A Type of Time-Symmetric Stochastic System and Related Games

This paper is concerned with a type of time-symmetric stochastic system, namely the socalled forward–backward doubly stochastic differential equations (FBDSDEs), in which the forward equations are delayed doubly stochastic differential equations (SDEs) and the backward equations are anticipated backward doubly SDEs. Under some monotonicity assumptions, the existence and uniqueness of measurable solutions to FBDSDEs are obtained. The future development of many processes depends on both their current state and historical state, and these processes can usually be represented by stochastic differential systems with time delay. Therefore, a class of nonzero sum differential game for doubly stochastic systems with time delay is studied in this paper. A necessary condition for the open-loop Nash equilibrium point of the Pontriagin-type maximum principle are established, and a sufficient condition for the Nash equilibrium point is obtained. Furthermore, the above results are applied to the study of nonzero sum differential games for linear quadratic backward doubly stochastic systems with delay. Based on the solution of FBDSDEs, an explicit expression of Nash equilibrium points for such game problems is established.

In 2003, Peng and Shi [22] introduced the following time-symmetric fully coupled forward-backward stochastic systems: which are the so-called forward-backward doubly stochastic differential equations (FBDS-DEs). The forward and backward equations in Equation (2) are the BDSDE in Equation (1) with stochastic integrals in different directions. Therefore, the FBDSDE in Equation (2) is established to provide a more general framework of fully coupled forward-backward stochastic differential equations. Under some monotone assumptions, Peng and Shi [22] obtained the unique solvability of FBDSDEs (2). Zhu et al. [23,24] have extended the results in [22] to different dimensions and the weaker monotonic assumptions, and gave the probabilistic interpretation for the solutions to SPDEs combined with algebra equations. Zhang and Shi [25] and Shi and Zhu [26] studied the stochastic control problem of FBDSDEs. Game theory has penetrated into economic theory and attracted more and more research. It was first proposed by Von Neumann and Morgenstern [27]. Nash [28] has done groundbreaking work on non-cooperative games and presents the concept of a classic Nash equilibrium. Zhao et al. [29] studied the optimal investment and reinsurance of insurers in default securities under a mean-variance criterion in the jump-diffusion risk model. Many papers on stochastic differential game problems driven by backward stochastic differential equations have been published (see [30][31][32]). The differential game problem for forward-backward doubly stochastic differential equations was addressed in [33]. However, the future evolution of a lot of processes depends not only on their current state, but also on their historical state, and these processes can usually be characterized by stochastic differential equations with time delay. The optimal control problem for stochastic differential equations with delay was discussed in [34][35][36][37][38][39]. The nonzero sum differential game of the stochastic differential delay equation was studied in [40,41]. Shen and Zeng [42] researched the optimal investment and reinsurance with time delay for insurers under a mean-variance criterion.
The extra noise {B(t)} in Equation (1) can be regarded as some additional financial information that is not disclosed to the public in practice, such as in the derivative securities market, but is available to some investors. Arriojas et al. [43] and Kazmerchuk et al. [44] obtained the option pricing formula with time delay based on the stock price process with time delay. As far as we know, there is little discussion about differential games of doubly stochastic systems with delay. In this article, we will discuss this direction, that is, the following nonzero sum differential game driven by doubly stochastic systems with time delay. The control system is where (y(·), z(·)) ∈ R n × R n×d is the state process pair, 0 < δ < T is a constant time delay parameter, and y δ (t) = y(t − δ), z δ (t) = z(t − δ). We denote J 1 (v(·)) and J 2 (v(·)), v(·) = (v 1 (·), v 2 (·)), which are the cost functionals corresponding to the players 1 and 2: Our task is to find (u 1 (·), u 2 (·)) ∈ U 1 × U 2 such that To figure out the above nonzero sum differential game problem, it is natural to involve the adjoint equation, which is a kind of anticipated BDSDE (see [45,46]). It is therefore necessary to explore the following general FBDSDE with the forward equation being a delayed doubly SDE and the backward equation being the anticipated BDSDE: Our work differs from the above in the following distinctions. First of all, we introduce a time-symmetric stochastic system, which generalizes the results in [22] to a more general case: forward doubly stochastic differential equations (SDEs) with delay as forward equations and anticipated backward doubly stochastic differential equations as backward equations. Secondly, we investigate the problem of a nonzero sum differential game driven by doubly stochastic systems with time delay, which enriches the types of stochastic delayed differential game problems. Finally, we explore the linear quadratic (LQ) games for a doubly stochastic system with time delay, and use the solution of the above general FBDSDE to give an explicit expression of the unique equilibrium point.
The structure of this paper is as follows. We give the framework of the doubly stochastic games with delay and a preliminary view on the general FBDSDE in Section 2. We set up a necessary condition for the open-loop Nash equilibrium of such games to form a Pontryagin maximum principle in Section 3. Section 4 is devoted to the verification theorem of a sufficient condition for Nash equilibrium. In order to visually demonstrate the above results, the nonzero sum differential game for LQ double stochastic delay systems is studied in Section 5. By using the results of our FBDSDE, the explicit representation of Nash equilibrium points for LQ game problems is obtained. For the convenience of the reader, we present the skeleton of the proof on uniqueness and existence for the general FBDSDE in Section 6. Finally, we conclude this article with a summary.

