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

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.

1. Introduction

1.1. Basic Notations

Throughout this paper, we denote by 𝐑^𝑛 the space of 𝑛-dimensional Euclidean space, by 𝐑^𝑛×𝑑 the space of 𝑛×𝑑 matrices, and by 𝒮^𝑛 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,{ℱ_𝑡}_𝑡≥0,𝐏)be a complete filtered probability space satisfying the usual conditions, where the filtration {ℱ_𝑡}_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 measure 𝑁 on 𝐄×𝐑₊, where 𝐄⊂𝐑 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 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.

Let 𝑇>0 be fixed and 𝐔 be a nonempty subset of 𝐑^𝑘. We denote 𝕄=𝐑^𝑛×𝐑^𝑚×𝐑^𝑚×𝐑^𝑚. Any generic point in 𝕄 is denoted by 𝜃≡(𝑥,𝑦,𝑧,𝑐). Let 𝒰[0,𝑇] be the set of all ℱ_𝑡-predictable processes 𝑢∶[0,𝑇]×Ω→𝐔 such that sup_{0≤𝑡≤𝑇}𝔼|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;𝐑^𝑚) or ℒ² the set of integrable functions 𝑐∶𝐄→𝐑^𝑚 with norm ‖𝑐(𝑒)‖²_ℒ2∶=∫_𝐄|𝑐(𝑒)|²𝜋(𝑑𝑒)<∞. We define𝐿²_ℱ𝑇Ω;𝐑^𝑚∶=𝜉∶Ω⟶𝐑^𝑚∣𝜉isℱ_𝑇-measurable,𝔼𝜉²,𝐿<∞_2ℱ[0,𝑇];𝐑^𝑚∶=𝜑∶[0,𝑇]×Ω⟶𝐑^𝑚∣𝜑isℱ_𝑡-adapted,𝔼^𝑇₀𝜑(𝑡)²,𝐿𝑑𝑡<∞_2ℱ,𝑝[0,𝑇];𝐑^𝑚∶=𝜑∶[0,𝑇]×Ω⟶𝐑^𝑚∣𝜑isℱ_𝑡-predictable,𝔼^𝑇₀𝜑(𝑡)²,𝐹𝑑𝑡<∞_2𝑝[0,𝑇];𝐑^𝑚∶=𝜑∶[0,𝑇]×Ω×𝐄⟶𝐑^𝑚∣𝜑isℱ_𝑡-predictable,suchthat,𝔼^𝑇_0𝐄𝜑(𝑡,𝑒)²𝜋(𝑑𝑒)𝑑𝑡<∞. (1.1) We denoteℳ²[0,𝑇]∶=𝐿²_ℱ[0,𝑇];𝐑^𝑛×𝐿²_ℱ[0,𝑇];𝐑^𝑚×𝐿²_ℱ,𝑝[0,𝑇];𝐑^𝑚×𝐹²_𝑝[0,𝑇];𝐑^𝑚. (1.2) 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‖Θ(⋅)‖²_ℳ2[]0,𝑇∶=𝔼sup_{[]𝑡∈0,𝑇}|𝑥(𝑡)|²+sup_{[]𝑡∈0,𝑇}𝑦(𝑡)²+^𝑇₀|𝑧(𝑡)|²𝑑𝑡+𝔼^𝑇_0𝐄|𝑐(𝑡,𝑒)|²𝜋(𝑑𝑒)𝑑𝑡. (1.3)

1.2. Formulation of the Optimal Control Problem and Basic Assumptions

For any 𝑢(⋅)∈𝒰[0,𝑇] and 𝑎∈𝐑^𝑛, we consider the following fully coupled forward-backward stochastic control system:(𝑡)(𝑡)(𝑡)(𝑡)()(𝑡)(𝑡)(𝑡)(𝑡)()(𝑡)(𝑡)+𝑑𝑥=𝑏𝑡,𝑥,𝑦,𝑧,𝑐𝑡,⋅,𝑢𝑑𝑡+𝜎𝑡,𝑥,𝑦,𝑧,𝑐𝑡,⋅,𝑢𝑑𝑊_𝐄𝑔()()(𝑡)()(𝑡)𝑁(),(𝑡)(𝑡)(𝑡)(𝑡)()(𝑡(𝑡)(𝑡)−𝑡,𝑥𝑡−,𝑦𝑡−,𝑧,𝑐𝑡,⋅,𝑢,𝑒𝑑𝑒𝑑𝑡−𝑑𝑦=𝑓𝑡,𝑥,𝑦,𝑧,𝑐𝑡,⋅,𝑢)𝑑𝑡−𝑧𝑑𝑊_𝐄𝑐()𝑁(),𝑥(0)(𝑇)(𝑥(𝑇,𝑡,𝑒𝑑𝑒𝑑𝑡=𝑎,𝑦=ℎ)) (1.4) with the cost functional given by𝐽(𝑢(⋅))=𝔼^𝑇₀𝑙𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)𝑑𝑡+𝜙(𝑥(𝑇))+𝛾𝑦(0). (1.5) Here, Θ(⋅)≡(𝑥(⋅),𝑦(⋅),𝑧(⋅),𝑐(⋅,⋅)) takes value in 𝕄=𝐑^𝑛×𝐑^𝑚×𝐑^𝑚×𝐑^𝑚 and𝑏,𝜎∶[0,𝑇]×𝕄×𝐔⟶𝐑^𝑛,𝑔∶[0,𝑇]×𝕄×𝐔×𝐄⟶𝐑^𝑛,𝑓∶[0,𝑇]×𝕄×𝐔⟶𝐑^𝑚,ℎ∶𝐑^𝑛⟶𝐑^𝑚,𝜙∶𝐑^𝑛⟶𝐑,𝛾∶𝐑^𝑚⟶𝐑,𝑙∶[0,𝑇]×𝕄×𝐔⟶𝐑. (1.6)

For any 𝑢(⋅)∈𝒰[0,𝑇] and 𝑎∈𝐑^𝑛, 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 𝐽(𝑢(⋅))=inf_{𝑢(⋅)[]∈𝒰0,𝑇}𝐽(𝑢(⋅)). (1.7)

Any 𝑢(⋅)∈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 any 𝑢(⋅)∈𝒰[0,𝑇] and 𝑎∈𝐑^𝑛, FBSDEJ (1.4) admits a unique adapted solution Θ(⋅)≡(𝑥(⋅),𝑦(⋅),𝑧(⋅),𝑐(⋅,⋅))∈ℳ²[0,𝑇]. Moreover, the following estimate holds: ‖Θ(⋅)‖_ℳ2[0,𝑇]≡𝔼sup_{𝑡∈[0,𝑇]}|𝑥(𝑡)|²+sup_{𝑡∈[0,𝑇]}𝑦(𝑡)²+^𝑇₀|𝑧(𝑡)|²𝑑𝑡+𝔼^𝑇_0𝐄|𝑐(𝑡,𝑒)|²𝜋()𝑎|𝑑𝑒𝑑𝑡≤𝐶𝔼|²+|ℎ(0)|²+^𝑇₀|𝑏(𝑡,0,𝑢(𝑡))|𝑑𝑡²+^𝑇₀𝑓(𝑡,0,𝑢(𝑡))𝑑𝑡²+^𝑇₀|𝜎(𝑡,0,𝑢(𝑡))|²𝑑𝑡+^𝑇_0𝐄𝑔(𝑡,0,𝑢(𝑡),𝑒)²𝜋(𝑑𝑒)𝑑𝑡. (1.8)Further, if Θ(⋅)≡(𝑥(⋅),𝑦(⋅),𝑧(⋅),𝑐(̃⋅,⋅))∈ℳ²[0,𝑇]  is the unique adapted solution of (1.4) with ℎ(⋅)  and 𝑢(⋅)  replaced by ℎ∶𝐑^𝑛→𝐑^𝑚  and 𝑢(⋅)∈𝒰[0,𝑇], respectively, then the following stability estimate holds: Θ(⋅)−Θ(⋅)_ℳ2[0,𝑇]≡𝔼sup_{𝑡∈[0,𝑇]}|𝑥(𝑡)−𝑥(𝑡)|²+sup_{𝑡∈[0,𝑇]}𝑦(𝑡)−𝑦(𝑡)²+^𝑇₀|𝑧(𝑡)−𝑧(𝑡)|²𝑑𝑡+𝔼^𝑇_0𝐄|𝑐(𝑡,𝑒)−𝑐̃(𝑡,𝑒)|²𝜋()ℎ(𝑥(𝑇((𝑇𝑑𝑒𝑑𝑡≤𝐶𝔼|))−ℎ𝑥))|²+^𝑇₀𝑏(𝑡,Θ(𝑡),𝑢(𝑡))−𝑏𝑡,Θ(𝑡),𝑢(𝑡)𝑑𝑡²+^𝑇₀𝑓(𝑡,Θ(𝑡),𝑢(𝑡))−𝑓𝑡,Θ(𝑡),𝑢(𝑡)𝑑𝑡²+^𝑇₀𝜎(𝑡,Θ(𝑡),𝑢(𝑡))−𝜎𝑡,Θ(𝑡),𝑢(𝑡)²+𝑑𝑡_𝑇0𝐄𝑔(𝑡,Θ(𝑡),𝑢(𝑡),𝑒)−𝑔𝑡,Θ(𝑡),𝑢(𝑡),𝑒²𝜋(𝑑𝑒)𝑑𝑡. (1.9) 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 by forward-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 for discontinuous 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 developed reduction 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 C 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 C.

The rest of this paper is organized as follows. In Section 2, 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, maps 𝑡↦(𝑏(𝑡,𝜃,𝑢),𝜎(𝑡,𝜃,𝑢),𝑔(𝑡,𝜃,𝑢,𝑒),𝑓(𝑡,𝜃,𝑢)) is ℱ_𝑡-𝑝𝑟𝑜𝑔𝑟𝑒𝑠𝑠𝑖𝑣𝑒𝑙𝑦  measurable and there exists a constant 𝐿>0such that𝑏𝑡,𝑥,𝑦,𝑧,𝑐,𝑢−𝑏𝑡,𝑥,𝑦,𝑧,𝑐,𝑢²+𝜎𝑡,𝑥,𝑦,𝑧,𝑐,𝑢−𝜎𝑡,𝑥,𝑦,𝑧,𝑐,𝑢²+𝑔𝑡,𝑥,𝑦,𝑧,𝑐,𝑢,𝑒−𝑔𝑡,𝑥,𝑦,𝑧,𝑐,𝑢,𝑒²_ℒ2+𝑓𝑡,𝑥,𝑦,𝑧,𝑐,𝑢−𝑓𝑡,𝑥,𝑦,𝑧,𝑐,𝑢²≤𝐿|𝑥−𝑥|²+𝑦−𝑦²,∀(𝑡,𝑧,𝑐,𝑢)∈[0,𝑇]×𝐑^𝑚×𝐑^𝑛×𝐔,𝑥,𝑦,𝑥,𝑦∈𝐑^𝑛×𝐑^𝑚𝑏,𝑎.𝑠.,𝑡,𝑥,𝑦,𝑧,𝑐,𝑢²+𝜎𝑡,𝑥,𝑦,𝑧,𝑐,𝑢²+𝑔𝑡,𝑥,𝑦,𝑧,𝑐,𝑢,𝑒²_ℒ2+𝑓𝑡,𝑥,𝑦,𝑧,𝑐,𝑢²≤𝐿1+|𝑥|²+𝑦²,∀(𝑡,𝑧,𝑐,𝑢)∈[0,𝑇]×𝐑^𝑚×𝐑^𝑛×𝐔,𝑥,𝑦∈𝐑^𝑛×𝐑^𝑚,𝑎.𝑠. (2.1)(H2)Maps 𝜃↦(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 in 𝜃∈𝕄, and continuous in 𝑢∈𝐔.(H3)The map ℎ is twice differentiable with the derivatives up to the second order being uniformly bounded and uniformly Lipschitz continuous; the maps 𝜙,𝛾 are twice differentiable with the derivatives up to the second order being Lipschitz continuous; the map 𝜃↦𝑙(𝑡,𝜃,𝑢) is twice differentiable with the derivatives up to the second order being Lipschitz continuous in 𝜃∈𝕄, and continuous in 𝑢∈𝐔.

Next, to simplify our presentation, we now introduce some abbreviation notations. First, we make the following convention: for any differentiable map 𝑎≡(𝑎¹,𝑎²,…,𝑎^𝑙)^⊤∶𝐑^𝑛→𝐑^𝑙, let𝑎_𝑥=𝑎¹_𝑥1𝑎¹_𝑥2⋯𝑎¹_𝑥𝑛𝑎²_𝑥1𝑎²_𝑥2⋯𝑎²_𝑥𝑛𝑎⋮⋮⋱⋮_𝑙𝑥1𝑎^𝑙_𝑥2⋯𝑎^𝑙_𝑥𝑛∶𝐑^𝑛⟶𝐑^𝑙×𝑛. (2.2) In particular, for 𝑙=1,𝑎_𝑥=(𝑎_𝑥1,𝑎_𝑥2,…,𝑎_𝑥𝑛)∈𝐑^1×𝑛 is an 𝑛-dimensional row vector. Also, for any twice differentiable function 𝑎∶𝐑^𝑛→𝐑, the Hessian matrix is given by𝑎_𝑥𝑥≡𝑎^⊤_𝑥𝑥=𝑎_𝑥1𝑥1𝑎_𝑥1𝑥2⋯𝑎_{𝑥1𝑥𝑛}𝑎_𝑥2𝑥1𝑎_𝑥2𝑥2⋯𝑎_{𝑥2𝑥𝑛}𝑎⋮⋮⋱⋮_{𝑥𝑛𝑥1}𝑎_{𝑥𝑛𝑥2}⋯𝑎_{𝑥𝑛𝑥𝑛}∶𝐑^𝑛⟶𝒮^𝑛, (2.3) hereafter, 𝒮^𝑛 stands for the set of all real 𝑛×𝑛 symmetric matrices. On the other hand, for a twice differentiable function 𝑎∶𝐑^𝑛×𝐑^𝑚→𝐑 (denoting (𝑥,𝑦)∈𝐑^𝑛×𝐑^𝑚), we denote𝑎_𝑥𝑦≡𝑎^⊤_𝑥𝑦=𝑎_𝑥1𝑦1𝑎_𝑥1𝑦2⋯𝑎_{𝑥1𝑦𝑚}𝑎_𝑥2𝑦1𝑎_𝑥2𝑦2⋯𝑎_{𝑥2𝑦𝑚}𝑎⋮⋮⋱⋮_{𝑥𝑛𝑦1}𝑎_{𝑥𝑛𝑦2}⋯𝑎_{𝑥𝑛𝑦𝑚}∶𝐑^𝑛×𝐑^𝑚⟶𝐑^𝑛×𝑚. (2.4)

