Linear Quadratic Nonzero Sum Differential Games with Asymmetric Information

This paper studies a linear quadratic nonzero sum differential game problem with asymmetric information. Compared with the existing literature, a distinct feature is that the information available to players is asymmetric. Nash equilibrium points are obtained for several classes of asymmetric information by stochastic maximum principle and technique of completion square. The systems of some Riccati equations and forward-backward stochastic filtering equations are introduced and the existence and uniqueness of the solutions are proved. Finally, the unique Nash equilibrium point for each class of asymmetric information is represented in a feedback form of the optimal filtering of the state, through the solutions of the Riccati equations.


Introduction
Throughout this article, we denote by   the -dimensional Euclidean space,  × the collection of  ×  matrices.The superscript * denotes the transpose of vectors or matrices.Let (Ω, F, (F  ), ) be a complete filtered probability space in which F  denotes a natural filtration generated by a three dimensional standard Brownian motion ( 1 (),  2 (),  3 ()), F = F  , and  > 0 be a fixed time horizon.For a given Euclidean space, we denote by ⟨⋅, ⋅⟩ (resp., | ⋅ |) the scalar product (resp., norm).We also denote by L 2 F  (0, ; ) the space of all -valued, F  -adapted and square integrable processes, by L 2 F  (Ω; ) the space of all -valued, F  -measurable and square integrable random variables, by L 2 (0, ; ) the space of all -valued functions satisfying ∫  0 |()| 2  < ∞, and by () 2 the square of ().For the sake of simplicity, we set ( This work is interested in linear quadratic (LQ, for short) non-zero sum differential game with asymmetric information.For simplicity, we only study the case of two players.Let us now begin to specify the problem.Consider the following one-dimensional stochastic differential equation (SDE, for short) and cost functionals of the form Here , where G  = G   and   are nonempty convex subsets of  ( = 1, 2).Each element of U  is called an open-loop admissible control for Player  ( = 1, 2).And U 1 × U 2 is said to be the set of open-loop admissible controls for the players.
Suppose each player hopes to minimize her/his cost functional . In this study, the problem is, under the setting of asymmetric information, to look for ( 1 (⋅),  2 (⋅)) ∈ U 1 × U 2 which is called the Nash equilibrium point of the game, such that We call the problem above an LQ non-zero sum differential game with asymmetric information.For simplicity, we denote it by Problem (LQ NZSDG).
The LQ problems constitute an extremely important class of optimal control or differential game problems, since they can model many problems in applications, and also reasonably approximate nonlinear control or game problems.On the other hand, there also exist so called partial and asymmetric information problems in real world.For example, investors only partially know the information from security market (see [1,2]); in many situations, "insider trading" maybe exist, which means that the insider has access to material and non-public information about the security and the available information is asymmetric between the insider and the common trader (see, e.g., [3,4]); the principal faces information asymmetric and risk with regards to whether the agent has effectively completed a contract, when a principal hires an agent to perform specific duties (see, e.g., [5,6]).For more information about LQ control or game problems, the interested readers may refer the following partial list of the works including [7][8][9][10][11][12][13] with complete information, and [14] with partial information, and the references therein.
It is very important and meaningful to find explicit Nash equilibrium points for differential game problems.When the available information is partial or asymmetric, we need to derive the corresponding optimal filtering of the states and adjoint variables which will be used to represent the Nash equilibrium points.It is very difficult to obtain the equations satisfied by the optimal filtering when the available information is asymmetric for Player 1 and Player 2. Up till now, it seems that there has been no literature about LQ differential games with asymmetric information G 1  and G 2  .However, in case where G   ( = 1, 2) are chosen as certain special forms, we can still derive the filtering equations and then obtain the explicit form of the Nash equilibrium point.In the sequel, we will study Problem (LQ NZSDG) under the following four classes of asymmetric information: and G 2  = F 2,3  ; that is, the two players possess the common partial information and G 2  = F 2  ; that is, Player 1 possesses more information than Player 2; (iii) G 1  = F  and G 2  = F 2  ; that is, Player 1 possesses the full information and Player 2 possesses the partial informaion; and G 2  = F 3  ; that is, the two players possess the mutually independent information.
In Section 3, we will point out that some other cases similar to (i)-(iv) can be also solved by the same idea and method.To our knowledge, this paper is the first try to study LQ nonzero sum differential games in the setting of the asymmetric information.
The rest of this paper is organized as follows.In Section 2, we introduce some preliminaries which will be used to derive the forward-backward filtering equations and prove the corresponding existence and uniqueness of the solutions.In Section 3, we obtain the unique explicit Nash equilibrium point for each class of asymmetric information above.We also introduce some Riccati equations and represent the unique Nash equilibrium point in a feedback form of the optimal filtering of the state with respect to the corresponding asymmetric information, through the solutions of the Riccati equations.Some conclusions are given in Section 4.

