Nonzero Sum Differential Game of Mean-Field BSDEs with Jumps under Partial Information

Abstract

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.

1. Introduction

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 [1] and then summed up and developed by Basar and Olsder [2], Yeung and Petrosyan [3], and so forth. Berkovitz [4], Fleming [5], Elliott and Kalton [6], and Friedman [7] established the foundations for zero sum differential games and Varaiya [8] and Elliott and Davis [9] for stochastic differential games. Next, the advances in stochastic differential games continue to appear over a large number of fields. Please refer to Hamadène [10], Hamadène et al. [11], Altman [12], Wu and Yu [13], Yu and Ji [14], and Wang and Yu [15] 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 {ℱ_𝑡}_𝑡≥0. Recently, An and Øksendal [16, 17] and An et al. [18] established a maximum principle for partial information differential games of stochastic differential equations with jump (SDEJ). Wang and Yu [19] 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 [20] and Hui and Xiao [21] established a maximum principle for differential games of forward-backward SDE under partial information. Øksendal and Sulem [22] 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 [22] 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 (Ω,ℱ,𝑃) be a completed probability space. We suppose that the filtration {ℱ_𝑡}_𝑡≥0 is generated by the following two mutually independent processes: a 𝑑-dimensional standard Brownian motion {𝐵(𝑡)}_𝑡≥0 and a Poisson random measure 𝑁 on ℝ₊×𝐸, where 𝐸⊂ℝ^𝑙 is a nonempty open set equipped with its Borel field ℬ(𝐸), with compensator 𝑁(𝑑𝑒𝑑𝑡)=𝜋(𝑑𝑒)𝑑𝑡, such that 𝑁(𝐴×[0,𝑡])=(𝑁−𝑁)(𝐴×[0,𝑡])_𝑡≥0 is a martingale for ∀𝐴∈ℬ(𝐸) satisfying 𝜋(𝐴)<∞. 𝜋 is assumed to be a 𝜎-finite measure on (𝐸,ℬ(𝐸)) and called the characteristic measure. Let 𝒩 denote the class of 𝑃-null elements of ℱ. For each 𝑡∈[0,𝑇], we define ℱ_𝑡=ℱ^𝑊_𝑡∨ℱ^𝑁_𝑡, where for any process {𝜂(𝑡)},ℱ^𝜂_𝑠,𝑡=𝜎{𝜂(𝑟)−𝜂(𝑠);𝑠≤𝑟≤𝑡}∨𝒩, ℱ^𝜂_𝑡=ℱ^𝜂_0,𝑡.

Let (Ω²,ℱ²,𝑃²)=(Ω×Ω,ℱ⊗ℱ,𝑃⊗𝑃) be the completion of the product probability space of the above (Ω,ℱ,𝑃) with itself, where we define ℱ²_𝑡=ℱ_𝑡⊗ℱ_𝑡 with 𝑡∈[0,𝑇] and ℱ_𝑡⊗ℱ_𝑡 being the completion of ℱ_𝑡×ℱ_𝑡. It is worthy of noting that any random variable 𝜉=𝜉(𝜔) defined on Ω can be extended naturally to Ω² as 𝜉^′(𝜔,𝜔^′)=𝜉(𝜔) with (𝜔,𝜔^′)∈Ω². For 𝐻=ℝ^𝑛 and so on, let 𝐿¹(Ω²,ℱ²,𝑃²;𝐻) be the set of random variable 𝜉∶Ω²→𝐻 which is ℱ²-measurable such that 𝔼²|𝜉|≡∫_Ω2|𝜉(𝜔^′,𝜔)|𝑃(𝑑𝜔^′)𝑃(𝑑𝜔)<∞. For any ∈𝐿¹(Ω²,ℱ²,𝑃²;𝐻), we denote 𝔼^′𝜂(𝜔,⋅)≐_Ω𝜂𝜔,𝜔^′𝑃𝑑𝜔^′. (1) Particularly, for example, if 𝜂₁(𝜔,𝜔^′)=𝜂₁(𝜔^′), then 𝔼^′𝜂₁=_Ω𝜂₁𝜔^′𝑃𝑑𝜔^′=𝔼𝜂₁. (2)

We introduce the following notations: 𝑀²0,𝑇;ℝ^𝑛=𝑣(𝑡,𝜔)∶𝑣(𝑡,𝜔)isanℝ^𝑛ℱ-valued,_𝑡-measurableprocesssuchthat𝔼^𝑇₀|𝑣(𝑡,𝜔)|²,𝐹𝑑𝑡<∞_2𝑁0,𝑇;ℝ^𝑛=𝑟(𝑡,𝑒,𝜔)∶𝑟(𝑡,𝑒,𝜔)isanℝ^𝑛ℱ-valued,_𝑡-measurableprocesssuchthat𝔼^𝑇_0𝐄|𝑟(𝑡,𝑒,𝜔)|²𝜋(),𝐿𝑑𝑒𝑑𝑡<∞_2𝜋(⋅)ℝ^𝑛=𝑟(𝑒)∶𝑟(𝑒)isanℝ^𝑛ℬ(𝐄)‖𝑟‖=-valued,-measurablefunctionsuchthat_𝐄|𝑟(𝑒)|²𝜋(𝑑𝑒)^1/2,𝐿<∞₂Ω,ℱ_𝑇,𝑃;ℝ^𝑛=𝜉∶𝜉isanℝ^𝑛ℱ-valued,_𝑇𝜉-measurablerandomvariablesuchthat𝔼²<∞. (3) We use the usual inner product ⟨⋅,⋅⟩ and Euclidean norm |⋅| in ℝ^𝑛, ℝ^𝑛×𝑑, and ℝ^𝑛×𝑙. The notation “𝑇” appearing in the superscripts denotes the transpose of a matrix. All the equalities and inequalities mentioned in this paper are in the sense of 𝑑𝑡×𝑑𝑃 almost surely on [0,𝑇]×Ω.

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 𝑛 players. Let us now give a detailed formulation of the problem. Consider the following MF-BSDEJ: −𝑑𝑦^𝑣(𝑡)=𝔼^′𝑓𝑡,𝑦^𝑣(𝑡),𝑧^𝑣(𝑡),𝑟^𝑣(𝑡,⋅),𝑦^𝑣(𝑡)^′,𝑧^𝑣(𝑡)^′,𝑟^𝑣(𝑡,⋅)^′,𝑣(𝑡)𝑑𝑡−𝑧^𝑣(𝑡)𝑑𝑊(𝑡)−_𝐄𝑟^𝑣()𝑁(),𝑦𝑡,𝑒𝑑𝑒𝑑𝑡_𝑣(𝑇)=𝜉, (4) where 𝜉∈𝐿²(Ω,ℱ_𝑇,𝑃;ℝ^𝑛), 𝑓∶[0,𝑇]×ℝ^𝑛×ℝ^𝑛×𝑑×𝐿²_𝜋(⋅)(ℝ^𝑛)×ℝ^𝑛×ℝ^𝑛×𝑑×𝐿²_𝜋(⋅)(ℝ^𝑛)×ℝ^𝑘1×ℝ^𝑘2→ℝ^𝑛, 𝑣₁(⋅) and 𝑣₂(⋅) are the control processes of Player 1 and Player 2, and 𝑣(⋅)=(𝑣₁(⋅),𝑣₂(⋅)). 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 𝜉 at the terminal time 𝑇.

To study our problem, we give some assumptions on 𝑣₁(⋅), 𝑣₂(⋅), and 𝑓. Let 𝑈_𝑖 be a nonempty convex subset of ℝ^𝑘𝑖(𝑖=1,2) and ℰ^𝑖_𝑡⊆ℱ_𝑡(𝑖=1,2) a given subfiltration which represents the information available to Player 𝑖 at time 𝑡∈[0,𝑇], respectively. Now we introduce the admissible control set 𝒰_𝑖=𝑣_𝑖∶[0,𝑇]×Ω⟶𝑈_𝑖∣𝑣_𝑖isℰ^𝑖_𝑡𝔼-adapted,_𝑇0𝑣_𝑖(𝑡)²𝑑𝑡<∞,𝑖=1,2. (5) Each element of 𝒰_𝑖 is called an open-loop admissible control for Player 𝑖(𝑖=1,2). And 𝒰₁×𝒰₂ is called the set of open-loop admissible controls for the players.

We assume that(H1)𝑓 is continuously differentiable with respect to (𝑦,𝑧,𝑟,𝑦^′,𝑧^′,𝑟^′,𝑣₁,𝑣₂). Moreover, the norm of 𝑓_𝑦, 𝑓_𝑧, 𝑓_𝑟, 𝑓_𝑦′, 𝑓_𝑧′, 𝑓_𝑟′, 𝑓_𝑣1, 𝑓_𝑣2 is bounded by 𝑐>0.

