Research Article A Variational Formula for Nonzero-Sum ...downloads.hindawi.com › journals › mpe...

10
Research Article A Variational Formula for Nonzero-Sum Stochastic Differential Games of FBSDEs and Applications Maoning Tang Department of Mathematical Sciences, Huzhou University, Zhejiang 313000, China Correspondence should be addressed to Maoning Tang; [email protected] Received 4 December 2013; Revised 20 March 2014; Accepted 20 March 2014; Published 28 April 2014 Academic Editor: Gerhard-Wilhelm Weber Copyright © 2014 Maoning Tang. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A nonzero-sum stochastic differential game problem is investigated for fully coupled forward-backward stochastic differential equations (FBSDEs in short) where the control domain is not necessarily convex. A variational formula for the cost functional in a given spike perturbation direction of control processes is derived by the Hamiltonian and associated adjoint systems. As an application, a global stochastic maximum principleof Pontryagin’s type for open-loop Nash equilibrium points is established. Finally, an example of a linear quadratic nonzero-sum game problem is presented to illustrate that the theories may have interesting practical applications and the corresponding Nash equilibrium point is characterized by the optimality system. Here the optimality system is a fully coupled FBSDE with double dimensions (DFBSDEs in short) which consists of the state equation, the adjoint equation, and the optimality conditions. 1. Introduction Bismut [1] first investigated linear backward stochastic differ- ential equations (BSDEs in short) as the adjoint equation of the forward stochastic system. e existence and uniqueness of BSDEs with nonlinear generators under Lipschitz condi- tion were first proved by Pardoux and Peng [2] in 1990. Since then, the theory of BSDEs has extensive applications in both mathematical finance and stochastic control. Forward-backward stochastic differential equations (FBSDEs in short) consist of forward stochastic differential equations (SDEs in short) of Itˆ o type and BSDEs of Pardoux- Peng. e main motivations of studying FBSDEs mainly come from stochastic control theories and practical applica- tions of finance. In the stochastic optimal control problem, FBSDEs arise as the Hamilton system which is composed of the optimality conditions, the adjoint equation, and the state equation; see, for example, [3]. In mathematical finance, FBSDEs can be formulated as the price equations of financial assets in model uncertainty and risk minimizing strategy for economic management problems; see, for example, [4, 5]. It now becomes more clear that certain important problems in mathematical economics and mathematical finance, especially in the optimization problem, can be formulated to be forward-backward stochastic system. It is well known that in an optimal control problem, there is single control and single criterion to be optimized. And the so-called differential game is to generalize single control and single criterion in the optimal control problem to two controls and two criteria, one for each player. Each player attempts to control the state of the system so as to achieve his goal. e optimal control problem for forward- backward stochastic system is extensively studied; see, for example, [6–12] and references therein. But, to the best of our knowledge, very little work has been published to discuss the maximum principle of stochastic differential games for forward-backward stochastic systems. In 2012, Hui and Xiao [13] established the maximum principle of differential games for forward-backward stochastic systems under convex con- trol domain by means of a convex variation method and a duality technique. And in 2012, Øksendal and Sulem [5] studied optimal control problems with jumps under model uncertainty and partial information. ey rewrite such problems as stochastic differential games of FBSDEs and obtained the corresponding stochastic maximum principle Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2014, Article ID 283418, 9 pages http://dx.doi.org/10.1155/2014/283418

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Research ArticleA Variational Formula for Nonzero-Sum Stochastic DifferentialGames of FBSDEs and Applications

Maoning Tang

Department of Mathematical Sciences Huzhou University Zhejiang 313000 China

Correspondence should be addressed to Maoning Tang tangmaoninghutczjcn

Received 4 December 2013 Revised 20 March 2014 Accepted 20 March 2014 Published 28 April 2014

Academic Editor Gerhard-WilhelmWeber

Copyright copy 2014 Maoning TangThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A nonzero-sum stochastic differential game problem is investigated for fully coupled forward-backward stochastic differentialequations (FBSDEs in short) where the control domain is not necessarily convex A variational formula for the cost functionalin a given spike perturbation direction of control processes is derived by the Hamiltonian and associated adjoint systems Asan application a global stochastic maximum principleof Pontryaginrsquos type for open-loop Nash equilibrium points is establishedFinally an example of a linear quadratic nonzero-sum game problem is presented to illustrate that the theories may have interestingpractical applications and the corresponding Nash equilibrium point is characterized by the optimality system Here the optimalitysystem is a fully coupled FBSDE with double dimensions (DFBSDEs in short) which consists of the state equation the adjointequation and the optimality conditions

1 Introduction

Bismut [1] first investigated linear backward stochastic differ-ential equations (BSDEs in short) as the adjoint equation ofthe forward stochastic system The existence and uniquenessof BSDEs with nonlinear generators under Lipschitz condi-tion were first proved by Pardoux and Peng [2] in 1990 Sincethen the theory of BSDEs has extensive applications in bothmathematical finance and stochastic control

Forward-backward stochastic differential equations(FBSDEs in short) consist of forward stochastic differentialequations (SDEs in short) of Ito type and BSDEs of Pardoux-Peng The main motivations of studying FBSDEs mainlycome from stochastic control theories and practical applica-tions of finance In the stochastic optimal control problemFBSDEs arise as the Hamilton system which is composedof the optimality conditions the adjoint equation and thestate equation see for example [3] In mathematical financeFBSDEs can be formulated as the price equations of financialassets in model uncertainty and risk minimizing strategy foreconomic management problems see for example [4 5]It now becomes more clear that certain important problemsin mathematical economics and mathematical finance

especially in the optimization problem can be formulated tobe forward-backward stochastic system

It is well known that in an optimal control problemthere is single control and single criterion to be optimizedAnd the so-called differential game is to generalize singlecontrol and single criterion in the optimal control problemto two controls and two criteria one for each player Eachplayer attempts to control the state of the system so as toachieve his goal The optimal control problem for forward-backward stochastic system is extensively studied see forexample [6ndash12] and references therein But to the best ofour knowledge very little work has been published to discussthe maximum principle of stochastic differential games forforward-backward stochastic systems In 2012 Hui and Xiao[13] established the maximum principle of differential gamesfor forward-backward stochastic systems under convex con-trol domain by means of a convex variation method anda duality technique And in 2012 Oslashksendal and Sulem[5] studied optimal control problems with jumps undermodel uncertainty and partial informationThey rewrite suchproblems as stochastic differential games of FBSDEs andobtained the corresponding stochastic maximum principle

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 283418 9 pageshttpdxdoiorg1011552014283418

2 Mathematical Problems in Engineering

and verification theorem under convex control domainDifferent from the decoupled forward-backward stochasticsystem studied in [5 13] Tang [14] studied differential gamesfor fully coupled FBSDEs under convex control domainand established the local maximum principle and the veri-fication theorem In this paper we will also consider two-person nonzero differential games of fully coupled forward-backward stochastic systems Different from [5 13 14] thecontrol domain discussed in this paper is not necessarilyconvex The main contribution of our paper is to get directlya variation formula for the cost functional in a given spikeperturbation direction of control processes in terms of theHamiltonian and the associated adjoint system which is alinear FBSDEs and neither the variational systems nor thecorresponding Taylor type expansions of the state processand the cost functional will be considered As an applicationa global stochastic maximum principle for open-loop Nashequilibrium points is established And as a result a linearquadratic nonzero-sum game problem is studied to illustratethat the established theories and the corresponding Nashequilibrium point are characterized by the optimality system

The paper is organized as follows In Section 2 we formu-late the problem and give various assumptions used through-out the paper In Section 3 we obtain the representation forthe difference of the performance functional in terms of theHamilton and adjoint processes In Section 4 we use therepresentation in Section 3 to derive a representation for thevariation and ldquodirectional derivativerdquo of the difference of theperformance functional along with spike variation Section 5is devoted to deriving the global stochastic maximum prin-ciples by the ldquodirectional derivativerdquo formula established inSection 4 A linear quadratic nonzero-sum game problem isstudied in Section 6 In Section 7 we conclude this paper

2 Formulation of the Problem

Let (ΩF F119905119905ge0 119875) be a complete probability space on

which a 119889-dimensional standard Brownian motion 119861(sdot) isdefined with F

119905119905ge0

being its natural filtration augmentedby all 119875-null sets in F Let 119879 gt 0 be a fixed time horizonLet 119864 be a Euclidean space The inner product in 119864 isdenoted by ⟨sdot sdot⟩ and the norm in 119864 is denoted by | sdot | Wefurther introduce some other spaces that will be used in thepaper Denote by 1198712(ΩF

119879 119875 119864) the set of all 119864-valuedF

119879-

measurable random variable 120578 such that E|120578|2 lt infin Denoteby1198722F(0 119879 119864) the set of all 119864-valuedF

119905-adapted stochastic

processes 120593(119905) 119905 isin [0 119879] which satisfy Eint119879

0|120593(119905)|2119889119905 lt

infin Denote by S2F(0 119879 119864) set of all 119864-valued F119905-adapted

continuous stochastic processes 120593(119905) 119905 isin [0 119879] whichsatisfy Esup

0le119905le119879|120593(119905)|2 lt infin Finally we define M2[0 119879] =

1198782F(0 119879R119899)times1198782F(0 119879R

119898)times1198722F(0 119879R119898times119889)ThenM2[0 119879]

is a Banach space with respect to the norm sdot M2 given byΘ(sdot)

2

M2 = Esup0le119905le119879

|119909(119905)|2+Esup0le119905le119879

|119910(119905)|2+Eint119879

0|119911(119905)|2119889119905

for Θ(sdot) = (119909(sdot) 119910(sdot) 119911(sdot)) isin M2[0 119879]In the following we specify two-person nonzero-sum

differential game problem of fully coupled forward-backward

stochastic systems More precisely for 119886 isin R119899 we consider afully coupled nonlinear FBSDE

119909 (119905) = 119886 + int119905

0

119887 (119904 119909 (119904) 119910 (119904) 119911 (119904) 1199061(119904) 1199062(119904)) 119889119904

+ int119905

0

120590 (119904 119909 (119904) 119910 (119904) 119911 (119904)) 119889119861 (119904)

119910 (119905) = ℎ (119909 (119879)) + int119879

119905

119891 (119904 119909 (119904) 119910 (119904) 119911 (119904) 1199061(119904) 1199062(119904)) 119889119904

minus int119879

119905

119911 (119904) 119889119861 (119904)

(1)

The processes 1199061(sdot) and 119906

2(sdot) in the system (1) are the open-

loop control processes which present the controls of the twoplayers

For each one of the two players there is a cost functional

119869119894(1199061(sdot) 1199062(sdot))

= 119864 [int119879

0

119897119894(119905 119909 (119905) 119910 (119905) 119911 (119905) 119906

1(119905) 1199062(119905)) 119889119905

+ 120601119894(119909 (119879)) + 120574

119894(119910 (0)) ] 119894 = 1 2

(2)

In the above 119887 [0 119879]timesR119899timesR119898timesR119898times119889times1198801times1198802rarr R119899

120590 [0 119879] times R119899 times R119898 times R119898times119889 rarr R119899times119889 119891 [0 119879] times

R119899 times R119898 times R119898times119889 times 1198801times 1198802

rarr R119898 ℎ R119899 rarr R119898and 119897119894 [0 119879] times R119899 times R119898 times R119898times119889 times 119880

1times 1198802rarr 119877 120601

119894

R119899 rarr R 120574119894 R119898 rarr R are given Borel measurable mapping

(119894 = 1 2) Here 1198801sub R1198961 and 119880

2sub R1198962 are nonempty

Borel subsets The admissible control process (1199061(sdot) 1199062(sdot))

is defined as a F119905-adapted process with values in 119880

1times 1198802

such that sup0le119905le119879

E[|1199061(119905)|2 + |119906

2(119905)|2] lt infin The set of all

admissible control processes is denoted byA1timesA2

Before giving the basic assumptions on the coefficientsthroughout this paper we first introduce some abbreviationsLet 119866 be a given 119898 times 119899 full-rank matrix Denote 120579 = (

119909

119910

119911)

and 119860(119905 120579) = (minus119866lowast119891

119866119887

119866120590

) where 119866lowast is the transpose matrix of

119866 For all 120579 = (119909 119910 119911) and 120579 = (119909 119910 119911) we denote 119909 = 119909 minus 119909119910 = 119910 minus 119910 = 119911 minus 119911

Assumption 1 (i) The mappings 119891 119887 120590 and ℎ are con-tinuously differentiable with respect to (119909 119910 119911) and thecorresponding derivatives are bounded Moreover 119891 119887 and120590 are bounded by (1+ |119909|+ |119910|+ |119906

1| + |1199062|) and ℎ is bounded

by (1 + |119909|)(ii) The mappings 119891 119887 120590 and ℎ satisfy the following

Monotonicity conditions

⟨119860 (119905 120579 119906) minus 119860 (119905 120579 119906) 120579 minus 120579⟩

le minus1205731|119866119909|2minus 1205732[1003816100381610038161003816119866lowast11991010038161003816100381610038162

+1003816100381610038161003816119866lowast10038161003816100381610038162

]

⟨ℎ (119909) minus ℎ (119909) 119909 minus 119909⟩ ge 1205831|119866119909|2

(3)

Mathematical Problems in Engineering 3

or

⟨119860 (119905 120579 119906) minus 119860 (119905 120579 119906) 120579 minus 120579⟩

ge 1205731|119866119909|2+ 1205732(1003816100381610038161003816119866lowast11991010038161003816100381610038162

+1003816100381610038161003816119866lowast10038161003816100381610038162

)

⟨ℎ (119909) minus ℎ (119909) 119909 minus 119909⟩ le minus1205831|119866119909|2

(4)

Here 1205731 1205732 and 120583

1are given nonnegative constants with 120573

1+

1205732gt 0 120573

2+ 1205831gt 0 Moreover we have 120573

1gt 0 120583

1gt 0 (resp

1205732gt 0) when119898 gt 119899 (resp119898 lt 119899)(iii) The mappings 119897

119894 120601119894 and 120574

119894are continuously differen-

tiable with respect to (119909 119910 119911) (119894 = 1 2) And 119897119894is bounded by

(1 + |119909|2 + |119910|2 + |119911|2 + |1199061|2 + |119906

2|2) And the derivatives of

119897119894with respect to (119909 119910 119911) (119894 = 1 2) are bounded by (1 + |119909| +

|119910| + |119911| + |1199061| + |1199062|) And 120601

119894and ℎ119894are bounded by (1 + |119909|2)

and (1 + |119910|2) respectively And the derivatives of 120601119894and ℎ

119894

with respect to 119909 and 119910 are bounded by (1 + |119909|) and (1 + |119910|)respectively (119894 = 1 2)

Under Assumption 1 we see that for any given admissiblecontrol 119906(sdot) = (119906

1(sdot) 1199062(sdot)) isin A

1timesA2 the system (1) admits

a unique solution (119909119906(sdot) 119910119906(sdot) 119911119906(sdot)) isin M2 (see [15])Then wecall (119909119906(sdot) 119910119906(sdot) 119911119906(sdot)) or (119909(sdot) 119910(sdot) 119911(sdot)) if its dependence onadmissible control 119906(sdot) is clear from context the state processcorresponding to the control process 119906(sdot) = (119906

1(sdot) 1199062(sdot)) and

(119906(sdot) 119909(sdot) 119910(sdot) 119911(sdot)) the admissible pairThenwe can pose the following two-person nonzero-sum

stochastic differential game problem

Problem 2 Find an open-loop admissible control(1199061(sdot) 1199062(sdot)) isin A

1timesA2such that

1198691(1199061(sdot) 1199062(sdot)) = inf

1199061(sdot)isinA1

1198691(1199061(sdot) 1199062(sdot)) (5)

1198692(1199061(sdot) 1199062(sdot)) = inf

1199062(sdot)isinA

2

1198692(1199061(sdot) 1199062(sdot)) (6)

Any (1199061(sdot) 1199062(sdot)) isin A

1timesA2satisfying the above is called

an open-loop Nash equilibrium point of Problem 2 Suchan admissible control allows two players to play individualoptimal control strategies simultaneously

3 Representation for Difference ofthe Cost Functional

This section is devoted to establishing a representation for thedifference of the cost functional according to Hamiltonianand the adjoint processes of Problem 2

To simplify our notation for any admissible control119906(sdot) = (119906

1(sdot) 1199062(sdot)) we write Θ(119905) = (119909(119905) 119910(119905) 119911(119905)) as

the corresponding state process We define the Hamiltonianfunctions 119867

119894 [0 119879] times R119899 times R119898 times R119898times119889 times U

1times U2times R119899 times

R119899times119889 timesR119898 rarr R by

119867119894(119905 119909 119910 119911 119906

1 1199062 119901 119902 119896) = ⟨119896 minus119891 (119905 119909 119910 119911 119906

1 1199062)⟩

+ ⟨119901 119887 (119905 119909 119910 119911 1199061 1199062)⟩

+ 119897119894(119905 119909 119910 119911 119906

1 1199062)

+ ⟨119902 120590 (119905 119909 119910 119911)⟩ 119894 = 1 2

(7)

For any admissible pair (119906(sdot) Θ(sdot)) we define the corre-sponding adjoint process Λ119906

119894(sdot) = (119896119906

119894(sdot) 119901119906119894(sdot) 119902119906119894(sdot))(119894 = 1 2)

as the solution to the following FBSDEs

119889119896119894(119905) = minus119867

119894119910(119905) 119889119905 minus 119867

119894119911(119905) 119889119861 (119905)

119889119901119894(119905) = minus119867

119894119909(119905) 119889119905 + 119902

119894(119905) 119889119861 (119905)

119896119894(0) = minus120574

119894119910(119910 (0))

119901119894(119879) = minusℎ

lowast

119909(119909 (119879)) 119896

119894(119879) + 120601

119894119909(119909 (119879))

119894 = 1 2

(8)

where we have used the short hand notation

119867119894119910(119905) = 119867

119894119910(119905 119906 (119905) Θ (119905) Λ

119906

119894(119905)) 119894 = 12 (9)

And similarly we can define119867119894119909(119905) and119867

119894119911(119905)

Under Assumptions 1 it is easy to see thatthe above adjoint equations have unique solutionΛ119906

119894(sdot) = (119896119906

119894(sdot) 119901119906119894(sdot) 119902119906119894(sdot)) isin M2[0 119879] 119894 = 1 2

Now let (119906120576(sdot) Θ120576(sdot)) = (1199061205761(sdot) 1199061205762(sdot) 119909120576(sdot) 119910120576(sdot) 119911120576(sdot)) be

another admissible pair In the following we will give thepresentation for the difference 119869

119894(119906120576(sdot)) minus 119869

119894(119906(sdot)) (119894 = 1 2)

in terms of the Hamiltonian119867119894(119894 = 1 2) and adjoint process

Λ119906119894(sdot) (119894 = 1 2) associated with the admissible control pair

(119906(sdot) Θ119906(sdot)) as well as other relevant expressions We stateour result as follows

Theorem 3 Under Assumptions 1 one has the representationfor the difference of the cost functional as follows sdot(119910120576(0) minus119910(0))(119910120576(0))]

119869119894(119906120576

1(sdot) 119906120576

2(sdot)) minus 119869

119894(1199061(sdot) 1199062(sdot))

= Eint119879

0

[119867119894(119905 Θ (119905) 119906

120576(119905) Λ

119906

119894(119905))

minus 119867119894(119905 Θ (119905) 119906 (119905) Λ

119906

119894(119905))

+ 119867119894(119905 Θ120576(119905) 119906120576(119905) Λ

119906

119894(119905))

minus 119867119894(119905 Θ (119905) 119906

120576(119905) Λ

119906

119894(119905))

minus ⟨119867119894119909(119905) 119909120576(119905) minus 119909 (119905)⟩

minus ⟨119867119894119910(119905) 119910120576(119905) minus 119910 (119905)⟩

minus ⟨119867119894119911(119905) 119911120576(119905) minus 119911 (119905)⟩] 119889119905

4 Mathematical Problems in Engineering

+ E [120574119894(119910120576(0)) minus 120574

119894(119910 (0)) minus 120574

119894119910(119910 (0))

sdot (119910120576(0) minus 119910 (0)) (119910

120576(0)) ]

+ E [120601119894(119909120598(119879)) minus 120601

119894(119909 (119879)) minus 120601

119894119909(119909 (119879))

sdot (119909120576(119879) minus 119909 (119879))]

minus E [⟨ℎ (119909120598(119879)) minus ℎ (119909 (119879)) 119896

119906

119894(119879)⟩

minus ⟨ℎ119909(119909 (119879)) (119909

120576(119879) minus 119909 (119879)) 119896

119906

119894(119879)⟩]

(10)

Proof First from the definition of 119867 (see (7)) it is easy tocheck that

119867119894(119905 Θ120576(119905) 119906120576(119905) Λ

119906

119894(119905)) minus 119867

119894(119905 Θ (119905) 119906 (119905) Λ

119906

119894(119905))

= 119867119894(119905 Θ (119905) 119906

120576(119905) Λ

119906

119894(sdot)) minus 119867

119894(119905 Θ (119905) 119906 (119905) Λ

119906

119894(119905))

+ 119867119894(119905 Θ120576(119905) 119906120576(119905) Λ

119906

119894(119905))

minus 119867119894(119905 Θ (119905) 119906

120576(119905) Λ

119906

119894(119905)) 119894 = 1 2

(11)

From (1) we know that

119909120576(119905) minus 119909 (119905) = int

119905

0

[119887 (119904 Θ120576(119904) 119906120576(119904))

minus 119887 (119904 Θ (119904) 119906 (119904))] 119889119904

+ int119905

0

[120590 (119904 Θ120576(119904) 119906120576(119904))

minus 120590 (119904 Θ (119904) 119906 (119904))] 119889119882 (119904)

119910120576(119905) minus 119910 (119905) = ℎ (119909

120598(119879)) minus ℎ (119909 (119879))

+ int119879

119905

[119891 (119904 Θ120576(119904) 119906120576(119904))

minus 119891 (119904 Θ (119904) 119906 (119904))] 119889119904

minus int119879

119905

[119911120576(119904) minus 119911 (119904)] 119889119882 (119904)

(12)

Furthermore recalling (8) and applying Ito formula to⟨119896119906119894(119905) 119910120576(119905) minus 119910(119905)⟩ + ⟨119901119906

119894(119905) 119909120576(119905) minus 119909(119905)⟩ we deduce that

Eint119879

0

⟨119896119906

119894(119905) minus (119891 (119905 Θ

120576(119905) 119906120576(119905) )

minus119891 (119905 Θ (119905) 119906 (119905)))⟩ 119889119905

+ 119864int119879

0

⟨119901119906

119894(119905) 119887 (119905 Θ

120576(119905) 119906120576(119905))

minus 119887 (119905 Θ (119905) 119906 (119905))⟩ 119889119905

+ 119864int119879

0

⟨119902119906

119894(119905) 120590 (119905 Θ

120576(119905) 119906120576(119905))

minus 120590 (119905 Θ (119905) 119906 (119905))⟩ 119889119905

= Eint119879

0

[⟨119867119894119909(119905) 119909120576(119905) minus 119909 (119905)⟩

+ ⟨119867119894119910(119905) 119910120576(119905) minus 119910 (119905)⟩

+ ⟨119867119894119911(119905) 119911120576(119905) minus 119911 (119905)⟩] 119889119905

+ E [120574119894119910(119910 (0)) sdot (119910

120576(0) minus 119910 (0))]

+ E [120601119894119909(119909 (119879)) sdot (119909

120576(119879) minus 119909 (119879))]

+ E [⟨ℎ (119909120576(119879)) minus ℎ (119909 (119879)) 119896

119906

119894(119879)⟩

minus ⟨ℎ119909(119909 (119879)) (119909

120576(119879) minus 119909 (119879)) 119896

119906

119894(119879)⟩]

119894 = 1 2

(13)

On the other hand by the definition of theHamilton function119867 (see (7)) we deduce that

119869119894(119906120576

1(sdot) 119906120576

2(sdot)) minus 119869

119894(1199061(sdot) 1199062(sdot))

= Eint119879

0

[119897119894(119905 Θ120576(119905) 119906120576(119905)) minus 119897

119894(119905 Θ (119905) 119906 (119905))] 119889119905

+ E [120574119894(119910120576(0)) minus 120574

119894(119910 (0))]

+ 119864 [120601119894(119909120598(119879)) minus 120601

119894(119909 (119879))]

= Eint119879

0

[119867119894(119905 Θ120576(119905) 119906120576(119905) Λ

119906

119894(119905))

minus119867119894(119905 Θ (119905) 119906 (119905) Λ

119906

119894(119905))] 119889119905

minus Eint119879

0

⟨119896119906

119894(119905) minus (119891 (119905 Θ

120576(119905) 119906120576(119905))

minus119891 (119905 Θ (119905) 119906 (119905)))⟩ 119889119905

minus Eint119879

0

⟨119901119906

119894(119905) 119887 (119905 Θ

120576(119905) 119906120576(119905))

minus119887 (119905 Θ (119905) 119906 (119905))⟩ 119889119905

minus Eint119879

0

⟨119902119906

119894(119905) 120590 (119905 Θ

120576(119905) 119906120576(119905))

minus120590 (119905 Θ (119905) 119906 (119905)) ⟩ 119889119905

+ E [120574119894(119910120576(0)) minus 120574

119894(119910 (0))]

+ 119864 [120601119894(119909120598(119879)) minus 120601

119894(119909 (119879))]

(14)

Now putting (11) and (13) into (14) we deduce that (10)holds The proof is complete

Mathematical Problems in Engineering 5

4 A Variational Formula for StochasticDifferential Games

In this section we will obtain a directional derivative ata given admissible control process in some given controlprocess direction The choice of the given control processdirection depends on the convexity of the control domain1198801times 1198802 If the control domain 119880

1times 1198802is convex a classical

way of treating such a problem consists of using the convexperturbation method More precisely if (119906

1(sdot) 1199062(sdot)) is a

given admissible control and (V1(sdot) V2(sdot)) is an arbitrary

given admissible control we can define a convex perturbedadmissible control as

119906120576

119894(sdot) = 119906

119894(sdot) + 120576 (V

119894(sdot) minus 119906

119894(sdot)) 119894 = 1 2 (15)

where 120576 is a sufficiently small positive constant Then onecan prove the cost functional 119869

