochastics vol 15 no 4 pp 785ndash818 2011

[17] F Bagherya S Haademb B Oslashksendal and I Turpina ldquoOptimalstopping and stochastic control differential games for jumpdiffusionsrdquo Stochastics vol 85 no 1 pp 85ndash97 2013

[18] B Oslashksendal and A Sulem Applied Stochastic Control of JumpDiffusions Springer Berlin Germany 2nd edition 2007

[19] Q Lin R Loxton K L Teo and Y H Wu ldquoA new computa-tional method for a class of free terminal time optimal controlproblemsrdquo Pacific Journal of Optimization vol 7 no 1 pp 63ndash81 2011

[20] Q Lin R Loxton K L Teo and Y H Wu ldquoOptimal controlcomputation for nonlinear systems with state-dependent stop-ping criteriardquo Automatica vol 48 no 9 pp 2116ndash2129 2012

[21] C Jiang Q Lin C Yu K L Teo and G-R Duan ldquoAn exactpenalty method for free terminal time optimal control problemwith continuous inequality constraintsrdquo Journal of OptimizationTheory and Applications vol 154 no 1 pp 30ndash53 2012

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

Fix an open solvency set S sub R119896 Let

120591S = inf 119905 gt 0 119884 (119905) notin S (10)

be the bankruptcy time 120591S is the first time at which thestochastic process 119884(119905) exits the solvency set S Similaroptimal control problems in which the terminal time isgoverned by a stopping criterion are considered in [19ndash21] inthe deterministic case

Let 119891119894 R119896 times 119870 rarr R and 119892

119894 R119896 rarr R be given

functions for 119894 = 1 2 Let A be a family of admissiblecontrols contained in the set of 119906(sdot) such that (9) has a uniquestrong solution and

119864119910[int

120591S

0

1003816100381610038161003816119891119894 (119884119905 119906119905)1003816100381610038161003816 119889119905] lt infin 119894 = 1 2 (11)

for all 119910 isin S where 119864119910 denotes expectation given that119884(0) =119910 isin R119896 Denote by Γ the set of all stopping times 120591 le 120591SMoreover we assume that

the family 119892minus

119894(119884120591) 120591 isin Γ is uniformly integrable

for 119894 = 1 2

(12)

Then for 119906 isin A and 120591 isin Γ we define the performancefunctionals as follows

J119910

119894(119906 120591) = 119864

119910[int

120591

0

119891119894(119884 (119905) 119906 (119905)) 119889119905 + 119892

119894(119884 (120591))]

119894 = 1 2

(13)

We interpret 119892119894(119884(120591)) as 0 if 120591 = infin We may regardJ

119910

1(119906 120591)

as the payoff to the controller who controls 119906 andJ1199102(119906 120591) as

the payoff to the stopper who decides 120591

Definition 2 (nash equilibrium) A pair (119906lowast 120591lowast) isin A times Γ iscalled a Nash equilibrium for the stochastic differential game(9) and (13) if the following holds

J119910

1(119906lowast 120591lowast) ge J

119910

1(119906 120591lowast) forall119906 isin U 119910 isin S (14)

J119910

2(119906lowast 120591lowast) ge J

119910

2(119906lowast 120591) forall120591 isin Γ 119910 isin S (15)

Condition (14) states that if the stopper chooses 120591lowast it isoptimal for the controller to use the control 119906lowast Similarlycondition (15) states that if the controller uses 119906lowast it is optimalfor the stopper to decide 120591lowast Thus (119906lowast 120591lowast) is an equilibriumpoint in the sense that there is no reason for each individualplayer to deviate from it as long as the other player does not

We restrict ourselves to Markov controls that is weassume that 119906

0(119905) =

0(119884(119905)) 119906

1(119905 119911) =

1(119884(119905) 119911) As

customary we do not distinguish between 1199060and

0 1199061and

1 Then the controls 119906

0and 119906

1can simply be identified with

functions 0(119910) and

1(119910 119911) where 119910 isin S and 119911 isin R119896

When the control 119906 is Markovian the correspondingprocess 119884(119906)(119905) becomes a Markov process with the generator119860119906 of 120601 isin 1198622(R119896) given by

119860120601 (119910) =

119896

sum

119894=1

120572119894(119910 1199060(119910))

120597120601

120597119910119894

(119910)

+1

2

119896

sum

119894119895=1

(120573120573119879)119894119895(119910 1199060(119910))

1205972120601

120597119910119894120597119910119895

(119910)

+

119896

sum

119895=1

intR

120601 (119910 + 120579119895(119910 1199061(119910 119911) 119911)) minus 120601 (119910)

minusnabla120601 (119910) 120579119895(119910 1199061(119910 119911) 119911) ] (119889119911)

(16)

where nabla120601 = (1205971206011205971199101 120597120601120597119910

119896) is the gradient of 120601 and 120579119895

is the 119895th column of the 119896 times 119896matrix 120579Now we can state the main result of this section

Theorem 3 (verification theorem for game (9) and (13))Suppose there exist two functions 120601

119894 S rarr R 119894 = 1 2 such

that

(i) 120601119894isin 1198621(S0)⋂119862(S) 119894 = 1 2 whereS is the closure of

S and S0 is the interior of S(ii) 120601119894ge 119892119894on S 119894 = 1 2

Define the following continuation regions

119863119894= 119910 isin S 120601

119894(119910) gt 119892

119894(119910) 119894 = 1 2 (17)

Suppose 119884(119905) = 119884119906(119905) spends 0 time on 120597119863

119894as 119894 =

1 2 that is(iii) 119864119910[int120591119878

0120594120597119863119894(119884(119905))119889119905] = 0 for all 119910 isin S 119906 isin U 119894 = 1 2

(iv) 120597119863119894is a Lipschitz surface 119894 = 1 2

(v) 120601119894isin 1198622(S120597119863

119894)The second-order derivatives of 120601

119894are

locally bounded near 120597119863119894 respectively 119894 = 1 2

(vi) 1198631= 1198632= 119863

(vii) there exists isin A such that for 119894 = 1 2

119860120601119894(119910) + 119891

119894(119910 (119910))

= sup119906isinA

119860119906120601119894(119910) + 119891

119894(119910 119906 (119910))

= 0 119910 isin 119863

le 0 119910 isin S 119863

(18)

(viii) 119864119910[|120601119894(119884120591)| + int120591

0|119860119906120601119894(119884119906(119905))|119889119905] lt infin 119894 = 1 2 for all

119906 isin U 120591 isin ΓFor 119906 isin A define

120591119863= 120591119906

119863= inf 119905 gt 0 119884

119906(119905) notin 119863 lt infin (19)

and in particular

120591119863= 120591

119863= inf 119905 gt 0 119884

(119905) notin 119863 lt infin (20)

4 Abstract and Applied Analysis

Suppose that(ix) the family 120601

119894(119884(120591)) 120591 isin Γ 120591 le 120591

119863 is uniformly

integrable for all 119906 isin A and 119910 isin S 119894 = 1 2

Then (120591119863 ) isin Γ timesA is a Nash equilibrium for game (9) (13)

and

1206011(119910) = sup

119906isinA

J119906120591119863

1(119910) = J

120591119863

1(119910) (21)

1206012(119910) = sup

120591isinΓ

J120591

2(119910) = J

120591119863

2(119910) (22)

Proof From (i) (iv) and (v) we may assume by an approx-imation theorem (see Theorem 21 in [18]) that 120601

119894isin 1198622(S)

119894 = 1 2For a given 119906 isin A we define with 119884(119905) = 119884

119906(119905)

120591119863= 120591119906

119863= inf 119905 gt 0 119884

119906(119905) notin 119863 (23)

In particular let be as in (vii) Then

120591119863= 120591

119863= inf 119905 gt 0 119884

(119905) notin 119863 (24)

We first prove that (21) holds Let 120591119863isin Γ be as in (24) For

arbitrary 119906 isin A by (vii) and the Dynkinrsquos formula for jumpdiffusion (see Theorem 124 in [18]) we have

1206011(119910) = 119864

119910[int

120591119863and119898

0

minus1198601199061206011(119884 (119905)) 119889119905 + 120601

1(119884 (120591119863and 119898))]

ge 119864119910[int

120591119863and119898

0

1198911(119884 (119905) 119906 (119905)) 119889119905 + 120601

1(119884 (120591119863and 119898))]

(25)

where119898 = 1 2 Therefore by (11) (12) (i) (ii) (viii) andthe Fatou lemma

1206011(119910)

ge lim inf119898rarrinfin

119864119910[int

120591119863and119898

0

1198911(119884 (119905) 119906 (119905)) 119889119905+120601

1(119884 (120591119863and 119898))]

ge 119864119910[int

120591119863

0

1198911(119884 (119905) 119906 (119905)) 119889119905 + 119892

1(119884 (120591119863))]

= J119910

1(119906 120591119863)

(26)

Since this holds for all 119906 isin A we have

1206011(119910) ge sup

119906isinA

J119910

1(119906 120591119863) (27)

In particular applying the Dynkinrsquos formula to 119906 = we getan equality that is

1206011(119910) = 119864

119910[int

120591119863and119898

0

minus1198601206011( (119905)) 119889119905 + 120601

1( (120591119863and 119898))]

= 119864119910[int

120591119863and119898

0

1198911( (119905) (119905)) 119889119905 + 120601

1( (120591119863and 119898))]

(28)

where (119905) = 119884(119905) and119898 = 1 2 Hence we deduce that

1206011(119910) = J

119910

1( 120591119863) (29)

Since we always have

J119910

1( 120591119863) le sup119906isinA

J119910

1(119906 120591119863) (30)

we conclude by combining (27) (29) and (30) that

1206011(119910) = J

119910

1( 120591119863) = sup119906isinA

J119910

1(119906 120591119863) (31)

which is (21)Next we prove that (22) holds Let isin A be as in (vii)