Preliminary Results
In this section, we are going to introduce two lemmas, which will be often used later.First, we present existence and uniqueness for the solutions of the forward-backward stochastic differential equation (FBSDE, for short), whose dynamics is described by Here (⋅) satisfies an (forward) SDE, ((⋅), (⋅)) satisfies a backward stochastic differential equation, (⋅) is a dimensional standard Brownian motion, (, , ) takes value in   ×   ×  × , and , , , and  are the mappings with suitable sizes.We introduce the notations and make the following assumption.
We also make the following assumption.
The following lemma is from the monograph by Chung [15] (see the example, Section 9.2).Lemma 3. If F 1 ,F 2 , and F 3 are three -algebras, and

Nash Equilibrium Point
In this section, we will derive the explicit form of the Nash equilibrium point for Problem (LQ NZSDG), applying stochastic maximum principle for partial information optimal control problem and the technique of complete square.Further, we also introduce the Riccati equations and represent the Nash equilibrium point as a feedback of the optimal filters x, x, and , through the solutions to the Riccati equations.
We first introduce two LQ stochastic control problems with two pieces of general asymmetric information G 1  and G 2  which is closely related to problem (LQ NZSDG).
Problem (LQSC1): subject to Problem (LQSC2): Mathematical Problems in Engineering subject to We can check that is a Nash equilibrium point, then, from the definition of Nash equilibrium point (see ( 5)), we can conclude that  1 (resp.,  2 ) is an optimal control for Problem (LQSC1) (resp., Problem (LQSC2)).Appealing to the stochastic maximum principle under partial information (see [16], Remark 2.1 with the drift coefficient of the observation equation being zero and convex control domain, or [17], Theorem 3.1 with nonrandom jumps), we can derive the following necessary conditions of the optimal controls for Problem (LQSC1) and Problem (LQSC2).Lemma 4. If  1 (resp.,  2 ) is an optimal control for Problem (LQSC1) (resp., Problem (LQSC2)), then we have where (, ( It is obvious that ( 1 ,  2 ) ∈ U 1 × U 2 is a candidate Nash equilibrium point for Problem (LQ NZSDG).We will prove ( 1 ,  2 ) is exactly a Nash equilibrium point in the sequel.Proof.For any V 1 (⋅) ∈ U 1 , we have where We apply Itô's formula to  1 ( V 1 , 2 − ) and get Then, because  1 and  1 are nonnegative, and  1 is positive, we have Similarly, for any V 2 (⋅) ∈ U 2 , we also have Therefore, we can conclude that ( 1 ,  2 ) in ( 14) is a Nash equilibrium point for Problem (LQ NZSDG) indeed.
Combining Lemmas 4 and 5, we obtain the following theorem.Note that, under the two pieces of general asymmetric information G 1  and G 2  , the optimal filtering E(  () | G   ) ( = 1, 2) is very abstract which leads to the difficulty in finding the filtering equations satisfied by E(  () | G   ) ( = 1, 2).In the following, we begin to study Problem (LQ NZSDG) under several classes of particular asymmetric information.Though the chosen observable information is a bit special, the mathematical deductions are still highly complicated, and the derived results are interesting and meaningful.

