Forward Backward Stochastic Differential Equation Games with Delay and Noisy Memory
aa r X i v : . [ m a t h . O C ] J un Forward backward stochastic differentialequation games with delay and noisy memory
K. R. Dahl † October 9, 2018
Abstract
The goal of this paper is to study a stochastic game connected to asystem of forward backward stochastic differential equations (FBSDEs)involving delay and noisy memory. We derive sufficient and necessarymaximum principles for a set of controls for the players to be a Nashequilibrium in the game. Furthermore, we study a corresponding FB-SDE involving Malliavin derivatives. This kind of equation has not beenstudied before. The maximum principles give conditions for determiningthe Nash equilibrium of the game. We use this to derive a closed formNash equilibrium for an economic model where the players maximize theirconsumption with respect to recursive utility.
Key words:
Forward backward stochastic differential equations. Stochas-tic game. Delay. Noisy memory.
AMS subject classification: † Department of Mathematics, University of Oslo, Pb. 1053 Blindern, 0316 Oslo, [email protected]
The aim of this paper is to study a stochastic game between two players. Thegame is based on a forward stochastic differential equation (SDE) for the process X . In applications to economy, this process can be thought of as the marketsituation, e.g. the financial market, the housing market or the oil market. ThisSDE includes two kinds of memory of the past; regular memory and noisymemory. Regular memory (also called delay, see f. ex. the survey paper byIvanov et al. [7]) means that the SDE can depend on previous values of theprocess X . That is, for some given δ > , X ( t ) depends on X ( t − δ ) . For moreon stochastic delay differential equations and optimal control with delay, seeØksendal et al [18] and Agram and Øksendal [3]. In constrast, noisy memorymeans that the SDE may involve an Itô integral over previous values of theprocess, so for δ > , X ( t ) depends on R tt − δ X ( s ) dB ( s ) where { B ( s ) } s ∈ [0 ,T ] is aBrownian motion. For more on noisy memory, see Dahl et al. [6].Connected to this SDE are two backward stochastic differential equations(BSDEs). These BSDEs are connected to the SDE in the sense that they dependon { X ( t ) } t ∈ [0 ,T ] , as well as the delay and noisy memory of this process. Hence,this forms an FBSDE system. Each of these BSDEs corresponds to one of theplayers in the stochastic game; corresponding to player i = 1 , is a BSDE inthe process { W i ( t ) } t ∈ [0 ,T ] . The length of memory can be different for the twoplayers, so for i = 1 , , player i has memory span δ i . The players may alsohave different levels of information, which is included in the model by having(potentially) different filtrations {E ( i ) t } t ∈ [0 ,T ] , i = 1 , .Each of the players aim to find an optimal control u i which maximizes theirpersonal performance (objective) function, J i . Seminal work in stochastic op-timal control has been done by Krylov and his students, see e.g. Krylov [8]and [9]. The performance function of each of the agents will be defined in sucha way that it depends on the player’s profit rate, the market process X andthe process W i coming from the player’s BSDE (more on this in Section 2,2BSDE games with delay and noisy memoryequation (3)). This kind of problem, where both players maximize their perfor-mance which depends on an FBSDE, is called an FBSDE stochastic game, andhas been studied by e.g. Øksendal and Sulem [14]. However, they do not includememory in their model. We study conditions for a pair of controls ( u , u ) tobe a Nash equilibrium for such a stochastic game. That is, we would like to de-termine controls such that the players cannot benefit by changing their actions.In order to do so, we derive sufficient and necessary maximum principles givingconditions for a control to be Nash optimal. This is done in Sections 3 and4. Maximum principles for forward backward stochastic differential equations(FBSDEs) have been studied by Wang and Wu [23] as well as Øksendal andSulem [14], but these papers do not consider a stochastic game.In connection with these maximum principles, there are adjoint equations(see e.g. Øksendal [12] for an introduction to sotchastic maximum principlesand adjoint equations, or Øksendal and Sulem [15] for maximum principles andadjoint equations where delay is involved). In our case, these adjoint equationsare a system of coupled forward backward stochastic differential equations in-volving Malliavin derivatives (see Di Nunno et al. [5] for more on Malliavinderivatives). To the best of our knowledge, such equations have not been stud-ied before. In Section 5 we study a slightly simplified version of these adjointFBSDEs, and establish a connection between these equations and a system ofFBSDEs without Malliavin derivatives. Finally, in Section 6, we apply our re-sults to a specific example in order to determine the optimal consumption withrespect to recursive utility. Let (Ω , F , P ) be a probability space, and let B ( t ) , t ∈ [0 , T ] be a Brownianmotion in this space. Also, let ˜ N ( t, · ) be an independent compensated Poissonrandom measure. Let ( F t ) t ∈ [0 ,T ] be the P -augmented filtration generated by B ( t ) and ˜ N ( t, · ) . 3BSDE games with delay and noisy memoryWe will consider a game between two players: player and player . Let u i ( t ) be the control process chosen by player i = 1 , , and denote u ( t ) = ( u ( t ) , u ( t )) .Let A i , i = 1 , , denote the set of admissible controls for player i and A = A × A .We consider a controlled forward stochastic differential equation for a process X ( t ) = X u ( t, ω ) , ω ∈ Ω , t ∈ [0 , T ] determining the market situation (in thefollowing, we omit the ω for notational ease unless it is important to highlightits dependence): dX ( t ) = b ( t, X ( t ) , Y ( t ) , Λ ( t ) , u ( t ) , ω ) dt + σ ( t, X ( t ) , Y ( t ) , Λ ( t ) , u ( t ) , ω ) dB ( t )+ R R γ ( t − , X ( t − ) , Y ( t − ) , Λ ( t − ) , u ( t − ) , ζ, ω ) ˜ N ( dt, dζ ) X (0) = x (1)where Y ( t ) = ( Y ( t ) , Y ( t )) , Λ ( t ) = (Λ ( t ) , Λ ( t )) , and Y i ( t ) := X ( t − δ i ) , Λ i ( t ) := R tt − δ i X ( s ) dB ( s ) , and δ i ≥ for i = 1 , . The superscript t − meansthat we are taking the left limit of the process is question (that is, the valuebefore a potential jump at time t ), see Øksendal and Sulem [13] for more onthis.Here, the delay processes Y i , and the noisy memory processes Λ i correspondto player i = 1 , respectively. Hence, the two players may have memories fordifferent time intervals, depending on the values of δ i . Also, b : [0 , T ] × R × R × R × A × Ω → R ,σ : [0 , T ] × R × R × R × A × Ω → R ,γ : [0 , T ] × R × R × R × A × Ω → R are predictable functions such that for each u ∈ A the SDE (1) has a uniquesolution. 4BSDE games with delay and noisy memory Remark 2.1
Existence and uniqueness of solution for the SDE (1) is guaran-teed under certain, fairly unrestrictive, assumptions on the coefficient functions,see Dahl et al. [6], Assumption 1, for conditions ensuring existence and unique-ness of solution to (1) . This can be seen by viewing equation (1) as a stochasticfunctional differential equation.