For 120591 isin Γ by the Dynkinrsquos formula and (vii) we have

119864119910[1206012( (120591119898))] = 120601

2(119910) + 119864

119910[int

120591119898

0

1198601206012( (119905)) 119889119905]

le 1206012(119910) minus 119864

119910[int

120591119898

0

1198912( (119905) (119905)) 119889119905]

(32)

where 120591119898= 120591 and 119898119898 = 1 2

Letting119898 rarr infin gives by (11) (12) (i) (ii) (viii) and theFatou Lemma

1206012(119910) ge lim inf

119898rarrinfin119864119910[int

120591and119898

0

1198912( (119905) (119905)) 119889119905 + 120601

2( (120591119898))]

ge 119864119910[int

120591

0

1198912( (119905) (119905)) 119889119905 + 119892

2( (120591)) 120594

120591ltinfin]

= J119910

2( 120591)

(33)

The inequality (33) holds for all 120591 isin Γ Then we have

1206012(119910) ge sup

120591isinΓ

J119910

2( 120591) (34)

Similarly applying the above argument to the pair ( 120591119863) we

get an equality in (34) that is

1206012(119910) = J

119910

2( 120591119863) (35)

We always have

J119910

2( 120591119863) le sup120591isinΓ

J119910

2( 120591) (36)

Therefore combining (34) (35) and (36) we get

1206012(119910) = J

119910

2( 120591119863) = sup120591isinΓ

J119910

2( 120591) (37)

which is (22) The proof is completed

Abstract and Applied Analysis 5

3 An Example

In this section we come back to Example 1 and useTheorem 3to study the solutions of game (4) and (8) Here and in thefollowing all the processes are assumed to be one-dimensionfor simplicity

To put game (4) and (8) into the framework of Section 2we define the process 119884(119905) = (119884

0(119905) 1198841(119905)) 119884(0) = 119910 = (119904 119910

1)

by

1198891198840(119905) = 119889119905 119884

0(0) = 119904 isin R

1198891198841(119905) = 119889119883 (119905)

= 1198841(119905) [(1 minus 119906 (119905)) 119903 (119905) + 119906 (119905) 119887 (119905)] 119889119905

+ 119906 (119905) 120590 (119905) 119889119861 (119905)

+119906 (119905minus) intR

120574 (119905 119911) (119889119905 119889119911)

1198841(0) = 119909 = 119910

1

(38)

Then the performance functionals to the controller (6) andthe stopper (7) can be formulated as follows

J119910

1(119906 120591) = 119864

1199041199101 [119890minus120575(119904+120591)

1198801(119862 (1198841(120591)))

minusint

120591

0

119890minus120575(119904+119905)

ℎ (1198840(119905) 1198841(119905) 119906 (119905)) 119889119905]

J119910

2(119906 120591) = 119864

1199041199101 [119890minus120575(119904+120591)

1198802(1198841(120591) minus 119862 (119884

1(120591)))]

(39)

In this case the generator 119860119906 in (16) has the form

119860119906120601 (119904 119909)

=120597120601

120597119904+ [(1 minus 119906) 119903119909 + 119906119887119909]

120597120601

120597119909+1

2119909211990621205902 1205972120601

1205971199092

+ intR0

120601 (119904 119909 + 119909119906120574 (119911)) minus 120601 (119904 119909) minus 119909119906120574 (119911)120597120601

120597119909

times ] (119889119911)

(40)

To obtain a possible Nash equilibrium ( 120591) isin A times Γ forgame (4) and (8) according to Theorem 3 it is necessary tofind a subset 119863 of S = R2

+= [0infin)

2 and 120601119894(119904 119909) 119894 = 1 2

such that

(i) 1206011(119904 119909) = 119890

minus1205751199041198801(119862(119909)) and 120601

2(119904 119909) = 119890

minus1205751199041198802(119909 minus

119862(119909)) for all (119904 119909) isin 119863(ii) 1206011(119904 119909) ge 119890

minus1205751199041198801(119862(119909)) and 120601

2(119904 119909) ge 119890

minus1205751199041198802(119909 minus

119862(119909)) for all (119904 119909) isin S(iii) 119860119906120601

1(119904 119909) minus ℎ(119904 119909 119906) le 0 and 119860119906120601

2(119904 119909) le 0 for all

(119904 119909) isin S 119863 and 119906(iv) there exists such that 119860120601

1(119904 119909) minus ℎ(119904 119909 ) =

1198601206012(119904 119909) = 0 for all (119904 119909) isin 119863

Imposing the first-order condition on 1198601199061206011(119904 119909) minus

ℎ(119904 119909 119906) and 1198601199061206012(119904 119909) we get the following equations for

the optimal control processes

(119887119909 minus 119903119909)1205971206011

120597119909(119904 119909) + 119909

2120590212059721206011

1205971199092(119904 119909)

+ intR

119903120574 (119911) [1205971206011

120597119909(119904 119909 + 119903120574 (119911)) minus

1205971206011

120597119909(119904 119909)]

minus120597ℎ

120597119906(119904 119909 ) = 0

(119887119909 minus 119903119909)1205971206012

120597119909(119904 119909) + 119909

2120590212059721206012

1205971199092(119904 119909)

+ intR

119903120574 (119911) [1205971206012

120597119909(119904 119909 + 119903120574 (119911)) minus

1205971206012

120597119909(119904 119909)] = 0

(41)

With as in (41) we put

119860120601119894(119904 119909)

=120597120601119894

120597119904+ [(1 minus ) 119903119909 + 119887119909]

120597120601119894

120597119909+1

2119909221205902 1205972120601119894

1205971199092

+ intR0

120601119894(119904 119909 + 119909120574 (119911)) minus 120601

119894(119904 119909) minus 119909120574 (119911)

120597120601119894

120597119909

times ] (119889119911)

(42)

Thus we may reduce game (4) and (8) to the problem ofsolving a family of nonlinear variational-integro inequalitiesWe summarize as follows

Theorem 4 Suppose there exist satisfying (41) and two 1198621-functions 120601

119894 119894 = 1 2 such that

(1)

119863 = (119904 119909) 1206012(119904 119909) gt 119890

minus1205751199041198802(119909 minus 119862 (119909))

= (119904 119909) 1206011(119904 119909) gt 119890

minus1205751199041198801(119862 (119909))

(43)

(2) 120601119894isin 1198622(119863) 119894 = 1 2

(3) 1206012(119904 119909) = 119890

minus1205751199041198802(119909 minus 119862(119909)) and 120601

1(119904 119909) =

119890minus1205751199041198801(119862(119909)) for all (119904 119909) isin S 119863

(4) 1198601199061206011(119904 119909) minus ℎ(119904 119909 119906) le 0 and 119860119906120601

2(119904 119909) le 0 for all

(119904 119909) isin S 119863 and for all 119906 isin A(5) 119860120601

1(119904 119909) minus ℎ(119904 119909 ) = 0 for all (119904 119909) isin 119863 where

1198601206011is given by (42)

(6) 1198601206012(119904 119909) = 0 for all (119904 119909) isin 119863 where 119860120601

2is given

by (42)

Then the pair ( 120591) is a Nash equilibrium of the stochasticdifferential game (4) and (8) where

120591 = inf 119905 gt 0 119884(119905) notin 119863 (44)

6 Abstract and Applied Analysis

Moreover the corresponding equilibrium performances are

1206011(119904 119909) = J

(119904119909)

1( 120591)

1206012(119904 119909) = J

(119904119909)

2( 120591)

(45)

In this paper we will not discuss general solutions of thisfamily of nonlinear variational-integro inequalities Insteadwe discuss a solution in special case when

120574 (119905 119911) = 0 ℎ (119904 119909 119906) =1199062

2 (46)

Let us try the functions 120601119894 119894 = 1 2 of the form

120601119894(119904 119909) = 119890

minus120575119904120595119894(119909) (47)

and a continuation region119863 of the form

119863 = (119904 119909) 119909 lt 1199090 for some 119909

0gt 0 (48)

Then we have

119860119906120601119894(119904 119909) = 119890

minus120575119904119860119906120595119894(119909) (49)

where119860119906120595119894(119909) = minus 120575120595

119894(119909) + [(1 minus 119906) 119903119909 + 119906119887119909] 120595

1015840

119894(119909)

+1

211990921199062120590212059510158401015840

119894(119909)

(50)

By conditions (1) and (3) in Theorem 4 we get

1205951(119909) = 119880

1(119862 (119909)) 119909 ge 119909

0

1205951(119909) gt 119880

1(119862 (119909)) 0 lt 119909 lt 119909

0

1205952(119909) = 119880

2(119909 minus 119862 (119909)) 119909 ge 119909

0

1205952(119909) gt 119880

2(119909 minus 119862 (119909)) 0 lt 119909 lt 119909

0

(51)

From conditions (4) (5) and (6) ofTheorem 4 we get thecandidate for the optimal control as follows

= Argmax119906isinA

1198601199061205951(119909) minus

1199062

2

= Argmax119906isinA

minus 1205751205951(119909) + [(1 minus 119906) 119903119909 + 119906119887119909] 120595

1015840

1(119909)

+1

211990921199062120590212059510158401015840

1(119909) minus

1199062

2

=(119887119909 minus 119903119909) 120595

1015840

1(119909)

1 minus 1199092120590212059510158401015840

1(119909)

(52)

= Argmax119906isinA

1198601199061205952(119909)

= Argmax119906isinA

minus 1205751205952(119909) + [(1 minus 119906) 119903119909 + 119906119887119909] 120595

1015840

2(119909)

+1

211990921199062120590212059510158401015840

2(119909)

=minus (119887119909 minus 119903119909) 120595

1015840

2(119909)