Case 1: G
and G 2  = F 2,3  .In this case, from the notations defined by (1), we have Hereinafter, we simply call ŷ1 and  2 the optimal filters of  1 and  2 , respectively, if there is no ambiguity from the notations and context.Then Theorem 6 can be rewritten as follows.
Theorem 8. ( 1 ,  2 ) is a Nash equilibrium point for Problem (LQ NZSDG) if and only if ( 1 ,  2 ) has the following form: where (, ( We can see that (22a)-( 22d) is a very complicated FBSDE.First, (forward) SDE (22a) is one dimensional and the combination of BSDEs (22b) and (22c) is two dimensional, which is more intricate than the case of forward SDE and BSDE with the same dimension.Second, the drift terms and terminal conditions in (22b) and (22c) contain .Finally, the drift term in (22a) contains the optimal filter ŷ1 (resp.,  2 ) of  1 (resp.,  2 ) with respect to F 1,2  (resp., F 2,3  ), whose dynamics has not been known.Now it is the position to seek the dynamics of ŷ1 () and  2 () which will be used to construct the analytical representation of the Nash equilibrium point.Applying Lemma 5.4 in Xiong [18] and Lemma 3, we obtain the optimal filters of  and  1 in (22a) and (22b) with respect to F 1,2  for Player 1 which satisfies Similarly, we can obtain the optimal filters of  and  2 in (22a) and (22c) with respect to F 2,3  for Player 2 which satisfies Note that (23a) and (24a) involve the optimal filter ỹ of   with respect to . We can derive that ỹ2 and ỹ1 together with the optimal filter x of  satisfy Note that (23a)-(25d) are coupled forward-backward stochastic filtering equations.It is remarkable that the filtering equations are essentially different from the classical ones of SDEs, and the main reason is that BSDEs are included in the equations.To our best knowledge, this class of filtering equations is originally found by Huang et al. [19] when they studied the partial information control problems of backward stochastic systems.This class of filtering equations is later also discussed when some authors investigated the optimal control or differential games of partial informatio in BSDEs or FBSDEs (see [20][21][22][23][24][25][26]).
In the following, the Riccati equations are introduced, and the Nash equilibrium point is represented in a feedback of the optimal filters x, x, and .Hereinafter, we suppose the assumption (H 3 ) always holds. Set where   and   are undetermined deterministic functions on [0, ] satisfying   () =   and   () = 0.
Let  =  1 +  2 , and then we have where  is the solution to (37).Note that ODE (41) has a unique solution .Introduce two another auxiliary equations: where  1 ,  2 , and  are the solutions to (38), (39), and (41), respectively.Similarly, we can prove that (33b) and (36b) also have unique solutions  1 and  2 satisfying Based on the arguments above, we can derive the analytical expressions for  1 ,  2 ,  1 ,  2 , , and .Then (25a) can be rewritten as with Γ   = exp{∫ equations with respect to the asymmetric information G 1  and G 2  are introduced and the existence and uniqueness of the solutions are proved.Finally, the corresponding unique Nash equilibrium point is represented in a feedback form of the optimal filtering of the state, through the solutions of some Riccati equations.

Remark 7 .
If (15a)-(15d) has a unique solution, then Problem (LQ NZSDG) has a unique Nash equilibrium point.If (15a)-(15d) have many solutions, then Problem (LQ NZSDG) may have many Nash equilibrium points.If (22a)-(22d) have no solution, Problem (LQ NZSDG) has no Nash equilibrium point.The existence and uniqueness of the Nash equilibrium point for Problem (LQ NZSDG) are equivalent to the existence and uniqueness of (15a)-(15d).