119894(1199061(sdot) 1199062(sdot)) is Gateaux dif-

ferentiable at 119906119894(sdot) in the direction V

119894(sdot) minus 119906

119894(sdot) (119894 = 1 2)

and get a local stochastic maximum principle for open-loop Nash equilibrium points see for example [13 14]Different from [13 14] our control domain in the presentpaper is not necessarily convex so the convex perturbedcontrol 119906120576

119894(sdot) may no longer be admissible and the convex

perturbation method cannot be used to obtain the corre-sponding variational formula and maximum principle Aclassical way of treating the nonconvex control domainconsists of using the spike variations perturbation methodMore precisely let (119906(sdot) Θ(sdot)) = (119906

1(sdot) 1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot))

be any given admissible pair with the corresponding adjointprocess Λ119906

119894(sdot) = (119896119906

119894(sdot) 119901119906119894(sdot) 119902119906119894(sdot)) isin M2[0 119879] 119894 = 1 2 We

define the following spike variations

119906120591120576

119894(119905) =

119906119894(119905) if 119905 isin [120591 120591 + 120576]

119906119894(119905) otherwise

(16)

with fixed 120591 isin [0 119879) sufficiently small positive 120576 and anygiven admissible control 119906

119894(sdot) isin A

1 119894 = 1 2

Now we state the following variational formula for thecost functional (2) associated with the spike variation (16) ina unified way

Theorem 4 Under Assumption 1 one has a variational for-mula for the cost functional (5) and (6) as follows

119889

1198891205761198691(119906120591120576

1(sdot) 1199062(sdot))

10038161003816100381610038161003816100381610038161003816120576=0

= lim120576rarr0

1198691(1199061205911205761(sdot) 1199062(sdot)) minus 119869

1(1199061(sdot) 1199062(sdot))

120576

= E [1198671(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(120591))

minus 1198671(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(119905))]

(17)

119889

1198891205761198692(1199061(sdot) 119906120591120576

2(sdot))

10038161003816100381610038161003816100381610038161003816120576=0

= lim120576rarr0

1198692(1199061(sdot) 119906120591120576

2(sdot)) minus 119869

2(1199061(sdot) 1199062(sdot))

120576

= E [1198672(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(120591))

minus 1198672(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(120591))]

(18)

Proof We first prove the equality (17) Let Θ1120576(sdot) =

(1199091120576(sdot) 1199101120576(sdot) 1199111120576(sdot)) be the state process corresponding tothe admissible control (119906120591120576

1(sdot) 1199062(sdot)) Under Assumption 1

by the continuous dependence theory of FBSDEs (seeProposition 32 in [16]) we have

10038171003817100381710038171003817Θ1120576(sdot) minus Θ

119906(sdot)10038171003817100381710038171003817M2

le 1198621205762 (19)

InTheorem 3 replacing (119906120576(sdot) Θ120576(sdot)) by (1199061205911205761(sdot) 1199062(sdot) Θ1120576(sdot))

we have

1198691(119906120591120576

1(sdot) 1199062(sdot)) minus 119869

1(1199061(sdot) 1199062(sdot))

= Eint119879

0

[119867119894(119905 Θ (119905) 119906

120591120576

1(119905) 1199062(119905) Λ

119906

119894(119905))

minus 119867119894(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

119906

119894(119905))

+ 119867119894(119905 Θ120576(119905) 119906120591120576

1(119905) 1199062(119905) Λ

119906

119894(119905))

minus 119867119894(119905 Θ (119905) 119906

120591120576

1(119905) 1199062(119905) Λ

119906

119894(119905))

minus ⟨119867119894119909(119905) 119909120576(119905) minus 119909 (119905)⟩

minus ⟨119867119894119910(119905) 119910120576(119905) minus 119910 (119905)⟩

minus ⟨119867119894119911(119905) 119911120576(119905) minus 119911 (119905)⟩ ] 119889119905

+ E [120574119894(119910120576(0)) minus 120574

119894(119910 (0)) minus 120574

119894119910(119910 (0))

sdot (119910120576(0) minus 119910 (0))]

+ E [120601119894(119909120598(119879)) minus 120601

119894(119909 (119879)) minus 120601

119894119909(119909 (119879))

sdot (119909120576(119879) minus 119909 (119879))]

minus E [⟨ℎ (119909120598(119879)) minus ℎ (119909 (119879)) 119896

119906

119894(119879)⟩

minus ⟨ℎ119909(119909 (119879)) (119909

120576(119879) minus 119909 (119879)) 119896

119906

119894(119879)⟩]

(20)

Combining (19) and Assumption 1 by Taylor Expansions on119867 and the dominated convergence theorem from (20) weconclude that

1198691(119906120591120576

1(sdot) 1199062(sdot)) minus 119869

1(1199061(sdot) 1199062(sdot))

= Eint119879

0

[1198671(119905 Θ (119905) 119906

120591120576

1(119905) 1199062(119905) Λ

119906

1(119905))

minus1198671(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

119906

1(119905))] 119889119905 + 119900 (120576)

6 Mathematical Problems in Engineering

= 120576E [1198671(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(120591))

minus1198671(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(119905))] + 119900 (120576)

(21)

which imply that (17) holdsSimilarly we can prove that (18) holds The proof is

complete

5 Stochastic Maximum Principle

In this section applying the variational formulas (17) and (18)we will state and prove the global maximum principle for theNash equilibrium points of Problem 2

Theorem 5 Under Assumption 1 let 119906(sdot) = (1199061(sdot) 1199062(sdot)) be

a Nash equilibrium point of Problem 2 with the state processΘ(sdot) = (119909(sdot) 119910(sdot) 119911(sdot)) Let Λ

119894(sdot) = (119901

119894(sdot) 119902119894(sdot) 119896119894(sdot)) (119894 = 1 2)

be the unique solution of the adjoint equation (8) correspondingto (119906(sdot) Θ(sdot)) Then

1198671(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

1(120591))

= min1199061isin1198801

1198671(120591 Θ (120591) 119906

1 1199062(120591) Λ

1(120591))

(22)

1198672(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

2(120591))

= min1199062isin1198802

1198672(120591 Θ (120591) 119906

1(120591) 119906

2 Λ2(120591))

(23)

hold for ae (120591 120596) isin [0 119879] times Ω

Proof Since 119906(sdot) = (1199061(sdot) 1199062(sdot)) is a Nash equilibrium point

of Problem 2 by (5) we have

1198691(119906120591120576

1(sdot) 1199062(sdot)) minus 119869

1(1199061(sdot) 1199062(sdot)) ge 0 (24)

Using the notation inTheorem 3 for any arbitrary admissiblecontrol (119906

1(sdot) 1199062(sdot)) and 119905 isin [0 119879] we have

E [1198671(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(120591))

minus1198671(119905 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(120591))] ge 0

(25)

which implies that (22) holdsSimilarly we can prove that (23) holds The proof is

complete

6 An Example Linear Quadratic Case

In this section we work out an example of linear quadraticnonzero-sum differential games to illustrate our stochastic

maximum principle More precisely consider the follow-ing one-dimensional linear fully coupled forward-backwardstochastic system

119909 (119905) = 119886 + int119905

0

[119860 (119904) 119909 (119904) minus 119861 (119904) 119910 (119904) + 1198631(119904) 1199061(119904)

+1198632(119904) 1199062(119904)] 119889119904

+ int119905

0

[119866 (119904) 119909 (119904)] 119889119861 (119904)

119910 (119905) = 119872119909 (119879) + int119879

119905

[119860 (119904) 119910 (119904) + 119867 (119904) 119909 (119904) + 119866 (119904) 119911 (119904)

+ 1198681(119904) 1199061(119904) + 119868

2(119904) 1199062(119904)] 119889119904

minus int119879

119905

119911 (119904) 119889119861 (119904)

(26)

with the quadratic cost functional

119869119894(1199061(sdot) 1199062(sdot)) = 119864 [int

119879

0

119871119894(119905) 1199062

119894(119905) 119889119905

+ 1198761198941199092(119879) + 119873

1198941199102(0) ] 119894 = 1 2

(27)

where 119860 119861 1198631 1198632 119866 119867 119868

1 1198682 1198711 and 119871

2are one-

dimensional deterministic bounded measurable functionsand119872119873

111987321198761 and119876

2are constants Also assume 119861 ge 0

1198731ge 0 119873

2ge 0 119876

1ge 0 119876

2ge 0 119872 ge 120573 119871

1ge 120573 119871

2ge 120573

119867 gt 120573 where 120573 gt 0 is a positive constantUnder the above assumptions on the coefficients of

(26) and (27) it is easy to check that for any admissible(1199061(sdot) 1199062(sdot)) isin A

1times 1198602 the state system (26) has a unique

solution and the corresponding stochastic differential gameproblem is well defined For this case the corresponding 119867becomes

119867119894(119905 119909 119910 119911 119906

1 1199062 119901119894 119902119894 119896119894)

= minus119896119894[119860 (119905) 119910 + 119867 (119905) 119909 + 119866 (119905) 119911 + 119868

1(119905) 1199061+ 1198682(119905) 1199062]

+ 119901119894[119860 (119905) 119909 minus 119861 (119905) 119910 + 119863

1(119905) 1199061+ 1198632(119905) 1199062(119904)]

+ 119902119894119866 (119905) 119909 + 119871

119894(119905) 1199062

119894(119905) 119894 = 1 2

(28)

The corresponding adjoint equation associated with anadmissible control pair (119906

1(sdot) 1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot)) becomes

119896119894(119905) = minus2119873

119894119910 (0) + int

119905

0

[119860 (119904) 119896119894(119904) + 119861 (119904) 119901

119894(119904)] 119889119904

+ int119905

0

119866 (119904) 119896119894(119904) 119889119861 (119904)

Mathematical Problems in Engineering 7

119901119894(119905) = minus119872119896

119894(119879) + 2119876

119894119909 (119879)

+ int119879

119905

[119860 (119904) 119901119894(119904) + 119866 (119904) 119902

119894(119904) minus 119867 (119904) 119896

119894(119904)] 119889119904

minus int119879

119905

119902119894(119904) 119889119861 (119904) 119894 = 1 2

(29)

It is easy to check that the state system (26) has a uniquesolution (119901119894(sdot) 119902119894(sdot) 119896119894(sdot)) 119894 = 1 2

Suppose that (1199061(sdot) 1199062(sdot)) is a Nash equilibrium point

By the maximum principle (see Theorem 5) putting theoptimality conditions (22) and (23) the corresponding stateequation (26) and the adjoint equations (28) associated with(1199061(sdot) 1199062(sdot)) together we obtain the following optimality

system for a Nash equilibrium point

119909 (119905) = 119886 + int119905

0

[119860 (119904) 119909 (119904) minus 119861 (119904) 119910 (119904) + 1198631(119904) 1199061(119904)

+1198632(119904) 1199062(119904)] 119889119904

+ int119905

0

119866 (119904) 119909 (119904) 119889119861 (119904)

119910 (119905) = 119872119909 (119879) + int119879

119905

[119860 (119904) 119910 (119904) + 119867 (119904) 119909 (119904) + 119866 (119904) 119911 (119904)

+ 1198681(119904) 1199061(119904) + 119868

2(119904) 1199062(119904)] 119889119904

minus int119879

119905

119911 (119904) 119889119861 (119904)

119896119894(119905) = minus2119873

119894119910 (0) + int

119905

0

[119860 (119904) 119896119894(119904) + 119861 (119904) 119901

119894(119904)] 119889119904

+ int119905

0

119866 (119904) 119896119894(119904) 119889119861 (119904)

119901119894(119905) = minus119872119896

119894(119879) + 2119876

119894119909 (119879)

+ int119879

119905

[119860 (119904) 119901119894(119904) + 119866 (119904) 119902

119894(119904) minus 119867 (119904) 119896

119894(119904)] 119889119904

minus int119879

119905

119902119894(119904) 119889119861 (119904)

1198671(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

1(119905))

= min1199061isin1198801

1198671(119905 Θ (119905) 119906

1 1199062(119905) Λ

1(119905))

1198672(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

2(119905))

= min1199062isin1198802

1198672(119905 Θ (119905) 119906

1(119905) 1199062 Λ2(119905))

119905 isin [0 119879] 119894 = 1 2

(30)

This is called coupled forward-backward stochastic differ-ential equationswith double dimensions (DFBSDE for short)

Note that the coupling comes from the two last relations(which is the maximum condition) We also refer to Yu [17]for the general theory of this kind equationThe8-tuple (119906

1(sdot)

1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot) 119901(sdot) 119902(sdot) 119896(sdot)) of F-adapted processes

satisfying the above is called an adapted solution of (30)We now look at the sufficiency of the existence of a Nashequilibrium point

Theorem 6 Suppose that (1199061(sdot) 1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot) 119901(sdot)

119902(sdot) 119896(sdot)) is an adapted solution to DFBSDE (30) Then(1199061(sdot) 1199062(sdot)) is a Nash equilibrium point

Proof Let (1199061(sdot) 1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot) 119901(sdot) 119902(sdot) 119896(sdot)) be an

adapted solution to DFBSDE (30) For any admissible controlpair (119906

1(sdot) 1199062(sdot)) isin A

1times 1198602 fromTheorem 3 we have

119869119894(1199061(sdot) 1199062(sdot)) minus 119869

119894(1199061(sdot) 1199062(sdot))

= Eint119879

0

[1198671(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

119906

119894(119905))

minus119867119894(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

119906

119894(119905)) ] 119889119905

+ E [1198731(119910(11990611199062)(0))2

minus 1198731(119910 (0))

2

minus21198731119910 (0) (119910

(11990611199062)(0) minus 119910 (0)) ]

+ E [1198761(119909(11990611199062)(119879))2

minus 11987611199092(119879)

minus 211987611199092(119879) sdot (119909

(11990611199062)(119879) minus 119909 (119879)) ]

= Eint119879

0

[1198671(119905 Θ (119905) 119906

120576(119905) Λ

119906

119894(119905))

minus1198671(119905 Θ (119905) 119906 (119905) Λ

119906

1(119905))] 119889119905

+ E [1198731(119910(11990611199062)(0) minus 119910 (0))

2

]

+ E [1198761(119909(11990611199062)(119879) minus 119909 (119879))

2

]

ge 0

(31)

which implies that

1198691(1199061(sdot) 1199062(sdot)) = inf

1199061(sdot)isinA

1

1198691(1199061(sdot) 1199062(sdot)) (32)

Similarly we can get

1198692(1199061(sdot) 1199062(sdot)) = inf

1199062(sdot)isinA

2

1198692(1199061(sdot) 1199062(sdot)) (33)

Therefore (1199061(sdot) 1199062(sdot)) is a Nash equilibrium point

Proposition 7 Suppose that 1198801times 1198802

= R times R and(1199061(sdot) 1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot) 119901(sdot) 119902(sdot) 119896(sdot)) is an adapted solu-

tion to DFBSDE (30) Then (1199061(sdot) 1199062(sdot)) has the following

representation

119906119894(119905) =

1

2119871119894(119905)

[119896 (119905) 119868119894(119905) minus 119901

119894(119905) 119863119894(119905)] 119894 = 1 2 (34)

8 Mathematical Problems in Engineering

Proof From Theorem 6 we deduce that (1199061(sdot) 1199062(sdot)) is a

Nash equilibrium point Then from 1198801times 1198802= R times R and

the optimality conditions (22) and (23) it follows that

11986711199061

(119905 Θ (119905) 1199061(119905) 1199062(119905) Λ

1(119905)) = 0

11986721199062

(119905 Θ (119905) 1199061(119905) 1199062(119905) Λ

2(119905)) = 0

(35)

which imply that

119906119894(119905) =

1

2119871119894(119905)

[119896 (119905) 119868119894(119905) minus 119901

119894(119905) 119863119894(119905)] 119894 = 1 2 (36)

The proof is complete

Remark 8 In summary DFBSDE (30) completely character-izes the Nash equilibrium point Therefore solving the dif-ferential game problem is equivalent to solving the DFBSDE(30) Moreover a candidate equilibrium point can be givenby (34) We refer the reader to Yu [17] for the theory ofsolvability toDFBSDE (30) Since the linear quadratic controlproblem is an important and fascinating class of stochasticcontrol ones and the theoretical results of this problem havelots of significant impacts on a wide range of engineeringmanaging and financial applications in the future we willfocus on the study of financial application as the reviewerssuggest

7 Conclusion

In this paper a two-person nonzero-sum differential gameis studied for a fully coupled forward-backward stochasticsystem with the control process 119906(sdot) not appearing in theforward diffusion term but the control domain not neces-sarily convex For this case we obtain a variation formulafor the cost functional As an application the maximumprinciple for open-loop Nash equilibrium points is estab-lished Finally we work out an example of linear quadraticnonzero-sum differential games to illustrate our stochasticmaximum principle As the reviewers suggest our systemdiscussed in this paper may be extended to the discontinuoussystem such Markov-switching jump-diffusion system forwhich the optimal control problems and differential gameshave extensive applications in industry and finance (seeeg [18 19] and the references therein) Some investigationson this topic will be studied and carried out in our futurepublications

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was partially supported by the National Nat-ural Science Foundation of China (11101140 11301177)the China Postdoctoral Science Foundation (2011M5007212012T50391) and theNatural Science Foundation of ZhejiangProvince (nos Y6110775 and Y6110789)

References

[1] J-M Bismut ldquoConjugate convex functions in optimal stochas-tic controlrdquo Journal of Mathematical Analysis and Applicationsvol 44 no 3 pp 384ndash404 1973

[2] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990

[3] S Tang ldquoGeneral linear quadratic optimal stochastic controlproblems with random coefficients linear stochastic Hamiltonsystems and backward stochastic Riccati equationsrdquo SIAMJournal on Control and Optimization vol 42 no 1 pp 53ndash752003

[4] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward-backward stochastic differential equationswith jumpsrdquo SIAM Journal on Control and Optimization vol48 no 5 pp 2945ndash2976 200910

[5] B Oslashksendal and A Sulem ldquoForward backward stochasticdifferential games and stochastic control under model uncer-taintyrdquo Journal of Optimization Theory and Applications 2012

[6] W S Xu ldquoStochastic maximum principle for optimal controlproblem of forward and backward systemrdquo Australian Mathe-matical Society Journal B vol 37 no 2 pp 172ndash185 1995

[7] Z Wu ldquoMaximum principle for optimal control problem offully coupled forward-backward stochastic systemsrdquo SystemsScience and Mathematical Sciences vol 11 no 3 pp 249ndash2591998

[8] J-T Shi and Z Wu ldquoThe maximum principle for fully coupledforward-backward stochastic control systemrdquo Acta AutomaticaSinica vol 32 no 2 pp 161ndash169 2006

[9] QMeng ldquoAmaximumprinciple for optimal control problem offully coupled forward-backward stochastic systems with partialinformationrdquo Science in China A vol 52 no 7 pp 1579ndash15882009

[10] J Shi and Z Wu ldquoMaximum principle for forward-backwardstochastic control system with random jumps and applicationsto financerdquo Journal of Systems Science amp Complexity vol 23 no2 pp 219ndash231 2010

[11] Z Wu ldquoA general maximum principle for optimal control offorward-backward stochastic systemsrdquo Automatica vol 49 no5 pp 1473ndash1480 2013

[12] J Yong ldquoOptimality variational principle for controlledforward-backward stochastic differential equations with mixedinitial-terminal conditionsrdquo SIAM Journal on Control andOptimization vol 48 no 6 pp 4119ndash4156 2010

[13] E C M Hui and H Xiao ldquoMaximum principle for differentialgames of forward-backward stochastic systems with applica-tionsrdquo Journal of Mathematical Analysis and Applications vol386 no 1 pp 412ndash427 2012

[14] M Tang ldquoMaximum principle for non-zero sum stochasticd-ifferential games of fully coupled forward-backward stochasticsystemsrdquo preprint

[15] S Peng and ZWu ldquoFully coupled forward-backward stochasticdifferential equations and applications to optimal controlrdquoSIAM Journal on Control and Optimization vol 37 no 3 pp825ndash843 1999

[16] Q Lin ldquoOptimal control of coupled forward-backward stochas-tic system with jumps and related hamilton-jacobi-bellmanequationsrdquo httparxivorgabs11114642

[17] Z Yu ldquoLinear-quadratic optimal control and nonzero-sum dif-ferential game of forward-backward stochastic systemrdquo AsianJournal of Control vol 14 no 1 pp 173ndash185 2012

Mathematical Problems in Engineering 9

[18] B Z Temocin and G-W Weber ldquoOptimal control of stochastichybrid systemwith jumps a numerical approximationrdquo Journalof Computational and Applied Mathematics vol 259 pp 443ndash451 2014

[19] N Azevedo D Pinheiro and G W Weber ldquoDynamic pro-gramming for a Markov-switching jumpCdiffusionrdquo Journal ofComputational andAppliedMathematics vol 267 pp 1ndash19 2014

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Stochastic AnalysisInternational Journal of

Page 2: Research Article A Variational Formula for Nonzero-Sum ...downloads.hindawi.com › journals › mpe › 2014 › 283418.pdf · the forward stochastic system. e existence and uniqueness

2 Mathematical Problems in Engineering

and verification theorem under convex control domainDifferent from the decoupled forward-backward stochasticsystem studied in [5 13] Tang [14] studied differential gamesfor fully coupled FBSDEs under convex control domainand established the local maximum principle and the veri-fication theorem In this paper we will also consider two-person nonzero differential games of fully coupled forward-backward stochastic systems Different from [5 13 14] thecontrol domain discussed in this paper is not necessarilyconvex The main contribution of our paper is to get directlya variation formula for the cost functional in a given spikeperturbation direction of control processes in terms of theHamiltonian and the associated adjoint system which is alinear FBSDEs and neither the variational systems nor thecorresponding Taylor type expansions of the state processand the cost functional will be considered As an applicationa global stochastic maximum principle for open-loop Nashequilibrium points is established And as a result a linearquadratic nonzero-sum game problem is studied to illustratethat the established theories and the corresponding Nashequilibrium point are characterized by the optimality system

The paper is organized as follows In Section 2 we formu-late the problem and give various assumptions used through-out the paper In Section 3 we obtain the representation forthe difference of the performance functional in terms of theHamilton and adjoint processes In Section 4 we use therepresentation in Section 3 to derive a representation for thevariation and ldquodirectional derivativerdquo of the difference of theperformance functional along with spike variation Section 5is devoted to deriving the global stochastic maximum prin-ciples by the ldquodirectional derivativerdquo formula established inSection 4 A linear quadratic nonzero-sum game problem isstudied in Section 6 In Section 7 we conclude this paper

2 Formulation of the Problem

Let (ΩF F119905119905ge0 119875) be a complete probability space on

which a 119889-dimensional standard Brownian motion 119861(sdot) isdefined with F

119905119905ge0

being its natural filtration augmentedby all 119875-null sets in F Let 119879 gt 0 be a fixed time horizonLet 119864 be a Euclidean space The inner product in 119864 isdenoted by ⟨sdot sdot⟩ and the norm in 119864 is denoted by | sdot | Wefurther introduce some other spaces that will be used in thepaper Denote by 1198712(ΩF

119879 119875 119864) the set of all 119864-valuedF

119879-

measurable random variable 120578 such that E|120578|2 lt infin Denoteby1198722F(0 119879 119864) the set of all 119864-valuedF

119905-adapted stochastic

processes 120593(119905) 119905 isin [0 119879] which satisfy Eint119879

0|120593(119905)|2119889119905 lt

infin Denote by S2F(0 119879 119864) set of all 119864-valued F119905-adapted

continuous stochastic processes 120593(119905) 119905 isin [0 119879] whichsatisfy Esup

0le119905le119879|120593(119905)|2 lt infin Finally we define M2[0 119879] =

1198782F(0 119879R119899)times1198782F(0 119879R

119898)times1198722F(0 119879R119898times119889)ThenM2[0 119879]

is a Banach space with respect to the norm sdot M2 given byΘ(sdot)

2

M2 = Esup0le119905le119879

|119909(119905)|2+Esup0le119905le119879

|119910(119905)|2+Eint119879

0|119911(119905)|2119889119905

for Θ(sdot) = (119909(sdot) 119910(sdot) 119911(sdot)) isin M2[0 119879]In the following we specify two-person nonzero-sum

differential game problem of fully coupled forward-backward

stochastic systems More precisely for 119886 isin R119899 we consider afully coupled nonlinear FBSDE

119909 (119905) = 119886 + int119905

0

119887 (119904 119909 (119904) 119910 (119904) 119911 (119904) 1199061(119904) 1199062(119904)) 119889119904

+ int119905

0

120590 (119904 119909 (119904) 119910 (119904) 119911 (119904)) 119889119861 (119904)

119910 (119905) = ℎ (119909 (119879)) + int119879

119905

119891 (119904 119909 (119904) 119910 (119904) 119911 (119904) 1199061(119904) 1199062(119904)) 119889119904

minus int119879

119905

119911 (119904) 119889119861 (119904)

(1)

The processes 1199061(sdot) and 119906

2(sdot) in the system (1) are the open-

loop control processes which present the controls of the twoplayers

For each one of the two players there is a cost functional

119869119894(1199061(sdot) 1199062(sdot))

= 119864 [int119879

0

119897119894(119905 119909 (119905) 119910 (119905) 119911 (119905) 119906

1(119905) 1199062(119905)) 119889119905

+ 120601119894(119909 (119879)) + 120574

119894(119910 (0)) ] 119894 = 1 2

(2)

In the above 119887 [0 119879]timesR119899timesR119898timesR119898times119889times1198801times1198802rarr R119899

120590 [0 119879] times R119899 times R119898 times R119898times119889 rarr R119899times119889 119891 [0 119879] times

R119899 times R119898 times R119898times119889 times 1198801times 1198802

rarr R119898 ℎ R119899 rarr R119898and 119897119894 [0 119879] times R119899 times R119898 times R119898times119889 times 119880

1times 1198802rarr 119877 120601

119894

R119899 rarr R 120574119894 R119898 rarr R are given Borel measurable mapping

(119894 = 1 2) Here 1198801sub R1198961 and 119880

2sub R1198962 are nonempty

Borel subsets The admissible control process (1199061(sdot) 1199062(sdot))

is defined as a F119905-adapted process with values in 119880

1times 1198802

such that sup0le119905le119879

E[|1199061(119905)|2 + |119906

2(119905)|2] lt infin The set of all

admissible control processes is denoted byA1timesA2

Before giving the basic assumptions on the coefficientsthroughout this paper we first introduce some abbreviationsLet 119866 be a given 119898 times 119899 full-rank matrix Denote 120579 = (

