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

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.

Nevertheless, in the aforementioned control and game problems, it is assumed that the information is fully observed.
is does not make sense in real life. In general, only partial information is available for controllers in most cases. e latest research studies on the partially observed optimal control issues of stochastic differential systems were given by [26][27][28][29][30][31][32]. e partially observed game issues of stochastic systems were studied by [33][34][35][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. is 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. irdly, 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. e 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.

Statement of the Problems
roughout our article, (Ω, F, F t t ≥ 0 , P) is a complete filtered probability space, on which three mutually independent one-dimensional standard Brownian motions W(t), Y 1 (t), and Y 2 (t) are defined. Let F W t , F 1 t , and F 2 t be the natural filtrations generated by W(·), Y 1 (·), and Y 2 (·), respectively. We ]. e finite time duration is defined by T > 0, and the constant time delays are defined by 0 < δ, δ 1 , δ 2 < T, respectively. e expectation on (Ω, F, P) is denoted by E, and the conditional expectation under F t is denoted by In R and R n×d , 〈·, ·〉 is the usual inner product and |·| is the Euclidean norm. e 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 dt × dP almost surely on [0, T] × Ω.
We introduce the following notations: Let the nonempty set U i ∈ R(i � 1, 2) be convex and the admissible control set be defined as the following: Every element in U i is known as an admissible control to Player i(i � 1, 2). And U 1 × U 2 is known as the admissible control set to the players.
is 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: where and f: , and ψ 2 (·) ∈ L 2 F (−δ 2 , 0; R) are the initial paths of y, v 1 , andv 2 , respectively. We suppose that ψ 1 (t) ∈ F 1 0 and ψ 2 (t) ∈ F 2 0 are measurable continuous functions such that E  e control processes for Player 1 and Player 2 are v 1 (·) and v 2 (·), and v(·) � (v 1 (·), v 2 (·)). 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 ξ in the end time T.
Suppose that the two participants cannot directly observe the state processes y v 1 ,v 2 (·), but they can be aware of related noisy processes Y 1 (·) and Y 2 (·) of y v 1 ,v 2 (·), which are described as follows: where W 1 (·) and W 2 (·) are R-valued stochastic processes depending on v 1 (·) and v 2 (·) and h i : Mathematical Problems in Engineering Now, if (H1) is true and both v 1 (·) and v 2 (·) are admissible controls, then BSDDE (3) has a unique solution [14]). Define Obviously, Z v 1 ,v 2 (t) satisfies the subsequent SDE: Hence, by (H1) and Girsanov's theorem, we obtain a three-dimensional Brownian motion (W(·), Making sure to accomplish the target ξ, each player owns his individual interest, which is the cost functional as follows: where E v 1 ,v 2 is the expectation on (Ω, F, P v 1 ,v 2 ) and We also assume for i � 1, 2, , and their partial derivatives Assume that every player wants to minimize the cost functional J i (v 1 (·), v 2 (·)) by picking the appropriate admissible control v i (·)(i � 1, 2). en, our partially observed nonzero-sum stochastic differential game problem is to find out a pair of admissible controls (u 1 (·), u 2 (·)) ∈ U 1 × U 2 such that Obviously, cost functional (8) can be converted to So, the original problem (10) is the same thing as minimizing (11) over (v 1 (·), v 2 (·)) ∈ U 1 × U 2 subject to (3) and (7). For the sake of convenience, we refer to the above game problem as Problem (POBNZ). If an admissible control u(·) � (u 1 (·), u 2 (·)) 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 (y(·), z(·)) � (y u (·), z u (·)).

Lemma 1. Assume (H1) and (H2) are true. en,
Since (u 1 (·), u 2 (·)) is a Nash equilibrium point, then Mathematical Problems in Engineering From this and Lemma 1, we have the following. Lemma 2. Assume (H1) and (H2) are true. en, we get the following variational inequality: Our Hamiltonian function H i : , is defined as follows: and its derivatives.
We note that the adjoint equation to (19) is a BSDE, whose solution is (P i (·), Q 1i (·), Q 2i (·)): 22) and the adjoint equation to (14) is an SDE, whose solution is p i (·): 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 eorem 2.2 in [14]. Based on variational inequality (20), we set out the main result of this section.
Proof. For i � 1, using Itô's formula to 〈y 1 1 (t), p 1 (t)〉 + 〈Γ(t), P 1 (t)〉, from variational equations (14) and (15), variational inequality (20), and adjoint equations (22) and (23), we obtain Mathematical Problems in Engineering 5 Because v 1 (t) satisfies u 1 (t) + v 1 (t) ∈ U 1 , we have is implies that Now, assume that F is an arbitrary element of σ-algebra Obviously, w 1 is an admissible control. Using the above inequality to w 1 , we obtain which implies that Similar to the aforementioned method, we can get the other inequality for any v 2 ∈ U 2 . e proof of eorem 1 is completed.
So, we come to the expected conclusion. e proof is completed.

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: