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Research Article | Open Access
Nonzero Sum Differential Game of Mean-Field BSDEs with Jumps under Partial Information
This paper is concerned with a kind of nonzero sum differential game of mean-field backward stochastic differential equations with jump (MF-BSDEJ), in which the coefficient contains not only the state process but also its marginal distribution. Moreover, the cost functional is also of mean-field type. It is required that the control is adapted to a subfiltration of the filtration generated by the underlying Brownian motion and Poisson random measure. We establish a necessary condition in the form of maximum principle with Pontryagin’s type for open-loop Nash equilibrium point of this type of partial information game and then give a verification theorem which is a sufficient condition for Nash equilibrium point. The theoretical results are applied to study a partial information linear-quadratic (LQ) game.
Game theory had been an active area of research and a useful tool in many applications, particularly in biology and economics. The study of differential games was originally stated by Isaacs  and then summed up and developed by Basar and Olsder , Yeung and Petrosyan , and so forth. Berkovitz , Fleming , Elliott and Kalton , and Friedman  established the foundations for zero sum differential games and Varaiya  and Elliott and Davis  for stochastic differential games. Next, the advances in stochastic differential games continue to appear over a large number of fields. Please refer to Hamadène , Hamadène et al. , Altman , Wu and Yu , Yu and Ji , and Wang and Yu  for more information.
For the partial information two-person zero sum (or nonzero sum) stochastic differential games, the objective is to find a saddle point (or equilibrium point) for which the controller has less information than the complete information filtration . Recently, An and Øksendal [16, 17] and An et al.  established a maximum principle for partial information differential games of stochastic differential equations with jump (SDEJ). Wang and Yu  developed some results for optimal control of BSDEs and established a maximum principle for partial information differential games of backward stochastic differential equations (BSDEs). They established a necessary condition in the form of maximum principle with Pontryagin’s type for open-loop Nash equilibrium point of this type of partial information game and gave a verification theorem which is a sufficient condition for Nash equilibrium point. Meng and Tang  and Hui and Xiao  established a maximum principle for differential games of forward-backward SDE under partial information. Øksendal and Sulem  established a general maximum principle for forward-backward stochastic differential games for Itô-Lévy processes with partial information and applied the theory to optimal portfolio and consumption problems under model uncertainty, in markets modeled by Itô-Lévy processes.
To the best of our knowledge, there are few results about the partial information differential games of the discontinuous mean-field backward stochastic system. In the present paper we will research this topic. This paper is concerned with a new kind of nonzero sum differential game of mean-field backward stochastic differential equations with jump (MF-BSDEJ) under partial information. It is required that the control is adapted to a subfiltration of the filtration generated by the underlying Brownian motion and Poisson random measure. We establish a necessary condition in the form of maximum principle with Pontryagin’s type for open-loop Nash equilibrium point of this type of partial information game and then give a verification theorem which is a sufficient condition for Nash equilibrium point. We note that the state system and the cost function in  are not mean-field, and the game systems in [15, 19] are BSDEs. The theoretical results are applied to study a partial information linear-quadratic (LQ) game.
The rest of this paper is organized as follows. In Section 2, we state our partial information differential game of MF-BSDEJ and the main assumptions. Section 3 is devoted to the necessary optimality conditions. In Section 4, we obtain the sufficient maximum principle of differential game of MF-BSDEJ under partial information. In Section 5, we give a partial information linear-quadratic (LQ) game as example to show the applications of our theoretical results.
2. Statement of the Problems
Let be a completed probability space. We suppose that the filtration is generated by the following two mutually independent processes: a -dimensional standard Brownian motion and a Poisson random measure on , where is a nonempty open set equipped with its Borel field , with compensator , such that is a martingale for satisfying . is assumed to be a -finite measure on and called the characteristic measure. Let denote the class of -null elements of . For each , we define , where for any process , .
Let be the completion of the product probability space of the above with itself, where we define with and being the completion of . It is worthy of noting that any random variable defined on can be extended naturally to as with . For and so on, let be the set of random variable which is -measurable such that . For any , we denote Particularly, for example, if , then
We introduce the following notations: We use the usual inner product and Euclidean norm in , , and . The notation “” appearing in the superscripts denotes the transpose of a matrix. All the equalities and inequalities mentioned in this paper are in the sense of almost surely on .
This work is interested in a class of partial information nonzero sum differential games of MF-BSDEJ, which is inspirited by some interesting financial phenomena. For simplicity, we only consider the case of two players, which is similar for players. Let us now give a detailed formulation of the problem. Consider the following MF-BSDEJ: where , , and are the control processes of Player 1 and Player 2, and . We always use the subscript 1 (resp., the subscript 2) to characterize the variables corresponding to Player 1 (resp., Player 2). The mean-field backward game system (4) has the meaning that the two players work together to achieve a goal at the terminal time .
