# Necessary Conditions for Optimal Control of Forward-Backward Stochastic Systems with Random Jumps

Authors: Jingtao Shi
Journal: International Journal of Stochastic Analysis
 (2012)

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## Abstract

This paper deals with the general optimal control problem for fully coupled forward-backward stochastic differential equations with random jumps (FBSDEJs). The control domain is not assumed to be convex, and the control variable appears in both diffusion and jump coefficients of the forward equation. Necessary conditions of Pontraygin's type for the optimal controls are derived by means of spike variation technique and Ekeland variational principle. A linear quadratic stochastic optimal control problem is discussed as an illustrating example.


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## Body

## 1. Introduction

### 1.1. Basic Notations

Throughout this paper, we denote byRn the space of n-dimensional Euclidean space, by Rn×d the space of n×d matrices, and by 𝒮n the space of n×n symmetric matrices. 〈·,·〉 and |·| denote the scalar product and norm in the Euclidean space, respectively. ⊤ appearing in the superscripts denotes the transpose of a matrix.Let(Ω,ℱ,{ℱt}t≥0,P)be a complete filtered probability space satisfying the usual conditions, where the filtration {ℱt}t≥0 is generated by the following two mutually independent processes:(i)
a one-dimensional standard Brownian motion{W(t)}t≥0;(ii)
a Poisson random measureN on E×R+, where E⊂R is a nonempty open set equipped with its Borel field ℬ(E), with compensator N̂(dedt)=π(de)dt, such that Ñ(A×[0,t])=(N-N̂)(A×[0,t])t≥0 is a martingale for all A∈ℬ(E) satisfying π(A)<∞.  π is assumed to be a σ-finite measure on (E,ℬ(E)) and is called the characteristic measure.LetT>0 be fixed and U be a nonempty subset of Rk. We denote 𝕄=Rn×Rm×Rm×Rm. Any generic point in 𝕄 is denoted by θ≡(x,y,z,c). Let 𝒰[0,T] be the set of all ℱt-predictable processes u:[0,T]×Ω→U such that sup0≤t≤T𝔼|u(t)|i<+∞,foralli=1,2,…. Any u(·)∈𝒰[0,T] is called an admissible control process. We denote by ℒ2(E,ℬ(E),π;Rm) or ℒ2 the set of integrable functions c:E→Rm with norm ∥c(e)∥ℒ22:=∫E|c(e)|2π(de)<∞. We define(1.1)LFT2(Ω;Rm):={ξ:Ω⟶Rm∣ξisFT-measurable,E|ξ|2<∞},LF2([0,T];Rm):={φ:[0,T]×Ω⟶Rm∣φisFt-adapted,E∫0T|φ(t)|2dt<∞},LF,p2([0,T];Rm):={φ̂:[0,T]×Ω⟶Rm∣φ̂isFt-predictable,E∫0T|φ̂(t)|2dt<∞},Fp2([0,T];Rm):={∫0Tφ̃:[0,T]×Ω×E⟶Rm∣φ̃isFt-predictable,suchthat,E∫0T∫E|φ̃(t,e)|2π(de)dt<∞}.
We denote(1.2)M2[0,T]:=LF2([0,T];Rn)×LF2([0,T];Rm)×LF,p2([0,T];Rm)×Fp2([0,T];Rm).
Clearly, ℳ2[0,T] is a Banach space. Any process in ℳ2[0,T] is denoted by Θ(·)≡(x(·),y(·),z(·),c(·,·)), whose norm is defined by(1.3)‖Θ(⋅)‖M2[0,T]2:=E[sup⁡t∈[0,T]|x(t)|2+sup⁡t∈[0,T]|y(t)|2+∫0T|z(t)|2dt+E∫0T∫E|c(t,e)|2π(de)dt].


### 1.2. Formulation of the Optimal Control Problem and Basic Assumptions

For anyu(·)∈𝒰[0,T] and a∈Rn, we consider the following fully coupled forward-backward stochastic control system:(1.4)dx(t)=b(t,x(t),y(t),z(t),c(t,⋅),u(t))dt+σ(t,x(t),y(t),z(t),c(t,⋅),u(t))dW(t)+∫Eg(t,x(t-),y(t-),z(t),c(t,⋅),u(t),e)Ñ(dedt),-dy(t)=f(t,x(t),y(t),z(t),c(t,⋅),u(t))dt-z(t)dW(t)-∫Ec(t,e)Ñ(dedt),x(0)=a,y(T)=h(x(T)),
with the cost functional given by(1.5)J(u(⋅))=E[∫0Tl(t,x(t),y(t),z(t),c(t,⋅),u(t))dt+ϕ(x(T))+γ(y(0))].
Here, Θ(·)≡(x(·),y(·),z(·),c(·,·)) takes value in 𝕄=Rn×Rm×Rm×Rm and(1.6)b,σ:[0,T]×M×U⟶Rn,g:[0,T]×M×U×E⟶Rn,f:[0,T]×M×U⟶Rm,h:Rn⟶Rm,ϕ:Rn⟶R,γ:Rm⟶R,l:[0,T]×M×U⟶R.For anyu(·)∈𝒰[0,T] and a∈Rn, we refer to Θ(·)≡(x(·),y(·),z(·),c(·,·)) as the state process corresponding to the admissible control u(·) if FBSDEJ (1.4) admits a unique adapted solution. For controlled FBSDEJ (1.4) with cost functional (1.5), we consider the following.Problem C. Find u¯(·)∈𝒰[0,T], such that
(1.7)J(u¯(⋅))=inf⁡u(⋅)∈U[0,T]J(u(⋅)).Anyu¯(·)∈𝒰[0,T] satisfying (1.7) is called an optimal control process of Problem C, and the corresponding state process, denoted by Θ¯(·)≡(x¯(·),y¯(·),z¯(·),c¯(·,·)), is called optimal state process. We also refer to (x¯(·),y¯(·),z¯(·),c¯(·,·),u¯(·)) as an optimal 5-tuple of Problem C.Our main goal in this paper is to derive some necessary conditions for the optimal control of Problem C, which is called the stochastic maximum principle of Pontraygin’s type. For this target, we first introduce the following basic assumption throughout this paper.(H0)For anyu(·)∈𝒰[0,T]anda∈Rn,FBSDEJ(1.4) admits a unique adapted solutionΘ(·)≡(x(·),y(·),z(·),c(·,·))∈ℳ2[0,T]. Moreover, the following estimate holds:(1.8)‖Θ(⋅)‖M2[0,T]≡E[sup⁡t∈[0,T]|x(t)|2+sup⁡t∈[0,T]|y(t)|2+∫0T|z(t)|2dt+E∫0T∫E|c(t,e)|2π(de)dt]≤CE[|a|2+|h(0)|2+(∫0T|b(t,0,u(t))|dt)2+(∫0T|f(t,0,u(t))|dt)2+∫0T|σ(t,0,u(t))|2dt+∫0T∫E|g(t,0,u(t),e)|2π(de)dt(∫0T|b(t,0,u(t))|dt)2].Further, ifΘ̃(·)≡(x̃(·),ỹ(·),z̃(·),c̃(·,·))∈ℳ2[0,T]  is the unique adapted solution of (1.4) withh(·)  andu(·)  replaced byh̃:Rn→Rm  andũ(·)∈𝒰[0,T], respectively, then the following stability estimate holds:(1.9)‖Θ(⋅)-Θ̃(⋅)‖M2[0,T]≡E[sup⁡t∈[0,T]|x(t)-x̃(t)|2+sup⁡t∈[0,T]|y(t)-ỹ(t)|2+∫0T|z(t)-z̃(t)|2dt+E∫0T∫E|c(t,e)-c̃(t,e)|2π(de)dt]≤CE[|h(x(T))-h(x̃(T))|2+(∫0T|b(t,Θ(t),u(t))-b(t,Θ̃(t),ũ(t))|dt)2+(∫0T|f(t,Θ(t),u(t))-f(t,Θ̃(t),ũ(t))|dt)2+∫0T|σ(t,Θ(t),u(t))-σ(t,Θ̃(t),ũ(t))|2dt+∫0T∫E|g(t,Θ(t),u(t),e)-g(t,Θ̃(t),ũ(t),e)|2π(de)dt(∫0T|b(t,Θ(t),u(t))-b(t,Θ̃(t),ũ(t))|dt)2].
By adopting the ideas from Wu [1], we know that under certain G-monotonicity conditions for the coefficients, the existence and uniqueness of FBSDEJ (1.4) guaranteed which leads to hypothesis (H0). Our main goal in this paper is to derive some necessary conditions for the optimal control of Problem C. Hence, we impose the well-posedness of the state equation (1.4) as an assumption to avoid some technicalities not closely to our main results.


### 1.3. Developments of Stochastic Optimal Control and Contributions of the Paper

It is well known that the optimal control problem is one of the central themes of modern control sciences. Necessary conditions for the optimal control of the (forward) continuous stochastic control system, that is, the so-called stochastic maximum principle of Pontraygin’s type, were extensively studied since early 1960s. When Brownian motion is the unique noise source, Peng [2] (see also Yong and Zhou [3]) obtained the maximum principle for the general case, that is, the control variable entering the diffusion coefficient and control domain being not convex.Forward-backward stochastic control systems where the controlled systems are described byforward-backward stochastic differential equations (FBSDEs) are widely used in mathematical economics and mathematical finance, which includes the usual forward SDEs as a special case. They are encountered in stochastic recursive utility optimization problems (see [4–8]) and principal-agent problems (see [9, 10]). Peng [11] first obtained the necessary conditions for optimal control for the partially coupling case when the control domain is convex. And then Xu [12] studied the nonconvex control domain case and obtained the corresponding necessary conditions. But he needs to assume that the diffusion coefficient in the forward control system does not contain the control variable. Ji and Zhou [8] applied the Ekeland variational principle to establish a maximum principle for a partially coupled forward-backward stochastic control system, while the forward state is constrained in a convex set at the terminal time. Wu [13] recently established a general maximum principle for optimal control problems derived by forward-backward stochastic systems, where control domains are nonconvex and forward diffusion coefficients explicitly depend on control variables. Moreover, some financial optimization problems for large investors (see [14–16]) and some asset pricing problems with forward-backward differential utility (see [7]) directly lead to fully coupled FBSDEs. Wu [17] first (see also Meng [18]) obtained the necessary conditions for optimal control of fully coupled forward-backward stochastic control systems when the control domain is convex. And then Shi and Wu [19] studied the nonconvex control domain case and obtained the corresponding necessary conditions under some G-monotonic assumptions. But they also (similar to Xu [12]) need to assume that the control variable does not appear in the diffusion coefficient of the forward equation. Very recently, Yong [20] completely solved the problem of finding necessary conditions for optimal control of fully coupled FBSDEs. He considered an optimal control problem for general coupled FBSDEs with mixed initial-terminal conditions and derived the necessary conditions for the optimal controls when the control domain is not assumed to be convex and the control variable appears in the diffusion coefficient of the forward equation.However, recently more and more research attentions are drawn towards the optimal control problem fordiscontinuous stochastic systems or stochastic systems with random jumps. The reason is clear for its applicable aspect. For example, there is compelling evidence that the dynamics of prices of financial instruments exhibit jumps that cannot be adequately captured solely by diffusion processes (i.e., processes satisfying some Itô-type stochastic differential equations (SDE for short), see Merton [21] and Cont and Tankov [22]. Several empirical studies demonstrate the existence of jumps in stock markets, the foreign exchange market, and bond markets. Jumps constitute also a key feature in the description of credit risk sensitive instruments. Therefore, models that incorporate jumps have become increasingly popular in finance and several areas of science and engineering, this leads to paying more attention to stochastic differential equations with jumps (SDEJs for short). As consequences, optimal control problems involving systems of SDEJs are widely studied. Situ [23] first obtained the maximum principle for (forward) stochastic control system with jumps but the jump coefficient does not contain the control variable. Tang and Li [24] discussed a more general case, where the control is allowed into both diffusion and jump coefficients, and the control domain is not necessarily convex; also some general state constraints are imposed. Maximum principles for forward-backward stochastic control systems with random jumps are studied in Øksendal and Sulem [25], Shi and Wu [26], where the FBSDEJs are partially coupled and the control domains are convex. Recently, Shi [27] obtained the necessary condition of optimal control as well as a sufficient condition of optimality under the assumption that the diffusion and jump coefficients do not contain the control variable and the control domain need not be convex. Necessary conditions for fully coupled forward-backward stochastic control systems with random jumps were studied in Shi and Wu [28] (see also Meng and Sun [29]) where the control domains are convex.In this paper, we consider the general optimal control problem for fully coupled FBSDEJ (1.4). Here, by the word “general” we mean the allowance of the control variable into both diffusion and jump coefficients of the forward equation and the control domain is not assumed to be convex. It is well worth mentioning that the idea of second-order spike variation technique, developed by Peng [2], plays an important role in deriving the necessary conditions of general stochastic optimal control of jump-diffusion process in Tang and Li [24]. Following the standard idea of deriving necessary conditions for optimal control processes, due to the fact the control domain U is not assumed to be convex, one needs to use spike variation for the control process and then to try obtaining a Taylor-type expansion for the state process (x(·),y(·),z(·),c(·,·)) and the cost functional (1.5) with respect to the spike variation of the control process, followed by some suitable duality relations to get a maximum principle of Pontryagin-type. However, the derivation of Taylor expansion of the state process (x(·),y(·),z(·),c(·,·)) with respect to the spike variation of the control process is technically difficult. The main reasons are that both (z(·),c(·,·)) and u(·) appear in the diffusion and jump coefficients of the forward equation of (1.4) and that the regularity/integrability of the continuous martingale process z(·) and the discontinuous martingale process c(·,·) (as part of the state process) seems to be not enough in the case when a second-order expansion is necessary. Note that in [25–29], due to the special structure of the problems, the second-order expansion is not necessary when one tries to derive the necessary conditions for optimal controls.We overcome the above difficulty by the newly developedreduction method by Wu [13] and Yong [20] in the continuous case, independently. In fact, some ideas have been proposed early in Kohlmann and Zhou [30] and Ma and Yong [31]. Let us make it more precise. We first introduce the controlled initial value problem for a system of SDEJs, where (x(·),y(·)) is regarded as the state process and (z(·),c(·,·),u(·)) is regarded as the control process. Next, we regard the original terminal condition y(T)=h(x(T)) as terminal state constraint and then we translate Problem C into a high-dimensional reduced optimal control problem described by standard SDEJ with state constraint (see Problem Ĉ in Section 3). The advantage of this reduced optimal control problem is that one needs not much regularity/integrability of processes (z(·),c(·,·)) since it is treated as a control process. We apply the Ekeland variational principle to deal with this high-dimensional reduced optimal control problem with state constraint. Finally, necessary conditions for the optimal control of Problem C are derived by the equivalence of Problems C and Ĉ.The rest of this paper is organized as follows. In Section2, under some suitable assumptions, we give the main result of this paper, together with some discussions on special cases. In Section 3, after we make the reduction of our optimal control problems for FBSDEJs Problem C, a proof of our main result will be presented. Section 4 is devoted to a linear quadratic stochastic optimal control problem as an illustrating example. Finally, in Section 5 we give the concluding remarks and compare our theorem with some existing results.


## 1.1. Basic Notations

Throughout this paper, we denote byRn the space of n-dimensional Euclidean space, by Rn×d the space of n×d matrices, and by 𝒮n the space of n×n symmetric matrices. 〈·,·〉 and |·| denote the scalar product and norm in the Euclidean space, respectively. ⊤ appearing in the superscripts denotes the transpose of a matrix.Let(Ω,ℱ,{ℱt}t≥0,P)be a complete filtered probability space satisfying the usual conditions, where the filtration {ℱt}t≥0 is generated by the following two mutually independent processes:(i)
a one-dimensional standard Brownian motion{W(t)}t≥0;(ii)
a Poisson random measureN on E×R+, where E⊂R is a nonempty open set equipped with its Borel field ℬ(E), with compensator N̂(dedt)=π(de)dt, such that Ñ(A×[0,t])=(N-N̂)(A×[0,t])t≥0 is a martingale for all A∈ℬ(E) satisfying π(A)<∞.  π is assumed to be a σ-finite measure on (E,ℬ(E)) and is called the characteristic measure.LetT>0 be fixed and U be a nonempty subset of Rk. We denote 𝕄=Rn×Rm×Rm×Rm. Any generic point in 𝕄 is denoted by θ≡(x,y,z,c). Let 𝒰[0,T] be the set of all ℱt-predictable processes u:[0,T]×Ω→U such that sup0≤t≤T𝔼|u(t)|i<+∞,foralli=1,2,…. Any u(·)∈𝒰[0,T] is called an admissible control process. We denote by ℒ2(E,ℬ(E),π;Rm) or ℒ2 the set of integrable functions c:E→Rm with norm ∥c(e)∥ℒ22:=∫E|c(e)|2π(de)<∞. We define(1.1)LFT2(Ω;Rm):={ξ:Ω⟶Rm∣ξisFT-measurable,E|ξ|2<∞},LF2([0,T];Rm):={φ:[0,T]×Ω⟶Rm∣φisFt-adapted,E∫0T|φ(t)|2dt<∞},LF,p2([0,T];Rm):={φ̂:[0,T]×Ω⟶Rm∣φ̂isFt-predictable,E∫0T|φ̂(t)|2dt<∞},Fp2([0,T];Rm):={∫0Tφ̃:[0,T]×Ω×E⟶Rm∣φ̃isFt-predictable,suchthat,E∫0T∫E|φ̃(t,e)|2π(de)dt<∞}.
We denote(1.2)M2[0,T]:=LF2([0,T];Rn)×LF2([0,T];Rm)×LF,p2([0,T];Rm)×Fp2([0,T];Rm).
Clearly, ℳ2[0,T] is a Banach space. Any process in ℳ2[0,T] is denoted by Θ(·)≡(x(·),y(·),z(·),c(·,·)), whose norm is defined by(1.3)‖Θ(⋅)‖M2[0,T]2:=E[sup⁡t∈[0,T]|x(t)|2+sup⁡t∈[0,T]|y(t)|2+∫0T|z(t)|2dt+E∫0T∫E|c(t,e)|2π(de)dt].