In addition to this, the players (potentially) have different levels of informa-tion, represented by different subfiltrations E ( i ) t ⊆ F t for all t ∈ [0 , T ] , i = 1 , .For i = 1 , , let g i ( · , x, y, Λ , w i , z i , k i ( · ) , u, ω ) be a given predictable process,and let h i ( x, ω ) be an F T -measurable function. Associated to the FSDE (1),we have a pair of backward stochastic differential equations (BSDEs) in theunknown stochastic processes ( W i , Z i , K i ) , i = 1 , : dW i ( t ) = − g i ( t, X ( t ) , Y ( t ) , Λ ( t ) , W i ( t ) , Z i ( t ) , K i ( t, · ) , u ( t ) , ω ) dt + Z i ( t ) dB ( t ) + R R K i ( t, ζ ) ˜ N ( dt, dζ ) W i ( T ) = h i ( X ( T ) , ω ) . (2)Note that these BSDEs are coupled to the SDE (1) due to the dependencyon X . Also, the BSDEs depend on the memory of the market process X , due tothe dependency on the processes Y and Λ . However, equation (2) is a standardBSDE, hence the conditions for existence and uniqueness of solution is wellknown, see e.g. Pardoux and Peng [20].For i = 1 , , let f i : [0 , T ] × R × R × R × A × Ω → R , ϕ i : R → R , ψ i : R → R be functions representing a profit rate, bequest function and risk evaluation.Then, the performance function of each player i = 1 , is defined by: J i ( u ) = E [ Z T f i ( t, X u ( t ) , Y ui ( t ) , Λ ui ( t ) , u i ( t )) dt + ϕ i ( X u ( T )) + ψ i ( W ui (0))] (3)where we must assume all conditions necessary for the integrals and the expec-5BSDE games with delay and noisy memorytation to exist.Also, note that the performance J i of player i is a function of the control u ( t ) = ( u ( t ) , u ( t )) , which is determined by both players. Therefore, thisproblem setting specifies a stochastic game.A pair of controls (ˆ u , ˆ u ) is called a Nash equilibrium for this stochasticgame if the following holds: J ( u , ˆ u ) ≤ J (ˆ u , ˆ u ) for all u ∈ A J (ˆ u , u ) ≤ J (ˆ u , ˆ u ) for all u ∈ A . (4)In words, this means that in the Nash equilibrium, neither player would liketo change their control.Assume there exists a Nash equilibrium for this forward-backward stochas-tic differential (FBSDE) game with delay and noisy memory. We would like tofind this Nash equilibrium, and we will do so by proving sufficient and neces-sary maximum principles for this problem. Therefore, we define a Hamiltonianfunction for each player i = 1 , as follows: H i ( t, x, y , Λ , w i , z i , k i , u , u , λ i , p i , q i , r i ) = f i ( t, x, y i , Λ i , u i )+ λ i g i ( t, x, y , Λ , w i , z i , k i , u , u ) + p i b ( t, x, y , Λ , u , u )+ q i σ ( t, x, y , Λ , u , u ) + R R r i ( ζ ) γ ( t, x, y , Λ , u , u , ζ ) ν ( dζ ) . (5)Assume H i is C in x, y , y , Λ , Λ , w i , z i , k i , u , u for i = 1 , . In thefollowing, for ease of notation, we will use the abbreviation H i ( t ) = H i ( t, x, y , Λ , w i , z i , k i , u , u , λ i , p i , q i , r i ) For i = 1 , , we define a system of FBSDEs associated to these Hamiltoniansin the unknown adjoint processes ( λ i , p i , q i , r i ) :6BSDE games with delay and noisy memoryFSDE in λ i (which depends on p i , q i , r i ): dλ i ( t ) = ∂H i ∂w i ( t ) dt + ∂H i ∂z i ( t ) dB ( t ) + R R ∇ k i ( H i ( t, ζ )) ˜ N ( dt, dζ ) λ i (0) = ψ ′ i ( W i (0)) . (6)where ∇ k i ( H i ( t, ζ )) is the Fréchet derivative of H i at k i , see the appendix inØksendal and Sulem [14] for a closer explanation of this gradient.We also define a BSDE in p i , q i , r i , which depends on λ i : dp i ( t ) = E [ µ i ( t ) |F t ] dt + q i ( t ) dB ( t ) + R R r i ( t, ζ ) ˜ N ( dt, dζ ) p i ( T ) = ϕ ′ i ( X ( T )) + h ′ i ( X ( T )) λ i ( T ) (7)where µ i ( t ) = − ∂H i ∂x ( t ) − ∂H i ∂y i ( t + δ i ) [0 ,T − δ i ] ( t ) − Z t + δ i t D t [ ∂H i ∂ Λ i ( s ) [0 ,T ] ( s ) ds ] and D t [ · ] denotes the Malliavin derivative (see Remark 2.2). Note that theconditional expectation in (7) is well defined by the extension of the Malliavinderivative introduced by Aase et al. [1], see Remark 2.2. Equations (6)-(7) forman FBSDE-system involving Malliavin derivatives. To the best of our knowledge,such systems have not been studied before. Remark 2.2
We refer to Nualart [11], Sanz-Solè [22] and Di Nunno et al. [5]for information about the Malliavin derivative D t for Brownian motion B ( t ) and, more generally, Lévy processes. In Aase et al. [1], D t was extended fromthe space D , to L ( P ) , where D , denotes the classical space of Malliavindifferentiable F T -measurable random variables. The extension is such that forall F ∈ L ( F T , P ) , the following holds: ( i ) D t F ∈ ( S ) ∗ , where ( S ) ∗ ⊇ L ( P ) denotes the Hida space of stochasticdistributions, ( ii ) the map ( t, ω ) → E [ D t F |F t ] belongs to L ( F T , λ × P ) , where λ denotes the Lebesgue measure on [0 , T ] .Moreover, the following generalized Clark-Ocone theorem holds: ( iii ) F = E [ F ] + Z T E [ D t F |F t ] dB ( t ) . (8) See [1], Theorem 3.11, and also [5], Theorem 6.35.Notice that by combining Itô’s isometry with the Clark-Ocone theorem, weobtain E h Z T E [ D t F |F t ] dt i = E h(cid:16) Z T E [ D t F |F t ] dB ( t ) (cid:17) i = E [( F − E [ F ] )] (9) ( iv ) As observed in Agram et al. [2], we can also apply the Clark-Ocone theoremto show the following generalized duality formula:Let F ∈ L ( F T , P ) and let ϕ ( t ) ∈ L ( λ × P ) be adapted. Then E h F Z T ϕ ( t ) dB ( t ) i = E h Z T E [ D t F |F t ] ϕ ( t ) dt i (10) Remark 2.3
Note that equation (6) is linear in λ i , and hence, if p i , q i , r i weregiven, it could be solved by using the Itô formula. However, this solution willdepend on the processes X, Y i , Λ i and W i , so in order to find an explicit solutionfor λ i , we must also solve the coupled FBSDE system (1) - (2) .The BSDE 7 is linear in p i , and hence, if λ i was given, it would be possibleto find a unique solution to this equation by using e.g. Proposition 6.2.1 inPham [19] or Theorem 1.7 in Øksendal and Sulem [17]. However, as for theadjoint SDE (6) , this solution will depend on the coupled FBSDE system (1) - (2) . In the remaining part of the paper, we will prove a sufficient (Section 3)and a necessary maximum principle (Section 4) for this kind of FBSDE gamewith delay and noisy memory. Then, we will study existence and uniqueness ofsolutions of the FBSDE system (6)-(7) (Section 5). Finally, we will present an8BSDE games with delay and noisy memoryexample which illustrates our results: optimal consumption rate with respect torecursive utility (see Section 6).