Now, let (𝑥(⋅),𝑦(⋅),𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅)) be an optimal 5-tuple of Problem C. For 𝑎(⋅)=𝑏(⋅),𝜎(⋅),𝑓(⋅),𝑙(⋅), we denote𝑎_𝑥(𝑡,⋅)=𝑎_𝑥𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡),𝑎_𝑦(𝑡,⋅)=𝑎_𝑦𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡),𝑎_𝑧(𝑡,⋅)=𝑎_𝑧𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡),𝑎_𝑐(𝑡,⋅)=𝑎_𝑐𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡), (2.5) and let𝑔_𝑥(𝑡,𝑒)=𝑔_𝑥𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,𝑒),𝑢(𝑡),𝑒,𝑔_𝑦(𝑡,𝑒)=𝑔_𝑦𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,𝑒),𝑢(𝑡),𝑒,𝑔_𝑧(𝑡,𝑒)=𝑔_𝑧𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,𝑒),𝑢(𝑡),𝑒,𝑔_𝑐(𝑡,𝑒)=𝑔_𝑐𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,𝑒),𝑢(𝑡),𝑒. (2.6) We also introduce𝐵_𝑋(𝑡,⋅)=𝑏_𝑥(𝑡,⋅)𝑏_𝑦()−𝑡,⋅𝑓_𝑥(𝑡,⋅)−𝑓_𝑦(),𝑡,⋅Σ_𝑋(𝑡,⋅)=𝜎_𝑥(𝑡,⋅)𝜎_𝑦(),𝑡,⋅00Υ_𝑋(𝑡,𝑒)=𝑔_𝑥(𝑡,𝑒)𝑔_𝑦().𝑡,𝑒00 (2.7) For 𝑎(⋅)=𝑏^𝑖(⋅),𝜎^𝑖(⋅),𝑓^𝑗(⋅),𝑙(⋅),1≤𝑖≤𝑛,1≤𝑗≤𝑚, let𝑎_𝑥𝑥(𝑡,⋅)=𝑎_𝑥𝑥𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡),𝑎_𝑦𝑦(𝑡,⋅)=𝑎_𝑦𝑦𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡),𝑎_𝑥𝑦(𝑡,⋅)=𝑎_𝑥𝑦𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡),𝑎_𝑦𝑥(𝑡,⋅)=𝑎_𝑦𝑥t,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡), (2.8) and let𝑔_𝑥𝑥(𝑡,𝑒)=𝑔_𝑥𝑥𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡),𝑒,𝑔_𝑦𝑦(𝑡,𝑒)=𝑔_𝑦𝑦𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡),𝑒,𝑔_𝑥𝑦(𝑡,𝑒)=𝑔_𝑥𝑦𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡),𝑒,𝑔_𝑦𝑥(𝑡,𝑒)=𝑔_𝑦𝑥𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡),𝑒. (2.9)

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(⋅,⋅))∈𝐿²_ℱ([0,𝑇];𝐑^𝑛)×𝐿²_ℱ([0,𝑇];𝐑^𝑚)×𝐿²_ℱ,𝑝([0,𝑇];𝐑^𝑛)×𝐹²_𝑝([0,𝑇];𝐑^𝑛) to the following FBSDEJ: −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)(𝑡)−𝑡,⋅)𝑑𝑡−𝑘𝑑𝑊_𝐄𝑟()𝑁(),(𝑡)=−𝑡,𝑒dedt𝑑𝑞𝑏_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,⋅)𝑑𝑡𝑏_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)+𝑡,⋅)𝑑𝑊_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(𝑁(),𝑝(𝑇)𝑡,𝑒)𝑑𝑒𝑑𝑡=−𝜙_𝑥(𝑥(𝑇))−ℎ_𝑥(𝑥(𝑇))^⊤𝑞(𝑇),𝑞(0)=𝔼𝛾_𝑦𝑦₀. (2.10) Let (𝑃(⋅),𝑄(⋅),𝐾(⋅,⋅))∈𝐿²_ℱ([0,𝑇];𝒮^𝑛+𝑚)×𝐿²_ℱ,𝑝([0,𝑇];𝒮^𝑛+𝑚)×𝐹²_𝑝([0,𝑇];𝒮^𝑛+𝑚) be the unique adapted solution to the following matrix-valued BSDEJ: −𝑑𝑃(𝑡)=𝐵_𝑋(𝑡,⋅)^⊤𝑃(𝑡)+𝑃(𝑡)𝐵_𝑋(𝑡,⋅)+Σ_𝑋(𝑡,⋅)^⊤𝑃(𝑡)Σ_𝑋(𝑡,⋅)+Σ_𝑋(𝑡,⋅)^⊤𝑄(𝑡)+𝑄(𝑡)Σ_𝑋()+𝑡,⋅_𝐄Υ_𝑋(𝑡,𝑒)^⊤𝑃(𝑡)Υ_𝑋(𝑡,𝑒)+Υ_𝑋(𝑡,𝑒)^⊤𝐾(𝑡,𝑒)Υ_𝑋()+𝑡,𝑒Υ_𝑋(𝑡,𝑒)^⊤𝐾(𝑡,𝑒)+𝐾(𝑡,𝑒)Υ_𝑋(𝑡,𝑒)𝜋(𝑑𝑒)+𝐻_𝑋𝑋((𝑡)(𝑡)−𝑡,⋅)𝑑𝑡−𝑄𝑑𝑊_𝐄𝐾()𝑁(),𝑃(𝑇)=𝑡,𝑒𝑑𝑒𝑑𝑡−𝜙_𝑥𝑥(𝑥(𝑇))−ℎ_𝑥𝑥(𝑥(𝑇))^⊤𝑞(𝑇)0,00 (2.11) 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×𝐑^𝑛×𝐑^𝑚×𝐑^𝑛×𝐑^𝑛→𝐑 is defined as follows: 𝐻(⋅)(⋅)(⋅)+(⋅)+(⋅)+𝑡,𝑥,𝑦,𝑧,𝑐,𝑢,𝑝,𝑞,𝑘,𝑟∶=𝑝,𝑏t,𝑥,𝑦,𝑧,𝑐,𝑢𝑞,−𝑓𝑡,𝑥,𝑦,𝑧,𝑐,𝑢𝑘,𝜎𝑡,𝑥,𝑦,𝑧,𝑐,𝑢_𝐄𝑟(𝑒),𝑔𝑡,𝑥,𝑦,𝑧,𝑐(𝑒),𝑢,𝑒𝜋(de)−𝑙𝑡,𝑥,𝑦,𝑧,𝑐(⋅),𝑢. (2.12) Then, 𝐻𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)(𝑡)(𝑡)(𝑡)()−,𝑝,𝑞,𝑘,𝑟𝑡,⋅𝐻𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐()(𝑡)(𝑡)(𝑡)()−1𝑡,⋅,𝑢,𝑝,𝑞,𝑘,𝑟𝑡,⋅2𝜎𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)−𝜎𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢^⊤𝑃₁(𝑡)×𝜎𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)−𝜎𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐()−1𝑡,⋅,𝑢2_𝐄𝑔𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,𝑒),𝑢(𝑡),𝑒−𝑔𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐𝑡,𝑔,𝑢,𝑒^⊤×P₁(𝑡)+𝐾₁(×𝑔𝑡,𝑒)𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,𝑒),𝑢(𝑡),𝑒−𝑔𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,𝑒),𝑢,𝑒𝜋(de)≥0,∀𝑢∈𝐔,𝑎.𝑒.,𝑎.𝑠., (2.13) with 𝑃(⋅)≡^{𝑃1(⋅)𝑃2}_𝑃^(⋅)_{2(⋅)⊤𝑃3(⋅)}, and 𝐾(⋅,⋅)≡^{𝐾1(⋅,⋅)𝐾2}_𝐾^(⋅,⋅)_{2(⋅,⋅)⊤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}_𝑄^(⋅)_{2(⋅)⊤𝑄3(⋅)}: −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,⋅_𝐄𝑔_𝑥(𝑡,𝑒)^⊤𝑃₁(𝑡)𝑔_𝑥(𝑡,𝑒)+𝑔_𝑥(𝑡,𝑒)^⊤𝐾₁(𝑡,𝑒)𝑔_𝑥()+𝑡,𝑒𝑔_𝑥(𝑡,𝑒)^⊤𝐾₁(𝑡,𝑒)+𝐾₁(𝑡,𝑒)𝑔_𝑥(𝑡,𝑒)𝜋(𝑑𝑒)+𝐻_𝑥𝑥(𝑡,⋅)𝑑𝑡−𝑄₁(𝑡)𝑑𝑊(𝑡)−_𝐄𝐾₁()𝑁(),𝑃𝑡,𝑒𝑑𝑒𝑑𝑡₁(𝑇)=−𝜙_𝑥𝑥(𝑥(𝑇))−ℎ_𝑥𝑥(𝑥(𝑇))^⊤𝑞(𝑇),−𝑑𝑃₂(𝑡)=𝑃₁(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,⋅_𝐄𝑔_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()𝑁(),𝑃𝑡,𝑒𝑑𝑒𝑑𝑡₂(𝑇)=0,−𝑑𝑃₃(𝑡)=𝑏_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,⋅_𝐄𝑔_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,⋅)𝑑𝑡−𝑄₃(𝑡)𝑑𝑊(𝑡)−_𝐄𝐾₃()𝑁(),𝑃𝑡,𝑒𝑑𝑒𝑑𝑡₃(𝑇)=0. (2.14)(2.15)(2.16)