Now, if both 𝑣₁(⋅) and 𝑣₂(⋅) are admissible controls and assumption (H1) holds, then MF-BSDEJ (4) admits a unique solution (𝑦^𝑣1,𝑣2(⋅),𝑧^𝑣1,𝑣2(⋅),𝑟^𝑣1,𝑣2(⋅,⋅))∈𝑀²(0,𝑇;ℝ^𝑛)×𝑀²(0,𝑇;ℝ^𝑛×𝑑)×𝐹²_𝑁(0,𝑇;ℝ^𝑛) (see Shen and Siu [23]). Ensuring to achieve the goal 𝜉, the players have their own benefits, which are described by the following cost functionals: 𝐽_𝑖(𝑣(⋅))=𝔼^𝑇₀𝔼^′𝑙_𝑖𝑡,𝑦^𝑣(𝑡),𝑧^𝑣(𝑡),𝑟^𝑣(𝑡,⋅),𝑦^𝑣(𝑡)^′,𝑧^𝑣(𝑡)^′,𝑟^𝑣(𝑡,⋅)^′,𝑣(𝑡)𝑑𝑡+Φ_𝑖𝑦^𝑣(0), (6) where 𝑣(⋅)=(𝑣₁(⋅),𝑣₂(⋅)),  𝑙_𝑖∶[0,𝑇]×ℝ^𝑛×ℝ^𝑛×𝑑×𝐿²_𝜋(⋅)(ℝ^𝑛)×ℝ^𝑛×ℝ^𝑛×𝑑×𝐿²_𝜋(⋅)(ℝ^𝑛)×ℝ^𝑘1×ℝ^𝑘2→ℝ,Φ_𝑖: ℝ^𝑛→ℝ,    𝑖=1,2, satisfying the condition 𝔼^𝑇₀𝔼^′𝑙_𝑖𝑡,𝑦^𝑣(𝑡),𝑧^𝑣(𝑡),𝑟^𝑣(𝑡,⋅),𝑦^𝑣(𝑡)^′,𝑧^𝑣(𝑡)^′,𝑟^𝑣(𝑡,⋅)^′(𝑡)+Φ,𝑣𝑑𝑡_𝑖𝑦^𝑣(0)<∞,𝑖=1,2. (7) We also assume that(H2)𝑙_𝑖 is continuously differentiable in (𝑦,𝑧,𝑟,𝑦^′,𝑧^′,𝑟^′,𝑣₁,𝑣₂) and its partial derivatives are continuous in (𝑦,𝑧,𝑟,𝑦^′,𝑧^′,𝑟^′,𝑣₁,𝑣₂) and bounded by 𝑐(1+|𝑦|+|𝑧|+‖𝑟‖+|𝑦^′|+|𝑧^′|+‖𝑟^′‖+|𝑣₁|+|𝑣₂|). Moreover, Φ_𝑖 is continuously differentiable and Φ_𝑖𝑦 is bounded by 𝑐(1+|𝑦|).

Suppose each player hopes to minimize her/his cost functional 𝐽_𝑖(𝑣₁(⋅),𝑣₂(⋅)) by selecting an appropriate admissible control 𝑣_𝑖(⋅)(𝑖=1,2). Then the problem is to find a pair of admissible controls (𝑢₁(⋅),𝑢₂(⋅))∈𝒰₁×𝒰₂ such that 𝐽₁𝑢₁(⋅),𝑢₂(⋅)=min_{𝑣1(⋅)∈𝒰1}𝐽₁𝑣₁(⋅),𝑢₂(⋅),𝐽₂𝑢₁(⋅),𝑢₂(⋅)=min_{𝑣2(⋅)∈𝒰2}𝐽₂𝑢₁(⋅),𝑣₂(⋅). (8) 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 𝑢(⋅)=(𝑢₁(⋅),𝑢₂(⋅)) satisfying (8), then we call it an equilibrium point of Problem BNZ and denote the corresponding state trajectory by (𝑦(⋅),𝑧(⋅),𝑟(⋅,⋅))=(𝑦^𝑢(⋅),𝑧^𝑢(⋅),𝑟^𝑢(⋅,⋅)).

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 𝑢(⋅)=(𝑢₁(⋅),𝑢₂(⋅)) be an equilibrium point of Problem BNZ and let (𝑦(⋅),𝑧(⋅),𝑟(⋅,⋅)) be the corresponding optimal trajectory. Let (𝑣₁(⋅),𝑣₂(⋅)) be such that (𝑢₁(⋅)+𝑣₁(⋅),𝑢₂(⋅)+𝑣₂(⋅))∈𝒰₁×𝒰₂. Since 𝒰₁ and 𝒰₂ are convex, for any 0≤𝜌≤1, (𝑢^𝜌₁(⋅),𝑢^𝜌₂(⋅)) = (𝑢₁(⋅)+𝜌𝑣₁(⋅),𝑢₁(⋅)+𝜌𝑣₁(⋅)) is also in 𝒰₁×𝒰₂. As illustrated before, we denote by (𝑦^𝑢𝜌1(⋅),𝑧^𝑢𝜌1(⋅),𝑟^𝑢𝜌1(⋅,⋅)) and (𝑦^𝑢𝜌2(⋅),𝑧^𝑢𝜌2(⋅),𝑟^𝑢𝜌2(⋅,⋅)) the corresponding state trajectories of game system (4) along with the controls (𝑢^𝜌₁(⋅),𝑢₂(⋅)) and (𝑢₁(⋅),𝑢^𝜌₂(⋅)).

For convenience, we introduce the notations 𝜑(𝑡,⋅)=𝜑𝑡,𝑦(𝑡),𝑧(𝑡),𝑟(𝑡,⋅),𝑦^′(𝑡),𝑧^′(𝑡),𝑟^′(𝑡,⋅),𝑢₁(𝑡),𝑢₂(𝑡),𝜑^𝑣(𝑡,⋅)=𝜑𝑡,𝑦(𝑡),𝑧(𝑡),𝑟(𝑡,⋅),𝑦^′(𝑡),𝑧^′(𝑡),𝑟^′(𝑡,⋅),𝑣₁(𝑡),𝑣₂(𝑡,𝜑_𝑢𝜌1)(𝑡,⋅)=𝜑𝑡,𝑦(𝑡),𝑧(𝑡),𝑟(𝑡,⋅),𝑦^′(𝑡),𝑧^′(𝑡),𝑟^′(𝑡,⋅),𝑢^𝜌₁(𝑡),𝑢₂(𝑡),𝜑^𝑢𝜌2(𝑡,⋅)=𝜑𝑡,𝑦(𝑡),𝑧(𝑡),𝑟(𝑡,⋅),𝑦^′(𝑡),𝑧^′(𝑡),𝑟^′(𝑡,⋅),𝑢₁(𝑡),𝑢^𝜌₂(𝑡), (9) where 𝜑 denotes one of 𝑓 and 𝑙.

We introduce the variational equations as follows: −𝑑𝑦¹_𝑖(𝑡)=𝔼^′𝑓_𝑦(𝑡,⋅)𝑦¹_𝑖(𝑡)+𝑓_𝑧(𝑡,⋅)𝑧¹_𝑖(𝑡)+_𝐄𝑓_𝑟(𝑡,𝑒)𝑟¹_𝑖(𝑡,𝑒)𝜋(𝑑𝑒)+𝑓_𝑦′(𝑡,⋅)𝑦¹_𝑖(𝑡)^′+𝑓_𝑧′(𝑡,⋅)𝑧¹_𝑖(𝑡)^′+_𝐄𝑓_𝑟′(𝑡,𝑒)𝑟¹_𝑖(𝑡,𝑒)^′𝜋(𝑑𝑒)+𝑓_𝑣𝑖(𝑡,⋅)𝑣_𝑖(𝑡)𝑑𝑡−𝑧¹_𝑖(𝑡)𝑑𝑊(𝑡)−_𝐄𝑟¹_𝑖()𝑁(),𝑦𝑡,𝑒𝑑𝑒𝑑𝑡_1𝑖(𝑇)=0,(𝑖=1,2). (10) By (H1), it is easy to know that (10) admits unique adapted solution (𝑦¹_𝑖(𝑡),𝑧¹_𝑖(𝑡),𝑟¹_𝑖(𝑡,⋅)) ∈𝑀²(0,𝑇;ℝ^𝑛) × 𝑀²(0,𝑇;ℝ^𝑛×𝑑)×𝐹²_𝑁(0,𝑇;ℝ^𝑛), 𝑖=1,2.

For 𝑡∈[0,𝑇], 𝜌>0, we set 𝑦^𝜌_𝑖(𝑡)=𝑦^𝑢𝜌𝑖(𝑡)−𝑦(𝑡)𝜌−𝑦¹_𝑖(𝑡),𝑧^𝜌_𝑖(𝑡)=𝑧^𝑢𝜌𝑖(𝑡)−𝑧(𝑡)𝜌−𝑧¹_𝑖(𝑡),𝑟̃^𝜌_𝑖()=𝑟𝑡,⋅^𝑢𝜌𝑖(𝑡,⋅)−𝑟(𝑡,⋅)𝜌−𝑟¹_𝑖(𝑡,⋅),(𝑖=1,2). (11) We have the following.