We prove a sufficient maximum principle which roughly states that under con-cavity conditions, a control (ˆ u , ˆ u ) satisfying a conditional maximum principleand an L -condition is a Nash equilibrium for the stochastic game. Theorem 3.1
Let ˆ u ∈ A and ˆ u ∈ A with corresponding solutions ˆ X ( t ) , ˆ Y i ( t ) , ˆΛ i ( t ) , ˆ W i ( t ) , ˆ Z i ( t ) , ˆ K i ( t ) , ˆ λ i ( t ) , ˆ p i ( t ) , ˆ q i ( t ) , ˆ r i ( t, ζ ) of the FSDE (1) , the BSDE (2) , andthe FBSDE system (6) - (7) for i = 1 , . Also, assume that: • (Concavity I) The functions x → h i ( x ) , x → ϕ i ( x ) , x → ψ i ( x ) are concavefor i = 1 , . • (The conditional maximum principle) ess sup v ∈A E [ H ( t, ˆ X ( t ) , ˆ Y ( t ) , ˆ Λ ( t ) , ˆ W ( t ) , ˆ Z ( t ) , ˆ K ( t, · ) ,v, ˆ u ( t ) , ˆ λ ( t ) , ˆ p ( t ) , ˆ q ( t ) , ˆ r ( t, · )) |E (1) t ]= E [ H ( t, ˆ X ( t ) , ˆ Y ( t ) , ˆ Λ ( t ) , ˆ W ( t ) , ˆ Z ( t ) , ˆ K ( t, · ) , ˆ u ( t ) , ˆ u ( t ) , ˆ λ ( t ) , ˆ p ( t ) , ˆ q ( t ) , ˆ r ( t, · )) |E (1) t ] and similarly ess sup v ∈A E [ H ( t, ˆ X ( t ) , ˆ Y ( t ) , ˆ Λ ( t ) , ˆ W ( t ) , ˆ Z ( t ) , ˆ K ( t, · ) , ˆ u , v, ˆ λ ( t ) , ˆ p ( t ) , ˆ q ( t ) , ˆ r ( t, · )) |E (2) t ]= E [ H ( t, ˆ X ( t ) , ˆ Y ( t ) , ˆ Λ ( t ) , ˆ W ( t ) , ˆ Z ( t ) , ˆ K ( t, · ) , ˆ u ( t ) , ˆ u ( t ) , ˆ λ ( t ) , ˆ p ( t ) , ˆ q ( t ) , ˆ r ( t, · )) |E (2) t ] . • (Concavity II) The functions ˆ H ( t, x, y , Λ , w , z , k ):= ess sup v ∈A E [ H ( t, x, y , ˆ y , Λ , ˆΛ , w , z , k , v, ˆ u , ˆ λ , ˆ p , ˆ q , ˆ r ) |E (1) t ] and ˆ H ( t, x, y , Λ , w , z , k ):= ess sup v ∈A E [ H ( t, x, ˆ y , y , ˆΛ , Λ , w , z , k , ˆ u , v, ˆ λ , ˆ p , ˆ q , ˆ r ) |E (2) t ] are concave for all t a.s. • Finally, assume that the following L conditions hold: E [ R T n ˆ p i ( t ) h(cid:0) σ ( t ) − ˆ σ ( t ) (cid:1) + R R (cid:0) r i ( t, ζ ) − ˆ r i ( t, ζ ) (cid:1) ν ( dζ ) i + (cid:0) X ( t ) − ˆ X ( t ) (cid:1) [ˆ q i ( t ) + R R ˆ r i ( t, ζ ) ν ( dζ )]+ (cid:0) Y i ( t ) − ˆ Y i ( t ) (cid:1) [( ∂ ˆ H i ∂z ) ( t ) + R R ||∇ k ˆ H i ( t, ζ ) || ν ( dζ )]+ˆ λ i ( t )[ (cid:0) Λ i ( t ) − ˆΛ i ( t ) (cid:1) + R R ( K i ( t, ζ ) − ˆ K i ( t, ζ )) ν ( dζ )] o ] < ∞ for i = 1 , .Then, (ˆ u , ˆ u ) is a Nash equilibrium.Proof. We would like to show that J ( u , ˆ u ) ≤ J (ˆ u , ˆ u ) for all u ∈ A .Choose u ∈ A . By the definition of the performance function J , δ := J ( u , ˆ u ) − J (ˆ u , ˆ u ) = I + I + I where I = E [ Z T { f ( t, x, y, Λ , u ) − f ( t, ˆ x, ˆ y, ˆΛ , ˆ u ) } dt ] ,I = E [ ϕ ( X ( T )) − ϕ ( ˆ X ( T ))] , I = E [ ψ ( W (0)) − ψ ( ˆ W (0))] . Note that from the definition of the Hamiltonian, I = E [ R T { H ( t ) − ˆ H ( t ) − ˆ λ ( t )( g ( t ) − ˆ g ( t )) − ˆ p ( t )( b ( t ) − ˆ b ( t )) − ˆ q ( t )( σ ( t ) − ˆ σ ( t )) − R R ˆ r ( t, ζ )( γ ( t, ζ ) − ˆ γ ( t, ζ ) ν ( dζ )) } dt ] (11)where we have used the abbreviation ˆ H ( t ) := H ( t, ˆ X ( t ) , ˆ Y ( t ) , ˆ Λ ( t ) , ˆ W ( t ) , ˆ Z ( t ) , ˆ K ( t, · ) , ˆ u , ˆ λ , ˆ p , ˆ q , ˆ r , ω ) and corresponding abbreviations for H ( t ) , b ( t ) , ˆ b ( t ) , σ, ˆ σ ( t ) , γ ( t ) and ˆ γ ( t ) .Also, I = E [ ϕ ( X ( T )) − ϕ ( ˆ X ( T ))] ≤ E [ ϕ ′ ( ˆ X ( T ))( X ( T ) − ˆ X ( T ))]= E [(ˆ p ( T ) − h ′ ( ˆ X ( T ))ˆ λ ( T ))( X ( T ) − ˆ X ( T ))]= E [ˆ p ( T )( X ( T ) − ˆ X ( T ))] − E [ˆ λ ( T ) h ′ ( ˆ X ( T ))( X ( T ) − ˆ X ( T ))]= E [ R T ˆ p ( t )( dX ( t ) − d ˆ X ( t )) + R T ( X ( t ) − ˆ X ( t )) d ˆ p ( t )+ R T ˆ q ( t )( σ ( t ) − ˆ σ ( t )) dt + R T R R ˆ r ( t, ζ )( γ ( t, ζ ) − ˆ γ ( t, ζ )) ν ( dζ ) dt ] − E [ˆ λ ( T ) h ′ ( ˆ X ( T ))( X ( T ) − ˆ X ( T ))]= E [ R T ˆ p ( t )( b ( t ) − ˆ b ( t )) dt + R T ( X ( t ) − ˆ X ( t ))( − ∂ ˆ H ∂x ( t ) − ∂ ˆ H ∂y ( t + δ ) [0 ,T − δ ] ( t ) + R t + δ t D t [ − ∂ ˆ H ∂ Λ ( s )] [0 ,T ] ( s ) ds ) dt + R T ˆ q ( t )( σ ( t ) − ˆ σ ( t )) dt + R T R R ˆ r ( t, ζ )( γ ( t, ζ ) − ˆ γ ( t, ζ )) ν ( dζ ) dt ]] − E [ˆ λ ( T ) h ′ ( ˆ X ( T ))( X ( T ) − ˆ X ( T ))] (12)where the first inequality follows from the concavity of ϕ , the second equalityfollows from equation (7), the fourth equality from Itô’s product rule applied to11BSDE games with delay and noisy memory ˆ p X and ˆ p ˆ X , the fifth equality follows from equation (7), the double expecta-tion rule and equation (1).Also, note that I = E [ ψ ( W (0)) − ψ ( ˆ W (0))] ≤ E [ ψ ′ ( ˆ W (0))( W (0) − ˆ W (0))]= E [ˆ λ ( T )( W ( T ) − ˆ W ( T ))] − { E [ R T ( W ( t ) − ˆ W ( t )) d ˆ λ ( t )+ R T ˆ λ ( t )( dW ( t ) − d ˆ W ( t )) + R T ∂ ˆ H ∂z ( t )( Z ( t ) − ˆ Z ( t )) dt + R T R R ∇ k ˆ H ( t )( K ( t ) − ˆ K ( t )) ν ( dζ ) dt ] } = E [ˆ λ ( T )( h ( X ( T )) − h ( ˆ X ( T )))] − { E [ R T ∂ ˆ H ∂w ( t )( W ( t ) − ˆ W ( t )) dt + R T ˆ λ ( t )( − g ( t ) + ˆ g ( t )) dt + R T ∂ ˆ H ∂z ( t )( Z ( t ) − ˆ Z ( t )) dt + R T R R ∇ k ˆ H ( t )( K ( t ) − ˆ K ( t )) ν ( dζ ) dt ] }≤ E [ˆ λ ( T ) h ′ ( ˆ X ( T ))( X ( T ) − ˆ X ( T ))] − { E [ R T ∂ ˆ H ∂w ( t )( W ( t ) − ˆ W ( t )) dt + R T ˆ λ ( t )( − g ( t ) + ˆ g ( t )) dt + R T ∂ ˆ H ∂z ( t )( Z ( t ) − ˆ Z ( t )) dt + R T R R ∇ k ˆ H ( t )( K ( t ) − ˆ K ( t )) ν ( dζ ) dt ] } (13)where the first inequality follows from the concavity of ψ , the second equalityfollows from equation (6), the third equality follows from Itô’s product ruleapplied to ˆ λ Y and ˆ λ ˆ Y , the fourth equality follows from equation (2) as wellas equation (6). The final inequality follows from the concavity of h and that ˆ λ ( T ) ≥ .Hence, 12BSDE games with delay and noisy memory ∆ = I + I + I ≤ E [ R T { H ( t ) − ˆ H ( t ) − (cid:16) ∂ ˆ H ∂x ( t ) + ∂ ˆ H ∂y ( t + δ ) [0 ,T − δ ] ( t )+ R t + δ t D t [ ∂ ˆ H ∂ Λ ( s )] [0 ,T ] ( s ) ds (cid:17) ( X ( t ) − ˆ X ( t )) dt }− R T (cid:8) ∂ ˆ H ∂w ( t )( W ( t ) − ˆ W ( t )) + ∂ ˆ H ∂z ( t )( Z ( t ) − ˆ Z ( t ))+ R R ∇ k ˆ H ( t )( K ( t, ζ ) − ˆ K ( t, ζ )) ν ( dζ ) (cid:9) dt ] . (14)Note that by changing the order of integration and using the duality formulafor Malliavin derivatives (see Di Nunno et al. [5]), we get: E h R T ∂ ˆ H ∂ Λ ( s ) (cid:0) Λ ( s ) − ˆΛ ( s ) (cid:1) ds i = E h R T ∂ ˆ H ∂ Λ ( s ) R ss − δ (cid:0) X ( t ) − ˆ X ( t ) (cid:1) dB ( t ) ds i = R T E h ∂ ˆ H ∂ Λ ( s ) R ss − δ (cid:0) X ( t ) − ˆ X ( t ) (cid:1) dB ( t ) i ds = R T E [ R ss − δ E [ D t ( ∂ ˆ H ∂ Λ ( s )) |F t ] (cid:0) X ( t ) − ˆ X ( t ) (cid:1) dt ] ds = E [ R T R t + δ t E [ D t ( ∂ ˆ H ∂ Λ ( s )) |F t ] [0 ,T ] ( s ) ds ( X ( t ) − ˆ X ( t )) dt i = E [ R T R t + δ t D t ( ∂ ˆ H ∂ Λ ( s )) [0 ,T ] ( s ) ds ( X ( t ) − ˆ X ( t )) dt i . (15)Also, note that E h R T ∂ ˆ H∂y ( t ) (cid:0) Y ( t ) − ˆ Y ( t ) (cid:1) dt i = E h R T ∂ ˆ H∂y ( t ) (cid:0) X ( t − δ ) − ˆ X ( t − δ ) (cid:1) dt i = E h R T ∂ ˆ H∂y ( t + δ ) [0 ,T − δ ] ( t ) (cid:0) X ( t ) − ˆ X ( t ) (cid:1) dt i . (16)Hence, by the inequality (14) combined with equations (15) and (16),13BSDE games with delay and noisy memory ∆ ≤ E [ R T { H ( t ) − ˆ H ( t ) − ∂ ˆ H ∂x ( t )( X ( t ) − ˆ X ( t )) − ∂ ˆ H ∂y ( t )( Y ( t ) − ˆ Y ( t )) − ∂ ˆ H ∂ Λ ( t )(Λ ( t ) − ˆΛ ( t )) dt − ∂ ˆ H ∂w ( t )( W ( t ) − ˆ W ( t )) − ∂ ˆ H ∂z ( t )( Z ( t ) − ˆ Z ( t ))+ R R ∇ k ˆ H ( t )( K ( t, ζ ) − ˆ K ( t, ζ )) ν ( dζ ) } dt ] . (17)By assumption, ˆ H is concave, so it is superdifferentiable ∗ (see Rockafel-lar [21]) at the point ~x := ( ˆ X, ˆ Y , ˆΛ , ˆ W , ˆ Z , ˆ K ) . Thus, there exists a super-gradient ~a := ( a , a , a , a , a , a ( · )) such that for all ~y := ( x, y, Λ , w, z, k ) , thefollowing holds: ˆ H ( ~x ) + ~a · ( ~y − ~x ) ≥ ˆ H ( ~y ) . (18)Define φ ( x, y, Λ , w, z, k ) := ˆ H ( x, y, Λ , w, z, k ) − ˆ H ( ˆ X, ˆ Y , ˆΛ , ˆ W , ˆ Z , ˆ K ) −{ a ( x − ˆ X ) + a ( y − ˆ Y ) + a (Λ − Λ ) + a ( w − ˆ W ) + a ( z − ˆ Z )+ R R a ( ζ )( k − ˆ K ) ν ( dζ )) } . (19)Then, by equation (18) φ ( x, y, Λ , w, z, k ) ≤ for all x, y, Λ , w, z, k,φ ( ˆ X, ˆ Y , ˆΛ , ˆ W , ˆ Z , ˆ K ) = 0 (by definition) . (20)Therefore, by differentiating equation (19) and using equation (20), we findthat ∗ Defined similarly as subdifferentiability for convex functions. a = ∂ ˆ H ∂x ( ˆ X, ˆ Y , ˆΛ , ˆ W , ˆ Z , ˆ K ) = ∂ ˆ H ∂x a = ∂ ˆ H ∂y ( ˆ X, ˆ Y , ˆΛ , ˆ W , ˆ Z , ˆ K ) = ∂ ˆ H ∂y a = ∂ ˆ H ∂ Λ ( ˆ X, ˆ Y , ˆΛ , ˆ W , ˆ Z , ˆ K ) = ∂ ˆ H ∂ Λ a = ∂ ˆ H ∂w ( ˆ X, ˆ Y , ˆΛ , ˆ W , ˆ Z , ˆ K ) = ∂ ˆ H ∂w a = ∂ ˆ H ∂z ( ˆ X, ˆ Y , ˆΛ , ˆ W , ˆ Z , ˆ K ) = ∂ ˆ H ∂z a = ∇ k ˆ H ( ˆ X, ˆ Y , ˆΛ , ˆ W , ˆ Z , ˆ K ) = ∇ k ˆ H . Therefore, it follows from this, equation (17) and equation (20) that ∆ = φ ( X ( t ) , Y ( t ) , Λ ( t ) , W ( t ) , Z ( t ) , K ( t, · )) ≤ where the final inequality follows since ˆ H is concave.This means that J ( u , ˆ u ) ≤ J (ˆ u , ˆ u ) for all u ∈ A .In a similar way, one can prove that J (ˆ u , u ) ≤ J (ˆ u , ˆ u ) for all u ∈ A .This completes the proof that (ˆ u , ˆ u ) is a Nash-equilibrium. (cid:3) In the