## 1.2. Formulation of the Optimal Control Problem and Basic Assumptions

For anyu(·)∈𝒰[0,T] and a∈Rn, we consider the following fully coupled forward-backward stochastic control system:(1.4)dx(t)=b(t,x(t),y(t),z(t),c(t,⋅),u(t))dt+σ(t,x(t),y(t),z(t),c(t,⋅),u(t))dW(t)+∫Eg(t,x(t-),y(t-),z(t),c(t,⋅),u(t),e)Ñ(dedt),-dy(t)=f(t,x(t),y(t),z(t),c(t,⋅),u(t))dt-z(t)dW(t)-∫Ec(t,e)Ñ(dedt),x(0)=a,y(T)=h(x(T)),
with the cost functional given by(1.5)J(u(⋅))=E[∫0Tl(t,x(t),y(t),z(t),c(t,⋅),u(t))dt+ϕ(x(T))+γ(y(0))].
Here, Θ(·)≡(x(·),y(·),z(·),c(·,·)) takes value in 𝕄=Rn×Rm×Rm×Rm and(1.6)b,σ:[0,T]×M×U⟶Rn,g:[0,T]×M×U×E⟶Rn,f:[0,T]×M×U⟶Rm,h:Rn⟶Rm,ϕ:Rn⟶R,γ:Rm⟶R,l:[0,T]×M×U⟶R.For anyu(·)∈𝒰[0,T] and a∈Rn, we refer to Θ(·)≡(x(·),y(·),z(·),c(·,·)) as the state process corresponding to the admissible control u(·) if FBSDEJ (1.4) admits a unique adapted solution. For controlled FBSDEJ (1.4) with cost functional (1.5), we consider the following.Problem C. Find u¯(·)∈𝒰[0,T], such that
(1.7)J(u¯(⋅))=inf⁡u(⋅)∈U[0,T]J(u(⋅)).Anyu¯(·)∈𝒰[0,T] satisfying (1.7) is called an optimal control process of Problem C, and the corresponding state process, denoted by Θ¯(·)≡(x¯(·),y¯(·),z¯(·),c¯(·,·)), is called optimal state process. We also refer to (x¯(·),y¯(·),z¯(·),c¯(·,·),u¯(·)) as an optimal 5-tuple of Problem C.Our main goal in this paper is to derive some necessary conditions for the optimal control of Problem C, which is called the stochastic maximum principle of Pontraygin’s type. For this target, we first introduce the following basic assumption throughout this paper.(H0)For anyu(·)∈𝒰[0,T]anda∈Rn,FBSDEJ(1.4) admits a unique adapted solutionΘ(·)≡(x(·),y(·),z(·),c(·,·))∈ℳ2[0,T]. Moreover, the following estimate holds:(1.8)‖Θ(⋅)‖M2[0,T]≡E[sup⁡t∈[0,T]|x(t)|2+sup⁡t∈[0,T]|y(t)|2+∫0T|z(t)|2dt+E∫0T∫E|c(t,e)|2π(de)dt]≤CE[|a|2+|h(0)|2+(∫0T|b(t,0,u(t))|dt)2+(∫0T|f(t,0,u(t))|dt)2+∫0T|σ(t,0,u(t))|2dt+∫0T∫E|g(t,0,u(t),e)|2π(de)dt(∫0T|b(t,0,u(t))|dt)2].Further, ifΘ̃(·)≡(x̃(·),ỹ(·),z̃(·),c̃(·,·))∈ℳ2[0,T]  is the unique adapted solution of (1.4) withh(·)  andu(·)  replaced byh̃:Rn→Rm  andũ(·)∈𝒰[0,T], respectively, then the following stability estimate holds:(1.9)‖Θ(⋅)-Θ̃(⋅)‖M2[0,T]≡E[sup⁡t∈[0,T]|x(t)-x̃(t)|2+sup⁡t∈[0,T]|y(t)-ỹ(t)|2+∫0T|z(t)-z̃(t)|2dt+E∫0T∫E|c(t,e)-c̃(t,e)|2π(de)dt]≤CE[|h(x(T))-h(x̃(T))|2+(∫0T|b(t,Θ(t),u(t))-b(t,Θ̃(t),ũ(t))|dt)2+(∫0T|f(t,Θ(t),u(t))-f(t,Θ̃(t),ũ(t))|dt)2+∫0T|σ(t,Θ(t),u(t))-σ(t,Θ̃(t),ũ(t))|2dt+∫0T∫E|g(t,Θ(t),u(t),e)-g(t,Θ̃(t),ũ(t),e)|2π(de)dt(∫0T|b(t,Θ(t),u(t))-b(t,Θ̃(t),ũ(t))|dt)2].
By adopting the ideas from Wu [1], we know that under certain G-monotonicity conditions for the coefficients, the existence and uniqueness of FBSDEJ (1.4) guaranteed which leads to hypothesis (H0). Our main goal in this paper is to derive some necessary conditions for the optimal control of Problem C. Hence, we impose the well-posedness of the state equation (1.4) as an assumption to avoid some technicalities not closely to our main results.


## 1.3. Developments of Stochastic Optimal Control and Contributions of the Paper

It is well known that the optimal control problem is one of the central themes of modern control sciences. Necessary conditions for the optimal control of the (forward) continuous stochastic control system, that is, the so-called stochastic maximum principle of Pontraygin’s type, were extensively studied since early 1960s. When Brownian motion is the unique noise source, Peng [2] (see also Yong and Zhou [3]) obtained the maximum principle for the general case, that is, the control variable entering the diffusion coefficient and control domain being not convex.Forward-backward stochastic control systems where the controlled systems are described byforward-backward stochastic differential equations (FBSDEs) are widely used in mathematical economics and mathematical finance, which includes the usual forward SDEs as a special case. They are encountered in stochastic recursive utility optimization problems (see [4–8]) and principal-agent problems (see [9, 10]). Peng [11] first obtained the necessary conditions for optimal control for the partially coupling case when the control domain is convex. And then Xu [12] studied the nonconvex control domain case and obtained the corresponding necessary conditions. But he needs to assume that the diffusion coefficient in the forward control system does not contain the control variable. Ji and Zhou [8] applied the Ekeland variational principle to establish a maximum principle for a partially coupled forward-backward stochastic control system, while the forward state is constrained in a convex set at the terminal time. Wu [13] recently established a general maximum principle for optimal control problems derived by forward-backward stochastic systems, where control domains are nonconvex and forward diffusion coefficients explicitly depend on control variables. Moreover, some financial optimization problems for large investors (see [14–16]) and some asset pricing problems with forward-backward differential utility (see [7]) directly lead to fully coupled FBSDEs. Wu [17] first (see also Meng [18]) obtained the necessary conditions for optimal control of fully coupled forward-backward stochastic control systems when the control domain is convex. And then Shi and Wu [19] studied the nonconvex control domain case and obtained the corresponding necessary conditions under some G-monotonic assumptions. But they also (similar to Xu [12]) need to assume that the control variable does not appear in the diffusion coefficient of the forward equation. Very recently, Yong [20] completely solved the problem of finding necessary conditions for optimal control of fully coupled FBSDEs. He considered an optimal control problem for general coupled FBSDEs with mixed initial-terminal conditions and derived the necessary conditions for the optimal controls when the control domain is not assumed to be convex and the control variable appears in the diffusion coefficient of the forward equation.However, recently more and more research attentions are drawn towards the optimal control problem fordiscontinuous stochastic systems or stochastic systems with random jumps. The reason is clear for its applicable aspect. For example, there is compelling evidence that the dynamics of prices of financial instruments exhibit jumps that cannot be adequately captured solely by diffusion processes (i.e., processes satisfying some Itô-type stochastic differential equations (SDE for short), see Merton [21] and Cont and Tankov [22]. Several empirical studies demonstrate the existence of jumps in stock markets, the foreign exchange market, and bond markets. Jumps constitute also a key feature in the description of credit risk sensitive instruments. Therefore, models that incorporate jumps have become increasingly popular in finance and several areas of science and engineering, this leads to paying more attention to stochastic differential equations with jumps (SDEJs for short). As consequences, optimal control problems involving systems of SDEJs are widely studied. Situ [23] first obtained the maximum principle for (forward) stochastic control system with jumps but the jump coefficient does not contain the control variable. Tang and Li [24] discussed a more general case, where the control is allowed into both diffusion and jump coefficients, and the control domain is not necessarily convex; also some general state constraints are imposed. Maximum principles for forward-backward stochastic control systems with random jumps are studied in Øksendal and Sulem [25], Shi and Wu [26], where the FBSDEJs are partially coupled and the control domains are convex. Recently, Shi [27] obtained the necessary condition of optimal control as well as a sufficient condition of optimality under the assumption that the diffusion and jump coefficients do not contain the control variable and the control domain need not be convex. Necessary conditions for fully coupled forward-backward stochastic control systems with random jumps were studied in Shi and Wu [28] (see also Meng and Sun [29]) where the control domains are convex.In this paper, we consider the general optimal control problem for fully coupled FBSDEJ (1.4). Here, by the word “general” we mean the allowance of the control variable into both diffusion and jump coefficients of the forward equation and the control domain is not assumed to be convex. It is well worth mentioning that the idea of second-order spike variation technique, developed by Peng [2], plays an important role in deriving the necessary conditions of general stochastic optimal control of jump-diffusion process in Tang and Li [24]. Following the standard idea of deriving necessary conditions for optimal control processes, due to the fact the control domain U is not assumed to be convex, one needs to use spike variation for the control process and then to try obtaining a Taylor-type expansion for the state process (x(·),y(·),z(·),c(·,·)) and the cost functional (1.5) with respect to the spike variation of the control process, followed by some suitable duality relations to get a maximum principle of Pontryagin-type. However, the derivation of Taylor expansion of the state process (x(·),y(·),z(·),c(·,·)) with respect to the spike variation of the control process is technically difficult. The main reasons are that both (z(·),c(·,·)) and u(·) appear in the diffusion and jump coefficients of the forward equation of (1.4) and that the regularity/integrability of the continuous martingale process z(·) and the discontinuous martingale process c(·,·) (as part of the state process) seems to be not enough in the case when a second-order expansion is necessary. Note that in [25–29], due to the special structure of the problems, the second-order expansion is not necessary when one tries to derive the necessary conditions for optimal controls.We overcome the above difficulty by the newly developedreduction method by Wu [13] and Yong [20] in the continuous case, independently. In fact, some ideas have been proposed early in Kohlmann and Zhou [30] and Ma and Yong [31]. Let us make it more precise. We first introduce the controlled initial value problem for a system of SDEJs, where (x(·),y(·)) is regarded as the state process and (z(·),c(·,·),u(·)) is regarded as the control process. Next, we regard the original terminal condition y(T)=h(x(T)) as terminal state constraint and then we translate Problem C into a high-dimensional reduced optimal control problem described by standard SDEJ with state constraint (see Problem Ĉ in Section 3). The advantage of this reduced optimal control problem is that one needs not much regularity/integrability of processes (z(·),c(·,·)) since it is treated as a control process. We apply the Ekeland variational principle to deal with this high-dimensional reduced optimal control problem with state constraint. Finally, necessary conditions for the optimal control of Problem C are derived by the equivalence of Problems C and Ĉ.The rest of this paper is organized as follows. In Section2, under some suitable assumptions, we give the main result of this paper, together with some discussions on special cases. In Section 3, after we make the reduction of our optimal control problems for FBSDEJs Problem C, a proof of our main result will be presented. Section 4 is devoted to a linear quadratic stochastic optimal control problem as an illustrating example. Finally, in Section 5 we give the concluding remarks and compare our theorem with some existing results.