Lemma 1. Let assumptions (H1) and (H2) hold. Then, for 𝑖=1,2, lim_𝜌→0sup_{0≤𝑡≤𝑇}𝔼𝑦^𝜌_𝑖(𝑡)²=0,lim_𝜌→0𝔼^𝑇₀𝑧^𝜌_𝑖(𝑡)²𝑑𝑡=0,lim_𝜌→0𝔼^𝑇₀𝑟̃^𝜌_𝑖(𝑡)²𝑑𝑡=0. (12)(13)

Proof. For 𝑖=1, we have −𝑑𝑦^𝜌₁(𝑡)=1𝜌𝔼^′𝑓^𝑢𝜌1(𝑡,⋅)−𝑓(𝑡,⋅)−𝔼^′𝑓_𝑦(𝑡,⋅)𝑦¹₁(𝑡)+𝑓_𝑧(𝑡,⋅)𝑧¹₁(𝑡)+_𝐄𝑓_𝑟(𝑡,𝑒)𝑟¹₁(𝑡,𝑒)𝜋(𝑑𝑒)+𝑓_𝑦′(𝑡,⋅)𝑦¹₁(𝑡)^′+𝑓_𝑧′(𝑡,⋅)𝑧¹₁(𝑡)^′+_𝐄𝑓_𝑟′(𝑡,𝑒)𝑟¹₁(𝑡,𝑒)^′𝜋(𝑑𝑒)+𝑓_𝑣1(𝑡,⋅)𝑣₁(𝑡)−𝑧𝑑𝑡_𝜌1(𝑡)𝑑𝑊(𝑡)−_𝐄𝑟̃^𝜌₁()𝑁(),𝑦𝑡,𝑒𝑑𝑒𝑑𝑡_𝜌1(𝑇)=0, (14) or −𝑑𝑦^𝜌₁(𝑡)=𝔼^′𝐴^𝜌₁(𝑡,⋅)𝑦^𝜌₁(𝑡)+𝐵^𝜌₁(𝑡,⋅)𝑧^𝜌₁(𝑡)+_𝐄𝐶^𝜌₁(𝑡,𝑒)𝑟̃^𝜌₁(𝑡,𝑒)𝜋(𝑑𝑒)+𝐷^𝜌₁(𝑡,⋅)𝑦^𝜌₁(𝑡)^′+𝐸^𝜌₁(𝑡,⋅)𝑧^𝜌₁(𝑡)^′+_𝐄𝐹^𝜌₁(𝑡,𝑒)𝑟̃^𝜌₁(𝑡,𝑒)^′𝜋(𝑑𝑒)+𝐺^𝜌₁(−𝑧𝑡,⋅)𝑑𝑡_𝜌1(𝑡)𝑑𝑊(𝑡)−_𝐄𝑟̃^𝜌₁()𝑁(),𝑦𝑡,𝑒𝑑𝑒𝑑𝑡_𝜌1(𝑇)=0, (15) where we denote (Θ)=𝑡,𝑦(𝑡)+𝜆𝜌𝑦¹₁(𝑡)+𝑦^𝜌₁(𝑡)(𝑡)𝑧,𝑧+𝜆𝜌¹₁(𝑡)+𝑧^𝜌₁(𝑡)()𝑟,𝑟𝑡,⋅+𝜆𝜌¹₁(𝑡,⋅)+𝑟̃^𝜌₁(𝑡,⋅),𝑦(𝑡)^′+𝜆𝜌𝑦¹₁(𝑡)^′+𝑦^𝜌₁(𝑡)^′,(𝑧(𝑡))^′+𝜆𝜌𝑧¹₁(𝑡)^′+𝑧^𝜌₁(𝑡)^′,(𝑟(𝑡,⋅))^′+𝜆𝜌𝑟¹₁(𝑡,⋅)^′+𝑟̃^𝜌₁(𝑡,⋅)^′,𝑢₁(𝑡)+𝜆𝜌𝑣₁(𝑡),𝑢₂(𝑡),𝐴^𝜌₁(𝑡,⋅)=¹₀𝑓_𝑦(Θ)𝑑𝜆,𝐵^𝜌₁(𝑡,⋅)=¹₀𝑓_𝑧(Θ)𝐶𝑑𝜆,_𝜌1(𝑡,⋅)=¹₀𝑓_𝑟(Θ)𝑑𝜆,𝐷^𝜌₁(𝑡,⋅)=¹₀𝑓_𝑦′(Θ)𝐸𝑑𝜆,_𝜌1(𝑡,⋅)=¹₀𝑓_𝑧′(Θ)𝑑𝜆,𝐹^𝜌₁(𝑡,⋅)=¹₀𝑓_𝑟′(Θ)𝐺𝑑𝜆,_𝜌1()=𝑡,⋅₁₀𝑓_𝑣1(Θ)−𝑓_𝑣1(𝑡,⋅)𝑣₁(𝑡)+𝐴𝑑𝜆_𝜌1(𝑡,⋅)−𝑓_𝑦(𝑡,⋅)𝑦¹₁(𝑡)+𝐵^𝜌₁(𝑡,⋅)−𝑓_𝑧(𝑡,⋅)𝑧¹₁(𝑡)+𝐶^𝜌₁(𝑡,⋅)−𝑓_𝑟(𝑡,⋅)𝑟¹₁()+𝐷𝑡,⋅_𝜌1(𝑡,⋅)−𝑓_𝑦′(𝑡,⋅)𝑦¹₁(𝑡)^′+𝐸^𝜌₁(𝑡,⋅)−𝑓_𝑧′(𝑡,⋅)𝑧¹₁(𝑡)^′+𝐹^𝜌₁(𝑡,⋅)−𝑓_𝑟′(𝑡,⋅)𝑟¹₁(𝑡,⋅)^′. (16) Applying Itô’s formula to |𝑦^𝜌₁(𝑡)|² on [𝑡,𝑇], by virtue of (H1), we get 𝔼𝑦^𝜌₁(𝑡)²+𝔼^𝑇_𝑡𝑧^𝜌₁(𝑠)²+𝑟̃^𝜌₁(𝑠)²𝑑𝑠=2𝔼𝔼^′𝑇_𝑡𝐄𝑦^𝜌₁(𝑠),𝐴^𝜌₁(𝑠,⋅)𝑦^𝜌₁(𝑠)+𝐵^𝜌₁(𝑠,⋅)𝑧^𝜌₁(𝑠)+_𝐄𝐶^𝜌₁(𝑠,𝑒)𝑟̃^𝜌₁(𝑠,𝑒)𝜋(𝑑𝑒)+𝐷^𝜌₁(𝑠,⋅)𝑦^𝜌₁(𝑠)^′+𝐸^𝜌₁(𝑠,⋅)𝑧^𝜌₁(𝑠)^′+_𝐄𝐹^𝜌₁(𝑠,𝑒)𝑟^𝜌₁̃(𝑠,𝑒)^′𝜋(𝑑𝑒)+𝐺^𝜌₁(𝑠,⋅)𝜋(𝑑𝑒)𝑑𝑠≤𝐶₀𝔼^𝑇_𝑡𝑦^𝜌₁(𝑠)²+1𝑑𝑠2𝔼^𝑇_𝑡𝑧^𝜌₁(𝑠)²+𝑟̃^𝜌₁(𝑠)²𝑑𝑠+𝐶₁𝛼𝔼^𝑇_𝑡𝐺^𝜌₁(𝑠)²𝑑𝑠. (17) By Gronwall’s inequality, we easily obtain the desired result. Similarly, we can show that the conclusion holds for 𝑖=2.

Since (𝑢₁(⋅),𝑢₂(⋅)) is an equilibrium point of Problem BNZ, then 𝜌⁻¹𝐽₁𝑢^𝜌₁(⋅),𝑢₂(⋅)−𝐽₁𝑢₁(⋅),𝑢₂(⋅)≥0,𝜌⁻¹𝐽₂𝑢₁(⋅),𝑢^𝜌₂(⋅)−𝐽₂𝑢₁(⋅),𝑢₂(⋅)≥0. (18)(19)

From this and Lemma 1, we have the following variational inequality.

Lemma 2. Let assumption (H1) hold. Then, 𝔼^𝑇₀𝔼^′𝑙_𝑖𝑦(𝑡,⋅)𝑦¹_𝑖(𝑡)+𝑙_𝑖𝑧(𝑡,⋅)𝑧¹_𝑖(𝑡)+_𝐄𝑙_𝑖𝑟(𝑡)𝑟¹_𝑖(𝑡,𝑒)𝜋(𝑑𝑒)+𝑙_𝑖𝑦′(𝑡,⋅)𝑦¹_𝑖(𝑡)^′+𝑙_𝑖𝑧′(𝑡,⋅)𝑧¹_𝑖(𝑡)^′+_𝐄𝑙_𝑖𝑟′(𝑡)𝑟¹_𝑖(𝑡,𝑒)^′𝜋(𝑑𝑒)+𝑙_𝑖𝑣𝑖(𝑡,⋅)𝑣_𝑖(𝑡Φ)𝑑𝑡+𝔼_𝑖𝑦𝑦(0)𝑦¹_𝑖(0)≥0,(𝑖=1,2). (20)

Proof. For 𝑖=1, from (12), we derive 𝜌⁻¹Φ₁𝑦^𝑢𝜌1(0)−Φ₁𝑦(0)=𝜌⁻¹𝔼¹₀Φ_1𝑦𝑦(0)+𝜆𝑦^𝑢𝜌1(0)(0)×𝑦−𝑦_𝑢𝜌1(0)(0)Φ−𝑦𝑑𝜆⟶𝔼_1𝑦𝑦(0)𝑦¹₁(0),𝜌⟶0. (21) Similarly, we have 𝜌⁻¹𝔼^𝑇₀𝔼^′𝑙^𝑢𝜌1₁(𝑡,⋅)−𝑙₁(𝑡,⋅)𝑑𝑡⟶𝔼^𝑇₀𝔼^′𝑙_1𝑦(𝑡,⋅)𝑦¹₁(𝑡)+𝑙_1𝑧(𝑡,⋅)𝑧¹₁(𝑡)+_𝐄𝑙_1𝑟(𝑡)𝑟¹₁(𝑡,𝑒)𝜋(𝑑𝑒)+𝑙_1𝑦′(𝑡,⋅)𝑦¹₁(𝑡)^′+𝑙_1𝑧′(𝑡,⋅)𝑧¹₁(𝑡)^′+_𝐄𝑙_1𝑟′(𝑡)𝑟¹₁(𝑡,𝑒)^′𝜋(𝑑𝑒)+𝑙_1𝑣1(𝑡,⋅)𝑣₁(𝑡)𝑑𝑡,𝜌⟶0. (22) Let 𝜌→0 in (18); then, it follows that, for 𝑖=1, (20) holds. Similarly, we can show that the conclusion holds for 𝑖=2.