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: 𝑑𝑥(𝑡)=𝑏(𝑡,𝑥(𝑡),𝑢(𝑡))𝑑𝑡+𝜎(𝑡,𝑥(𝑡),𝑢(𝑡))𝑑𝑊(𝑡)+_𝐄𝑔(()(𝑡))𝑁(),𝑥(0)𝑡,𝑥𝑡−,𝑢,𝑒𝑑𝑒𝑑𝑡=𝑎, (2.17) with the cost functional 𝐽(𝑢(⋅))=𝔼^𝑇₀𝑙(𝑡,𝑥(𝑡),𝑢(𝑡))𝑑𝑡+𝜙(𝑥(𝑇)). (2.18)
In this case, 𝑏(⋅)()(⋅)(),𝑔(⋅)()(⋅)𝑙(⋅)()𝑦𝑡,𝑥,𝑦,𝑧,𝑐,𝑢=𝑏𝑡,𝑥,𝑢,𝜎𝑡,𝑥,𝑦,𝑧,𝑐,𝑢=𝜎𝑡,𝑥,𝑢𝑡,𝑥,𝑦,𝑧,𝑐,𝑢,⋅=𝑔𝑡,𝑥,𝑢,⋅,𝑓𝑡,𝑥,𝑦,𝑧,𝑐,𝑢=0,𝑡,𝑥,𝑦,𝑧,𝑐,𝑢,⋅=𝑙𝑡,𝑥,𝑢,𝛾=0. (2.19) Hence, we have, by some direct computation/observation, 𝑦(⋅)(⋅)()(⋅)𝑃=𝑧=𝑐⋅,⋅=0,𝑞=0,₂(⋅)=𝑄₂(⋅)=𝐾₂(⋅,⋅)=0,𝑃₃(⋅)=𝑄₃(⋅)=𝐾₃(⋅,⋅)=0. (2.20) Consequently, (2.10) and (2.11) (or equivalently, (2.14)) become −𝑑𝑝(𝑡)=𝑏_𝑥(𝑡,⋅)^⊤𝑝(𝑡)+𝜎_𝑥(𝑡,⋅)^⊤𝑘(t)+_𝐄𝑔_𝑥(𝑡,𝑒)^⊤𝑟(𝑡,𝑒)𝜋(𝑑𝑒)−𝑙_𝑥((𝑡)(𝑡)−𝑡,⋅)𝑑𝑡−𝑘𝑑𝑊_𝐄𝑟()𝑁(),𝑝(𝑇)𝑡,𝑒𝑑𝑒𝑑𝑡=𝜙_𝑥(𝑥(𝑇)),−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,⋅_𝐄𝑔_𝑥(𝑡,𝑒)^⊤𝑃₁(𝑡)𝑔_𝑥(𝑡,𝑒)+𝑔_𝑥(𝑡,𝑒)^⊤𝐾₁(𝑡,𝑒)𝑔_𝑥()+𝑡,𝑒𝑔_𝑥(𝑡,𝑒)^⊤𝐾₁(𝑡,𝑒)+𝐾₁(𝑡,𝑒)𝑔_𝑥(𝑡,𝑒)𝜋(𝑑𝑒)+𝐻_𝑥𝑥(𝑡,⋅)𝑑𝑡−𝑄₁(𝑡)𝑑𝑊(𝑡)−_𝐄𝐾₁()𝑁(),𝑃𝑡,𝑒𝑑𝑒𝑑𝑡₁(𝑇)=−𝜙_𝑥𝑥(𝑥(𝑇)), (2.21)(2.22) where the Hamiltonian function 𝐻∶[0,𝑇]×𝐑^𝑛×𝐔×𝐑^𝑛×𝐑^𝑛×𝐑^𝑛→𝐑 is defined as 𝐻(⋅)(+⟨(+𝑡,𝑥,𝑢,𝑝,𝑘,𝑟∶=𝑝,𝑏𝑡,𝑥,𝑢)𝑘,𝜎𝑡,𝑥,𝑢)⟩_𝐄𝑟(𝑒),𝑔(𝑡,𝑥,𝑢,𝑒)𝜋(𝑑𝑒)−𝑙(𝑡,𝑥,𝑢),a.s. (2.23) The maximum condition (2.13) is reduced to the following: 𝐻𝑡,𝑥(𝑡),𝑢(𝑡),𝑝(𝑡),𝑘(𝑡),𝑟(𝑡,⋅)−𝐻𝑡,𝑥(𝑡)(𝑡)(𝑡)()−1,𝑢,𝑝,𝑘,𝑟𝑡,⋅2(𝜎(𝑡,𝑥(𝑡),𝑢(𝑡))−𝜎(𝑡,𝑥(𝑡),𝑢))^⊤𝑃₁(𝑡)(𝜎(𝑡,𝑥(𝑡),𝑢(𝑡))−𝜎(𝑡,𝑥(𝑡)−1,𝑢))2_𝐄𝑔(𝑡,𝑥(𝑡),𝑢(𝑡),𝑒)−𝑔(𝑡,𝑥(𝑡),𝑢,𝑒)^⊤𝑃₁(𝑡)+𝐾₁(×𝑔(𝑡,𝑒)𝑡,𝑥(𝑡),𝑢(𝑡),𝑒)−𝑔(𝑡,𝑥(𝑡),𝑢,𝑒)𝜋(𝑑𝑒)≥0,∀𝑢∈𝐔,a.e.,a.s. (2.24) 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: −𝑑𝑦(𝑡)=𝑓𝑡,𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)𝑑𝑡−𝑧(𝑡)𝑑𝑊(𝑡)−_𝐄𝑐()𝑁(),𝑦(𝑇)𝑡,𝑒𝑑𝑒𝑑𝑡=ℎ_𝑇, (2.25) with the cost functional 𝐽(𝑢(⋅))=𝔼^𝑇₀𝑙𝑡,𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)𝑑𝑡+𝛾𝑦(0). (2.26) In this case, 𝑏(⋅)(⋅)(⋅)𝑓(⋅)(⋅)(⋅)(⋅)(𝑥)𝑡,𝑥,𝑦,𝑧,𝑐,𝑢=𝜎𝑡,𝑥,𝑦,𝑧,𝑐,𝑢=𝑔𝑡,𝑥,𝑦,𝑧,𝑐,𝑢,⋅=0,𝑡,𝑥,𝑦,𝑧,𝑐,𝑢=𝑓𝑡,𝑦,𝑧,𝑐,𝑢,𝑙𝑡,𝑥,𝑦,𝑧,𝑐,𝑢=𝑔𝑡,𝑦,𝑧,𝑐,𝑢,⋅,𝜙=0. (2.27) Hence, we have 𝑥(⋅)(⋅)(⋅)()𝑃=0,𝑝=𝑘=𝑟⋅,⋅=0,₁(⋅)=𝑄₁(⋅)=𝐾₁(⋅,⋅)=0,𝑃₂(⋅)=𝑄₂(⋅)=𝐾₂(⋅,⋅)=0. (2.28) Consequently, (2.10) and (2.11) (or equivalently, (2.16)) become 𝑑𝑞(𝑡)=𝑓_𝑦(𝑡,⋅)^⊤𝑞(𝑡)+𝑙_𝑦(𝑡,⋅)𝑑𝑡−𝑓_𝑧(𝑡,⋅)^⊤𝑞(𝑡)+𝑙_𝑧((𝑡)−𝑡,⋅)𝑑𝑊_𝐄𝑓_𝑐(𝑡,𝑒)^⊤𝑞(𝑡)+𝑙_𝑐(𝑁(),𝑞(0)𝑡,𝑒)𝑑𝑒𝑑𝑡=𝔼𝛾_𝑦𝑦₀;−𝑑𝑃₃(𝑡)=−𝑓_𝑦(𝑡,⋅)^⊤𝑃₃(𝑡)−𝑃₃(𝑡)𝑓_𝑦(𝑡,⋅)+_𝐄𝑔_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,⋅)𝑑𝑡−𝑄₃(𝑡)𝑑𝑊(𝑡)−_𝐄𝐾₃()𝑁(),𝑃𝑡,𝑒𝑑𝑒𝑑𝑡₃(𝑇)=0. (2.29) The maximum condition (2.13) is reduced to the following: 𝑞(𝑡),𝑓𝑡,𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)−𝑓𝑡,𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢+𝑙𝑡,𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)−𝑙𝑡,𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢≤0,∀𝑢∈𝐔,a.e.,a.s. (2.30) 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: 𝑑𝑥(𝑡)=𝑏(𝑡,𝑥(𝑡),𝑢(𝑡))𝑑𝑡+𝜎(𝑡,𝑥(𝑡),𝑢(𝑡))𝑑𝑊(𝑡)+_𝐄𝑔(()(𝑡))𝑁(),(𝑡)(𝑡)(𝑡)(𝑡)()(𝑡)(𝑡)(𝑡)𝑡,𝑥𝑡−,𝑢,𝑒𝑑𝑒𝑑𝑡−𝑑𝑦=𝑓𝑡,𝑥,𝑦,𝑧,𝑐𝑡,⋅,𝑢𝑑𝑡−𝑧𝑑𝑊−_𝐄𝑐()𝑁(),𝑥(0)(𝑇)(𝑥(𝑇,𝑡,𝑒𝑑𝑒𝑑𝑡=𝑎,𝑦=ℎ)) (2.31) with the cost functional 𝐽(𝑢(⋅))=𝔼^𝑇₀𝑙𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)𝑑𝑡+𝜙(𝑥(𝑇))+𝛾𝑦(0). (2.32)
In this case, we have 𝑏(⋅)()(⋅)(),𝑔(⋅)().𝑡,𝑥,𝑦,𝑧,𝑐,𝑢=𝑏𝑡,𝑥,𝑢,𝜎𝑡,𝑥,𝑦,𝑧,𝑐,𝑢=𝜎𝑡,𝑥,𝑢𝑡,𝑥,𝑦,𝑧,𝑐,𝑢,⋅=𝑔𝑡,𝑥,𝑢,⋅ (2.33) Consequently, (2.10) becomes −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)(𝑡)−𝑡,⋅)𝑑𝑡−𝑘𝑑𝑊_𝐄𝑟()𝑁(),(𝑡)=𝑡,𝑒𝑑𝑒𝑑𝑡𝑑𝑞𝑓_𝑦(𝑡,⋅)^⊤𝑞(𝑡)+𝑙_𝑦(𝑡,⋅)𝑑𝑡−𝑓_𝑧(𝑡,⋅)^⊤𝑞(𝑡)+𝑙_𝑧((𝑡)−𝑡,⋅)𝑑𝑊_𝐄𝑓_𝑐(𝑡,𝑒)^⊤𝑞(𝑡)+𝑙_𝑐(𝑁(),𝑝(𝑇)𝑡,𝑒)𝑑𝑒𝑑𝑡=𝜙_𝑥(𝑥(𝑇))+ℎ_𝑥(𝑥(𝑇))^⊤𝑞(𝑇),𝑞(0)=𝔼𝛾_𝑦𝑦₀, (2.34) and (2.14), (2.15), (2.16) become −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,⋅_𝐄𝑔_𝑥(𝑡,𝑒)^⊤𝑃₁(𝑡)𝑔_𝑥(𝑡,𝑒)+𝑔_𝑥(𝑡,𝑒)^⊤𝐾₁(𝑡,𝑒)𝑔_𝑥()+𝑡,𝑒𝑔_𝑥(𝑡,𝑒)^⊤𝐾₁(𝑡,𝑒)+𝐾₁(𝑡,𝑒)𝑔_𝑥(𝑡,𝑒)𝜋(𝑑𝑒)+𝐻_𝑥𝑥(𝑡,⋅)𝑑𝑡−𝑄₁(𝑡)𝑑𝑊(𝑡)−_𝐄𝐾₁()𝑁(),𝑃𝑡,𝑒𝑑𝑒𝑑𝑡₁(𝑇)=−𝜙_𝑥𝑥(𝑥(𝑇))−ℎ_𝑥𝑥(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,⋅_𝐄𝑔_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()𝑁(),𝑃𝑡,𝑒𝑑𝑒𝑑𝑡₂(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()𝑁(),𝑃𝑡,𝑒𝑑𝑒𝑑𝑡₃(𝑇)=0, (2.35) where the Hamiltonian function 𝐻∶[0,𝑇]×𝕄×𝐔×𝐑^𝑛×𝐑^𝑚×𝐑^𝑛×𝐑^𝑛→𝐑 is defined as follows: 𝐻(⋅)(⋅)(+(⋅)+⟨(+𝑡,𝑥,𝑦,𝑧,𝑐,𝑢,𝑝,𝑞,𝑘,𝑟∶=𝑝,𝑏𝑡,𝑥,𝑢)𝑞,−𝑓𝑡,𝑥,𝑦,𝑧,𝑐,𝑢𝑘,𝜎𝑡,𝑥,𝑢)⟩_𝐄𝑟(𝑒),𝑔(𝑡,𝑥,𝑢,𝑒)𝜋(𝑑𝑒)−𝑙𝑡,𝑥,𝑦,𝑧,𝑐(⋅),𝑢. (2.36) 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: (𝑡)(𝑡)(𝑡)(𝑡)()(𝑡)(𝑡)(𝑡)(𝑡)()(𝑡)(𝑡)+𝑑𝑥=𝑏𝑡,𝑥,𝑦,𝑧,𝑐𝑡,⋅,𝑢𝑑𝑡+𝜎𝑡,𝑥,𝑦,𝑧,𝑐𝑡,⋅,𝑢𝑑𝑊_𝐄𝑔()()(𝑡)()(𝑡)𝑁(),(𝑡)(𝑡)(𝑡)(𝑡)()(𝑡(𝑡)(𝑡)+𝑡,𝑥𝑡−,𝑦𝑡−,𝑧,𝑐𝑡,⋅,𝑢,𝑒𝑑𝑒𝑑𝑡𝑑𝑦=−𝑓𝑡,𝑥,𝑦,𝑧,𝑐𝑡,⋅,𝑢)𝑑𝑡+𝑧𝑑𝑊_𝐄𝑐()𝑁(),𝑥(0)(0)𝑡,𝑒𝑑𝑒𝑑𝑡=𝑎,𝑦=𝑦₀, (3.1) 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(⋅))∈𝐿²_ℱ,𝑝([0,𝑇];𝐑^𝑚)×𝐹²_𝑝([0,𝑇];𝐑^𝑚)×𝒰[0,𝑇] and 𝑦₀∈𝐑^𝑚, there exists a unique strong solution 𝑥(⋅),𝑦(⋅)≡𝑥⋅,𝑦₀,𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅),𝑦⋅,𝑦₀,𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅)∈𝐿²_ℱ[0,𝑇];𝐑^𝑛×𝐿²_ℱ[0,𝑇];𝐑^𝑚 (3.2) to (3.1), depending on the 4-tuple (𝑦₀,𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅)). Next, we regard the original terminal condition as the terminal state constraint: 𝑦(𝑇)=ℎ(𝑥(𝑇)). (3.3)

Let 𝒜 be the set of all 4-tuples (𝑦₀,𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅))∈𝐑^𝑚×𝐿²_ℱ,𝑝([0,𝑇];𝐑^𝑚)×𝐹²_𝑝([0,𝑇];𝐑^𝑚)×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(⋅,⋅))∈𝐑^𝑚×𝐿²_ℱ,𝑝([0,𝑇];𝐑^𝑚)×𝐹²_𝑝([0,𝑇];𝐑^𝑚) such that state equation (3.1) admits a unique state process (𝑥(⋅),𝑦(⋅))∈𝐿²_ℱ([0,𝑇];𝐑^𝑛)×𝐿²_ℱ([0,𝑇];𝐑^𝑚) satisfying the state constraint (3.3). Hence, (H0) and (H1) imply 𝒜≠∅. We rewrite the cost functional (1.5) as follows:𝐽𝑦₀,𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅)=𝔼^𝑇₀𝑙𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)𝑑𝑡+𝜙(𝑥(𝑇))+𝛾𝑦₀. (3.4)

Now, we can pose the following.

Problem 𝐶. Find (𝑦₀,𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅))∈𝒜, such that𝐽𝑦₀,𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅)=inf_{(𝑦0(⋅),𝑧,𝑐(⋅,⋅),𝑢(⋅))∈𝒜}𝐽𝑦₀,𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅). (3.5)

We, respectively, refer to (𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅)) as an optimal control process, to (𝑥(⋅),𝑦(⋅)) as the corresponding optimal state process, and to (𝑦₀,𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅)) as an optimal 4-tuple of Problem C.

It is clear that Problems C and C are equivalent. The advantage of Problem C 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 𝑓(𝑢)≤inf_𝑣∈𝑉𝑓(𝑣)+𝜌, (3.6) then there exists 𝑢_𝜌∈𝑉, satisfying the following:(i)𝑓(𝑢_𝜌)≤𝑓(𝑢), (ii)𝑑(𝑢,𝑢_𝜌)≤√𝜌, (iii)forall𝑣∈𝑉,𝑓(𝑣)+√𝜌𝑑(𝑣,𝑢_𝜌)≥𝑓(𝑢_𝜌).

Let (𝑦₀,𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅)) be an optimal 4-tuple of Problem C, with the corresponding optimal state process (𝑥(⋅),𝑦(⋅)). For any 𝛿>0, we define𝐽^𝛿𝑦₀(⋅)()(⋅)≐𝐽𝑦,𝑧,𝑐⋅,⋅,𝑢₀,𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅)−𝐽𝑦₀,𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅)+𝛿²+𝔼𝑦(𝑇)−ℎ(𝑥(𝑇))^21/2∀𝑦₀,𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅)∈𝐑^𝑚×𝐿²_ℱ,𝑝[0,𝑇];𝐑^𝑚×𝐹²_𝑝[0,𝑇];𝐑^𝑚×𝒰[0,𝑇]. (3.7)

If we define the Ekeland’s distance on 𝒰[0,𝑇] by𝑑_𝐸(𝑢(⋅),𝑢(⋅))≐_Ω|{𝑡∈[0,𝑇],𝑢(𝑡)≠𝑢(𝑡)}|𝑑𝐏,∀𝑢(⋅),𝑢(⋅)∈𝒰[0,𝑇], (3.8)with |𝐴| being the Lebesgue measure of set 𝐴⊆[0,𝑇], then 𝐿²_ℱ,𝑝([0,𝑇];𝐑^𝑚)×𝐹²_𝑝([0,𝑇];𝐑^𝑚)×𝒰[0,𝑇] is a complete metric space under the following metric:‖𝑧(⋅)−𝑧(⋅)‖²+‖𝑐(⋅,⋅)−𝑐̃(⋅,⋅)‖²_𝜋+𝑑_𝐸(𝑢(⋅),𝑢(⋅))^21/2,∀(𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅)),(𝑧(⋅),𝑐̃(⋅,⋅),𝑢(⋅))∈𝐿²_ℱ,𝑝[0,𝑇];𝐑^𝑚×𝐹²_𝑝[0,𝑇];𝐑^𝑚×𝒰[0,𝑇], (3.9)where ‖𝑧(⋅)‖²≐𝔼∫^𝑇₀|𝑧(𝑡)|²𝑑𝑡, ‖𝑐(⋅,⋅)‖²_𝜋≐𝔼∫^𝑇₀∫_𝐄|𝑐(𝑡,𝑒)|²𝜋(𝑑𝑒)𝑑𝑡, and (𝑦₀,𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅))↦𝐽^𝛿(𝑦₀,𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅)) is continuous. Also it is clear that𝐽^𝛿𝑦₀(⋅)()(⋅)𝑦,𝑧,𝑐⋅,⋅,𝑢>0,∀₀,𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅)∈𝐑^𝑚×𝐿²_ℱ,𝑝[0,𝑇];𝐑^𝑚×𝐹²_𝑝[0,𝑇];𝐑^𝑚[],𝐽×𝒰0,𝑇_𝛿𝑦₀,𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅)=𝛿≤inf_{(𝑦0,𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅))∈𝐑𝑚×𝐿2ℱ,𝑝([0,𝑇];𝐑𝑚)×𝐹2𝑝([0,𝑇];𝐑𝑚)×𝒰[0,𝑇]}𝐽^𝛿𝑦₀,𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅)+𝛿. (3.10) Hence, by Lemma 3.1, there exists a 4-tuple (𝑦^𝛿₀,𝑧^𝛿(⋅),𝑐^𝛿(⋅,⋅),𝑢^𝛿(⋅))∈𝐑^𝑚×𝐿²_ℱ,𝑝([0,𝑇];𝐑^𝑚)×𝐹²_𝑝([0,𝑇];𝐑^𝑚)×𝒰[0,𝑇] such that(i)𝐽^𝛿𝑦^𝛿₀,𝑧^𝛿(⋅),𝑐^𝛿(⋅,⋅),𝑢^𝛿(⋅)≤𝐽^𝛿𝑦₀,𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅)(𝑧=𝛿,ii)^𝛿(⋅)−𝑧(⋅)²+𝑐^𝛿(⋅,⋅)−𝑐(⋅,⋅)²_𝜋+𝑑_𝐸𝑢^𝛿(⋅),𝑢(⋅)²()−√≤𝛿,iii𝛿𝑧^𝛿(⋅)−𝑧(⋅)²+𝑐^𝛿(⋅,⋅)−𝑐(⋅,⋅)²_𝜋+𝑑_𝐸𝑢^𝛿(⋅),𝑢(⋅)^21/2≤𝐽^𝛿𝑦₀,𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅)−𝐽^𝛿𝑦^𝛿₀,𝑧^𝛿(⋅),𝑐^𝛿(⋅,⋅),𝑢^𝛿(⋅),∀𝑦₀,𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅)∈𝐑^𝑚×𝐿²_ℱ,𝑝[0,𝑇];𝐑^𝑚×𝐹²_𝑝[0,𝑇];𝐑^𝑚×𝒰[0,𝑇]. (3.11) Thus, (𝑦^𝛿₀,𝑧^𝛿(⋅),𝑐^𝛿(⋅,⋅),𝑢^𝛿(⋅)) is a global minimum point of the following penalized cost functional𝐽^𝛿𝑦₀,𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅)+√𝛿𝑧^𝛿(⋅)−𝑧(⋅)²+𝑐^𝛿(⋅,⋅)−𝑐(⋅,⋅)²_𝜋+𝑑_𝐸𝑢^𝛿(⋅),𝑢(⋅)^21/2. (3.12) In other words, if we pose a penalized optimal control problem with the state constraint (3.3) and the cost functional (3.12), then (𝑦^𝛿₀,𝑧^𝛿(⋅),𝑐^𝛿(⋅,⋅),𝑢^𝛿(⋅)) is an optimal 4-tuple of the problem. Note that this problem does not have state constraints, and the optimal 4-tuple (𝑦^𝛿₀,𝑧^𝛿(⋅),𝑐^𝛿(⋅,⋅),𝑢^𝛿(⋅)) approaches (𝑦₀,𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅)) as 𝛿→0.