## 2. The Main Result and Some Special Cases

In this section, under some suitable assumptions, we will state the necessary conditions for the optimal control of our Problem C. Also, some interesting special cases will be discussed.Let us introduce the following further assumptions beyond (H0).(H1)
For any(θ,u)∈𝕄×U, mapst↦(b(t,θ,u),σ(t,θ,u),g(t,θ,u,e),f(t,θ,u)) is ℱt-progressively  measurable and there exists a constantL>0such that(2.1)|b(t,x,y,z,c,u)-b(t,x̃,ỹ,z,c,u)|2+|σ(t,x,y,z,c,u)-σ(t,x̃,ỹ,z,c,u)|2+‖g(t,x,y,z,c,u,e)-g(t,x̃,ỹ,z,c,u,e)‖L22+|f(t,x,y,z,c,u)-f(t,x̃,ỹ,z,c,u)|2≤L(|x-x̃|2+|y-ỹ|2),∀(t,z,c,u)∈[0,T]×Rm×Rn×U,(x,y),(x̃,ỹ)∈Rn×Rm,a.s.,|b(t,x,y,z,c,u)|2+|σ(t,x,y,z,c,u)|2+‖g(t,x,y,z,c,u,e)‖L22+|f(t,x,y,z,c,u)|2≤L(1+|x|2+|y|2),∀(t,z,c,u)∈[0,T]×Rm×Rn×U,(x,y)∈Rn×Rm,a.s.(H2)
Mapsθ↦(b(t,θ,u),σ(t,θ,u),g(t,θ,u,e),f(t,θ,u))is twice continuously differential, with the (partial) derivatives up to the second order being uniformly bounded, Lipschitz continuous inθ∈𝕄, and continuous inu∈U.(H3)
The maphis twice differentiable with the derivatives up to the second order being uniformly bounded and uniformly Lipschitz continuous; the mapsϕ,γare twice differentiable with the derivatives up to the second order being Lipschitz continuous; the mapθ↦l(t,θ,u)is twice differentiable with the derivatives up to the second order being Lipschitz continuous inθ∈𝕄, and continuous inu∈U.Next, to simplify our presentation, we now introduce some abbreviation notations. First, we make the following convention: for any differentiable mapa≡(a1,a2,…,al)⊤:Rn→Rl, let(2.2)ax=(ax11ax21⋯axn1ax12ax22⋯axn2⋮⋮⋱⋮ax1lax2l⋯axnl):Rn⟶Rl×n.
In particular, for l=1,ax=(ax1,ax2,…,axn)∈R1×n is an n-dimensional row vector. Also, for any twice differentiable function a:Rn→R, the Hessian matrix is given by(2.3)axx≡(ax⊤)x=(ax1x1ax1x2⋯ax1xnax2x1ax2x2⋯ax2xn⋮⋮⋱⋮axnx1axnx2⋯axnxn):Rn⟶Sn,
hereafter, 𝒮n stands for the set of all real n×n symmetric matrices. On the other hand, for a twice differentiable function a:Rn×Rm→R (denoting (x,y)∈Rn×Rm), we denote(2.4)axy≡(ax⊤)y=(ax1y1ax1y2⋯ax1ymax2y1ax2y2⋯ax2ym⋮⋮⋱⋮axny1axny2⋯axnym):Rn×Rm⟶Rn×m.Now, let(x¯(·),y¯(·),z¯(·),c¯(·,·),u¯(·)) be an optimal 5-tuple of Problem C. For a(·)=b(·),σ(·),f(·),l(·), we denote(2.5)a¯x(t,⋅)=ax(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t)),a¯y(t,⋅)=ay(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t)),a¯z(t,⋅)=az(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t)),a¯c(t,⋅)=ac(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t)),
and let(2.6)g¯x(t,e)=gx(t,x¯(t),y¯(t),z¯(t),c¯(t,e),u¯(t),e),g¯y(t,e)=gy(t,x¯(t),y¯(t),z¯(t),c¯(t,e),u¯(t),e),g¯z(t,e)=gz(t,x¯(t),y¯(t),z¯(t),c¯(t,e),u¯(t),e),g¯c(t,e)=gc(t,x¯(t),y¯(t),z¯(t),c¯(t,e),u¯(t),e).
We also introduce(2.7)B¯X(t,⋅)=(b¯x(t,⋅)b¯y(t,⋅)-f¯x(t,⋅)-f¯y(t,⋅)),Σ¯X(t,⋅)=(σ¯x(t,⋅)σ¯y(t,⋅)00),Υ¯X(t,e)=(g¯x(t,e)g¯y(t,e)00).
For a(·)=bi(·),σi(·),fj(·),l(·),1≤i≤n,1≤j≤m, let(2.8)a¯xx(t,⋅)=axx(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t)),a¯yy(t,⋅)=ayy(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t)),a¯xy(t,⋅)=axy(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t)),a¯yx(t,⋅)=ayx(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t)),
and let(2.9)g¯xx(t,e)=gxx(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t),e),g¯yy(t,e)=gyy(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t),e),g¯xy(t,e)=gxy(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t),e),g¯yx(t,e)=gyx(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t),e).Our main result in this paper is the following theorem for Problem C.Theorem 2.1.
Suppose that  (H0)–(H3) hold, and let(x¯(·),y¯(·),z¯(·),c¯(·,·),u¯(·)) be an optimal 5-tuple of  Problem C. Then there exists a unique adapted solution  (p(·),q(·),k(·),r(·,·))∈Lℱ2([0,T];Rn)×Lℱ2([0,T];Rm)×Lℱ,p2([0,T];Rn)×Fp2([0,T];Rn) to the following FBSDEJ:
(2.10)-dp(t)=[b¯x(t,⋅)⊤p(t)-f¯x(t,⋅)⊤q(t)+σ¯x(t,⋅)⊤k(t)+∫Eg¯x(t,e)⊤r(t,e)π(de)-l¯x(t,⋅)]dt-k(t)dW(t)-∫Er(t,e)Ñ(dedt),dq(t)=[-b¯y(t,⋅)⊤p(t)+f¯y(t,⋅)⊤q(t)-σ¯y(t,⋅)⊤k(t)-∫Eg¯y(t,e)⊤r(t,e)π(de)+l¯y(t,⋅)]dt+[∫Eg¯z-b¯z(t,⋅)⊤p(t)+f¯z(t,⋅)⊤q(t)-σ¯z(t,⋅)⊤k(t)-∫Eg¯z(t,e)⊤r(t,e)π(de)+l¯z(t,⋅)]dW(t)+∫E[-b¯c(t,e)⊤p(t-)+f¯c(t,e)⊤q(t-)-σ¯c(t,e)⊤k(t)-g¯c(t,e)⊤r(t,e)+l¯c(t,e)]Ñ(dedt),p(T)=-ϕx(x¯(T))-hx(x¯(T))⊤q(T),q(0)=Eγy(y¯0).
Let (P(·),Q(·),K(·,·))∈Lℱ2([0,T];𝒮n+m)×Lℱ,p2([0,T];𝒮n+m)×Fp2([0,T];𝒮n+m) be the unique adapted solution to the following matrix-valued BSDEJ:
(2.11)-dP(t)={[(t,e)Υ¯X]B¯X(t,⋅)⊤P(t)+P(t)B¯X(t,⋅)+ΣX¯(t,⋅)⊤P(t)ΣX¯(t,⋅)+ΣX¯(t,⋅)⊤Q(t)+Q(t)ΣX¯(t,⋅)+∫E[Υ¯X(t,e)⊤P(t)Υ¯X(t,e)+Υ¯X(t,e)⊤K(t,e)Υ¯X(t,e)+Υ¯X(t,e)⊤K(t,e)+K(t,e)Υ¯X(t,e)]π(de)+H¯XX(t,⋅)ΣX¯}dt-Q(t)dW(t)-∫EK(t,e)Ñ(dedt),P(T)=(-ϕxx(x¯(T))-hxx(x¯(T))⊤q(T)000),
where H¯XX(t,·)=H¯XX(t,x¯(t),y¯(t),z¯(t),c¯(t,·),u¯(t),p(t),q(t),k(t),r(t,·)), and the Hamiltonian function H¯:[0,T]×𝕄×U×Rn×Rm×Rn×Rn→R is defined as follows:
(2.12)H¯(t,x,y,z,c(⋅),u,p,q,k,r(⋅)):=〈p,b(t,x,y,z,c(⋅),u)〉+〈q,-f(t,x,y,z,c(⋅),u)〉+〈k,σ(t,x,y,z,c(⋅),u)〉+∫E〈r(e),g(t,x,y,z,c(e),u,e)〉π(de)-l(t,x,y,z,c(⋅),u).
Then,
(2.13)H¯(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t),p(t),q(t),k(t),r(t,⋅))-H¯(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u,p(t),q(t),k(t),r(t,⋅))-12(σ(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t))-σ(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u))⊤P1(t)×(σ(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t))-σ(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u))-12∫E(g(t,x¯(t),y¯(t),z¯(t),c¯(t,e),u¯(t),e)-g(t,x¯(t),y¯(t),z¯(t),c¯(t,g),u,e))⊤×[P1(t)+K1(t,e)]×(g(t,x¯(t),y¯(t),z¯(t),c¯(t,e),u¯(t),e)-g(t,x¯(t),y¯(t),z¯(t),c¯(t,e),u,e))π(de)≥0,∀u∈U,a.e.,a.s.,
with P(·)≡(P1(·)P2(·)P2(·)⊤P3(·)), and K(·,·)≡(K1(·,·)K2(·,·)K2(·,·)⊤K3(·,·)).Remark 2.2.
In fact, the second-order adjoint equation (2.11) can be split into the following three BSDEJs if we add that Q(·)≡(Q1(·)Q2(·)Q2(·)⊤Q3(·)):
(2.14)-dP1(t)={∫Eb¯x(t,⋅)⊤P1(t)+P1(t)b¯x(t,⋅)+σ¯x(t,⋅)⊤P1(t)σ¯x(t,⋅)+σ¯x(t,⋅)⊤Q1(t)+Q1(t)σ¯x(t,⋅)-f¯x(t,⋅)⊤P2(t)-P2(t)f¯x(t,⋅)+∫E[g¯x(t,e)⊤P1(t)g¯x(t,e)+g¯x(t,e)⊤K1(t,e)g¯x(t,e)+g¯x(t,e)⊤K1(t,e)+K1(t,e)g¯x(t,e)]π(de)+H¯xx(t,⋅)∫E}dt-Q1(t)dW(t)-∫EK1(t,e)Ñ(dedt),P1(T)=-ϕxx(x¯(T))-hxx(x¯(T))⊤q(T),(2.15)-dP2(t)={∫EP1(t)b¯y(t,⋅)+b¯x(t,⋅)P2(t)+σ¯x(t,⋅)⊤P1(t)σ¯y(t,⋅)⊤+σ¯x(t,⋅)Q2(t)+Q1(t)⊤σ¯y(t,⋅)-f¯x(t,⋅)P3(t)-P2(t)⊤f¯y(t,⋅)+∫E[g¯x(t,e)⊤P1(t)g¯y(t,e)⊤+g¯x(t,e)⊤K1(t,e)g¯y(t,e)⊤+g¯x(t,e)K2(t,e)+K1(t,e)g¯y(t,e)⊤]π(de)+H¯xy(t,⋅)∫E}dt-Q2(t)dW(t)-∫EK2(t,e)Ñ(dedt),P2(T)=0,(2.16)-dP3(t)={∫Eb¯y(t,⋅)⊤P2(t)+P2(t)b¯y(t,⋅)+σ¯y(t,⋅)P1(t)σ¯y(t,⋅)⊤+σ¯y(t,⋅)⊤Q2(t)+Q2(t)σ¯y(t,⋅)-f¯y(t,⋅)⊤P3(t)-P3(t)f¯y(t,⋅)+∫E[g¯y(t,e)P1(t)g¯y(t,e)⊤+g¯y(t,e)K1(t,e)g¯y(t,e)⊤+g¯y(t,e)⊤K2(t,e)+K2(t,e)g¯y(t,e)]π(de)+H¯yy(t,⋅)∫E}dt-Q3(t)dW(t)-∫EK3(t,e)Ñ(dedt),P3(T)=0.Note that this kind of three second-order adjoint equations appears in Wu [13] whereas not in Yong [20].Let us now look at some special cases. It can be seen that our theorem recovers the known results.(1) A Classical Stochastic Optimal Control Problem with Random Jumps
Consider a controlled SDEJ:(2.17)dx(t)=b(t,x(t),u(t))dt+σ(t,x(t),u(t))dW(t)+∫Eg(t,x(t-),u(t),e)Ñ(dedt),x(0)=a,
with the cost functional
(2.18)J(u(⋅))=E[∫0Tl(t,x(t),u(t))dt+ϕ(x(T))].
In this case,(2.19)b(t,x,y,z,c(⋅),u)=b(t,x,u),σ(t,x,y,z,c(⋅),u)=σ(t,x,u),g(t,x,y,z,c(⋅),u,⋅)=g(t,x,u,⋅),f(t,x,y,z,c(⋅),u)=0,l(t,x,y,z,c(⋅),u,⋅)=l(t,x,u),γ(y)=0.
Hence, we have, by some direct computation/observation,
(2.20)y(⋅)=z(⋅)=c(⋅,⋅)=0,q(⋅)=0,P2(⋅)=Q2(⋅)=K2(⋅,⋅)=0,P3(⋅)=Q3(⋅)=K3(⋅,⋅)=0.
Consequently, (2.10) and (2.11) (or equivalently, (2.14)) become
(2.21)-dp(t)=[b¯x(t,⋅)⊤p(t)+σ¯x(t,⋅)⊤k(t)+∫Eg¯x(t,e)⊤r(t,e)π(de)-l¯x(t,⋅)]dt-k(t)dW(t)-∫Er(t,e)Ñ(dedt),p(T)=ϕx(x¯(T)),(2.22)-dP1(t)={∫Eb¯x(t,⋅)⊤P1(t)+P1(t)b¯x(t,⋅)+σ¯x(t,⋅)⊤P1(t)σ¯x(t,⋅)+σ¯x(t,⋅)⊤Q1(t)+Q1(t)σ¯x(t,⋅)+∫E[g¯x(t,e)⊤P1(t)g¯x(t,e)+g¯x(t,e)⊤K1(t,e)g¯x(t,e)+g¯x(t,e)⊤K1(t,e)+K1(t,e)g¯x(t,e)]π(de)+H¯xx(t,⋅)∫E}dt-Q1(t)dW(t)-∫EK1(t,e)Ñ(dedt),P1(T)=-ϕxx(x¯(T)),
where the Hamiltonian function H¯:[0,T]×Rn×U×Rn×Rn×Rn→R is defined as
(2.23)H¯(t,x,u,p,k,r(⋅)):=〈p,b(t,x,u)〉+〈k,σ(t,x,u)〉+∫E〈r(e),g(t,x,u,e)〉π(de)-l(t,x,u),a.s.
The maximum condition (2.13) is reduced to the following:
(2.24)H¯(t,x¯(t),u¯(t),p(t),k(t),r(t,⋅))-H¯(t,x¯(t),u,p(t),k(t),r(t,⋅))-12(σ(t,x¯(t),u¯(t))-σ(t,x¯(t),u))⊤P1(t)(σ(t,x¯(t),u¯(t))-σ(t,x¯(t),u))-12∫E(g(t,x¯(t),u¯(t),e)-g(t,x¯(t),u,e))⊤[P1(t)+K1(t,e)]×(g(t,x¯(t),u¯(t),e)-g(t,x¯(t),u,e))π(de)≥0,∀u∈U,a.e.,a.s.
These are the necessary conditions for the stochastic optimal control problem with random jumps (see (2.37) of Tang and Li [24]). When the jump coefficient g is independent of u(·), our result is reduced to Situ [23]. Where there are no random jumps, our result recovers the well-known maximum principle for the classical stochastic optimal control problem of Peng [2] and Yong and Zhou [3]. And when U is convex, the classical result of Bensoussan [32] is recovered.(2) A Stochastic Optimal Control Problem for BSDEJs with Random Jumps
Consider a controlled BSDEJ:(2.25)-dy(t)=f(t,y(t),z(t),c(t,⋅),u(t))dt-z(t)dW(t)-∫Ec(t,e)Ñ(dedt),y(T)=hT,
with the cost functional
(2.26)J(u(⋅))=E[∫0Tl(t,y(t),z(t),c(t,⋅),u(t))dt+γ(y(0))].
In this case,
(2.27)b(t,x,y,z,c(⋅),u)=σ(t,x,y,z,c(⋅),u)=g(t,x,y,z,c(⋅),u,⋅)=0,f(t,x,y,z,c(⋅),u)=f(t,y,z,c(⋅),u),l(t,x,y,z,c(⋅),u)=g(t,y,z,c(⋅),u,⋅),ϕ(x)=0.
Hence, we have
(2.28)x(⋅)=0,p(⋅)=k(⋅)=r(⋅,⋅)=0,P1(⋅)=Q1(⋅)=K1(⋅,⋅)=0,P2(⋅)=Q2(⋅)=K2(⋅,⋅)=0.
Consequently, (2.10) and (2.11) (or equivalently, (2.16)) become
(2.29)dq(t)=[f¯y(t,⋅)⊤q(t)+l¯y(t,⋅)]dt-[f¯z(t,⋅)⊤q(t)+l¯z(t,⋅)]dW(t)-∫E[f¯c(t,e)⊤q(t)+l¯c(t,e)]Ñ(dedt),q(0)=Eγy(y¯0);-dP3(t)={[g¯y(t,e)⊤]-f¯y(t,⋅)⊤P3(t)-P3(t)f¯y(t,⋅)+∫E[g¯y(t,e)P1(t)g¯y(t,e)⊤+g¯y(t,e)K1(t,e)g¯y(t,e)⊤+g¯y(t,e)⊤K2(t,e)+K2(t,e)g¯y(t,e)]π(de)-q(t)⊤f¯yy(t,⋅)-l¯yy(t,⋅)∫E}dt-Q3(t)dW(t)-∫EK3(t,e)Ñ(dedt),P3(T)=0.
The maximum condition (2.13) is reduced to the following:
(2.30)〈q(t),f(t,y¯(t),z¯(t),c¯(t,⋅),u¯(t))-f(t,y¯(t),z¯(t),c¯(t,⋅),u)〉+l(t,y¯(t),z¯(t),c¯(t,⋅),u¯(t))-l(t,y¯(t),z¯(t),c¯(t,⋅),u)≤0,∀u∈U,a.e.,a.s.
As we know, this new result has not been published elsewhere. When there are no random jumps, our result partially recovers that of Dokuchaev and Zhou [33].(3) A Stochastic Optimal Control Problem for FBSDEJs with Random Jumps
Consider a controlled FBSDEJ:(2.31)dx(t)=b(t,x(t),u(t))dt+σ(t,x(t),u(t))dW(t)+∫Eg(t,x(t-),u(t),e)Ñ(dedt),-dy(t)=f(t,x(t),y(t),z(t),c(t,⋅),u(t))dt-z(t)dW(t)-∫Ec(t,e)Ñ(dedt),x(0)=a,y(T)=h(x(T)),
with the cost functional
(2.32)J(u(⋅))=E[∫0Tl(t,x(t),y(t),z(t),c(t,⋅),u(t))dt+ϕ(x(T))+γ(y(0))].
In this case, we have(2.33)b(t,x,y,z,c(⋅),u)=b(t,x,u),σ(t,x,y,z,c(⋅),u)=σ(t,x,u),g(t,x,y,z,c(⋅),u,⋅)=g(t,x,u,⋅).
Consequently, (2.10) becomes
(2.34)-dp(t)=[∫Eg¯xb¯x(t,⋅)⊤p(t)-f¯x(t,⋅)⊤q(t)+σ¯x(t,⋅)⊤k(t)+∫Eg¯x(t,e)⊤r(t,e)π(de)-l¯x(t,⋅)]dt-k(t)dW(t)-∫Er(t,e)Ñ(dedt),dq(t)=[f¯y(t,⋅)⊤q(t)+l¯y(t,⋅)]dt-[f¯z(t,⋅)⊤q(t)+l¯z(t,⋅)]dW(t)-∫E[f¯c(t,e)⊤q(t)+l¯c(t,e)]Ñ(dedt),p(T)=ϕx(x¯(T))+hx(x¯(T))⊤q(T),q(0)=Eγy(y¯0),
and (2.14), (2.15), (2.16) become
(2.35)-dP1(t)={b¯x(t,⋅)⊤P1(t)+P1(t)b¯x(t,⋅)+σ¯x(t,⋅)⊤P1(t)σ¯x(t,⋅)+σ¯x(t,⋅)⊤Q1(t)+Q1(t)σ¯x(t,⋅)-f¯x(t,⋅)⊤P2(t)-P2(t)f¯x(t,⋅)+∫E[g¯x(t,e)⊤P1(t)g¯x(t,e)+g¯x(t,e)⊤K1(t,e)g¯x(t,e)+g¯x(t,e)⊤K1(t,e)+K1(t,e)g¯x(t,e)]π(de)+H¯xx(t,⋅)}dt-Q1(t)dW(t)-∫EK1(t,e)Ñ(dedt),P1(T)=-ϕxx(x¯(T))-hxx(x¯(T))⊤q(T),-dP2(t)={b¯x(t,⋅)P2(t)+σ¯x(t,⋅)Q2(t)-f¯x(t,⋅)P3(t)-P2(t)⊤f¯y(t,⋅)+∫E[g¯x(t,e)⊤P1(t)g¯y(t,e)⊤+g¯x(t,e)⊤K1(t,e)g¯y(t,e)⊤+g¯x(t,e)K2(t,e)+K1(t,e)g¯y(t,e)⊤]π(de)+H¯xy(t,⋅)}dt-Q2(t)dW(t)-∫EK2(t,e)Ñ(dedt),P2(T)=0,-dP3(t)={-f¯y(t,⋅)⊤P3(t)-P3(t)f¯y(t,⋅)-q(t)⊤f¯yy(t,⋅)-l¯yy(t,⋅)}dt-Q3(t)dW(t)-∫EK3(t,e)Ñ(dedt),P3(T)=0,
where the Hamiltonian function H¯:[0,T]×𝕄×U×Rn×Rm×Rn×Rn→R is defined as follows:
(2.36)H¯(t,x,y,z,c(⋅),u,p,q,k,r(⋅)):=〈p,b(t,x,u)〉+〈q,-f(t,x,y,z,c(⋅),u)〉+〈k,σ(t,x,u)〉+∫E〈r(e),g(t,x,u,e)〉π(de)-l(t,x,y,z,c(⋅),u).
The maximum condition (2.13) remains the same. When U is convex, we essentially recover the results in Shi and Wu [26] and Øksendal and Sulem [25] (partial information case), and when σ,g are independent of u(·) and U is not assumed to be convex, our result is reduced to that of Shi [27]. Note that in both of these cases, our second adjoint equations are new.
When there are no random jumps, our result partially recovers that of Wu [13]. Note that in (3.27) of [13], some additional parameters have to be involved and to be determined. And when U is convex, our result is reduced to that of Peng [11] and Wang and Wu [34] (partial observation case). Result of Xu [12] is recovered when σ is independent of u(·) and U is not assumed to be convex.(4) A Stochastic Optimal Control Problem for Fully Coupled FBSDEJs with Random Jumps
Consider the controlled fully coupled FBSDEJ (1.4) with the cost functional (1.5). When U is convex, we essentially recover the results in Shi and Wu [28] and Meng and Sun [29] (partial information case). When there are no random jumps, our result becomes a special case of Yong [20], because in [20] the author considered the mixed initial-terminal conditions, and some additional necessary conditions for the optimal control are derived. When σ is independent of u(·), our result recovers those of Shi and Wu [19, 35] (partial observation case). And when U is convex, our result is reduced to that of Wu [17].


