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Research Article | Open Access
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.
Throughout this article, we denote by the -dimensional Euclidean space, the collection of matrices. The superscript denotes the transpose of vectors or matrices. Let be a complete filtered probability space in which denotes a natural filtration generated by a three dimensional standard Brownian motion , , and be a fixed time horizon. For a given Euclidean space, we denote by (resp., ) the scalar product (resp., norm). We also denote by the space of all -valued, -adapted and square integrable processes, by the space of all -valued, -measurable and square integrable random variables, by the space of all -valued functions satisfying , and by 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 , , , , , , and are bounded and deterministic functions in , and are bounded, nonnegative and deterministic functions in , and are bounded, positive and deterministic functions in , and and are two nonnegative constants. Hereinafter, we omit all dependence on time variable of all processes or deterministic functions if there is no risk of ambiguity from the context for the notational simplicity; and are the control processes of Player 1 and Player 2, respectively. We always use the subscript 1 (resp., the subscript 2) to characterize the control variable corresponding to Player 1 (resp., Player 2) and use the notation to denote the dependence of the state on the control variable .
Let denote the full information up to time and be a given sub-filtration which represents the information available to Player at time . If and , we call the available information partial or incomplete for Player . If , we call the available information asymmetric for Player 1 and Player 2. Now we introduce the admissible control set where and are nonempty convex subsets of . Each element of is called an open-loop admissible control for Player . And is said to be the set of open-loop admissible controls for the players.
Suppose each player hopes to minimize her/his cost functional by selecting a suitable admissible control . In this study, the problem is, under the setting of asymmetric information, to look for 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 [7–13] with complete information, and  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 and . However, in case where 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:(i) and ; that is, the two players possess the common partial information ;(ii) and ; that is, Player 1 possesses more information than Player 2;(iii) and ; that is, Player 1 possesses the full information and Player 2 possesses the partial informaion;(iv) and ; 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.
2. 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. and are uniformly Lipschitz continuous with respect to their variables; for each , is in ; for every , , , and .
We also make the following assumption.The functions and satisfy the monotonic conditions: where , , and are given nonnegative constants satisfying , .
Then we have the following lemma, which is a direct deduction of Theorem 1 in Wu and Yu  with no random jumps.
Lemma 1. If the assumptions and hold, then (6) has a unique triple ,.
Remark 2. If we assume and all functions are deterministic, then (6) is reduced to a forward-backward ordinary differential equation (ODE, for short): We define the notation , . If , , , and satisfy the assumptions and with replaced by and is uniformly bounded, then (9) has a unique solution .
The following lemma is from the monograph by Chung  (see the example, Section 9.2).
Lemma 3. If , , and are three -algebras, and is independent of , then, for any integrable random variable , we have .
3. 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 , , and , through the solutions to the Riccati equations.
We first introduce two LQ stochastic control problems with two pieces of general asymmetric information and which is closely related to problem (LQ NZSDG).
Problem (LQSC1): subject to
Problem (LQSC2): subject to
We can check that , , and hold. If is a Nash equilibrium point, then, from the definition of Nash equilibrium point (see (5)), we can conclude that (resp., ) is an optimal control for Problem (LQSC1) (resp., Problem (LQSC2)). Appealing to the stochastic maximum principle under partial information (see , Remark 2.1 with the drift coefficient of the observation equation being zero and convex control domain, or , 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 (resp., ) is an optimal control for Problem (LQSC1) (resp., Problem (LQSC2)), then we have where is a solution to the following FBSDE:
It is obvious that is a candidate Nash equilibrium point for Problem (LQ NZSDG). We will prove is exactly a Nash equilibrium point in the sequel.
Lemma 5. in (21) is indeed a Nash equilibrium point for Problem (LQ NZSDG).
Proof. For any , we have where We apply Itô’s formula to and get Then, because and are nonnegative, and is positive, we have Similarly, for any , we also have Therefore, we can conclude that in (14) is a Nash equilibrium point for Problem (LQ NZSDG) indeed.
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).
Note that, under the two pieces of general asymmetric information and , the optimal filtering is very abstract which leads to the difficulty in finding the filtering equations satisfied by . 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.
3.1. Case 1: and
In this case, from the notations defined by (1), we have and . Hereinafter, we simply call and the optimal filters of and , respectively, if there is no ambiguity from the notations and context. Then Theorem 6 can be rewritten as follows.
Theorem 8. is a Nash equilibrium point for Problem (LQ NZSDG) if and only if has the following form: where is a solution to the following FBSDE:
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 (resp., ) of (resp., ) with respect to (resp., ), whose dynamics has not been known.
Now it is the position to seek the dynamics of and which will be used to construct the analytical representation of the Nash equilibrium point. Applying Lemma 5.4 in Xiong  and Lemma 3, we obtain the optimal filters of and in (22a) and (22b) with respect to for Player 1 which satisfiesSimilarly, we can obtain the optimal filters of and in (22a) and (22c) with respect to for Player 2 which satisfiesNote that (23a) and (24a) involve the optimal filter of with respect to ; that is, . We can derive that and together with the optimal filter 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.  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 [20–26]).