119909

119910

119911)

and 119860(119905 120579) = (minus119866lowast119891

119866119887

119866120590

) where 119866lowast is the transpose matrix of

119866 For all 120579 = (119909 119910 119911) and 120579 = (119909 119910 119911) we denote 119909 = 119909 minus 119909119910 = 119910 minus 119910 = 119911 minus 119911

Assumption 1 (i) The mappings 119891 119887 120590 and ℎ are con-tinuously differentiable with respect to (119909 119910 119911) and thecorresponding derivatives are bounded Moreover 119891 119887 and120590 are bounded by (1+ |119909|+ |119910|+ |119906

1| + |1199062|) and ℎ is bounded

by (1 + |119909|)(ii) The mappings 119891 119887 120590 and ℎ satisfy the following

Monotonicity conditions

⟨119860 (119905 120579 119906) minus 119860 (119905 120579 119906) 120579 minus 120579⟩

le minus1205731|119866119909|2minus 1205732[1003816100381610038161003816119866lowast11991010038161003816100381610038162

+1003816100381610038161003816119866lowast10038161003816100381610038162

]

⟨ℎ (119909) minus ℎ (119909) 119909 minus 119909⟩ ge 1205831|119866119909|2

(3)

Mathematical Problems in Engineering 3

or

⟨119860 (119905 120579 119906) minus 119860 (119905 120579 119906) 120579 minus 120579⟩

ge 1205731|119866119909|2+ 1205732(1003816100381610038161003816119866lowast11991010038161003816100381610038162

+1003816100381610038161003816119866lowast10038161003816100381610038162

)

⟨ℎ (119909) minus ℎ (119909) 119909 minus 119909⟩ le minus1205831|119866119909|2

(4)

Here 1205731 1205732 and 120583

1are given nonnegative constants with 120573

1+

1205732gt 0 120573

2+ 1205831gt 0 Moreover we have 120573

1gt 0 120583

1gt 0 (resp

1205732gt 0) when119898 gt 119899 (resp119898 lt 119899)(iii) The mappings 119897

119894 120601119894 and 120574

119894are continuously differen-

tiable with respect to (119909 119910 119911) (119894 = 1 2) And 119897119894is bounded by

(1 + |119909|2 + |119910|2 + |119911|2 + |1199061|2 + |119906

2|2) And the derivatives of

119897119894with respect to (119909 119910 119911) (119894 = 1 2) are bounded by (1 + |119909| +

|119910| + |119911| + |1199061| + |1199062|) And 120601

119894and ℎ119894are bounded by (1 + |119909|2)

and (1 + |119910|2) respectively And the derivatives of 120601119894and ℎ

119894

with respect to 119909 and 119910 are bounded by (1 + |119909|) and (1 + |119910|)respectively (119894 = 1 2)

Under Assumption 1 we see that for any given admissiblecontrol 119906(sdot) = (119906

1(sdot) 1199062(sdot)) isin A

1timesA2 the system (1) admits

a unique solution (119909119906(sdot) 119910119906(sdot) 119911119906(sdot)) isin M2 (see [15])Then wecall (119909119906(sdot) 119910119906(sdot) 119911119906(sdot)) or (119909(sdot) 119910(sdot) 119911(sdot)) if its dependence onadmissible control 119906(sdot) is clear from context the state processcorresponding to the control process 119906(sdot) = (119906

1(sdot) 1199062(sdot)) and

(119906(sdot) 119909(sdot) 119910(sdot) 119911(sdot)) the admissible pairThenwe can pose the following two-person nonzero-sum

stochastic differential game problem

Problem 2 Find an open-loop admissible control(1199061(sdot) 1199062(sdot)) isin A

1timesA2such that

1198691(1199061(sdot) 1199062(sdot)) = inf

1199061(sdot)isinA1

1198691(1199061(sdot) 1199062(sdot)) (5)

1198692(1199061(sdot) 1199062(sdot)) = inf

1199062(sdot)isinA

2

1198692(1199061(sdot) 1199062(sdot)) (6)

Any (1199061(sdot) 1199062(sdot)) isin A

1timesA2satisfying the above is called

an open-loop Nash equilibrium point of Problem 2 Suchan admissible control allows two players to play individualoptimal control strategies simultaneously

3 Representation for Difference ofthe Cost Functional

This section is devoted to establishing a representation for thedifference of the cost functional according to Hamiltonianand the adjoint processes of Problem 2

To simplify our notation for any admissible control119906(sdot) = (119906

1(sdot) 1199062(sdot)) we write Θ(119905) = (119909(119905) 119910(119905) 119911(119905)) as

the corresponding state process We define the Hamiltonianfunctions 119867

119894 [0 119879] times R119899 times R119898 times R119898times119889 times U

1times U2times R119899 times

R119899times119889 timesR119898 rarr R by

119867119894(119905 119909 119910 119911 119906

1 1199062 119901 119902 119896) = ⟨119896 minus119891 (119905 119909 119910 119911 119906

1 1199062)⟩

+ ⟨119901 119887 (119905 119909 119910 119911 1199061 1199062)⟩

+ 119897119894(119905 119909 119910 119911 119906

1 1199062)

+ ⟨119902 120590 (119905 119909 119910 119911)⟩ 119894 = 1 2

(7)

For any admissible pair (119906(sdot) Θ(sdot)) we define the corre-sponding adjoint process Λ119906

119894(sdot) = (119896119906

119894(sdot) 119901119906119894(sdot) 119902119906119894(sdot))(119894 = 1 2)

as the solution to the following FBSDEs

119889119896119894(119905) = minus119867

119894119910(119905) 119889119905 minus 119867

119894119911(119905) 119889119861 (119905)

119889119901119894(119905) = minus119867

119894119909(119905) 119889119905 + 119902

119894(119905) 119889119861 (119905)

119896119894(0) = minus120574

119894119910(119910 (0))

119901119894(119879) = minusℎ

lowast

119909(119909 (119879)) 119896

119894(119879) + 120601

119894119909(119909 (119879))

119894 = 1 2

(8)

where we have used the short hand notation

119867119894119910(119905) = 119867

119894119910(119905 119906 (119905) Θ (119905) Λ

119906

119894(119905)) 119894 = 12 (9)

And similarly we can define119867119894119909(119905) and119867

119894119911(119905)

Under Assumptions 1 it is easy to see thatthe above adjoint equations have unique solutionΛ119906

119894(sdot) = (119896119906

119894(sdot) 119901119906119894(sdot) 119902119906119894(sdot)) isin M2[0 119879] 119894 = 1 2

Now let (119906120576(sdot) Θ120576(sdot)) = (1199061205761(sdot) 1199061205762(sdot) 119909120576(sdot) 119910120576(sdot) 119911120576(sdot)) be

another admissible pair In the following we will give thepresentation for the difference 119869

119894(119906120576(sdot)) minus 119869

119894(119906(sdot)) (119894 = 1 2)

in terms of the Hamiltonian119867119894(119894 = 1 2) and adjoint process

Λ119906119894(sdot) (119894 = 1 2) associated with the admissible control pair

(119906(sdot) Θ119906(sdot)) as well as other relevant expressions We stateour result as follows

Theorem 3 Under Assumptions 1 one has the representationfor the difference of the cost functional as follows sdot(119910120576(0) minus119910(0))(119910120576(0))]

119869119894(119906120576

1(sdot) 119906120576

2(sdot)) minus 119869

119894(1199061(sdot) 1199062(sdot))

= Eint119879

0

[119867119894(119905 Θ (119905) 119906

120576(119905) Λ

119906

119894(119905))

minus 119867119894(119905 Θ (119905) 119906 (119905) Λ

119906

119894(119905))

+ 119867119894(119905 Θ120576(119905) 119906120576(119905) Λ

119906

119894(119905))

minus 119867119894(119905 Θ (119905) 119906

120576(119905) Λ

119906

119894(119905))

minus ⟨119867119894119909(119905) 119909120576(119905) minus 119909 (119905)⟩

minus ⟨119867119894119910(119905) 119910120576(119905) minus 119910 (119905)⟩

minus ⟨119867119894119911(119905) 119911120576(119905) minus 119911 (119905)⟩] 119889119905

4 Mathematical Problems in Engineering

+ E [120574119894(119910120576(0)) minus 120574

119894(119910 (0)) minus 120574

119894119910(119910 (0))

sdot (119910120576(0) minus 119910 (0)) (119910

120576(0)) ]

+ E [120601119894(119909120598(119879)) minus 120601

119894(119909 (119879)) minus 120601

119894119909(119909 (119879))

sdot (119909120576(119879) minus 119909 (119879))]

minus E [⟨ℎ (119909120598(119879)) minus ℎ (119909 (119879)) 119896

119906

119894(119879)⟩

minus ⟨ℎ119909(119909 (119879)) (119909

120576(119879) minus 119909 (119879)) 119896

119906

119894(119879)⟩]

(10)

Proof First from the definition of 119867 (see (7)) it is easy tocheck that

119867119894(119905 Θ120576(119905) 119906120576(119905) Λ

119906

119894(119905)) minus 119867

119894(119905 Θ (119905) 119906 (119905) Λ

119906

119894(119905))

= 119867119894(119905 Θ (119905) 119906

120576(119905) Λ

119906

119894(sdot)) minus 119867

119894(119905 Θ (119905) 119906 (119905) Λ

119906

119894(119905))

+ 119867119894(119905 Θ120576(119905) 119906120576(119905) Λ

119906

119894(119905))

minus 119867119894(119905 Θ (119905) 119906

120576(119905) Λ

119906

119894(119905)) 119894 = 1 2

(11)

From (1) we know that

119909120576(119905) minus 119909 (119905) = int

119905

0

[119887 (119904 Θ120576(119904) 119906120576(119904))

minus 119887 (119904 Θ (119904) 119906 (119904))] 119889119904

+ int119905

0

[120590 (119904 Θ120576(119904) 119906120576(119904))

minus 120590 (119904 Θ (119904) 119906 (119904))] 119889119882 (119904)

119910120576(119905) minus 119910 (119905) = ℎ (119909

120598(119879)) minus ℎ (119909 (119879))

+ int119879

119905

[119891 (119904 Θ120576(119904) 119906120576(119904))

minus 119891 (119904 Θ (119904) 119906 (119904))] 119889119904

minus int119879

119905

[119911120576(119904) minus 119911 (119904)] 119889119882 (119904)

(12)

Furthermore recalling (8) and applying Ito formula to⟨119896119906119894(119905) 119910120576(119905) minus 119910(119905)⟩ + ⟨119901119906

119894(119905) 119909120576(119905) minus 119909(119905)⟩ we deduce that

Eint119879

0

⟨119896119906

119894(119905) minus (119891 (119905 Θ

120576(119905) 119906120576(119905) )

minus119891 (119905 Θ (119905) 119906 (119905)))⟩ 119889119905

+ 119864int119879

0

⟨119901119906

119894(119905) 119887 (119905 Θ

120576(119905) 119906120576(119905))

minus 119887 (119905 Θ (119905) 119906 (119905))⟩ 119889119905

+ 119864int119879

0

⟨119902119906

119894(119905) 120590 (119905 Θ

120576(119905) 119906120576(119905))

minus 120590 (119905 Θ (119905) 119906 (119905))⟩ 119889119905

= Eint119879

0

[⟨119867119894119909(119905) 119909120576(119905) minus 119909 (119905)⟩

+ ⟨119867119894119910(119905) 119910120576(119905) minus 119910 (119905)⟩

+ ⟨119867119894119911(119905) 119911120576(119905) minus 119911 (119905)⟩] 119889119905

+ E [120574119894119910(119910 (0)) sdot (119910

120576(0) minus 119910 (0))]

+ E [120601119894119909(119909 (119879)) sdot (119909

120576(119879) minus 119909 (119879))]

+ E [⟨ℎ (119909120576(119879)) minus ℎ (119909 (119879)) 119896

119906

119894(119879)⟩

minus ⟨ℎ119909(119909 (119879)) (119909

120576(119879) minus 119909 (119879)) 119896

119906

119894(119879)⟩]

119894 = 1 2

(13)

On the other hand by the definition of theHamilton function119867 (see (7)) we deduce that

119869119894(119906120576

1(sdot) 119906120576

2(sdot)) minus 119869

119894(1199061(sdot) 1199062(sdot))

= Eint119879

0

[119897119894(119905 Θ120576(119905) 119906120576(119905)) minus 119897

119894(119905 Θ (119905) 119906 (119905))] 119889119905

+ E [120574119894(119910120576(0)) minus 120574

119894(119910 (0))]

+ 119864 [120601119894(119909120598(119879)) minus 120601

119894(119909 (119879))]

= Eint119879

0

[119867119894(119905 Θ120576(119905) 119906120576(119905) Λ

119906

119894(119905))

minus119867119894(119905 Θ (119905) 119906 (119905) Λ

119906

119894(119905))] 119889119905

minus Eint119879

0

⟨119896119906

119894(119905) minus (119891 (119905 Θ

120576(119905) 119906120576(119905))

minus119891 (119905 Θ (119905) 119906 (119905)))⟩ 119889119905

minus Eint119879

0

⟨119901119906

119894(119905) 119887 (119905 Θ

120576(119905) 119906120576(119905))

minus119887 (119905 Θ (119905) 119906 (119905))⟩ 119889119905

minus Eint119879

0

⟨119902119906

119894(119905) 120590 (119905 Θ

120576(119905) 119906120576(119905))

minus120590 (119905 Θ (119905) 119906 (119905)) ⟩ 119889119905

+ E [120574119894(119910120576(0)) minus 120574

119894(119910 (0))]

+ 119864 [120601119894(119909120598(119879)) minus 120601

119894(119909 (119879))]

(14)

Now putting (11) and (13) into (14) we deduce that (10)holds The proof is complete

Mathematical Problems in Engineering 5

4 A Variational Formula for StochasticDifferential Games

In this section we will obtain a directional derivative ata given admissible control process in some given controlprocess direction The choice of the given control processdirection depends on the convexity of the control domain1198801times 1198802 If the control domain 119880

1times 1198802is convex a classical

way of treating such a problem consists of using the convexperturbation method More precisely if (119906

1(sdot) 1199062(sdot)) is a

given admissible control and (V1(sdot) V2(sdot)) is an arbitrary

given admissible control we can define a convex perturbedadmissible control as

119906120576

119894(sdot) = 119906

119894(sdot) + 120576 (V

119894(sdot) minus 119906

119894(sdot)) 119894 = 1 2 (15)

where 120576 is a sufficiently small positive constant Then onecan prove the cost functional 119869

119894(1199061(sdot) 1199062(sdot)) is Gateaux dif-

ferentiable at 119906119894(sdot) in the direction V

119894(sdot) minus 119906

119894(sdot) (119894 = 1 2)

and get a local stochastic maximum principle for open-loop Nash equilibrium points see for example [13 14]Different from [13 14] our control domain in the presentpaper is not necessarily convex so the convex perturbedcontrol 119906120576

119894(sdot) may no longer be admissible and the convex

perturbation method cannot be used to obtain the corre-sponding variational formula and maximum principle Aclassical way of treating the nonconvex control domainconsists of using the spike variations perturbation methodMore precisely let (119906(sdot) Θ(sdot)) = (119906

1(sdot) 1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot))

be any given admissible pair with the corresponding adjointprocess Λ119906

119894(sdot) = (119896119906

119894(sdot) 119901119906119894(sdot) 119902119906119894(sdot)) isin M2[0 119879] 119894 = 1 2 We

define the following spike variations

119906120591120576

119894(119905) =

119906119894(119905) if 119905 isin [120591 120591 + 120576]

119906119894(119905) otherwise

(16)

with fixed 120591 isin [0 119879) sufficiently small positive 120576 and anygiven admissible control 119906

119894(sdot) isin A

1 119894 = 1 2

Now we state the following variational formula for thecost functional (2) associated with the spike variation (16) ina unified way

Theorem 4 Under Assumption 1 one has a variational for-mula for the cost functional (5) and (6) as follows

119889

1198891205761198691(119906120591120576

1(sdot) 1199062(sdot))

10038161003816100381610038161003816100381610038161003816120576=0

= lim120576rarr0

1198691(1199061205911205761(sdot) 1199062(sdot)) minus 119869

1(1199061(sdot) 1199062(sdot))

120576

= E [1198671(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(120591))

minus 1198671(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(119905))]

(17)

119889

1198891205761198692(1199061(sdot) 119906120591120576

2(sdot))

10038161003816100381610038161003816100381610038161003816120576=0

= lim120576rarr0

1198692(1199061(sdot) 119906120591120576

2(sdot)) minus 119869

2(1199061(sdot) 1199062(sdot))

120576

= E [1198672(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(120591))

minus 1198672(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(120591))]

(18)

Proof We first prove the equality (17) Let Θ1120576(sdot) =

(1199091120576(sdot) 1199101120576(sdot) 1199111120576(sdot)) be the state process corresponding tothe admissible control (119906120591120576

1(sdot) 1199062(sdot)) Under Assumption 1

by the continuous dependence theory of FBSDEs (seeProposition 32 in [16]) we have

10038171003817100381710038171003817Θ1120576(sdot) minus Θ

119906(sdot)10038171003817100381710038171003817M2

le 1198621205762 (19)

InTheorem 3 replacing (119906120576(sdot) Θ120576(sdot)) by (1199061205911205761(sdot) 1199062(sdot) Θ1120576(sdot))

we have

1198691(119906120591120576

1(sdot) 1199062(sdot)) minus 119869

1(1199061(sdot) 1199062(sdot))

= Eint119879

0

[119867119894(119905 Θ (119905) 119906

120591120576

1(119905) 1199062(119905) Λ

119906

119894(119905))

minus 119867119894(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

119906

119894(119905))

+ 119867119894(119905 Θ120576(119905) 119906120591120576

1(119905) 1199062(119905) Λ

119906

119894(119905))

minus 119867119894(119905 Θ (119905) 119906

120591120576

1(119905) 1199062(119905) Λ

119906

119894(119905))

minus ⟨119867119894119909(119905) 119909120576(119905) minus 119909 (119905)⟩

minus ⟨119867119894119910(119905) 119910120576(119905) minus 119910 (119905)⟩

minus ⟨119867119894119911(119905) 119911120576(119905) minus 119911 (119905)⟩ ] 119889119905

+ E [120574119894(119910120576(0)) minus 120574

119894(119910 (0)) minus 120574

119894119910(119910 (0))

sdot (119910120576(0) minus 119910 (0))]

+ E [120601119894(119909120598(119879)) minus 120601

119894(119909 (119879)) minus 120601

119894119909(119909 (119879))

sdot (119909120576(119879) minus 119909 (119879))]

minus E [⟨ℎ (119909120598(119879)) minus ℎ (119909 (119879)) 119896

119906

119894(119879)⟩

minus ⟨ℎ119909(119909 (119879)) (119909

120576(119879) minus 119909 (119879)) 119896

119906

119894(119879)⟩]

(20)

Combining (19) and Assumption 1 by Taylor Expansions on119867 and the dominated convergence theorem from (20) weconclude that

1198691(119906120591120576

1(sdot) 1199062(sdot)) minus 119869

1(1199061(sdot) 1199062(sdot))

= Eint119879

0

[1198671(119905 Θ (119905) 119906

120591120576

1(119905) 1199062(119905) Λ

119906

1(119905))

minus1198671(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

119906

1(119905))] 119889119905 + 119900 (120576)

6 Mathematical Problems in Engineering

= 120576E [1198671(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(120591))

minus1198671(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(119905))] + 119900 (120576)

(21)

which imply that (17) holdsSimilarly we can prove that (18) holds The proof is

complete

5 Stochastic Maximum Principle

In this section applying the variational formulas (17) and (18)we will state and prove the global maximum principle for theNash equilibrium points of Problem 2

Theorem 5 Under Assumption 1 let 119906(sdot) = (1199061(sdot) 1199062(sdot)) be

a Nash equilibrium point of Problem 2 with the state processΘ(sdot) = (119909(sdot) 119910(sdot) 119911(sdot)) Let Λ

119894(sdot) = (119901

119894(sdot) 119902119894(sdot) 119896119894(sdot)) (119894 = 1 2)

be the unique solution of the adjoint equation (8) correspondingto (119906(sdot) Θ(sdot)) Then

1198671(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

1(120591))

= min1199061isin1198801

1198671(120591 Θ (120591) 119906

1 1199062(120591) Λ

1(120591))

(22)

1198672(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

2(120591))

= min1199062isin1198802

1198672(120591 Θ (120591) 119906

1(120591) 119906

2 Λ2(120591))

(23)

hold for ae (120591 120596) isin [0 119879] times Ω

Proof Since 119906(sdot) = (1199061(sdot) 1199062(sdot)) is a Nash equilibrium point

of Problem 2 by (5) we have

1198691(119906120591120576

1(sdot) 1199062(sdot)) minus 119869

1(1199061(sdot) 1199062(sdot)) ge 0 (24)

Using the notation inTheorem 3 for any arbitrary admissiblecontrol (119906

1(sdot) 1199062(sdot)) and 119905 isin [0 119879] we have

E [1198671(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(120591))

minus1198671(119905 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(120591))] ge 0

(25)

which implies that (22) holdsSimilarly we can prove that (23) holds The proof is

complete

6 An Example Linear Quadratic Case

In this section we work out an example of linear quadraticnonzero-sum differential games to illustrate our stochastic

maximum principle More precisely consider the follow-ing one-dimensional linear fully coupled forward-backwardstochastic system

119909 (119905) = 119886 + int119905

0

[119860 (119904) 119909 (119904) minus 119861 (119904) 119910 (119904) + 1198631(119904) 1199061(119904)

+1198632(119904) 1199062(119904)] 119889119904

+ int119905

0

[119866 (119904) 119909 (119904)] 119889119861 (119904)

119910 (119905) = 119872119909 (119879) + int119879

119905

[119860 (119904) 119910 (119904) + 119867 (119904) 119909 (119904) + 119866 (119904) 119911 (119904)

+ 1198681(119904) 1199061(119904) + 119868

2(119904) 1199062(119904)] 119889119904

minus int119879

119905

119911 (119904) 119889119861 (119904)

(26)

with the quadratic cost functional

119869119894(1199061(sdot) 1199062(sdot)) = 119864 [int

119879

0

119871119894(119905) 1199062

119894(119905) 119889119905

+ 1198761198941199092(119879) + 119873

1198941199102(0) ] 119894 = 1 2

(27)

where 119860 119861 1198631 1198632 119866 119867 119868

1 1198682 1198711 and 119871

2are one-

dimensional deterministic bounded measurable functionsand119872119873

111987321198761 and119876

2are constants Also assume 119861 ge 0

1198731ge 0 119873

2ge 0 119876

1ge 0 119876

2ge 0 119872 ge 120573 119871

1ge 120573 119871

2ge 120573

119867 gt 120573 where 120573 gt 0 is a positive constantUnder the above assumptions on the coefficients of

(26) and (27) it is easy to check that for any admissible(1199061(sdot) 1199062(sdot)) isin A

1times 1198602 the state system (26) has a unique

solution and the corresponding stochastic differential gameproblem is well defined For this case the corresponding 119867becomes

119867119894(119905 119909 119910 119911 119906

1 1199062 119901119894 119902119894 119896119894)

= minus119896119894[119860 (119905) 119910 + 119867 (119905) 119909 + 119866 (119905) 119911 + 119868

1(119905) 1199061+ 1198682(119905) 1199062]

+ 119901119894[119860 (119905) 119909 minus 119861 (119905) 119910 + 119863

1(119905) 1199061+ 1198632(119905) 1199062(119904)]

+ 119902119894119866 (119905) 119909 + 119871

119894(119905) 1199062

119894(119905) 119894 = 1 2

(28)

The corresponding adjoint equation associated with anadmissible control pair (119906

1(sdot) 1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot)) becomes

119896119894(119905) = minus2119873

119894119910 (0) + int

119905

0

[119860 (119904) 119896119894(119904) + 119861 (119904) 119901

119894(119904)] 119889119904

+ int119905

0

119866 (119904) 119896119894(119904) 119889119861 (119904)

Mathematical Problems in Engineering 7

119901119894(119905) = minus119872119896

119894(119879) + 2119876

119894119909 (119879)

+ int119879

119905

[119860 (119904) 119901119894(119904) + 119866 (119904) 119902

119894(119904) minus 119867 (119904) 119896

119894(119904)] 119889119904

minus int119879

119905

119902119894(119904) 119889119861 (119904) 119894 = 1 2

(29)

It is easy to check that the state system (26) has a uniquesolution (119901119894(sdot) 119902119894(sdot) 119896119894(sdot)) 119894 = 1 2

Suppose that (1199061(sdot) 1199062(sdot)) is a Nash equilibrium point

By the maximum principle (see Theorem 5) putting theoptimality conditions (22) and (23) the corresponding stateequation (26) and the adjoint equations (28) associated with(1199061(sdot) 1199062(sdot)) together we obtain the following optimality

system for a Nash equilibrium point

119909 (119905) = 119886 + int119905

0

[119860 (119904) 119909 (119904) minus 119861 (119904) 119910 (119904) + 1198631(119904) 1199061(119904)

+1198632(119904) 1199062(119904)] 119889119904

+ int119905

0

119866 (119904) 119909 (119904) 119889119861 (119904)

119910 (119905) = 119872119909 (119879) + int119879

119905

[119860 (119904) 119910 (119904) + 119867 (119904) 119909 (119904) + 119866 (119904) 119911 (119904)

+ 1198681(119904) 1199061(119904) + 119868

2(119904) 1199062(119904)] 119889119904

minus int119879

119905

119911 (119904) 119889119861 (119904)

119896119894(119905) = minus2119873

119894119910 (0) + int

119905

0

[119860 (119904) 119896119894(119904) + 119861 (119904) 119901

119894(119904)] 119889119904

+ int119905

0

119866 (119904) 119896119894(119904) 119889119861 (119904)

119901119894(119905) = minus119872119896

119894(119879) + 2119876

119894119909 (119879)

+ int119879

119905

[119860 (119904) 119901119894(119904) + 119866 (119904) 119902

119894(119904) minus 119867 (119904) 119896

119894(119904)] 119889119904

minus int119879

119905

119902119894(119904) 119889119861 (119904)