## 3. Problem Reduction and the Proof of the Main Theorem

This section is devoted to the proof of our main theorem. The proof is lengthy and technical. Therefore, we divide it into several steps to make the idea clear.Step 1 (problem reduction).
Consider the following initial value problem for a control system of SDEJs:(3.1)dx(t)=b(t,x(t),y(t),z(t),c(t,⋅),u(t))dt+σ(t,x(t),y(t),z(t),c(t,⋅),u(t))dW(t)+∫Eg(t,x(t-),y(t-),z(t),c(t,⋅),u(t),e)Ñ(dedt),dy(t)=-f(t,x(t),y(t),z(t),c(t,⋅),u(t))dt+z(t)dW(t)+∫Ec(t,e)Ñ(dedt),x(0)=a,y(0)=y0,
where (x(·),y(·)) is regarded as the state process and (z(·),c(·,·),u(·)) as the control process.
It is standard that (see [36]), for any (z(·),c(·,·),u(·))∈Lℱ,p2([0,T];Rm)×Fp2([0,T];Rm)×𝒰[0,T] and y0∈Rm, there exists a unique strong solution
(3.2)(x(⋅),y(⋅))≡(x(⋅,y0,z(⋅),c(⋅,⋅),u(⋅)),y(⋅,y0,z(⋅),c(⋅,⋅),u(⋅)))∈LF2([0,T];Rn)×LF2([0,T];Rm)
to (3.1), depending on the 4-tuple (y0,z(·),c(·,·),u(·)). Next, we regard the original terminal condition as the terminal state constraint:
(3.3)y(T)=h(x(T)).Let𝒜 be the set of all 4-tuples (y0,z(·),c(·,·),u(·))∈Rm×Lℱ,p2([0,T];Rm)×Fp2([0,T];Rm)×𝒰[0,T] such that the unique corresponding state process (x(·),y(·)) satisfies the constraint (3.3). Note that hypothesis (H0) implies that, for any u(·)∈𝒰[0,T], there exists a unique 4-tuple (y0,z(·),c(·,·))∈Rm×Lℱ,p2([0,T];Rm)×Fp2([0,T];Rm) such that state equation (3.1) admits a unique state process (x(·),y(·))∈Lℱ2([0,T];Rn)×Lℱ2([0,T];Rm) satisfying the state constraint (3.3). Hence, (H0) and (H1) imply 𝒜≠∅. We rewrite the cost functional (1.5) as follows:(3.4)J(y0,z(⋅),c(⋅,⋅),u(⋅))=E[∫0Tl(t,x(t),y(t),z(t),c(t,⋅),u(t))dt+ϕ(x(T))+γ(y0)].Now, we can pose the following.ProblemĈ. Find (y¯0,z¯(·),c¯(·,·),u¯(·))∈𝒜, such that(3.5)J(y¯0,z¯(⋅),c¯(⋅,⋅),u¯(⋅))=inf⁡(y0,z(⋅),c(⋅,⋅),u(⋅))∈AJ(y0,z(⋅),c(⋅,⋅),u(⋅)).We, respectively, refer to(z¯(·),c¯(·,·),u¯(·)) as an optimal control process, to (x¯(·),y¯(·)) as the corresponding optimal state process, and to (y¯0,z¯(·),c¯(·,·),u¯(·)) as an optimal 4-tuple of Problem Ĉ.It is clear that Problems C andĈ are equivalent. The advantage of Problem Ĉ is that one does not need much regularity/integrability on (z(·),c(·,·)) since it is treated as part of a control process; the disadvantage is that one has to treat terminal constraint (3.3).Step 2 (applying Ekeland’s variational principle).
For convenience, let us cite Ekeland’s variational principle, whose proof can be found in Ekeland [37] or Yong and Zhou [3].Lemma 3.1.
Let(V,d(·,·)) be a complete metric space, and let  f(·):V→R be lower-semicontinuous, bounded below. If for all  ρ>0 there exists  u∈V satisfying
(3.6)f(u)≤inf⁡v∈Vf(v)+ρ,
then there exists uρ∈V, satisfying the following:(i)
f(uρ)≤f(u),(ii)
d(u,uρ)≤ρ,(iii)
forallv∈V,f(v)+ρd(v,uρ)≥f(uρ).Let(y¯0,z¯(·),c¯(·,·),u¯(·)) be an optimal 4-tuple of Problem Ĉ, with the corresponding optimal state process (x¯(·),y¯(·)). For any δ>0, we define(3.7)Jδ(y0,z(⋅),c(⋅,⋅),u(⋅))≐{[J(y0,z(⋅),c(⋅,⋅),u(⋅))-J(y¯0,z¯(⋅),c¯(⋅,⋅),u¯(⋅))+δ]2+E|y(T)-h(x(T))|2}1/2∀(y0,z(⋅),c(⋅,⋅),u(⋅))∈Rm×LF,p2([0,T];Rm)×Fp2([0,T];Rm)×U[0,T].If we define the Ekeland’s distance on𝒰[0,T] by(3.8)dE(u(⋅),ũ(⋅))≐∫Ω|{t∈[0,T],u(t)≠ũ(t)}|dP,∀u(⋅),ũ(⋅)∈U[0,T],with |A| being the Lebesgue measure of set A⊆[0,T], then Lℱ,p2([0,T];Rm)×Fp2([0,T];Rm)×𝒰[0,T] is a complete metric space under the following metric:(3.9)[‖z(⋅)-z̃(⋅)‖2+‖c(⋅,⋅)-c̃(⋅,⋅)‖π2+dE(u(⋅),ũ(⋅))2]1/2,∀(z(⋅),c(⋅,⋅),u(⋅)),(z̃(⋅),c̃(⋅,⋅),ũ(⋅))∈LF,p2([0,T];Rm)×Fp2([0,T];Rm)×U[0,T],where ∥z(·)∥2≐𝔼∫0T|z(t)|2dt, ∥c(·,·)∥π2≐𝔼∫0T∫E|c(t,e)|2π(de)dt, and (y0,z(·),c(·,·),u(·))↦Jδ(y0,z(·),c(·,·),u(·)) is continuous. Also it is clear that(3.10)Jδ(y0,z(⋅),c(⋅,⋅),u(⋅))>0,∀(y0,z(⋅),c(⋅,⋅),u(⋅))∈Rm×LF,p2([0,T];Rm)×Fp2([0,T];Rm)×U[0,T],Jδ(y¯0,z¯(⋅),c¯(⋅,⋅),u¯(⋅))=δ≤inf⁡(y0,z(⋅),c(⋅,⋅),u(⋅))∈Rm×LF,p2([0,T];Rm)×Fp2([0,T];Rm)×U[0,T]Jδ(y0,z(⋅),c(⋅,⋅),u(⋅))+δ.
Hence, by Lemma 3.1, there exists a 4-tuple (y0δ,zδ(·),cδ(·,·),uδ(·))∈Rm×Lℱ,p2([0,T];Rm)×Fp2([0,T];Rm)×𝒰[0,T] such that(3.11)(i)Jδ(y0δ,zδ(⋅),cδ(⋅,⋅),uδ(⋅))≤Jδ(y¯0,z¯(⋅),c¯(⋅,⋅),u¯(⋅))=δ,(ii)‖zδ(⋅)-z¯(⋅)‖2+‖cδ(⋅,⋅)-c¯(⋅,⋅)‖π2+dE(uδ(⋅),u¯(⋅))2≤δ,(iii)-δ[‖zδ(⋅)-z¯(⋅)‖2+‖cδ(⋅,⋅)-c¯(⋅,⋅)‖π2+dE(uδ(⋅),u¯(⋅))2]1/2≤Jδ(y0,z(⋅),c(⋅,⋅),u(⋅))-Jδ(y0δ,zδ(⋅),cδ(⋅,⋅),uδ(⋅)),∀(y0,z(⋅),c(⋅,⋅),u(⋅))∈Rm×LF,p2([0,T];Rm)×Fp2([0,T];Rm)×U[0,T].
Thus, (y0δ,zδ(·),cδ(·,·),uδ(·)) is a global minimum point of the following penalized cost functional(3.12)Jδ(y0,z(⋅),c(⋅,⋅),u(⋅))+δ[‖zδ(⋅)-z¯(⋅)‖2+‖cδ(⋅,⋅)-c¯(⋅,⋅)‖π2+dE(uδ(⋅),u¯(⋅))2]1/2.
In other words, if we pose a penalized optimal control problem with the state constraint (3.3) and the cost functional (3.12), then (y0δ,zδ(·),cδ(·,·),uδ(·)) is an optimal 4-tuple of the problem. Note that this problem does not have state constraints, and the optimal 4-tuple (y0δ,zδ(·),cδ(·,·),uδ(·)) approaches (y¯0,z¯(·),c¯(·,·),u¯(·)) as δ→0.Step 3 (nontriviality of the multiplier).
Note that the state process(xδ(·),yδ(·)) corresponding to the 4-tuple (y0δ,zδ(·),cδ(·,·),uδ(·)) satisfies the following SDEJ:
(3.13)dxδ(t)=b(t,xδ(t),yδ(t),zδ(t),cδ(t,⋅),uδ(t))dt+σ(t,xδ(t),yδ(t),zδ(t),cδ(t,⋅),uδ(t))dW(t)+∫Eg(t,xδ(t-),yδ(t-),zδ(t),cδ(t,⋅),uδ(t),e)Ñ(dedt),dyδ(t)=-f(t,xδ(t),yδ(t),zδ(t),cδ(t,⋅),uδ(t))dt+zδ(t)dW(t)+∫Ecδ(t,e)Ñ(dedt),xδ(0)=a,yδ(0)=y0δ,
with
(3.14)E|yδ(T)-h(xδ(T))|2≤Jδ(y0δ,zδ(⋅),cδ(⋅,⋅),uδ(⋅))2≤Jδ(y¯0,z¯(⋅),c¯(⋅,⋅),u¯(⋅))2=δ2⟶0,asδ⟶0.We now regardΘδ(·)≡(xδ(·),yδ(·),zδ(·),cδ(·,·)) as the unique solution to the following FBSDEJ:(3.15)dxδ(t)=b(t,xδ(t),yδ(t),zδ(t),cδ(t,⋅),uδ(t))dt+σ(t,xδ(t),yδ(t),zδ(t),cδ(t,⋅),uδ(t))dW(t)+∫Eg(t,xδ(t-),yδ(t-),zδ(t),cδ(t,⋅),uδ(t),e)Ñ(dedt),-dyδ(t)=f(t,xδ(t),yδ(t),zδ(t),cδ(t,⋅),uδ(t))dt-zδ(t)dW(t)-∫Ecδ(t,e)Ñ(dedt),xδ(0)=a,yδ(T)=h(yδ(T))+hTδ,
with(3.16)hTδ≡yδ(T)-h(xδ(T)).Next, for anyηT∈LℱT2(Ω;Rm), with(3.17)E|ηT|2≤1,
by (H0), the following FBSDEJ(3.18)dxδ,ɛ(t)=b(t,xδ,ɛ(t),yδ,ɛ(t),zδ,ɛ(t),cδ,ɛ(t,⋅),uδ(t))dt+σ(t,xδ,ɛ(t),yδ,ɛ(t),zδ,ɛ(t),cδ,ɛ(t,⋅),uδ(t))dW(t)+∫Eg(t,xδ,ɛ(t-),yδ,ɛ(t-),zδ,ɛ(t),cδ,ɛ(t,⋅),uδ(t),e)Ñ(dedt),-dyδ,ɛ(t)=f(t,xδ,ɛ(t),yδ,ɛ(t),zδ,ɛ(t),cδ,ɛ(t,⋅),uδ(t))dt-zδ,ɛ(t)dW(t)-∫Ecδ,ɛ(t,e)Ñ(dedt),xδ,ɛ(0)=a,yδ,ɛ(T)=h(yδ,ɛ(T))+hTδ+ɛηT,
admits a unique adapted solution Θδ,ɛ(·)≡(xδ,ɛ(·),yδ,ɛ(·),zδ,ɛ(·),cδ,ɛ(·,·)). Note that FBSDEJ (3.18) is nothing but FBSDEJ (3.15) with only hTδ replaced by hTδ+ɛηT, and uδ(·) remains unchanged. Thus, by (1.9), we have(3.19)‖Θδ,ɛ(⋅)-Θδ(⋅)‖M2[0,T]≤Kɛ2,asɛ⟶0,
which implies(3.20)lim⁡ɛ→0(|yδ,ɛ(0)-yδ(0)|2+E∫0T|zδ,ɛ(t)-zδ(t)|2dt+E∫0T∫E|cδ,ɛ(t,e)-cδ(t,e)|2π(de)dt)=0.
Hence, with y0δ,ɛ=yδ,ɛ(0), by taking (y0,z(·),c(·,·),u(·))=(y0δ,ɛ,zδ,ɛ(·),cδ,ɛ(·,·),uδ(·)) in the last relation in (3.11), we have
(3.21)-Kδɛ≤Jδ(y0δ,ɛ,zδ,ɛ(⋅),cδ,ɛ(⋅,⋅),uδ(⋅))-Jδ(y0δ,zδ(⋅),cδ(⋅,⋅),uδ(⋅))=Jδ(y0δ,ɛ,zδ,ɛ(⋅),cδ,ɛ(⋅,⋅),uδ(⋅))2-Jδ(y0δ,zδ(⋅),cδ(⋅,⋅),uδ(⋅))2Jδ(y0δ,ɛ,zδ,ɛ(⋅),cδ,ɛ(⋅,⋅),uδ(⋅))+Jδ(y0δ,zδ(⋅),cδ(⋅,⋅),uδ(⋅))=[J(y0δ,ɛ,zδ,ɛ(⋅),cδ,ɛ(⋅,⋅),uδ(⋅))-J(y¯0,z¯(⋅),c¯(⋅,⋅),u¯(⋅))+δ]2Jδ(y0δ,ɛ,zδ,ɛ(⋅),cδ,ɛ(⋅,⋅),uδ(⋅))+Jδ(y0δ,zδ(⋅),cδ(⋅,⋅),uδ(⋅))-[J(y0δ,zδ(⋅),cδ(⋅,⋅),uδ(⋅))-J(y¯0,z¯(⋅),c¯(⋅,⋅),u¯(⋅))+δ]2Jδ(y0δ,ɛ,zδ,ɛ(⋅),cδ,ɛ(⋅,⋅),uδ(⋅))+Jδ(y0δ,zδ(⋅),cδ(⋅,⋅),uδ(⋅))+E[|yδ,ɛ(T)-h(xδ,ɛ(T))|2-|yδ(T)-h(xδ(T))|2]Jδ(y0δ,ɛ,zδ,ɛ(⋅),cδ,ɛ(⋅,⋅),uδ(⋅))+Jδ(y0δ,zδ(⋅),cδ(⋅,⋅),uδ(⋅))≡Φ0δ,ɛ[J(y0δ,ɛ,zδ,ɛ(⋅),cδ,ɛ(⋅,⋅),uδ(⋅))-J(y0δ,zδ(⋅),cδ(⋅,⋅),uδ(⋅))]+E〈ΦTδ,ɛ,yδ,ɛ(T)-h(xδ,ɛ(T))-hTδ〉≡(Φ0δ+o(1))[J(y0δ,ɛ,zδ,ɛ(⋅),cδ,ɛ(⋅,⋅),uδ(⋅))-J(y0δ,zδ(⋅),cδ(⋅,⋅),uδ(⋅))]+ɛE[〈ΦTδ+o(1),ηT〉],