1199092120590212059510158401015840

2(119909)

(53)

Let120595119894(119909) =

119894(119909) on 0 lt 119909 lt 119909

0 119894 = 1 2 By condition (5)

in Theorem 4 we have 1198601(119909) minus

22 = 0 for 0 lt 119909 lt 119909

0

Substituting (52) into 1198601199061(119909) minus 119906

22 = 0 we obtain

minus 1205751(119909) + [119903119909 +

(119887119909 minus 119903119909)21015840

1(119909)

1 minus 1199092120590210158401015840

1(119909)

] 1015840

1(119909)

+

[(119887119909 minus 119903119909) 1015840

1(119909)]2

2(1 minus 1199092120590210158401015840

1(119909))2[1199092120590210158401015840

1(119909) minus 1] = 0

(54)

Similarly we obtain1198602(119909) = 0 for 0 lt 119909 lt 119909

0by condition

(6) in Theorem 4 And we substitute (53) in 1198601199062(119909) = 0 to

get

minus 1205752(119909) + 119903119909

1015840

2(119909) +

[(119887119909 minus 119903119909) 1015840

2(119909)]2

1 minus 1199092120590210158401015840

2(119909)

+

[119909120590 (119887119909 minus 119903119909) 1015840

2(119909)]2

10158401015840

2(119909)

2(1 minus 1199092120590210158401015840

2(119909))2

= 0

(55)

Therefore we conclude that

1205951(119909) =

1198801(119862 (119909)) 119909 ge 119909

0

1(119909) 0 lt 119909 lt 119909

0

(56)

1205952(119909) =

1198802(119909 minus 119862 (119909)) 119909 ge 119909

0

2(119909) 0 lt 119909 lt 119909

0

(57)

where 1(119909) and

2(119909) are the solutions of (56) and (57)

respectivelyAccording to Theorem 4 we use the continuity and

differentiability of 120595119894at 119909 = 119909

0to determine 119909

0 119894 = 1 2 that

is

1205951(1199090) = 1198801(119862 (1199090))

1205951015840

1(1199090) = 119880

1015840

1(119862 (1199090)) 1198621015840(1199090)

1205952(1199090) = 1198802(1199090minus 119862 (119909

0))

1205951015840

2(1199090) = 119880

1015840

2(1199090minus 119862 (119909

0)) (1 minus 119862

1015840(1199090))

(58)

At the end of this section we summarize the above resultsin the following theorem

Theorem 5 Let 120595119894 119894 = 1 2 and let 119909

0be the solutions of

equations (56)ndash(58) Then the pair ( 120591) given by

=(119887119909 minus 119903119909) 120595

1015840

1(119909)

1 minus 1199092120590212059510158401015840

1(119909)

=minus (119887119909 minus 119903119909) 120595

1015840

2(119909)

1199092120590212059510158401015840

2(119909)

120591 = inf 119905 gt 0 119909 ge 1199090

(59)

is a Nash equilibrium of game (4) and (8) The correspondingequilibrium performances are

120601119894(119904 119909) = 119890

minus120575119904120595119894(119909) 119894 = 1 2 (60)

Abstract and Applied Analysis 7

4 Conclusion

A verification theorem is obtained for the general stochasticdifferential game between a controller and a stopper Inthe special case of quadratic cost we use this theorem tocharacterize the Nash equilibrium However the question ofthe existence and uniqueness of Nash equilibrium for thegame remains open It will be considered in our subsequentwork

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China under Grant 61273022 and 11171050

References

[1] J Bertoin Levy Processes Cambridge University Press Cam-bridge UK 1996

[2] D Applebaum Levy Processes and Stochastic Calculus Cam-bridge University Press Cambridge UK 2003

[3] S Mataramvura and B Oslashksendal ldquoRisk minimizing portfoliosand HJBI equations for stochastic differential gamesrdquo Stochas-tics vol 80 no 4 pp 317ndash337 2008

[4] R J Elliott and T K Siu ldquoOn risk minimizing portfolios underaMarkovian regime-switching Black-Scholes economyrdquoAnnalsof Operations Research vol 176 pp 271ndash291 2010

[5] G Wang and Z Wu ldquoMean-variance hedging and forward-backward stochastic differential filtering equationsrdquo Abstractand Applied Analysis vol 2011 Article ID 310910 20 pages 2011

[6] I Karatzas and W Sudderth ldquoStochastic games of control andstopping for a linear diffusionrdquo in Random Walk SequentialAnalysis and Related Topics A Festschrift in Honor of YS ChowA Hsiung Z L Ying and C H Zhang Eds pp 100ndash117WorldScientific Publishers 2006

[7] S Hamadene ldquoMixed zero-sum stochastic differential gameand American game optionsrdquo SIAM Journal on Control andOptimization vol 45 no 2 pp 496ndash518 2006

[8] M K Ghosh M K S Rao and D Sheetal ldquoDifferential gamesof mixed type with control and stopping timesrdquo NonlinearDifferential Equations and Applications vol 16 no 2 pp 143ndash158 2009

[9] M K Ghosh and K S Mallikarjuna Rao ldquoExistence of value instochastic differential games of mixed typerdquo Stochastic Analysisand Applications vol 30 no 5 pp 895ndash905 2012

[10] I Karatzas and Q Li ldquoBSDE approach to non-zero-sumstochastic differential games of control and stoppingrdquo inStochastic Processes Finance and Control pp 105ndash153 WorldScientific Publishers 2012

[11] J-P Lepeltier ldquoOn a general zero-sum stochastic game withstopping strategy for one player and continuous strategy for theotherrdquo Probability and Mathematical Statistics vol 6 no 1 pp43ndash50 1985

[12] I Karatzas and W D Sudderth ldquoThe controller-and-stoppergame for a linear diffusionrdquo The Annals of Probability vol 29no 3 pp 1111ndash1127 2001

[13] AWeerasinghe ldquoA controller and a stopper gamewith degener-ate variance controlrdquo Electronic Communications in Probabilityvol 11 pp 89ndash99 2006

[14] I Karatzas and I-M Zamfirescu ldquoMartingale approach tostochastic differential games of control and stoppingrdquo TheAnnals of Probability vol 36 no 4 pp 1495ndash1527 2008

[15] E Bayraktar I Karatzas and S Yao ldquoOptimal stopping fordynamic convex risk measuresrdquo Illinois Journal of Mathematicsvol 54 no 3 pp 1025ndash1067 2010

[16] E Bayraktar and V R Young ldquoProving regularity of theminimal probability of ruin via a game of stopping and controlrdquoFinance and Stochastics vol 15 no 4 pp 785ndash818 2011

[17] F Bagherya S Haademb B Oslashksendal and I Turpina ldquoOptimalstopping and stochastic control differential games for jumpdiffusionsrdquo Stochastics vol 85 no 1 pp 85ndash97 2013

[18] B Oslashksendal and A Sulem Applied Stochastic Control of JumpDiffusions Springer Berlin Germany 2nd edition 2007

[19] Q Lin R Loxton K L Teo and Y H Wu ldquoA new computa-tional method for a class of free terminal time optimal controlproblemsrdquo Pacific Journal of Optimization vol 7 no 1 pp 63ndash81 2011

[20] Q Lin R Loxton K L Teo and Y H Wu ldquoOptimal controlcomputation for nonlinear systems with state-dependent stop-ping criteriardquo Automatica vol 48 no 9 pp 2116ndash2129 2012

[21] C Jiang Q Lin C Yu K L Teo and G-R Duan ldquoAn exactpenalty method for free terminal time optimal control problemwith continuous inequality constraintsrdquo Journal of OptimizationTheory and Applications vol 154 no 1 pp 30ndash53 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

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International Journal of

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

Suppose that(ix) the family 120601

119894(119884(120591)) 120591 isin Γ 120591 le 120591

119863 is uniformly

integrable for all 119906 isin A and 119910 isin S 119894 = 1 2

Then (120591119863 ) isin Γ timesA is a Nash equilibrium for game (9) (13)

and

1206011(119910) = sup

119906isinA

J119906120591119863

1(119910) = J

120591119863

1(119910) (21)

1206012(119910) = sup

120591isinΓ

J120591

2(119910) = J

120591119863

2(119910) (22)

Proof From (i) (iv) and (v) we may assume by an approx-imation theorem (see Theorem 21 in [18]) that 120601

119894isin 1198622(S)

119894 = 1 2For a given 119906 isin A we define with 119884(119905) = 119884

119906(119905)

120591119863= 120591119906

119863= inf 119905 gt 0 119884

119906(119905) notin 119863 (23)

In particular let be as in (vii) Then

120591119863= 120591

119863= inf 119905 gt 0 119884

(119905) notin 119863 (24)

We first prove that (21) holds Let 120591119863isin Γ be as in (24) For

arbitrary 119906 isin A by (vii) and the Dynkinrsquos formula for jumpdiffusion (see Theorem 124 in [18]) we have

1206011(119910) = 119864

119910[int

120591119863and119898

0

minus1198601199061206011(119884 (119905)) 119889119905 + 120601

1(119884 (120591119863and 119898))]

ge 119864119910[int

120591119863and119898

0

1198911(119884 (119905) 119906 (119905)) 119889119905 + 120601

1(119884 (120591119863and 119898))]

(25)

where119898 = 1 2 Therefore by (11) (12) (i) (ii) (viii) andthe Fatou lemma

1206011(119910)

ge lim inf119898rarrinfin

119864119910[int

120591119863and119898

0

1198911(119884 (119905) 119906 (119905)) 119889119905+120601

1(119884 (120591119863and 119898))]

ge 119864119910[int

120591119863

0

1198911(119884 (119905) 119906 (119905)) 119889119905 + 119892

1(119884 (120591119863))]