1198671(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

1(119905))

= min1199061isin1198801

1198671(119905 Θ (119905) 119906

1 1199062(119905) Λ

1(119905))

1198672(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

2(119905))

= min1199062isin1198802

1198672(119905 Θ (119905) 119906

1(119905) 1199062 Λ2(119905))

119905 isin [0 119879] 119894 = 1 2

(30)

This is called coupled forward-backward stochastic differ-ential equationswith double dimensions (DFBSDE for short)

Note that the coupling comes from the two last relations(which is the maximum condition) We also refer to Yu [17]for the general theory of this kind equationThe8-tuple (119906

1(sdot)

1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot) 119901(sdot) 119902(sdot) 119896(sdot)) of F-adapted processes

satisfying the above is called an adapted solution of (30)We now look at the sufficiency of the existence of a Nashequilibrium point

Theorem 6 Suppose that (1199061(sdot) 1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot) 119901(sdot)

119902(sdot) 119896(sdot)) is an adapted solution to DFBSDE (30) Then(1199061(sdot) 1199062(sdot)) is a Nash equilibrium point

Proof Let (1199061(sdot) 1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot) 119901(sdot) 119902(sdot) 119896(sdot)) be an

adapted solution to DFBSDE (30) For any admissible controlpair (119906

1(sdot) 1199062(sdot)) isin A

1times 1198602 fromTheorem 3 we have

119869119894(1199061(sdot) 1199062(sdot)) minus 119869

119894(1199061(sdot) 1199062(sdot))

= Eint119879

0

[1198671(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

119906

119894(119905))

minus119867119894(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

119906

119894(119905)) ] 119889119905

+ E [1198731(119910(11990611199062)(0))2

minus 1198731(119910 (0))

2

minus21198731119910 (0) (119910

(11990611199062)(0) minus 119910 (0)) ]

+ E [1198761(119909(11990611199062)(119879))2

minus 11987611199092(119879)

minus 211987611199092(119879) sdot (119909

(11990611199062)(119879) minus 119909 (119879)) ]

= Eint119879

0

[1198671(119905 Θ (119905) 119906

120576(119905) Λ

119906

119894(119905))

minus1198671(119905 Θ (119905) 119906 (119905) Λ

119906

1(119905))] 119889119905

+ E [1198731(119910(11990611199062)(0) minus 119910 (0))

2

]

+ E [1198761(119909(11990611199062)(119879) minus 119909 (119879))

2

]

ge 0

(31)

which implies that

1198691(1199061(sdot) 1199062(sdot)) = inf

1199061(sdot)isinA

1

1198691(1199061(sdot) 1199062(sdot)) (32)

Similarly we can get

1198692(1199061(sdot) 1199062(sdot)) = inf

1199062(sdot)isinA

2

1198692(1199061(sdot) 1199062(sdot)) (33)

Therefore (1199061(sdot) 1199062(sdot)) is a Nash equilibrium point

Proposition 7 Suppose that 1198801times 1198802

= R times R and(1199061(sdot) 1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot) 119901(sdot) 119902(sdot) 119896(sdot)) is an adapted solu-

tion to DFBSDE (30) Then (1199061(sdot) 1199062(sdot)) has the following

representation

119906119894(119905) =

1

2119871119894(119905)

[119896 (119905) 119868119894(119905) minus 119901

119894(119905) 119863119894(119905)] 119894 = 1 2 (34)

8 Mathematical Problems in Engineering

Proof From Theorem 6 we deduce that (1199061(sdot) 1199062(sdot)) is a

Nash equilibrium point Then from 1198801times 1198802= R times R and

the optimality conditions (22) and (23) it follows that

11986711199061

(119905 Θ (119905) 1199061(119905) 1199062(119905) Λ

1(119905)) = 0

11986721199062

(119905 Θ (119905) 1199061(119905) 1199062(119905) Λ

2(119905)) = 0

(35)

which imply that

119906119894(119905) =

1

2119871119894(119905)

[119896 (119905) 119868119894(119905) minus 119901

119894(119905) 119863119894(119905)] 119894 = 1 2 (36)

The proof is complete

Remark 8 In summary DFBSDE (30) completely character-izes the Nash equilibrium point Therefore solving the dif-ferential game problem is equivalent to solving the DFBSDE(30) Moreover a candidate equilibrium point can be givenby (34) We refer the reader to Yu [17] for the theory ofsolvability toDFBSDE (30) Since the linear quadratic controlproblem is an important and fascinating class of stochasticcontrol ones and the theoretical results of this problem havelots of significant impacts on a wide range of engineeringmanaging and financial applications in the future we willfocus on the study of financial application as the reviewerssuggest

7 Conclusion

In this paper a two-person nonzero-sum differential gameis studied for a fully coupled forward-backward stochasticsystem with the control process 119906(sdot) not appearing in theforward diffusion term but the control domain not neces-sarily convex For this case we obtain a variation formulafor the cost functional As an application the maximumprinciple for open-loop Nash equilibrium points is estab-lished Finally we work out an example of linear quadraticnonzero-sum differential games to illustrate our stochasticmaximum principle As the reviewers suggest our systemdiscussed in this paper may be extended to the discontinuoussystem such Markov-switching jump-diffusion system forwhich the optimal control problems and differential gameshave extensive applications in industry and finance (seeeg [18 19] and the references therein) Some investigationson this topic will be studied and carried out in our futurepublications

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was partially supported by the National Nat-ural Science Foundation of China (11101140 11301177)the China Postdoctoral Science Foundation (2011M5007212012T50391) and theNatural Science Foundation of ZhejiangProvince (nos Y6110775 and Y6110789)

References

[1] J-M Bismut ldquoConjugate convex functions in optimal stochas-tic controlrdquo Journal of Mathematical Analysis and Applicationsvol 44 no 3 pp 384ndash404 1973

[2] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990

[3] S Tang ldquoGeneral linear quadratic optimal stochastic controlproblems with random coefficients linear stochastic Hamiltonsystems and backward stochastic Riccati equationsrdquo SIAMJournal on Control and Optimization vol 42 no 1 pp 53ndash752003

[4] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward-backward stochastic differential equationswith jumpsrdquo SIAM Journal on Control and Optimization vol48 no 5 pp 2945ndash2976 200910

[5] B Oslashksendal and A Sulem ldquoForward backward stochasticdifferential games and stochastic control under model uncer-taintyrdquo Journal of Optimization Theory and Applications 2012

[6] W S Xu ldquoStochastic maximum principle for optimal controlproblem of forward and backward systemrdquo Australian Mathe-matical Society Journal B vol 37 no 2 pp 172ndash185 1995

[7] Z Wu ldquoMaximum principle for optimal control problem offully coupled forward-backward stochastic systemsrdquo SystemsScience and Mathematical Sciences vol 11 no 3 pp 249ndash2591998

[8] J-T Shi and Z Wu ldquoThe maximum principle for fully coupledforward-backward stochastic control systemrdquo Acta AutomaticaSinica vol 32 no 2 pp 161ndash169 2006

[9] QMeng ldquoAmaximumprinciple for optimal control problem offully coupled forward-backward stochastic systems with partialinformationrdquo Science in China A vol 52 no 7 pp 1579ndash15882009

[10] J Shi and Z Wu ldquoMaximum principle for forward-backwardstochastic control system with random jumps and applicationsto financerdquo Journal of Systems Science amp Complexity vol 23 no2 pp 219ndash231 2010

[11] Z Wu ldquoA general maximum principle for optimal control offorward-backward stochastic systemsrdquo Automatica vol 49 no5 pp 1473ndash1480 2013

[12] J Yong ldquoOptimality variational principle for controlledforward-backward stochastic differential equations with mixedinitial-terminal conditionsrdquo SIAM Journal on Control andOptimization vol 48 no 6 pp 4119ndash4156 2010

[13] E C M Hui and H Xiao ldquoMaximum principle for differentialgames of forward-backward stochastic systems with applica-tionsrdquo Journal of Mathematical Analysis and Applications vol386 no 1 pp 412ndash427 2012

[14] M Tang ldquoMaximum principle for non-zero sum stochasticd-ifferential games of fully coupled forward-backward stochasticsystemsrdquo preprint

[15] S Peng and ZWu ldquoFully coupled forward-backward stochasticdifferential equations and applications to optimal controlrdquoSIAM Journal on Control and Optimization vol 37 no 3 pp825ndash843 1999

[16] Q Lin ldquoOptimal control of coupled forward-backward stochas-tic system with jumps and related hamilton-jacobi-bellmanequationsrdquo httparxivorgabs11114642

[17] Z Yu ldquoLinear-quadratic optimal control and nonzero-sum dif-ferential game of forward-backward stochastic systemrdquo AsianJournal of Control vol 14 no 1 pp 173ndash185 2012

Mathematical Problems in Engineering 9

[18] B Z Temocin and G-W Weber ldquoOptimal control of stochastichybrid systemwith jumps a numerical approximationrdquo Journalof Computational and Applied Mathematics vol 259 pp 443ndash451 2014

[19] N Azevedo D Pinheiro and G W Weber ldquoDynamic pro-gramming for a Markov-switching jumpCdiffusionrdquo Journal ofComputational andAppliedMathematics vol 267 pp 1ndash19 2014

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

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Page 3: Research Article A Variational Formula for Nonzero-Sum ...downloads.hindawi.com › journals › mpe › 2014 › 283418.pdf · the forward stochastic system. e existence and uniqueness

Mathematical Problems in Engineering 3

or

⟨119860 (119905 120579 119906) minus 119860 (119905 120579 119906) 120579 minus 120579⟩

ge 1205731|119866119909|2+ 1205732(1003816100381610038161003816119866lowast11991010038161003816100381610038162

+1003816100381610038161003816119866lowast10038161003816100381610038162

)

⟨ℎ (119909) minus ℎ (119909) 119909 minus 119909⟩ le minus1205831|119866119909|2

(4)

Here 1205731 1205732 and 120583

1are given nonnegative constants with 120573

1+

1205732gt 0 120573

2+ 1205831gt 0 Moreover we have 120573

1gt 0 120583

1gt 0 (resp

1205732gt 0) when119898 gt 119899 (resp119898 lt 119899)(iii) The mappings 119897

119894 120601119894 and 120574

119894are continuously differen-

tiable with respect to (119909 119910 119911) (119894 = 1 2) And 119897119894is bounded by

(1 + |119909|2 + |119910|2 + |119911|2 + |1199061|2 + |119906

2|2) And the derivatives of

119897119894with respect to (119909 119910 119911) (119894 = 1 2) are bounded by (1 + |119909| +

|119910| + |119911| + |1199061| + |1199062|) And 120601

119894and ℎ119894are bounded by (1 + |119909|2)

and (1 + |119910|2) respectively And the derivatives of 120601119894and ℎ

119894

with respect to 119909 and 119910 are bounded by (1 + |119909|) and (1 + |119910|)respectively (119894 = 1 2)

Under Assumption 1 we see that for any given admissiblecontrol 119906(sdot) = (119906

1(sdot) 1199062(sdot)) isin A

1timesA2 the system (1) admits

a unique solution (119909119906(sdot) 119910119906(sdot) 119911119906(sdot)) isin M2 (see [15])Then wecall (119909119906(sdot) 119910119906(sdot) 119911119906(sdot)) or (119909(sdot) 119910(sdot) 119911(sdot)) if its dependence onadmissible control 119906(sdot) is clear from context the state processcorresponding to the control process 119906(sdot) = (119906

1(sdot) 1199062(sdot)) and

(119906(sdot) 119909(sdot) 119910(sdot) 119911(sdot)) the admissible pairThenwe can pose the following two-person nonzero-sum

stochastic differential game problem

Problem 2 Find an open-loop admissible control(1199061(sdot) 1199062(sdot)) isin A

1timesA2such that

1198691(1199061(sdot) 1199062(sdot)) = inf

1199061(sdot)isinA1

1198691(1199061(sdot) 1199062(sdot)) (5)

1198692(1199061(sdot) 1199062(sdot)) = inf

1199062(sdot)isinA

2

1198692(1199061(sdot) 1199062(sdot)) (6)

Any (1199061(sdot) 1199062(sdot)) isin A

1timesA2satisfying the above is called

an open-loop Nash equilibrium point of Problem 2 Suchan admissible control allows two players to play individualoptimal control strategies simultaneously

3 Representation for Difference ofthe Cost Functional

This section is devoted to establishing a representation for thedifference of the cost functional according to Hamiltonianand the adjoint processes of Problem 2

To simplify our notation for any admissible control119906(sdot) = (119906

1(sdot) 1199062(sdot)) we write Θ(119905) = (119909(119905) 119910(119905) 119911(119905)) as

the corresponding state process We define the Hamiltonianfunctions 119867

119894 [0 119879] times R119899 times R119898 times R119898times119889 times U

1times U2times R119899 times

R119899times119889 timesR119898 rarr R by

119867119894(119905 119909 119910 119911 119906

1 1199062 119901 119902 119896) = ⟨119896 minus119891 (119905 119909 119910 119911 119906

1 1199062)⟩

+ ⟨119901 119887 (119905 119909 119910 119911 1199061 1199062)⟩

+ 119897119894(119905 119909 119910 119911 119906

1 1199062)

+ ⟨119902 120590 (119905 119909 119910 119911)⟩ 119894 = 1 2

(7)

For any admissible pair (119906(sdot) Θ(sdot)) we define the corre-sponding adjoint process Λ119906

119894(sdot) = (119896119906

119894(sdot) 119901119906119894(sdot) 119902119906119894(sdot))(119894 = 1 2)

as the solution to the following FBSDEs

119889119896119894(119905) = minus119867

119894119910(119905) 119889119905 minus 119867

119894119911(119905) 119889119861 (119905)

119889119901119894(119905) = minus119867

119894119909(119905) 119889119905 + 119902

119894(119905) 119889119861 (119905)

119896119894(0) = minus120574

119894119910(119910 (0))

119901119894(119879) = minusℎ

lowast

119909(119909 (119879)) 119896

119894(119879) + 120601

119894119909(119909 (119879))

119894 = 1 2

(8)

where we have used the short hand notation

119867119894119910(119905) = 119867

119894119910(119905 119906 (119905) Θ (119905) Λ

119906

119894(119905)) 119894 = 12 (9)

And similarly we can define119867119894119909(119905) and119867

119894119911(119905)

Under Assumptions 1 it is easy to see thatthe above adjoint equations have unique solutionΛ119906

119894(sdot) = (119896119906

119894(sdot) 119901119906119894(sdot) 119902119906119894(sdot)) isin M2[0 119879] 119894 = 1 2

Now let (119906120576(sdot) Θ120576(sdot)) = (1199061205761(sdot) 1199061205762(sdot) 119909120576(sdot) 119910120576(sdot) 119911120576(sdot)) be

another admissible pair In the following we will give thepresentation for the difference 119869

119894(119906120576(sdot)) minus 119869

119894(119906(sdot)) (119894 = 1 2)

in terms of the Hamiltonian119867119894(119894 = 1 2) and adjoint process

Λ119906119894(sdot) (119894 = 1 2) associated with the admissible control pair

(119906(sdot) Θ119906(sdot)) as well as other relevant expressions We stateour result as follows

Theorem 3 Under Assumptions 1 one has the representationfor the difference of the cost functional as follows sdot(119910120576(0) minus119910(0))(119910120576(0))]

119869119894(119906120576

1(sdot) 119906120576

2(sdot)) minus 119869

119894(1199061(sdot) 1199062(sdot))

= Eint119879

0

[119867119894(119905 Θ (119905) 119906

120576(119905) Λ

119906

119894(119905))

minus 119867119894(119905 Θ (119905) 119906 (119905) Λ

119906

119894(119905))

+ 119867119894(119905 Θ120576(119905) 119906120576(119905) Λ

119906

119894(119905))

minus 119867119894(119905 Θ (119905) 119906

120576(119905) Λ

119906

119894(119905))

minus ⟨119867119894119909(119905) 119909120576(119905) minus 119909 (119905)⟩

minus ⟨119867119894119910(119905) 119910120576(119905) minus 119910 (119905)⟩

minus ⟨119867119894119911(119905) 119911120576(119905) minus 119911 (119905)⟩] 119889119905

4 Mathematical Problems in Engineering

+ E [120574119894(119910120576(0)) minus 120574

119894(119910 (0)) minus 120574

119894119910(119910 (0))

sdot (119910120576(0) minus 119910 (0)) (119910

120576(0)) ]

+ E [120601119894(119909120598(119879)) minus 120601

119894(119909 (119879)) minus 120601

119894119909(119909 (119879))

sdot (119909120576(119879) minus 119909 (119879))]

minus E [⟨ℎ (119909120598(119879)) minus ℎ (119909 (119879)) 119896

119906

119894(119879)⟩

minus ⟨ℎ119909(119909 (119879)) (119909

120576(119879) minus 119909 (119879)) 119896

119906

119894(119879)⟩]

(10)

Proof First from the definition of 119867 (see (7)) it is easy tocheck that

119867119894(119905 Θ120576(119905) 119906120576(119905) Λ

119906

119894(119905)) minus 119867

119894(119905 Θ (119905) 119906 (119905) Λ

119906

119894(119905))

= 119867119894(119905 Θ (119905) 119906

120576(119905) Λ

119906

119894(sdot)) minus 119867

119894(119905 Θ (119905) 119906 (119905) Λ

119906

119894(119905))

+ 119867119894(119905 Θ120576(119905) 119906120576(119905) Λ

119906

119894(119905))

minus 119867119894(119905 Θ (119905) 119906

120576(119905) Λ

119906

119894(119905)) 119894 = 1 2

(11)

From (1) we know that

119909120576(119905) minus 119909 (119905) = int

119905

0

[119887 (119904 Θ120576(119904) 119906120576(119904))

minus 119887 (119904 Θ (119904) 119906 (119904))] 119889119904

+ int119905

0

[120590 (119904 Θ120576(119904) 119906120576(119904))

minus 120590 (119904 Θ (119904) 119906 (119904))] 119889119882 (119904)

119910120576(119905) minus 119910 (119905) = ℎ (119909

120598(119879)) minus ℎ (119909 (119879))

+ int119879

119905

[119891 (119904 Θ120576(119904) 119906120576(119904))

minus 119891 (119904 Θ (119904) 119906 (119904))] 119889119904

minus int119879

119905

[119911120576(119904) minus 119911 (119904)] 119889119882 (119904)

(12)

Furthermore recalling (8) and applying Ito formula to⟨119896119906119894(119905) 119910120576(119905) minus 119910(119905)⟩ + ⟨119901119906

119894(119905) 119909120576(119905) minus 119909(119905)⟩ we deduce that

Eint119879

0

⟨119896119906

119894(119905) minus (119891 (119905 Θ

120576(119905) 119906120576(119905) )

minus119891 (119905 Θ (119905) 119906 (119905)))⟩ 119889119905

+ 119864int119879

0

⟨119901119906

119894(119905) 119887 (119905 Θ

120576(119905) 119906120576(119905))

minus 119887 (119905 Θ (119905) 119906 (119905))⟩ 119889119905

+ 119864int119879

0

⟨119902119906

119894(119905) 120590 (119905 Θ

120576(119905) 119906120576(119905))

minus 120590 (119905 Θ (119905) 119906 (119905))⟩ 119889119905

= Eint119879

0

[⟨119867119894119909(119905) 119909120576(119905) minus 119909 (119905)⟩

+ ⟨119867119894119910(119905) 119910120576(119905) minus 119910 (119905)⟩

+ ⟨119867119894119911(119905) 119911120576(119905) minus 119911 (119905)⟩] 119889119905

+ E [120574119894119910(119910 (0)) sdot (119910

120576(0) minus 119910 (0))]

+ E [120601119894119909(119909 (119879)) sdot (119909

120576(119879) minus 119909 (119879))]

+ E [⟨ℎ (119909120576(119879)) minus ℎ (119909 (119879)) 119896

119906

119894(119879)⟩

minus ⟨ℎ119909(119909 (119879)) (119909

120576(119879) minus 119909 (119879)) 119896

119906

119894(119879)⟩]

119894 = 1 2

(13)

On the other hand by the definition of theHamilton function119867 (see (7)) we deduce that

119869119894(119906120576

1(sdot) 119906120576

2(sdot)) minus 119869

119894(1199061(sdot) 1199062(sdot))

= Eint119879

0

[119897119894(119905 Θ120576(119905) 119906120576(119905)) minus 119897

119894(119905 Θ (119905) 119906 (119905))] 119889119905

+ E [120574119894(119910120576(0)) minus 120574

119894(119910 (0))]

+ 119864 [120601119894(119909120598(119879)) minus 120601

119894(119909 (119879))]

= Eint119879

0

[119867119894(119905 Θ120576(119905) 119906120576(119905) Λ

119906

119894(119905))

minus119867119894(119905 Θ (119905) 119906 (119905) Λ

119906

119894(119905))] 119889119905

minus Eint119879

0

⟨119896119906

119894(119905) minus (119891 (119905 Θ

120576(119905) 119906120576(119905))

minus119891 (119905 Θ (119905) 119906 (119905)))⟩ 119889119905

minus Eint119879

0

⟨119901119906

119894(119905) 119887 (119905 Θ

120576(119905) 119906120576(119905))

minus119887 (119905 Θ (119905) 119906 (119905))⟩ 119889119905

minus Eint119879

0

⟨119902119906

119894(119905) 120590 (119905 Θ

120576(119905) 119906120576(119905))

minus120590 (119905 Θ (119905) 119906 (119905)) ⟩ 119889119905

+ E [120574119894(119910120576(0)) minus 120574

119894(119910 (0))]

+ 119864 [120601119894(119909120598(119879)) minus 120601

119894(119909 (119879))]

(14)

Now putting (11) and (13) into (14) we deduce that (10)holds The proof is complete

Mathematical Problems in Engineering 5

4 A Variational Formula for StochasticDifferential Games

In this section we will obtain a directional derivative ata given admissible control process in some given controlprocess direction The choice of the given control processdirection depends on the convexity of the control domain1198801times 1198802 If the control domain 119880

1times 1198802is convex a classical

way of treating such a problem consists of using the convexperturbation method More precisely if (119906

1(sdot) 1199062(sdot)) is a

given admissible control and (V1(sdot) V2(sdot)) is an arbitrary

given admissible control we can define a convex perturbedadmissible control as

119906120576

119894(sdot) = 119906

119894(sdot) + 120576 (V

119894(sdot) minus 119906

119894(sdot)) 119894 = 1 2 (15)

where 120576 is a sufficiently small positive constant Then onecan prove the cost functional 119869

119894(1199061(sdot) 1199062(sdot)) is Gateaux dif-

ferentiable at 119906119894(sdot) in the direction V

119894(sdot) minus 119906

119894(sdot) (119894 = 1 2)

and get a local stochastic maximum principle for open-loop Nash equilibrium points see for example [13 14]Different from [13 14] our control domain in the presentpaper is not necessarily convex so the convex perturbedcontrol 119906120576

119894(sdot) may no longer be admissible and the convex

perturbation method cannot be used to obtain the corre-sponding variational formula and maximum principle Aclassical way of treating the nonconvex control domainconsists of using the spike variations perturbation methodMore precisely let (119906(sdot) Θ(sdot)) = (119906

1(sdot) 1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot))

be any given admissible pair with the corresponding adjointprocess Λ119906

119894(sdot) = (119896119906

119894(sdot) 119901119906119894(sdot) 119902119906119894(sdot)) isin M2[0 119879] 119894 = 1 2 We

define the following spike variations

119906120591120576

119894(119905) =

119906119894(119905) if 119905 isin [120591 120591 + 120576]

119906119894(119905) otherwise

(16)

with fixed 120591 isin [0 119879) sufficiently small positive 120576 and anygiven admissible control 119906

119894(sdot) isin A

1 119894 = 1 2

Now we state the following variational formula for thecost functional (2) associated with the spike variation (16) ina unified way

Theorem 4 Under Assumption 1 one has a variational for-mula for the cost functional (5) and (6) as follows

119889

1198891205761198691(119906120591120576

1(sdot) 1199062(sdot))

10038161003816100381610038161003816100381610038161003816120576=0

= lim120576rarr0

1198691(1199061205911205761(sdot) 1199062(sdot)) minus 119869

1(1199061(sdot) 1199062(sdot))

120576

= E [1198671(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(120591))

minus 1198671(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(119905))]

(17)

119889

1198891205761198692(1199061(sdot) 119906120591120576

2(sdot))

10038161003816100381610038161003816100381610038161003816120576=0

= lim120576rarr0

1198692(1199061(sdot) 119906120591120576

2(sdot)) minus 119869

2(1199061(sdot) 1199062(sdot))

120576

= E [1198672(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(120591))

minus 1198672(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(120591))]

(18)

Proof We first prove the equality (17) Let Θ1120576(sdot) =

(1199091120576(sdot) 1199101120576(sdot) 1199111120576(sdot)) be the state process corresponding tothe admissible control (119906120591120576

1(sdot) 1199062(sdot)) Under Assumption 1

by the continuous dependence theory of FBSDEs (seeProposition 32 in [16]) we have

10038171003817100381710038171003817Θ1120576(sdot) minus Θ

119906(sdot)10038171003817100381710038171003817M2

le 1198621205762 (19)

InTheorem 3 replacing (119906120576(sdot) Θ120576(sdot)) by (1199061205911205761(sdot) 1199062(sdot) Θ1120576(sdot))

we have

1198691(119906120591120576

1(sdot) 1199062(sdot)) minus 119869

1(1199061(sdot) 1199062(sdot))

= Eint119879

0

[119867119894(119905 Θ (119905) 119906

120591120576

1(119905) 1199062(119905) Λ

119906

119894(119905))

minus 119867119894(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

119906

119894(119905))

+ 119867119894(119905 Θ120576(119905) 119906120591120576

1(119905) 1199062(119905) Λ

119906

119894(119905))

minus 119867119894(119905 Θ (119905) 119906

120591120576

1(119905) 1199062(119905) Λ

119906

119894(119905))

minus ⟨119867119894119909(119905) 119909120576(119905) minus 119909 (119905)⟩

minus ⟨119867119894119910(119905) 119910120576(119905) minus 119910 (119905)⟩

minus ⟨119867119894119911(119905) 119911120576(119905) minus 119911 (119905)⟩ ] 119889119905