where o(1) stands for certain scalars or vectors that go to 0 as ɛ→0,(3.22)Φ0δ,ɛ≐2Jδ(y0δ,ɛ,zδ,ɛ(⋅),cδ,ɛ(⋅,⋅),uδ(⋅))+Jδ(y0δ,zδ(⋅),cδ(⋅,⋅),uδ(⋅))×{∫01[β(J(y0δ,ɛ,zδ,ɛ(⋅),cδ,ɛ(⋅,⋅),uδ(⋅))-J(y¯0,z¯(⋅),c¯(⋅,⋅),u¯(⋅)))+(1-β)(J(y0δ,zδ(⋅),cδ(⋅,⋅),uδ(⋅))-J(y¯0,z¯(⋅),c¯(⋅,⋅),u¯(⋅)))]dβ+δ∫01},ΦTδ,ɛ≐yδ,ɛ(T)-h(xδ,ɛ(T))+yδ(T)-h(xδ(T))Jδ(y0δ,ɛ,zδ,ɛ(⋅),cδ,ɛ(⋅,⋅),uδ(⋅))+Jδ(y0δ,zδ(⋅),cδ(⋅,⋅),uδ(⋅)),(3.23)Φ0δ≐J(y0δ,zδ(⋅),cδ(⋅,⋅),uδ(⋅))-J(y¯0,z¯(⋅),c¯(⋅,⋅),u¯(⋅))+δJδ(y0δ,zδ(⋅),cδ(⋅,⋅),uδ(⋅))∈[0,1],ΦTδ≐yδ(T)-h(xδ(T))Jδ(y0δ,zδ(⋅),cδ(⋅,⋅),uδ(⋅))∈LFT2(Ω;Rm).We point out that(Φ0δ,ΦTδ) is independent of ηT, and(3.24)Φ0δ≥0,|Φ0δ|2+E|ΦTδ|2=1.
Then there is a subsequence of (Φ0δ,ΦTδ), still denoted by (Φ0δ,ΦTδ), such that(3.25)lim⁡δ→0(Φ0δ,ΦTδ)=(Φ0,ΦT),with|Φ0|2+E|ΦT|2=1.We claim that(3.26)Φ0≠0.
To show this, let us observe that by (3.19), one has
(3.27)(Φ0δ+o(1))[J(y0δ,ɛ,zδ,ɛ(⋅),cδ,ɛ(⋅,⋅),uδ(⋅))-Jδ(y0δ,zδ(⋅),cδ(⋅,⋅),uδ(⋅))]≤ɛK(Φ0δ+o(1)).
Hence, dividing by ɛ in (3.21) and then letting ɛ→0 yields(3.28)-Kδ≤K|Φ0δ|+E〈ΦTδ,ηT〉,with the constantK independent of ηT. Now, if Φ0=0, then the above leads to(3.29)-rδ≤E〈ΦTδ,ηT〉,∀ηT∈LFT2(Ω;Rm),E|ηT|2≤1,withrδ→0, as δ→0, uniformly in 𝔼|ηT|2≤1. Then,(3.30)(E|ΦTδ|2)1/2=sup⁡E|ηT|2≤1E〈ΦTδ,ηT〉≤|rδ|⟶0,asδ⟶0.Thus, forδ>0 small enough, we must have(3.31)1=|Φ0δ|2+E|ΦTδ|2<1.
It is a contradiction which proves (3.26).We refer to(Φ0,ΦT) as the Lagrange multiplier of the corresponding optimal 4-tuple (y¯0,z¯(·),c¯(·,·),u¯(·)). Equation (3.26) shows the nontriviality of the Lagrange multiplier (Φ0,ΦT).Step 4 (spike variations).
For notational simplicity, we now denote(3.32)X=(xy),v(⋅)=(zc(⋅)u),X0=(ay0),XT=X(T)=(x(T)y(T)),B(t,X,v(⋅))=(b(t,x,y,z,c(⋅),u)-f(t,x,y,z,c(⋅),u)),Σ(t,X,v(⋅))=(σ(t,x,y,z,c(⋅),u)z),Υ(t,X,v(⋅))=(g(t,x,y,z,c(⋅),u,⋅)c(⋅)),Γ(X0,XT)=ϕ(xT)+γ(y0),Π(X0,XT)=(0yT-h(xT)),Rl=Rn×Rm,V[0,T]=LF,p2([0,T];Rm)×Fp2([0,T];Rm)×U[0,T],H=Rl×LFT2(Ω;Rl)≡Rl×Xl2,H0=Rn×LFT2(Ω;Rm)≡Rn×Xm2.
Consequently,
(3.33)J(y0,z(⋅),c(⋅,⋅),u(⋅))=J(y0,v(⋅,⋅)),Jδ(y0,z(⋅),c(⋅,⋅),u(⋅))=Jδ(y0,v(⋅,⋅)).Note thatℋ and ℋ0 are Hilbert spaces. We identify ℋ*=ℋ and ℋ0*=ℋ0. Also, from the above,(3.34)Γ:H⟶R,Π:H⟶H0.
We denote the gradient DΓ and the Hessian D2Γ of Γ as follows:(3.35)DΓ(X0,XT)≡(ΓX0(X0,XT),ΓXT(X0,XT))∈L(H;R)≡H*=H,D2Γ(X0,XT)≡(ΓX0X0(X0,XT)ΓX0XT(X0,XT)ΓXTX0(X0,XT)ΓXTXT(X0,XT))∈Ls(H;H),
where ℒ(ℋ1;ℋ2) is the set of all linear bounded operators from ℋ1 to ℋ2, and ℒs(ℋ;ℋ) is the set of all linear bounded self-adjoint operators from ℋ to itself. Clearly,(3.36)ΓX0(X0,XT)=(0,γy(y0))⊤∈Rl,ΓXT(X0,XT)=(ϕx(xT),0)⊤∈Xl2,ΓX0X0(X0,XT)=(000γyy(y0))∈Sm+n,ΓX0XT(X0,XT)=(0000)∈L(Xl2;Rl),ΓXTX0(X0,XT)=(0000)∈L(Rl;Xl2),ΓXTXT(X0,XT)=(ϕxx(xT)000)∈L(Xl2;Xl2).Also, by the Fréchet differentiability, forΠ:ℋ→ℋ0, we have(3.37)DΠ(X0,XT)≡(ΠX0(X0,XT),ΠXT(X0,XT))∈L(H;H0),D2Π(X0,XT)≡(ΠX0X0(X0,XT)ΠX0XT(X0,XT)ΠXTX0(X0,XT)ΠXTXT(X0,XT))∈L(H;L(H;H0)),To make the above more precise, let us take anyΦ̂≡(Φ̂0,Φ̂T)∈ℋ0. Then,(3.38)〈Π(X0,XT),Φ̂〉=〈yT-h(xT),Φ̂T〉.Thus,(3.39)[DΠ(X0,XT)Φ̂]≡D[〈Π(X0,XT),Φ̂〉]=(〈Π(X0,XT),Φ̂〉X0,〈Π(X0,XT),Φ̂〉XT),≡(ΠX0(X0,XT)Φ̂,ΠXT(X0,XT)Φ̂)∈L(H;R),with(3.40)ΠX0(X0,XT)Φ̂≡〈Π(X0,XT),Φ̂〉X0=(0,0),ΠXT(X0,XT)Φ̂≡〈Π(X0,XT),Φ̂〉XT=(-hx(xT)⊤Φ̂T,Φ̂T);[D2Π(X0,XT)Φ̂]≡D2[〈Π(X0,XT),Φ̂〉]=(〈Π(X0,XT),Φ̂〉X0X0〈Π(X0,XT),Φ̂〉X0XT〈Π(X0,XT),Φ̂〉XTX0〈Π(X0,XT),Φ̂〉XTXT)≡(ΠX0X0(X0,XT)Φ̂ΠX0XT(X0,XT)Φ̂ΠXTX0(X0,XT)Φ̂ΠXTXT(X0,XT)Φ̂)∈Ls(H;H),with(3.41)ΠX0X0(X0,XT)Φ̂≡〈Π(X0,XT),Φ̂〉X0X0=(0000),ΠXTX0(X0,XT)Φ̂≡〈Π(X0,XT),Φ̂〉XTX0=(0000),ΠX0XT(X0,XT)Φ̂≡〈Π(X0,XT),Φ̂〉X0XT=(0000),ΠXTXT(X0,XT)Φ̂≡〈Π(X0,XT),Φ̂〉XTXT=(-hxx(xT)Φ̂T000).Next, for fixedX0∈Rl,v(·)∈𝒱[0,T] and any ɛ∈(0,1), let(3.42)X0δ,ɛ≐X0δ+(0ɛy0),(3.43)vδ,ɛ(t,⋅)={vδ(t,⋅),t∈[0,T]∖Sɛ,v(t,⋅),t∈Sɛ,
for some measurable set Sɛ⊆[0,T] with |Sɛ|=ɛT. It is clear that vδ,ɛ(·,·)∈𝒱[0,T].LetXδ,ɛ(·) be the state process of (3.1) corresponding to (X0δ,ɛ,vδ,ɛ(·,·)). Further, let X1δ,ɛ(·) and X2δ,ɛ(·) be, respectively, the solutions to the following SDEJs:(3.44)dX1δ,ɛ(t)=BXδ(t,⋅)X1δ,ɛ(t)dt+[ΣXδ+ΔΣδ(t,⋅)ISɛ(t)]dW(t)+∫E[ΥXδ(t,e)X1δ,ɛ(t-)+ΔΥδ(t,e)ISɛ(t)]Ñ(dedt),X1δ,ɛ(0)=(0ɛy0);(3.45)dX2δ,ɛ(t)=[BXδ(t,⋅)X2δ,ɛ(t)+ΔBδ(t,⋅)ISɛ(t)+12BXXδ(t,⋅)X1δ,ɛ(t)2]dt+[ΣXδ(t,⋅)X2δ,ɛ(t)+ΔΣXδ(t,⋅)X1δ,ɛ(t)ISɛ(t)+12ΣXXδ(t,⋅)X1δ,ɛ(t)2]dW(t)+∫E[12ΥXδ(t,e)X2δ,ɛ(t-)+ΔΥXδ(t,e)X1δ,ɛ(t-)ISɛ(t)+12ΥXXδ(t,e)X1δ,ɛ(t-)2]Ñ(dedt),X2δ,ɛ(0)=0,
where ISɛ(·) denotes the indicator function of the set Sɛ, and for any X∈Rl,(3.46)BXδ(t,⋅)=BX(t,Xδ(t),vδ(t,⋅)),ΣXδ(t,⋅)=ΣX(t,Xδ(t),vδ(t,⋅)),ΥXδ(t,⋅)=ΥX(t,Xδ(t),vδ(t,⋅)),ΔBδ(t,⋅)=B(t,Xδ(t),v(t,⋅))-B(t,Xδ(t),vδ(t,⋅)),ΔΣδ(t,⋅)=Σ(t,Xδ(t),v(t,⋅))-Σ(t,Xδ(t),vδ(t,⋅)),ΔΥδ(t,⋅)=Υ(t,Xδ(t),v(t,⋅))-Υ(t,Xδ(t),vδ(t,⋅)),ΔΣXδ(t,⋅)=ΣX(t,Xδ(t),v(t,⋅))-ΣX(t,Xδ(t),vδ(t,⋅)),ΔΥXδ(t,⋅)=ΥX(t,Xδ(t),v(t,⋅))-ΥX(t,Xδ(t),vδ(t,⋅)),BXXδ(t,⋅)X2=(〈BXX1,δ(t,⋅)X,X〉〈BXX2,δ(t,⋅)X,X〉⋮〈BXXl,δ(t,⋅)X,X〉),BXXi,δ(t,⋅)=BXXi(t,Xδ(t),vδ(t,⋅)),1≤i≤l,ΣXXδ(t,⋅)X2=(〈ΣXX1,δ(t,⋅)X,X〉〈ΣXX2,δ(t,⋅)X,X〉⋮〈ΣXXl,δ(t,⋅)X,X〉),ΣXXi,δ(t,⋅)=ΣXXi(t,Xδ(t),vδ(t,⋅)),1≤i≤l,ΥXXδ(t,⋅)X2=(〈ΥXX1,δ(t,⋅)X,X〉〈ΥXX2,δ(t,⋅)X,X〉⋮〈ΥXXl,δ(t,⋅)X,X〉),ΥXXi,δ(t,⋅)=ΥXXi(t,Xδ(t),vδ(t,⋅)),1≤i≤l.We point out that everything is reduced to the classical optimal control problems for SDEJs. Then by Lemma  2.1 of Tang and Li [24], the following estimates hold for any integers k≥1:
(3.47)sup⁡0≤t≤TE|X1δ,ɛ(t)|2k+sup⁡0≤t≤TE|Xδ,ɛ(t)-Xδ(t)|2k≤Cɛk,sup⁡0≤t≤TE|X2δ,ɛ(t)|2k+sup⁡0≤t≤TE|Xδ,ɛ(t)-Xδ(t)-X1δ,ɛ(t)|2k≤Cɛ2k,sup⁡0≤t≤TE|Xδ,ɛ(t)-Xδ(t)-X1δ,ɛ(t)-X2δ,ɛ(t)|2k=o(ɛ2k).Now, from the last relation in (3.11), noting (3.19), we obtain
(3.48)-δ[ɛy0+ɛ(T+K)]≤Jδ(y0δ,ɛ,vδ,ɛ(⋅,⋅))-Jδ(y0δ,vδ(⋅,⋅))=Jδ(y0δ,ɛ,vδ,ɛ(⋅,⋅))2-Jδ(y0δ,vδ(⋅,⋅))2Jδ(y0δ,ɛ,vδ,ɛ(⋅,⋅))+Jδ(y0δ,vδ(⋅,⋅))=[J(y0δ,ɛ,vδ,ɛ(⋅,⋅))-J(y¯0,v¯(⋅,⋅))+δ]2-[J(y0δ,vδ(⋅,⋅))-J(y¯0,v¯(⋅,⋅))+δ]2Jδ(y0δ,ɛ,vδ,ɛ(⋅,⋅))+Jδ(y0δ,vδ(⋅,⋅))+E[Π(X0δ,ɛ,Xδ,ɛ(T))2-Π(X0δ,Xδ(T))2]Jδ(y0δ,ɛ,vδ,ɛ(⋅,⋅))+Jδ(y0δ,vδ(⋅,⋅))≡Φ0δ,ɛ[J(y0δ,ɛ,vδ,ɛ(⋅,⋅))-J(y0δ,vδ(⋅,⋅))]+E〈(0ΦTδ,ɛ),Π(X0δ,ɛ,Xδ,ɛ(T))-Π(X0δ,Xδ(T))〉≡(Φ0δ+o(1))[J(y0δ,ɛ,vδ,ɛ(⋅,⋅))-J(y0δ,vδ(⋅,⋅))]+E〈(0ΦTδ+o(1)),Π(X0δ,ɛ,Xδ,ɛ(T))-Π(X0δ,Xδ(T))〉,
where (Φ0δ,ΦTδ) is defined in (3.25) and (Φ0δ,ɛ,ΦTδ,ɛ) is defined similarly to (3.22), with X0δ,ɛ=(ay0δ,ɛ) replaced by (3.42). We have shown that along a sequence,(3.49)lim⁡δ→0(Φ0δ,ΦTδ)=(Φ0,ΦT),withΦ0≠0.Note that(3.50)J(y0δ,ɛ,vδ,ɛ(⋅,⋅))-J(y0δ,vδ(⋅,⋅))=E∫0T[l(t,Xδ,ɛ(t),vδ,ɛ(t,⋅))-l(t,Xδ(t),vδ(t,⋅))]dt+E[Γ(X0δ,ɛ,Xδ,ɛ(T))-Γ(X0δ,Xδ(T))]≡I1+I2.We first deal with termI1. We have
(3.51)I1=E∫0T[l(t,Xδ,ɛ(t),vδ,ɛ(t,⋅))-l(t,Xδ(t)+X1δ,ɛ(t)+X2δ,ɛ(t),vδ,ɛ(t,⋅))]dt+E∫0T[l(t,Xδ(t)+X1δ,ɛ(t)+X2δ,ɛ(t),vδ,ɛ(t,⋅))-l(t,Xδ(t)+X1δ,ɛ(t)+X2δ,ɛ(t),vδ(t,⋅))]dt+E∫0T[l(t,Xδ(t)+X1δ,ɛ(t)+X2δ,ɛ(t),vδ(t,⋅))-l(t,Xδ(t),vδ(t,⋅))]dt=E∫0T[〈lX(t,Xδ(t)+X1δ,ɛ(t)+X2δ,ɛ(t),vδ,ɛ(t,⋅)),Xδ,ɛ(t)-Xδ(t)-X1δ,ɛ(t)-X2δ,ɛ(t)〉]dt+12E∫0T〈lXX(t,Xδ(t)+X1δ,ɛ(t)+X2δ,ɛ(t),vδ,ɛ(t,⋅))×(Xδ,ɛ(t)-Xδ(t)-X1δ,ɛ(t)-X2δ,ɛ(t)),Xδ,ɛ(t)-Xδ(t)-X1δ,ɛ(t)-X2δ,ɛ(t)〉dt+E∫0T〈D2lδ,ɛ,1,2(Xδ,ɛ(t)-Xδ(t)-X1δ,ɛ(t)-X2δ,ɛ(t)),Xδ,ɛ(t)-Xδ(t)-X1δ,ɛ(t)-X2δ,ɛ(t)〉dt+E∫0T[l(t,Xδ(t),vδ,ɛ(t,⋅))-l(t,Xδ(t),vδ(t,⋅))]dt+E∫0T〈lX(t,Xδ(t),vδ,ɛ(t,⋅))-lX(t,Xδ(t),vδ(t,⋅)),X1δ,ɛ(t)+X2δ,ɛ(t)〉dt+12E∫0T〈[lXX(t,Xδ(t),vδ,ɛ(t,⋅))-lXX(t,Xδ(t),vδ(t,⋅))]×(X1δ,ɛ(t)+X2δ,ɛ(t)),X1δ,ɛ(t)+X2δ,ɛ(t)〉dt+E∫0T〈D2lδ,ɛ(X1δ,ɛ(t)+X2δ,ɛ(t)),X1δ,ɛ(t)+X2δ,ɛ(t)〉dt+E∫0T〈lX(t,Xδ(t),vδ(t,⋅)),X1δ,ɛ(t)+X2δ,ɛ(t)〉dt+12E∫0T〈lXX(t,Xδ(t),vδ(t,⋅))(X1δ,ɛ(t)+X2δ,ɛ(t)),X1δ,ɛ(t)+X2δ,ɛ(t)〉dt+E∫0T〈D2lδ(X1δ,ɛ(t)+X2δ,ɛ(t)),X1δ,ɛ(t)+X2δ,ɛ(t)〉dt,