We define the Hamiltonian function 𝐻_𝑖∶[0,𝑇]×ℝ^𝑛×ℝ^𝑛×𝑑×𝐿²_𝜋(⋅)(ℝ^𝑛)×ℝ^𝑛×ℝ^𝑛×𝑑×𝐿²_𝜋(⋅)(ℝ^𝑛)×ℝ^𝑘1×ℝ^𝑘2 × ℝ^𝑛→ℝ, 𝑖=1,2, as follows: 𝐻_𝑖𝑡,𝑦,𝑧,𝑟,𝑦^′,𝑧^′,𝑟^′,𝑣₁,𝑣₂,𝑝_𝑖=−𝑓𝑡,𝑦,𝑧,𝑟(⋅),𝑦^′,𝑧^′,𝑟^′(⋅),𝑣₁,𝑣₂,𝑝_𝑖+𝑙_𝑖𝑡,𝑦,𝑧,𝑟(⋅),𝑦^′,𝑧^′,𝑟^′(⋅),𝑣₁,𝑣₂,𝑖=1,2. (23) Let 𝐻_𝑖(𝑡,⋅)=𝐻_𝑖𝑡,𝑦,𝑧,𝑟,𝑦^′,𝑧^′,𝑟^′,𝑢₁,𝑢₂,𝑝_𝑖,𝐻^𝑣1,𝑣2_𝑖(𝑡,⋅)=𝐻_𝑖𝑡,𝑦,𝑧,𝑟,𝑦^′,𝑧^′,𝑟^′,𝑣₁,𝑣₂,𝑝_𝑖,𝑖=1,2. (24)

We introduce the following adjoint equation: 𝑑𝑝^𝑣1,𝑣2_𝑖(𝑡)=−𝔼^′𝐻^𝑣1,𝑣2_𝑖𝑦(𝑡,⋅)+𝐻^𝑣1,𝑣2_𝑖𝑦′(𝑡,⋅)𝑑𝑡−𝔼^′𝐻^𝑣1,𝑣2_𝑖𝑧(𝑡,⋅)+𝐻^𝑣1,𝑣2_𝑖𝑧′((𝑡)−𝑡,⋅)𝑑𝑊_𝐄𝔼^′𝐻^𝑣1,𝑣2_𝑖𝑟(𝑡,𝑒)+𝐻^𝑣1,𝑣2_𝑖𝑟′(𝑁(),𝑝𝑡,𝑒)𝑑𝑒𝑑𝑡_{𝑣1,𝑣2𝑖}(0)=−Φ_𝑖𝑦𝑦(0),(𝑖=1,2). (25)

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 (𝑢₁(⋅),𝑢₂(⋅)) is an equilibrium point of Problem BNZ and (𝑦(⋅),𝑧(⋅),𝑟(⋅,⋅)) is the corresponding state trajectory. Then one has that 𝔼𝐻_1𝑣1(𝑡,⋅),𝑣₁−𝑢₁(𝑡)∣ℰ¹_𝑡𝔼𝐻≥0_2𝑣2(𝑡,⋅),𝑣₂−𝑢₂(𝑡)∣ℰ²_𝑡≥0, (26) hold for any (𝑣₁,𝑣₂)∈𝑈₁×𝑈₂,𝑎.𝑒.,𝑎.𝑠., where 𝑝_𝑖(⋅)(𝑖=1,2) is the solution of the adjoint equation (25).

Proof. For 𝑖=1, applying Itô’s formula to ⟨𝑦¹₁(𝑡),𝑝₁(𝑡)⟩, we obtain 𝔼^𝑇₀𝔼^′𝑙_1𝑦(𝑡,⋅)𝑦¹₁(𝑡)+𝑙_1𝑧(𝑡,⋅)𝑧¹₁(𝑡)+_𝐄𝑙_1𝑟(𝑡,𝑒)𝑟¹₁(𝑡,𝑒)𝜋(𝑑𝑒)+𝑙_1𝑦′(𝑡,⋅)𝑦¹₁(𝑡)^′+𝑙_1𝑧′(𝑡,⋅)𝑧¹₁(𝑡)^′+_𝐄𝑙_1𝑟′(𝑡,𝑒)𝑟¹₁(𝑡,𝑒)^′𝜋(𝑑𝑒)+𝑙_1𝑣1(𝑡)𝑣₁(𝑡)𝑑𝑡+𝔼Φ_1𝑦𝑦(0)𝑦¹₁(0)=𝔼−𝑓^𝑇_𝑣1(𝑡)𝑝₁(𝑡)+𝑙_1𝑣1(𝑡),𝑣₁(𝑡)𝑑𝑡. (27) From Lemma 2, it follows that we have 𝔼^𝑇₀𝐻_1𝑣1(𝑡,⋅),𝑣₁(𝑡)𝑑𝑡≥0. (28) Because 𝑣₁(𝑡) satisfies 𝑢₁(𝑡)+𝑣₁(𝑡)∈𝒰₁, we have 𝔼^𝑇₀𝐻_1𝑣1(𝑡,⋅),𝑣₁−𝑢₁(𝑡)𝑑𝑡≥0,∀𝑣₁∈𝑈₁. (29) This implies that 𝔼𝐻_1𝑣1(𝑡,⋅),𝑣₁−𝑢₁(𝑡)≥0,∀𝑣₁∈𝑈₁. (30) Now, let 𝑣₁(𝑡)∈𝑈₁ be a deterministic element and let 𝐹 be an arbitrary element of the 𝜎-algebra ℰ¹_𝑡. And set 𝑤₁(𝑡)=𝑣₁(𝑡)𝟏_𝐹+𝑢₁(𝑡)𝟏_Ω−𝐹. (31) It is obvious that 𝑤₁ is an admissible control.
Applying the above inequality with 𝑤₁, we get 𝔼𝟏_𝐹𝐻_1𝑣1(𝑡,⋅),𝑣₁−𝑢₁(𝑡)≥0,∀𝐹∈ℰ¹_𝑡, (32) which implies that 𝔼𝐻_1𝑣1(𝑡,⋅),𝑣₁−𝑢₁(𝑡)∣ℰ¹_𝑡≥0,∀𝑣₁∈𝑈₁,a.e.,a.s. (33) Proceeding in the same way as the above proof, we can show that the other inequality holds for any 𝑣₂∈𝑈₂. 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 (𝑦(𝑡),𝑧(𝑡),𝑟(𝑡,⋅) ,𝑢₁(𝑡),𝑢₂(𝑡)) be a quintuple satisfying (4) and suppose that there exists a solution 𝑝_𝑖(𝑡) of the corresponding adjoint forward SDE (25). We assume that(H3)for 𝑖=1,2, for all 𝑡∈[0,𝑇], 𝐻_𝑖(𝑡,𝑦,𝑧,𝑟,𝑦^′,𝑧^′,𝑟^′,𝑣₁,𝑣₂,𝑝_𝑖) is convex in (𝑦,𝑧,𝑟,𝑦^′,𝑧^′,𝑟^′,𝑣₁,𝑣₂), and Φ_𝑖(𝑦) is convex in 𝑦.Let 𝐻_𝑖(𝑡,⋅)=𝐻_𝑖𝑡,𝑦(𝑡),𝑧(𝑡),𝑟(𝑡,⋅),𝑦^′(𝑡),𝑧^′(𝑡),𝑟^′(𝑡,⋅),𝑢₁(𝑡),𝑢₂(𝑡),𝑝_𝑖(𝑡),𝐻^𝑣1_𝑖(𝑡,⋅)=𝐻_𝑖𝑡,𝑦(𝑡),𝑧(𝑡),𝑟(𝑡,⋅),𝑦^′(𝑡),𝑧^′(𝑡),𝑟^′(𝑡,⋅),𝑣₁(𝑡),𝑢₂(𝑡),𝑝_𝑖(𝑡,𝐻_𝑣2𝑖)(𝑡,⋅)=𝐻_𝑖𝑡,𝑦(𝑡),𝑧(𝑡),𝑟(𝑡,⋅),𝑦^′(𝑡),𝑧^′(𝑡),𝑟^′(𝑡,⋅),𝑢₁(𝑡),𝑣₂(𝑡),𝑝_𝑖(𝑡)𝜑()(𝑡)(𝑡)(),𝑖=1,2,𝑡,⋅=𝜑𝑡,𝑦,𝑧,𝑟𝑡,⋅,𝑦^′(𝑡),𝑧^′(𝑡),𝑟^′(𝑡,⋅),𝑢₁(𝑡),𝑢₂(𝑡),𝜑^𝑣1(𝑡,⋅)=𝜑𝑡,𝑦(𝑡),𝑧(𝑡),𝑟(𝑡,⋅),𝑦^′(𝑡),𝑧^′(𝑡),𝑟^′(𝑡,⋅),𝑣₁(𝑡),𝑢₂(𝑡),𝜑^𝑣2(𝑡,⋅)=𝜑𝑡,𝑦(𝑡),𝑧(𝑡),𝑟(𝑡,⋅),𝑦^′(𝑡),𝑧^′(𝑡),𝑟^′(𝑡,⋅),𝑢₁(𝑡),𝑣₂(𝑡), (34) where 𝜑 denotes one of 𝑓 and 𝑙.