+ E [120574119894(119910120576(0)) minus 120574

119894(119910 (0)) minus 120574

119894119910(119910 (0))

sdot (119910120576(0) minus 119910 (0))]

+ E [120601119894(119909120598(119879)) minus 120601

119894(119909 (119879)) minus 120601

119894119909(119909 (119879))

sdot (119909120576(119879) minus 119909 (119879))]

minus E [⟨ℎ (119909120598(119879)) minus ℎ (119909 (119879)) 119896

119906

119894(119879)⟩

minus ⟨ℎ119909(119909 (119879)) (119909

120576(119879) minus 119909 (119879)) 119896

119906

119894(119879)⟩]

(20)

Combining (19) and Assumption 1 by Taylor Expansions on119867 and the dominated convergence theorem from (20) weconclude that

1198691(119906120591120576

1(sdot) 1199062(sdot)) minus 119869

1(1199061(sdot) 1199062(sdot))

= Eint119879

0

[1198671(119905 Θ (119905) 119906

120591120576

1(119905) 1199062(119905) Λ

119906

1(119905))

minus1198671(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

119906

1(119905))] 119889119905 + 119900 (120576)

6 Mathematical Problems in Engineering

= 120576E [1198671(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(120591))

minus1198671(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(119905))] + 119900 (120576)

(21)

which imply that (17) holdsSimilarly we can prove that (18) holds The proof is

complete

5 Stochastic Maximum Principle

In this section applying the variational formulas (17) and (18)we will state and prove the global maximum principle for theNash equilibrium points of Problem 2

Theorem 5 Under Assumption 1 let 119906(sdot) = (1199061(sdot) 1199062(sdot)) be

a Nash equilibrium point of Problem 2 with the state processΘ(sdot) = (119909(sdot) 119910(sdot) 119911(sdot)) Let Λ

119894(sdot) = (119901

119894(sdot) 119902119894(sdot) 119896119894(sdot)) (119894 = 1 2)

be the unique solution of the adjoint equation (8) correspondingto (119906(sdot) Θ(sdot)) Then

1198671(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

1(120591))

= min1199061isin1198801

1198671(120591 Θ (120591) 119906

1 1199062(120591) Λ

1(120591))

(22)

1198672(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

2(120591))

= min1199062isin1198802

1198672(120591 Θ (120591) 119906

1(120591) 119906

2 Λ2(120591))

(23)

hold for ae (120591 120596) isin [0 119879] times Ω

Proof Since 119906(sdot) = (1199061(sdot) 1199062(sdot)) is a Nash equilibrium point

of Problem 2 by (5) we have

1198691(119906120591120576

1(sdot) 1199062(sdot)) minus 119869

1(1199061(sdot) 1199062(sdot)) ge 0 (24)

Using the notation inTheorem 3 for any arbitrary admissiblecontrol (119906

1(sdot) 1199062(sdot)) and 119905 isin [0 119879] we have

E [1198671(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(120591))

minus1198671(119905 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(120591))] ge 0

(25)

which implies that (22) holdsSimilarly we can prove that (23) holds The proof is

complete

6 An Example Linear Quadratic Case

In this section we work out an example of linear quadraticnonzero-sum differential games to illustrate our stochastic

maximum principle More precisely consider the follow-ing one-dimensional linear fully coupled forward-backwardstochastic system

119909 (119905) = 119886 + int119905

0

[119860 (119904) 119909 (119904) minus 119861 (119904) 119910 (119904) + 1198631(119904) 1199061(119904)

+1198632(119904) 1199062(119904)] 119889119904

+ int119905

0

[119866 (119904) 119909 (119904)] 119889119861 (119904)

119910 (119905) = 119872119909 (119879) + int119879

119905

[119860 (119904) 119910 (119904) + 119867 (119904) 119909 (119904) + 119866 (119904) 119911 (119904)

+ 1198681(119904) 1199061(119904) + 119868

2(119904) 1199062(119904)] 119889119904

minus int119879

119905

119911 (119904) 119889119861 (119904)

(26)

with the quadratic cost functional

119869119894(1199061(sdot) 1199062(sdot)) = 119864 [int

119879

0

119871119894(119905) 1199062

119894(119905) 119889119905

+ 1198761198941199092(119879) + 119873

1198941199102(0) ] 119894 = 1 2

(27)

where 119860 119861 1198631 1198632 119866 119867 119868

1 1198682 1198711 and 119871

2are one-

dimensional deterministic bounded measurable functionsand119872119873

111987321198761 and119876

2are constants Also assume 119861 ge 0

1198731ge 0 119873

2ge 0 119876

1ge 0 119876

2ge 0 119872 ge 120573 119871

1ge 120573 119871

2ge 120573

119867 gt 120573 where 120573 gt 0 is a positive constantUnder the above assumptions on the coefficients of

(26) and (27) it is easy to check that for any admissible(1199061(sdot) 1199062(sdot)) isin A

1times 1198602 the state system (26) has a unique

solution and the corresponding stochastic differential gameproblem is well defined For this case the corresponding 119867becomes

119867119894(119905 119909 119910 119911 119906

1 1199062 119901119894 119902119894 119896119894)

= minus119896119894[119860 (119905) 119910 + 119867 (119905) 119909 + 119866 (119905) 119911 + 119868

1(119905) 1199061+ 1198682(119905) 1199062]

+ 119901119894[119860 (119905) 119909 minus 119861 (119905) 119910 + 119863

1(119905) 1199061+ 1198632(119905) 1199062(119904)]

+ 119902119894119866 (119905) 119909 + 119871

119894(119905) 1199062

119894(119905) 119894 = 1 2

(28)

The corresponding adjoint equation associated with anadmissible control pair (119906

1(sdot) 1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot)) becomes

119896119894(119905) = minus2119873

119894119910 (0) + int

119905

0

[119860 (119904) 119896119894(119904) + 119861 (119904) 119901

119894(119904)] 119889119904

+ int119905

0

119866 (119904) 119896119894(119904) 119889119861 (119904)

Mathematical Problems in Engineering 7

119901119894(119905) = minus119872119896

119894(119879) + 2119876

119894119909 (119879)

+ int119879

119905

[119860 (119904) 119901119894(119904) + 119866 (119904) 119902

119894(119904) minus 119867 (119904) 119896

119894(119904)] 119889119904

minus int119879

119905

119902119894(119904) 119889119861 (119904) 119894 = 1 2

(29)

It is easy to check that the state system (26) has a uniquesolution (119901119894(sdot) 119902119894(sdot) 119896119894(sdot)) 119894 = 1 2

Suppose that (1199061(sdot) 1199062(sdot)) is a Nash equilibrium point

By the maximum principle (see Theorem 5) putting theoptimality conditions (22) and (23) the corresponding stateequation (26) and the adjoint equations (28) associated with(1199061(sdot) 1199062(sdot)) together we obtain the following optimality

system for a Nash equilibrium point

119909 (119905) = 119886 + int119905

0

[119860 (119904) 119909 (119904) minus 119861 (119904) 119910 (119904) + 1198631(119904) 1199061(119904)

+1198632(119904) 1199062(119904)] 119889119904

+ int119905

0

119866 (119904) 119909 (119904) 119889119861 (119904)

119910 (119905) = 119872119909 (119879) + int119879

119905

[119860 (119904) 119910 (119904) + 119867 (119904) 119909 (119904) + 119866 (119904) 119911 (119904)

+ 1198681(119904) 1199061(119904) + 119868

2(119904) 1199062(119904)] 119889119904

minus int119879

119905

119911 (119904) 119889119861 (119904)

119896119894(119905) = minus2119873

119894119910 (0) + int

119905

0

[119860 (119904) 119896119894(119904) + 119861 (119904) 119901

119894(119904)] 119889119904

+ int119905

0

119866 (119904) 119896119894(119904) 119889119861 (119904)

119901119894(119905) = minus119872119896

119894(119879) + 2119876

119894119909 (119879)

+ int119879

119905

[119860 (119904) 119901119894(119904) + 119866 (119904) 119902

119894(119904) minus 119867 (119904) 119896

119894(119904)] 119889119904

minus int119879

119905

119902119894(119904) 119889119861 (119904)

1198671(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

1(119905))

= min1199061isin1198801

1198671(119905 Θ (119905) 119906

1 1199062(119905) Λ

1(119905))

1198672(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

2(119905))

= min1199062isin1198802

1198672(119905 Θ (119905) 119906

1(119905) 1199062 Λ2(119905))

119905 isin [0 119879] 119894 = 1 2

(30)

This is called coupled forward-backward stochastic differ-ential equationswith double dimensions (DFBSDE for short)

Note that the coupling comes from the two last relations(which is the maximum condition) We also refer to Yu [17]for the general theory of this kind equationThe8-tuple (119906

1(sdot)

1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot) 119901(sdot) 119902(sdot) 119896(sdot)) of F-adapted processes

satisfying the above is called an adapted solution of (30)We now look at the sufficiency of the existence of a Nashequilibrium point

Theorem 6 Suppose that (1199061(sdot) 1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot) 119901(sdot)

119902(sdot) 119896(sdot)) is an adapted solution to DFBSDE (30) Then(1199061(sdot) 1199062(sdot)) is a Nash equilibrium point

Proof Let (1199061(sdot) 1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot) 119901(sdot) 119902(sdot) 119896(sdot)) be an

adapted solution to DFBSDE (30) For any admissible controlpair (119906

1(sdot) 1199062(sdot)) isin A

1times 1198602 fromTheorem 3 we have

119869119894(1199061(sdot) 1199062(sdot)) minus 119869

119894(1199061(sdot) 1199062(sdot))

= Eint119879

0

[1198671(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

119906

119894(119905))

minus119867119894(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

119906

119894(119905)) ] 119889119905

+ E [1198731(119910(11990611199062)(0))2

minus 1198731(119910 (0))

2

minus21198731119910 (0) (119910

(11990611199062)(0) minus 119910 (0)) ]

+ E [1198761(119909(11990611199062)(119879))2

minus 11987611199092(119879)

minus 211987611199092(119879) sdot (119909

(11990611199062)(119879) minus 119909 (119879)) ]

= Eint119879

0

[1198671(119905 Θ (119905) 119906

120576(119905) Λ

119906

119894(119905))

minus1198671(119905 Θ (119905) 119906 (119905) Λ

119906

1(119905))] 119889119905

+ E [1198731(119910(11990611199062)(0) minus 119910 (0))

2

]

+ E [1198761(119909(11990611199062)(119879) minus 119909 (119879))

2

]

ge 0

(31)

which implies that

1198691(1199061(sdot) 1199062(sdot)) = inf

1199061(sdot)isinA

1

1198691(1199061(sdot) 1199062(sdot)) (32)

Similarly we can get

1198692(1199061(sdot) 1199062(sdot)) = inf

1199062(sdot)isinA

2

1198692(1199061(sdot) 1199062(sdot)) (33)

Therefore (1199061(sdot) 1199062(sdot)) is a Nash equilibrium point

Proposition 7 Suppose that 1198801times 1198802

= R times R and(1199061(sdot) 1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot) 119901(sdot) 119902(sdot) 119896(sdot)) is an adapted solu-

tion to DFBSDE (30) Then (1199061(sdot) 1199062(sdot)) has the following

representation

119906119894(119905) =

1

2119871119894(119905)

[119896 (119905) 119868119894(119905) minus 119901

119894(119905) 119863119894(119905)] 119894 = 1 2 (34)

8 Mathematical Problems in Engineering

Proof From Theorem 6 we deduce that (1199061(sdot) 1199062(sdot)) is a

Nash equilibrium point Then from 1198801times 1198802= R times R and

the optimality conditions (22) and (23) it follows that

11986711199061

(119905 Θ (119905) 1199061(119905) 1199062(119905) Λ

1(119905)) = 0

11986721199062

(119905 Θ (119905) 1199061(119905) 1199062(119905) Λ

2(119905)) = 0

(35)

which imply that

119906119894(119905) =

1

2119871119894(119905)

[119896 (119905) 119868119894(119905) minus 119901

119894(119905) 119863119894(119905)] 119894 = 1 2 (36)

The proof is complete

Remark 8 In summary DFBSDE (30) completely character-izes the Nash equilibrium point Therefore solving the dif-ferential game problem is equivalent to solving the DFBSDE(30) Moreover a candidate equilibrium point can be givenby (34) We refer the reader to Yu [17] for the theory ofsolvability toDFBSDE (30) Since the linear quadratic controlproblem is an important and fascinating class of stochasticcontrol ones and the theoretical results of this problem havelots of significant impacts on a wide range of engineeringmanaging and financial applications in the future we willfocus on the study of financial application as the reviewerssuggest

7 Conclusion

In this paper a two-person nonzero-sum differential gameis studied for a fully coupled forward-backward stochasticsystem with the control process 119906(sdot) not appearing in theforward diffusion term but the control domain not neces-sarily convex For this case we obtain a variation formulafor the cost functional As an application the maximumprinciple for open-loop Nash equilibrium points is estab-lished Finally we work out an example of linear quadraticnonzero-sum differential games to illustrate our stochasticmaximum principle As the reviewers suggest our systemdiscussed in this paper may be extended to the discontinuoussystem such Markov-switching jump-diffusion system forwhich the optimal control problems and differential gameshave extensive applications in industry and finance (seeeg [18 19] and the references therein) Some investigationson this topic will be studied and carried out in our futurepublications

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was partially supported by the National Nat-ural Science Foundation of China (11101140 11301177)the China Postdoctoral Science Foundation (2011M5007212012T50391) and theNatural Science Foundation of ZhejiangProvince (nos Y6110775 and Y6110789)

References

[1] J-M Bismut ldquoConjugate convex functions in optimal stochas-tic controlrdquo Journal of Mathematical Analysis and Applicationsvol 44 no 3 pp 384ndash404 1973

[2] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990

[3] S Tang ldquoGeneral linear quadratic optimal stochastic controlproblems with random coefficients linear stochastic Hamiltonsystems and backward stochastic Riccati equationsrdquo SIAMJournal on Control and Optimization vol 42 no 1 pp 53ndash752003

[4] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward-backward stochastic differential equationswith jumpsrdquo SIAM Journal on Control and Optimization vol48 no 5 pp 2945ndash2976 200910

[5] B Oslashksendal and A Sulem ldquoForward backward stochasticdifferential games and stochastic control under model uncer-taintyrdquo Journal of Optimization Theory and Applications 2012

[6] W S Xu ldquoStochastic maximum principle for optimal controlproblem of forward and backward systemrdquo Australian Mathe-matical Society Journal B vol 37 no 2 pp 172ndash185 1995

[7] Z Wu ldquoMaximum principle for optimal control problem offully coupled forward-backward stochastic systemsrdquo SystemsScience and Mathematical Sciences vol 11 no 3 pp 249ndash2591998

[8] J-T Shi and Z Wu ldquoThe maximum principle for fully coupledforward-backward stochastic control systemrdquo Acta AutomaticaSinica vol 32 no 2 pp 161ndash169 2006

[9] QMeng ldquoAmaximumprinciple for optimal control problem offully coupled forward-backward stochastic systems with partialinformationrdquo Science in China A vol 52 no 7 pp 1579ndash15882009

[10] J Shi and Z Wu ldquoMaximum principle for forward-backwardstochastic control system with random jumps and applicationsto financerdquo Journal of Systems Science amp Complexity vol 23 no2 pp 219ndash231 2010

[11] Z Wu ldquoA general maximum principle for optimal control offorward-backward stochastic systemsrdquo Automatica vol 49 no5 pp 1473ndash1480 2013

[12] J Yong ldquoOptimality variational principle for controlledforward-backward stochastic differential equations with mixedinitial-terminal conditionsrdquo SIAM Journal on Control andOptimization vol 48 no 6 pp 4119ndash4156 2010

[13] E C M Hui and H Xiao ldquoMaximum principle for differentialgames of forward-backward stochastic systems with applica-tionsrdquo Journal of Mathematical Analysis and Applications vol386 no 1 pp 412ndash427 2012

[14] M Tang ldquoMaximum principle for non-zero sum stochasticd-ifferential games of fully coupled forward-backward stochasticsystemsrdquo preprint

[15] S Peng and ZWu ldquoFully coupled forward-backward stochasticdifferential equations and applications to optimal controlrdquoSIAM Journal on Control and Optimization vol 37 no 3 pp825ndash843 1999

[16] Q Lin ldquoOptimal control of coupled forward-backward stochas-tic system with jumps and related hamilton-jacobi-bellmanequationsrdquo httparxivorgabs11114642

[17] Z Yu ldquoLinear-quadratic optimal control and nonzero-sum dif-ferential game of forward-backward stochastic systemrdquo AsianJournal of Control vol 14 no 1 pp 173ndash185 2012

Mathematical Problems in Engineering 9

[18] B Z Temocin and G-W Weber ldquoOptimal control of stochastichybrid systemwith jumps a numerical approximationrdquo Journalof Computational and Applied Mathematics vol 259 pp 443ndash451 2014

[19] N Azevedo D Pinheiro and G W Weber ldquoDynamic pro-gramming for a Markov-switching jumpCdiffusionrdquo Journal ofComputational andAppliedMathematics vol 267 pp 1ndash19 2014

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

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 4: Research Article A Variational Formula for Nonzero-Sum ...downloads.hindawi.com › journals › mpe › 2014 › 283418.pdf · the forward stochastic system. e existence and uniqueness

4 Mathematical Problems in Engineering

+ E [120574119894(119910120576(0)) minus 120574

119894(119910 (0)) minus 120574

119894119910(119910 (0))

sdot (119910120576(0) minus 119910 (0)) (119910

120576(0)) ]

+ E [120601119894(119909120598(119879)) minus 120601

119894(119909 (119879)) minus 120601

119894119909(119909 (119879))

sdot (119909120576(119879) minus 119909 (119879))]

minus E [⟨ℎ (119909120598(119879)) minus ℎ (119909 (119879)) 119896

119906

119894(119879)⟩

minus ⟨ℎ119909(119909 (119879)) (119909

120576(119879) minus 119909 (119879)) 119896

119906

119894(119879)⟩]

(10)

Proof First from the definition of 119867 (see (7)) it is easy tocheck that

119867119894(119905 Θ120576(119905) 119906120576(119905) Λ

119906

119894(119905)) minus 119867

119894(119905 Θ (119905) 119906 (119905) Λ

119906

119894(119905))

= 119867119894(119905 Θ (119905) 119906

120576(119905) Λ

119906

119894(sdot)) minus 119867

119894(119905 Θ (119905) 119906 (119905) Λ

119906

119894(119905))

+ 119867119894(119905 Θ120576(119905) 119906120576(119905) Λ

119906

119894(119905))

minus 119867119894(119905 Θ (119905) 119906

120576(119905) Λ

119906

119894(119905)) 119894 = 1 2

(11)

From (1) we know that

119909120576(119905) minus 119909 (119905) = int

119905

0

[119887 (119904 Θ120576(119904) 119906120576(119904))

minus 119887 (119904 Θ (119904) 119906 (119904))] 119889119904

+ int119905

0

[120590 (119904 Θ120576(119904) 119906120576(119904))

minus 120590 (119904 Θ (119904) 119906 (119904))] 119889119882 (119904)

119910120576(119905) minus 119910 (119905) = ℎ (119909

120598(119879)) minus ℎ (119909 (119879))

+ int119879

119905

[119891 (119904 Θ120576(119904) 119906120576(119904))

minus 119891 (119904 Θ (119904) 119906 (119904))] 119889119904

minus int119879

119905

[119911120576(119904) minus 119911 (119904)] 119889119882 (119904)

(12)

Furthermore recalling (8) and applying Ito formula to⟨119896119906119894(119905) 119910120576(119905) minus 119910(119905)⟩ + ⟨119901119906

119894(119905) 119909120576(119905) minus 119909(119905)⟩ we deduce that

Eint119879

0

⟨119896119906

119894(119905) minus (119891 (119905 Θ

120576(119905) 119906120576(119905) )

minus119891 (119905 Θ (119905) 119906 (119905)))⟩ 119889119905

+ 119864int119879

0

⟨119901119906

119894(119905) 119887 (119905 Θ

120576(119905) 119906120576(119905))

minus 119887 (119905 Θ (119905) 119906 (119905))⟩ 119889119905

+ 119864int119879

0

⟨119902119906

119894(119905) 120590 (119905 Θ

120576(119905) 119906120576(119905))

minus 120590 (119905 Θ (119905) 119906 (119905))⟩ 119889119905

= Eint119879

0

[⟨119867119894119909(119905) 119909120576(119905) minus 119909 (119905)⟩

+ ⟨119867119894119910(119905) 119910120576(119905) minus 119910 (119905)⟩

+ ⟨119867119894119911(119905) 119911120576(119905) minus 119911 (119905)⟩] 119889119905

+ E [120574119894119910(119910 (0)) sdot (119910

120576(0) minus 119910 (0))]

+ E [120601119894119909(119909 (119879)) sdot (119909

120576(119879) minus 119909 (119879))]

+ E [⟨ℎ (119909120576(119879)) minus ℎ (119909 (119879)) 119896

119906

119894(119879)⟩

minus ⟨ℎ119909(119909 (119879)) (119909

120576(119879) minus 119909 (119879)) 119896

119906

119894(119879)⟩]

119894 = 1 2

(13)

On the other hand by the definition of theHamilton function119867 (see (7)) we deduce that

119869119894(119906120576

1(sdot) 119906120576

2(sdot)) minus 119869

119894(1199061(sdot) 1199062(sdot))

= Eint119879

0

[119897119894(119905 Θ120576(119905) 119906120576(119905)) minus 119897

119894(119905 Θ (119905) 119906 (119905))] 119889119905

+ E [120574119894(119910120576(0)) minus 120574

119894(119910 (0))]

+ 119864 [120601119894(119909120598(119879)) minus 120601

119894(119909 (119879))]

= Eint119879

0

[119867119894(119905 Θ120576(119905) 119906120576(119905) Λ

119906

119894(119905))

minus119867119894(119905 Θ (119905) 119906 (119905) Λ

119906

119894(119905))] 119889119905

minus Eint119879

0

⟨119896119906

119894(119905) minus (119891 (119905 Θ

120576(119905) 119906120576(119905))

minus119891 (119905 Θ (119905) 119906 (119905)))⟩ 119889119905

minus Eint119879

0

⟨119901119906

119894(119905) 119887 (119905 Θ

120576(119905) 119906120576(119905))

minus119887 (119905 Θ (119905) 119906 (119905))⟩ 119889119905

minus Eint119879

0

⟨119902119906

119894(119905) 120590 (119905 Θ

120576(119905) 119906120576(119905))

minus120590 (119905 Θ (119905) 119906 (119905)) ⟩ 119889119905

+ E [120574119894(119910120576(0)) minus 120574

119894(119910 (0))]

+ 119864 [120601119894(119909120598(119879)) minus 120601

119894(119909 (119879))]

(14)

Now putting (11) and (13) into (14) we deduce that (10)holds The proof is complete

Mathematical Problems in Engineering 5

4 A Variational Formula for StochasticDifferential Games

In this section we will obtain a directional derivative ata given admissible control process in some given controlprocess direction The choice of the given control processdirection depends on the convexity of the control domain1198801times 1198802 If the control domain 119880

1times 1198802is convex a classical

way of treating such a problem consists of using the convexperturbation method More precisely if (119906

1(sdot) 1199062(sdot)) is a

given admissible control and (V1(sdot) V2(sdot)) is an arbitrary

given admissible control we can define a convex perturbedadmissible control as

119906120576

119894(sdot) = 119906

119894(sdot) + 120576 (V

119894(sdot) minus 119906

119894(sdot)) 119894 = 1 2 (15)

where 120576 is a sufficiently small positive constant Then onecan prove the cost functional 119869

119894(1199061(sdot) 1199062(sdot)) is Gateaux dif-

ferentiable at 119906119894(sdot) in the direction V

119894(sdot) minus 119906

119894(sdot) (119894 = 1 2)

and get a local stochastic maximum principle for open-loop Nash equilibrium points see for example [13 14]Different from [13 14] our control domain in the presentpaper is not necessarily convex so the convex perturbedcontrol 119906120576

119894(sdot) may no longer be admissible and the convex

perturbation method cannot be used to obtain the corre-sponding variational formula and maximum principle Aclassical way of treating the nonconvex control domainconsists of using the spike variations perturbation methodMore precisely let (119906(sdot) Θ(sdot)) = (119906

1(sdot) 1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot))

be any given admissible pair with the corresponding adjointprocess Λ119906

119894(sdot) = (119896119906

119894(sdot) 119901119906119894(sdot) 119902119906119894(sdot)) isin M2[0 119879] 119894 = 1 2 We

define the following spike variations

119906120591120576

119894(119905) =

119906119894(119905) if 119905 isin [120591 120591 + 120576]

119906119894(119905) otherwise

(16)

with fixed 120591 isin [0 119879) sufficiently small positive 120576 and anygiven admissible control 119906

119894(sdot) isin A

1 119894 = 1 2

Now we state the following variational formula for thecost functional (2) associated with the spike variation (16) ina unified way

Theorem 4 Under Assumption 1 one has a variational for-mula for the cost functional (5) and (6) as follows

119889

1198891205761198691(119906120591120576

1(sdot) 1199062(sdot))

10038161003816100381610038161003816100381610038161003816120576=0

= lim120576rarr0

1198691(1199061205911205761(sdot) 1199062(sdot)) minus 119869

1(1199061(sdot) 1199062(sdot))

120576

= E [1198671(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(120591))

minus 1198671(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(119905))]

(17)

119889

1198891205761198692(1199061(sdot) 119906120591120576

2(sdot))

10038161003816100381610038161003816100381610038161003816120576=0

= lim120576rarr0

1198692(1199061(sdot) 119906120591120576

2(sdot)) minus 119869

2(1199061(sdot) 1199062(sdot))

120576

= E [1198672(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(120591))

minus 1198672(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(120591))]

(18)

Proof We first prove the equality (17) Let Θ1120576(sdot) =

(1199091120576(sdot) 1199101120576(sdot) 1199111120576(sdot)) be the state process corresponding tothe admissible control (119906120591120576

1(sdot) 1199062(sdot)) Under Assumption 1

by the continuous dependence theory of FBSDEs (seeProposition 32 in [16]) we have

10038171003817100381710038171003817Θ1120576(sdot) minus Θ

119906(sdot)10038171003817100381710038171003817M2

le 1198621205762 (19)