with
(3.52)D2lδ,ɛ,1,2≐∫01β[lXX(t,β(Xδ(t)+X1δ,ɛ(t)+X2δ,ɛ(t))+(1-β)X1δ,ɛ(t),vδ(t,⋅))-lXX(t,Xδ(t)+X1δ,ɛ(t)+X2δ,ɛ(t),vδ(t,⋅))]dβ,D2lδ,ɛ≐∫01β[lXX(t,βXδ(t)+(1-β)(Xδ(t)+X1δ,ɛ(t)+X2δ,ɛ(t)),vδ,ɛ(t,⋅))-lXX(t,βXδ(t)+(1-β)(Xδ(t)+X1δ,ɛ(t)+X2δ,ɛ(t)),vδ(t,⋅))-lXX(t,Xδ(t),vδ,ɛ(t,⋅))+lXX(t,Xδ(t),vδ(t,⋅))]dβ,D2lδ≐∫01β[lXX(t,βXδ(t)+(1-β)(Xδ(t)+X1δ,ɛ(t)+X2δ,ɛ(t)),vδ(t,⋅))-lXX(t,Xδ(t),vδ(t,⋅))]dβ.Next we deal with termI2. Similarly, we have(3.53)I2=E{DΓδ(X0δ,ɛ-X0δXδ,ɛ(T)-Xδ(T))+12〈D2Γδ(X0δ,ɛ-X0δXδ,ɛ(T)-Xδ(T)),(X0δ,ɛ-X0δXδ,ɛ(T)-Xδ(T))〉+〈D2Γδ,ɛ(X0δ,ɛ-X0δXδ,ɛ(T)-Xδ(T)),(X0δ,ɛ-X0δXδ,ɛ(T)-Xδ(T))〉}≡E{(X0δ,ɛ-X0δXδ,ɛ(T)-Xδ(T))ΓX0δ(X0δ,ɛ-X0δ)+ΓXTδ(Xδ,ɛ(T)-Xδ(T))+12〈ΓX0X0δ(X0δ,ɛ-X0δ),X0δ,ɛ-X0δ〉+12〈ΓXTXTδ(Xδ,ɛ(T)-Xδ(T)),Xδ,ɛ(T)-Xδ(T)〉+〈D2Γδ,ɛ(X0δ,ɛ-X0δXδ,ɛ(T)-Xδ(T)),(X0δ,ɛ-X0δXδ,ɛ(T)-Xδ(T))〉},
with
(3.54)DΓδ≐(ΓX0δ,ΓXTδ)≡(ΓX0(X0δ,Xδ(T)),ΓXT(X0δ,Xδ(T))),D2Γδ≐(ΓX0X0δ00ΓXTXTδ)≡(ΓX0X0(X0δ,Xδ(T))00ΓXTXT(X0δ,Xδ(T))),D2Γδ,ɛ≐∫01β[D2Γ(βX0δ+(1-β)X0δ,ɛ,βXTδ+(1-β)XTδ,ɛ)-D2Γ(X0δ,Xδ(T))]dβ.Also, we have(3.55)E〈(0ΦTδ,ɛ),Π(X0δ,ɛ,Xδ,ɛ(T))-Π(X0δ,Xδ(T))〉≡E〈(0ΦTδ,ɛ),Π(0,Xδ,ɛ(T))-Π(0,Xδ(T))〉=E{〈ΠXT(0,Xδ(T))(0ΦTδ,ɛ),Xδ,ɛ(T)-Xδ(T)〉+12〈ΠXTXT(0,Xδ(T))(0ΦTδ,ɛ)(Xδ,ɛ(T)-Xδ(T)),Xδ,ɛ(T)-Xδ(T)〉+〈D2Πδ,ɛ(0ΦTδ,ɛ)(Xδ,ɛ(T)-Xδ(T)),Xδ,ɛ(T)-Xδ(T)〉},
with(3.56)D2Πδ,ɛ≐∫01β[ΠXTXT(0,βXδ(T)+(1-β)Xδ,ɛ(T))-ΠXTXT(0,Xδ(T))]dβ.Under assumptions (H2), (H3), we have(3.57)|Φ0δ,ɛD2lδ,ɛ,1,2|≤K|Xδ,ɛ(t)-Xδ(t)-X1δ,ɛ(t)-X2δ,ɛ(t)|,|Φ0δ,ɛD2lδ,ɛ|+|Φ0δ,ɛD2lδ|≤K|X1δ,ɛ(t)+X2δ,ɛ(t)|,|Φ0δ,ɛD2Γδ,ɛ|+|D2Πδ,ɛ(0ΦTδ,ɛ)|≤K(|X0δ,ɛ-X0δ|+|Xδ,ɛ(T)-Xδ(T)|).Hence, by (3.47), we have
(3.58)E{|〈D2Πδ,ɛ(0ΦTδ,ɛ)(Xδ,ɛ(T)-Xδ(T)),Xδ,ɛ(T)-Xδ(T)〉||〈Φ0δ,ɛD2lδ,ɛ,1,2(Xδ,ɛ(t)-Xδ(t)-X1δ,ɛ(t)-X2δ,ɛ(t)),Xδ,ɛ(t)-Xδ(t)-X1δ,ɛ(t)-X2δ,ɛ(t)〉|+|〈Φ0δ,ɛD2lδ,ɛ(X1δ,ɛ(t)+X2δ,ɛ(t)),X1δ,ɛ(t)+X2δ,ɛ(t)〉|+|〈Φ0δ,ɛD2lδ(X1δ,ɛ(t)+X2δ,ɛ(t)),X1δ,ɛ(t)+X2δ,ɛ(t)〉|+|Φ0δ,ɛ〈D2Γδ,ɛ(X0δ,ɛ-X0δXδ,ɛ(T)-Xδ(T)),(X0δ,ɛ-X0δXδ,ɛ(T)-Xδ(T))〉|+|〈D2Πδ,ɛ(0ΦTδ,ɛ)(Xδ,ɛ(T)-Xδ(T)),Xδ,ɛ(T)-Xδ(T)〉|}≤KE(|Xδ,ɛ(t)-Xδ(t)-X1δ,ɛ(t)-X2δ,ɛ(t)|3+|X1δ,ɛ(t)+X2δ,ɛ(t)|3+|X0δ,ɛ-X0δ|3+|Xδ,ɛ(T)-Xδ(T)|3)≤Kɛ3/2.Consequently, combining (3.50) with (3.55), from (3.48), we can derive that(3.59)-δ[ɛy0+ɛ(T+K)]≤Φ0δ,ɛ[J(y0δ,ɛ,vδ,ɛ(⋅,⋅))-J(y0δ,vδ(⋅,⋅))]+E〈(0ΦTδ,ɛ),Π(0,Xδ,ɛ(T))-Π(0,Xδ(T))〉=Φ0δ,ɛ{E∫0T[12l(t,Xδ(t),vδ,ɛ(t,⋅))-l(t,Xδ(t),vδ(t,⋅))+〈lX(t,Xδ(t),vδ(t,⋅)),X1δ,ɛ(t)+X2δ,ɛ(t)〉+12〈lXX(t,Xδ(t),vδ(t,⋅))X1δ,ɛ(t),X1δ,ɛ(t)〉]dt+E[12ΓX0(X0δ,Xδ(T))(X0δ,ɛ-X0δ)+12〈ΓX0X0(X0δ,Xδ(T))(X0δ,ɛ-X0δ),X0δ,ɛ-X0δ〉+ΓXT(X0δ,Xδ(T))(Xδ,ɛ(T)-Xδ(T))+12〈ΓXTXT(X0δ,Xδ(T))(Xδ,ɛ(T)-Xδ(T)),Xδ,ɛ(T)-Xδ(T)〉]∫0T}+E{〈ΠXT(0,Xδ(T))(0ΦTδ,ɛ),Xδ,ɛ(T)-Xδ(T)〉+12〈ΠXTXT(0,Xδ(T))(0ΦTδ,ɛ)(Xδ,ɛ(T)-Xδ(T)),Xδ,ɛ(T)-Xδ(T)〉}+O(ɛ3/2)=E{∫0T{12Φ0δ,ɛ[l(t,Xδ(t),vδ,ɛ(t,⋅))-l(t,Xδ(t),vδ(t,⋅))]+Φ0δ,ɛ〈lX(t,Xδ(t),vδ(t,⋅)),X1δ,ɛ(t)+X2δ,ɛ(t)〉+12Φ0δ,ɛ〈lXX(t,Xδ(t),vδ(t,⋅))X1δ,ɛ(t),X1δ,ɛ(t)〉}dt+ɛ〈Φ0δ,ɛΓX0(X0δ,Xδ(T)),(0y0)〉+ɛ2〈Φ0δ,ɛΓX0X0(X0δ,Xδ(T))(0y0),(0y0)〉+〈Φ0δ,ɛΓXT(X0δ,Xδ(T))+ΠXT(0,Xδ(T))(0ΦTδ,ɛ),X1δ,ɛ(T)+X2δ,ɛ(T)〉+12〈[Φ0δ,ɛΓXTXT(X0δ,Xδ(T))+ΠXTXT(0,Xδ(T))(0ΦTδ,ɛ)]X1δ,ɛ(T),X1δ,ɛ(T)〉}+O(ɛ3/2).Step 5 (duality).
Let(Φ̃δ,ɛ(·),Ψ̃δ,ɛ(·),Ξ̃δ,ɛ(·,·)) be the adapted solution to the following BSDEJ:
(3.60)-dΦ̃δ,ɛ(t)=[BXδ(t,⋅)⊤Φ̃δ,ɛ(t)+ΣXδ(t,⋅)⊤Ψ̃δ,ɛ(t)+∫EΥXδ(t,e)⊤Ξ̃δ,ɛ(t,e)π(de)-Φ0δ,ɛlXδ(t,⋅)]dt-Ψ̃δ,ɛ(t)dW(t)-∫EΞ̃δ,ɛ(t,e)Ñ(dedt),dΦ̃δ,ɛ(T)=-[Φ0δ,ɛΓXT(X0δ,Xδ(T))+ΠXT(0,Xδ(T))(0ΦTδ,ɛ)],
where lXδ(t,·)≐lX(t,Xδ(t),vδ(t,·)). Then, as (ɛ,δ)→(0,0), (Φ̃δ,ɛ(·),Ψ̃δ,ɛ(·),Ξ̃δ,ɛ(·,·)) goes to (Φ̃(·),Ψ̃(·),Ξ̃(·,·)), which is the adapted solution to the following BSDEJ:
(3.61)-dΦ̃(t)=[B¯X(t,⋅)⊤Φ̃(t)+ΣX̅(t,⋅)⊤Ψ̃(t)+∫EΥ¯X(t,e)⊤Ξ̃(t,e)π(de)-Φ0l¯X(t,⋅)]dt-Ψ̃(t)dW(t)-∫EΞ̃(t,e)Ñ(dedt),dΦ̃(T)=-[Φ0ΓXT(X¯0,X¯(T))+ΠXT(0,X¯(T))(0ΦT)],
where
(3.62)B¯X(t,⋅)≐BX(t,X¯(t),v¯(t,⋅)),ΣX̅(t,⋅)≐ΣX(t,X¯(t),v¯(t,⋅)),Υ¯X(t,⋅)≐BX(t,X¯(t),v¯(t,⋅)),l¯X(t,⋅)≐lX(t,X¯(t),v¯(t,⋅)),(Φ0,ΦT) is determined by (3.25).Applying Itô’s formula to〈Φ̃δ,ɛ(·),X1δ,ɛ(·)+X2δ,ɛ(·)〉, we can get(3.63)-E〈Φ0δ,ɛΓXT(X0δ,Xδ(T))+ΠXT(0,Xδ(T))(0ΦTδ,ɛ),X1δ,ɛ(T)+X2δ,ɛ(T)〉=E〈Φ̃δ,ɛ(T),X1δ,ɛ(T)+X2δ,ɛ(T)〉=E〈Φ̃δ,ɛ(0),(0ɛy0)〉+E∫0T{〈∫EΥXδ-BXδ(t,⋅)⊤Φ̃δ,ɛ(t)-ΣXδ(t,⋅)⊤Ψ̃δ,ɛ(t)-∫EΥXδ(t,e)⊤Ξ̃δ,ɛ(t,e)π(de)+Φ0δ,ɛlXδ(t,⋅),X1δ,ɛ(t)+X2δ,ɛ(t)〉+〈Φ̃δ,ɛ(t),BXδ(t,⋅)⊤(X1δ,ɛ(t)+X2δ,ɛ(t))+ΔBδ(t,⋅)ISɛ(t)+12BXXδ(t,⋅)X1δ,ɛ(t)2〉+〈12Ψ̃δ,ɛ(t),ΣXδ(t,⋅)⊤(X1δ,ɛ(t)+X2δ,ɛ(t))+(ΔΣXδ(t,⋅)X1δ,ɛ(t)+ΔΣδ(t,⋅))ISɛ(t)+12ΣXXδ(t,⋅)X1δ,ɛ(t)2〉+∫E〈12Ξ̃δ,ɛ(t,e),ΥXδ(t,e)⊤(X1δ,ɛ(t)+X2δ,ɛ(t))+(ΔΥXδ(t,e)X1δ,ɛ(t)+ΔΥδ(t,e))ISɛ(t)+12ΥXXδ(t,e)X1δ,ɛ(t)2〉π(de)Σδ(t,⋅)}dt=E〈Φ̃δ,ɛ(0),(0ɛy0)〉+E∫0T{∫E〈Φ̃δ,ɛ(t),ΔBδ(t,⋅)ISɛ(t)+12BXXδ(t,⋅)X1δ,ɛ(t)2〉+〈Φ0δ,ɛlXδ(t,⋅),X1δ,ɛ(t)+X2δ,ɛ(t)〉+〈Ψ̃δ,ɛ(t),ΔΣδ(t,⋅)ISɛ(t)+12ΣXXδ(t,⋅)X1δ,ɛ(t)2〉+∫E〈Ξ̃δ,ɛ(t,e),ΔΥδ(t,e)ISɛ(t)+12ΥXXδ(t,e)X1δ,ɛ(t)2〉π(de)∫E}dt+O(ɛ3/2).
Then by (3.59), we have(3.64)-δ[ɛy0+ɛ(T+K)]≤E∫0T{1212Φ0δ,ɛ[l(t,Xδ(t),vδ,ɛ(t,⋅))-l(t,Xδ(t),vδ(t,⋅))]+Φ0δ,ɛ〈lXδ(t,⋅),X1δ,ɛ(t)+X2δ,ɛ(t)〉+12Φ0δ,ɛ〈lXXδ(t,⋅)X1δ,ɛ(t),X1δ,ɛ(t)〉∫0T}dt-E〈Φ̃δ,ɛ(0),(0ɛy0)〉-E∫0T{∫E〈Φ̃δ,ɛ(t),ΔBδ(t,⋅)ISɛ(t)+12BXXδ(t,⋅)X1δ,ɛ(t)2〉-〈Φ0δ,ɛlXδ(t,⋅),X1δ,ɛ(t)+X2δ,ɛ(t)〉+〈Ψ̃δ,ɛ(t),ΔΣδ(t,⋅)ISɛ(t)+12ΣXXδ(t,⋅)X1δ,ɛ(t)2〉+∫E〈Ξ̃δ,ɛ(t,e),ΔΥδ(t,e)ISɛ(t)+12ΥXXδ(t,e)X1δ,ɛ(t)2〉π(de)∫E}dt+E{ΠXTXT(0,Xδ(T))(0ΦTδ,ɛ)ɛ〈Φ0δ,ɛΓX0(X0δ,Xδ(T)),(0y0)〉+ɛ2〈Φ0δ,ɛΓX0X0(X0δ,Xδ(T))(0y0),(0y0)〉+12〈[(0ΦTδ,ɛ)Φ0δ,ɛΓXTXT(X0δ,Xδ(T))+ΠXTXT(0,Xδ(T))(0ΦTδ,ɛ)]X1δ,ɛ(T),X1δ,ɛ(T)〉}+O(ɛ3/2)=E∫0T{1212Φ0δ,ɛ[l(t,Xδ(t),vδ,ɛ(t,⋅))-l(t,Xδ(t),vδ(t,⋅))]+12Φ0δ,ɛ〈lXXδ(t,⋅)X1δ,ɛ(t),X1δ,ɛ(t)〉∫0T}dt-E∫0T{Σδ(t,⋅)ISɛ〈Φ̃δ,ɛ(t),ΔBδ(t,⋅)ISɛ(t)+12BXXδ(t,⋅)X1δ,ɛ(t)2〉+〈Ψ̃δ,ɛ(t),ΔΣδ(t,⋅)ISɛ(t)+12ΣXXδ(t,⋅)X1δ,ɛ(t)2〉+∫E〈Ξ̃δ,ɛ(t,e),ΔΥδ(t,e)ISɛ(t)+12ΥXXδ(t,e)X1δ,ɛ(t)2〉π(de)Σδ(t,⋅)ISɛ}dt+E{ΠXTXT(0,Xδ(T))(0ΦTδ,ɛ)ɛ〈Φ0δ,ɛΓX0(X0δ,Xδ(T))-Φ̃δ,ɛ(0),(0y0)〉+ɛ2〈Φ0δ,ɛΓX0X0(X0δ,Xδ(T))(0y0),(0y0)〉+12〈[(0ΦTδ,ɛ)ΠXTXT(0,Xδ(T))Φ0δ,ɛΓXTXT(X0δ,Xδ(T))+ΠXTXT(0,Xδ(T))(0ΦTδ,ɛ)]X1δ,ɛ(T),X1δ,ɛ(T)〉}+O(ɛ3/2).
Note that Yδ,ɛ(·)≐X1δ,ɛ(·)X1δ,ɛ(·)⊤ satisfies
(3.65)dYδ,ɛ(t)={[ΥXδ(t,e)⊤]BXδ(t,⋅)Yδ,ɛ(t)+Yδ,ɛ(t)BXδ(t,⋅)⊤+ΣXδ(t,⋅)Yδ,ɛ(t)ΣXδ(t,⋅)⊤+[ΔΣδ(t,⋅)ΔΣδ(t,⋅)⊤+ΣXδ(t,⋅)X1δ,ɛ(t)ΔΣδ(t,⋅)⊤+ΔΣδ(t,⋅)X1δ,ɛ(t)ΣXδ(t,⋅)⊤]ISɛ(t)+∫E{ΣXδ(t,⋅)ΥXδ(t,e)Yδ,ɛ(t)ΥXδ(t,e)⊤+[ΥXδ(t,e)X1δ,ɛ(t)ΔΥδ(t,e)⊤+ΔΥδ(t,e)X1δ,ɛ(t)ΥXδ(t,e)⊤+ΔΥδ(t,e)ΔΥδ(t,⋅)⊤]ISɛ(t)ΥXδ(t,e)⊤}π(de)ΣXδ(t,⋅)}dt+{ΣXδ(t,⋅)Yδ,ɛ(t)+Yδ,ɛ(t)ΣXδ(t,⋅)⊤+[X1δ,ɛ(t)ΔΣδ(t,⋅)⊤+ΔΣδ(t,⋅)X1δ,ɛ(t)⊤]ISɛ(t)}dW(t)+∫E{ΥXδ(t,e)Yδ,ɛ(t-)+Yδ,ɛ(t-)ΥXδ(t,e)⊤+ΥXδ(t,e)Yδ,ɛ(t-)ΥXδ(t,e)⊤+[X1δ,ɛ(t-)ΔΥδ(t,e)⊤+ΔΥδ(t,e)X1δ,ɛ(t-)⊤+ΥXδ(t,e)X1δ,ɛ(t-)ΔΥδ(t,e)⊤+ΔΥδ(t,e)X1δ,ɛ(t-)ΥXδ(t,e)⊤+ΔΥδ(t,e)ΔΥδ(t,e)⊤]ISɛ(t)}Ñ(dedt),Yδ,ɛ(0)=(000ɛy0y0⊤).
Now, let (Pδ,ɛ(·),Qδ,ɛ(·),Kδ,ɛ(·,·)) be the adapted solution to the following BSDEJ:(3.66)-dPδ,ɛ(t)={HXXδ,ɛ,Φ0δ,ɛBXδ(t,⋅)⊤Pδ,ɛ(t)+Pδ,ɛ(t)BXδ(t,⋅)+ΣXδ(t,⋅)⊤Pδ,ɛ(t)ΣXδ(t,⋅)+ΣXδ(t,⋅)⊤Qδ,ɛ(t)+Qδ,ɛ(t)ΣXδ(t,⋅)+∫E[ΥXδ(t,e)⊤Pδ,ɛ(t)ΥXδ(t,e)+ΥXδ(t,e)⊤Kδ,ɛ(t,e)ΥXδ(t,e)+ΥXδ(t,e)⊤Kδ,ɛ(t,e)+Kδ,ɛ(t,e)ΥXδ(t,e)]π(de)+HXXδ,ɛ,Φ0δ,ɛ(t,⋅)ΣXδ(t,⋅)⊤}dt-Qδ,ɛ(t)dW(t)-∫EKδ,ɛ(t,e)Ñ(dedt),dPδ,ɛ(T)=-[Φ0δ,ɛΓXTXT(X0δ,Xδ(T))+ΠXTXT(0,Xδ(T))(0ΦTδ,ɛ)],
where(3.67)HXXδ,ɛ,Φ0δ,ɛ(t,⋅)≐HXX(t,Φ0δ,ɛ,Xδ(t),vδ(t,⋅),Φ̃δ,ɛ(t),Ψ̃δ,ɛ(t),Ξ̃δ,ɛ(t,⋅)),
with HXX(t,λ,X,v(·),Φ̃,Ψ̃,Ξ̃(·)) being the Hessian of the following(3.68)H(t,λ,X,v(⋅),Φ̃,Ψ̃,Ξ̃(⋅))≐〈Φ̃,B(t,X,v(⋅))〉+〈Ψ̃,Σ(t,X,v(⋅))〉+∫E〈Ξ̃(e),Υ(t,X,v(e))〉π(de)-λl(t,X,v(⋅)).