Notations and Formulation of Problems
Suppose (Ω, F , P) is a probability space, and [0, T] is a fixed arbitrarily large time duration throughout this paper. Let {W(t); 0 ≤ t ≤ T} and {B(t); 0 ≤ t ≤ T} be two mutually independent standard Brownian motions defined on (Ω, F , P), with values in R d and R l , respectively. Let N denote the class of P-null elements of F . For each Note that the collection {F t , t ∈ [0, T]} is neither increasing nor decreasing, and it does not produce a filtration. E denotes the expectation on (Ω, F , P). E F t := E[·|F t ] denotes the conditional expectation under F t . We use the usual inner product ·, · and Euclidean norm |·| in R n , R m , R m×l and R n×d . The symbol " " that appears on the superscript indicates the transpose of the matrix. All the equations and inequalities mentioned in this paper are in the sense of dt × dP almost surely on [0, T] × Ω. We introduce the following notations: We take into account the following controlled doubly stochastic differential systems with delay: where (y(·), z(·)) ∈ R n × R n×d is the state process pair, 0 < δ < T is a constant time delay parameter, and y are given functions, and φ(·), ψ(·) ∈ L 2 F (−δ, 0; R n ) are the initial paths of y, z, respectively. Let U i be a nonempty convex subset of R i and v i (·) be the control process of player i, i = 1, 2. We denote by U i the set of U i -valued control processes v i ∈ L 2 F (0, T; R k i ) and it is called the admissible control set for player i, i = 1, 2. Each element of U i is called an (open-loop) admissible control for player i, i = 1, 2. In addition, U = U 1 × U 2 is called the set of admissible controls for the two players.
Assume that each participant wants to minimize her/his cost functional J i (v 1 (·), v 2 (·)) by selecting an appropriate admissible control v i (·)(i = 1, 2). Then the problem is to find a pair of admissible controls (u 1 (·), u 2 (·)) ∈ U 1 × U 2 such that We call the above problem a doubly stochastic differential game with time delay. For simplicity's sake, let us write it as Problem (A). If we can find an admissible control u(·) = (u 1 (·), u 2 (·)) satisfying Equation (4), then we call it an equilibrium point of Problem (A) and denote the corresponding state trajectory by (y(·), z(·)) = (y u (·), z u (·)).