InTheorem 3 replacing (119906120576(sdot) Θ120576(sdot)) by (1199061205911205761(sdot) 1199062(sdot) Θ1120576(sdot))

we have

1198691(119906120591120576

1(sdot) 1199062(sdot)) minus 119869

1(1199061(sdot) 1199062(sdot))

= Eint119879

0

[119867119894(119905 Θ (119905) 119906

120591120576

1(119905) 1199062(119905) Λ

119906

119894(119905))

minus 119867119894(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

119906

119894(119905))

+ 119867119894(119905 Θ120576(119905) 119906120591120576

1(119905) 1199062(119905) Λ

119906

119894(119905))

minus 119867119894(119905 Θ (119905) 119906

120591120576

1(119905) 1199062(119905) Λ

119906

119894(119905))

minus ⟨119867119894119909(119905) 119909120576(119905) minus 119909 (119905)⟩

minus ⟨119867119894119910(119905) 119910120576(119905) minus 119910 (119905)⟩

minus ⟨119867119894119911(119905) 119911120576(119905) minus 119911 (119905)⟩ ] 119889119905

+ E [120574119894(119910120576(0)) minus 120574

119894(119910 (0)) minus 120574

119894119910(119910 (0))

sdot (119910120576(0) minus 119910 (0))]

+ E [120601119894(119909120598(119879)) minus 120601

119894(119909 (119879)) minus 120601

119894119909(119909 (119879))

sdot (119909120576(119879) minus 119909 (119879))]

minus E [⟨ℎ (119909120598(119879)) minus ℎ (119909 (119879)) 119896

119906

119894(119879)⟩

minus ⟨ℎ119909(119909 (119879)) (119909

120576(119879) minus 119909 (119879)) 119896

119906

119894(119879)⟩]

(20)

Combining (19) and Assumption 1 by Taylor Expansions on119867 and the dominated convergence theorem from (20) weconclude that

1198691(119906120591120576

1(sdot) 1199062(sdot)) minus 119869

1(1199061(sdot) 1199062(sdot))

= Eint119879

0

[1198671(119905 Θ (119905) 119906

120591120576

1(119905) 1199062(119905) Λ

119906

1(119905))

minus1198671(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

119906

1(119905))] 119889119905 + 119900 (120576)

6 Mathematical Problems in Engineering

= 120576E [1198671(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(120591))

minus1198671(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(119905))] + 119900 (120576)

(21)

which imply that (17) holdsSimilarly we can prove that (18) holds The proof is

complete

5 Stochastic Maximum Principle

In this section applying the variational formulas (17) and (18)we will state and prove the global maximum principle for theNash equilibrium points of Problem 2

Theorem 5 Under Assumption 1 let 119906(sdot) = (1199061(sdot) 1199062(sdot)) be

a Nash equilibrium point of Problem 2 with the state processΘ(sdot) = (119909(sdot) 119910(sdot) 119911(sdot)) Let Λ

119894(sdot) = (119901

119894(sdot) 119902119894(sdot) 119896119894(sdot)) (119894 = 1 2)

be the unique solution of the adjoint equation (8) correspondingto (119906(sdot) Θ(sdot)) Then

1198671(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

1(120591))

= min1199061isin1198801

1198671(120591 Θ (120591) 119906

1 1199062(120591) Λ

1(120591))

(22)

1198672(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

2(120591))

= min1199062isin1198802

1198672(120591 Θ (120591) 119906

1(120591) 119906

2 Λ2(120591))

(23)

hold for ae (120591 120596) isin [0 119879] times Ω

Proof Since 119906(sdot) = (1199061(sdot) 1199062(sdot)) is a Nash equilibrium point

of Problem 2 by (5) we have

1198691(119906120591120576

1(sdot) 1199062(sdot)) minus 119869

1(1199061(sdot) 1199062(sdot)) ge 0 (24)

Using the notation inTheorem 3 for any arbitrary admissiblecontrol (119906

1(sdot) 1199062(sdot)) and 119905 isin [0 119879] we have

E [1198671(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(120591))

minus1198671(119905 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(120591))] ge 0

(25)

which implies that (22) holdsSimilarly we can prove that (23) holds The proof is

complete

6 An Example Linear Quadratic Case

In this section we work out an example of linear quadraticnonzero-sum differential games to illustrate our stochastic

maximum principle More precisely consider the follow-ing one-dimensional linear fully coupled forward-backwardstochastic system

119909 (119905) = 119886 + int119905

0

[119860 (119904) 119909 (119904) minus 119861 (119904) 119910 (119904) + 1198631(119904) 1199061(119904)

+1198632(119904) 1199062(119904)] 119889119904

+ int119905

0

[119866 (119904) 119909 (119904)] 119889119861 (119904)

119910 (119905) = 119872119909 (119879) + int119879

119905

[119860 (119904) 119910 (119904) + 119867 (119904) 119909 (119904) + 119866 (119904) 119911 (119904)

+ 1198681(119904) 1199061(119904) + 119868

2(119904) 1199062(119904)] 119889119904

minus int119879

119905

119911 (119904) 119889119861 (119904)

(26)

with the quadratic cost functional

119869119894(1199061(sdot) 1199062(sdot)) = 119864 [int

119879

0

119871119894(119905) 1199062

119894(119905) 119889119905

+ 1198761198941199092(119879) + 119873

1198941199102(0) ] 119894 = 1 2

(27)

where 119860 119861 1198631 1198632 119866 119867 119868

1 1198682 1198711 and 119871

2are one-

dimensional deterministic bounded measurable functionsand119872119873

111987321198761 and119876

2are constants Also assume 119861 ge 0

1198731ge 0 119873

2ge 0 119876

1ge 0 119876

2ge 0 119872 ge 120573 119871

1ge 120573 119871

2ge 120573

119867 gt 120573 where 120573 gt 0 is a positive constantUnder the above assumptions on the coefficients of

(26) and (27) it is easy to check that for any admissible(1199061(sdot) 1199062(sdot)) isin A

1times 1198602 the state system (26) has a unique

solution and the corresponding stochastic differential gameproblem is well defined For this case the corresponding 119867becomes

119867119894(119905 119909 119910 119911 119906

1 1199062 119901119894 119902119894 119896119894)

= minus119896119894[119860 (119905) 119910 + 119867 (119905) 119909 + 119866 (119905) 119911 + 119868

1(119905) 1199061+ 1198682(119905) 1199062]

+ 119901119894[119860 (119905) 119909 minus 119861 (119905) 119910 + 119863

1(119905) 1199061+ 1198632(119905) 1199062(119904)]

+ 119902119894119866 (119905) 119909 + 119871

119894(119905) 1199062

119894(119905) 119894 = 1 2

(28)

The corresponding adjoint equation associated with anadmissible control pair (119906

1(sdot) 1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot)) becomes

119896119894(119905) = minus2119873

119894119910 (0) + int

119905

0

[119860 (119904) 119896119894(119904) + 119861 (119904) 119901

119894(119904)] 119889119904

+ int119905

0

119866 (119904) 119896119894(119904) 119889119861 (119904)

Mathematical Problems in Engineering 7

119901119894(119905) = minus119872119896

119894(119879) + 2119876

119894119909 (119879)

+ int119879

119905

[119860 (119904) 119901119894(119904) + 119866 (119904) 119902

119894(119904) minus 119867 (119904) 119896

119894(119904)] 119889119904

minus int119879

119905

119902119894(119904) 119889119861 (119904) 119894 = 1 2

(29)

It is easy to check that the state system (26) has a uniquesolution (119901119894(sdot) 119902119894(sdot) 119896119894(sdot)) 119894 = 1 2

Suppose that (1199061(sdot) 1199062(sdot)) is a Nash equilibrium point

By the maximum principle (see Theorem 5) putting theoptimality conditions (22) and (23) the corresponding stateequation (26) and the adjoint equations (28) associated with(1199061(sdot) 1199062(sdot)) together we obtain the following optimality

system for a Nash equilibrium point

119909 (119905) = 119886 + int119905

0

[119860 (119904) 119909 (119904) minus 119861 (119904) 119910 (119904) + 1198631(119904) 1199061(119904)

+1198632(119904) 1199062(119904)] 119889119904

+ int119905

0

119866 (119904) 119909 (119904) 119889119861 (119904)

119910 (119905) = 119872119909 (119879) + int119879

119905

[119860 (119904) 119910 (119904) + 119867 (119904) 119909 (119904) + 119866 (119904) 119911 (119904)

+ 1198681(119904) 1199061(119904) + 119868

2(119904) 1199062(119904)] 119889119904

minus int119879

119905

119911 (119904) 119889119861 (119904)

119896119894(119905) = minus2119873

119894119910 (0) + int

119905

0

[119860 (119904) 119896119894(119904) + 119861 (119904) 119901

119894(119904)] 119889119904

+ int119905

0

119866 (119904) 119896119894(119904) 119889119861 (119904)

119901119894(119905) = minus119872119896

119894(119879) + 2119876

119894119909 (119879)

+ int119879

119905

[119860 (119904) 119901119894(119904) + 119866 (119904) 119902

119894(119904) minus 119867 (119904) 119896

119894(119904)] 119889119904

minus int119879

119905

119902119894(119904) 119889119861 (119904)

1198671(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

1(119905))

= min1199061isin1198801

1198671(119905 Θ (119905) 119906

1 1199062(119905) Λ

1(119905))

1198672(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

2(119905))

= min1199062isin1198802

1198672(119905 Θ (119905) 119906

1(119905) 1199062 Λ2(119905))

119905 isin [0 119879] 119894 = 1 2

(30)

This is called coupled forward-backward stochastic differ-ential equationswith double dimensions (DFBSDE for short)

Note that the coupling comes from the two last relations(which is the maximum condition) We also refer to Yu [17]for the general theory of this kind equationThe8-tuple (119906

1(sdot)

1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot) 119901(sdot) 119902(sdot) 119896(sdot)) of F-adapted processes

satisfying the above is called an adapted solution of (30)We now look at the sufficiency of the existence of a Nashequilibrium point

Theorem 6 Suppose that (1199061(sdot) 1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot) 119901(sdot)

119902(sdot) 119896(sdot)) is an adapted solution to DFBSDE (30) Then(1199061(sdot) 1199062(sdot)) is a Nash equilibrium point

Proof Let (1199061(sdot) 1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot) 119901(sdot) 119902(sdot) 119896(sdot)) be an

adapted solution to DFBSDE (30) For any admissible controlpair (119906

1(sdot) 1199062(sdot)) isin A

1times 1198602 fromTheorem 3 we have

119869119894(1199061(sdot) 1199062(sdot)) minus 119869

119894(1199061(sdot) 1199062(sdot))

= Eint119879

0

[1198671(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

119906

119894(119905))

minus119867119894(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

119906

119894(119905)) ] 119889119905

+ E [1198731(119910(11990611199062)(0))2

minus 1198731(119910 (0))

2

minus21198731119910 (0) (119910

(11990611199062)(0) minus 119910 (0)) ]

+ E [1198761(119909(11990611199062)(119879))2

minus 11987611199092(119879)

minus 211987611199092(119879) sdot (119909

(11990611199062)(119879) minus 119909 (119879)) ]

= Eint119879

0

[1198671(119905 Θ (119905) 119906

120576(119905) Λ

119906

119894(119905))

minus1198671(119905 Θ (119905) 119906 (119905) Λ

119906

1(119905))] 119889119905

+ E [1198731(119910(11990611199062)(0) minus 119910 (0))

2

]

+ E [1198761(119909(11990611199062)(119879) minus 119909 (119879))

2

]

ge 0

(31)

which implies that

1198691(1199061(sdot) 1199062(sdot)) = inf

1199061(sdot)isinA

1

1198691(1199061(sdot) 1199062(sdot)) (32)

Similarly we can get

1198692(1199061(sdot) 1199062(sdot)) = inf

1199062(sdot)isinA

2

1198692(1199061(sdot) 1199062(sdot)) (33)

Therefore (1199061(sdot) 1199062(sdot)) is a Nash equilibrium point

Proposition 7 Suppose that 1198801times 1198802

= R times R and(1199061(sdot) 1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot) 119901(sdot) 119902(sdot) 119896(sdot)) is an adapted solu-

tion to DFBSDE (30) Then (1199061(sdot) 1199062(sdot)) has the following

representation

119906119894(119905) =

1

2119871119894(119905)

[119896 (119905) 119868119894(119905) minus 119901

119894(119905) 119863119894(119905)] 119894 = 1 2 (34)

8 Mathematical Problems in Engineering

Proof From Theorem 6 we deduce that (1199061(sdot) 1199062(sdot)) is a

Nash equilibrium point Then from 1198801times 1198802= R times R and

the optimality conditions (22) and (23) it follows that

11986711199061

(119905 Θ (119905) 1199061(119905) 1199062(119905) Λ

1(119905)) = 0

11986721199062

(119905 Θ (119905) 1199061(119905) 1199062(119905) Λ

2(119905)) = 0

(35)

which imply that

119906119894(119905) =

1

2119871119894(119905)

[119896 (119905) 119868119894(119905) minus 119901

119894(119905) 119863119894(119905)] 119894 = 1 2 (36)

The proof is complete

Remark 8 In summary DFBSDE (30) completely character-izes the Nash equilibrium point Therefore solving the dif-ferential game problem is equivalent to solving the DFBSDE(30) Moreover a candidate equilibrium point can be givenby (34) We refer the reader to Yu [17] for the theory ofsolvability toDFBSDE (30) Since the linear quadratic controlproblem is an important and fascinating class of stochasticcontrol ones and the theoretical results of this problem havelots of significant impacts on a wide range of engineeringmanaging and financial applications in the future we willfocus on the study of financial application as the reviewerssuggest

7 Conclusion

In this paper a two-person nonzero-sum differential gameis studied for a fully coupled forward-backward stochasticsystem with the control process 119906(sdot) not appearing in theforward diffusion term but the control domain not neces-sarily convex For this case we obtain a variation formulafor the cost functional As an application the maximumprinciple for open-loop Nash equilibrium points is estab-lished Finally we work out an example of linear quadraticnonzero-sum differential games to illustrate our stochasticmaximum principle As the reviewers suggest our systemdiscussed in this paper may be extended to the discontinuoussystem such Markov-switching jump-diffusion system forwhich the optimal control problems and differential gameshave extensive applications in industry and finance (seeeg [18 19] and the references therein) Some investigationson this topic will be studied and carried out in our futurepublications

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was partially supported by the National Nat-ural Science Foundation of China (11101140 11301177)the China Postdoctoral Science Foundation (2011M5007212012T50391) and theNatural Science Foundation of ZhejiangProvince (nos Y6110775 and Y6110789)

References

[1] J-M Bismut ldquoConjugate convex functions in optimal stochas-tic controlrdquo Journal of Mathematical Analysis and Applicationsvol 44 no 3 pp 384ndash404 1973

[2] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990

[3] S Tang ldquoGeneral linear quadratic optimal stochastic controlproblems with random coefficients linear stochastic Hamiltonsystems and backward stochastic Riccati equationsrdquo SIAMJournal on Control and Optimization vol 42 no 1 pp 53ndash752003

[4] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward-backward stochastic differential equationswith jumpsrdquo SIAM Journal on Control and Optimization vol48 no 5 pp 2945ndash2976 200910

[5] B Oslashksendal and A Sulem ldquoForward backward stochasticdifferential games and stochastic control under model uncer-taintyrdquo Journal of Optimization Theory and Applications 2012

[6] W S Xu ldquoStochastic maximum principle for optimal controlproblem of forward and backward systemrdquo Australian Mathe-matical Society Journal B vol 37 no 2 pp 172ndash185 1995

[7] Z Wu ldquoMaximum principle for optimal control problem offully coupled forward-backward stochastic systemsrdquo SystemsScience and Mathematical Sciences vol 11 no 3 pp 249ndash2591998

[8] J-T Shi and Z Wu ldquoThe maximum principle for fully coupledforward-backward stochastic control systemrdquo Acta AutomaticaSinica vol 32 no 2 pp 161ndash169 2006

[9] QMeng ldquoAmaximumprinciple for optimal control problem offully coupled forward-backward stochastic systems with partialinformationrdquo Science in China A vol 52 no 7 pp 1579ndash15882009

[10] J Shi and Z Wu ldquoMaximum principle for forward-backwardstochastic control system with random jumps and applicationsto financerdquo Journal of Systems Science amp Complexity vol 23 no2 pp 219ndash231 2010

[11] Z Wu ldquoA general maximum principle for optimal control offorward-backward stochastic systemsrdquo Automatica vol 49 no5 pp 1473ndash1480 2013

[12] J Yong ldquoOptimality variational principle for controlledforward-backward stochastic differential equations with mixedinitial-terminal conditionsrdquo SIAM Journal on Control andOptimization vol 48 no 6 pp 4119ndash4156 2010

[13] E C M Hui and H Xiao ldquoMaximum principle for differentialgames of forward-backward stochastic systems with applica-tionsrdquo Journal of Mathematical Analysis and Applications vol386 no 1 pp 412ndash427 2012

[14] M Tang ldquoMaximum principle for non-zero sum stochasticd-ifferential games of fully coupled forward-backward stochasticsystemsrdquo preprint

[15] S Peng and ZWu ldquoFully coupled forward-backward stochasticdifferential equations and applications to optimal controlrdquoSIAM Journal on Control and Optimization vol 37 no 3 pp825ndash843 1999

[16] Q Lin ldquoOptimal control of coupled forward-backward stochas-tic system with jumps and related hamilton-jacobi-bellmanequationsrdquo httparxivorgabs11114642

[17] Z Yu ldquoLinear-quadratic optimal control and nonzero-sum dif-ferential game of forward-backward stochastic systemrdquo AsianJournal of Control vol 14 no 1 pp 173ndash185 2012

Mathematical Problems in Engineering 9

[18] B Z Temocin and G-W Weber ldquoOptimal control of stochastichybrid systemwith jumps a numerical approximationrdquo Journalof Computational and Applied Mathematics vol 259 pp 443ndash451 2014

[19] N Azevedo D Pinheiro and G W Weber ldquoDynamic pro-gramming for a Markov-switching jumpCdiffusionrdquo Journal ofComputational andAppliedMathematics vol 267 pp 1ndash19 2014

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

Page 5: Research Article A Variational Formula for Nonzero-Sum ...downloads.hindawi.com › journals › mpe › 2014 › 283418.pdf · the forward stochastic system. e existence and uniqueness

Mathematical Problems in Engineering 5

4 A Variational Formula for StochasticDifferential Games

In this section we will obtain a directional derivative ata given admissible control process in some given controlprocess direction The choice of the given control processdirection depends on the convexity of the control domain1198801times 1198802 If the control domain 119880

1times 1198802is convex a classical

way of treating such a problem consists of using the convexperturbation method More precisely if (119906

1(sdot) 1199062(sdot)) is a

given admissible control and (V1(sdot) V2(sdot)) is an arbitrary

given admissible control we can define a convex perturbedadmissible control as

119906120576

119894(sdot) = 119906

119894(sdot) + 120576 (V

119894(sdot) minus 119906

119894(sdot)) 119894 = 1 2 (15)

where 120576 is a sufficiently small positive constant Then onecan prove the cost functional 119869

119894(1199061(sdot) 1199062(sdot)) is Gateaux dif-

ferentiable at 119906119894(sdot) in the direction V

119894(sdot) minus 119906

119894(sdot) (119894 = 1 2)

and get a local stochastic maximum principle for open-loop Nash equilibrium points see for example [13 14]Different from [13 14] our control domain in the presentpaper is not necessarily convex so the convex perturbedcontrol 119906120576

119894(sdot) may no longer be admissible and the convex

perturbation method cannot be used to obtain the corre-sponding variational formula and maximum principle Aclassical way of treating the nonconvex control domainconsists of using the spike variations perturbation methodMore precisely let (119906(sdot) Θ(sdot)) = (119906

1(sdot) 1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot))

be any given admissible pair with the corresponding adjointprocess Λ119906

119894(sdot) = (119896119906

119894(sdot) 119901119906119894(sdot) 119902119906119894(sdot)) isin M2[0 119879] 119894 = 1 2 We

define the following spike variations

119906120591120576

119894(119905) =

119906119894(119905) if 119905 isin [120591 120591 + 120576]

119906119894(119905) otherwise

(16)

with fixed 120591 isin [0 119879) sufficiently small positive 120576 and anygiven admissible control 119906

119894(sdot) isin A

1 119894 = 1 2

Now we state the following variational formula for thecost functional (2) associated with the spike variation (16) ina unified way

Theorem 4 Under Assumption 1 one has a variational for-mula for the cost functional (5) and (6) as follows

119889

1198891205761198691(119906120591120576

1(sdot) 1199062(sdot))

10038161003816100381610038161003816100381610038161003816120576=0

= lim120576rarr0

1198691(1199061205911205761(sdot) 1199062(sdot)) minus 119869

1(1199061(sdot) 1199062(sdot))

120576

= E [1198671(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(120591))

minus 1198671(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(119905))]

(17)

119889

1198891205761198692(1199061(sdot) 119906120591120576

2(sdot))

10038161003816100381610038161003816100381610038161003816120576=0

= lim120576rarr0

1198692(1199061(sdot) 119906120591120576

2(sdot)) minus 119869

2(1199061(sdot) 1199062(sdot))

120576

= E [1198672(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(120591))

minus 1198672(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(120591))]

(18)

Proof We first prove the equality (17) Let Θ1120576(sdot) =

(1199091120576(sdot) 1199101120576(sdot) 1199111120576(sdot)) be the state process corresponding tothe admissible control (119906120591120576

1(sdot) 1199062(sdot)) Under Assumption 1

by the continuous dependence theory of FBSDEs (seeProposition 32 in [16]) we have

10038171003817100381710038171003817Θ1120576(sdot) minus Θ

119906(sdot)10038171003817100381710038171003817M2

le 1198621205762 (19)

InTheorem 3 replacing (119906120576(sdot) Θ120576(sdot)) by (1199061205911205761(sdot) 1199062(sdot) Θ1120576(sdot))

we have

1198691(119906120591120576

1(sdot) 1199062(sdot)) minus 119869

1(1199061(sdot) 1199062(sdot))

= Eint119879

0

[119867119894(119905 Θ (119905) 119906

120591120576

1(119905) 1199062(119905) Λ

119906

119894(119905))

minus 119867119894(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

119906

119894(119905))

+ 119867119894(119905 Θ120576(119905) 119906120591120576

1(119905) 1199062(119905) Λ

119906

119894(119905))

minus 119867119894(119905 Θ (119905) 119906

120591120576

1(119905) 1199062(119905) Λ

119906

119894(119905))

minus ⟨119867119894119909(119905) 119909120576(119905) minus 119909 (119905)⟩

minus ⟨119867119894119910(119905) 119910120576(119905) minus 119910 (119905)⟩

minus ⟨119867119894119911(119905) 119911120576(119905) minus 119911 (119905)⟩ ] 119889119905

+ E [120574119894(119910120576(0)) minus 120574

119894(119910 (0)) minus 120574

119894119910(119910 (0))

sdot (119910120576(0) minus 119910 (0))]

+ E [120601119894(119909120598(119879)) minus 120601

119894(119909 (119879)) minus 120601

119894119909(119909 (119879))

sdot (119909120576(119879) minus 119909 (119879))]

minus E [⟨ℎ (119909120598(119879)) minus ℎ (119909 (119879)) 119896

119906

119894(119879)⟩

minus ⟨ℎ119909(119909 (119879)) (119909

120576(119879) minus 119909 (119879)) 119896

119906

119894(119879)⟩]

(20)

Combining (19) and Assumption 1 by Taylor Expansions on119867 and the dominated convergence theorem from (20) weconclude that

1198691(119906120591120576

1(sdot) 1199062(sdot)) minus 119869

1(1199061(sdot) 1199062(sdot))

= Eint119879

0

[1198671(119905 Θ (119905) 119906

120591120576

1(119905) 1199062(119905) Λ

119906

1(119905))

minus1198671(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

119906

1(119905))] 119889119905 + 119900 (120576)

6 Mathematical Problems in Engineering

= 120576E [1198671(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(120591))

minus1198671(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(119905))] + 119900 (120576)

(21)

which imply that (17) holdsSimilarly we can prove that (18) holds The proof is

complete

5 Stochastic Maximum Principle

In this section applying the variational formulas (17) and (18)we will state and prove the global maximum principle for theNash equilibrium points of Problem 2

Theorem 5 Under Assumption 1 let 119906(sdot) = (1199061(sdot) 1199062(sdot)) be

a Nash equilibrium point of Problem 2 with the state processΘ(sdot) = (119909(sdot) 119910(sdot) 119911(sdot)) Let Λ

119894(sdot) = (119901

119894(sdot) 119902119894(sdot) 119896119894(sdot)) (119894 = 1 2)

be the unique solution of the adjoint equation (8) correspondingto (119906(sdot) Θ(sdot)) Then

1198671(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

1(120591))

= min1199061isin1198801

1198671(120591 Θ (120591) 119906

1 1199062(120591) Λ

1(120591))

(22)

1198672(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

2(120591))

= min1199062isin1198802

1198672(120591 Θ (120591) 119906

1(120591) 119906

2 Λ2(120591))

(23)

hold for ae (120591 120596) isin [0 119879] times Ω

Proof Since 119906(sdot) = (1199061(sdot) 1199062(sdot)) is a Nash equilibrium point

of Problem 2 by (5) we have

1198691(119906120591120576

1(sdot) 1199062(sdot)) minus 119869

1(1199061(sdot) 1199062(sdot)) ge 0 (24)

Using the notation inTheorem 3 for any arbitrary admissiblecontrol (119906

1(sdot) 1199062(sdot)) and 119905 isin [0 119879] we have

E [1198671(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(120591))

minus1198671(119905 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(120591))] ge 0

(25)

which implies that (22) holdsSimilarly we can prove that (23) holds The proof is

complete

6 An Example Linear Quadratic Case

In this section we work out an example of linear quadraticnonzero-sum differential games to illustrate our stochastic

maximum principle More precisely consider the follow-ing one-dimensional linear fully coupled forward-backwardstochastic system

119909 (119905) = 119886 + int119905

0

[119860 (119904) 119909 (119904) minus 119861 (119904) 119910 (119904) + 1198631(119904) 1199061(119904)

+1198632(119904) 1199062(119904)] 119889119904

+ int119905

0

[119866 (119904) 119909 (119904)] 119889119861 (119904)