= J119910

1(119906 120591119863)

(26)

Since this holds for all 119906 isin A we have

1206011(119910) ge sup

119906isinA

J119910

1(119906 120591119863) (27)

In particular applying the Dynkinrsquos formula to 119906 = we getan equality that is

1206011(119910) = 119864

119910[int

120591119863and119898

0

minus1198601206011( (119905)) 119889119905 + 120601

1( (120591119863and 119898))]

= 119864119910[int

120591119863and119898

0

1198911( (119905) (119905)) 119889119905 + 120601

1( (120591119863and 119898))]

(28)

where (119905) = 119884(119905) and119898 = 1 2 Hence we deduce that

1206011(119910) = J

119910

1( 120591119863) (29)

Since we always have

J119910

1( 120591119863) le sup119906isinA

J119910

1(119906 120591119863) (30)

we conclude by combining (27) (29) and (30) that

1206011(119910) = J

119910

1( 120591119863) = sup119906isinA

J119910

1(119906 120591119863) (31)

which is (21)Next we prove that (22) holds Let isin A be as in (vii)

For 120591 isin Γ by the Dynkinrsquos formula and (vii) we have

119864119910[1206012( (120591119898))] = 120601

2(119910) + 119864

119910[int

120591119898

0

1198601206012( (119905)) 119889119905]

le 1206012(119910) minus 119864

119910[int

120591119898

0

1198912( (119905) (119905)) 119889119905]

(32)

where 120591119898= 120591 and 119898119898 = 1 2

Letting119898 rarr infin gives by (11) (12) (i) (ii) (viii) and theFatou Lemma

1206012(119910) ge lim inf

119898rarrinfin119864119910[int

120591and119898

0

1198912( (119905) (119905)) 119889119905 + 120601

2( (120591119898))]

ge 119864119910[int

120591

0

1198912( (119905) (119905)) 119889119905 + 119892

2( (120591)) 120594

120591ltinfin]

= J119910

2( 120591)

(33)

The inequality (33) holds for all 120591 isin Γ Then we have

1206012(119910) ge sup

120591isinΓ

J119910

2( 120591) (34)

Similarly applying the above argument to the pair ( 120591119863) we

get an equality in (34) that is

1206012(119910) = J

119910

2( 120591119863) (35)

We always have

J119910

2( 120591119863) le sup120591isinΓ

J119910

2( 120591) (36)

Therefore combining (34) (35) and (36) we get

1206012(119910) = J

119910

2( 120591119863) = sup120591isinΓ

J119910

2( 120591) (37)

which is (22) The proof is completed

Abstract and Applied Analysis 5

3 An Example

In this section we come back to Example 1 and useTheorem 3to study the solutions of game (4) and (8) Here and in thefollowing all the processes are assumed to be one-dimensionfor simplicity

To put game (4) and (8) into the framework of Section 2we define the process 119884(119905) = (119884

0(119905) 1198841(119905)) 119884(0) = 119910 = (119904 119910

1)

by

1198891198840(119905) = 119889119905 119884

0(0) = 119904 isin R

1198891198841(119905) = 119889119883 (119905)

= 1198841(119905) [(1 minus 119906 (119905)) 119903 (119905) + 119906 (119905) 119887 (119905)] 119889119905

+ 119906 (119905) 120590 (119905) 119889119861 (119905)

+119906 (119905minus) intR

120574 (119905 119911) (119889119905 119889119911)

1198841(0) = 119909 = 119910

1

(38)

Then the performance functionals to the controller (6) andthe stopper (7) can be formulated as follows

J119910

1(119906 120591) = 119864

1199041199101 [119890minus120575(119904+120591)

1198801(119862 (1198841(120591)))

minusint

120591

0

119890minus120575(119904+119905)

ℎ (1198840(119905) 1198841(119905) 119906 (119905)) 119889119905]

J119910

2(119906 120591) = 119864

1199041199101 [119890minus120575(119904+120591)

1198802(1198841(120591) minus 119862 (119884

1(120591)))]

(39)

In this case the generator 119860119906 in (16) has the form

119860119906120601 (119904 119909)

=120597120601

120597119904+ [(1 minus 119906) 119903119909 + 119906119887119909]

120597120601

120597119909+1

2119909211990621205902 1205972120601

1205971199092

+ intR0

120601 (119904 119909 + 119909119906120574 (119911)) minus 120601 (119904 119909) minus 119909119906120574 (119911)120597120601

120597119909

times ] (119889119911)

(40)

To obtain a possible Nash equilibrium ( 120591) isin A times Γ forgame (4) and (8) according to Theorem 3 it is necessary tofind a subset 119863 of S = R2

+= [0infin)

2 and 120601119894(119904 119909) 119894 = 1 2

such that

(i) 1206011(119904 119909) = 119890

minus1205751199041198801(119862(119909)) and 120601

2(119904 119909) = 119890

minus1205751199041198802(119909 minus

119862(119909)) for all (119904 119909) isin 119863(ii) 1206011(119904 119909) ge 119890

minus1205751199041198801(119862(119909)) and 120601

2(119904 119909) ge 119890

minus1205751199041198802(119909 minus

119862(119909)) for all (119904 119909) isin S(iii) 119860119906120601

1(119904 119909) minus ℎ(119904 119909 119906) le 0 and 119860119906120601

2(119904 119909) le 0 for all

(119904 119909) isin S 119863 and 119906(iv) there exists such that 119860120601

1(119904 119909) minus ℎ(119904 119909 ) =

1198601206012(119904 119909) = 0 for all (119904 119909) isin 119863

Imposing the first-order condition on 1198601199061206011(119904 119909) minus

ℎ(119904 119909 119906) and 1198601199061206012(119904 119909) we get the following equations for

the optimal control processes

(119887119909 minus 119903119909)1205971206011

120597119909(119904 119909) + 119909

2120590212059721206011

1205971199092(119904 119909)

+ intR

119903120574 (119911) [1205971206011

120597119909(119904 119909 + 119903120574 (119911)) minus

1205971206011

120597119909(119904 119909)]

minus120597ℎ

120597119906(119904 119909 ) = 0

(119887119909 minus 119903119909)1205971206012

120597119909(119904 119909) + 119909

2120590212059721206012

1205971199092(119904 119909)

+ intR

119903120574 (119911) [1205971206012

120597119909(119904 119909 + 119903120574 (119911)) minus

1205971206012

120597119909(119904 119909)] = 0

(41)

With as in (41) we put

119860120601119894(119904 119909)

=120597120601119894

120597119904+ [(1 minus ) 119903119909 + 119887119909]

120597120601119894

120597119909+1

2119909221205902 1205972120601119894

1205971199092

+ intR0

120601119894(119904 119909 + 119909120574 (119911)) minus 120601

119894(119904 119909) minus 119909120574 (119911)

120597120601119894

120597119909

times ] (119889119911)

(42)

Thus we may reduce game (4) and (8) to the problem ofsolving a family of nonlinear variational-integro inequalitiesWe summarize as follows

Theorem 4 Suppose there exist satisfying (41) and two 1198621-functions 120601

119894 119894 = 1 2 such that

(1)

119863 = (119904 119909) 1206012(119904 119909) gt 119890

minus1205751199041198802(119909 minus 119862 (119909))

= (119904 119909) 1206011(119904 119909) gt 119890

minus1205751199041198801(119862 (119909))

(43)

(2) 120601119894isin 1198622(119863) 119894 = 1 2

(3) 1206012(119904 119909) = 119890

minus1205751199041198802(119909 minus 119862(119909)) and 120601

1(119904 119909) =

119890minus1205751199041198801(119862(119909)) for all (119904 119909) isin S 119863

(4) 1198601199061206011(119904 119909) minus ℎ(119904 119909 119906) le 0 and 119860119906120601

2(119904 119909) le 0 for all

(119904 119909) isin S 119863 and for all 119906 isin A(5) 119860120601

1(119904 119909) minus ℎ(119904 119909 ) = 0 for all (119904 119909) isin 119863 where

1198601206011is given by (42)

(6) 1198601206012(119904 119909) = 0 for all (119904 119909) isin 119863 where 119860120601

2is given

by (42)

Then the pair ( 120591) is a Nash equilibrium of the stochasticdifferential game (4) and (8) where

120591 = inf 119905 gt 0 119884(119905) notin 119863 (44)

6 Abstract and Applied Analysis

Moreover the corresponding equilibrium performances are

1206011(119904 119909) = J

(119904119909)

1( 120591)

1206012(119904 119909) = J

(119904119909)

2( 120591)

(45)

In this paper we will not discuss general solutions of thisfamily of nonlinear variational-integro inequalities Insteadwe discuss a solution in special case when

120574 (119905 119911) = 0 ℎ (119904 119909 119906) =1199062

2 (46)

Let us try the functions 120601119894 119894 = 1 2 of the form

120601119894(119904 119909) = 119890

minus120575119904120595119894(119909) (47)

and a continuation region119863 of the form

119863 = (119904 119909) 119909 lt 1199090 for some 119909

0gt 0 (48)

Then we have

119860119906120601119894(119904 119909) = 119890

minus120575119904119860119906120595119894(119909) (49)

where119860119906120595119894(119909) = minus 120575120595

119894(119909) + [(1 minus 119906) 119903119909 + 119906119887119909] 120595

1015840

119894(119909)

+1

211990921199062120590212059510158401015840

119894(119909)

(50)

By conditions (1) and (3) in Theorem 4 we get

1205951(119909) = 119880

1(119862 (119909)) 119909 ge 119909

0

1205951(119909) gt 119880

1(119862 (119909)) 0 lt 119909 lt 119909

0

1205952(119909) = 119880

2(119909 minus 119862 (119909)) 119909 ge 119909

0

1205952(119909) gt 119880

2(119909 minus 119862 (119909)) 0 lt 119909 lt 119909

0

(51)