Step 3 (nontriviality of the multiplier). Note that the state process (𝑥^𝛿(⋅),𝑦^𝛿(⋅)) corresponding to the 4-tuple (𝑦^𝛿₀,𝑧^𝛿(⋅),𝑐^𝛿(⋅,⋅),𝑢^𝛿(⋅)) satisfies the following SDEJ: 𝑑𝑥^𝛿(𝑡)=𝑏𝑡,𝑥^𝛿(𝑡),𝑦^𝛿(𝑡),𝑧^𝛿(𝑡),𝑐^𝛿(𝑡,⋅),𝑢^𝛿(𝑡)𝑑𝑡+𝜎𝑡,𝑥^𝛿(𝑡),𝑦^𝛿(𝑡),𝑧^𝛿(𝑡),𝑐^𝛿(𝑡,⋅),𝑢^𝛿(𝑡)(𝑡)+𝑑𝑊_𝐄𝑔𝑡,𝑥^𝛿(𝑡−),𝑦^𝛿(𝑡−),𝑧^𝛿(𝑡),𝑐^𝛿(𝑡,⋅),𝑢^𝛿(𝑡),𝑒𝑁(𝑑𝑒𝑑𝑡),𝑑𝑦^𝛿(𝑡)=−𝑓𝑡,𝑥^𝛿(𝑡),𝑦^𝛿(𝑡),𝑧^𝛿(𝑡),𝑐^𝛿(𝑡,⋅),𝑢^𝛿(𝑡)𝑑𝑡+𝑧^𝛿(𝑡)𝑑𝑊(𝑡)+_𝐄𝑐^𝛿()𝑁(),𝑥𝑡,𝑒𝑑𝑒𝑑𝑡_𝛿(0)=𝑎,𝑦^𝛿(0)=𝑦^𝛿₀, (3.13) with 𝔼𝑦^𝛿(𝑇)−ℎ𝑥^𝛿(𝑇)²≤𝐽^𝛿𝑦^𝛿₀,𝑧^𝛿(⋅),𝑐^𝛿(⋅,⋅),𝑢^𝛿(⋅)²≤𝐽^𝛿𝑦₀,𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅)²=𝛿²⟶0,as𝛿⟶0. (3.14)

We now regard Θ^𝛿(⋅)≡(𝑥^𝛿(⋅),𝑦^𝛿(⋅),𝑧^𝛿(⋅),𝑐^𝛿(⋅,⋅)) as the unique solution to the following FBSDEJ:𝑑𝑥^𝛿(𝑡)=𝑏𝑡,𝑥^𝛿(𝑡),𝑦^𝛿(𝑡),𝑧^𝛿(𝑡),𝑐^𝛿(𝑡,⋅),𝑢^𝛿(𝑡)𝑑𝑡+𝜎𝑡,𝑥^𝛿(𝑡),𝑦^𝛿(𝑡),𝑧^𝛿(𝑡),𝑐^𝛿(𝑡,⋅),𝑢^𝛿(𝑡)(𝑡)+𝑑𝑊_𝐄𝑔𝑡,𝑥^𝛿(𝑡−),𝑦^𝛿(𝑡−),𝑧^𝛿(𝑡),𝑐^𝛿(𝑡,⋅),𝑢^𝛿(𝑡),𝑒𝑁(𝑑𝑒𝑑𝑡),−𝑑𝑦^𝛿(𝑡)=𝑓𝑡,𝑥^𝛿(𝑡),𝑦^𝛿(𝑡),𝑧^𝛿(𝑡),𝑐^𝛿(𝑡,⋅),𝑢^𝛿(𝑡)𝑑𝑡−𝑧^𝛿(𝑡)𝑑𝑊(𝑡)−_𝐄𝑐^𝛿()𝑁(),𝑥𝑡,𝑒𝑑𝑒𝑑𝑡_𝛿(0)=𝑎,𝑦^𝛿(𝑇)=ℎ𝑦^𝛿(𝑇)+ℎ^𝛿_𝑇, (3.15) withℎ^𝛿_𝑇≡𝑦^𝛿(𝑇)−ℎ𝑥^𝛿(𝑇). (3.16)

Next, for any 𝜂_𝑇∈𝐿²_ℱ𝑇(Ω;𝐑^𝑚), with𝔼𝜂_𝑇²≤1, (3.17) by (H0), the following FBSDEJ𝑑𝑥^𝛿,𝜀(𝑡)=𝑏𝑡,𝑥^𝛿,𝜀(𝑡),𝑦^𝛿,𝜀(𝑡),𝑧^𝛿,𝜀(𝑡),𝑐^𝛿,𝜀(𝑡,⋅),𝑢^𝛿(𝑡)𝑑𝑡+𝜎𝑡,𝑥^𝛿,𝜀(𝑡),𝑦^𝛿,𝜀(𝑡),𝑧^𝛿,𝜀(𝑡),𝑐^𝛿,𝜀(𝑡,⋅),𝑢^𝛿(𝑡)(𝑡)+𝑑𝑊_𝐄𝑔𝑡,𝑥^𝛿,𝜀(𝑡−),𝑦^𝛿,𝜀(𝑡−),𝑧^𝛿,𝜀(𝑡),𝑐^𝛿,𝜀(𝑡,⋅),𝑢^𝛿(𝑡),𝑒𝑁(𝑑𝑒𝑑𝑡),−𝑑𝑦^𝛿,𝜀(𝑡)=𝑓𝑡,𝑥^𝛿,𝜀(𝑡),𝑦^𝛿,𝜀(𝑡),𝑧^𝛿,𝜀(𝑡),𝑐^𝛿,𝜀(𝑡,⋅),𝑢^𝛿(𝑡)𝑑𝑡−𝑧^𝛿,𝜀(𝑡)𝑑𝑊(𝑡)−_𝐄𝑐^𝛿,𝜀()𝑁(),𝑥𝑡,𝑒𝑑𝑒𝑑𝑡_𝛿,𝜀(0)=𝑎,𝑦^𝛿,𝜀(𝑇)=ℎ𝑦^𝛿,𝜀(𝑇)+ℎ^𝛿_𝑇+𝜀𝜂_𝑇, (3.18) admits a unique adapted solution Θ^𝛿,𝜀(⋅)≡(𝑥^𝛿,𝜀(⋅),𝑦^𝛿,𝜀(⋅),𝑧^𝛿,𝜀(⋅),𝑐^𝛿,𝜀(⋅,⋅)). Note that FBSDEJ (3.18) is nothing but FBSDEJ (3.15) with only ℎ^𝛿_𝑇 replaced by ℎ^𝛿_𝑇+𝜀𝜂_𝑇, and 𝑢^𝛿(⋅) remains unchanged. Thus, by (1.9), we haveΘ^𝛿,𝜀(⋅)−Θ^𝛿(⋅)_ℳ2[0,𝑇]≤𝐾𝜀²,as𝜀⟶0, (3.19) which implieslim_𝜀→0𝑦^𝛿,𝜀(0)−𝑦^𝛿(0)²+𝔼^𝑇₀𝑧^𝛿,𝜀(𝑡)−𝑧^𝛿(𝑡)²𝑑𝑡+𝔼^𝑇_0𝐄𝑐^𝛿,𝜀(𝑡,𝑒)−𝑐^𝛿(𝑡,𝑒)²𝜋(𝑑𝑒)𝑑𝑡=0. (3.20) Hence, with 𝑦₀^𝛿,𝜀=𝑦^𝛿,𝜀(0), by taking (𝑦₀,𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅))=(𝑦₀^𝛿,𝜀,𝑧^𝛿,𝜀(⋅),𝑐^𝛿,𝜀(⋅,⋅),𝑢^𝛿(⋅)) in the last relation in (3.11), we have −𝐾√𝛿𝜀≤𝐽^𝛿𝑦₀^𝛿,𝜀,𝑧^𝛿,𝜀(⋅),𝑐^𝛿,𝜀(⋅,⋅),𝑢^𝛿(⋅)−𝐽^𝛿𝑦^𝛿₀,𝑧^𝛿(⋅),𝑐^𝛿(⋅,⋅),𝑢^𝛿(⋅)=𝐽^𝛿𝑦₀^𝛿,𝜀,𝑧^𝛿,𝜀(⋅),𝑐^𝛿,𝜀(⋅,⋅),𝑢^𝛿(⋅)²−𝐽^𝛿𝑦^𝛿₀,𝑧^𝛿(⋅),𝑐^𝛿(⋅,⋅),𝑢^𝛿(⋅)²𝐽^𝛿𝑦₀^𝛿,𝜀,𝑧^𝛿,𝜀(⋅),𝑐^𝛿,𝜀(⋅,⋅),𝑢^𝛿(⋅)+𝐽^𝛿𝑦^𝛿₀,𝑧^𝛿(⋅),𝑐^𝛿(⋅,⋅),𝑢^𝛿(⋅=𝐽𝑦_0𝛿,𝜀),𝑧^𝛿,𝜀(⋅),𝑐^𝛿,𝜀(⋅,⋅),𝑢^𝛿(⋅)−𝐽𝑦₀,𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅)+𝛿²𝐽^𝛿𝑦₀^𝛿,𝜀,𝑧^𝛿,𝜀(⋅),𝑐^𝛿,𝜀(⋅,⋅),𝑢^𝛿(⋅)+𝐽^𝛿𝑦^𝛿₀,𝑧^𝛿(⋅),𝑐^𝛿(⋅,⋅),𝑢^𝛿(⋅)−𝐽𝑦^𝛿₀,𝑧^𝛿(⋅),𝑐^𝛿(⋅,⋅),𝑢^𝛿(⋅)−𝐽𝑦₀,𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅)+𝛿²𝐽^𝛿𝑦₀^𝛿,𝜀,𝑧^𝛿,𝜀(⋅),𝑐^𝛿,𝜀(⋅,⋅),𝑢^𝛿(⋅)+𝐽^𝛿𝑦^𝛿₀,𝑧^𝛿(⋅),𝑐^𝛿(⋅,⋅),𝑢^𝛿(⋅)+𝔼𝑦^𝛿,𝜀(𝑇)−ℎ𝑥^𝛿,𝜀(𝑇)²−𝑦^𝛿(𝑇)−ℎ𝑥^𝛿(𝑇)²𝐽^𝛿𝑦₀^𝛿,𝜀,𝑧^𝛿,𝜀(⋅),𝑐^𝛿,𝜀(⋅,⋅),𝑢^𝛿(⋅)+𝐽^𝛿𝑦^𝛿₀,𝑧^𝛿(⋅),𝑐^𝛿(⋅,⋅),𝑢^𝛿(⋅)≡Φ₀^𝛿,𝜀𝐽𝑦₀^𝛿,𝜀,𝑧^𝛿,𝜀(⋅),𝑐^𝛿,𝜀(⋅,⋅),𝑢^𝛿(⋅)−𝐽𝑦^𝛿₀,𝑧^𝛿(⋅),𝑐^𝛿(⋅,⋅),𝑢^𝛿(⋅Φ)+𝔼_𝑇^𝛿,𝜀,𝑦^𝛿,𝜀(𝑇)−ℎ𝑥^𝛿,𝜀(𝑇)−ℎ^𝛿_𝑇≡Φ^𝛿₀+𝑜(1)𝐽𝑦₀^𝛿,𝜀,𝑧^𝛿,𝜀(⋅),𝑐^𝛿,𝜀(⋅,⋅),𝑢^𝛿(⋅)−𝐽𝑦^𝛿₀,𝑧^𝛿(⋅),𝑐^𝛿(⋅,⋅),𝑢^𝛿(⋅)+𝜀𝔼Φ^𝛿_𝑇+𝑜(1),𝜂_𝑇, (3.21) where 𝑜(1) stands for certain scalars or vectors that go to 0 as 𝜀→0,Φ₀^𝛿,𝜀≐2𝐽^𝛿𝑦₀^𝛿,𝜀,𝑧^𝛿,𝜀(⋅),𝑐^𝛿,𝜀(⋅,⋅),𝑢^𝛿(⋅)+𝐽^𝛿𝑦^𝛿₀,𝑧^𝛿(⋅),𝑐^𝛿(⋅,⋅),𝑢^𝛿(⋅)×¹₀𝛽𝐽𝑦₀^𝛿,𝜀,𝑧^𝛿,𝜀(⋅),𝑐^𝛿,𝜀(⋅,⋅),𝑢^𝛿(⋅)−𝐽𝑦₀,𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅)+1−𝛽𝐽𝑦^𝛿₀,𝑧^𝛿(⋅),𝑐^𝛿(⋅,⋅),𝑢^𝛿(⋅)−𝐽𝑦₀,𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅),Φ𝑑𝛽+𝛿_{𝑇𝛿,𝜀}≐𝑦^𝛿,𝜀(𝑇)−ℎ𝑥^𝛿,𝜀(𝑇)+𝑦^𝛿(𝑇)−ℎ𝑥^𝛿(𝑇)𝐽^𝛿𝑦₀^𝛿,𝜀,𝑧^𝛿,𝜀(⋅),𝑐^𝛿,𝜀(⋅,⋅),𝑢^𝛿(⋅)+𝐽^𝛿𝑦^𝛿₀,𝑧^𝛿(⋅),𝑐^𝛿(⋅,⋅),𝑢^𝛿(⋅),Φ^𝛿₀≐𝐽𝑦^𝛿₀,𝑧^𝛿(⋅),𝑐^𝛿(⋅,⋅),𝑢^𝛿(⋅)−𝐽𝑦₀,𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅)+𝛿𝐽^𝛿𝑦^𝛿₀,𝑧^𝛿(⋅),𝑐^𝛿(⋅,⋅),𝑢^𝛿(⋅)∈[],Φ0,1_𝛿𝑇≐𝑦^𝛿(𝑇)−ℎ𝑥^𝛿(𝑇)𝐽^𝛿𝑦^𝛿₀,𝑧^𝛿(⋅),𝑐^𝛿(⋅,⋅),𝑢^𝛿(⋅)∈𝐿²_ℱ𝑇Ω;𝐑^𝑚. (3.22)(3.23)

We point out that (Φ^𝛿₀,Φ^𝛿_𝑇) is independent of 𝜂_𝑇, andΦ^𝛿₀≥0,Φ^𝛿₀²+𝐄Φ^𝛿_𝑇²=1. (3.24) Then there is a subsequence of (Φ^𝛿₀,Φ^𝛿_𝑇), still denoted by (Φ^𝛿₀,Φ^𝛿_𝑇), such thatlim_𝛿→0Φ^𝛿₀,Φ^𝛿_𝑇=Φ₀,Φ_𝑇,withΦ₀²+𝐄Φ_𝑇²=1. (3.25)

We claim thatΦ₀≠0. (3.26) To show this, let us observe that by (3.19), one has Φ^𝛿₀+𝑜(1)𝐽𝑦₀^𝛿,𝜀,𝑧^𝛿,𝜀(⋅),𝑐^𝛿,𝜀(⋅,⋅),𝑢^𝛿(⋅)−𝐽^𝛿𝑦^𝛿₀,𝑧^𝛿(⋅),𝑐^𝛿(⋅,⋅),𝑢^𝛿(⋅)≤𝜀𝐾Φ^𝛿₀+𝑜(1). (3.27) Hence, dividing by 𝜀 in (3.21) and then letting 𝜀→0 yields−𝐾√𝛿≤𝐾Φ^𝛿₀+𝔼Φ^𝛿_𝑇,𝜂_𝑇, (3.28)

with the constant 𝐾 independent of 𝜂_𝑇. Now, if Φ₀=0, then the above leads to−𝑟^𝛿≤𝔼Φ^𝛿_𝑇,𝜂_𝑇,∀𝜂_𝑇∈𝐿²_ℱ𝑇Ω;𝐑^𝑚,𝔼𝜂_𝑇²≤1, (3.29)

with 𝑟^𝛿→0, as 𝛿→0, uniformly in 𝔼|𝜂_𝑇|²≤1. Then,𝔼Φ^𝛿_𝑇^21/2=sup_{𝔼|𝜂𝑇|2≤1}𝔼Φ^𝛿_𝑇,𝜂_𝑇≤𝑟^𝛿⟶0,as𝛿⟶0. (3.30)

Thus, for 𝛿>0 small enough, we must have1=Φ^𝛿₀²+𝐄Φ^𝛿_𝑇²<1. (3.31) It is a contradiction which proves (3.26).

We refer to (Φ₀,Φ_𝑇) as the Lagrange multiplier of the corresponding optimal 4-tuple (𝑦₀,𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅)). Equation (3.26) shows the nontriviality of the Lagrange multiplier (Φ₀,Φ_𝑇).

Step 4 (spike variations). For notational simplicity, we now denote 𝑥𝑦(⋅)=𝑧𝑐(⋅)𝑢𝑋=,𝑣,𝑋₀=𝑎𝑦₀,𝑋_𝑇(𝑇)=𝑥(𝑇)𝑦(𝑇),𝐵((⋅=𝑏(⋅)(⋅),Σ((⋅=𝜎(⋅)𝑧,Υ((⋅=𝑔(⋅)𝑐(⋅),Γ𝑋=𝑋𝑡,𝑋,𝑣))𝑡,𝑥,𝑦,𝑧,𝑐,𝑢−𝑓𝑡,𝑥,𝑦,𝑧,𝑐,𝑢𝑡,𝑋,𝑣))𝑡,𝑥,𝑦,𝑧,𝑐,𝑢𝑡,𝑋,𝑣))𝑡,𝑥,𝑦,𝑧,𝑐,𝑢,⋅₀,𝑋_𝑇=𝜙𝑥_𝑇+𝛾𝑦₀,Π𝑋₀,𝑋_𝑇=0𝑦_𝑇−ℎ𝑥_𝑇,𝐑^𝑙=𝐑^𝑛×𝐑^𝑚,𝒱[0,𝑇]=𝐿²_ℱ,𝑝[0,𝑇];𝐑^𝑚×𝐹²_𝑝[0,𝑇];𝐑^𝑚×𝒰[0,𝑇],ℋ=𝐑^𝑙×𝐿²_ℱ𝑇Ω;𝐑^𝑙≡𝐑^𝑙×𝒳²_𝑙,ℋ₀=𝐑^𝑛×𝐿²_ℱ𝑇Ω;𝐑^𝑚≡𝐑^𝑛×𝒳²_𝑚. (3.32) Consequently, 𝐽𝑦₀,𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅)=𝐽𝑦₀(),𝐽,𝑣⋅,⋅_𝛿𝑦₀,𝑧(⋅),𝑐(⋅,⋅),𝑢(⋅)=𝐽^𝛿𝑦₀,𝑣(⋅,⋅). (3.33)

Note that ℋ and ℋ₀ are Hilbert spaces. We identify ℋ^∗=ℋ and ℋ^∗₀=ℋ₀. Also, from the above,Γ∶ℋ⟶𝐑,Π∶ℋ⟶ℋ₀. (3.34) We denote the gradient 𝐷Γ and the Hessian 𝐷²Γ of Γ as follows:𝐷Γ𝑋₀,𝑋_𝑇≡Γ_𝑋0𝑋₀,𝑋_𝑇,Γ_𝑋𝑇𝑋₀,𝑋_𝑇∈ℒ(ℋ;𝐑)≡ℋ^∗𝐷=ℋ,₂Γ𝑋₀,𝑋_𝑇≡Γ_𝑋0𝑋0𝑋₀,𝑋_𝑇Γ_{𝑋0𝑋𝑇}𝑋₀,𝑋_𝑇Γ_{𝑋𝑇𝑋0}𝑋₀,𝑋_𝑇Γ_{𝑋𝑇𝑋𝑇}𝑋₀,𝑋_𝑇∈ℒ_𝑠(ℋ;ℋ), (3.35) where ℒ(ℋ₁;ℋ₂) is the set of all linear bounded operators from ℋ₁ to ℋ₂, and ℒ_𝑠(ℋ;ℋ) is the set of all linear bounded self-adjoint operators from ℋ to itself. Clearly,Γ_𝑋0𝑋₀,𝑋_𝑇=0,𝛾_𝑦𝑦₀^⊤∈𝐑^𝑙,Γ_𝑋𝑇𝑋₀,𝑋_𝑇=𝜙_𝑥𝑥_𝑇,0^⊤∈𝒳²_𝑙,Γ_𝑋0𝑋0𝑋₀,𝑋_𝑇=000𝛾_𝑦𝑦𝑦₀∈𝒮^𝑚+𝑛,Γ_{𝑋0𝑋𝑇}𝑋₀,𝑋_𝑇=𝒳0000∈ℒ²_𝑙;𝐑^𝑙,Γ_{𝑋𝑇𝑋0}𝑋₀,𝑋_𝑇=𝐑0000∈ℒ^𝑙;𝒳²_𝑙,Γ_{𝑋𝑇𝑋𝑇}𝑋₀,𝑋_𝑇=𝜙_𝑥𝑥𝑥_𝑇0𝒳00∈ℒ²_𝑙;𝒳²_𝑙. (3.36)

Also, by the Fréchet differentiability, for Π∶ℋ→ℋ₀, we have𝐷Π𝑋₀,𝑋_𝑇≡Π_𝑋0𝑋₀,𝑋_𝑇,Π_𝑋𝑇𝑋₀,𝑋_𝑇∈ℒℋ;ℋ₀,𝐷²Π𝑋₀,𝑋_𝑇≡Π_𝑋0𝑋0𝑋₀,𝑋_𝑇Π_{𝑋0𝑋𝑇}𝑋₀,𝑋_𝑇Π_{𝑋𝑇𝑋0}𝑋₀,𝑋_𝑇Π_{𝑋𝑇𝑋𝑇}𝑋₀,𝑋_𝑇∈ℒℋ;ℒℋ;ℋ₀, (3.37)

To make the above more precise, let us take any Φ≡(Φ₀,Φ_𝑇)∈ℋ₀. Then,Π𝑋₀,𝑋_𝑇,Φ=𝑦_𝑇−ℎ𝑥_𝑇,Φ_𝑇. (3.38)

Thus,𝐷Π𝑋₀,𝑋_𝑇Φ≡𝐷Π𝑋₀,𝑋_𝑇,Φ=Π𝑋₀,𝑋_𝑇,Φ_𝑋0,Π𝑋₀,𝑋_𝑇,Φ_𝑋𝑇,≡Π_𝑋0𝑋₀,𝑋_𝑇Φ,Π_𝑋𝑇𝑋₀,𝑋_𝑇Φ∈ℒ(ℋ;𝐑), (3.39)

withΠ_𝑋0𝑋₀,𝑋_𝑇Φ≡Π𝑋₀,𝑋_𝑇,Φ_𝑋0=(),Π0,0_𝑋𝑇𝑋₀,𝑋_𝑇Φ≡Π𝑋₀,𝑋_𝑇,Φ_𝑋𝑇=−ℎ_𝑥𝑥_𝑇^⊤Φ_𝑇,Φ_𝑇;𝐷²Π𝑋₀,𝑋_𝑇Φ≡𝐷²Π𝑋₀,𝑋_𝑇,Φ=Π𝑋₀,𝑋_𝑇,Φ_𝑋0𝑋0Π𝑋₀,𝑋_𝑇,Φ_{𝑋0𝑋𝑇}Π𝑋₀,𝑋_𝑇,Φ_{𝑋𝑇𝑋0}Π𝑋₀,𝑋_𝑇,Φ_{𝑋𝑇𝑋𝑇}≡Π_𝑋0𝑋0𝑋₀,𝑋_𝑇ΦΠ_{𝑋0𝑋𝑇}𝑋₀,𝑋_𝑇ΦΠ_{𝑋𝑇𝑋0}𝑋₀,𝑋_𝑇ΦΠ_{𝑋𝑇𝑋𝑇}𝑋₀,𝑋_𝑇Φ∈ℒ_𝑠(ℋ;ℋ), (3.40)

withΠ_𝑋0𝑋0𝑋₀,𝑋_𝑇Φ≡Π𝑋₀,𝑋_𝑇,Φ_𝑋0𝑋0=,Π0000_{𝑋𝑇𝑋0}𝑋₀,𝑋_𝑇Φ≡Π𝑋₀,𝑋_𝑇,Φ_{𝑋𝑇𝑋0}=,Π0000_{𝑋0𝑋𝑇}𝑋₀,𝑋_𝑇Φ≡Π𝑋₀,𝑋_𝑇,Φ_{𝑋0𝑋𝑇}=,Π0000_{𝑋𝑇𝑋𝑇}𝑋₀,𝑋_𝑇Φ≡Π𝑋₀,𝑋_𝑇,Φ_{𝑋𝑇𝑋𝑇}=−ℎ_𝑥𝑥𝑥_𝑇Φ_𝑇0.00 (3.41)

Next, for fixed 𝑋₀∈𝐑^𝑙,𝑣(⋅)∈𝒱[0,𝑇] and any 𝜀∈(0,1), let𝑋₀^𝛿,𝜀≐𝑋^𝛿₀+0√𝜀𝑦₀𝑣_𝛿,𝜀,()=𝑣𝑡,⋅^𝛿(𝑡,⋅),𝑡∈[0,𝑇]⧵𝑆_𝜀,𝑣(𝑡,⋅),𝑡∈𝑆_𝜀, (3.42)(3.43) for some measurable set 𝑆_𝜀⊆[0,𝑇] with |𝑆_𝜀|=𝜀𝑇. It is clear that 𝑣^𝛿,𝜀(⋅,⋅)∈𝒱[0,𝑇].

Let 𝑋^𝛿,𝜀(⋅) be the state process of (3.1) corresponding to (𝑋₀^𝛿,𝜀,𝑣^𝛿,𝜀(⋅,⋅)). Further, let 𝑋₁^𝛿,𝜀(⋅) and 𝑋₂^𝛿,𝜀(⋅) be, respectively, the solutions to the following SDEJs:𝑑𝑋₁^𝛿,𝜀(𝑡)=𝐵^𝛿_𝑋(𝑡,⋅)𝑋₁^𝛿,𝜀(𝑡)𝑑𝑡+Σ^𝛿_𝑋+ΔΣ^𝛿(𝑡,⋅)𝐼_𝑆𝜀(𝑡(𝑡)+)𝑑𝑊_𝐄Υ^𝛿_𝑋(𝑡,𝑒)𝑋₁^𝛿,𝜀(𝑡−)+ΔΥ^𝛿(𝑡,𝑒)𝐼_𝑆𝜀(𝑡𝑁(),𝑋)𝑑𝑒𝑑𝑡_1𝛿,𝜀(0)=0√𝜀𝑦₀;𝑑𝑋₂^𝛿,𝜀(𝑡)=𝐵^𝛿_𝑋(𝑡,⋅)𝑋₂^𝛿,𝜀(𝑡)+Δ𝐵^𝛿(𝑡,⋅)𝐼_𝑆𝜀(𝑡)+12𝐵^𝛿_𝑋𝑋(𝑡,⋅)𝑋₁^𝛿,𝜀(𝑡)²+Σ𝑑𝑡_𝛿𝑋(𝑡,⋅)𝑋₂^𝛿,𝜀(𝑡)+ΔΣ^𝛿_𝑋(𝑡,⋅)𝑋₁^𝛿,𝜀(𝑡)𝐼_𝑆𝜀(𝑡)+12Σ^𝛿_𝑋𝑋(𝑡,⋅)𝑋₁^𝛿,𝜀(𝑡)²(𝑡)+𝑑𝑊_𝐄Υ^𝛿_𝑋(𝑡,𝑒)𝑋₂^𝛿,𝜀(𝑡−)+ΔΥ^𝛿_𝑋(𝑡,𝑒)𝑋₁^𝛿,𝜀(𝑡−)𝐼_𝑆𝜀(𝑡)+12Υ^𝛿_𝑋𝑋(𝑡,𝑒)𝑋₁^𝛿,𝜀(𝑡−)²𝑁(),𝑋𝑑𝑒𝑑𝑡_2𝛿,𝜀(0)=0, (3.44)(3.45) where 𝐼_𝑆𝜀(⋅) denotes the indicator function of the set 𝑆_𝜀, and for any 𝑋∈𝐑^𝑙,𝐵^𝛿_𝑋(𝑡,⋅)=𝐵_𝑋𝑡,𝑋^𝛿(𝑡),𝑣^𝛿(),Σ𝑡,⋅_𝛿𝑋(𝑡,⋅)=Σ_𝑋𝑡,𝑋^𝛿(𝑡),𝑣^𝛿(),Υ𝑡,⋅_𝛿𝑋(𝑡,⋅)=Υ_𝑋𝑡,𝑋^𝛿(𝑡),𝑣^𝛿(𝑡,⋅),Δ𝐵^𝛿(𝑡,⋅)=𝐵𝑡,𝑋^𝛿(𝑡),𝑣(𝑡,⋅)−𝐵𝑡,𝑋^𝛿(𝑡),𝑣^𝛿(𝑡,⋅),ΔΣ^𝛿(𝑡,⋅)=Σ𝑡,𝑋^𝛿(𝑡),𝑣(𝑡,⋅)−Σ𝑡,𝑋^𝛿(𝑡),𝑣^𝛿(𝑡,⋅),ΔΥ^𝛿(𝑡,⋅)=Υ𝑡,𝑋^𝛿(𝑡),𝑣(𝑡,⋅)−Υ𝑡,𝑋^𝛿(𝑡),𝑣^𝛿(𝑡,⋅),ΔΣ^𝛿_𝑋(𝑡,⋅)=Σ_𝑋𝑡,𝑋^𝛿(𝑡),𝑣(𝑡,⋅)−Σ_𝑋𝑡,𝑋^𝛿(𝑡),𝑣^𝛿(𝑡,⋅),ΔΥ^𝛿_𝑋(𝑡,⋅)=Υ_𝑋𝑡,𝑋^𝛿(𝑡),𝑣(𝑡,⋅)−Υ_𝑋𝑡,𝑋^𝛿(𝑡),𝑣^𝛿(),𝐵𝑡,⋅_𝛿𝑋𝑋(𝑡,⋅)𝑋²=𝐵^1,𝛿_𝑋𝑋()𝐵𝑡,⋅𝑋,𝑋_{2,𝛿𝑋𝑋}()⋮𝐵𝑡,⋅X,𝑋_{𝑙,𝛿𝑋𝑋}(𝑡,⋅)𝑋,𝑋,𝐵^𝑖,𝛿_𝑋𝑋(𝑡,⋅)=𝐵^𝑖_𝑋𝑋𝑡,𝑋^𝛿(𝑡),𝑣^𝛿()Σ𝑡,⋅,1≤𝑖≤𝑙,_𝛿𝑋𝑋(𝑡,⋅)𝑋²=Σ^1,𝛿_𝑋𝑋()Σ𝑡,⋅𝑋,𝑋_{2,𝛿𝑋𝑋}()⋮Σ𝑡,⋅𝑋,𝑋_{𝑙,𝛿𝑋𝑋}(𝑡,⋅)𝑋,𝑋,Σ^𝑖,𝛿_𝑋𝑋(𝑡,⋅)=Σ^𝑖_𝑋𝑋𝑡,𝑋^𝛿(𝑡),𝑣^𝛿()Υ𝑡,⋅,1≤𝑖≤𝑙,_𝛿𝑋𝑋(𝑡,⋅)𝑋²=Υ^1,𝛿_𝑋𝑋()Υ𝑡,⋅𝑋,𝑋_{2,𝛿𝑋𝑋}()⋮Υ𝑡,⋅𝑋,𝑋_{𝑙,𝛿𝑋𝑋}(𝑡,⋅)𝑋,𝑋,Υ^𝑖,𝛿_𝑋𝑋(𝑡,⋅)=Υ^𝑖_𝑋𝑋𝑡,𝑋^𝛿(𝑡),𝑣^𝛿(𝑡,⋅),1≤𝑖≤𝑙. (3.46)