Necessary Maximum Principle
For convex admissible control sets, the classical method to obtain the necessary optimality condition is the convex perturbation method. Let u(·) = (u 1 (·), u 2 (·)) be an equilibrium point of Problem (A) and (y(·), z(·)) be the corresponding optimal trajectory.
We have the following: Lemma 1. Let the hypotheses (H1) and (H2) be true. Then, for i = 1, 2, lim Using Itô's formula to ỹ Applying Grownwall's inequalities, we can easily get the desired result. Again, we can prove that for i = 2. The proof is complete.

Remark 1.
It is easy to see that the adjoint Equation (12) above is a linear anticipated BDSDE, then the unique solvability of Equation (12) can be guaranteed by theorem 3.2 in [45] and theorem 2.4 in [46].
Proof of Theorem 2. For i = 1. Using Itô's formula to p 1 (t), y 1 1 (t) , we obtain Combining the initial conditions and the termination conditions, we get Then, we get According to Lemma 2, we have At present, take an arbitrary element F of σ-algebra F t , and set Obviously, w(t) is an admissible control. We apply the inequality in Equation (13) to w(t), and get which contains that The expression within the conditional expectation is F t -measurable, so the result follows. Following the above proof, we can prove that the other inequality is true for any v 2 ∈ U 2 . The proof is completed.

Sufficient Maximum Principle
In this section, the sufficient maximum principle for Problem (A) is investigated. Let (y(t), z(t), u 1 (t), u 2 (t)) be a quintuple satisfying Equation (3), and suppose there exists a solution (p i (t), q i (t)) of the corresponding adjoint forward SDE (12). We assume that:

Now we put into use Itô's formula to p 1 (t), y v 1 (t) − y(t) on [0, T], and get
Then, we have Based on the convexity of H 1 with respect to (y, z, y δ , z δ , v 1 , v 2 ), we achieve Noticing the fact that Then, we get Finally, by the necessary optimality conditions in Equation (14), we obtain This implies that In the same way Hence, the desired conclusion is drawn. The proof is completed.

Applications in LQ Doubly Stochastic Games with Delay
In this section, our maximal principle is used for the nonzero sum differential game problem of LQ doubly stochastic systems with delay. To simplify the notation, let us assume that d = l = 1. The control system is where (ξ(·), η(·)) ∈ L 2 F (−δ, T; R n ) is the initial path of (y, z). A,Ā, B,B, C,C, D,D are n × n bounded matrices, v 1 (t) and v 2 (t), t ∈ [0, T] are two admissible control processes, i.e., F t -measurable squareintegrable processes taking values in R k . E 1 and E 2 are n × k bounded matrices. We denote J 1 (v 1 (·), v 2 (·)) and J 2 (v 1 (·), v 2 (·)), which are the cost functionals corresponding to the players 1 and 2: where M i (t), R i (t), Q i , i = 1, 2 are n × n non-negative symmetric bounded matrices, and N i (t), i = 1, 2 are k × k positive symmetric bounded matrices and the inverse N −1 i (t), i = 1, 2 are also bounded. Our task is to find (u 1 (·), u 2 (·)) ∈ R k × R k such that We need the following assumption: Ā, B,B, C,C, D,D, and i = 1, 2. Now, with the help of the above general FBDSDE, the explicit expression for the Nash equilibrium point of the above game problem can be obtained.

Theorem 4. The mapping
(u 1 (t), u 2 (t)) = (N −1 is one Nash equilibrium point for the above game problems in Equations (16)- (18), where (y(t), z(t), p 1 (t), p 2 (t), q 1 (t), q 2 (t)) is the solution of the following general FBDSDE: Similar to [31,48], the proof of Theorem 4 is easy to give, and we have therefore excluded it. For sake of clarity, we give the following Problem (S), which is a special case of Problem (A). To simplify the notation, let us assume that n = d = l = k = 1. The control system is where (ξ(·), η(·)) ∈ L 2 F (−δ, T; R) is the initial path of (y, z). v 1 (t) and v 2 (t), t ∈ [0, T] are two admissible control processes, i.e., F t -measurable square-integrable processes taking values in R. We denote J 1 (v 1 (·), v 2 (·)) and J 2 (v 1 (·), v 2 (·)), which are the cost functionals corresponding to the players 1 and 2: Our task is to find (u 1 (·), u 2 (·)) ∈ R k × R k such that Then the Hamiltonian functions are where (p 1 (t), p 2 (t), q 1 (t), q 2 (t)) is the solution of the following adjoint equations: It is easy to see that the above equation is the anticipated BDSDE, which is solvable theorem 3.2 in [45] and theorem 2.4 in [46]. From the maximum principle, we get that (u 1 (t), u 2 (t)) = (p 1 (t), p 2 (t)), t ∈ [0, T] is one Nash equilibrium point for the above game in Equations (16)-(18).