From conditions (4) (5) and (6) ofTheorem 4 we get thecandidate for the optimal control as follows

= Argmax119906isinA

1198601199061205951(119909) minus

1199062

2

= Argmax119906isinA

minus 1205751205951(119909) + [(1 minus 119906) 119903119909 + 119906119887119909] 120595

1015840

1(119909)

+1

211990921199062120590212059510158401015840

1(119909) minus

1199062

2

=(119887119909 minus 119903119909) 120595

1015840

1(119909)

1 minus 1199092120590212059510158401015840

1(119909)

(52)

= Argmax119906isinA

1198601199061205952(119909)

= Argmax119906isinA

minus 1205751205952(119909) + [(1 minus 119906) 119903119909 + 119906119887119909] 120595

1015840

2(119909)

+1

211990921199062120590212059510158401015840

2(119909)

=minus (119887119909 minus 119903119909) 120595

1015840

2(119909)

1199092120590212059510158401015840

2(119909)

(53)

Let120595119894(119909) =

119894(119909) on 0 lt 119909 lt 119909

0 119894 = 1 2 By condition (5)

in Theorem 4 we have 1198601(119909) minus

22 = 0 for 0 lt 119909 lt 119909

0

Substituting (52) into 1198601199061(119909) minus 119906

22 = 0 we obtain

minus 1205751(119909) + [119903119909 +

(119887119909 minus 119903119909)21015840

1(119909)

1 minus 1199092120590210158401015840

1(119909)

] 1015840

1(119909)

+

[(119887119909 minus 119903119909) 1015840

1(119909)]2

2(1 minus 1199092120590210158401015840

1(119909))2[1199092120590210158401015840

1(119909) minus 1] = 0

(54)

Similarly we obtain1198602(119909) = 0 for 0 lt 119909 lt 119909

0by condition

(6) in Theorem 4 And we substitute (53) in 1198601199062(119909) = 0 to

get

minus 1205752(119909) + 119903119909

1015840

2(119909) +

[(119887119909 minus 119903119909) 1015840

2(119909)]2

1 minus 1199092120590210158401015840

2(119909)

+

[119909120590 (119887119909 minus 119903119909) 1015840

2(119909)]2

10158401015840

2(119909)

2(1 minus 1199092120590210158401015840

2(119909))2

= 0

(55)

Therefore we conclude that

1205951(119909) =

1198801(119862 (119909)) 119909 ge 119909

0

1(119909) 0 lt 119909 lt 119909

0

(56)

1205952(119909) =

1198802(119909 minus 119862 (119909)) 119909 ge 119909

0

2(119909) 0 lt 119909 lt 119909

0

(57)

where 1(119909) and

2(119909) are the solutions of (56) and (57)

respectivelyAccording to Theorem 4 we use the continuity and

differentiability of 120595119894at 119909 = 119909

0to determine 119909

0 119894 = 1 2 that

is

1205951(1199090) = 1198801(119862 (1199090))

1205951015840

1(1199090) = 119880

1015840

1(119862 (1199090)) 1198621015840(1199090)

1205952(1199090) = 1198802(1199090minus 119862 (119909

0))

1205951015840

2(1199090) = 119880

1015840

2(1199090minus 119862 (119909

0)) (1 minus 119862

1015840(1199090))

(58)

At the end of this section we summarize the above resultsin the following theorem

Theorem 5 Let 120595119894 119894 = 1 2 and let 119909

0be the solutions of

equations (56)ndash(58) Then the pair ( 120591) given by

=(119887119909 minus 119903119909) 120595

1015840

1(119909)

1 minus 1199092120590212059510158401015840

1(119909)

=minus (119887119909 minus 119903119909) 120595

1015840

2(119909)

1199092120590212059510158401015840

2(119909)

120591 = inf 119905 gt 0 119909 ge 1199090

(59)

is a Nash equilibrium of game (4) and (8) The correspondingequilibrium performances are

120601119894(119904 119909) = 119890

minus120575119904120595119894(119909) 119894 = 1 2 (60)

Abstract and Applied Analysis 7

4 Conclusion

A verification theorem is obtained for the general stochasticdifferential game between a controller and a stopper Inthe special case of quadratic cost we use this theorem tocharacterize the Nash equilibrium However the question ofthe existence and uniqueness of Nash equilibrium for thegame remains open It will be considered in our subsequentwork

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China under Grant 61273022 and 11171050

References

[1] J Bertoin Levy Processes Cambridge University Press Cam-bridge UK 1996

[2] D Applebaum Levy Processes and Stochastic Calculus Cam-bridge University Press Cambridge UK 2003

[3] S Mataramvura and B Oslashksendal ldquoRisk minimizing portfoliosand HJBI equations for stochastic differential gamesrdquo Stochas-tics vol 80 no 4 pp 317ndash337 2008

[4] R J Elliott and T K Siu ldquoOn risk minimizing portfolios underaMarkovian regime-switching Black-Scholes economyrdquoAnnalsof Operations Research vol 176 pp 271ndash291 2010

[5] G Wang and Z Wu ldquoMean-variance hedging and forward-backward stochastic differential filtering equationsrdquo Abstractand Applied Analysis vol 2011 Article ID 310910 20 pages 2011

[6] I Karatzas and W Sudderth ldquoStochastic games of control andstopping for a linear diffusionrdquo in Random Walk SequentialAnalysis and Related Topics A Festschrift in Honor of YS ChowA Hsiung Z L Ying and C H Zhang Eds pp 100ndash117WorldScientific Publishers 2006

[7] S Hamadene ldquoMixed zero-sum stochastic differential gameand American game optionsrdquo SIAM Journal on Control andOptimization vol 45 no 2 pp 496ndash518 2006

[8] M K Ghosh M K S Rao and D Sheetal ldquoDifferential gamesof mixed type with control and stopping timesrdquo NonlinearDifferential Equations and Applications vol 16 no 2 pp 143ndash158 2009

[9] M K Ghosh and K S Mallikarjuna Rao ldquoExistence of value instochastic differential games of mixed typerdquo Stochastic Analysisand Applications vol 30 no 5 pp 895ndash905 2012

[10] I Karatzas and Q Li ldquoBSDE approach to non-zero-sumstochastic differential games of control and stoppingrdquo inStochastic Processes Finance and Control pp 105ndash153 WorldScientific Publishers 2012

[11] J-P Lepeltier ldquoOn a general zero-sum stochastic game withstopping strategy for one player and continuous strategy for theotherrdquo Probability and Mathematical Statistics vol 6 no 1 pp43ndash50 1985

[12] I Karatzas and W D Sudderth ldquoThe controller-and-stoppergame for a linear diffusionrdquo The Annals of Probability vol 29no 3 pp 1111ndash1127 2001

[13] AWeerasinghe ldquoA controller and a stopper gamewith degener-ate variance controlrdquo Electronic Communications in Probabilityvol 11 pp 89ndash99 2006

[14] I Karatzas and I-M Zamfirescu ldquoMartingale approach tostochastic differential games of control and stoppingrdquo TheAnnals of Probability vol 36 no 4 pp 1495ndash1527 2008

[15] E Bayraktar I Karatzas and S Yao ldquoOptimal stopping fordynamic convex risk measuresrdquo Illinois Journal of Mathematicsvol 54 no 3 pp 1025ndash1067 2010

[16] E Bayraktar and V R Young ldquoProving regularity of theminimal probability of ruin via a game of stopping and controlrdquoFinance and Stochastics vol 15 no 4 pp 785ndash818 2011

[17] F Bagherya S Haademb B Oslashksendal and I Turpina ldquoOptimalstopping and stochastic control differential games for jumpdiffusionsrdquo Stochastics vol 85 no 1 pp 85ndash97 2013

[18] B Oslashksendal and A Sulem Applied Stochastic Control of JumpDiffusions Springer Berlin Germany 2nd edition 2007

[19] Q Lin R Loxton K L Teo and Y H Wu ldquoA new computa-tional method for a class of free terminal time optimal controlproblemsrdquo Pacific Journal of Optimization vol 7 no 1 pp 63ndash81 2011

[20] Q Lin R Loxton K L Teo and Y H Wu ldquoOptimal controlcomputation for nonlinear systems with state-dependent stop-ping criteriardquo Automatica vol 48 no 9 pp 2116ndash2129 2012

[21] C Jiang Q Lin C Yu K L Teo and G-R Duan ldquoAn exactpenalty method for free terminal time optimal control problemwith continuous inequality constraintsrdquo Journal of OptimizationTheory and Applications vol 154 no 1 pp 30ndash53 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

3 An Example

In this section we come back to Example 1 and useTheorem 3to study the solutions of game (4) and (8) Here and in thefollowing all the processes are assumed to be one-dimensionfor simplicity

To put game (4) and (8) into the framework of Section 2we define the process 119884(119905) = (119884

0(119905) 1198841(119905)) 119884(0) = 119910 = (119904 119910

1)

by

1198891198840(119905) = 119889119905 119884

0(0) = 119904 isin R

1198891198841(119905) = 119889119883 (119905)

= 1198841(119905) [(1 minus 119906 (119905)) 119903 (119905) + 119906 (119905) 119887 (119905)] 119889119905

+ 119906 (119905) 120590 (119905) 119889119861 (119905)

+119906 (119905minus) intR

120574 (119905 119911) (119889119905 119889119911)

1198841(0) = 119909 = 119910

1

(38)

Then the performance functionals to the controller (6) andthe stopper (7) can be formulated as follows

J119910

1(119906 120591) = 119864

1199041199101 [119890minus120575(119904+120591)

1198801(119862 (1198841(120591)))

minusint

120591

0

119890minus120575(119904+119905)