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: sup_{0≤𝑡≤𝑇}𝔼𝑋₁^𝛿,𝜀(𝑡)^2𝑘+sup_{0≤𝑡≤𝑇}𝔼𝑋^𝛿,𝜀(𝑡)−𝑋^𝛿(𝑡)^2𝑘≤𝐶𝜀^𝑘,sup_{0≤𝑡≤𝑇}𝔼𝑋₂^𝛿,𝜀(𝑡)^2𝑘+sup_{0≤𝑡≤𝑇}𝔼𝑋^𝛿,𝜀(𝑡)−𝑋^𝛿(𝑡)−𝑋₁^𝛿,𝜀(𝑡)^2𝑘≤𝐶𝜀^2𝑘,sup_{0≤𝑡≤𝑇}𝔼𝑋^𝛿,𝜀(𝑡)−𝑋^𝛿(𝑡)−𝑋₁^𝛿,𝜀(𝑡)−𝑋₂^𝛿,𝜀(𝑡)^2𝑘=𝑜𝜀^2𝑘. (3.47)

Now, from the last relation in (3.11), noting (3.19), we obtain −√𝛿√𝜀𝑦₀+𝜀(𝑇+𝐾)≤𝐽^𝛿𝑦₀^𝛿,𝜀,𝑣^𝛿,𝜀(⋅,⋅)−𝐽^𝛿𝑦^𝛿₀,𝑣^𝛿()=𝐽⋅,⋅_𝛿𝑦₀^𝛿,𝜀,𝑣^𝛿,𝜀(⋅,⋅)²−𝐽^𝛿𝑦^𝛿₀,𝑣^𝛿(⋅,⋅)²𝐽^𝛿𝑦₀^𝛿,𝜀,𝑣^𝛿,𝜀(⋅,⋅)+𝐽^𝛿𝑦^𝛿₀,𝑣^𝛿()=𝐽𝑦⋅,⋅_0𝛿,𝜀,𝑣^𝛿,𝜀(⋅,⋅)−𝐽𝑦₀,𝑣(⋅,⋅)+𝛿²−𝐽𝑦^𝛿₀,𝑣^𝛿(⋅,⋅)−𝐽𝑦₀,𝑣(⋅,⋅)+𝛿²𝐽^𝛿𝑦₀^𝛿,𝜀,𝑣^𝛿,𝜀(⋅,⋅)+𝐽^𝛿𝑦^𝛿₀,𝑣^𝛿()+𝔼Π𝑋⋅,⋅_0𝛿,𝜀,𝑋^𝛿,𝜀(𝑇)²−Π𝑋^𝛿₀,𝑋^𝛿(𝑇)²𝐽^𝛿𝑦₀^𝛿,𝜀,𝑣^𝛿,𝜀(⋅,⋅)+𝐽^𝛿𝑦^𝛿₀,𝑣^𝛿(⋅,⋅)≡Φ₀^𝛿,𝜀𝐽𝑦₀^𝛿,𝜀,𝑣^𝛿,𝜀(⋅,⋅)−𝐽𝑦^𝛿₀,𝑣^𝛿()0Φ⋅,⋅+𝔼_{𝑇𝛿,𝜀},Π𝑋₀^𝛿,𝜀,𝑋^𝛿,𝜀(𝑇)−Π𝑋^𝛿₀,𝑋^𝛿(𝑇)≡Φ^𝛿₀+𝑜(1)𝐽𝑦₀^𝛿,𝜀,𝑣^𝛿,𝜀(⋅,⋅)−𝐽𝑦^𝛿₀,𝑣^𝛿()0Φ⋅,⋅+𝔼_𝛿𝑇(1)𝑋+𝑜,Π₀^𝛿,𝜀,𝑋^𝛿,𝜀(𝑇)−Π𝑋^𝛿₀,𝑋^𝛿(𝑇), (3.48) where (Φ^𝛿₀,Φ^𝛿_𝑇) is defined in (3.25) and (Φ₀^𝛿,𝜀,Φ_𝑇^𝛿,𝜀) is defined similarly to (3.22), with 𝑋₀^𝛿,𝜀=^𝑎_𝑦0^𝛿,𝜀 replaced by (3.42). We have shown that along a sequence,lim_𝛿→0Φ^𝛿₀,Φ^𝛿_𝑇=Φ₀,Φ_𝑇,withΦ₀≠0. (3.49)

Note that𝐽𝑦₀^𝛿,𝜀,𝑣^𝛿,𝜀(⋅,⋅)−𝐽𝑦^𝛿₀,𝑣^𝛿(⋅,⋅)=𝔼^𝑇₀𝑙𝑡,𝑋^𝛿,𝜀(𝑡),𝑣^𝛿,𝜀(𝑡,⋅)−𝑙𝑡,𝑋^𝛿(𝑡),𝑣^𝛿()Γ𝑋𝑡,⋅𝑑𝑡+𝔼₀^𝛿,𝜀,𝑋^𝛿,𝜀(𝑇)−Γ𝑋^𝛿₀,𝑋^𝛿(𝑇)≡𝐼₁+𝐼₂. (3.50)

We first deal with term 𝐼₁. We have 𝐼₁=𝔼^𝑇₀𝑙𝑡,𝑋^𝛿,𝜀(𝑡),𝑣^𝛿,𝜀(𝑡,⋅)−𝑙𝑡,𝑋^𝛿(𝑡)+𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡),𝑣^𝛿,𝜀(𝑡,⋅)𝑑𝑡+𝔼^𝑇₀𝑙𝑡,𝑋^𝛿(𝑡)+𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡),𝑣^𝛿,𝜀(𝑡,⋅)−𝑙𝑡,𝑋^𝛿(𝑡)+𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡),𝑣^𝛿(𝑡,⋅)𝑑𝑡+𝔼^𝑇₀𝑙𝑡,𝑋^𝛿(𝑡)+𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡),𝑣^𝛿(𝑡,⋅)−𝑙𝑡,𝑋^𝛿(𝑡),𝑣^𝛿(𝑡,⋅)𝑑𝑡=𝔼^𝑇₀𝑙_𝑋𝑡,𝑋^𝛿(𝑡)+𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡),𝑣^𝛿,𝜀(𝑡,⋅),𝑋^𝛿,𝜀(𝑡)−𝑋^𝛿(𝑡)−𝑋₁^𝛿,𝜀(𝑡)−𝑋₂^𝛿,𝜀(𝑡+1)𝑑𝑡2𝔼^𝑇₀𝑙_𝑋𝑋𝑡,𝑋^𝛿(𝑡)+𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡),𝑣^𝛿,𝜀()×𝑋𝑡,⋅_𝛿,𝜀(𝑡)−𝑋^𝛿(𝑡)−𝑋₁^𝛿,𝜀(𝑡)−𝑋₂^𝛿,𝜀(𝑡),𝑋^𝛿,𝜀(𝑡)−𝑋^𝛿(𝑡)−𝑋₁^𝛿,𝜀(𝑡)−𝑋₂^𝛿,𝜀(𝑡)𝑑𝑡+𝔼^𝑇₀𝐷²𝑙^{𝛿,𝜀,1,2}𝑋^𝛿,𝜀(𝑡)−𝑋^𝛿(𝑡)−𝑋₁^𝛿,𝜀(𝑡)−𝑋₂^𝛿,𝜀(𝑡),𝑋^𝛿,𝜀(𝑡)−𝑋^𝛿(𝑡)−𝑋₁^𝛿,𝜀(𝑡)−𝑋₂^𝛿,𝜀(𝑡)𝑑𝑡+𝔼^𝑇₀𝑙𝑡,𝑋^𝛿(𝑡),𝑣^𝛿,𝜀(𝑡,⋅)−𝑙𝑡,𝑋^𝛿(𝑡),𝑣^𝛿(𝑡,⋅)𝑑𝑡+𝔼^𝑇₀𝑙_𝑋𝑡,𝑋^𝛿(𝑡),𝑣^𝛿,𝜀(𝑡,⋅)−𝑙_𝑋𝑡,𝑋^𝛿(𝑡),𝑣^𝛿(𝑡,⋅),𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡+1)𝑑𝑡2𝔼^𝑇₀𝑙_𝑋𝑋𝑡,𝑋^𝛿(𝑡),𝑣^𝛿,𝜀(𝑡,⋅)−𝑙_𝑋𝑋𝑡,𝑋^𝛿(𝑡),𝑣^𝛿(×𝑋𝑡,⋅)_1𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡),𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡)𝑑𝑡+𝔼^𝑇₀𝐷²𝑙^𝛿,𝜀𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡),𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡)𝑑𝑡+𝔼^𝑇₀𝑙_𝑋𝑡,𝑋^𝛿(𝑡),𝑣^𝛿(𝑡,⋅),𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡+1)𝑑𝑡2𝔼^𝑇₀𝑙_𝑋𝑋𝑡,𝑋^𝛿(𝑡),𝑣^𝛿(𝑡,⋅)𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡),𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡)𝑑𝑡+𝔼^𝑇₀𝐷²𝑙^𝛿𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡),𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡)𝑑𝑡, (3.51) with 𝐷²𝑙^{𝛿,𝜀,1,2}≐¹₀𝛽𝑙_𝑋𝑋𝑡,𝛽𝑋^𝛿(𝑡)+𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡)+1−𝛽𝑋₁^𝛿,𝜀(𝑡),𝑣^𝛿(𝑡,⋅)−𝑙_𝑋𝑋𝑡,𝑋^𝛿(𝑡)+𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡),𝑣^𝛿()𝐷𝑡,⋅𝑑𝛽,₂𝑙^𝛿,𝜀≐¹₀𝛽𝑙_𝑋𝑋𝑡,𝛽𝑋^𝛿(𝑡)+1−𝛽𝑋^𝛿(𝑡)+𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡),𝑣^𝛿,𝜀(𝑡,⋅)−𝑙_𝑋𝑋𝑡,𝛽𝑋^𝛿(𝑡)+1−𝛽𝑋^𝛿(𝑡)+𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡),𝑣^𝛿(𝑡,⋅)−𝑙_𝑋𝑋𝑡,𝑋^𝛿(𝑡),𝑣^𝛿,𝜀(𝑡,⋅)+𝑙_𝑋𝑋𝑡,𝑋^𝛿(𝑡),𝑣^𝛿()𝐷𝑡,⋅𝑑𝛽,₂𝑙^𝛿≐¹₀𝛽𝑙_𝑋𝑋𝑡,𝛽𝑋^𝛿(𝑡)+1−𝛽𝑋^𝛿(𝑡)+𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡),𝑣^𝛿(𝑡,⋅)−𝑙_𝑋𝑋𝑡,𝑋^𝛿(𝑡),𝑣^𝛿(𝑡,⋅)𝑑𝛽. (3.52)

Next we deal with term 𝐼₂. Similarly, we have𝐼₂=𝔼𝐷Γ^𝛿𝑋₀^𝛿,𝜀−𝑋^𝛿₀𝑋^𝛿,𝜀(𝑇)−𝑋^𝛿(𝑇)+12𝐷²Γ^𝛿𝑋₀^𝛿,𝜀−𝑋^𝛿₀𝑋^𝛿,𝜀(𝑇)−𝑋^𝛿(𝑇),𝑋₀^𝛿,𝜀−𝑋^𝛿₀𝑋^𝛿,𝜀(𝑇)−𝑋^𝛿(𝑇)+𝐷²Γ^𝛿,𝜀𝑋₀^𝛿,𝜀−𝑋^𝛿₀𝑋^𝛿,𝜀(𝑇)−𝑋^𝛿(𝑇),𝑋₀^𝛿,𝜀−𝑋^𝛿₀𝑋^𝛿,𝜀(𝑇)−𝑋^𝛿(𝑇)≡𝔼Γ^𝛿_𝑋0𝑋₀^𝛿,𝜀−𝑋^𝛿₀+Γ^𝛿_𝑋𝑇𝑋^𝛿,𝜀(𝑇)−𝑋^𝛿(𝑇)+12Γ^𝛿_𝑋0𝑋0𝑋₀^𝛿,𝜀−𝑋^𝛿₀,𝑋₀^𝛿,𝜀−𝑋^𝛿₀+12Γ^𝛿_{𝑋𝑇𝑋𝑇}𝑋^𝛿,𝜀(𝑇)−𝑋^𝛿(𝑇),𝑋^𝛿,𝜀(𝑇)−𝑋^𝛿(𝑇+𝐷₂)Γ^𝛿,𝜀𝑋₀^𝛿,𝜀−𝑋^𝛿₀𝑋^𝛿,𝜀(𝑇)−𝑋^𝛿(𝑇),𝑋₀^𝛿,𝜀−𝑋^𝛿₀𝑋^𝛿,𝜀(𝑇)−𝑋^𝛿(𝑇), (3.53) with 𝐷Γ^𝛿≐Γ^𝛿_𝑋0,Γ^𝛿_𝑋𝑇≡Γ_𝑋0𝑋^𝛿₀,𝑋^𝛿(𝑇),Γ_𝑋𝑇𝑋^𝛿₀,𝑋^𝛿(𝑇),𝐷²Γ^𝛿≐Γ^𝛿_𝑋0𝑋000Γ^𝛿_{𝑋𝑇𝑋𝑇}≡Γ_𝑋0𝑋0𝑋^𝛿₀,𝑋^𝛿(𝑇)00Γ_{𝑋𝑇𝑋𝑇}𝑋^𝛿₀,𝑋^𝛿(𝑇),𝐷²Γ^𝛿,𝜀≐¹₀𝛽𝐷²Γ𝛽𝑋^𝛿₀+1−𝛽𝑋₀^𝛿,𝜀,𝛽𝑋^𝛿_𝑇+1−𝛽𝑋_𝑇^𝛿,𝜀−𝐷²Γ𝑋^𝛿₀,𝑋^𝛿(𝑇)𝑑𝛽. (3.54)