The Proof of Theorem 1
Proof of Theorem 1. Since the initial path of (y, z) in [−δ, 0] and the terminal conditions and trajectories of (p, q) in [T, T + δ] are given in advance, we only need to consider (y t , p t , z t , q t ), 0 ≤ t ≤ T.
Similarly to the above arguments, the desired result can be obtained easily in the case n = m. The uniqueness is proved.
The proof of the existence is a combination of the above technique and a priori estimate technique introduced by Peng [49]. We divide the proof of existence into three cases: m > n, m < n and m = n.
Applying Itô's formula to H y, p on [0, T], it follows that with some constant C > 0. Hereafter, C will be some generic constant, which can be different from line to line and depends only on the Lipschitz constants k, λ, µ 1 , β 1 , H and T. It is obvious that On the other hand, for the difference of the solutions ( p, q) = (p − p , q − q ), we apply a standard method of estimation. Applying Itô's formula to | p(t)| 2 on [t, T], we have Combining the estimates in Equations (22) and (23), for a sufficiently large constant C > 0, we have We now choose γ 0 = 1 2C . It is clear that, for each fixed γ ∈ [0, γ 0 ], the mapping I α 0 +γ is contractive in the sense that Thus this mapping has a unique fixed point U = (y, p, z, q) Case 2 If m < n, then µ 2 > 0. We consider the following equations When α = 1, the existence of the solution of Equation (24) implies clearly that of Equation (16). Due to the existence and uniqueness of the anticipated BDSDE (see [45,46]), when α = 0, we know that Equation (24) is uniquely solvable. By the same techniques, we can also prove the following lemma similar to Lemma 3.
Now we give the proof of the existence of Theorem 1.

Conclusions
The future evolution of a lot of processes depends not only on their current state, but also on their historical state, and these processes can usually be characterized by stochastic differential equations with time delay. In this article, we have discussed a class of differential games driven by doubly stochastic systems with time delay. To deal with the above nonzero sum differential game problem, it is natural to involve the adjoint equation, which is a kind of anticipated BDSDE. It is therefore necessary to explore a kind of general FBDSDE with the forward equation being a delayed doubly SDE and the backward equation being an anticipated BDSDE, which are so-called time-symmetry stochastic systems. This kind of FBDSDE covers a lot of the previous results, which promotes the results in [35] to doubly stochastic integrals, and extends the results in [23] to the case that involves the time delay and anticipation. We have adopted the convex variational method, and established a necessary condition and a sufficient condition for the equilibrium point of the game. In the LQ game problem, the state equation and the adjoint equation are completely coupled, then a class of linear FBDSDE is constructed, in which the forward equation is an anticipated forward doubly SDE and the backward equation is a delayed backward doubly SDE. By means of the unique solvability of the FBDSDE, the explicit expression for the Nash equilibrium point of the LQ game is obtained. Many financial and economic phenomena can be modeled by the LQ model, and we expect that the LQ game driven by doubly stochastic systems with time delay can be widely applied in these fields.
Notwithstanding that we are committed to the above game problem, we are also able to progress some consequences of optimal control for BDSDEs with time delay, for example Xu and Han [19,20].