ℎ (1198840(119905) 1198841(119905) 119906 (119905)) 119889119905]

J119910

2(119906 120591) = 119864

1199041199101 [119890minus120575(119904+120591)

1198802(1198841(120591) minus 119862 (119884

1(120591)))]

(39)

In this case the generator 119860119906 in (16) has the form

119860119906120601 (119904 119909)

=120597120601

120597119904+ [(1 minus 119906) 119903119909 + 119906119887119909]

120597120601

120597119909+1

2119909211990621205902 1205972120601

1205971199092

+ intR0

120601 (119904 119909 + 119909119906120574 (119911)) minus 120601 (119904 119909) minus 119909119906120574 (119911)120597120601

120597119909

times ] (119889119911)

(40)

To obtain a possible Nash equilibrium ( 120591) isin A times Γ forgame (4) and (8) according to Theorem 3 it is necessary tofind a subset 119863 of S = R2

+= [0infin)

2 and 120601119894(119904 119909) 119894 = 1 2

such that

(i) 1206011(119904 119909) = 119890

minus1205751199041198801(119862(119909)) and 120601

2(119904 119909) = 119890

minus1205751199041198802(119909 minus

119862(119909)) for all (119904 119909) isin 119863(ii) 1206011(119904 119909) ge 119890

minus1205751199041198801(119862(119909)) and 120601

2(119904 119909) ge 119890

minus1205751199041198802(119909 minus

119862(119909)) for all (119904 119909) isin S(iii) 119860119906120601

1(119904 119909) minus ℎ(119904 119909 119906) le 0 and 119860119906120601

2(119904 119909) le 0 for all

(119904 119909) isin S 119863 and 119906(iv) there exists such that 119860120601

1(119904 119909) minus ℎ(119904 119909 ) =

1198601206012(119904 119909) = 0 for all (119904 119909) isin 119863

Imposing the first-order condition on 1198601199061206011(119904 119909) minus

ℎ(119904 119909 119906) and 1198601199061206012(119904 119909) we get the following equations for

the optimal control processes

(119887119909 minus 119903119909)1205971206011

120597119909(119904 119909) + 119909

2120590212059721206011

1205971199092(119904 119909)

+ intR

119903120574 (119911) [1205971206011

120597119909(119904 119909 + 119903120574 (119911)) minus

1205971206011

120597119909(119904 119909)]

minus120597ℎ

120597119906(119904 119909 ) = 0

(119887119909 minus 119903119909)1205971206012

120597119909(119904 119909) + 119909

2120590212059721206012

1205971199092(119904 119909)

+ intR

119903120574 (119911) [1205971206012

120597119909(119904 119909 + 119903120574 (119911)) minus

1205971206012

120597119909(119904 119909)] = 0

(41)

With as in (41) we put

119860120601119894(119904 119909)

=120597120601119894

120597119904+ [(1 minus ) 119903119909 + 119887119909]

120597120601119894

120597119909+1

2119909221205902 1205972120601119894

1205971199092

+ intR0

120601119894(119904 119909 + 119909120574 (119911)) minus 120601

119894(119904 119909) minus 119909120574 (119911)

120597120601119894

120597119909

times ] (119889119911)

(42)

Thus we may reduce game (4) and (8) to the problem ofsolving a family of nonlinear variational-integro inequalitiesWe summarize as follows

Theorem 4 Suppose there exist satisfying (41) and two 1198621-functions 120601

119894 119894 = 1 2 such that

(1)

119863 = (119904 119909) 1206012(119904 119909) gt 119890

minus1205751199041198802(119909 minus 119862 (119909))

= (119904 119909) 1206011(119904 119909) gt 119890

minus1205751199041198801(119862 (119909))

(43)

(2) 120601119894isin 1198622(119863) 119894 = 1 2

(3) 1206012(119904 119909) = 119890

minus1205751199041198802(119909 minus 119862(119909)) and 120601

1(119904 119909) =

119890minus1205751199041198801(119862(119909)) for all (119904 119909) isin S 119863

(4) 1198601199061206011(119904 119909) minus ℎ(119904 119909 119906) le 0 and 119860119906120601

2(119904 119909) le 0 for all

(119904 119909) isin S 119863 and for all 119906 isin A(5) 119860120601

1(119904 119909) minus ℎ(119904 119909 ) = 0 for all (119904 119909) isin 119863 where

1198601206011is given by (42)

(6) 1198601206012(119904 119909) = 0 for all (119904 119909) isin 119863 where 119860120601

2is given

by (42)

Then the pair ( 120591) is a Nash equilibrium of the stochasticdifferential game (4) and (8) where

120591 = inf 119905 gt 0 119884(119905) notin 119863 (44)

6 Abstract and Applied Analysis

Moreover the corresponding equilibrium performances are

1206011(119904 119909) = J

(119904119909)

1( 120591)

1206012(119904 119909) = J

(119904119909)

2( 120591)

(45)

In this paper we will not discuss general solutions of thisfamily of nonlinear variational-integro inequalities Insteadwe discuss a solution in special case when

120574 (119905 119911) = 0 ℎ (119904 119909 119906) =1199062

2 (46)

Let us try the functions 120601119894 119894 = 1 2 of the form

120601119894(119904 119909) = 119890

minus120575119904120595119894(119909) (47)

and a continuation region119863 of the form

119863 = (119904 119909) 119909 lt 1199090 for some 119909

0gt 0 (48)

Then we have

119860119906120601119894(119904 119909) = 119890

minus120575119904119860119906120595119894(119909) (49)

where119860119906120595119894(119909) = minus 120575120595

119894(119909) + [(1 minus 119906) 119903119909 + 119906119887119909] 120595

1015840

119894(119909)

+1

211990921199062120590212059510158401015840

119894(119909)

(50)

By conditions (1) and (3) in Theorem 4 we get

1205951(119909) = 119880

1(119862 (119909)) 119909 ge 119909

0

1205951(119909) gt 119880

1(119862 (119909)) 0 lt 119909 lt 119909

0

1205952(119909) = 119880

2(119909 minus 119862 (119909)) 119909 ge 119909

0

1205952(119909) gt 119880

2(119909 minus 119862 (119909)) 0 lt 119909 lt 119909

0

(51)

From conditions (4) (5) and (6) ofTheorem 4 we get thecandidate for the optimal control as follows

= Argmax119906isinA

1198601199061205951(119909) minus

1199062

2

= Argmax119906isinA

minus 1205751205951(119909) + [(1 minus 119906) 119903119909 + 119906119887119909] 120595

1015840

1(119909)

+1

211990921199062120590212059510158401015840

1(119909) minus

1199062

2

=(119887119909 minus 119903119909) 120595

1015840

1(119909)

1 minus 1199092120590212059510158401015840

1(119909)

(52)

= Argmax119906isinA

1198601199061205952(119909)

= Argmax119906isinA

minus 1205751205952(119909) + [(1 minus 119906) 119903119909 + 119906119887119909] 120595

1015840

2(119909)

+1

211990921199062120590212059510158401015840

2(119909)

=minus (119887119909 minus 119903119909) 120595

1015840

2(119909)

1199092120590212059510158401015840

2(119909)

(53)

Let120595119894(119909) =

119894(119909) on 0 lt 119909 lt 119909

0 119894 = 1 2 By condition (5)

in Theorem 4 we have 1198601(119909) minus

22 = 0 for 0 lt 119909 lt 119909

0

Substituting (52) into 1198601199061(119909) minus 119906

22 = 0 we obtain

minus 1205751(119909) + [119903119909 +

(119887119909 minus 119903119909)21015840

1(119909)

1 minus 1199092120590210158401015840

1(119909)

] 1015840

1(119909)

+

[(119887119909 minus 119903119909) 1015840

1(119909)]2

2(1 minus 1199092120590210158401015840

1(119909))2[1199092120590210158401015840

1(119909) minus 1] = 0

(54)

Similarly we obtain1198602(119909) = 0 for 0 lt 119909 lt 119909

0by condition

(6) in Theorem 4 And we substitute (53) in 1198601199062(119909) = 0 to

get

minus 1205752(119909) + 119903119909

1015840

2(119909) +

[(119887119909 minus 119903119909) 1015840

2(119909)]2

1 minus 1199092120590210158401015840

2(119909)

+

[119909120590 (119887119909 minus 119903119909) 1015840

2(119909)]2

10158401015840

2(119909)

2(1 minus 1199092120590210158401015840

2(119909))2

= 0

(55)

Therefore we conclude that

1205951(119909) =

1198801(119862 (119909)) 119909 ge 119909

0

1(119909) 0 lt 119909 lt 119909

0

(56)

1205952(119909) =

1198802(119909 minus 119862 (119909)) 119909 ge 119909

0

2(119909) 0 lt 119909 lt 119909

0

(57)

where 1(119909) and

2(119909) are the solutions of (56) and (57)

respectivelyAccording to Theorem 4 we use the continuity and

differentiability of 120595119894at 119909 = 119909

0to determine 119909

0 119894 = 1 2 that

is

1205951(1199090) = 1198801(119862 (1199090))

1205951015840

1(1199090) = 119880

1015840

1(119862 (1199090)) 1198621015840(1199090)

1205952(1199090) = 1198802(1199090minus 119862 (119909

0))

1205951015840

2(1199090) = 119880

1015840

2(1199090minus 119862 (119909

0)) (1 minus 119862

1015840(1199090))

(58)

At the end of this section we summarize the above resultsin the following theorem

Theorem 5 Let 120595119894 119894 = 1 2 and let 119909

0be the solutions of

equations (56)ndash(58) Then the pair ( 120591) given by

=(119887119909 minus 119903119909) 120595

1015840

1(119909)

1 minus 1199092120590212059510158401015840

1(119909)

=minus (119887119909 minus 119903119909) 120595

1015840

2(119909)