Theorem 4 (partial information sufficient maximum principle). Assume that (H1)–(H3) are satisfied. Moreover, the following partial information maximum conditions hold: 𝔼𝐻₁(𝑡,⋅)∣ℰ¹_𝑡=min_{𝑣1∈𝒰1}𝔼𝐻^𝑣1₁(𝑡,⋅)∣ℰ¹_𝑡,𝔼𝐻₂(𝑡,⋅)∣ℰ²_𝑡=min_{𝑣2∈𝒰2}𝔼𝐻^𝑣2₂(𝑡,⋅)∣ℰ²_𝑡. (35)(36) Then (𝑢₁(⋅),𝑢₂(⋅)) is an equilibrium point of Problem BNZ.

Proof. For any 𝑣₁(⋅)∈𝒰₁, we consider 𝐽₁𝑣₁(⋅),𝑢₂(⋅)−𝐽₁𝑢₁(⋅),𝑢₂(⋅)=𝐈₁+𝐈₂, (37) where 𝐈₁=𝔼^𝑇₀𝔼^′𝑙^𝑣1₁(𝑡,⋅)−𝑙₁(𝐈𝑡,⋅)𝑑𝑡,₂=𝔼Φ₁𝑦^𝑣1(0)−Φ₁𝑦(0). (38) Now applying Itô’s formula to ⟨𝑝₁(𝑡),𝑦^𝑣1(𝑡)−𝑦(𝑡)⟩ on [0,𝑇], we get 𝔼Φ_1𝑦𝑦(0),𝑦^𝑣1(0)−𝑦(0)=𝔼^𝑇₀𝔼^′𝑦^𝑣1(𝑡)−𝑦(𝑡),−𝐻_1𝑦(+𝑦𝑡,⋅)_𝑣1(𝑡)^′−𝑦^′(𝑡),−𝐻_1𝑦′(𝑡,⋅)𝑑𝑡+𝔼^𝑇₀𝔼^′𝑧^𝑣1(𝑡)−𝑧(𝑡),−𝐻_1𝑧(+𝑧𝑡,⋅)_𝑣1(𝑡)^′−𝑧^′(𝑡),−𝐻_1𝑧′(𝑡,⋅)𝑑𝑡+𝔼^𝑇_0𝐄𝔼^′𝑟^𝑣1(𝑡,𝑒)−𝑟(𝑡,𝑒),−𝐻_1𝑟(+𝑟𝑡,𝑒)_𝑣1(𝑡,𝑒)^′−(𝑟(𝑡,𝑒))^′,−𝐻_1𝑟′(𝑡,𝑒)𝜋(𝑑𝑒)𝑑𝑡−𝔼^𝑇₀𝔼^′𝑝₁(𝑡),𝑓^𝑣1(𝑡,⋅)−𝑓(𝑡,⋅)𝑑𝑡. (39) Moreover, by virtue of (39) and convexity of Φ₁, it instantly follows that 𝐈₂≥𝔼Φ_1𝑦𝑦(0),𝑦^𝑣1(0)−𝑦(0)=−Ξ₁+Ξ₂, (40) where Ξ₁=𝔼^𝑇₀𝔼^′𝑦^𝑣1(𝑡)−𝑦(𝑡),𝐻_1𝑦(+𝑦𝑡,⋅)_𝑣1(𝑡)^′−𝑦^′(𝑡),𝐻_1𝑦′(𝑡,⋅)𝑑𝑡+𝔼^𝑇₀𝔼^′𝑧^𝑣1(𝑡)−𝑧(𝑡),𝐻_1𝑧(+𝑧𝑡,⋅)_𝑣1(𝑡)^′−𝑧^′(𝑡),𝐻_1𝑧′(𝑡,⋅)𝑑𝑡+𝔼^𝑇_0𝐄𝔼^′𝑟^𝑣1(𝑡,𝑒)−𝑟(𝑡,𝑒),𝐻_1𝑟(+𝑟𝑡,𝑒)_𝑣1(𝑡,𝑒)^′−(𝑟(𝑡,𝑒))^′,𝐻_1𝑟′(()Ξ𝑡,𝑒)×𝜋𝑑𝑒𝑑𝑡,₂=−𝔼^𝑇₀𝔼^′𝑝₁(𝑡),𝑓^𝑣1(𝑡,⋅)−𝑓(𝑡,⋅)𝑑𝑡. (41) Noting the definition of 𝐻₁ and 𝐈₁, we have 𝐈₁=𝔼^𝑇₀𝔼^′𝐻^𝑣1₁(𝑡,⋅)−𝐻₁(𝑡,⋅)𝑑𝑡+𝔼^𝑇₀𝔼^′𝑝₁(𝑡),𝑓^𝑣1(𝑡,⋅)−𝑓(𝑡,⋅)𝑑𝑡=Ξ₃−Ξ₂, (42) where Ξ₃=𝔼^𝑇₀𝔼^′𝐻^𝑣1₁(𝑡,⋅)−𝐻₁(𝑡,⋅)𝑑𝑡. (43) Using convexity of 𝐻₁(𝑡,𝑦,𝑧,𝑟,𝑦^′,𝑧^′,𝑟^′,𝑣₁,𝑣₂,𝑝₁) with respect to (𝑦,𝑧,𝑟,𝑦^′,𝑧^′,𝑟^′,𝑣₁,𝑣₂), we obtain 𝐻^𝑣1₁(𝑡,⋅)−𝐻₁(𝑡,⋅)≥𝐻_1𝑦(𝑡)𝑦^𝑣1(𝑡)−𝑦(𝑡)+𝐻_1𝑧(𝑡,⋅)𝑧^𝑣1(𝑡)(𝑡)+−𝑧_𝐄𝐻_1𝑟(𝑡,𝑒)𝑟^𝑣1(𝑡,𝑒)−𝑟(𝑡,𝑒)𝜋(𝑑𝑒)+𝐻_1𝑦′(𝑡,⋅)𝑦^𝑣1(𝑡)^′−𝑦^′(𝑡)+𝐻_1𝑧′(𝑡,⋅)𝑧^𝑣1(𝑡)^′−𝑧^′(𝑡)+_𝐄𝐻_1𝑟′(𝑡,𝑒)𝑟^𝑣1(𝑡,𝑒)^′−𝑟(𝑡,𝑒)𝜋(𝑑𝑒)+𝐻_1𝑢1(𝑡,⋅)𝑣₁(𝑡)−𝑢₁(𝑡). (44) Since 𝑣₁→𝔼[𝐻^𝑣1₁(𝑡,⋅)∣ℰ¹_𝑡],𝑣₁∈𝒰₁, is minimal for 𝑢₁(𝑡) and 𝑣₁(𝑡) and 𝑢₁(𝑡) are ℰ¹_𝑡-measurable, we get 𝔼𝐻_1𝑢1(𝑡,⋅)∣ℰ¹_𝑡𝑣₁(𝑡)−𝑢₁(𝑡)=𝔼𝐻_1𝑢1(𝑡,⋅)𝑣₁(𝑡)−𝑢₁(𝑡)∣ℰ¹_𝑡≥0. (45) Hence combining (43), (44), and (45), we obtain Ξ₃≥𝔼^𝑇₀𝔼^′𝑦^𝑣1(𝑡,⋅)−𝑦(𝑡),−𝐻_1𝑦(+𝑦𝑡,⋅)_𝑣1(𝑡)^′−𝑦^′(𝑡),−𝐻_1𝑦′(𝑡,⋅)𝑑𝑡+𝔼^𝑇₀𝔼^′𝑧^𝑣1(𝑡)−𝑧(𝑡),−𝐻_1𝑧(+𝑧𝑡,⋅)_𝑣1(𝑡)^′−𝑧^′(𝑡),−𝐻_1𝑧′(𝑡,⋅)𝑑𝑡+𝔼^𝑇_0𝐄𝔼^′𝑟^𝑣1(𝑡,𝑒)−𝑟(𝑡,𝑒),𝐻_1𝑟(+𝑟𝑡,𝑒)_𝑣1(𝑡,𝑒)^′−(𝑟(𝑡,𝑒))^′,𝐻_1𝑟′(()𝑡,𝑒)×𝜋𝑑𝑒𝑑𝑡=Ξ₁. (46) Therefore, it follows from (35), (40), and (46) that 𝐽₁𝑣₁(⋅),𝑢₂(⋅)−𝐽₁𝑢₁(⋅),𝑢₂(⋅)≥Ξ₃−Ξ₂−Ξ₁+Ξ₂≥Ξ₁−Ξ₂−Ξ₁+Ξ₂=0. (47) Then it implies that 𝐽₁𝑢₁(⋅),𝑢₂(⋅)=min_{𝑣1(⋅)∈𝒰1}𝐽₁𝑣₁(⋅),𝑢₂(⋅). (48)
In the same way 𝐽₂𝑢₁(⋅),𝑢₂(⋅)=min_{𝑣2(⋅)∈𝒰2}𝐽₂𝑢₁(⋅),𝑣₂(⋅). (49) 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 𝑛=𝑑=𝑘₁=𝑘₂=1, 𝑈₁=𝑈₂=ℝ, and ℰ¹_𝑡=ℰ²_𝑡=ℰ_𝑡⊆ℱ_𝑡.