following, we need some additional assumptions and notation: • For all t ∈ [0 , T ] and all bounded E i ( t ) -measurable random variables α i ( ω ) , the control β i ( t ) := ( t ,T ) ( t ) α i ( ω ) is in A i for i = 1 , . (21) • For all u i , β i ∈ A i with β i bounded, there exists κ i > such that thecontrol 15BSDE games with delay and noisy memory u i ( t ) + sβ i ( t ) for t ∈ [0 , T ] (22)belongs to A i for all s ∈ ( − κ i , κ i ) , i = 1 , . • Also, assume that the following derivative processes exist and belong to L ([0 , T ] × Ω) : x ( t ) = dds X ( u + sβ ,u ) ( t ) | s =0 ,y ( t ) = dds Y ( u + sβ ,u )1 ( t ) | s =0 , ˜Λ ( t ) = dds Λ ( u + sβ ,u )1 ( t ) | s =0 ,w ( t ) = dds W ( u + sβ ,u )1 ( t ) | s =0 ,z ( t ) = dds Z ( u + sβ ,u )1 ( t ) | s =0 ,k ( t ) = dds K ( u + sβ ,u )1 ( t ) | s =0 , (23)and similarly for x ( t ) = dds X ( u ,u + sβ ) ( t ) | s =0 etc. Notice that x i (0) = 0 for i = 1 , since X (0) = x .If these assumptions hold, we can prove a necessary maximum principle forour noisy memory FBSDE game. The proof of the following theorem is based onthe same idea as the proof of Theorem 2.2 in Øksendal and Sulem [14], howeverthe presence of noisy memory in our problem requires some extra care. Theorem 4.1
Suppose that u ∈ A with corresponding solutions X ( t ) , Y i ( t ) , Λ i ( t ) , W i ( t ) , Z i ( t ) , K i ( t, ζ ) , λ i ( t ) , p i ( t ) , q i ( t ) , r i ( t, ζ ) , i = 1 , , of equations (1) , (2) , (6) and (7) . Also, assume that conditions (21) - (23) hold. Then, the fol-lowing are equivalent: ( i ) ∂∂s J ( u + sβ , u ) | s =0 = ∂∂s J ( u , u + sβ ) | s =0 = 0 for all bounded β ∈A , β ∈ A . ( ii ) E [ ∂H ( t,X ( t ) , Y ( t ) , Λ ( t ) ,W ( t ) ,Z ( t ) ,K ( t, · ) ,v ,u ( t ) ,λ ( t ) ,p ( t ) ,q ( t ) ,r ( t, · )) ∂v ] | v = u ( t ) = E [ ∂H ( t,X ( t ) , Y ( t ) , Λ ( t ) ,W ( t ) ,Z ( t ) ,K ( t, · ) ,u ( t ) ,v ,λ ( t ) ,p ( t ) ,q ( t ) ,r ( t, · )) ∂v ] | v = u ( t ) = 0 .Proof. We only prove that ∂∂s J ( u + sβ , u ) | s =0 = 0 for all bounded β ∈ A is equivalent to E [ ∂H ( t,X ( t ) , Y ( t ) , Λ ( t ) ,W ( t ) ,Z ( t ) ,K ( t, · ) ,v ,u ( t ) ,λ ( t ) ,p ( t ) ,q ( t ) ,r ( t, · )) ∂v ] | v = u ( t ) = 0 . The remaining part of the theorem (i.e., the same statement for J and H )is proved in a similar way.Note that, by the definition of J and by interchanging differentiation andintegration, D := ∂∂s J ( u + sβ , u ) | s =0 = E [ R T { ∂f ∂x ( t ) x ( t ) + ∂f ∂y ( t ) y ( t ) + ∂f ∂ Λ ( t )˜Λ ( t ) ∂f ∂u ( t ) β ( t ) } dt + ϕ ′ ( X ( T )) x ( T ) + φ ′ ( W (0)) w (0)] . We study the different parts of D separately. First, by the Itô product rule,the adjoint BSDE (7) and the definition of x ( t ) ,17BSDE games with delay and noisy memory I := E [ ϕ ′ ( X ( T )) x ( T )]= E [ p ( T ) x ( T )] − E [ h ′ ( X ( T )) λ ( T ) x ( T )]= E [ p (0) x (0)] + E [ R T p ( t ) dx ( t ) + R T x ( t ) dp ( t )+ R T d [ p , x ]( t )] − E [ h ′ ( X ( T )) λ ( T ) x ( T )]= E [ R T p ( t ) (cid:0) ∂b∂x ( t ) x ( t ) + ∂b∂y ( t ) y ( t ) + ∂b∂ Λ ( t )˜Λ ( t ) + ∂b∂u ( t ) β ( t ) (cid:1) dt ]+ E [ R T x ( t ) E [ µ ( t ) |F t ] dt ]+ E [ R T q ( t ) (cid:0) ∂σ∂x ( t ) x ( t ) + ∂σ∂y ( t ) y ( t ) + ∂σ∂ Λ ˜Λ ( t ) + ∂σ∂u ( t ) β ( t ) (cid:1) dt ]+ E [ R T R R r ( t, ζ ) (cid:0) ∂γ∂x ( t ) x ( t ) + ∂γ∂y ( t ) y ( t ) + ∂γ∂ Λ ˜Λ ( t ) + ∂γ∂u ( t ) β ( t ) (cid:1) dν ( ζ ) dt ] − E [ h ′ ( X ( T )) λ ( T ) x ( T )] . (24)Also, by the FSDE (6), the BSDE (2), the definition of x ( t ) and the Itôproduct rule, I := E [ φ ′ ( W (0)) w (0)]= E [ λ (0) w (0)]= E [ λ ( T ) w ( T )] − E [ R T λ ( t ) dw ( t ) + R T w ( t ) dλ ( t )+ R T z ( t ) ∂H ∂z ( t ) dt + R T R R ∇ k H ( t, ζ ) k ( t, ζ ) ν ( dζ ) dt ]= E [ λ ( T ) h ′ ( X ( T )) x ( T )] + E [ R T λ ( t ) (cid:0) ∂g ∂x ( t ) x ( t ) + ∂g ∂y ( t ) y ( t )+ ∂g ∂ Λ ( t )˜Λ( t ) + ∂g ∂w ( t ) w ( t ) + ∂g ∂z ( t ) z ( t ) + ∇ k g ( t ) k ( t )+ ∂g ∂u ( t ) β ( t ) (cid:1) dt ] − E [ R T ∂H ∂w ( t ) w ( t ) dt ] − E [ R T z ( t ) ∂H ∂z ( t ) dt + R T R R ∇ k H ( t, ζ ) k ( t, ζ ) ν ( dζ ) dt ] . (25)By the definition of D as well as equations (24) and (25),18BSDE games with delay and noisy memory D = A + E [ R T β ( t ) (cid:0) ∂f ∂u ( t ) + ∂b∂u ( t ) p ( t ) + ∂σ∂u ( t ) q ( t ) + ∂γ∂u ( t ) r ( t )+ ∂g ∂u ( t ) λ ( t ) (cid:1) dt ] + E [ R T w ( t ) {− ∂H ∂w ( t ) + ∂g ∂w ( t ) λ ( t ) } dt + R T z ( t ) {− ∂H ∂z ( t ) + ∂g ∂x ( t ) λ ( t ) } dt + R T k ( t ) {−∇ k H ( t ) + ∇ k g ( t ) λ ( t ) } dt ] (26)where A := E [ R T x ( t ) { ∂f ∂x ( t ) + ∂b∂x ( t ) p ( t ) + E [ µ ( t ) |F t ] + ∂σ∂x ( t ) q ( t )+ ∂γ∂x ( t ) r ( t ) + ∂g ∂x ( t ) λ ( t ) } dt + R T y ( t ) { ∂f ∂y ( t ) + ∂b∂y ( t ) p ( t )+ ∂σ∂y ( t ) q ( t ) + ∂γ∂y ( t ) r ( t ) + ∂g ∂y ( t ) λ ( t ) } dt + R T ˜Λ ( t ) { ∂f ∂ Λ ( t )+ ∂b∂ Λ ( t ) p ( t ) + ∂σ∂ Λ ( t ) q ( t ) + ∂γ∂ Λ ( t ) r ( t ) + ∂g ∂ Λ ( t ) λ ( t ) } dt ]= E [ R T x ( t ) { ∂H ∂x ( t ) + E [ µ ( t ) |F t ] } dt ] + E [ R T y ( t ) ∂H ∂y ( t )]+ E [ R T ˜Λ ( t ) ∂H ∂ Λ ( t )] . (27)Then, by using the definition of the Hamiltonian H , see equation (5), wesee that everything inside the curly brackets in equation (26) is equal to zero.Hence, D = A + E [ Z T β ( t ) ∂H ∂u ( t ) dt ] . Recall that from the definitions of y and ˜Λ , y ( t ) = x ( t − δ ) and ˜Λ ( t ) = Z tt − δ x ( u ) dB ( u ) . This implies, by change of variables19BSDE games with delay and noisy memory E [ R T y ( t ) ∂H ∂y ( t )] = E [ R T x ( t − δ ) ∂H ∂y ( t ) dt ]= R T − δ − δ x ( u ) ∂H ∂y ( u + δ ) du ]= E [ R T x ( u ) [0 ,T − δ ] ( u ) ∂H ∂y ( u + δ ) du ] . Also, by the duality formula for Malliavin derivatives (see Di Nunno et al. [5])and changing the order of integration E [ R T ˜Λ ( t ) ∂H ∂ Λ ( t )] = E [ R T R tt − δ x ( u ) dB ( u ) ∂H ∂ Λ ( t ) dt ]= E [ R T R tt − δ E [ D u ( ∂H ∂ Λ ( t )) |F u ] x ( u ) du dt ]= E [ R T R u + δ u E [ D u ( ∂H ∂ Λ ( t )) |F u ] [0 ,T ] ( t ) dt x ( u ) du ] . But, from the definition of µ , E [ R T x ( t ) E [ µ ( t ) |F t ] dt ] = E [ R T E [ x ( t ) µ ( t ) |F t ] dt ]= E [ R T E [ x ( t ) {− ∂H ∂x ( t ) − ∂H ∂y ( t + δ ) [0 ,T − δ ] − R t + δ t D t [ ∂H ∂ Λ ( s )] [0 ,T ] ( s ) ds }|F t ] dt ] . So, by the rule of double expectation and the calculations above, A = 0 .This implies that D = E [ R T β ( t ) ∂H ∂u ( t ) dt ] , so ∂∂s J ( u + sβ , u ) | s =0 = E [ Z T β ( t ) ∂H ∂u ( t ) dt ] which was what we wanted to prove. (cid:3) In this section, we consider a slightly simplified version of the system of noisymemory FBSDEs in equations (6) and (7). Instead, consider the following noisymemory FBSDE :FSDE in λ , dλ ( t ) = ∂H∂w ( t ) dt + ∂H∂z ( t ) dB ( t ) + R R ∇ k H ( t, ζ ) ˜ N ( dt, dζ ) λ (0) = φ ′ ( W (0)) . (28)BSDE in p, q and r , dp ( t ) = − E [ µ ( t ) |F t ] dt + q ( t ) dB ( t ) + R R r ( t, ζ ) ˜ N ( dt, dζ ) p ( T ) = ϕ ′ ( X ( T )) + h ′ ( X ( T )) λ ( T ) (29)where H ( t, x, y , y , Λ , Λ , w, z, k, u , u , λ, p, q, r )= f ( t, x, y, Λ , u , u ) + λg ( t, x, y , y , Λ , Λ , w, z, k, u , u )+ pb ( t, x, y , y , Λ , Λ , u , u ) + qσ ( t, x, y , y , Λ , Λ , u , u )+ R R r ( ζ ) γ ( t, x, y , y , Λ , Λ , u , u , ζ ) ν ( dζ ) and µ ( t ) = ∂H∂x ( t ) + ∂H∂y ( t + δ ) [0 ,T − δ ] ( t ) + Z t + δt E [ D t [ ∂H∂ Λ ( s )] |F t ] [0 ,T ] ( s ) ds. Note that the set of equations (6) and (7) are two such systems such as(28)-(29) involving the same X process as well as the same controls u , u .Also, consider the following system consisting of an FSDE and two BSDEs:21BSDE games with delay and noisy memoryFSDE in λ , d ˜ λ ( t ) = ∂ H ∂w ( t ) dt + ∂ H ∂z ( t ) dB ( t ) + R R ∇ k H ( t, ζ ) ˜ N ( dt, dζ )˜ λ (0) = φ ′ ( W (0)) . (30)BSDE in p , q and r , dp ( t ) = − E [ µ ( t ) |F t ] dt + q ( t ) dB ( t ) + R R r ( t, ζ ) ˜ N ( dt, dζ ) p ( T ) = ϕ ′ ( X ( T )) + h ′ ( X ( T ))˜ λ ( T ) . (31)BSDE in p , q and r , dp ( t ) = − E [ µ ( t ) |F t ] dt + q ( t ) dB ( t ) + R R r ( t, ζ ) ˜ N ( dt, dζ ) p ( T ) = 0 (32)where H ( t, x, y , y , Λ , Λ , w, z, k, u , u , ˜ λ, p , p , q , q , r , r )= q ( t ) x + H ( t, x, y , y , Λ , Λ , w, z, k, u , u , ˜ λ, p , q , r ) , (33) µ ( t ) = q ( t ) + ∂H∂x ( t ) + ∂H∂y ( t + δ ) [0 ,T − δ ] ( t ) and µ ( t ) = ∂H∂ Λ ( t ) − ∂H∂ Λ ( t + δ ) [0 ,T − δ ] ( t ) . Note that ∂ H ∂ Λ ( t ) = ∂H∂ Λ ( t ) , ∂ H ∂ Λ ( t ) = q ( t ) + ∂H∂ Λ ( t ) and ∂ H ∂y ( t ) = ∂H∂y ( t ) . Hence,equations (28) and (30) are structurally equal.Then, by similar techniques as in Dahl et al. [6], we can show the followingtheorem: Theorem 5.1
Assume that ( p i , q i , r i ) for i = 1 , and ˜ λ solve the FBSDE sys- tem (30) - (32) . Define λ = ˜ λ , p ( t ) = p ( t ) , q ( t ) = q ( t ) and r ( t, · ) = r ( t, · ) andassume that E [ R T ( ∂H ( t ) ∂z ) ] dt < ∞ . Then, ( p, q, r, λ ) solves the noisy memoryFBSDE (28) - (29) and q ( t ) = Z t + δt E [ D t [ ∂H∂ Λ ( s )] |F t ] ds. Proof.