119910 (119905) = 119872119909 (119879) + int119879

119905

[119860 (119904) 119910 (119904) + 119867 (119904) 119909 (119904) + 119866 (119904) 119911 (119904)

+ 1198681(119904) 1199061(119904) + 119868

2(119904) 1199062(119904)] 119889119904

minus int119879

119905

119911 (119904) 119889119861 (119904)

(26)

with the quadratic cost functional

119869119894(1199061(sdot) 1199062(sdot)) = 119864 [int

119879

0

119871119894(119905) 1199062

119894(119905) 119889119905

+ 1198761198941199092(119879) + 119873

1198941199102(0) ] 119894 = 1 2

(27)

where 119860 119861 1198631 1198632 119866 119867 119868

1 1198682 1198711 and 119871

2are one-

dimensional deterministic bounded measurable functionsand119872119873

111987321198761 and119876

2are constants Also assume 119861 ge 0

1198731ge 0 119873

2ge 0 119876

1ge 0 119876

2ge 0 119872 ge 120573 119871

1ge 120573 119871

2ge 120573

119867 gt 120573 where 120573 gt 0 is a positive constantUnder the above assumptions on the coefficients of

(26) and (27) it is easy to check that for any admissible(1199061(sdot) 1199062(sdot)) isin A

1times 1198602 the state system (26) has a unique

solution and the corresponding stochastic differential gameproblem is well defined For this case the corresponding 119867becomes

119867119894(119905 119909 119910 119911 119906

1 1199062 119901119894 119902119894 119896119894)

= minus119896119894[119860 (119905) 119910 + 119867 (119905) 119909 + 119866 (119905) 119911 + 119868

1(119905) 1199061+ 1198682(119905) 1199062]

+ 119901119894[119860 (119905) 119909 minus 119861 (119905) 119910 + 119863

1(119905) 1199061+ 1198632(119905) 1199062(119904)]

+ 119902119894119866 (119905) 119909 + 119871

119894(119905) 1199062

119894(119905) 119894 = 1 2

(28)

The corresponding adjoint equation associated with anadmissible control pair (119906

1(sdot) 1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot)) becomes

119896119894(119905) = minus2119873

119894119910 (0) + int

119905

0

[119860 (119904) 119896119894(119904) + 119861 (119904) 119901

119894(119904)] 119889119904

+ int119905

0

119866 (119904) 119896119894(119904) 119889119861 (119904)

Mathematical Problems in Engineering 7

119901119894(119905) = minus119872119896

119894(119879) + 2119876

119894119909 (119879)

+ int119879

119905

[119860 (119904) 119901119894(119904) + 119866 (119904) 119902

119894(119904) minus 119867 (119904) 119896

119894(119904)] 119889119904

minus int119879

119905

119902119894(119904) 119889119861 (119904) 119894 = 1 2

(29)

It is easy to check that the state system (26) has a uniquesolution (119901119894(sdot) 119902119894(sdot) 119896119894(sdot)) 119894 = 1 2

Suppose that (1199061(sdot) 1199062(sdot)) is a Nash equilibrium point

By the maximum principle (see Theorem 5) putting theoptimality conditions (22) and (23) the corresponding stateequation (26) and the adjoint equations (28) associated with(1199061(sdot) 1199062(sdot)) together we obtain the following optimality

system for a Nash equilibrium point

119909 (119905) = 119886 + int119905

0

[119860 (119904) 119909 (119904) minus 119861 (119904) 119910 (119904) + 1198631(119904) 1199061(119904)

+1198632(119904) 1199062(119904)] 119889119904

+ int119905

0

119866 (119904) 119909 (119904) 119889119861 (119904)

119910 (119905) = 119872119909 (119879) + int119879

119905

[119860 (119904) 119910 (119904) + 119867 (119904) 119909 (119904) + 119866 (119904) 119911 (119904)

+ 1198681(119904) 1199061(119904) + 119868

2(119904) 1199062(119904)] 119889119904

minus int119879

119905

119911 (119904) 119889119861 (119904)

119896119894(119905) = minus2119873

119894119910 (0) + int

119905

0

[119860 (119904) 119896119894(119904) + 119861 (119904) 119901

119894(119904)] 119889119904

+ int119905

0

119866 (119904) 119896119894(119904) 119889119861 (119904)

119901119894(119905) = minus119872119896

119894(119879) + 2119876

119894119909 (119879)

+ int119879

119905

[119860 (119904) 119901119894(119904) + 119866 (119904) 119902

119894(119904) minus 119867 (119904) 119896

119894(119904)] 119889119904

minus int119879

119905

119902119894(119904) 119889119861 (119904)

1198671(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

1(119905))

= min1199061isin1198801

1198671(119905 Θ (119905) 119906

1 1199062(119905) Λ

1(119905))

1198672(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

2(119905))

= min1199062isin1198802

1198672(119905 Θ (119905) 119906

1(119905) 1199062 Λ2(119905))

119905 isin [0 119879] 119894 = 1 2

(30)

This is called coupled forward-backward stochastic differ-ential equationswith double dimensions (DFBSDE for short)

Note that the coupling comes from the two last relations(which is the maximum condition) We also refer to Yu [17]for the general theory of this kind equationThe8-tuple (119906

1(sdot)

1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot) 119901(sdot) 119902(sdot) 119896(sdot)) of F-adapted processes

satisfying the above is called an adapted solution of (30)We now look at the sufficiency of the existence of a Nashequilibrium point

Theorem 6 Suppose that (1199061(sdot) 1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot) 119901(sdot)

119902(sdot) 119896(sdot)) is an adapted solution to DFBSDE (30) Then(1199061(sdot) 1199062(sdot)) is a Nash equilibrium point

Proof Let (1199061(sdot) 1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot) 119901(sdot) 119902(sdot) 119896(sdot)) be an

adapted solution to DFBSDE (30) For any admissible controlpair (119906

1(sdot) 1199062(sdot)) isin A

1times 1198602 fromTheorem 3 we have

119869119894(1199061(sdot) 1199062(sdot)) minus 119869

119894(1199061(sdot) 1199062(sdot))

= Eint119879

0

[1198671(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

119906

119894(119905))

minus119867119894(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

119906

119894(119905)) ] 119889119905

+ E [1198731(119910(11990611199062)(0))2

minus 1198731(119910 (0))

2

minus21198731119910 (0) (119910

(11990611199062)(0) minus 119910 (0)) ]

+ E [1198761(119909(11990611199062)(119879))2

minus 11987611199092(119879)

minus 211987611199092(119879) sdot (119909

(11990611199062)(119879) minus 119909 (119879)) ]

= Eint119879

0

[1198671(119905 Θ (119905) 119906

120576(119905) Λ

119906

119894(119905))

minus1198671(119905 Θ (119905) 119906 (119905) Λ

119906

1(119905))] 119889119905

+ E [1198731(119910(11990611199062)(0) minus 119910 (0))

2

]

+ E [1198761(119909(11990611199062)(119879) minus 119909 (119879))

2

]

ge 0

(31)

which implies that

1198691(1199061(sdot) 1199062(sdot)) = inf

1199061(sdot)isinA

1

1198691(1199061(sdot) 1199062(sdot)) (32)

Similarly we can get

1198692(1199061(sdot) 1199062(sdot)) = inf

1199062(sdot)isinA

2

1198692(1199061(sdot) 1199062(sdot)) (33)

Therefore (1199061(sdot) 1199062(sdot)) is a Nash equilibrium point

Proposition 7 Suppose that 1198801times 1198802

= R times R and(1199061(sdot) 1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot) 119901(sdot) 119902(sdot) 119896(sdot)) is an adapted solu-

tion to DFBSDE (30) Then (1199061(sdot) 1199062(sdot)) has the following

representation

119906119894(119905) =

1

2119871119894(119905)

[119896 (119905) 119868119894(119905) minus 119901

119894(119905) 119863119894(119905)] 119894 = 1 2 (34)

8 Mathematical Problems in Engineering

Proof From Theorem 6 we deduce that (1199061(sdot) 1199062(sdot)) is a

Nash equilibrium point Then from 1198801times 1198802= R times R and

the optimality conditions (22) and (23) it follows that

11986711199061

(119905 Θ (119905) 1199061(119905) 1199062(119905) Λ

1(119905)) = 0

11986721199062

(119905 Θ (119905) 1199061(119905) 1199062(119905) Λ

2(119905)) = 0

(35)

which imply that

119906119894(119905) =

1

2119871119894(119905)

[119896 (119905) 119868119894(119905) minus 119901

119894(119905) 119863119894(119905)] 119894 = 1 2 (36)

The proof is complete

Remark 8 In summary DFBSDE (30) completely character-izes the Nash equilibrium point Therefore solving the dif-ferential game problem is equivalent to solving the DFBSDE(30) Moreover a candidate equilibrium point can be givenby (34) We refer the reader to Yu [17] for the theory ofsolvability toDFBSDE (30) Since the linear quadratic controlproblem is an important and fascinating class of stochasticcontrol ones and the theoretical results of this problem havelots of significant impacts on a wide range of engineeringmanaging and financial applications in the future we willfocus on the study of financial application as the reviewerssuggest

7 Conclusion

In this paper a two-person nonzero-sum differential gameis studied for a fully coupled forward-backward stochasticsystem with the control process 119906(sdot) not appearing in theforward diffusion term but the control domain not neces-sarily convex For this case we obtain a variation formulafor the cost functional As an application the maximumprinciple for open-loop Nash equilibrium points is estab-lished Finally we work out an example of linear quadraticnonzero-sum differential games to illustrate our stochasticmaximum principle As the reviewers suggest our systemdiscussed in this paper may be extended to the discontinuoussystem such Markov-switching jump-diffusion system forwhich the optimal control problems and differential gameshave extensive applications in industry and finance (seeeg [18 19] and the references therein) Some investigationson this topic will be studied and carried out in our futurepublications

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was partially supported by the National Nat-ural Science Foundation of China (11101140 11301177)the China Postdoctoral Science Foundation (2011M5007212012T50391) and theNatural Science Foundation of ZhejiangProvince (nos Y6110775 and Y6110789)

References

[1] J-M Bismut ldquoConjugate convex functions in optimal stochas-tic controlrdquo Journal of Mathematical Analysis and Applicationsvol 44 no 3 pp 384ndash404 1973

[2] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990

[3] S Tang ldquoGeneral linear quadratic optimal stochastic controlproblems with random coefficients linear stochastic Hamiltonsystems and backward stochastic Riccati equationsrdquo SIAMJournal on Control and Optimization vol 42 no 1 pp 53ndash752003

[4] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward-backward stochastic differential equationswith jumpsrdquo SIAM Journal on Control and Optimization vol48 no 5 pp 2945ndash2976 200910

[5] B Oslashksendal and A Sulem ldquoForward backward stochasticdifferential games and stochastic control under model uncer-taintyrdquo Journal of Optimization Theory and Applications 2012

[6] W S Xu ldquoStochastic maximum principle for optimal controlproblem of forward and backward systemrdquo Australian Mathe-matical Society Journal B vol 37 no 2 pp 172ndash185 1995

[7] Z Wu ldquoMaximum principle for optimal control problem offully coupled forward-backward stochastic systemsrdquo SystemsScience and Mathematical Sciences vol 11 no 3 pp 249ndash2591998

[8] J-T Shi and Z Wu ldquoThe maximum principle for fully coupledforward-backward stochastic control systemrdquo Acta AutomaticaSinica vol 32 no 2 pp 161ndash169 2006

[9] QMeng ldquoAmaximumprinciple for optimal control problem offully coupled forward-backward stochastic systems with partialinformationrdquo Science in China A vol 52 no 7 pp 1579ndash15882009

[10] J Shi and Z Wu ldquoMaximum principle for forward-backwardstochastic control system with random jumps and applicationsto financerdquo Journal of Systems Science amp Complexity vol 23 no2 pp 219ndash231 2010

[11] Z Wu ldquoA general maximum principle for optimal control offorward-backward stochastic systemsrdquo Automatica vol 49 no5 pp 1473ndash1480 2013

[12] J Yong ldquoOptimality variational principle for controlledforward-backward stochastic differential equations with mixedinitial-terminal conditionsrdquo SIAM Journal on Control andOptimization vol 48 no 6 pp 4119ndash4156 2010

[13] E C M Hui and H Xiao ldquoMaximum principle for differentialgames of forward-backward stochastic systems with applica-tionsrdquo Journal of Mathematical Analysis and Applications vol386 no 1 pp 412ndash427 2012

[14] M Tang ldquoMaximum principle for non-zero sum stochasticd-ifferential games of fully coupled forward-backward stochasticsystemsrdquo preprint

[15] S Peng and ZWu ldquoFully coupled forward-backward stochasticdifferential equations and applications to optimal controlrdquoSIAM Journal on Control and Optimization vol 37 no 3 pp825ndash843 1999

[16] Q Lin ldquoOptimal control of coupled forward-backward stochas-tic system with jumps and related hamilton-jacobi-bellmanequationsrdquo httparxivorgabs11114642

[17] Z Yu ldquoLinear-quadratic optimal control and nonzero-sum dif-ferential game of forward-backward stochastic systemrdquo AsianJournal of Control vol 14 no 1 pp 173ndash185 2012

Mathematical Problems in Engineering 9

[18] B Z Temocin and G-W Weber ldquoOptimal control of stochastichybrid systemwith jumps a numerical approximationrdquo Journalof Computational and Applied Mathematics vol 259 pp 443ndash451 2014

[19] N Azevedo D Pinheiro and G W Weber ldquoDynamic pro-gramming for a Markov-switching jumpCdiffusionrdquo Journal ofComputational andAppliedMathematics vol 267 pp 1ndash19 2014

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

Page 6: Research Article A Variational Formula for Nonzero-Sum ...downloads.hindawi.com › journals › mpe › 2014 › 283418.pdf · the forward stochastic system. e existence and uniqueness

6 Mathematical Problems in Engineering

= 120576E [1198671(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(120591))

minus1198671(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(119905))] + 119900 (120576)

(21)

which imply that (17) holdsSimilarly we can prove that (18) holds The proof is

complete

5 Stochastic Maximum Principle

In this section applying the variational formulas (17) and (18)we will state and prove the global maximum principle for theNash equilibrium points of Problem 2

Theorem 5 Under Assumption 1 let 119906(sdot) = (1199061(sdot) 1199062(sdot)) be

a Nash equilibrium point of Problem 2 with the state processΘ(sdot) = (119909(sdot) 119910(sdot) 119911(sdot)) Let Λ

119894(sdot) = (119901

119894(sdot) 119902119894(sdot) 119896119894(sdot)) (119894 = 1 2)

be the unique solution of the adjoint equation (8) correspondingto (119906(sdot) Θ(sdot)) Then

1198671(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

1(120591))

= min1199061isin1198801

1198671(120591 Θ (120591) 119906

1 1199062(120591) Λ

1(120591))

(22)

1198672(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

2(120591))

= min1199062isin1198802

1198672(120591 Θ (120591) 119906

1(120591) 119906

2 Λ2(120591))

(23)

hold for ae (120591 120596) isin [0 119879] times Ω

Proof Since 119906(sdot) = (1199061(sdot) 1199062(sdot)) is a Nash equilibrium point

of Problem 2 by (5) we have

1198691(119906120591120576

1(sdot) 1199062(sdot)) minus 119869

1(1199061(sdot) 1199062(sdot)) ge 0 (24)

Using the notation inTheorem 3 for any arbitrary admissiblecontrol (119906

1(sdot) 1199062(sdot)) and 119905 isin [0 119879] we have

E [1198671(120591 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(120591))

minus1198671(119905 Θ (120591) 119906

1(120591) 119906

2(120591) Λ

119906

1(120591))] ge 0

(25)

which implies that (22) holdsSimilarly we can prove that (23) holds The proof is

complete

6 An Example Linear Quadratic Case

In this section we work out an example of linear quadraticnonzero-sum differential games to illustrate our stochastic

maximum principle More precisely consider the follow-ing one-dimensional linear fully coupled forward-backwardstochastic system

119909 (119905) = 119886 + int119905

0

[119860 (119904) 119909 (119904) minus 119861 (119904) 119910 (119904) + 1198631(119904) 1199061(119904)

+1198632(119904) 1199062(119904)] 119889119904

+ int119905

0

[119866 (119904) 119909 (119904)] 119889119861 (119904)

119910 (119905) = 119872119909 (119879) + int119879

119905

[119860 (119904) 119910 (119904) + 119867 (119904) 119909 (119904) + 119866 (119904) 119911 (119904)

+ 1198681(119904) 1199061(119904) + 119868

2(119904) 1199062(119904)] 119889119904

minus int119879

119905

119911 (119904) 119889119861 (119904)

(26)

with the quadratic cost functional

119869119894(1199061(sdot) 1199062(sdot)) = 119864 [int

119879

0

119871119894(119905) 1199062

119894(119905) 119889119905

+ 1198761198941199092(119879) + 119873

1198941199102(0) ] 119894 = 1 2

(27)

where 119860 119861 1198631 1198632 119866 119867 119868

1 1198682 1198711 and 119871

2are one-

dimensional deterministic bounded measurable functionsand119872119873

111987321198761 and119876

2are constants Also assume 119861 ge 0

1198731ge 0 119873

2ge 0 119876

1ge 0 119876

2ge 0 119872 ge 120573 119871

1ge 120573 119871

2ge 120573

119867 gt 120573 where 120573 gt 0 is a positive constantUnder the above assumptions on the coefficients of

(26) and (27) it is easy to check that for any admissible(1199061(sdot) 1199062(sdot)) isin A

1times 1198602 the state system (26) has a unique

solution and the corresponding stochastic differential gameproblem is well defined For this case the corresponding 119867becomes

119867119894(119905 119909 119910 119911 119906

1 1199062 119901119894 119902119894 119896119894)

= minus119896119894[119860 (119905) 119910 + 119867 (119905) 119909 + 119866 (119905) 119911 + 119868

1(119905) 1199061+ 1198682(119905) 1199062]

+ 119901119894[119860 (119905) 119909 minus 119861 (119905) 119910 + 119863

1(119905) 1199061+ 1198632(119905) 1199062(119904)]

+ 119902119894119866 (119905) 119909 + 119871

119894(119905) 1199062

119894(119905) 119894 = 1 2

(28)

The corresponding adjoint equation associated with anadmissible control pair (119906

1(sdot) 1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot)) becomes

119896119894(119905) = minus2119873

119894119910 (0) + int

119905

0

[119860 (119904) 119896119894(119904) + 119861 (119904) 119901

119894(119904)] 119889119904

+ int119905

0

119866 (119904) 119896119894(119904) 119889119861 (119904)

Mathematical Problems in Engineering 7

119901119894(119905) = minus119872119896

119894(119879) + 2119876

119894119909 (119879)

+ int119879

119905

[119860 (119904) 119901119894(119904) + 119866 (119904) 119902

119894(119904) minus 119867 (119904) 119896

119894(119904)] 119889119904

minus int119879

119905

119902119894(119904) 119889119861 (119904) 119894 = 1 2

(29)

It is easy to check that the state system (26) has a uniquesolution (119901119894(sdot) 119902119894(sdot) 119896119894(sdot)) 119894 = 1 2

Suppose that (1199061(sdot) 1199062(sdot)) is a Nash equilibrium point

By the maximum principle (see Theorem 5) putting theoptimality conditions (22) and (23) the corresponding stateequation (26) and the adjoint equations (28) associated with(1199061(sdot) 1199062(sdot)) together we obtain the following optimality

system for a Nash equilibrium point

119909 (119905) = 119886 + int119905

0

[119860 (119904) 119909 (119904) minus 119861 (119904) 119910 (119904) + 1198631(119904) 1199061(119904)

+1198632(119904) 1199062(119904)] 119889119904

+ int119905

0

119866 (119904) 119909 (119904) 119889119861 (119904)

119910 (119905) = 119872119909 (119879) + int119879

119905

[119860 (119904) 119910 (119904) + 119867 (119904) 119909 (119904) + 119866 (119904) 119911 (119904)

+ 1198681(119904) 1199061(119904) + 119868

2(119904) 1199062(119904)] 119889119904

minus int119879

119905

119911 (119904) 119889119861 (119904)

119896119894(119905) = minus2119873

119894119910 (0) + int

119905

0

[119860 (119904) 119896119894(119904) + 119861 (119904) 119901

119894(119904)] 119889119904

+ int119905

0

119866 (119904) 119896119894(119904) 119889119861 (119904)

119901119894(119905) = minus119872119896

119894(119879) + 2119876

119894119909 (119879)

+ int119879

119905

[119860 (119904) 119901119894(119904) + 119866 (119904) 119902

119894(119904) minus 119867 (119904) 119896

119894(119904)] 119889119904

minus int119879

119905

119902119894(119904) 119889119861 (119904)

1198671(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

1(119905))

= min1199061isin1198801

1198671(119905 Θ (119905) 119906

1 1199062(119905) Λ

1(119905))

1198672(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

2(119905))

= min1199062isin1198802

1198672(119905 Θ (119905) 119906

1(119905) 1199062 Λ2(119905))

119905 isin [0 119879] 119894 = 1 2

(30)

This is called coupled forward-backward stochastic differ-ential equationswith double dimensions (DFBSDE for short)

Note that the coupling comes from the two last relations(which is the maximum condition) We also refer to Yu [17]for the general theory of this kind equationThe8-tuple (119906

1(sdot)

1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot) 119901(sdot) 119902(sdot) 119896(sdot)) of F-adapted processes

satisfying the above is called an adapted solution of (30)We now look at the sufficiency of the existence of a Nashequilibrium point

Theorem 6 Suppose that (1199061(sdot) 1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot) 119901(sdot)

119902(sdot) 119896(sdot)) is an adapted solution to DFBSDE (30) Then(1199061(sdot) 1199062(sdot)) is a Nash equilibrium point

Proof Let (1199061(sdot) 1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot) 119901(sdot) 119902(sdot) 119896(sdot)) be an

adapted solution to DFBSDE (30) For any admissible controlpair (119906

1(sdot) 1199062(sdot)) isin A

1times 1198602 fromTheorem 3 we have

119869119894(1199061(sdot) 1199062(sdot)) minus 119869

119894(1199061(sdot) 1199062(sdot))

= Eint119879

0

[1198671(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

119906

119894(119905))

minus119867119894(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

119906

119894(119905)) ] 119889119905

+ E [1198731(119910(11990611199062)(0))2

minus 1198731(119910 (0))

2

minus21198731119910 (0) (119910

(11990611199062)(0) minus 119910 (0)) ]

+ E [1198761(119909(11990611199062)(119879))2

minus 11987611199092(119879)

minus 211987611199092(119879) sdot (119909

(11990611199062)(119879) minus 119909 (119879)) ]

= Eint119879

0

[1198671(119905 Θ (119905) 119906

120576(119905) Λ

119906

119894(119905))

minus1198671(119905 Θ (119905) 119906 (119905) Λ

119906

1(119905))] 119889119905

+ E [1198731(119910(11990611199062)(0) minus 119910 (0))

2

]

+ E [1198761(119909(11990611199062)(119879) minus 119909 (119879))

2

]

ge 0

(31)

which implies that

1198691(1199061(sdot) 1199062(sdot)) = inf

1199061(sdot)isinA

1

1198691(1199061(sdot) 1199062(sdot)) (32)

Similarly we can get

1198692(1199061(sdot) 1199062(sdot)) = inf

1199062(sdot)isinA

2

1198692(1199061(sdot) 1199062(sdot)) (33)

Therefore (1199061(sdot) 1199062(sdot)) is a Nash equilibrium point

Proposition 7 Suppose that 1198801times 1198802

= R times R and(1199061(sdot) 1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot) 119901(sdot) 119902(sdot) 119896(sdot)) is an adapted solu-

tion to DFBSDE (30) Then (1199061(sdot) 1199062(sdot)) has the following

representation

119906119894(119905) =

1

2119871119894(119905)

[119896 (119905) 119868119894(119905) minus 119901

119894(119905) 119863119894(119905)] 119894 = 1 2 (34)

8 Mathematical Problems in Engineering

Proof From Theorem 6 we deduce that (1199061(sdot) 1199062(sdot)) is a

Nash equilibrium point Then from 1198801times 1198802= R times R and

the optimality conditions (22) and (23) it follows that

11986711199061

(119905 Θ (119905) 1199061(119905) 1199062(119905) Λ

1(119905)) = 0

11986721199062

(119905 Θ (119905) 1199061(119905) 1199062(119905) Λ

2(119905)) = 0

(35)

which imply that

119906119894(119905) =

1

2119871119894(119905)

[119896 (119905) 119868119894(119905) minus 119901

119894(119905) 119863119894(119905)] 119894 = 1 2 (36)

The proof is complete

Remark 8 In summary DFBSDE (30) completely character-izes the Nash equilibrium point Therefore solving the dif-ferential game problem is equivalent to solving the DFBSDE(30) Moreover a candidate equilibrium point can be givenby (34) We refer the reader to Yu [17] for the theory ofsolvability toDFBSDE (30) Since the linear quadratic controlproblem is an important and fascinating class of stochasticcontrol ones and the theoretical results of this problem havelots of significant impacts on a wide range of engineeringmanaging and financial applications in the future we willfocus on the study of financial application as the reviewerssuggest

7 Conclusion

In this paper a two-person nonzero-sum differential gameis studied for a fully coupled forward-backward stochasticsystem with the control process 119906(sdot) not appearing in theforward diffusion term but the control domain not neces-sarily convex For this case we obtain a variation formulafor the cost functional As an application the maximumprinciple for open-loop Nash equilibrium points is estab-lished Finally we work out an example of linear quadraticnonzero-sum differential games to illustrate our stochasticmaximum principle As the reviewers suggest our systemdiscussed in this paper may be extended to the discontinuoussystem such Markov-switching jump-diffusion system forwhich the optimal control problems and differential gameshave extensive applications in industry and finance (seeeg [18 19] and the references therein) Some investigationson this topic will be studied and carried out in our futurepublications

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was partially supported by the National Nat-ural Science Foundation of China (11101140 11301177)the China Postdoctoral Science Foundation (2011M5007212012T50391) and theNatural Science Foundation of ZhejiangProvince (nos Y6110775 and Y6110789)

References

[1] J-M Bismut ldquoConjugate convex functions in optimal stochas-tic controlrdquo Journal of Mathematical Analysis and Applicationsvol 44 no 3 pp 384ndash404 1973