Consider the following: −𝑑𝑦^𝑣1,𝑣2(𝑡)=𝔼^′𝐴(𝑡)𝑦^𝑣1,𝑣2(𝑡)+𝐶(𝑡)𝑧^𝑣1,𝑣2(𝑡)+𝐷(𝑡)𝑟^𝑣1,𝑣2()+𝑡,⋅𝐴(𝑡)𝑦^𝑣1,𝑣2(𝑡)^′+𝐶(𝑡)𝑧^𝑣1,𝑣2(𝑡)^′+𝐷(𝑡)𝑟^𝑣1,𝑣2(𝑡,⋅)^′+𝐵₁(𝑡)𝑣₁(𝑡)+𝐵₂(𝑡)𝑣₂(𝑡)𝑑𝑡−𝑧^𝑣1,𝑣2(𝑡)𝑑𝑊(𝑡)−_𝐄𝑟^𝑣1,𝑣2()𝑁(),𝑦𝑡,𝑒𝑑𝑒𝑑𝑡_𝑣1,𝑣2(𝑇)=𝜉. (50) The cost functional is 𝐽_𝑖𝑣₁(⋅),𝑣₂(⋅)=12𝔼^𝑇₀𝔼^′𝑀_𝑖(𝑡)𝑣²_𝑖(𝑡)+𝑁_𝑖(𝑡)𝑦^𝑣1,𝑣2(𝑡)²+𝑁_𝑖(𝑡)𝑦^𝑣1,𝑣2(𝑡)^′2𝑑𝑡+𝐿_𝑖𝑦^𝑣1,𝑣2(0)²,𝑖=1,2, (51) where constants 𝐿_𝑖≥0, 𝑖=1,2. Functions 𝐴(⋅), 𝐴(⋅), 𝐵₁(⋅), 𝐵₂(⋅), 𝐶(⋅), 𝐶(⋅), 𝐷(⋅), 𝐷(⋅) are bounded and deterministic; 𝑁_𝑖(⋅), 𝑁_𝑖(⋅), 𝑖=1,2, are nonnegative, bounded, and deterministic; 𝑀_𝑖(⋅), 𝑖=1,2, are positive, bounded, and deterministic; 𝑀_𝑖⁻¹(⋅), 𝑖=1,2, are also bounded. Our task is to find (𝑢₁(⋅),𝑢₂(⋅))∈𝒰₁×𝒰₂ such that 𝐽₁𝑢₁(⋅),𝑢₂(⋅)=min_{𝑣1(⋅)∈𝒰1}𝐽₁𝑣₁(⋅),𝑢₂(⋅),𝐽₂𝑢₁(⋅),𝑢₂(⋅)=min_{𝑣2(⋅)∈𝒰2}𝐽₂𝑢₁(⋅),𝑣₂(⋅). (52)

Theorem 5. The mapping 𝑢₁(𝑡)=𝑀₁⁻¹(𝑡)𝐵₁(𝑡)𝔼𝑝₁(𝑡)∣ℰ_𝑡,𝑢₂(𝑡)=𝑀₂⁻¹(𝑡)𝐵₂(𝑡)𝔼𝑝₂(𝑡)∣ℰ_𝑡, (53) is one Nash equilibrium point for the above game problem, where (𝑝₁(𝑡),𝑝₂(𝑡),𝑦(𝑡),𝑧(𝑡),𝑟(𝑡,⋅)) is the solution of the following mean-field forward-backward stochastic differential equations with jumps (MF-FBSDEJ): 𝑑𝑝_𝑖(𝑡)=𝔼^′𝐴(𝑡)𝑝_𝑖(𝑡)+𝐴(𝑡)𝑝^′_𝑖(𝑡)−𝑁_𝑖(𝑡)𝑦(𝑡)−𝑁_𝑖(𝑡)𝑦^′(𝑡)𝑑𝑡+𝔼^′𝐶(𝑡)𝑝_𝑖(𝑡)+𝐶(𝑡)𝑝^′_𝑖(𝑡(𝑡)+)𝑑𝑊_𝐄𝔼^′𝐷(𝑡)𝑝_𝑖(𝑡)+𝐷(𝑡)𝑝^′_𝑖(𝑡𝑁(),(𝑡))𝑑𝑒𝑑𝑡−𝑑𝑦=𝔼^′𝐴(𝑡)𝑦(𝑡)+𝐶(𝑡)𝑧(𝑡)+𝐷(𝑡)𝑟(𝑡,⋅)+𝐴(𝑡)𝑦^′(𝑡)+𝐶(𝑡)𝑧^′(𝑡)+𝐷(𝑡)𝑟^′(𝑡,⋅)+𝐵²₁(𝑡)𝑀₁⁻¹(𝑡)×𝔼𝑝₁(𝑡)∣ℰ_𝑡+𝐵²₂(𝑡)𝑀₂⁻¹(𝑡)𝔼𝑝₂(𝑡)∣ℰ_𝑡(𝑡)(𝑡)−𝑑𝑡−𝑧𝑑𝑊_𝐄𝑟()𝑁(),𝑝𝑡,𝑒𝑑𝑒𝑑𝑡_𝑖(0)=−𝐿_𝑖𝑦(0),𝑦(𝑇)=𝜉,𝑖=1,2. (54)