We introduce an assumption:(H3), which is needed in the following lemmas and theorems.
Proof. We first introduce another FBSDE: If is a solution to (25a)–(25d), then is a solution to (26), where On the other hand, if is a solution to (26), we introduce the following BSDE: From the existence and uniqueness of BSDE (see ), (28) has a unique solution with , . Further, we can check that is a solution to (25a)–(25d). In other words, the existence and uniqueness of (25a)–(25d) are equivalent to those of (26). It is easy to check that (26) satisfies the assumptions and . From Lemma 1, it has a unique solution . So do (25a)–(25d).
We observe that (25a)–(25d) are independent of (23a)–(23c) and (24a)–(24c). We can first solve (25a)–(25d) and derive the unique solution ,. Then we plug (resp., ) into (23a)–(23c) (resp., (24a)–(24c)). From Lemma 1, we have the following lemma.
After we obtain the unique solutions and by solving (23a)–(23c) and (24a)–(24c), respectively, from the existence and uniqueness of solutions of SDEs, we conclude that (22a) has a unique solution . Further, (22b) and (22c) also have unique solutions and , respectively. Then we can say that (22a)–(22d) have a unique solution, which implies the following theorem.
In the following, the Riccati equations are introduced, and the Nash equilibrium point is represented in a feedback of the optimal filters , , and . Hereinafter, we suppose the assumption always holds.
Set where and are undetermined deterministic functions on satisfying and .
Let . From , we have Since (37) is a standard Riccati equation, it has a unique solution . Introduce two auxiliary equations: where is the solution to (37). Obviously, ODEs (38) and (39) have unique solutions and , respectively. In addition, we can check that and in (33a) and (36a) are also the solutions to (38) and (39), respectively. From the uniqueness of solutions to (38) and (39), it follows that which implies in turn that (33a) and (36a) have the unique solutions to and .
Let , and then we have where is the solution to (37). Note that ODE (41) has a unique solution . Introduce two another auxiliary equations: where , and are the solutions to (38), (39), and (41), respectively. Similarly, we can prove that (33b) and (36b) also have unique solutions and satisfying
Based on the arguments above, we can derive the analytical expressions for , , , , , and . Then (25a) can be rewritten as which has a unique solution with −−+.
From the uniqueness of , , , , and , it follows that in (30) has a unique analytical expression.
Substituting in (30) into (23a)–(23c), we haveSet with . Applying Itô’s formula to in (47), we have which implies Substituting (47) and (49) into (46b) and comparing the drift and diffusion coefficients with (48), we conclude thatIt is clear that there exists a unique solution to (50a)–(50c). We denote and then, in terms of (47), (46a) can be rewritten as which has a unique solution: with .
Substituting in (30) into (24a)–(24c), we haveSet with . In the similar manner, we can deduce that satisfies which has a unique solution . We denote and then, in terms of (55), (54a) can be rewritten as which has a unique solution with .
Based on the arguments above, we derive the Nash equilibrium point which is represented in the feedback of the optimal filters , and of the state . Then Theorem 11 can be rewritten as follows.
Theorem 12. Under the assumption , Problem (LQ NZSDG) has a unique Nash equilibrium point denoted by where , , and are as shown in (45), (53), and (59), respectively, and and are uniquely determined by the systems of (50a)–(50c) and (56a)–(56c), respectively.
Remark 13. We introduce another assumption:are independent of .
We can check that when the assumption is replaced by , the foregoing lemmas and theorems still hold.
3.2. Case 2: and
In this case, we have and . Applying the similar methods shown in Section 3.1, we can obtain the following theorem.
Theorem 14. is a Nash equilibrium point for Problem (LQ NZSDG) if and only if
where is a solution of the following FBSDE:
Under the assumption , we can check that the filtering equations (23a)–(23c), (25a)–(25d), and the linear relations (30) and (47) still hold, and the systems of equations (33a), (33b), (36a), (36b), and (50a)–(50c) are still uniquely solvable. Then we have the following theorem.
Remark 16. In the cases similar to Case 2, such as and , , and , the corresponding results can be easily derived.
3.3. Case 3: and
In this case, we have and . Then we have the following theorem.
Theorem 17. is a Nash equilibrium point for Problem (LQ NZSDG) if and only if
where is a solution to the following FBSDE:
Under the assumption , we can check that still satisfies the filtering equations (25a)–(25d). From Section 3.1, we know that is shown as (45) and is uniquely represented by (30). Then (65a) can be rewritten asFrom Lemma 1, we can say that (66a)–(66c) has a unique solution . Further, the relation between and is as follows: where is the solution to (50a)–(50c), and with and defined by (51). Then we have the following theorem.
Remark 19. In the cases similar to Case 3, such as and , and , the corresponding results can be easily derived.
3.4. Case 4: and
In this case, we have and . Throughout this subsection, we make an additional assumption on (2):.
Theorem 20. is a Nash equilibrium point for Problem (LQ NZSDG) if and only if