Partially Observed Nonzero-Sum Differential Game of BSDEs with Delay and Applications

An, Qiguang; Zhu, Qingfeng

doi:https://doi.org/10.1155/2020/3518961

Mathematical Problems in Engineering

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Stochastic Process Theory and Its Applications

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Research Article | Open Access

Volume 2020 | Article ID 3518961 | https://doi.org/10.1155/2020/3518961

Partially Observed Nonzero-Sum Differential Game of BSDEs with Delay and Applications

Qiguang An¹and Qingfeng Zhu^1,2

Guest Editor: Zhimin Zhang

Received26 Apr 2020

Accepted01 Jun 2020

Published19 Jun 2020

Abstract

A class of partially observed nonzero-sum differential games for backward stochastic differential equations with time delays is studied, in which both game system and cost functional involve the time delays of state variables and control variables under each participant with different observation equations. A necessary condition (maximum principle) for the Nash equilibrium point to this kind of partially observed game is established, and a sufficient condition (verification theorem) for the Nash equilibrium point is given. A partially observed linear quadratic game is taken as an example to illustrate the application of the maximum principle.

1. Introduction

Game theory has penetrated into many fields of economics and attracted more and more attention. A series of studies on game theory was given by [1–9]. There have been many papers about the differential games driven by backward stochastic differential equations (BSDEs), such as [10, 11]. The prospective progress of many systems relies as much on their past history as on their current state. The optimal control problem of stochastic systems with time delays is studied by [12–22]. The game problem of stochastic systems with the time-delayed generator is discussed by [23–25].

Nevertheless, in the aforementioned control and game problems, it is assumed that the information is fully observed. This does not make sense in real life. In general, only partial information is available for controllers in most cases. The latest research studies on the partially observed optimal control issues of stochastic differential systems were given by [26–32]. The partially observed game issues of stochastic systems were studied by [33–36].

As far as we know, the results in regard to partially observed differential games corresponding to backward stochastic systems with time delays (BSDDE) are few. This problem will be investigated in this paper. Comparing the above results, our work differs in several aspects. Firstly, we research such a kind of partially observed differential game problem corresponding to the BSDDE, which enriches the game theory of backward stochastic systems. Secondly, under the circumstance of different observation equations for each participant, our controlled systems and utility functions include the delays of state variables and control variables. Thirdly, we study a class of linear quadratic (LQ) game corresponding to backward stochastic systems with the time-delayed generator and give the specific expression of the Nash equilibrium point.

The outline of this article is as follows. We present the main hypotheses and the partially observed differential game problem of BSDDE in Section 2. In Section 3, we obtain the necessary optimality conditions of the partially observed game of BSDDE. Section 4 is devoted to the sufficient maximum principle. In Section 5, we take a partially observed LQ game as an example to illustrate the application of our maximum principle. Section 6 is the conclusion of this paper.

2. Statement of the Problems

Throughout our article, is a complete filtered probability space, on which three mutually independent one-dimensional standard Brownian motions , and are defined. Let , and be the natural filtrations generated by , and , respectively. We set . For all , , which is the trivial -field, and . Set and , . The finite time duration is defined by , and the constant time delays are defined by , respectively. The expectation on is denoted by , and the conditional expectation under is denoted by . In and , is the usual inner product and is the Euclidean norm. The symbol “” that appears in the superscript represents the transpose of the matrix. In this article, all of the equalities and inequalities are in the sense of almost surely on .

We introduce the following notations:

Let the nonempty set be convex and the admissible control set be defined as the following:

Every element in is known as an admissible control to Player . And is known as the admissible control set to the players.

This work pays attention to a kind of partially observed games of BSDDE, which stems from some attractive financial scenarios. Now let us elaborate on the problem. Take into account the following BSDDE:whereand , , and , , and are the initial paths of , , and, respectively. We suppose that and are measurable continuous functions such that and . The control processes for Player 1 and Player 2 are and , and . Subscript 1 presents the variables to Player 1, and subscript 2 presents the variables to Player 2, respectively. BSDDE game system (3) means that the two players complete a common target in the end time .

Suppose that the two participants cannot directly observe the state processes , but they can be aware of related noisy processes and of , which are described as follows:where are -valued stochastic processes depending on and and , , are continuous functions.

We assume (H1) (i) and , are continuously differentiable with respect to (ii) , are bounded by

Now, if (H1) is true and both and are admissible controls, then BSDDE (3) has a unique solution (see [14]).

Define , where

Obviously, satisfies the subsequent SDE:

Hence, by (H1) and Girsanov’s theorem, we obtain a three-dimensional Brownian motion built on the probability space , in which is a probability measure.

Making sure to accomplish the target , each player owns his individual interest, which is the cost functional as follows:where is the expectation on and

We also assume for , (H2) (i) are continuously differentiable with respect to , and their partial derivatives are continuous in and bounded by (ii) are continuously differentiable, and are bounded by

Assume that every player wants to minimize the cost functional by picking the appropriate admissible control . Then, our partially observed nonzero-sum stochastic differential game problem is to find out a pair of admissible controls such that

Obviously, cost functional (8) can be converted to

So, the original problem (10) is the same thing as minimizing (11) over subject to (3) and (7). For the sake of convenience, we refer to the above game problem as Problem (POBNZ). If an admissible control which satisfied (10) can be found, then it is called as an equilibrium point of Problem (POBNZ), and the corresponding state trajectory is denoted by .

3. A Partially Observed Necessary Maximum Principle

In the case of a convex admissible control set, the convex perturbation method is the classical method to obtain the necessary optimality condition. Let the equilibrium point of Problem (POBNZ) be , and the corresponding optimal trajectory is . Let be such that . Since and are convex, for any , is also in . For the controls and , the corresponding state trajectories of game system (3) are denoted by and .

We introduce the subsequent symbols:where denotes one of , and

The variational equations are as follows:

From (H1), it is easy to see that (14) and (15) admit unique solutions and , respectively.

For and , we set

Similar to the arguments in Lemmas 3.1 and 3.2 in [34], it is easy to obtain subsequent Lemmas 1 and 2. Thus, we omit the details for simplicity.

Lemma 1. Assume (H1) and (H2) are true. Then,

Since is a Nash equilibrium point, then

Let . From Itô’s formula, we deduce

From this and Lemma 1, we have the following.

Lemma 2. Assume (H1) and (H2) are true. Then, we get the following variational inequality:

Our Hamiltonian function , is defined as follows:

Denote and its derivatives.

We note that the adjoint equation to (19) is a BSDE, whose solution is :and the adjoint equation to (14) is an SDE, whose solution is :

Remark 1. It is easy to see that equation (23) is a linear anticipated SDE. Under (H1) and (H2), the unique solvability of equation (23) is assured by Theorem 2.2 in [14].
Based on variational inequality (20), we set out the main result of this section.

Theorem 1 (partially observed necessary maximum principle). Assume (H1) and (H2) are true, an equilibrium point of Problem (POBNZ) is , the optimal trajectory is , and the solution of (7) is . Let , be the solution of (22) and be the solution of adjoint equation (23). Then, the following maximum principlefor any , a.e., , in which the Hamiltonian function is defined as (21).

Proof. For , using Itô’s formula to , from variational equations (14) and (15), variational inequality (20), and adjoint equations (22) and (23), we obtainBecause satisfies , we haveThis implies thatNow, assume that is an arbitrary element of -algebra and is a deterministic element. LetObviously, is an admissible control.
Using the above inequality to , we obtainwhich implies thatSimilar to the aforementioned method, we can get the other inequality for any . The proof of Theorem 1 is completed.

4. A Partially Observed Sufficient Maximum Principle

In this section, we explore a sufficient maximum principle to Problem (POBNZ). Let be a quintuple that satisfies (3), and assume that there is a solution corresponding to adjoint SDE (23). We assume the following: (H3) For , for all , is convex in , and is convex in

For , letwhere .

Theorem 2 (partially observed sufficient maximum principle). Assume (H1)–(H3) are true. Moreover, the maximum conditions (24) and (25) of the partial observation are true; then, is the equilibrium point to Problem (POBNZ).

Proof. For any , we considerUsing Itô’s formula to on , we deduceUsing Itô’s formula to on , from the convexity of , we haveThen, we haveBy the virtue of convexity of to , we deduceNotice the truth thatThen, we getFinally, by necessary optimality conditions (24), we obtainThen, it impliesIn the same way,So, we come to the expected conclusion. The proof is completed.

5. Application

In this section, we construct a partially observed LQ differential game with regard to backward stochastic systems with time delays. Using the classical filtering theory and the aforementioned theoretical results, we attempt to give a specific expression of the Nash equilibrium point. Let us think about the subsequent linear BSDDE:and the observation

We introduce the following cost functional:where constant , functions , are deterministic and bounded, and is bounded. Our partially observed nonzero-sum LQ differential game is to find out a pair of admissible controls satisfying

Similarly to [34], with the help of the necessary maximum principle (Theorem 1), we have the explicit expression to a Nash equilibrium point with regard to the above LQ game problem.

Theorem 3. For the above LQ game, we find out a Nash equilibrium point asin which satisfy the general FBSDE:

6. Conclusion

In this research, we have explored a class of partially observed game problem of the backward stochastic system with delay. More specially, based on the convex variational method, we establish the necessary and sufficient conditions with regard to Nash equilibrium in our game issue. The theoretical results of this paper are applied to an LQ game, for which the unique equilibrium point is expressed explicitly. On account of that the LQ model is usually used to depict many financial and economic phenomena, we expect that our LQ game result of BSDDEs can be widely used in these fields. As far as we know, the partially observed nonzero-sum backward game problem with the time-delay generator is firstly investigated in our paper.

Notwithstanding that we are committed to the above game problem, we are likewise able to progress some consequences of optimal control for BSDDEs, for example, [14, 34].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by National Key R&D Program of China (2018YFA0703900), the National Natural Science Foundation of China (11871309, 11671229, 11371226, and 11301298), the Natural Science Foundation of Shandong Province of China (ZR2019MA013 and ZR2018MG002), Special Funds of Taishan Scholar Project (tsqn20161041), Humanities and Social Sciences Project of the Ministry Education of China (19YJA910002), and Fostering Project of Dominant Discipline and Talent Team of Shandong Province Higher Education Institutions.

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Copyright © 2020 Qiguang An and Qingfeng Zhu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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