To study our problem, we give some assumptions on , , and . Let be a nonempty convex subset of and a given subfiltration which represents the information available to Player at time , respectively. Now we introduce the admissible control set Each element of is called an open-loop admissible control for Player . And is called the set of open-loop admissible controls for the players.
We assume that(H1) is continuously differentiable with respect to . Moreover, the norm of , , , , , , , is bounded by .
Now, if both and are admissible controls and assumption (H1) holds, then MF-BSDEJ (4) admits a unique solution (see Shen and Siu ). Ensuring to achieve the goal , the players have their own benefits, which are described by the following cost functionals: where , : , , satisfying the condition We also assume that(H2) is continuously differentiable in and its partial derivatives are continuous in and bounded by . Moreover, is continuously differentiable and is bounded by .
Suppose each player hopes to minimize her/his cost functional by selecting an appropriate admissible control . Then the problem is to find a pair of admissible controls such that We call the problem above a backward nonzero sum stochastic differential game, where the word backward means that the game system is described by a MF-BSDEJ. For simplicity, we denote it by Problem BNZ. If we can find an admissible control satisfying (8), then we call it an equilibrium point of Problem BNZ and denote the corresponding state trajectory by .
3. A Partial Information Necessary Maximum Principle
For the convex admissible control set, the classical way to derive necessary optimality conditions is to use the convex perturbation method. Let be an equilibrium point of Problem BNZ and let be the corresponding optimal trajectory. Let be such that . Since and are convex, for any , is also in . As illustrated before, we denote by and the corresponding state trajectories of game system (4) along with the controls and .
For convenience, we introduce the notations where denotes one of and .
We introduce the variational equations as follows: By (H1), it is easy to know that (10) admits unique adapted solution , .
For , , we set We have the following.
Lemma 1. Let assumptions (H1) and (H2) hold. Then, for ,
Proof. For , we have or where we denote Applying Itô’s formula to on , by virtue of (H1), we get By Gronwall’s inequality, we easily obtain the desired result. Similarly, we can show that the conclusion holds for .
Since is an equilibrium point of Problem BNZ, then
From this and Lemma 1, we have the following variational inequality.
Lemma 2. Let assumption (H1) hold. Then,
We define the Hamiltonian function , , as follows: Let
We introduce the following adjoint equation:
Starting from the variational inequality (20), we can now state necessary optimality conditions.
Theorem 3 (partial information necessary maximum principle). Suppose (H1) and (H2) hold. Suppose is an equilibrium point of Problem BNZ and is the corresponding state trajectory. Then one has that hold for any ., where is the solution of the adjoint equation (25).
Proof. For , applying Itô’s formula to , we obtain
From Lemma 2, it follows that we have
Because satisfies , we have
This implies that
Now, let be a deterministic element and let be an arbitrary element of the -algebra . And set
It is obvious that is an admissible control.
Applying the above inequality with , we get which implies that Proceeding in the same way as the above proof, we can show that the other inequality holds for any . Then the proof is completed.
4. A Partial Information Sufficient Maximum Principle
In this section, we investigate a sufficient maximum principle for Problem BNZ. Let be a quintuple satisfying (4) and suppose that there exists a solution of the corresponding adjoint forward SDE (25). We assume that(H3)for , for all , is convex in , and is convex in .Let where denotes one of and .
Theorem 4 (partial information sufficient maximum principle). Assume that (H1)–(H3) are satisfied. Moreover, the following partial information maximum conditions hold: Then is an equilibrium point of Problem BNZ.
Proof. For any , we consider
Now applying Itô’s formula to on , we get
Moreover, by virtue of (39) and convexity of , it instantly follows that
Noting the definition of and , we have
Using convexity of with respect to , we obtain
Since , is minimal for and and are -measurable, we get
Hence combining (43), (44), and (45), we obtain
Therefore, it follows from (35), (40), and (46) that
Then it implies that
In the same way Hence, we draw the desired conclusion. The proof is completed.
5. Application in a Partial Information LQ Case
In this section we work out an example of partial information linear-quadratic differential games of MF-BSDEJ to illustrate the application of the theoretical results. For notational simplification, we assume , , and .
Consider the following: The cost functional is where constants , . Functions , , , , , , , are bounded and deterministic; , , , are nonnegative, bounded, and deterministic; , , are positive, bounded, and deterministic; , , are also bounded. Our task is to find such that
Theorem 5. The mapping is one Nash equilibrium point for the above game problem, where is the solution of the following mean-field forward-backward stochastic differential equations with jumps (MF-FBSDEJ):