Also, we have𝔼0Φ_𝑇^𝛿,𝜀,Π𝑋₀^𝛿,𝜀,𝑋^𝛿,𝜀(𝑇)−Π𝑋^𝛿₀,𝑋^𝛿(𝑇)0Φ≡𝔼_{𝑇𝛿,𝜀},Π0,𝑋^𝛿,𝜀(𝑇)−Π0,𝑋^𝛿(𝑇)=𝔼Π_𝑋𝑇0,𝑋^𝛿(𝑇)0Φ_𝑇^𝛿,𝜀,𝑋^𝛿,𝜀(𝑇)−𝑋^𝛿(𝑇+1)2Π_{𝑋𝑇𝑋𝑇}0,𝑋^𝛿(𝑇)0Φ_𝑇^𝛿,𝜀𝑋^𝛿,𝜀(𝑇)−𝑋^𝛿(𝑇),𝑋^𝛿,𝜀(𝑇)−𝑋^𝛿(𝑇+𝐷₂)Π^𝛿,𝜀0Φ_𝑇^𝛿,𝜀𝑋^𝛿,𝜀(𝑇)−𝑋^𝛿(𝑇),𝑋^𝛿,𝜀(𝑇)−𝑋^𝛿(𝑇), (3.55) with𝐷²Π^𝛿,𝜀≐¹₀𝛽Π_{𝑋𝑇𝑋𝑇}0,𝛽𝑋^𝛿(𝑇)+1−𝛽𝑋^𝛿,𝜀(𝑇)−Π_{𝑋𝑇𝑋𝑇}0,𝑋^𝛿(𝑇)𝑑𝛽. (3.56)

Under assumptions (H2), (H3), we haveΦ₀^𝛿,𝜀𝐷²𝑙^{𝛿,𝜀,1,2}≤𝐾𝑋^𝛿,𝜀(𝑡)−𝑋^𝛿(𝑡)−𝑋₁^𝛿,𝜀(𝑡)−𝑋₂^𝛿,𝜀(𝑡,Φ_0𝛿,𝜀)𝐷²𝑙^𝛿,𝜀+Φ₀^𝛿,𝜀𝐷²𝑙^𝛿≤𝐾𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡,Φ_0𝛿,𝜀)𝐷²Γ^𝛿,𝜀+𝐷²Π^𝛿,𝜀0Φ_𝑇^𝛿,𝜀≤𝐾𝑋₀^𝛿,𝜀−𝑋^𝛿₀+𝑋^𝛿,𝜀(𝑇)−𝑋^𝛿(𝑇). (3.57)

Hence, by (3.47), we have 𝔼Φ₀^𝛿,𝜀𝐷²𝑙^{𝛿,𝜀,1,2}𝑋^𝛿,𝜀(𝑡)−𝑋^𝛿(𝑡)−𝑋₁^𝛿,𝜀(𝑡)−𝑋₂^𝛿,𝜀(𝑡),𝑋^𝛿,𝜀(𝑡)−𝑋^𝛿(𝑡)−𝑋₁^𝛿,𝜀(𝑡)−𝑋₂^𝛿,𝜀(𝑡+Φ_0𝛿,𝜀)𝐷²𝑙^𝛿,𝜀𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(t),𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡+Φ_0𝛿,𝜀)𝐷²𝑙^𝛿𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡),𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡+Φ_0𝛿,𝜀)𝐷²Γ^𝛿,𝜀𝑋₀^𝛿,𝜀−𝑋^𝛿₀𝑋^𝛿,𝜀(𝑇)−𝑋^𝛿(𝑇),𝑋₀^𝛿,𝜀−𝑋^𝛿₀𝑋^𝛿,𝜀(𝑇)−𝑋^𝛿(𝑇)+𝐷²Π^𝛿,𝜀0Φ_𝑇^𝛿,𝜀𝑋^𝛿,𝜀(𝑇)−𝑋^𝛿(𝑇),𝑋^𝛿,𝜀(𝑇)−𝑋^𝛿(𝑇𝑋)≤𝐾𝔼^𝛿,𝜀(𝑡)−𝑋^𝛿(𝑡)−𝑋₁^𝛿,𝜀(𝑡)−𝑋₂^𝛿,𝜀(𝑡)³+𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡)³+𝑋₀^𝛿,𝜀−𝑋^𝛿₀³+𝑋^𝛿,𝜀(𝑇)−𝑋^𝛿(𝑇)³≤𝐾𝜀^3/2. (3.58)

Consequently, combining (3.50) with (3.55), from (3.48), we can derive that−√𝛿√𝜀𝑦₀+𝜀(𝑇+𝐾)≤Φ₀^𝛿,𝜀𝐽𝑦₀^𝛿,𝜀,𝑣^𝛿,𝜀(⋅,⋅)−𝐽𝑦^𝛿₀,𝑣^𝛿()0Φ⋅,⋅+𝔼_{𝑇𝛿,𝜀},Π0,𝑋^𝛿,𝜀(𝑇)−Π0,𝑋^𝛿(𝑇)=Φ₀^𝛿,𝜀𝔼^𝑇₀𝑙𝑡,𝑋^𝛿(𝑡),𝑣^𝛿,𝜀(𝑡,⋅)−𝑙𝑡,𝑋^𝛿(𝑡),𝑣^𝛿()+𝑙𝑡,⋅_𝑋𝑡,𝑋^𝛿(𝑡),𝑣^𝛿(𝑡,⋅),𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡+1)2𝑙_𝑋𝑋𝑡,𝑋^𝛿(𝑡),𝑣^𝛿(𝑡,⋅)𝑋₁^𝛿,𝜀(𝑡),𝑋₁^𝛿,𝜀(𝑡Γ)𝑑𝑡+𝔼_𝑋0𝑋^𝛿₀,𝑋^𝛿(𝑇)𝑋₀^𝛿,𝜀−𝑋^𝛿₀+12Γ_𝑋0𝑋0𝑋^𝛿₀,𝑋^𝛿(𝑇)𝑋₀^𝛿,𝜀−𝑋^𝛿₀,𝑋₀^𝛿,𝜀−𝑋^𝛿₀+Γ_𝑋𝑇𝑋^𝛿₀,𝑋^𝛿(𝑇)𝑋^𝛿,𝜀(𝑇)−𝑋^𝛿(𝑇)+12Γ_{𝑋𝑇𝑋𝑇}𝑋^𝛿₀,𝑋^𝛿(𝑇)𝑋^𝛿,𝜀(𝑇)−𝑋^𝛿(𝑇),𝑋^𝛿,𝜀(𝑇)−𝑋^𝛿(𝑇Π)+𝔼_𝑋𝑇0,𝑋^𝛿(𝑇)0Φ_𝑇^𝛿,𝜀,𝑋^𝛿,𝜀(𝑇)−𝑋^𝛿(𝑇+1)2Π_{𝑋𝑇𝑋𝑇}0,𝑋^𝛿(𝑇)0Φ_𝑇^𝛿,𝜀𝑋^𝛿,𝜀(𝑇)−𝑋^𝛿(𝑇),𝑋^𝛿,𝜀(𝑇)−𝑋^𝛿(𝑇)+𝑂𝜀^3/2=𝔼^𝑇₀Φ₀^𝛿,𝜀𝑙𝑡,𝑋^𝛿(𝑡),𝑣^𝛿,𝜀(𝑡,⋅)−𝑙𝑡,𝑋^𝛿(𝑡),𝑣^𝛿(𝑡,⋅)+Φ₀^𝛿,𝜀𝑙_𝑋𝑡,𝑋^𝛿(𝑡),𝑣^𝛿(𝑡,⋅),𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡+1)2Φ₀^𝛿,𝜀𝑙_𝑋𝑋𝑡,𝑋^𝛿(𝑡),𝑣^𝛿(𝑡,⋅)𝑋₁^𝛿,𝜀(𝑡),𝑋₁^𝛿,𝜀(𝑡+√)𝑑𝑡𝜀Φ₀^𝛿,𝜀Γ_𝑋0𝑋^𝛿₀,𝑋^𝛿(𝑇),0𝑦₀+𝜀2Φ₀^𝛿,𝜀Γ_𝑋0𝑋0𝑋^𝛿₀,𝑋^𝛿(𝑇)0𝑦₀,0𝑦₀+Φ₀^𝛿,𝜀Γ_𝑋𝑇𝑋^𝛿₀,𝑋^𝛿(𝑇)+Π_𝑋𝑇0,𝑋^𝛿(𝑇)0Φ_𝑇^𝛿,𝜀,𝑋₁^𝛿,𝜀(𝑇)+𝑋₂^𝛿,𝜀(𝑇+1)2Φ₀^𝛿,𝜀Γ_{𝑋𝑇𝑋𝑇}𝑋^𝛿₀,𝑋^𝛿(𝑇)+Π_{𝑋𝑇𝑋𝑇}0,𝑋^𝛿(𝑇)0Φ_𝑇^𝛿,𝜀𝑋₁^𝛿,𝜀(𝑇),𝑋₁^𝛿,𝜀(𝑇𝜀)+𝑂^3/2. (3.59)

Step 5 (duality). Let (Φ^𝛿,𝜀(⋅),Ψ^𝛿,𝜀(⋅),Ξ^𝛿,𝜀(⋅,⋅)) be the adapted solution to the following BSDEJ: −𝑑Φ^𝛿,𝜀(𝑡)=𝐵^𝛿_𝑋(𝑡,⋅)^⊤Φ^𝛿,𝜀(𝑡)+Σ^𝛿_𝑋(𝑡,⋅)^⊤Ψ^𝛿,𝜀(𝑡)+_𝐄Υ^𝛿_𝑋(𝑡,𝑒)^⊤Ξ^𝛿,𝜀(𝑡,𝑒)𝜋(𝑑𝑒)−Φ₀^𝛿,𝜀𝑙^𝛿_𝑋(−Ψ𝑡,⋅)𝑑𝑡_𝛿,𝜀(𝑡)𝑑𝑊(𝑡)−_𝐄Ξ^𝛿,𝜀()𝑁(),𝑑Φ𝑡,𝑒𝑑𝑒𝑑𝑡_𝛿,𝜀(𝑇)=−Φ₀^𝛿,𝜀Γ_𝑋𝑇𝑋^𝛿₀,𝑋^𝛿(𝑇)+Π_𝑋𝑇0,𝑋^𝛿(𝑇0Φ)_{𝑇𝛿,𝜀}, (3.60) where 𝑙^𝛿_𝑋(𝑡,⋅)≐𝑙_𝑋(𝑡,𝑋^𝛿(𝑡),𝑣^𝛿(𝑡,⋅)). Then, as (𝜀,𝛿)→(0,0), (Φ^𝛿,𝜀(⋅),Ψ^𝛿,𝜀(⋅),Ξ^𝛿,𝜀(⋅,⋅)) goes to (Φ(⋅),Ψ(⋅),Ξ(⋅,⋅)), which is the adapted solution to the following BSDEJ: −𝑑Φ(𝑡)=𝐵_𝑋(𝑡,⋅)^⊤Φ(𝑡)+Σ_𝑋(𝑡,⋅)^⊤Ψ(𝑡)+_𝐄Υ_𝑋(𝑡,𝑒)^⊤Ξ(𝑡,𝑒)𝜋(𝑑𝑒)−Φ₀𝑙_𝑋(−Ψ(𝑡)(𝑡)−𝑡,⋅)𝑑𝑡𝑑𝑊_𝐄Ξ()𝑁(),𝑑Φ(𝑇)Φ𝑡,𝑒𝑑𝑒𝑑𝑡=−₀Γ_𝑋𝑇𝑋₀,𝑋(𝑇)+Π_𝑋𝑇0,𝑋(𝑇0Φ)_𝑇, (3.61) where 𝐵_𝑋(𝑡,⋅)≐𝐵_𝑋𝑡,𝑋(𝑡),𝑣(𝑡,⋅),Σ_𝑋(𝑡,⋅)≐Σ_𝑋𝑡,𝑋(𝑡),𝑣(𝑡,⋅),Υ_𝑋(𝑡,⋅)≐𝐵_𝑋𝑡,𝑋(𝑡),𝑣(𝑡,⋅),𝑙_𝑋(𝑡,⋅)≐𝑙_𝑋𝑡,𝑋(𝑡),𝑣(𝑡,⋅), (3.62)(Φ₀,Φ_𝑇) is determined by (3.25).