1199092120590212059510158401015840

2(119909)

120591 = inf 119905 gt 0 119909 ge 1199090

(59)

is a Nash equilibrium of game (4) and (8) The correspondingequilibrium performances are

120601119894(119904 119909) = 119890

minus120575119904120595119894(119909) 119894 = 1 2 (60)

Abstract and Applied Analysis 7

4 Conclusion

A verification theorem is obtained for the general stochasticdifferential game between a controller and a stopper Inthe special case of quadratic cost we use this theorem tocharacterize the Nash equilibrium However the question ofthe existence and uniqueness of Nash equilibrium for thegame remains open It will be considered in our subsequentwork

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China under Grant 61273022 and 11171050

References

[1] J Bertoin Levy Processes Cambridge University Press Cam-bridge UK 1996

[2] D Applebaum Levy Processes and Stochastic Calculus Cam-bridge University Press Cambridge UK 2003

[3] S Mataramvura and B Oslashksendal ldquoRisk minimizing portfoliosand HJBI equations for stochastic differential gamesrdquo Stochas-tics vol 80 no 4 pp 317ndash337 2008

[4] R J Elliott and T K Siu ldquoOn risk minimizing portfolios underaMarkovian regime-switching Black-Scholes economyrdquoAnnalsof Operations Research vol 176 pp 271ndash291 2010

[5] G Wang and Z Wu ldquoMean-variance hedging and forward-backward stochastic differential filtering equationsrdquo Abstractand Applied Analysis vol 2011 Article ID 310910 20 pages 2011

[6] I Karatzas and W Sudderth ldquoStochastic games of control andstopping for a linear diffusionrdquo in Random Walk SequentialAnalysis and Related Topics A Festschrift in Honor of YS ChowA Hsiung Z L Ying and C H Zhang Eds pp 100ndash117WorldScientific Publishers 2006

[7] S Hamadene ldquoMixed zero-sum stochastic differential gameand American game optionsrdquo SIAM Journal on Control andOptimization vol 45 no 2 pp 496ndash518 2006

[8] M K Ghosh M K S Rao and D Sheetal ldquoDifferential gamesof mixed type with control and stopping timesrdquo NonlinearDifferential Equations and Applications vol 16 no 2 pp 143ndash158 2009

[9] M K Ghosh and K S Mallikarjuna Rao ldquoExistence of value instochastic differential games of mixed typerdquo Stochastic Analysisand Applications vol 30 no 5 pp 895ndash905 2012

[10] I Karatzas and Q Li ldquoBSDE approach to non-zero-sumstochastic differential games of control and stoppingrdquo inStochastic Processes Finance and Control pp 105ndash153 WorldScientific Publishers 2012

[11] J-P Lepeltier ldquoOn a general zero-sum stochastic game withstopping strategy for one player and continuous strategy for theotherrdquo Probability and Mathematical Statistics vol 6 no 1 pp43ndash50 1985

[12] I Karatzas and W D Sudderth ldquoThe controller-and-stoppergame for a linear diffusionrdquo The Annals of Probability vol 29no 3 pp 1111ndash1127 2001

[13] AWeerasinghe ldquoA controller and a stopper gamewith degener-ate variance controlrdquo Electronic Communications in Probabilityvol 11 pp 89ndash99 2006

[14] I Karatzas and I-M Zamfirescu ldquoMartingale approach tostochastic differential games of control and stoppingrdquo TheAnnals of Probability vol 36 no 4 pp 1495ndash1527 2008

[15] E Bayraktar I Karatzas and S Yao ldquoOptimal stopping fordynamic convex risk measuresrdquo Illinois Journal of Mathematicsvol 54 no 3 pp 1025ndash1067 2010

[16] E Bayraktar and V R Young ldquoProving regularity of theminimal probability of ruin via a game of stopping and controlrdquoFinance and Stochastics vol 15 no 4 pp 785ndash818 2011

[17] F Bagherya S Haademb B Oslashksendal and I Turpina ldquoOptimalstopping and stochastic control differential games for jumpdiffusionsrdquo Stochastics vol 85 no 1 pp 85ndash97 2013

[18] B Oslashksendal and A Sulem Applied Stochastic Control of JumpDiffusions Springer Berlin Germany 2nd edition 2007

[19] Q Lin R Loxton K L Teo and Y H Wu ldquoA new computa-tional method for a class of free terminal time optimal controlproblemsrdquo Pacific Journal of Optimization vol 7 no 1 pp 63ndash81 2011

[20] Q Lin R Loxton K L Teo and Y H Wu ldquoOptimal controlcomputation for nonlinear systems with state-dependent stop-ping criteriardquo Automatica vol 48 no 9 pp 2116ndash2129 2012

[21] C Jiang Q Lin C Yu K L Teo and G-R Duan ldquoAn exactpenalty method for free terminal time optimal control problemwith continuous inequality constraintsrdquo Journal of OptimizationTheory and Applications vol 154 no 1 pp 30ndash53 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

Moreover the corresponding equilibrium performances are

1206011(119904 119909) = J

(119904119909)

1( 120591)

1206012(119904 119909) = J

(119904119909)

2( 120591)

(45)

In this paper we will not discuss general solutions of thisfamily of nonlinear variational-integro inequalities Insteadwe discuss a solution in special case when

120574 (119905 119911) = 0 ℎ (119904 119909 119906) =1199062

2 (46)

Let us try the functions 120601119894 119894 = 1 2 of the form

120601119894(119904 119909) = 119890

minus120575119904120595119894(119909) (47)

and a continuation region119863 of the form

119863 = (119904 119909) 119909 lt 1199090 for some 119909

0gt 0 (48)

Then we have

119860119906120601119894(119904 119909) = 119890

minus120575119904119860119906120595119894(119909) (49)

where119860119906120595119894(119909) = minus 120575120595

119894(119909) + [(1 minus 119906) 119903119909 + 119906119887119909] 120595

1015840

119894(119909)

+1

211990921199062120590212059510158401015840

119894(119909)

(50)

By conditions (1) and (3) in Theorem 4 we get

1205951(119909) = 119880

1(119862 (119909)) 119909 ge 119909

0

1205951(119909) gt 119880

1(119862 (119909)) 0 lt 119909 lt 119909

0

1205952(119909) = 119880

2(119909 minus 119862 (119909)) 119909 ge 119909

0

1205952(119909) gt 119880

2(119909 minus 119862 (119909)) 0 lt 119909 lt 119909

0

(51)

From conditions (4) (5) and (6) ofTheorem 4 we get thecandidate for the optimal control as follows

= Argmax119906isinA

1198601199061205951(119909) minus

1199062

2

= Argmax119906isinA

minus 1205751205951(119909) + [(1 minus 119906) 119903119909 + 119906119887119909] 120595

1015840

1(119909)

+1

211990921199062120590212059510158401015840

1(119909) minus

1199062

2

=(119887119909 minus 119903119909) 120595

1015840

1(119909)

1 minus 1199092120590212059510158401015840

1(119909)

(52)

= Argmax119906isinA

1198601199061205952(119909)

= Argmax119906isinA

minus 1205751205952(119909) + [(1 minus 119906) 119903119909 + 119906119887119909] 120595

1015840

2(119909)

+1

211990921199062120590212059510158401015840

2(119909)

=minus (119887119909 minus 119903119909) 120595

1015840

2(119909)

1199092120590212059510158401015840

2(119909)

(53)

Let120595119894(119909) =

119894(119909) on 0 lt 119909 lt 119909

0 119894 = 1 2 By condition (5)

in Theorem 4 we have 1198601(119909) minus

22 = 0 for 0 lt 119909 lt 119909

0

Substituting (52) into 1198601199061(119909) minus 119906

22 = 0 we obtain

minus 1205751(119909) + [119903119909 +

(119887119909 minus 119903119909)21015840

1(119909)

1 minus 1199092120590210158401015840

1(119909)

] 1015840

1(119909)

+

[(119887119909 minus 119903119909) 1015840

1(119909)]2

2(1 minus 1199092120590210158401015840

1(119909))2[1199092120590210158401015840

1(119909) minus 1] = 0

(54)

Similarly we obtain1198602(119909) = 0 for 0 lt 119909 lt 119909

0by condition

(6) in Theorem 4 And we substitute (53) in 1198601199062(119909) = 0 to

get

minus 1205752(119909) + 119903119909

1015840

2(119909) +

[(119887119909 minus 119903119909) 1015840

2(119909)]2

1 minus 1199092120590210158401015840

2(119909)

+

[119909120590 (119887119909 minus 119903119909) 1015840

2(119909)]2

10158401015840

2(119909)

2(1 minus 1199092120590210158401015840

2(119909))2

= 0

(55)

Therefore we conclude that

1205951(119909) =

1198801(119862 (119909)) 119909 ge 119909

0

1(119909) 0 lt 119909 lt 119909

0

(56)

1205952(119909) =

1198802(119909 minus 119862 (119909)) 119909 ge 119909

0

2(119909) 0 lt 119909 lt 119909

0

(57)

where 1(119909) and

2(119909) are the solutions of (56) and (57)

respectivelyAccording to Theorem 4 we use the continuity and

differentiability of 120595119894at 119909 = 119909

0to determine 119909

0 119894 = 1 2 that

is

1205951(1199090) = 1198801(119862 (1199090))

1205951015840

1(1199090) = 119880

1015840

1(119862 (1199090)) 1198621015840(1199090)

1205952(1199090) = 1198802(1199090minus 119862 (119909

0))

1205951015840

2(1199090) = 119880

1015840

2(1199090minus 119862 (119909

0)) (1 minus 119862

1015840(1199090))

(58)

At the end of this section we summarize the above resultsin the following theorem