Proof. We first prove the existence of the solution of (54). We set 𝜃(𝑡)=𝔼𝜃(𝑡)∣ℰ_𝑡,𝜃=𝑦,𝑧,𝑟,𝑦^′,𝑧^′,𝑟^′,𝑝₁,𝑝₂. (55) Similar to Lemma 5.4 of [24], the optimal filter (𝑦(𝑡),𝑧(𝑡),𝑟(̂𝑡,⋅),𝑦^′(𝑡),𝑧^′(𝑡),𝑟^′̂(𝑡,⋅),𝑝₁(𝑡),𝑝₂(𝑡)) of (𝑦(𝑡),𝑧(𝑡) , 𝑟(𝑡,⋅),𝑦^′(𝑡),𝑧^′(𝑡),𝑟^′(𝑡,⋅),𝑝₁(𝑡),𝑝₂(𝑡)) satisfies 𝑑𝑝_𝑖(𝑡)=𝔼^′𝐴(𝑡)𝑝_𝑖(𝑡)+𝐴(𝑡)𝑝^′_𝑖(𝑡)−𝑁_𝑖(𝑡)𝑦(𝑡)−𝑁_𝑖(𝑡)𝑦^′(𝑡)𝑑𝑡+𝔼^′𝐶(𝑡)𝑝_𝑖(𝑡)+𝐶(𝑡)𝑝^′_𝑖(𝑡(𝑡)+)𝑑𝑊_𝐄𝔼^′𝐷(𝑡)𝑝_𝑖(𝑡)+𝐷(𝑡)𝑝^′_𝑖(𝑡𝑁(),𝑦(𝑡))𝑑𝑒𝑑𝑡−𝑑=𝔼^′𝐴(𝑡)𝑦(𝑡)(𝑡)𝑧(𝑡)(𝑡)𝑟̂()++𝐶+𝐷𝑡,⋅𝐴(𝑡)𝑦^′(𝑡)+𝐶(𝑡)𝑧^′(𝑡)+𝐷(𝑡)𝑟^′̂(𝑡,⋅)+𝐵²₁(𝑡)𝑀₁⁻¹(𝑡)𝑝₁(𝑡)+𝐵²₂(𝑡)𝑀₂⁻¹(𝑡)𝑝₂(𝑡−𝑧(𝑡)(𝑡)−)𝑑𝑡𝑑𝑊_𝐄𝑟̂()𝑁(),𝑝𝑡,𝑒𝑑𝑒𝑑𝑡_𝑖(0)=−𝐿_𝑖𝑦(0),𝑦(𝑇)=𝔼𝜉∣ℰ_𝑇,𝑖=1,2. (56) Due to the above analysis, the candidate equilibrium point (𝑢₁(⋅),𝑢₂(⋅)) can be rewritten as 𝑢₁(𝑡)=𝑀₁⁻¹(𝑡)𝐵₁(𝑡)𝑝₁(𝑡),𝑢₂(𝑡)=𝑀₂⁻¹(𝑡)𝐵₂(𝑡)𝑝₂(𝑡), (57) where 𝑝_𝑖(𝑡)(𝑖=1,2) admits MF-FBSDEJ (56). We introduce a new MF-FBSDEJ: 𝑑𝑃(𝑡)=𝔼^′𝐴(𝑡)𝑃(𝑡)+𝐴(𝑡)𝑃^′(𝑡)−𝐵²₁(𝑡)𝑀₁⁻¹(𝑡)𝑁₁(𝑡)+𝐵²₂(𝑡)𝑀₂⁻¹(𝑡)𝑁₂(𝑡(𝑡)−𝐵)×𝑌²₁(𝑡)𝑀₁⁻¹(𝑡)𝑁₁(𝑡)+𝐵²₂(𝑡)𝑀₂⁻¹(𝑡)×𝑁₂(𝑡)𝑌^′(𝑡)𝑑𝑡+𝔼^′𝐶(𝑡)𝑃(𝑡)+𝐶(𝑡)𝑃^′(𝑡(𝑡)+)𝑑𝑊_𝐄𝔼^′𝐷(𝑡)𝑃(𝑡)+𝐷(𝑡)𝑃^′(𝑡𝑁(),(𝑡))𝑑𝑒𝑑𝑡−𝑑𝑌=𝔼^′[𝐴(𝑡)𝑌(𝑡)(𝑡)𝑍(𝑡)(𝑡)𝑅()++𝐶+𝐷𝑡,⋅𝐴(𝑡)𝑌^′(𝑡)+𝐶(𝑡)𝑍^′(𝑡)+𝐷(𝑡)𝑅^′()(𝑡(𝑡)(𝑡)−𝑡,⋅+𝑃)]𝑑𝑡−𝑍𝑑𝑊_𝐄𝑅()𝑁(),𝑃(0)𝐵𝑡,𝑒𝑑𝑒𝑑𝑡=−²₁(0)𝑀₁⁻¹(0)𝐿₁+𝐵²₂(0)𝑀₂⁻¹(0)𝐿₂𝑌(0),𝑌(𝑇)=𝜉. (58) Based on the analysis above, we can say the existence and uniqueness of MF-FBSDEJ (56) are equivalent to those of MF-FBSDEJ (58). It is easy to check that MF-FBSDEJ (58) satisfies assumptions (A3)–(A5) with 𝐻=1,  𝜇₁=1, and 𝜇₂=𝛽₂=0. According to Theorem 7 in Appendix, there exists a unique solution (𝑃(𝑡),𝑌(𝑡) , 𝑍(𝑡),𝑅(𝑡,⋅)) of (58); here, 𝑃(𝑡)=𝐵²₁(𝑡)𝑀₁⁻¹(𝑡)𝑝₁(𝑡)+𝐵²₂(𝑡)𝑀₂⁻¹(𝑡)𝑝₂(𝑡),𝑌(𝑡)=𝑦(𝑡),𝑍(𝑡)=𝑧(𝑡),𝑅(𝑡,⋅)=𝑟̂(𝑡,⋅). (59) Then there exists a unique solution (𝑝₁(𝑡),𝑝₂(𝑡),𝑦(𝑡),𝑧(𝑡),𝑟(̂𝑡,⋅)) of MF-FBSDEJ (56). Furthermore, there exists at most one equilibrium point for the underlying game.
Now we try to prove that (𝑢₁(⋅),𝑢₂(⋅)) is a Nash equilibrium point for our backward LQ game problem. We only prove 𝐽₁𝑢₁(⋅),𝑢₂(⋅)=min_{𝑣1(⋅)∈𝒰1}𝐽₁𝑣₁(⋅),𝑢₂(⋅). (60) It is similar to getting another inequality of (52). (𝑦^𝑣1(𝑡),𝑧^𝑣1(𝑡),𝑟^𝑣1(𝑡,⋅)) denotes the solution of the system −𝑑𝑦^𝑣1(𝑡)=𝔼^′𝐴(𝑡)𝑦^𝑣1(𝑡)+𝐶(𝑡)𝑧^𝑣1(𝑡)+𝐷(𝑡)𝑟^𝑣1()+𝑡,⋅𝐴(𝑡)𝑦^𝑣1(𝑡)^′+𝐶(𝑡)𝑧^𝑣1(𝑡)^′+𝐷(𝑡)𝑟^𝑣1(𝑡,⋅)^′+𝐵₁(𝑡)𝑣₁(𝑡)+𝐵₂(𝑡)𝑢₂(𝑡)𝑑𝑡−𝑧^𝑣1(𝑡)𝑑𝑊(𝑡)−_𝐄𝑟^𝑣1()𝑁(),𝑦𝑡,𝑒𝑑𝑒𝑑𝑡_𝑣1(𝑇)=𝜉. (61) Then 𝐽₁𝑣₁(⋅),𝑢₂(⋅)−𝐽₁𝑢₁(⋅),𝑢₂(⋅)=12𝔼^𝑇₀𝔼^′𝑀₁(𝑡)(𝑣₁(𝑡)−𝑢₁(𝑡))²+2𝑀₁(𝑡)𝑢₁(𝑡)𝑣₁(𝑡)−𝑢₁(𝑡)+𝑁₁(𝑡)(𝑦^𝑣1(𝑡)−𝑦(𝑡))²+2𝑁₁(𝑡)𝑦(𝑡)𝑦^𝑣1(𝑡)(𝑡)+−𝑦𝑁₁(𝑡)𝑦^𝑣1(𝑡)^′−𝑦^′(𝑡)²+2𝑁₁(𝑡)𝑦^′(𝑡)𝑦^𝑣1(𝑡)^′−𝑦^′(𝑡)𝑑𝑡+𝐿₁(𝑦^𝑣1(0)−𝑦^𝑢(0))²+2𝐿₁𝑦^𝑢(0)𝑦^𝑣(0)−𝑦^𝑢(0). (62) Applying Itô’s formula to (𝑦^𝑣1(𝑡)−𝑦(𝑡))𝑝₁(𝑡), we have 𝔼𝐿₁𝑦^𝑣(0)𝑦^𝑣(0)−𝑦^𝑢(0)=−𝔼^𝑇₀𝔼^′𝐵₁(𝑡)𝑝₁(𝑡)𝑣₁(𝑡)−𝑢₁(𝑡)+𝑁₁(𝑡)𝑦(𝑡)𝑦^𝑣1(𝑡)(𝑡)+−𝑦𝑁₁(𝑡)𝑦^′(𝑡)𝑦^𝑣1(𝑡)^′−𝑦^′(𝑡)𝑑𝑡. (63) As 𝑀₁(𝑡)>0,𝑁₁(𝑡)≥0,𝑁₁(𝑡)≥0, ∀𝑡∈[0,𝑇], 𝐿₁≥0, noting that 𝑢₁(𝑡)=𝑀₁⁻¹(𝑡)𝐵₁(𝑡)𝑝₁(𝑡), we have 𝐽₁𝑣₁(⋅),𝑢₂(⋅)−𝐽₁𝑢₁(⋅),𝑢₂(⋅)≥𝔼^𝑇₀𝔼^′𝑀₁(𝑡)𝑢₁(𝑡)−𝐵₁(𝑡)𝑝₁(𝑡)×𝑣₁(𝑡)−𝑢₁(𝑡)𝑑𝑡=0. (64) So (𝑢₁(𝑡),𝑢₂(𝑡))=(𝑀₁⁻¹(𝑡)𝐵₁(𝑡)𝑝₁(𝑡),𝑀₂⁻¹(𝑡)𝐵₂(𝑡)𝑝₂(𝑡)) is a Nash equilibrium point for our backward LQ nonzero sum differential game problem.

6. Appendix

For the sake of convenience and completeness, we cite the existence and uniqueness theorem of MF-BSDEJ obtained by Shen and Siu [23]. They studied the following MF-BSDEJ: −𝑑𝑦(𝑡)=𝔼^′𝑓𝑡,𝜉(𝑡),𝜉^′(𝑡)(𝑡)(𝑡)(𝑡),𝑣𝑑𝑡−𝑧𝑑𝑊−_𝐄𝑟()𝑁(),𝑦(𝑇)𝑡,𝑒𝑑𝑒𝑑𝑡=𝜉, (65) where (𝜉(𝑡),𝜉^′(𝑡))=(𝑦(⋅),𝑧(⋅),𝑟(⋅,⋅),𝑦^′(⋅),𝑧^′(⋅),𝑟^′(⋅,⋅))∈ℝ^𝑛×ℝ^𝑛×𝑑×𝐿²_𝜋(⋅)(ℝ^𝑛)×ℝ^𝑛×ℝ^𝑛×𝑑×𝐿²_𝜋(⋅)(ℝ^𝑛), 𝜉∈𝐿²(Ω,ℱ_𝑇,𝑃;ℝ^𝑛), is a random variable, and 𝑇>0; 𝑓∶[0,𝑇]×ℝ^𝑛×ℝ^𝑛×𝑑×𝐿²_𝜋(⋅)ℝ^𝑛×ℝ^𝑛×ℝ^𝑛×𝑑×𝐿²_𝜋(⋅)ℝ^𝑛⟶ℝ^𝑛. (66)

They assumed that(A1)for each 𝜉, 𝜉^′∈ℝ^𝑛×ℝ^𝑛×𝑑×𝐿²_𝜋(⋅)(ℝ^𝑛), 𝑓(⋅,𝜉,𝜉^′) is an ℱ_𝑡-measurable process defined on [0,𝑇] with 𝑓(⋅,0,0)∈𝑀²(0,𝑇;ℝ^𝑛×ℝ^𝑛×𝑑)×𝐹²_𝑁(0,𝑇;ℝ^𝑛);(A2)𝑓(𝑡,𝜁,𝜁^′) satisfies Lipschitz condition: there exists a constant 𝑙>0, such that 𝑓𝑡,𝜉,𝜉^′−𝑓𝑡,𝜉,𝜉^′≤𝑙𝜉−𝜉+𝜉^′−𝜉^′,∀𝜉=𝑦,𝑧,𝑟^𝑇,𝜉^′=𝑦^′,𝑧^′,𝑟^′𝑇,𝜉=𝑦,𝑧,𝑟^𝑇,𝜉^′=𝑦^′,𝑧^′,𝑟^′𝑇∈ℝ^𝑛×ℝ^𝑛×𝑑×𝐿²_𝜋(⋅)ℝ^𝑛,∀𝑡∈[0,𝑇]. (67)Based on the fixed-point theorem, Shen and Siu [23] obtained the following existence and uniqueness result.

Proposition 6. One assumes that (A1) and (A2) hold. Then MF-BSDEJ (65) has a unique solution (𝑦(𝑡),𝑧(𝑡),𝑟(𝑡,⋅))∈𝑀²(0,𝑇;ℝ^𝑛×ℝ^𝑛×𝑑)×𝐹²_𝑁(0,𝑇;ℝ^𝑛).