The jump terms do not make a difference here, so assume for sim-plicity that r = r = r = 0 everywhere.In general, we know that if dp ( t ) = − θ ( t, p , q ) dt + q ( t ) dB ( t ) , p ( T ) = F ,then q ( t ) = D t p ( t ) . (34)Now, note that the solution p of the BSDE (32) can be written p ( t ) = − E [ R Tt E [ µ ( s ) |F s ] ds |F t ]= − R Tt E [ µ ( s ) |F t ] ds = − R Tt E [ ∂H∂ Λ ( t ) − ∂H∂ Λ ( t + δ ) [0 ,T − δ ] ( t ) |F t ] ds = − R t + δt E [ ∂H ( s ) ∂ Λ |F t ] [0 ,T ] ( s ) ds where the equalities follow from Fubini’s theorem, the rule of double ex-pectation, the definition of µ and a change of variables. Hence, by equation(34): q ( t ) = D t p ( t )= D t [ R t + δt E [ ∂H ( s ) ∂ Λ |F t ] [0 ,T ] ( s )] ds = R t + δt E [ D t ( ∂H ( s ) ∂ Λ ) |F t ] [0 ,T ] ( s ) ds which is part of what we wanted to prove.By inserting this expression for q into the definition of µ , we see that23BSDE games with delay and noisy memory µ ( t ) = Z t + δt E [ D t [ ∂H ( s ) ∂ Λ ] |F t ] [0 ,T ] ( s ) ds + ∂H ( t ) ∂x + ∂H ( t + δ ) ∂y [0 ,T ] ( t + δ ) . Hence, we see that the BSDE (31) is the same as (29), so they have the samesolution. This completes the proof of the theorem. (cid:3)
We can also prove the following converse result.
Theorem 5.2 If p, q, r, λ solve the FBSDE (28) - (29) and we define ˜ λ = λ , p = p , q = q , r = r and p ( t ) = R t + δt E [ ∂H∂ Λ ( s ) |F t ] [0 ,T − δ ] ( s ) dsq ( t ) = R t + δt E [ D t [ ∂H∂ Λ ( s )] |F t ] [0 ,T − δ ] ( s ) dsr ( t, · ) = 0 . Then, ( p i , q i , r i ) for i = 1 , and ˜ λ solve the system of equations (30) - (32) .Proof. Again, the jump parts make no crucial difference, so we consider theno-jump situation for simplicity.It is clear that equation (30) holds from the assumptions above (from thedefinition of H , see (33)). Also, the BSDE (31) holds: Clearly, the terminalcondition holds, and by the computations in the proof of Theorem 5.1, theremaining part of equation (31) also holds. Therefore, it only remains to provethat the BSDE (32) holds.By the Itô isometry and the Clark-Ocone formula, E [ R T E [ D s ( ∂H ( r ) ∂ Λ ) |F s ] ds ] = E [( R T E [ D s ∂H ( r ) ∂ Λ |F s ] dB s ) ]= E [( ∂H∂ Λ ( r )) − E [ ∂H∂ Λ ( r )] ] . R T E [ R T E [ D s ( ∂H ( r ) ∂ Λ ) |F s ] ds ] dr = R T ( E [ ∂H∂ Λ ( r ) ] − E [ ∂H∂ Λ ( r )] ) dt < ∞ . Note that from the Clark-Ocone theorem, ∂H ( r ) ∂ Λ = E [ ∂H ( r ) ∂ Λ |F t ] + Z rt E [ D s ( ∂H ( r ) ∂ Λ ) |F s ] dB ( s ) . Therefore, by the definition of q in the theorem and the Fubini theorem R Tt q ( s ) dB ( s ) = R Tt R Tt E [ D s ( ∂H ( r ) ∂ Λ ) |F s ] [ s,s + δ ] ( r ) drdB ( s )= R Tt R Tt E [ D s ( ∂H ( r ) ∂ Λ ) |F s ] [ r − δ,r ] ( s ) dB ( s ) dr. By some algebra and the Clark-Ocone theorem (8), R Tt R Tt E [ D s ( ∂H ( r ) ∂ Λ ) |F s ] [ r − δ,r ] ( s ) dB ( s ) dr = R Tt R rr − δ E [ D s ( ∂H ( r ) ∂ Λ ) |F s ] dB ( s ) dr = R Tt ( ∂H ( r ) ∂ Λ − E [ ∂H ( r ) ∂ Λ |F r − δ ]) dr By splitting the integrals and using change of variables (twice) as well as somealgebra, = R Tt ∂H ( s ) ∂ Λ ds − R T − δt − δ E [ ∂H ( s + δ ) ∂ Λ |F s ] ds = R Tt ∂H ( s ) ∂ Λ ds − R Tt E [ ∂H ( s + δ ) ∂ Λ |F s ] [0 ,T − δ ] ( s ) ds − R t + δt E [ ∂H ( s ) ∂ Λ |F t ] [0 ,T − δ ] ( s ) ds = R Tt E [ ∂H ( s ) ∂ Λ − ∂H ( s + δ ) ∂ Λ [0 ,T − δ ] ( s ) |F s ] ds − p ( t ) . This proves that the BSDE (32) holds as well. (cid:3)
Now, we have expressed the solution of the Malliavin FBSDE via the solutionof the “double” FBSDE system (30)-(32). What kind of system of equations is25BSDE games with delay and noisy memorythis? The system consists of two connected BSDEs in ( p , q , r ) and ( p , q , r ) respectively, and these are again connected to a FBSDE in λ . However, fromequation (32) and the definition of µ , we see that the right hand side of (32)does not depend on p . Hence, the BSDE (32) can be rewritten dp ( t ) = h ( t, λ, p , q , r ( · )) dt + q ( t ) dB ( t ) + R R r ( t, ζ ) ˜ N ( dt, dζ ) p ( T ) = 0 . This can be solved to express p using λ, p , q and r ( · ) by letting q ( t ) = r ( t, · ) = 0 for all t and p ( t ) = E [ Z Tt h ( t, λ, p , q , r ( · )) dt |F t ] . Now, we can substitute this solution for p ( t ) into the FBSDE system (30)-(31). The resulting set of equations is a regular system of time advanced FBS-DEs with jumps. There are to the best of our knowledge, no general results onexistence and uniqueness of such systems of FBSDEs. However, if we simplifyby removing the jumps and there was no time-advanced part (i.e., no delayprocess Y i in the original FSDE (1)), there are some results by Ma et al. [10]. In this section, we apply the previous results to the problem of determining anoptimal consumption rate with respect to recursive utility (see also Øksendaland Sulem [16] and Dahl and Øksendal [4]). Let X ( t ) = X c ( t ) , where theconsumption rate c ( t ) is our control, and assume that26BSDE games with delay and noisy memory dX ( t ) = X ( t )[ µ ( t ) dt + σ ( t ) dB ( t ) + R R γ ( t, ζ ) ˜ N ( dt, dζ )] − [ c ( t ) + c ( t )] X ( t ) dt,X (0) = x > (35)and W i ( t ) is given by dW i ( t ) = − [ α i ( t ) W i ( t ) + η i ( t ) ln( Y i ( t )) + κ i ( t ) ln(Λ i ( t )) + ln( c i ( t ) X ( t ))]+ Z i ( t ) dB ( t ) + R R K i ( t, ζ ) ˜ N ( dt, dζ ) W i ( T ) = 0 . Let the performance functional be defined by J i ( c , c ) := W i (0) , i.e., J i is the recursive utility for player i . Also, assume that both players have fullinformation, so ( E ( i ) t ) t = ( F t ) t for i = 1 , .We would like to find a Nash equilibrium for this FBSDE game with delay.To do so we will use the maximum principle Theorem 3.1. Note that f i = ϕ i = h i = 0 and that ψ i ( w ) = w for i = 1 , . The Hamiltonians are: H i ( t, x, y , y , Λ , Λ , w i , z i , k i , c , c , λ i , p i , q i , r i ( ζ ))= λ i ( α i ( t ) w i + η i ( t ) ln( y i ) + ln( c i x ))+ p i ( xµ ( t ) − ( c + c ) x ) + q i σ ( t ) x + R R xr i ( ζ ) γ ( t, ζ ) ν ( dζ ) for i = 1 , . The adjoint BSDEs are dp i ( t ) = E [ µ i ( t ) |F t ] dt + q i ( t ) dB ( t ) + R R r i ( t, ζ ) ˜ N ( dt, dζ ) ,p i ( T ) = 0 where µ i ( t ) = − λ i ( t ) X ( t ) − λ i ( t + δ i ) η i ( t + δ i ) Y i ( t + δ i ) [0 ,T − δ i ] ( t ) − p i ( t )( µ ( t ) − ( c ( t ) + c ( t )))+ q i ( t ) σ ( t ) + R R r i ( t, ζ ) γ ( t, ζ ) ν ( dζ ) i = 1 , . Note that by the definition of Y i , Y i ( t + δ i ) = X ( { t + δ i }− δ i ) = X ( t ) .The adjoint BSDEs are linear, and the solutions are given by (see Øksendaland Sulem [17]) Γ i ( t ) p i ( t ) = E [ R Tt ( λ i ( s ) X ( s ) + λ i ( s + δ i ) η i ( s + δ i ) Y i ( s + δ i ) [0 ,T − δ i ] ( s ))Γ i ( s ) ds |F t ]= E [ R Tt ( λ i ( s ) X ( s ) + λ i ( s + δ i ) η i ( s + δ i ) X ( s ) [0 ,T − δ i ] ( s ))Γ i ( s ) ds |F t ] (36)where d Γ i ( t ) = Γ i ( t )[( µ ( t ) − ( c ( t ) + c ( t ))) dt + σ ( t ) dB ( t ) + R R γ ( t, ζ ) ˜ N ( dt, dζ )]Γ i (0) = 1 for i = 1 , . Note that by the SDE (35), x Γ i ( t ) = X ( t ) . (37)Hence, by combining equations (36) and (37), we see that X ( t ) p i ( t ) = E [ R Tt ( λ i ( s ) + λ i ( s + δ i ) η i ( s + δ i ) [0 ,T − δ i ] ( s )) ds |F t ] . (38)The adjoint FSDEs are dλ i ( t ) = λ i ( t ) α i ( t ) dtλ i (0) = 1 , for i = 1 , . These are (non-stochastic) differential equation with solution λ i ( t ) = exp( R t α i ( s ) ds ) for i = 1 , .We maximize H i with respect to c i . For i = 1 , , the first order condition is: ˆ c i ( t ) = λ i ( t ) p i ( t ) X ( t ) . c ∗ i ( t ) = λ i ( t ) E [ R Tt ( λ i ( s ) + λ i ( s + δ i ) η i ( s + δ i ) [0 ,T − δ i ] ( t )) ds |F t ] . where λ i ( t ) = exp( R t α i ( s ) ds ) for i = 1 , . In this paper, we have analyzed a two-player stochastic game connected to aset of FBSDEs involving delay and noisy memory of the market process. Wehave derived sufficient and necessary maximum principles for a set of controlsfor the two players to be a Nash equilibrium in this game. We have also studiedthe associated FBSDE involving Malliavin derivatives, and connected this to asystem of FBSDEs not involving Malliavin derivatives. Finally, we were able toderive a closed form Nash equilibrium solution to a game where the aim is tofind the optimal consumption with respect to recursive utility.
References [1] Aase, K., Øksendal, B., Privault, N. and Ubøe, J. 2000. White noise gener-alizations of the Clark-Haussmann-Ocone theorem with application to math-ematical finance,
Finance and Stochastics , 4 (4): 465-496.[2] Agram, N. and Øksendal, B. 2015. Malliavin calculus and optimal controlof stochastic Volterra equations,
Journal of Optimization Theory and Appli-cations , 3 (167): 1070-1094.[3] Agram, N. and Øksendal, B. 2014. Infinite horizon optimal control offorward-backward stochastic differential equations with delay,
Journal ofComputational and Applied Mathematics , 259 (B): 336-349.29BSDE games with delay and noisy memory[4] Dahl, K. and Øksendal, B. 2017. Singular recursive utility,
Stochastics , sofar only published online.[5] Di Nunno, G., Øksendal, B. and Proske, F. 2009.
Malliavin Calculus forLévy Processes with Applications to Finance , Springer, Berlin Heidelberg.[6] Dahl, K., Mohammed, S., Øksendal, B. and Røse, E. 2016. Optimal controlof systems with noisy memory and BSDEs with Malliavin derivatives,
Journalof Functional Analysis ∼ aswishch/sddesurvey.pdf.[8] Krylov, N. V. 2009. Controlled Diffusion Processes , Springer, Berlin Heidel-berg.[9] Krylov, N. V. 1972. Control of a Solution of a Stochastic Integral Equation
Theory of Probability and its Applications , 1 (17): 114-130.[10] Ma, J., Yin, H. and Zhang, J. 2012. On non-Markovian forward-backwardSDEs and backward stochastic PDEs.
Stochastic Processes and their Appli-cations , 12 (122): 3980-4004.[11] Nualart, D. 2006.
The Malliavin Calculus and Related Topics , Springer,Berlin Heidelberg.[12] Øksendal, B. 2013.
Stochastic differential equations: an introduction withapplications , Springer, Berlin Heidelberg.[13] Øksendal, B. and Sulem, A. 2007.
Applied Stochastic Control of Jump Dif-fusions , Springer, Berlin Heidelberg.[14] Øksendal, B. and Sulem, A. 2014. Forward-backward stochastic differentialgames and stochastic control under model uncertainty,
Journal of Optimiza-tion Theory and Applications , 22 (161): 22-55.30BSDE games with delay and noisy memory[15] Øksendal, B. and Sulem, A. 2000. A maximum principle for optimal con-trol of stochastic systems with delay, with applications to finance, Eds. J.M.Menaldi, E. Rofman and A. Sulem:
Optimal Control and Partial DifferentialEquations - Innovations and Applications , IOS Press, Amsterdam.[16] Øksendal, B. and Sulem, A. 2016. Optimal control of predictive mean-field equations and applications to finance, Eds. Benth F., Di Nunno G.:
Stochastics of Environmental and Financial Economics , Springer Proceedingsin Mathematics & Statistics, 138, Springer, Cham.[17] Øksendal, B. and Sulem, A. 2014. Risk minimization in financial marketsmodeled by Itô Lévy processes. Research report, Department of Mathematics,University of Oslo.[18] Øksendal, B., Sulem, A. and Zhang, T. 2011. Optimal control of stochasticdelay equations and time-advanced backward stochastic differential equations,
Advances in Applied Probability , 2 (43): 572-596.[19] Pham, H. 2009. Continuous-time Stochastic Control and Optimization withFinancial Applications,
Stochastic Modelling and Applied Probability , 61: 139-169.[20] Pardoux, E. and Peng, S. 1990. Adapted Solutions of Backward stochasticdifferential equation
Systems and Control Letters , 1(14): 55-61.[21] Rockafellar, R. T. 1970.
Convex Analysis , Princeton University Press,Princeton.[22] Sanz-Solè, M. 2005.
Malliavin Calculus . EPFL Press, Lausanne.[23] Wang, S. and Wu, Z. 2016. Stochastic maximum principle for optimal con-trol problems of forward-backward delay systems involving impulse controls,