Theorem 5 Let 120595119894 119894 = 1 2 and let 119909

0be the solutions of

equations (56)ndash(58) Then the pair ( 120591) given by

=(119887119909 minus 119903119909) 120595

1015840

1(119909)

1 minus 1199092120590212059510158401015840

1(119909)

=minus (119887119909 minus 119903119909) 120595

1015840

2(119909)

1199092120590212059510158401015840

2(119909)

120591 = inf 119905 gt 0 119909 ge 1199090

(59)

is a Nash equilibrium of game (4) and (8) The correspondingequilibrium performances are

120601119894(119904 119909) = 119890

minus120575119904120595119894(119909) 119894 = 1 2 (60)

Abstract and Applied Analysis 7

4 Conclusion

A verification theorem is obtained for the general stochasticdifferential game between a controller and a stopper Inthe special case of quadratic cost we use this theorem tocharacterize the Nash equilibrium However the question ofthe existence and uniqueness of Nash equilibrium for thegame remains open It will be considered in our subsequentwork

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China under Grant 61273022 and 11171050

References

[1] J Bertoin Levy Processes Cambridge University Press Cam-bridge UK 1996

[2] D Applebaum Levy Processes and Stochastic Calculus Cam-bridge University Press Cambridge UK 2003

[3] S Mataramvura and B Oslashksendal ldquoRisk minimizing portfoliosand HJBI equations for stochastic differential gamesrdquo Stochas-tics vol 80 no 4 pp 317ndash337 2008

[4] R J Elliott and T K Siu ldquoOn risk minimizing portfolios underaMarkovian regime-switching Black-Scholes economyrdquoAnnalsof Operations Research vol 176 pp 271ndash291 2010

[5] G Wang and Z Wu ldquoMean-variance hedging and forward-backward stochastic differential filtering equationsrdquo Abstractand Applied Analysis vol 2011 Article ID 310910 20 pages 2011

[6] I Karatzas and W Sudderth ldquoStochastic games of control andstopping for a linear diffusionrdquo in Random Walk SequentialAnalysis and Related Topics A Festschrift in Honor of YS ChowA Hsiung Z L Ying and C H Zhang Eds pp 100ndash117WorldScientific Publishers 2006

[7] S Hamadene ldquoMixed zero-sum stochastic differential gameand American game optionsrdquo SIAM Journal on Control andOptimization vol 45 no 2 pp 496ndash518 2006

[8] M K Ghosh M K S Rao and D Sheetal ldquoDifferential gamesof mixed type with control and stopping timesrdquo NonlinearDifferential Equations and Applications vol 16 no 2 pp 143ndash158 2009

[9] M K Ghosh and K S Mallikarjuna Rao ldquoExistence of value instochastic differential games of mixed typerdquo Stochastic Analysisand Applications vol 30 no 5 pp 895ndash905 2012

[10] I Karatzas and Q Li ldquoBSDE approach to non-zero-sumstochastic differential games of control and stoppingrdquo inStochastic Processes Finance and Control pp 105ndash153 WorldScientific Publishers 2012

[11] J-P Lepeltier ldquoOn a general zero-sum stochastic game withstopping strategy for one player and continuous strategy for theotherrdquo Probability and Mathematical Statistics vol 6 no 1 pp43ndash50 1985

[12] I Karatzas and W D Sudderth ldquoThe controller-and-stoppergame for a linear diffusionrdquo The Annals of Probability vol 29no 3 pp 1111ndash1127 2001

[13] AWeerasinghe ldquoA controller and a stopper gamewith degener-ate variance controlrdquo Electronic Communications in Probabilityvol 11 pp 89ndash99 2006

[14] I Karatzas and I-M Zamfirescu ldquoMartingale approach tostochastic differential games of control and stoppingrdquo TheAnnals of Probability vol 36 no 4 pp 1495ndash1527 2008

[15] E Bayraktar I Karatzas and S Yao ldquoOptimal stopping fordynamic convex risk measuresrdquo Illinois Journal of Mathematicsvol 54 no 3 pp 1025ndash1067 2010

[16] E Bayraktar and V R Young ldquoProving regularity of theminimal probability of ruin via a game of stopping and controlrdquoFinance and Stochastics vol 15 no 4 pp 785ndash818 2011

[17] F Bagherya S Haademb B Oslashksendal and I Turpina ldquoOptimalstopping and stochastic control differential games for jumpdiffusionsrdquo Stochastics vol 85 no 1 pp 85ndash97 2013

[18] B Oslashksendal and A Sulem Applied Stochastic Control of JumpDiffusions Springer Berlin Germany 2nd edition 2007

[19] Q Lin R Loxton K L Teo and Y H Wu ldquoA new computa-tional method for a class of free terminal time optimal controlproblemsrdquo Pacific Journal of Optimization vol 7 no 1 pp 63ndash81 2011

[20] Q Lin R Loxton K L Teo and Y H Wu ldquoOptimal controlcomputation for nonlinear systems with state-dependent stop-ping criteriardquo Automatica vol 48 no 9 pp 2116ndash2129 2012

[21] C Jiang Q Lin C Yu K L Teo and G-R Duan ldquoAn exactpenalty method for free terminal time optimal control problemwith continuous inequality constraintsrdquo Journal of OptimizationTheory and Applications vol 154 no 1 pp 30ndash53 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

4 Conclusion

A verification theorem is obtained for the general stochasticdifferential game between a controller and a stopper Inthe special case of quadratic cost we use this theorem tocharacterize the Nash equilibrium However the question ofthe existence and uniqueness of Nash equilibrium for thegame remains open It will be considered in our subsequentwork

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China under Grant 61273022 and 11171050

References

[1] J Bertoin Levy Processes Cambridge University Press Cam-bridge UK 1996

[2] D Applebaum Levy Processes and Stochastic Calculus Cam-bridge University Press Cambridge UK 2003

[3] S Mataramvura and B Oslashksendal ldquoRisk minimizing portfoliosand HJBI equations for stochastic differential gamesrdquo Stochas-tics vol 80 no 4 pp 317ndash337 2008

[4] R J Elliott and T K Siu ldquoOn risk minimizing portfolios underaMarkovian regime-switching Black-Scholes economyrdquoAnnalsof Operations Research vol 176 pp 271ndash291 2010

[5] G Wang and Z Wu ldquoMean-variance hedging and forward-backward stochastic differential filtering equationsrdquo Abstractand Applied Analysis vol 2011 Article ID 310910 20 pages 2011

[6] I Karatzas and W Sudderth ldquoStochastic games of control andstopping for a linear diffusionrdquo in Random Walk SequentialAnalysis and Related Topics A Festschrift in Honor of YS ChowA Hsiung Z L Ying and C H Zhang Eds pp 100ndash117WorldScientific Publishers 2006

[7] S Hamadene ldquoMixed zero-sum stochastic differential gameand American game optionsrdquo SIAM Journal on Control andOptimization vol 45 no 2 pp 496ndash518 2006

[8] M K Ghosh M K S Rao and D Sheetal ldquoDifferential gamesof mixed type with control and stopping timesrdquo NonlinearDifferential Equations and Applications vol 16 no 2 pp 143ndash158 2009

[9] M K Ghosh and K S Mallikarjuna Rao ldquoExistence of value instochastic differential games of mixed typerdquo Stochastic Analysisand Applications vol 30 no 5 pp 895ndash905 2012

[10] I Karatzas and Q Li ldquoBSDE approach to non-zero-sumstochastic differential games of control and stoppingrdquo inStochastic Processes Finance and Control pp 105ndash153 WorldScientific Publishers 2012

[11] J-P Lepeltier ldquoOn a general zero-sum stochastic game withstopping strategy for one player and continuous strategy for theotherrdquo Probability and Mathematical Statistics vol 6 no 1 pp43ndash50 1985

[12] I Karatzas and W D Sudderth ldquoThe controller-and-stoppergame for a linear diffusionrdquo The Annals of Probability vol 29no 3 pp 1111ndash1127 2001

[13] AWeerasinghe ldquoA controller and a stopper gamewith degener-ate variance controlrdquo Electronic Communications in Probabilityvol 11 pp 89ndash99 2006

[14] I Karatzas and I-M Zamfirescu ldquoMartingale approach tostochastic differential games of control and stoppingrdquo TheAnnals of Probability vol 36 no 4 pp 1495ndash1527 2008

[15] E Bayraktar I Karatzas and S Yao ldquoOptimal stopping fordynamic convex risk measuresrdquo Illinois Journal of Mathematicsvol 54 no 3 pp 1025ndash1067 2010

[16] E Bayraktar and V R Young ldquoProving regularity of theminimal probability of ruin via a game of stopping and controlrdquoFinance and Stochastics vol 15 no 4 pp 785ndash818 2011

[17] F Bagherya S Haademb B Oslashksendal and I Turpina ldquoOptimalstopping and stochastic control differential games for jumpdiffusionsrdquo Stochastics vol 85 no 1 pp 85ndash97 2013

[18] B Oslashksendal and A Sulem Applied Stochastic Control of JumpDiffusions Springer Berlin Germany 2nd edition 2007

[19] Q Lin R Loxton K L Teo and Y H Wu ldquoA new computa-tional method for a class of free terminal time optimal controlproblemsrdquo Pacific Journal of Optimization vol 7 no 1 pp 63ndash81 2011

[20] Q Lin R Loxton K L Teo and Y H Wu ldquoOptimal controlcomputation for nonlinear systems with state-dependent stop-ping criteriardquo Automatica vol 48 no 9 pp 2116ndash2129 2012

[21] C Jiang Q Lin C Yu K L Teo and G-R Duan ldquoAn exactpenalty method for free terminal time optimal control problemwith continuous inequality constraintsrdquo Journal of OptimizationTheory and Applications vol 154 no 1 pp 30ndash53 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of