[2] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990

[3] S Tang ldquoGeneral linear quadratic optimal stochastic controlproblems with random coefficients linear stochastic Hamiltonsystems and backward stochastic Riccati equationsrdquo SIAMJournal on Control and Optimization vol 42 no 1 pp 53ndash752003

[4] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward-backward stochastic differential equationswith jumpsrdquo SIAM Journal on Control and Optimization vol48 no 5 pp 2945ndash2976 200910

[5] B Oslashksendal and A Sulem ldquoForward backward stochasticdifferential games and stochastic control under model uncer-taintyrdquo Journal of Optimization Theory and Applications 2012

[6] W S Xu ldquoStochastic maximum principle for optimal controlproblem of forward and backward systemrdquo Australian Mathe-matical Society Journal B vol 37 no 2 pp 172ndash185 1995

[7] Z Wu ldquoMaximum principle for optimal control problem offully coupled forward-backward stochastic systemsrdquo SystemsScience and Mathematical Sciences vol 11 no 3 pp 249ndash2591998

[8] J-T Shi and Z Wu ldquoThe maximum principle for fully coupledforward-backward stochastic control systemrdquo Acta AutomaticaSinica vol 32 no 2 pp 161ndash169 2006

[9] QMeng ldquoAmaximumprinciple for optimal control problem offully coupled forward-backward stochastic systems with partialinformationrdquo Science in China A vol 52 no 7 pp 1579ndash15882009

[10] J Shi and Z Wu ldquoMaximum principle for forward-backwardstochastic control system with random jumps and applicationsto financerdquo Journal of Systems Science amp Complexity vol 23 no2 pp 219ndash231 2010

[11] Z Wu ldquoA general maximum principle for optimal control offorward-backward stochastic systemsrdquo Automatica vol 49 no5 pp 1473ndash1480 2013

[12] J Yong ldquoOptimality variational principle for controlledforward-backward stochastic differential equations with mixedinitial-terminal conditionsrdquo SIAM Journal on Control andOptimization vol 48 no 6 pp 4119ndash4156 2010

[13] E C M Hui and H Xiao ldquoMaximum principle for differentialgames of forward-backward stochastic systems with applica-tionsrdquo Journal of Mathematical Analysis and Applications vol386 no 1 pp 412ndash427 2012

[14] M Tang ldquoMaximum principle for non-zero sum stochasticd-ifferential games of fully coupled forward-backward stochasticsystemsrdquo preprint

[15] S Peng and ZWu ldquoFully coupled forward-backward stochasticdifferential equations and applications to optimal controlrdquoSIAM Journal on Control and Optimization vol 37 no 3 pp825ndash843 1999

[16] Q Lin ldquoOptimal control of coupled forward-backward stochas-tic system with jumps and related hamilton-jacobi-bellmanequationsrdquo httparxivorgabs11114642

[17] Z Yu ldquoLinear-quadratic optimal control and nonzero-sum dif-ferential game of forward-backward stochastic systemrdquo AsianJournal of Control vol 14 no 1 pp 173ndash185 2012

Mathematical Problems in Engineering 9

[18] B Z Temocin and G-W Weber ldquoOptimal control of stochastichybrid systemwith jumps a numerical approximationrdquo Journalof Computational and Applied Mathematics vol 259 pp 443ndash451 2014

[19] N Azevedo D Pinheiro and G W Weber ldquoDynamic pro-gramming for a Markov-switching jumpCdiffusionrdquo Journal ofComputational andAppliedMathematics vol 267 pp 1ndash19 2014

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

Page 7: Research Article A Variational Formula for Nonzero-Sum ...downloads.hindawi.com › journals › mpe › 2014 › 283418.pdf · the forward stochastic system. e existence and uniqueness

Mathematical Problems in Engineering 7

119901119894(119905) = minus119872119896

119894(119879) + 2119876

119894119909 (119879)

+ int119879

119905

[119860 (119904) 119901119894(119904) + 119866 (119904) 119902

119894(119904) minus 119867 (119904) 119896

119894(119904)] 119889119904

minus int119879

119905

119902119894(119904) 119889119861 (119904) 119894 = 1 2

(29)

It is easy to check that the state system (26) has a uniquesolution (119901119894(sdot) 119902119894(sdot) 119896119894(sdot)) 119894 = 1 2

Suppose that (1199061(sdot) 1199062(sdot)) is a Nash equilibrium point

By the maximum principle (see Theorem 5) putting theoptimality conditions (22) and (23) the corresponding stateequation (26) and the adjoint equations (28) associated with(1199061(sdot) 1199062(sdot)) together we obtain the following optimality

system for a Nash equilibrium point

119909 (119905) = 119886 + int119905

0

[119860 (119904) 119909 (119904) minus 119861 (119904) 119910 (119904) + 1198631(119904) 1199061(119904)

+1198632(119904) 1199062(119904)] 119889119904

+ int119905

0

119866 (119904) 119909 (119904) 119889119861 (119904)

119910 (119905) = 119872119909 (119879) + int119879

119905

[119860 (119904) 119910 (119904) + 119867 (119904) 119909 (119904) + 119866 (119904) 119911 (119904)

+ 1198681(119904) 1199061(119904) + 119868

2(119904) 1199062(119904)] 119889119904

minus int119879

119905

119911 (119904) 119889119861 (119904)

119896119894(119905) = minus2119873

119894119910 (0) + int

119905

0

[119860 (119904) 119896119894(119904) + 119861 (119904) 119901

119894(119904)] 119889119904

+ int119905

0

119866 (119904) 119896119894(119904) 119889119861 (119904)

119901119894(119905) = minus119872119896

119894(119879) + 2119876

119894119909 (119879)

+ int119879

119905

[119860 (119904) 119901119894(119904) + 119866 (119904) 119902

119894(119904) minus 119867 (119904) 119896

119894(119904)] 119889119904

minus int119879

119905

119902119894(119904) 119889119861 (119904)

1198671(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

1(119905))

= min1199061isin1198801

1198671(119905 Θ (119905) 119906

1 1199062(119905) Λ

1(119905))

1198672(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

2(119905))

= min1199062isin1198802

1198672(119905 Θ (119905) 119906

1(119905) 1199062 Λ2(119905))

119905 isin [0 119879] 119894 = 1 2

(30)

This is called coupled forward-backward stochastic differ-ential equationswith double dimensions (DFBSDE for short)

Note that the coupling comes from the two last relations(which is the maximum condition) We also refer to Yu [17]for the general theory of this kind equationThe8-tuple (119906

1(sdot)

1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot) 119901(sdot) 119902(sdot) 119896(sdot)) of F-adapted processes

satisfying the above is called an adapted solution of (30)We now look at the sufficiency of the existence of a Nashequilibrium point

Theorem 6 Suppose that (1199061(sdot) 1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot) 119901(sdot)

119902(sdot) 119896(sdot)) is an adapted solution to DFBSDE (30) Then(1199061(sdot) 1199062(sdot)) is a Nash equilibrium point

Proof Let (1199061(sdot) 1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot) 119901(sdot) 119902(sdot) 119896(sdot)) be an

adapted solution to DFBSDE (30) For any admissible controlpair (119906

1(sdot) 1199062(sdot)) isin A

1times 1198602 fromTheorem 3 we have

119869119894(1199061(sdot) 1199062(sdot)) minus 119869

119894(1199061(sdot) 1199062(sdot))

= Eint119879

0

[1198671(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

119906

119894(119905))

minus119867119894(119905 Θ (119905) 119906

1(119905) 1199062(119905) Λ

119906

119894(119905)) ] 119889119905

+ E [1198731(119910(11990611199062)(0))2

minus 1198731(119910 (0))

2

minus21198731119910 (0) (119910

(11990611199062)(0) minus 119910 (0)) ]

+ E [1198761(119909(11990611199062)(119879))2

minus 11987611199092(119879)

minus 211987611199092(119879) sdot (119909

(11990611199062)(119879) minus 119909 (119879)) ]

= Eint119879

0

[1198671(119905 Θ (119905) 119906

120576(119905) Λ

119906

119894(119905))

minus1198671(119905 Θ (119905) 119906 (119905) Λ

119906

1(119905))] 119889119905

+ E [1198731(119910(11990611199062)(0) minus 119910 (0))

2

]

+ E [1198761(119909(11990611199062)(119879) minus 119909 (119879))

2

]

ge 0

(31)

which implies that

1198691(1199061(sdot) 1199062(sdot)) = inf

1199061(sdot)isinA

1

1198691(1199061(sdot) 1199062(sdot)) (32)

Similarly we can get

1198692(1199061(sdot) 1199062(sdot)) = inf

1199062(sdot)isinA

2

1198692(1199061(sdot) 1199062(sdot)) (33)

Therefore (1199061(sdot) 1199062(sdot)) is a Nash equilibrium point

Proposition 7 Suppose that 1198801times 1198802

= R times R and(1199061(sdot) 1199062(sdot) 119909(sdot) 119910(sdot) 119911(sdot) 119901(sdot) 119902(sdot) 119896(sdot)) is an adapted solu-

tion to DFBSDE (30) Then (1199061(sdot) 1199062(sdot)) has the following

representation

119906119894(119905) =

1

2119871119894(119905)

[119896 (119905) 119868119894(119905) minus 119901

119894(119905) 119863119894(119905)] 119894 = 1 2 (34)

8 Mathematical Problems in Engineering

Proof From Theorem 6 we deduce that (1199061(sdot) 1199062(sdot)) is a

Nash equilibrium point Then from 1198801times 1198802= R times R and

the optimality conditions (22) and (23) it follows that

11986711199061

(119905 Θ (119905) 1199061(119905) 1199062(119905) Λ

1(119905)) = 0

11986721199062

(119905 Θ (119905) 1199061(119905) 1199062(119905) Λ

2(119905)) = 0

(35)

which imply that

119906119894(119905) =

1

2119871119894(119905)

[119896 (119905) 119868119894(119905) minus 119901

119894(119905) 119863119894(119905)] 119894 = 1 2 (36)

The proof is complete

Remark 8 In summary DFBSDE (30) completely character-izes the Nash equilibrium point Therefore solving the dif-ferential game problem is equivalent to solving the DFBSDE(30) Moreover a candidate equilibrium point can be givenby (34) We refer the reader to Yu [17] for the theory ofsolvability toDFBSDE (30) Since the linear quadratic controlproblem is an important and fascinating class of stochasticcontrol ones and the theoretical results of this problem havelots of significant impacts on a wide range of engineeringmanaging and financial applications in the future we willfocus on the study of financial application as the reviewerssuggest

7 Conclusion

In this paper a two-person nonzero-sum differential gameis studied for a fully coupled forward-backward stochasticsystem with the control process 119906(sdot) not appearing in theforward diffusion term but the control domain not neces-sarily convex For this case we obtain a variation formulafor the cost functional As an application the maximumprinciple for open-loop Nash equilibrium points is estab-lished Finally we work out an example of linear quadraticnonzero-sum differential games to illustrate our stochasticmaximum principle As the reviewers suggest our systemdiscussed in this paper may be extended to the discontinuoussystem such Markov-switching jump-diffusion system forwhich the optimal control problems and differential gameshave extensive applications in industry and finance (seeeg [18 19] and the references therein) Some investigationson this topic will be studied and carried out in our futurepublications

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was partially supported by the National Nat-ural Science Foundation of China (11101140 11301177)the China Postdoctoral Science Foundation (2011M5007212012T50391) and theNatural Science Foundation of ZhejiangProvince (nos Y6110775 and Y6110789)

References

[1] J-M Bismut ldquoConjugate convex functions in optimal stochas-tic controlrdquo Journal of Mathematical Analysis and Applicationsvol 44 no 3 pp 384ndash404 1973

[2] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990

[3] S Tang ldquoGeneral linear quadratic optimal stochastic controlproblems with random coefficients linear stochastic Hamiltonsystems and backward stochastic Riccati equationsrdquo SIAMJournal on Control and Optimization vol 42 no 1 pp 53ndash752003

[4] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward-backward stochastic differential equationswith jumpsrdquo SIAM Journal on Control and Optimization vol48 no 5 pp 2945ndash2976 200910

[5] B Oslashksendal and A Sulem ldquoForward backward stochasticdifferential games and stochastic control under model uncer-taintyrdquo Journal of Optimization Theory and Applications 2012

[6] W S Xu ldquoStochastic maximum principle for optimal controlproblem of forward and backward systemrdquo Australian Mathe-matical Society Journal B vol 37 no 2 pp 172ndash185 1995

[7] Z Wu ldquoMaximum principle for optimal control problem offully coupled forward-backward stochastic systemsrdquo SystemsScience and Mathematical Sciences vol 11 no 3 pp 249ndash2591998

[8] J-T Shi and Z Wu ldquoThe maximum principle for fully coupledforward-backward stochastic control systemrdquo Acta AutomaticaSinica vol 32 no 2 pp 161ndash169 2006

[9] QMeng ldquoAmaximumprinciple for optimal control problem offully coupled forward-backward stochastic systems with partialinformationrdquo Science in China A vol 52 no 7 pp 1579ndash15882009

[10] J Shi and Z Wu ldquoMaximum principle for forward-backwardstochastic control system with random jumps and applicationsto financerdquo Journal of Systems Science amp Complexity vol 23 no2 pp 219ndash231 2010

[11] Z Wu ldquoA general maximum principle for optimal control offorward-backward stochastic systemsrdquo Automatica vol 49 no5 pp 1473ndash1480 2013

[12] J Yong ldquoOptimality variational principle for controlledforward-backward stochastic differential equations with mixedinitial-terminal conditionsrdquo SIAM Journal on Control andOptimization vol 48 no 6 pp 4119ndash4156 2010

[13] E C M Hui and H Xiao ldquoMaximum principle for differentialgames of forward-backward stochastic systems with applica-tionsrdquo Journal of Mathematical Analysis and Applications vol386 no 1 pp 412ndash427 2012

[14] M Tang ldquoMaximum principle for non-zero sum stochasticd-ifferential games of fully coupled forward-backward stochasticsystemsrdquo preprint

[15] S Peng and ZWu ldquoFully coupled forward-backward stochasticdifferential equations and applications to optimal controlrdquoSIAM Journal on Control and Optimization vol 37 no 3 pp825ndash843 1999

[16] Q Lin ldquoOptimal control of coupled forward-backward stochas-tic system with jumps and related hamilton-jacobi-bellmanequationsrdquo httparxivorgabs11114642

[17] Z Yu ldquoLinear-quadratic optimal control and nonzero-sum dif-ferential game of forward-backward stochastic systemrdquo AsianJournal of Control vol 14 no 1 pp 173ndash185 2012

Mathematical Problems in Engineering 9

[18] B Z Temocin and G-W Weber ldquoOptimal control of stochastichybrid systemwith jumps a numerical approximationrdquo Journalof Computational and Applied Mathematics vol 259 pp 443ndash451 2014

[19] N Azevedo D Pinheiro and G W Weber ldquoDynamic pro-gramming for a Markov-switching jumpCdiffusionrdquo Journal ofComputational andAppliedMathematics vol 267 pp 1ndash19 2014

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

Page 8: Research Article A Variational Formula for Nonzero-Sum ...downloads.hindawi.com › journals › mpe › 2014 › 283418.pdf · the forward stochastic system. e existence and uniqueness

8 Mathematical Problems in Engineering

Proof From Theorem 6 we deduce that (1199061(sdot) 1199062(sdot)) is a

Nash equilibrium point Then from 1198801times 1198802= R times R and

the optimality conditions (22) and (23) it follows that

11986711199061

(119905 Θ (119905) 1199061(119905) 1199062(119905) Λ

1(119905)) = 0

11986721199062

(119905 Θ (119905) 1199061(119905) 1199062(119905) Λ

2(119905)) = 0

(35)

which imply that

119906119894(119905) =

1

2119871119894(119905)

[119896 (119905) 119868119894(119905) minus 119901

119894(119905) 119863119894(119905)] 119894 = 1 2 (36)

The proof is complete

Remark 8 In summary DFBSDE (30) completely character-izes the Nash equilibrium point Therefore solving the dif-ferential game problem is equivalent to solving the DFBSDE(30) Moreover a candidate equilibrium point can be givenby (34) We refer the reader to Yu [17] for the theory ofsolvability toDFBSDE (30) Since the linear quadratic controlproblem is an important and fascinating class of stochasticcontrol ones and the theoretical results of this problem havelots of significant impacts on a wide range of engineeringmanaging and financial applications in the future we willfocus on the study of financial application as the reviewerssuggest

7 Conclusion

In this paper a two-person nonzero-sum differential gameis studied for a fully coupled forward-backward stochasticsystem with the control process 119906(sdot) not appearing in theforward diffusion term but the control domain not neces-sarily convex For this case we obtain a variation formulafor the cost functional As an application the maximumprinciple for open-loop Nash equilibrium points is estab-lished Finally we work out an example of linear quadraticnonzero-sum differential games to illustrate our stochasticmaximum principle As the reviewers suggest our systemdiscussed in this paper may be extended to the discontinuoussystem such Markov-switching jump-diffusion system forwhich the optimal control problems and differential gameshave extensive applications in industry and finance (seeeg [18 19] and the references therein) Some investigationson this topic will be studied and carried out in our futurepublications

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was partially supported by the National Nat-ural Science Foundation of China (11101140 11301177)the China Postdoctoral Science Foundation (2011M5007212012T50391) and theNatural Science Foundation of ZhejiangProvince (nos Y6110775 and Y6110789)

References

[1] J-M Bismut ldquoConjugate convex functions in optimal stochas-tic controlrdquo Journal of Mathematical Analysis and Applicationsvol 44 no 3 pp 384ndash404 1973

[2] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990

[3] S Tang ldquoGeneral linear quadratic optimal stochastic controlproblems with random coefficients linear stochastic Hamiltonsystems and backward stochastic Riccati equationsrdquo SIAMJournal on Control and Optimization vol 42 no 1 pp 53ndash752003

[4] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward-backward stochastic differential equationswith jumpsrdquo SIAM Journal on Control and Optimization vol48 no 5 pp 2945ndash2976 200910

[5] B Oslashksendal and A Sulem ldquoForward backward stochasticdifferential games and stochastic control under model uncer-taintyrdquo Journal of Optimization Theory and Applications 2012

[6] W S Xu ldquoStochastic maximum principle for optimal controlproblem of forward and backward systemrdquo Australian Mathe-matical Society Journal B vol 37 no 2 pp 172ndash185 1995

[7] Z Wu ldquoMaximum principle for optimal control problem offully coupled forward-backward stochastic systemsrdquo SystemsScience and Mathematical Sciences vol 11 no 3 pp 249ndash2591998

[8] J-T Shi and Z Wu ldquoThe maximum principle for fully coupledforward-backward stochastic control systemrdquo Acta AutomaticaSinica vol 32 no 2 pp 161ndash169 2006

[9] QMeng ldquoAmaximumprinciple for optimal control problem offully coupled forward-backward stochastic systems with partialinformationrdquo Science in China A vol 52 no 7 pp 1579ndash15882009

[10] J Shi and Z Wu ldquoMaximum principle for forward-backwardstochastic control system with random jumps and applicationsto financerdquo Journal of Systems Science amp Complexity vol 23 no2 pp 219ndash231 2010

[11] Z Wu ldquoA general maximum principle for optimal control offorward-backward stochastic systemsrdquo Automatica vol 49 no5 pp 1473ndash1480 2013

[12] J Yong ldquoOptimality variational principle for controlledforward-backward stochastic differential equations with mixedinitial-terminal conditionsrdquo SIAM Journal on Control andOptimization vol 48 no 6 pp 4119ndash4156 2010

[13] E C M Hui and H Xiao ldquoMaximum principle for differentialgames of forward-backward stochastic systems with applica-tionsrdquo Journal of Mathematical Analysis and Applications vol386 no 1 pp 412ndash427 2012

[14] M Tang ldquoMaximum principle for non-zero sum stochasticd-ifferential games of fully coupled forward-backward stochasticsystemsrdquo preprint

[15] S Peng and ZWu ldquoFully coupled forward-backward stochasticdifferential equations and applications to optimal controlrdquoSIAM Journal on Control and Optimization vol 37 no 3 pp825ndash843 1999

[16] Q Lin ldquoOptimal control of coupled forward-backward stochas-tic system with jumps and related hamilton-jacobi-bellmanequationsrdquo httparxivorgabs11114642

[17] Z Yu ldquoLinear-quadratic optimal control and nonzero-sum dif-ferential game of forward-backward stochastic systemrdquo AsianJournal of Control vol 14 no 1 pp 173ndash185 2012

Mathematical Problems in Engineering 9

[18] B Z Temocin and G-W Weber ldquoOptimal control of stochastichybrid systemwith jumps a numerical approximationrdquo Journalof Computational and Applied Mathematics vol 259 pp 443ndash451 2014

[19] N Azevedo D Pinheiro and G W Weber ldquoDynamic pro-gramming for a Markov-switching jumpCdiffusionrdquo Journal ofComputational andAppliedMathematics vol 267 pp 1ndash19 2014

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

Page 9: Research Article A Variational Formula for Nonzero-Sum ...downloads.hindawi.com › journals › mpe › 2014 › 283418.pdf · the forward stochastic system. e existence and uniqueness

Mathematical Problems in Engineering 9

[18] B Z Temocin and G-W Weber ldquoOptimal control of stochastichybrid systemwith jumps a numerical approximationrdquo Journalof Computational and Applied Mathematics vol 259 pp 443ndash451 2014

[19] N Azevedo D Pinheiro and G W Weber ldquoDynamic pro-gramming for a Markov-switching jumpCdiffusionrdquo Journal ofComputational andAppliedMathematics vol 267 pp 1ndash19 2014

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

Page 10: Research Article A Variational Formula for Nonzero-Sum ...downloads.hindawi.com › journals › mpe › 2014 › 283418.pdf · the forward stochastic system. e existence and uniqueness

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


Stochastic Zero-sum and Nonzero-sum -regular Games

Stochastic Zero-sum and Nonzero-sum -regular Games

Uniqueness DaLuPa

Uniqueness DaLuPa

mygdidotorg.files.wordpress.com › 2011 › 06 › gdi-…  · Web view1NC Brain Drain 1/ 2. 1NC Brain Drain 2/ 3 **Uniqueness** 4. Uniqueness – NASA Hiring Declining 4. Uniqueness

mygdidotorg.files.wordpress.com › 2011 › 06 › gdi-…  · Web view1NC Brain Drain 1/ 2. 1NC Brain Drain 2/ 3 **Uniqueness** 4. Uniqueness – NASA Hiring Declining 4. Uniqueness

Psychotactics Uniqueness

Psychotactics Uniqueness

Andrea Merloni (EXC, MPE) A. Bongiorno (MPE), COSMOS + zCOSMOS teams

Andrea Merloni (EXC, MPE) A. Bongiorno (MPE), COSMOS + zCOSMOS teams

Analysis of Microstrip Patc, h Antennas With Nonzero ...

Analysis of Microstrip Patc, h Antennas With Nonzero ...

The Bible: The Bible: I. Uniqueness of Its Composition II. Uniqueness of Its Collection III. Uniqueness of Its Contents Copyright by Norman L. Geisler.

The Bible: The Bible: I. Uniqueness of Its Composition II. Uniqueness of Its Collection III. Uniqueness of Its Contents Copyright by Norman L. Geisler.

MATH1131 Mathematics 1A Algebraweb.maths.unsw.edu.au/.../math1131-alg-17s1/files/... · Definition a leadingrowis a nonzero row a leadingentryis the first nonzero entry in a leading

MATH1131 Mathematics 1A Algebraweb.maths.unsw.edu.au/.../math1131-alg-17s1/files/... · Definition a leadingrowis a nonzero row a leadingentryis the first nonzero entry in a leading

Seminar02 MPE

Seminar02 MPE

Uniqueness theorems for displacement fields with locally ...folias/wilcox/wilcox1.pdf · Uniqueness theorems for displacement fields 223 The role of existence and uniqueness theorems.

Uniqueness theorems for displacement fields with locally ...folias/wilcox/wilcox1.pdf · Uniqueness theorems for displacement fields 223 The role of existence and uniqueness theorems.

DEPARTMENT OF PHYSICAL EDUCATION & SPORTS SCIENCESigipess.du.ac.in/pdf/Syllabus/M.P.Ed_syllabus.pdf · MPE 0703 (xiii) Table Tennis MPE 0703 (xiv) Volleyball MPE 0703 (xv) Yoga MPE

DEPARTMENT OF PHYSICAL EDUCATION & SPORTS SCIENCESigipess.du.ac.in/pdf/Syllabus/M.P.Ed_syllabus.pdf · MPE 0703 (xiii) Table Tennis MPE 0703 (xiv) Volleyball MPE 0703 (xv) Yoga MPE

UNIQUENESS AND LOCALIZATION-I. ASSOCIATIVE AND NON ...hueckel/papers/BigoniHueckelNA.pdf · non-uniqueness thresiiolds in elastoplastic rate problems A sufiicient criterion for uniqueness

UNIQUENESS AND LOCALIZATION-I. ASSOCIATIVE AND NON ...hueckel/papers/BigoniHueckelNA.pdf · non-uniqueness thresiiolds in elastoplastic rate problems A sufiicient criterion for uniqueness

Uniqueness of cufflinks

Uniqueness of cufflinks

Docker: from zero to nonzero

Docker: from zero to nonzero

Uniqueness of Klonopin

Uniqueness of Klonopin

Multisensor Precipitation Estimator (MPE)In this presentation • What is MPE? • How has MPE come about? • What can MPE do now? • What are the upcoming improvements? MPE •

Multisensor Precipitation Estimator (MPE)In this presentation • What is MPE? • How has MPE come about? • What can MPE do now? • What are the upcoming improvements? MPE •

Browser Uniqueness

Browser Uniqueness

Existence and uniqueness for Volterra nonlinear integral ... · Existence and uniqueness for Volterra nonlinear integral ... I. G. Petrovski and R. ... Existence and uniqueness for

Existence and uniqueness for Volterra nonlinear integral ... · Existence and uniqueness for Volterra nonlinear integral ... I. G. Petrovski and R. ... Existence and uniqueness for

Modeling quantum fluid dynamics at nonzero temperatures

Modeling quantum fluid dynamics at nonzero temperatures

Executing on Purposeful Uniqueness

Executing on Purposeful Uniqueness