Applying Itô’s formula to [Pδ,ɛ(·)Yδ,ɛ(·)], we have(3.69)-E{〈[Φ0δ,ɛΓXTXT(X0δ,Xδ(T))+ΠXTXT(0,Xδ(T))(0ΦTδ,ɛ)]X1δ,ɛ(T),X1δ,ɛ(T)〉+〈Pδ,ɛ(0)(0ɛy0),(0ɛy0)(0ΦTδ,ɛ)〉}=E{tr⁡[Pδ,ɛ(T)Yδ,ɛ(T)]-tr⁡[Pδ,ɛ(0)Yδ,ɛ(0)]}=E∫0Ttr⁡{(0ΦTδ,ɛ)Pδ,ɛ(t)[ΔΣδ(t,⋅)ΔΣδ(t,⋅)⊤+ΣXδ(t,⋅)X1δ,ɛ(t)ΔΣδ(t,⋅)⊤+ΔΣδ(t,⋅)X1δ,ɛ(t)ΣXδ(t,⋅)⊤]ISɛ(t)+Qδ,ɛ(t)[X1δ,ɛ(t)ΔΣδ(t,⋅)⊤+ΔΣδ(t,⋅)X1δ,ɛ(t)⊤]ISɛ(t)-HXXδ,ɛ,Φ0δ,ɛ(t,⋅)Yδ,ɛ(t)+∫E{Σδ(t,⋅)⊤(0ΦTδ,ɛ)Pδ,ɛ(t)[ΥXδ(t,e)X1δ,ɛ(t)ΔΥδ(t,e)⊤+ΔΥδ(t,e)X1δ,ɛ(t)ΥXδ(t,e)⊤+ΔΥδ(t,e)ΔΥδ(t,e)⊤]ISɛ(t)+Kδ,ɛ(t,e)[X1δ,ɛ(t-)ΔΥδ(t,e)⊤+ΔΥδ(t,e)X1δ,ɛ(t-)⊤+ΥXδ(t,e)X1δ,ɛ(t-)ΔΥδ(t,e)⊤+ΔΥδ(t,e)X1δ,ɛ(t-)ΥXδ(t,e)⊤+ΔΥδ(t,e)ΔΥδ(t,e)⊤]ISɛ(t)(0ΦTδ,ɛ)}π(de)Σδ(t,⋅)⊤}dt=E∫0Ttr⁡{Pδ,ɛ(t)[ΔΣδ(t,⋅)ΔΣδ(t,⋅)⊤+∫EΔΥδ(t,e)ΔΥδ(t,e)⊤π(de)]ISɛ(t)+∫EKδ,ɛ(t,e)ΔΥδ(t,e)ΔΥδ(t,e)⊤ISɛ(t)π(de)-HXXδ,ɛ,Φ0δ,ɛ(t,⋅)Yδ,ɛ(t)[ΔΣδ(t,⋅)ΔΣδ(t,⋅)⊤+∫EΔΥδ(t,e)ΔΥδ(t,e)⊤π(de)]}dt+O(ɛ3/2)=E∫0Ttr⁡{HXXδ,ɛ,Φ0δ,ɛΔΣδ(t,⋅)⊤Pδ,ɛ(t)ΔΣδ(t,⋅)ISɛ(t)+∫EΔΥδ(t,e)⊤[Pδ,ɛ(t)+Kδ,ɛ(t,e)]ΔΥδ(t,e)ISɛ(t)π(de)-〈HXXδ,ɛ,Φ0δ,ɛ(t,⋅)X1δ,ɛ(t),X1δ,ɛ(t)〉Σδ(t,⋅)⊤}dt+O(ɛ3/2).
Consequently, from (3.64), we have
(3.70)-δ[ɛy0+ɛ(T+K)]≤E∫0T{[Pδ,ɛ(t)]Φ0δ,ɛ[l(t,Xδ(t),vδ,ɛ(t,⋅))-l(t,Xδ(t),vδ(t,⋅))]-〈Φ̃δ,ɛ(t),ΔBδ(t,⋅)〉-〈Ψ̃δ,ɛ(t),ΔΣδ(t,⋅)〉-∫E〈Ξ̃δ,ɛ(t,e),ΔΥδ(t,e)〉π(de)-12ΔΣδ(t,⋅)⊤Pδ,ɛ(t)ΔΣδ(t,⋅)-12∫EΔΥδ(t,e)⊤[Pδ,ɛ(t)+Kδ,ɛ(t,e)]ΔΥδ(t,e)π(de)}ISɛ(t)dt+E{ɛ〈Φ0δ,ɛΓX0(X0δ,Xδ(T))-Φ̃δ,ɛ(0),(0y0)〉+ɛ2〈(Φ0δ,ɛΓX0X0(X0δ,Xδ(T))-Pδ,ɛ(0))(0y0),(0y0)〉}+O(ɛ3/2).Step 6 (variational inequality).
In (3.70), dividing ɛ and then sending ɛ→0 followed by sending δ→0, we get
(3.71)0≤E〈Φ0ΓX0(X0,X(T))-Φ̃(0),(0y0)〉=E〈Φ0(0γy(y0))-Φ̃(0),(0y0)〉,∀y0∈Rm,
which implies
(3.72)Φ̃(0)=Φ0(0Eγy(y0)).Using a standard argument of Tang and Li [24] in (3.70), we have the following variational inequality:(3.73)Φ0[l(t,X¯(t),v(t,⋅))-l(t,X¯(t),v¯(t,⋅))]-〈Φ̃(t),B(t,X¯(t),v(t,⋅))-B(t,X¯(t),v¯(t,⋅))〉-〈Ψ̃(t),Σ(t,X¯(t),v(t,⋅))-Σ(t,X¯(t),v¯(t,⋅))〉-∫E〈Ξ̃(t,e),Υ(t,X¯(t),v(t,e))-Υ(t,X¯(t),v¯(t,e))〉π(de)-12[Σ(t,X¯(t),v(t,⋅))-Σ(t,X¯(t),v¯(t,⋅))]⊤P(t)[Σ(t,X¯(t),v(t,⋅))-Σ(t,X¯(t),v¯(t,⋅))]-12∫E[Υ(t,X¯(t),v(t,e))-Υ(t,X¯(t),v¯(t,e))]⊤[P(t)+K(t,e)]×[Υ(t,X¯(t),v(t,e))-Υ(t,X¯(t),v¯(t,e))]π(de)≥0,∀v(⋅),a.e.,a.s.Step 7 (finalizing the proof).
SinceΦ0≠0, by rescaling, we let Φ0=1. Then the first-order adjoint equation (3.61) reduces to
(3.74)-dΦ̃(t)=[B¯X(t,⋅)⊤Φ̃(t)+ΣX̅(t,⋅)⊤Ψ̃(t)+∫EΥ¯X(t,e)⊤Ξ̃(t,e)π(de)-l¯X(t,⋅)]dt-Ψ̃(t)dW(t)-∫EΞ̃(t,e)Ñ(dedt),Φ̃(T)=-[ΓXT(X¯0,X¯(T))+ΠXT(0,X¯(T))(0ΦT)].
Let (δ,ɛ)→0 in (3.66), then (Pδ,ɛ(·),Qδ,ɛ(·),Kδ,ɛ(·,·))→(P(·),Q(·),K(·,·)) which is the adapted solution to the following second-order adjoint equation:(3.75)-dP(t)={[Υ¯X(t,e)⊤]B¯X(t,⋅)⊤P(t)+P(t)B¯X(t,⋅)+ΣX¯(t,⋅)⊤P(t)ΣX¯(t,⋅)+ΣX¯(t,⋅)⊤Q(t)+Q(t)ΣX¯(t,⋅)+∫E[Υ¯X(t,e)⊤P(t)Υ¯X(t,e)+Υ¯X(t,e)⊤K(t,e)Υ¯X(t,e)+Υ¯X(t,e)⊤K(t,e)+K(t,e)Υ¯X(t,e)]π(de)+H¯XX(t,⋅)ΣX¯(t,⋅)}dt-Q(t)dW(t)-∫EK(t,e)Ñ(dedt),P(T)=-[ΓXTXT(X¯0,X¯(T))+ΠXTXT(0,X¯(T))(0ΦT)],
where(3.76)H¯XX(t,⋅)≐H¯XX(t,X¯(t),v¯(t,⋅),Φ̃(t),Ψ̃(t),Ξ̃(t,⋅)),
with H¯XX(t,X,v(·),Φ̃,Ψ̃,Ξ̃(·)) being the Hessian of the following(3.77)H¯(t,X,v(⋅),Φ̃,Ψ̃,Ξ̃(⋅))≐〈Φ̃,B(t,X,v(⋅))〉+〈Ψ̃,Σ(t,X,v(⋅))〉+∫E〈Ξ̃(e),Υ(t,X,v(e))〉π(de)-l(t,X,v(⋅)).
Also, from (3.72), we have(3.78)Φ̃(0)=(0Eγy(y0)).Note that(3.79)ΓXT(X¯0,X¯(T))=(ϕx(x¯(T))0),ΓXTXT(X¯0,X¯(T))=(ϕxx(x¯(T))000).
On the other hand, since(3.80)〈Π(0,X¯(T)),(0ΦT)〉=(y¯(T)-h(X¯(T)))ΦT,
we have(3.81)ΠXT(0,X¯(T))(0ΦT)=(-hx(x¯(T))⊤ΦTΦT),ΠXTXT(0,X¯(T))(0ΦT)=(-hxx(x¯(T))⊤ΦT000).Let(3.82)Φ̃(⋅)≡(p(⋅)q(⋅)),Ψ̃(⋅)≡(k(⋅)k̃(⋅)),Ξ̃(⋅,⋅)≡(r(⋅,⋅)r̃(⋅,⋅)).
Then it follows from (3.74), (3.78), and (3.82) that(3.83)(p(0)q(0))=(0Eγy(y0)),(p(T)q(T))=(-ϕx(x¯(T))+hx(x¯(T))⊤ΦT-ΦT).
By (3.82), we can rewrite the equations in (3.74) as(3.84)-d(p(t)q(t))={(b¯x(t,⋅)⊤-f¯x(t,⋅)⊤b¯y(t,⋅)⊤-f¯y(t,⋅)⊤)(p(t)q(t))+(σ¯x(t,⋅)⊤0σ¯y(t,⋅)⊤0)(k(t)k̃(t))+∫E(g¯x(t,e)⊤0g¯y(t,e)⊤0)(r(t,e)r̃(t,e))π(de)-(l¯x(t,⋅)l¯y(t,⋅))}dt-(k(t)k̃(t))dW(t)-∫E(r(t,e)r̃(t,e))Ñ(dedt),
and the variational inequality (3.73) now takes the form:
(3.85)l(t,x¯(t),y¯(t),z,c(⋅),u)-l(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t))-〈p(t),b(t,x¯(t),y¯(t),z,c(⋅),u)-b(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t))〉+〈q(t),f(t,x¯(t),y¯(t),z,c(⋅),u)-f(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t))〉-〈k(t),σ(t,x¯(t),y¯(t),z,c(⋅),u)-σ(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t))〉-∫E〈r(t,e),g(t,x¯(t),y¯(t),z,c(e),u)-g(t,x¯(t),y¯(t),z¯(t),c¯(t,e),u¯(t))〉π(de)-〈k̃(t),z-z¯(t)〉-∫E〈r̃(t,e),c(e)-c¯(t,e)〉π(de)-12(σ(t,x¯(t),y¯(t),z,c(⋅),u)-σ(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t))z-z¯(t))⊤P(t)×(σ(t,x¯(t),y¯(t),z,c(⋅),u)-σ(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t))z-z¯(t))-12∫E(g(t,x¯(t),y¯(t),z,c(e),u,e)-g(t,x¯(t),y¯(t),z¯(t),c¯(t,e),u¯(t),e)c(e)-c¯(t,e))⊤×[P(t)+K(t,e)]×(g(t,x¯(t),y¯(t),z,c(e),u,e)-g(t,x¯(t),y¯(t),z¯(t),c¯(t,e),u¯(t),e)c(e)-c¯(t,e))π(de)≥0,∀z,c(⋅),u,a.e.,a.s.
Thus, taking u=u¯(·), z=z¯(t)+ɛz0 and c(·)=c¯(t,·)+ɛc0(·), then dividing by ɛ and sending ɛ→0, we get
(3.86)0≤〈lz(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t)),z0〉-〈p(t),bz(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t))z0〉+〈q(t),fz(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t))z0〉-〈k(t),σz(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t))z0〉-∫E〈r(t,e),gz(t,x¯(t),y¯(t),z¯(t),c¯(t,e),u¯(t))z0〉-〈k̃(t),z0〉+〈lc(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t)),c0(⋅)〉-〈p(t),bc(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t))c0(⋅)〉+〈q(t),fc(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t))c0(⋅)〉-〈k(t),σc(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t))c0(⋅)〉-∫E〈r(t,e),gc(t,x¯(t),y¯(t),z¯(t),c¯(t,e),u¯(t))c0(e)〉-∫E〈r̃(t,e),c0(e)〉π(de),∀z0∈Rm,c0(⋅)∈L2(E,B(E),π;Rm).
Hence,(3.87)k̃(t)=lz(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t))-bz(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t))⊤p(t)+fz(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t))⊤q(t)-σz(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t))⊤k(t)-∫Egz(t,x¯(t),y¯(t),z¯(t),c¯(t,e),u¯(t))⊤r(t,e)π(de),r̃(t,⋅)=lc(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t))-bc(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t))⊤p(t)+fc(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t))⊤q(t)-σc(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t))⊤k(t)-∫Egc(t,x¯(t),y¯(t),z¯(t),c¯(t,e),u¯(t))⊤r(t,e)π(de).
Combining (3.83), (3.84) with (3.87), we arrive at (2.10).Next, we have(3.88)P(T)=-[ΓXTXT(X¯0,X¯(T))+ΠXTXT(0,X¯(T))(0ΦT)]=(-ϕxx(x¯(T))+hxx(x¯(T))⊤ΦT000)=(-ϕxx(x¯(T))-hxx(x¯(T))⊤q(T)000).
Then (2.11) follows from (3.75).Further, by takingz=z¯(·),c(·)=c¯(·,·) in the variational inequality (3.85), we obtain(3.89)l(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u)-l(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t))-〈p(t),b(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u)-b(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t))〉+〈q(t),f(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u)-f(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t))〉-〈k(t),σ(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u)-σ(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t))〉-∫E〈r(t,e),g(t,x¯(t),y¯(t),z¯(t),c¯(t,e),u,e)-g(t,x¯(t),y¯(t),z¯(t),c¯(t,e),u¯(t),e)〉π(de)-12(σ(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u)-σ(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t))0)⊤P(t)×(σ(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u)-σ(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u¯(t))0)-12∫E(g(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u,e)-g(t,x¯(t),y¯(t),z¯(t),c¯(t,e),u¯(t),e)0)⊤×[P(t)+K(t,e)]×(g(t,x¯(t),y¯(t),z¯(t),c¯(t,⋅),u,e)-g(t,x¯(t),y¯(t),z¯(t),c¯(t,e),u¯(t),e)0)π(de)≥0,∀u∈U,a.e.,a.s.,
which gives (2.13). This completes the proof of our main result—Theorem 2.1.