In the present paper we research the game problem of a MF-BSDEJ, so the game system and the adjoint equation constitute exactly a kind of initial coupled MF-FBSDEJ. Due to this, we give an existence and uniqueness theorem of MF-FBSDEJ. Consider the following MF-FBSDEJ: 𝑑𝑥(𝑡)=𝔼^′𝑏𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑟(𝑡,⋅),𝑥^′(𝑡),𝑦^′(𝑡),𝑧^′(𝑡),𝑟^′(𝑡,⋅)𝑑𝑡+𝔼^′𝜎𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑟(𝑡,⋅),𝑥^′(𝑡),𝑦^′(𝑡),𝑧^′(𝑡),𝑟^′()(𝑡)+𝑡,⋅𝑑𝑊_𝐄𝔼^′𝛾𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑟(𝑡,𝑒),𝑥^′(𝑡),𝑦^′(𝑡),𝑧^′(𝑡),𝑟^′()𝑁(),(𝑡)𝑡,𝑒,𝑒𝑑𝑒𝑑𝑡−𝑑𝑦=𝔼^′𝑓𝑡,𝑥(𝑡),𝑦(𝑡),𝑟(𝑡,⋅),𝑥^′(𝑡),𝑦^′(𝑡),𝑟^′()(𝑡)(𝑡)−𝑡,⋅𝑑𝑡−𝑧𝑑𝑊_𝐄𝑟()𝑁(),𝑥(0)𝑦(0)(𝑇)𝑡,𝑒𝑑𝑒𝑑𝑡=𝜓,𝑦=𝜉, (68) where (𝑥(⋅),𝑦(⋅),𝑧(⋅),𝑟(⋅,⋅),𝑥^′(⋅),𝑦^′(⋅),𝑧^′(⋅),𝑟^′(⋅,⋅))∈ℝ^𝑚×ℝ^𝑛×ℝ^𝑛×𝑑×𝐿²_𝜋(⋅)(ℝ^𝑛)×ℝ^𝑚×ℝ^𝑛×ℝ^𝑛×𝑑 × 𝐿²_𝜋(⋅)(ℝ^𝑛), 𝜉∈𝐿²(Ω,ℱ_𝑇,𝑃;ℝ^𝑛), is a random variable, and 𝑇>0; 𝑏∶Ω×[0,𝑇]×ℝ^𝑚×ℝ^𝑛×ℝ^𝑛×𝑑×𝐿²_𝜋(⋅)ℝ^𝑛×ℝ^𝑚×ℝ^𝑛×ℝ^𝑛×𝑑×𝐿²_𝜋(⋅)ℝ^𝑛⟶ℝ^𝑚,𝜎∶Ω×[0,𝑇]×ℝ^𝑚×ℝ^𝑛×ℝ^𝑛×𝑑×𝐿²_𝜋(⋅)ℝ^𝑛×ℝ^𝑚×ℝ^𝑛×ℝ^𝑛×𝑑×𝐿²_𝜋(⋅)ℝ^𝑛⟶ℝ^𝑚×𝑑,𝛾∶Ω×[0,𝑇]×ℝ^𝑚×ℝ^𝑛×ℝ^𝑛×𝑑×𝐿²_𝜋(⋅)ℝ^𝑛×ℝ^𝑚×ℝ^𝑛×ℝ^𝑛×𝑑×𝐿²_𝜋(⋅)ℝ^𝑛×𝐄⟶ℝ^𝑚,𝑓∶Ω×[0,𝑇]×ℝ^𝑚×ℝ^𝑛×ℝ^𝑛×𝑑×𝐿²_𝜋(⋅)ℝ^𝑛×ℝ^𝑚×ℝ^𝑛×ℝ^𝑛×𝑑×𝐿²_𝜋(⋅)ℝ^𝑛⟶ℝ^𝑛,𝜓∶ℝ^𝑛⟶ℝ^𝑚. (69) Given an 𝑛×𝑚 full-rank matrix 𝐻, let us introduce some notations 𝑥𝑦𝑧𝑟𝜁=,𝜁^′=𝑥^′𝑦^′𝑧^′𝑟^′,𝐴𝑡,𝜁,𝜁^′=−𝐻^𝑇𝑓𝐻𝑏𝐻𝜎𝐻𝛾𝑡,𝜁,𝜁^′, (70) where 𝐻𝜎=(𝐻𝜎₁⋯𝐻𝜎_𝑑). Assume that(A3)for each 𝜁, 𝜁^′∈ℝ^𝑚×ℝ^𝑛×ℝ^𝑛×𝑑×𝐿²_𝜋(⋅)(ℝ^𝑛), 𝐴(⋅,𝜁,𝜁^′) is an ℱ_𝑡-measurable process defined on [0,𝑇] with 𝐴(⋅,0,0)∈𝑀²(0,𝑇;ℝ^𝑚×ℝ^𝑛×ℝ^𝑛×𝑑)×𝐹²_𝑁(0,𝑇;ℝ^𝑛);(A4)𝐴(𝑡,𝜁,𝜁^′) and 𝜓(𝑦) satisfy Lipschitz conditions: there exists a constant 𝑘>0, such that 𝐴𝑡,𝜁,𝜁^′−𝐴𝑡,𝜁,𝜁^′≤𝑘𝜁−𝜁+𝜁^′−𝜁^′,∀𝜁=𝑥,𝑦,𝑧,𝑟^𝑇,𝜁^′=𝑥^′,𝑦^′,𝑧^′,𝑟^′𝑇,𝜁=𝑥,𝑦,𝑧,𝑟^𝑇,𝜁^′=𝑥^′,𝑦^′,𝑧^′,𝑟^′𝑇∈ℝ^𝑚×ℝ^𝑛×ℝ^𝑛×𝑑×𝐿²_𝜋(⋅)ℝ^𝑛,[],𝜓𝑦∀𝑡∈0,𝑇−𝜓𝑦≤𝑘𝑦−𝑦,∀𝑦,𝑦∈ℝ^𝑛; (71)(A5)𝐴(𝑡,𝜁,𝜁^′) and 𝜓(𝑦) satisfy monotonic conditions: 𝐴𝑡,𝜁,𝜁^′−𝐴𝑡,𝜁,𝜁^′,𝜁−𝜁≤−𝜇₁|𝐻(𝑥−𝑥)|²−𝜇₂𝐻^𝑇(𝑦−𝑦)²+𝐻^𝑇(𝑧−𝑧)²+𝐻^𝑇(𝑟−𝑟)²,∀𝜁=𝑥,𝑦,𝑧,𝑟^𝑇,𝜁^′=𝑥^′,𝑦^′,𝑧^′,𝑟^′𝑇,𝜁=𝑥,𝑦,𝑧,𝑟^𝑇,𝜁^′=𝑥^′,𝑦^′,𝑧^′,𝑟^′𝑇∈ℝ^𝑚×ℝ^𝑛×ℝ^𝑛×𝑑×𝐿²_𝜋(⋅)ℝ^𝑛,[].𝜓𝑦∀𝑡∈0,𝑇−𝜓𝑦,𝑦−𝑦≤−𝛽₂𝐻^𝑇𝑦−𝑦²,∀𝑦,𝑦∈ℝ^𝑛, (72) where 𝜇₁, 𝜇₂, and 𝛽₂ are given nonnegative constants with 𝜇₁+𝜇₂>0,𝜇₁+𝛽₂>0. Moreover we have 𝜇₁>0 (resp., 𝜇₂>0, 𝛽₂>0) when 𝑚<𝑛 (resp., 𝑚>𝑛).

By similar arguments of Yu and Ji [14], Wang and Yu [19], and Min et al. [25], we have the following existence and uniqueness theorem.

Theorem 7. One assumes that (A3), (A4), and (A5) hold. Then MF-FBSDEJ (68) has a unique solution (𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑟(𝑡,⋅))∈𝑀²(0,𝑇;ℝ^𝑚×ℝ^𝑛×ℝ^𝑛×𝑑)×𝐹²_𝑁(0,𝑇;ℝ^𝑛).

7. Conclusion

In this paper, we investigate a new differential game problem of mean-field BSDE with jump (MF-BSDEJ). Compared with the previous literature, our game systems are mean-field BSDE with jump and are under the framework of partial information. We established a maximum principle and a verification theorem for an equilibrium point of nonzero sum differential games. We also give a partial information linear-quadratic (LQ) game as example to show the applications of our theoretical results.

The subject issue studied in this paper possesses fine generality. Firstly, the mean-field BSDEJ game system covers many systems as its particular case. For example, if we drop the terms on jump or mean-field or both of them, then the MF-BSDEJ can be reduced to MF-BSDE or BSDEJ or BSDE. Secondly, if we suppose that ℰ_𝑡=ℱ_𝑡, for all 𝑡∈[0,𝑇], all the results are reduced to the case of full information. Thirdly, if the present nonzero sum stochastic differential game has only one player, the game problem is reduced to some related optimal control. Particularly, our results are a partial extension to differential games of full information BSDEs [15] and partial information BSDEs [19]. Finally, since many optimization and game problems in finance and economics can be associated with mean-field BSDE with jump, the outcomes of this paper can be widely applied in these areas.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the editors and the reviewers for their constructive comments and suggestions which helped them to improve this paper. This work was supported by the National Natural Science Foundation of China (Grant nos. 11371226, 11071145, 11301298, and 11231005), the Foundation for Innovative Research Groups of National Natural Science Foundation of China (Grant no. 11221061), the 111 Project (Grant no. B12023), and the Development Project of Shandong Province Science and Technology (Grant no. 2012GGB01241).