## 4. A Linear Quadratic Example

In this section, we briefly discuss a linear quadratic case, which serves as an illustrating example of our main result. Consider the following linear controlled FBSDEJ:(4.1)dx(t)=[b1x(t)+b2y(t)+b3z(t)+b4c(t,⋅)+b5u(t)]dt+[σ1x(t)+σ2y(t)+σ3z(t)+σ4c(t,⋅)+σ5u(t)]dW(t)+∫E[g1x(t-)+g2y(t-)+g3z(t)+g4c(t,e)+g5u(t)]Ñ(dedt),-dy(t)=[f1x(t)+f2y(t)+f3z(t)+f4c(t,⋅)+f5u(t)]dt-z(t)dW(t)-∫Ec(t,e)Ñ(dedt),x(0)=a,y(T)=h1x(T),
with the quadratic cost functional given by(4.2)J(u(⋅))=12E{∫0T[l1x2(t)+l2y2(t)+l3z2(t)+l4c2(t,⋅)+l5u2(t)]dt+ϕ1x2(T)+γ1y2(0)}.
In the above, bi,σi,gi,fi,li, i=1,…,5 and h1,ϕ1,γ1 are all real constants of R. The control domain U⊆R could be very arbitrary; in particular, U does not have to be a convex set. We let 𝒰[0,T] be the set of all ℱt-predictable processes u:[0,T]×Ω→U such that sup0≤t≤T𝔼|u(t)|i<+∞,∀i=1,2,….For anyu(·)∈𝒰[0,T], we denote(4.3)Ξ=(xyz),A(t,Ξ,c(⋅),u)=(-f1x-f2y-f3z-f4c(⋅)-f5ub1x+b2y+b3z+b4c(⋅)+b5uσ1x+σ2y+σ3z+σ4c(⋅)+σ5u),
and assume that (H4.1)(4.4)〈A(t,Ξ,c(⋅),u)-A(t,Ξ′,c′(⋅),u),Ξ-Ξ′〉+∫E〈g1(x-x′)+g2(y-y′)+g3(z-z′)+g4(c(e)-c′(e)),c(e)-c′(e)〉π(de)≤-β1|x-x′|2-β2(|y-y′|2+|z-z′|2+∫E|c(e)-c′(e)|2π(de)),h1≥μ1>0,
where β1,β2, and μ1 are given constants with β1>0,μ1>0,β2≥0. Then by Wu [1], there exists a unique adapted solution (x(·),y(·),z(·),c(·,·)) to (4.1).Now, suppose that the corresponding stochastic control problem admits an optimal 5-tuple(x¯(·),y¯(·),z¯(·),c¯(·,·),u¯(·)). Now let us look at Theorem 2.1. The first-order adjoint equation (2.10) reads(4.5)-dp(t)=[b1p(t)-f1q(t)+σ1k(t)+∫Eg1r(t,e)π(de)-l1x¯(t)]dt-k(t)dW(t)-∫Er(t,e)Ñ(dedt),dq(t)=[-b2p(t)+f2q(t)-σ2k(t)-∫Eg2r(t,e)π(de)+l2y¯(t)]dt+[-b3p(t)+f3q(t)-σ3k(t)-∫Eg3r(t,e)π(de)+l3z¯(t)]dW(t)+∫E[-b4p(t-)+f4q(t-)-σ4k(t)-g4r(t,e)+l4c¯(t,e)]Ñ(dedt),p(T)=-ϕ1x¯(T)-h1q(T),q(0)=γ1y¯0.
Similarly, we denote(4.6)ϑ=(qpk),B(t,ϑ,r(⋅))=(-b1p+f1q-σ1k-g1r(⋅)+l1x¯-b2p+f2q-σ2k-g2r(⋅)+l2y¯-b2p+f3q-σ3k-g3r(⋅)+l3z¯).
Since controlled FBSDEJ satisfies (H4.1), we can easily check that the above FBSDEJ (4.4) satisfies the following condition (H4.2):
(4.7)〈B(t,ϑ,r(⋅),u)-B(t,ϑ′,r′(⋅),u),ϑ-ϑ′〉+∫E〈-b4(p-p′)+f4(q-q′)-σ4(k-k′)-g4(r(e)-r′(e))+l4c¯(e),r(e)-r′(e)〉π(de)≥β1|q-q′|2+β2(|p-q′|2+|k-k′|2+∫E|r(e)-r′(e)|2π(de)),-h1≤-μ1,
where β1,β2, and μ1 are the same as in (H4.1). Then by Wu [38], FBSDEJ (4.4) admits a unique adapted solution (q(·),p(·),k(·),r(·,·)).The2×2 matrix-valued second-order adjoint equation (2.11) becomes(4.8)-dP(t)={B⊤P(t)+P(t)B+Σ⊤P(t)Σ+Σ⊤Q(t)+Q(t)Σ+Υ⊤P(t)Υ+∫E[Υ⊤K(t,e)Υ+Υ⊤K(t,e)+K(t,e)Υ]π(de)-LΣ⊤P(t)}dt-Q(t)dW(t)-∫EK(t,e)Ñ(dedt),P(T)=(-ϕ1000),
where(4.9)B=(b1b2-f1-f2),Σ=(σ1σ200),Υ=(g1g200),L=(l100l2).As before, (4.5) can be split into the following three scalar BSDEJs:
(4.10)-dP1(t)={∫E2b1P1(t)+σ12P1(t)+2σ1Q1(t)-2f1P2(t)+g12P1(t)+∫E[g12K1(t,e)+2g1K1(t,e)]π(de)-l1}dt-Q1(t)dW(t)-∫EK1(t,e)Ñ(dedt),P1(T)=-ϕ1,(4.11)-dP2(t)={∫Eb2P1(t)+b1P2(t)+σ1σ2P1(t)+σ1Q2(t)+σ2Q1(t)-f1P3(t)-f2P2(t)+g1g2P1(t)+∫E[g1g2K1(t,e)+g1K2(t,e)+g2K1(t,e)]π(de)}dt-Q2(t)dW(t)-∫EK2(t,e)Ñ(dedt),P2(T)=0,(4.12)-dP3(t)={∫E[g22K1(t,e)+2g2K2(t,e)]2b2P2(t)+σ22P1(t)+2σ2Q2(t)-2f2P3(t)+g22P1(t)+∫E[g22K1(t,e)+2g2K2(t,e)]π(de)-l2}dt-Q3(t)dW(t)-∫EK3(t,e)Ñ(dedt),P3(T)=0.Note that since all the coefficients in (4.5) are constants and the terminal condition is deterministic, we must have(4.13)Q(⋅)=0,K(⋅,⋅)=0,
and P(·) is deterministic. Consequently,(4.14)Ṗ(t)=-[B⊤P(t)+P(t)B+Σ⊤P(t)Σ+Υ⊤P(t)Υ-L],P(T)=(-ϕ1000).
And at the same time (4.7)–(4.10) reduce to(4.15)Ṗ1(t)=-(2b1+σ12+g12)P1(t)+2f2P2(t)+l1,P1(T)=-ϕ1,Ṗ2(t)=-(b2+σ1σ2-g1g2)P1(t)+(b1-f2)P2(t)-f1P3(t),P2(T)=0,Ṗ3(t)=-(σ22+g22)P1(t)-σ22P2(t)+2f2P3(t)-l2,P3(T)=0,
and Qi(·)=0,Ki(·,·)=0,i=1,2,3. Clearly, linear ODE (4.11) admits a unique explicit solution and then ODEs (4.12) do.Since the Hamiltonian function (2.12) reads(4.16)H(x,y,z,c(⋅),u,p,q,k,r(⋅))=(b1x+b2y+b3z+b4c(⋅)+b5u)p-(f1x+f2y+f3z+f4c(⋅)+f5u)q+(σ1x+σ2y+σ3z+σ4c(⋅)+σ5u)k+∫E(g1x+g2y+g3z+g4c(e)+g5u)r(e)π(de)-12(l1x2+l2y2+l3z2+l4c2(⋅)+l5u2),
the maximum condition (2.13) now takes the form (noting that K1(·,·)=0)(4.17)[b5p(t)-f5q(t)+σ5k(t)+∫Eg5r(t,e)π(de)][u¯(t)-u]-12l5[u¯2(t)-u2]-12[σ52+g52][u¯(t)-u]2P1(t)≥0,∀u∈U,t∈[0,T].
This implies that(4.18)b5p(t)-f5q(t)+σ5k(t)+∫Eg5r(t,e)π(de)≥12l5[u¯(t)+u]+12[σ52+g52][u¯(t)-u]P1(t),∀u∈U,u>u¯(t),t∈[0,T];b5p(t)-f5q(t)+σ5k(t)+∫Eg5r(t,e)π(de)≤12l5[u¯(t)+u]+12[σ52+g52][u¯(t)-u]P1(t),∀u∈U,u≤u¯(t),t∈[0,T].Now, let us look at an interesting special case. Noting that some related problems without random jumps were discussed by Shi and Wu [39] by a purely completion-of-squares technique.Suppose that(4.19)f1>0,b2≤0,σ3≤0,g4≤0,b1=f2,σ1=f3,g1=f4,σ2=-b3,g2=-b4,g3=-σ4.
It is very easy to check that the assumptions (H4.1) and (H4.2) hold. Now the first-order adjoint equation (4.4) reads(4.20)-dp(t)=[b1p(t)-f1q(t)+σ1k(t)+∫Eg1r(t,e)π(de)-l1x¯(t)]dt-k(t)dW(t)-∫Er(t,e)Ñ(dedt),dq(t)=[-b2p(t)+b1q(t)-σ2k(t)-∫Eg2r(t,e)π(de)+l2y¯(t)]dt+[σ2p(t)+σ1q(t)-σ3k(t)+∫Eσ4r(t,e)π(de)+l3z¯(t)]dW(t)+∫E[g2p(t-)+g1q(t-)-σ4k(t)-g4r(t,e)+l4c¯(t,e)]Ñ(dedt),p(T)=-ϕ1x¯(T)-h1q(T),q(0)=γ1y¯0,
and the second-order adjoint ODEs (4.12) reduce to(4.21)Ṗ1(t)=-(2b1+σ12+g12)P1(t)+2b1P2(t)+l1,P1(T)=-ϕ1,Ṗ2(t)=-(b2+σ1σ2-g1g2)P1(t)-f1P3(t),P2(T)=0,Ṗ3(t)=-(σ22+g22)P1(t)-σ22P2(t)+2b1P3(t)-l2,P3(T)=0.If, in addition,b1=σ2=0 (noting that this does not mean that the state equation (4.1) is decoupled), then (4.21) further reduce to(4.22)Ṗ1(t)=-(σ12+g12)P1(t)+l1,P1(T)=-ϕ1,(4.23)Ṗ2(t)=-(b2-g1g2)P1(t)-f1P3(t),P2(T)=0,(4.24)Ṗ3(t)=-g22P1(t)-l2,P3(T)=0.
We can first solve P1(·) explicitly from (4.22):(4.25)P1(t)=-e(σ12+g12)(T-t)[∫tTl1e-(σ12+g12)(T-s)ds+ϕ1],
and then solve P3(·) explicitly from (4.24). Finally, explicit P2(·) can be obtained from (4.23). The maximum condition (4.18) remains the same.However, we cannot obtain the explicit optimal control from the maximum condition (4.18) in general, since this depends heavily on the adjoint processes (p(·),q(·),k(·),r(·,·)). The explicit, adapted solution to fully coupled FBSDEJ such as (4.20) is an interesting and challenging open problem. And the explicit optimal control in its state feedback form is rather difficult to obtain even in some very simple cases when there are no random jumps. We will consider some relevant issues in the future work.


## 5. Concluding Remarks

In this paper, we have discussed a general optimal control problem for fully coupled forward-backward stochastic differential equations with random jumps (FBSDEJs). The control domain is not assumed to be convex, and the control variable appears in both diffusion and jump coefficients of the forward equation. Enlightened by Tang and Li [24], Wu [13], Yong [20], necessary conditions of Pontraygin’s type for the optimal controls are derived by means of spike variation technique and Ekeland variational principle. And the general maximum principle for forward stochastic control systems with random jumps [24] and the maximum principles for forward-backward stochastic control systems with random jumps [25, 26] are recovered in this paper. A linear quadratic stochastic optimal control problem is discussed as an illustrating example.



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*Source: 258674-2012-04-02.xml*