Applying Itô’s formula to ⟨Φ^𝛿,𝜀(⋅),𝑋₁^𝛿,𝜀(⋅)+𝑋₂^𝛿,𝜀(⋅)⟩, we can get−𝔼Φ₀^𝛿,𝜀Γ_𝑋𝑇𝑋^𝛿₀,𝑋^𝛿(𝑇)+Π_𝑋𝑇0,𝑋^𝛿(𝑇)0Φ_𝑇^𝛿,𝜀,𝑋₁^𝛿,𝜀(𝑇)+𝑋₂^𝛿,𝜀(𝑇Φ)=𝔼^𝛿,𝜀(𝑇),𝑋₁^𝛿,𝜀(𝑇)+𝑋₂^𝛿,𝜀(𝑇Φ)=𝔼^𝛿,𝜀(0),0√𝜀𝑦₀+𝔼^𝑇₀−𝐵^𝛿_𝑋(𝑡,⋅)^⊤Φ^𝛿,𝜀(𝑡)−Σ^𝛿_𝑋(𝑡,⋅)^⊤Ψ^𝛿,𝜀(𝑡)−_𝐄Υ^𝛿_𝑋(𝑡,𝑒)^⊤Ξ^𝛿,𝜀(𝑡,𝑒)𝜋(𝑑𝑒)+Φ₀^𝛿,𝜀𝑙^𝛿_𝑋(𝑡,⋅),𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡+Φ_𝛿,𝜀)(𝑡),𝐵^𝛿_𝑋(𝑡,⋅)^⊤𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡)+Δ𝐵^𝛿(𝑡,⋅)𝐼_𝑆𝜀(𝑡)+12𝐵^𝛿_𝑋𝑋(𝑡,⋅)𝑋₁^𝛿,𝜀(𝑡)²+Ψ^𝛿,𝜀(𝑡),Σ^𝛿_𝑋(𝑡,⋅)^⊤𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡)+ΔΣ^𝛿_𝑋(𝑡,⋅)𝑋₁^𝛿,𝜀(𝑡)+ΔΣ^𝛿(𝑡,⋅)𝐼_𝑆𝜀(𝑡)+12Σ^𝛿_𝑋𝑋(𝑡,⋅)𝑋₁^𝛿,𝜀(𝑡)²+_𝐄Ξ^𝛿,𝜀(𝑡,𝑒),Υ^𝛿_𝑋(𝑡,𝑒)^⊤𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡)+ΔΥ^𝛿_𝑋(𝑡,𝑒)𝑋₁^𝛿,𝜀(𝑡)+ΔΥ^𝛿(𝑡,𝑒)𝐼_𝑆𝜀(𝑡)+12Υ^𝛿_𝑋𝑋(𝑡,𝑒)𝑋₁^𝛿,𝜀(𝑡)²𝜋(Φ𝑑𝑒)𝑑𝑡=𝔼^𝛿,𝜀(0),0√𝜀𝑦₀+𝔼^𝑇₀Φ^𝛿,𝜀(𝑡),Δ𝐵^𝛿(𝑡,⋅)𝐼_𝑆𝜀(𝑡)+12𝐵^𝛿_𝑋𝑋(𝑡,⋅)𝑋₁^𝛿,𝜀(𝑡)²+Φ₀^𝛿,𝜀𝑙^𝛿_𝑋(𝑡,⋅),𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡+Ψ_𝛿,𝜀)(𝑡),ΔΣ^𝛿(𝑡,⋅)𝐼_𝑆𝜀(𝑡)+12Σ^𝛿_𝑋𝑋(𝑡,⋅)𝑋₁^𝛿,𝜀(𝑡)²+_𝐄Ξ^𝛿,𝜀(𝑡,𝑒),ΔΥ^𝛿(𝑡,𝑒)𝐼_𝑆𝜀(𝑡)+12Υ^𝛿_𝑋𝑋(𝑡,𝑒)𝑋₁^𝛿,𝜀(𝑡)²𝜋(𝜀𝑑𝑒)𝑑𝑡+𝑂^3/2. (3.63) Then by (3.59), we have−√𝛿√𝜀𝑦₀+𝜀(𝑇+𝐾)≤𝔼^𝑇₀Φ₀^𝛿,𝜀𝑙𝑡,𝑋^𝛿(𝑡),𝑣^𝛿,𝜀(𝑡,⋅)−𝑙𝑡,𝑋^𝛿(𝑡),𝑣^𝛿(𝑡,⋅)+Φ₀^𝛿,𝜀𝑙^𝛿_𝑋(𝑡,⋅),𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡+1)2Φ₀^𝛿,𝜀𝑙^𝛿_𝑋𝑋(𝑡,⋅)𝑋₁^𝛿,𝜀(𝑡),𝑋₁^𝛿,𝜀(𝑡Φ)𝑑𝑡−𝔼^𝛿,𝜀(0),0√𝜀𝑦₀−𝔼^𝑇₀Φ^𝛿,𝜀(𝑡),Δ𝐵^𝛿(𝑡,⋅)𝐼_𝑆𝜀(𝑡)+12𝐵^𝛿_𝑋𝑋(𝑡,⋅)𝑋₁^𝛿,𝜀(𝑡)²−Φ₀^𝛿,𝜀𝑙^𝛿_𝑋(𝑡,⋅),𝑋₁^𝛿,𝜀(𝑡)+𝑋₂^𝛿,𝜀(𝑡+Ψ_𝛿,𝜀)(𝑡),ΔΣ^𝛿(𝑡,⋅)𝐼_𝑆𝜀(𝑡)+12Σ^𝛿_𝑋𝑋(𝑡,⋅)𝑋₁^𝛿,𝜀(𝑡)²+_𝐄Ξ^𝛿,𝜀(𝑡,𝑒),ΔΥ^𝛿(𝑡,𝑒)𝐼_𝑆𝜀(𝑡)+12Υ^𝛿_𝑋𝑋(𝑡,𝑒)𝑋₁^𝛿,𝜀(𝑡)²𝜋(√𝑑𝑒)𝑑𝑡+𝔼𝜀Φ₀^𝛿,𝜀Γ_𝑋0𝑋^𝛿₀,𝑋^𝛿(𝑇,0𝑦)₀+𝜀2Φ₀^𝛿,𝜀Γ_𝑋0𝑋0𝑋^𝛿₀,𝑋^𝛿(𝑇)0𝑦₀,0𝑦₀+12Φ₀^𝛿,𝜀Γ_{𝑋𝑇𝑋𝑇}𝑋^𝛿₀,𝑋^𝛿(𝑇)+Π_{𝑋𝑇𝑋𝑇}0,𝑋^𝛿(𝑇)0Φ_𝑇^𝛿,𝜀𝑋₁^𝛿,𝜀(𝑇),𝑋₁^𝛿,𝜀(𝑇)+𝑂𝜀^3/2=𝔼^𝑇₀Φ₀^𝛿,𝜀𝑙𝑡,𝑋^𝛿(𝑡),𝑣^𝛿,𝜀(𝑡,⋅)−𝑙𝑡,𝑋^𝛿(𝑡),𝑣^𝛿()+1𝑡,⋅2Φ₀^𝛿,𝜀𝑙^𝛿_𝑋𝑋(𝑡,⋅)𝑋₁^𝛿,𝜀(𝑡),𝑋₁^𝛿,𝜀(𝑡)𝑑𝑡−𝔼^𝑇₀Φ^𝛿,𝜀(𝑡),Δ𝐵^𝛿(𝑡,⋅)𝐼_𝑆𝜀(𝑡)+12𝐵^𝛿_𝑋𝑋(𝑡,⋅)𝑋₁^𝛿,𝜀(𝑡)²+Ψ^𝛿,𝜀(𝑡),ΔΣ^𝛿(𝑡,⋅)𝐼_𝑆𝜀(𝑡)+12Σ^𝛿_𝑋𝑋(𝑡,⋅)𝑋₁^𝛿,𝜀(𝑡)²+_𝐄Ξ^𝛿,𝜀(𝑡,𝑒),ΔΥ^𝛿(𝑡,𝑒)𝐼_𝑆𝜀(𝑡)+12Υ^𝛿_𝑋𝑋(𝑡,𝑒)𝑋₁^𝛿,𝜀(𝑡)²𝜋(√𝑑𝑒)𝑑𝑡+𝔼𝜀Φ₀^𝛿,𝜀Γ_𝑋0𝑋^𝛿₀,𝑋^𝛿(𝑇)−Φ^𝛿,𝜀(0),0𝑦₀+𝜀2Φ₀^𝛿,𝜀Γ_𝑋0𝑋0𝑋^𝛿₀,𝑋^𝛿(𝑇)0𝑦₀,0𝑦₀+12Φ₀^𝛿,𝜀Γ_{𝑋𝑇𝑋𝑇}𝑋^𝛿₀,𝑋^𝛿(𝑇)+Π_{𝑋𝑇𝑋𝑇}0,𝑋^𝛿(𝑇)0Φ_𝑇^𝛿,𝜀𝑋₁^𝛿,𝜀(𝑇),𝑋₁^𝛿,𝜀(𝑇)+𝑂𝜀^3/2. (3.64) Note that 𝑌^𝛿,𝜀(⋅)≐𝑋₁^𝛿,𝜀(⋅)𝑋₁^𝛿,𝜀(⋅)^⊤ satisfies 𝑑𝑌^𝛿,𝜀(𝑡)=𝐵^𝛿_𝑋(𝑡,⋅)𝑌^𝛿,𝜀(𝑡)+𝑌^𝛿,𝜀(𝑡)𝐵^𝛿_𝑋(𝑡,⋅)^⊤+Σ^𝛿_𝑋(𝑡,⋅)𝑌^𝛿,𝜀(𝑡)Σ^𝛿_𝑋(𝑡,⋅)^⊤+ΔΣ^𝛿(𝑡,⋅)ΔΣ^𝛿(𝑡,⋅)^⊤+Σ^𝛿_𝑋(𝑡,⋅)𝑋₁^𝛿,𝜀(𝑡)ΔΣ^𝛿(𝑡,⋅)^⊤+ΔΣ^𝛿(𝑡,⋅)𝑋₁^𝛿,𝜀(𝑡)Σ^𝛿_𝑋(𝑡,⋅)^⊤𝐼_𝑆𝜀(𝑡)+_𝐄Υ^𝛿_𝑋(𝑡,𝑒)𝑌^𝛿,𝜀(𝑡)Υ^𝛿_𝑋(𝑡,𝑒)^⊤+Υ^𝛿_𝑋(𝑡,𝑒)𝑋₁^𝛿,𝜀(𝑡)ΔΥ^𝛿(𝑡,𝑒)^⊤+ΔΥ^𝛿(𝑡,𝑒)𝑋₁^𝛿,𝜀(𝑡)Υ^𝛿_𝑋(𝑡,𝑒)^⊤+ΔΥ^𝛿(𝑡,𝑒)ΔΥ^𝛿(𝑡,⋅)^⊤𝐼_𝑆𝜀(𝑡𝜋(+Σ)𝑑𝑒)𝑑𝑡_𝛿𝑋(𝑡,⋅)𝑌^𝛿,𝜀(𝑡)+𝑌^𝛿,𝜀(𝑡)Σ^𝛿_𝑋(𝑡,⋅)^⊤+𝑋₁^𝛿,𝜀(𝑡)ΔΣ^𝛿(𝑡,⋅)^⊤+ΔΣ^𝛿(𝑡,⋅)𝑋₁^𝛿,𝜀(𝑡)^⊤𝐼_𝑆𝜀(𝑡(𝑡)+)𝑑𝑊_𝐄Υ^𝛿_𝑋(𝑡,𝑒)𝑌^𝛿,𝜀(𝑡−)+𝑌^𝛿,𝜀(𝑡−)Υ^𝛿_𝑋(𝑡,𝑒)^⊤+Υ^𝛿_𝑋(𝑡,𝑒)𝑌^𝛿,𝜀(𝑡−)Υ^𝛿_𝑋(𝑡,𝑒)^⊤+𝑋₁^𝛿,𝜀(𝑡−)ΔΥ^𝛿(𝑡,𝑒)^⊤+ΔΥ^𝛿(𝑡,𝑒)𝑋₁^𝛿,𝜀(𝑡−)^⊤+Υ^𝛿_𝑋(𝑡,𝑒)𝑋₁^𝛿,𝜀(𝑡−)ΔΥ^𝛿(𝑡,𝑒)^⊤+ΔΥ^𝛿(𝑡,𝑒)𝑋₁^𝛿,𝜀(𝑡−)Υ^𝛿_𝑋(𝑡,𝑒)^⊤+ΔΥ^𝛿(𝑡,𝑒)ΔΥ^𝛿(𝑡,𝑒)^⊤𝐼_𝑆𝜀(𝑡𝑁(),𝑌)𝑑𝑒𝑑𝑡_𝛿,𝜀(0)=000𝜀𝑦₀𝑦^⊤₀. (3.65) Now, let (𝑃^𝛿,𝜀(⋅),𝑄^𝛿,𝜀(⋅),𝐾^𝛿,𝜀(⋅,⋅)) be the adapted solution to the following BSDEJ:−𝑑𝑃^𝛿,𝜀(𝑡)=𝐵^𝛿_𝑋(𝑡,⋅)^⊤𝑃^𝛿,𝜀(𝑡)+𝑃^𝛿,𝜀(𝑡)𝐵^𝛿_𝑋(𝑡,⋅)+Σ^𝛿_𝑋(𝑡,⋅)^⊤𝑃^𝛿,𝜀(𝑡)Σ^𝛿_𝑋(𝑡,⋅)+Σ^𝛿_𝑋(𝑡,⋅)^⊤𝑄^𝛿,𝜀(𝑡)+𝑄^𝛿,𝜀(𝑡)Σ^𝛿_𝑋(𝑡,⋅)+_𝐄Υ^𝛿_𝑋(𝑡,𝑒)^⊤𝑃^𝛿,𝜀(𝑡)Υ^𝛿_𝑋(𝑡,𝑒)+Υ^𝛿_𝑋(𝑡,𝑒)^⊤𝐾^𝛿,𝜀(𝑡,𝑒)Υ^𝛿_𝑋(𝑡,𝑒)+Υ^𝛿_𝑋(𝑡,𝑒)^⊤𝐾^𝛿,𝜀(𝑡,𝑒)+𝐾^𝛿,𝜀(𝑡,𝑒)Υ^𝛿_𝑋(𝑡,𝑒)𝜋(𝑑𝑒)+𝐻^{𝛿,𝜀,Φ0𝛿,𝜀}_𝑋𝑋(𝑡,⋅)𝑑𝑡−𝑄^𝛿,𝜀(𝑡)𝑑𝑊(𝑡)−_𝐄𝐾^𝛿,𝜀(𝑡,𝑒)𝑁(𝑑𝑒𝑑𝑡),𝑑𝑃^𝛿,𝜀(𝑇)=−Φ₀^𝛿,𝜀Γ_{𝑋𝑇𝑋𝑇}𝑋^𝛿₀,𝑋^𝛿(𝑇)+Π_{𝑋𝑇𝑋𝑇}0,𝑋^𝛿(𝑇)0Φ_𝑇^𝛿,𝜀, (3.66) where𝐻^{𝛿,𝜀,Φ0𝛿,𝜀}_𝑋𝑋(𝑡,⋅)≐𝐻_𝑋𝑋𝑡,Φ₀^𝛿,𝜀,𝑋^𝛿(𝑡),𝑣^𝛿(𝑡,⋅),Φ^𝛿,𝜀(𝑡),Ψ^𝛿,𝜀(𝑡),Ξ^𝛿,𝜀(𝑡,⋅), (3.67) with 𝐻_𝑋𝑋(𝑡,𝜆,𝑋,𝑣(⋅),Φ,Ψ,Ξ(⋅)) being the Hessian of the following𝐻(⋅),Ξ(⋅)≐((⋅+((⋅+𝑡,𝜆,𝑋,𝑣Φ,Ψ,Φ,𝐵𝑡,𝑋,𝑣))Ψ,Σ𝑡,𝑋,𝑣))_𝐄Ξ(𝑒),Υ(𝑡,𝑋,𝑣(𝑒))𝜋(𝑑𝑒)−𝜆𝑙(𝑡,𝑋,𝑣(⋅)). (3.68) Applying Itô’s formula to [𝑃^𝛿,𝜀(⋅)𝑌^𝛿,𝜀(⋅)], we have−𝔼Φ₀^𝛿,𝜀Γ_{𝑋𝑇𝑋𝑇}𝑋^𝛿₀,𝑋^𝛿(𝑇)+Π_{𝑋𝑇𝑋𝑇}0,𝑋^𝛿(𝑇)0Φ_𝑇^𝛿,𝜀𝑋₁^𝛿,𝜀(𝑇),𝑋₁^𝛿,𝜀(𝑇+𝑃_𝛿,𝜀)(0)0√𝜀𝑦₀,0√𝜀𝑦₀=𝔼tr𝑃^𝛿,𝜀(𝑇)𝑌^𝛿,𝜀(𝑇)−tr𝑃^𝛿,𝜀(0)𝑌^𝛿,𝜀(0)=𝔼^𝑇₀tr𝑃^𝛿,𝜀(𝑡)ΔΣ^𝛿(𝑡,⋅)ΔΣ^𝛿(𝑡,⋅)^⊤+Σ^𝛿_𝑋(𝑡,⋅)𝑋₁^𝛿,𝜀(𝑡)ΔΣ^𝛿(𝑡,⋅)^⊤+ΔΣ^𝛿(𝑡,⋅)𝑋₁^𝛿,𝜀(𝑡)Σ^𝛿_𝑋(𝑡,⋅)^⊤𝐼_𝑆𝜀(𝑡)+𝑄^𝛿,𝜀(𝑡)𝑋₁^𝛿,𝜀(𝑡)ΔΣ^𝛿(𝑡,⋅)^⊤+ΔΣ^𝛿(𝑡,⋅)𝑋₁^𝛿,𝜀(𝑡)^⊤𝐼_𝑆𝜀(𝑡)−𝐻^{𝛿,𝜀,Φ0𝛿,𝜀}_𝑋𝑋(𝑡,⋅)𝑌^𝛿,𝜀(𝑡)+_𝐄𝑃^𝛿,𝜀(𝑡)Υ^𝛿_𝑋(𝑡,𝑒)𝑋₁^𝛿,𝜀(𝑡)ΔΥ^𝛿(𝑡,𝑒)^⊤+ΔΥ^𝛿(𝑡,𝑒)𝑋₁^𝛿,𝜀(𝑡)Υ^𝛿_𝑋(𝑡,𝑒)^⊤+ΔΥ^𝛿(𝑡,𝑒)ΔΥ^𝛿(𝑡,𝑒)^⊤𝐼_𝑆𝜀(𝑡)+𝐾^𝛿,𝜀(𝑡,𝑒)𝑋₁^𝛿,𝜀(𝑡−)ΔΥ^𝛿(𝑡,𝑒)^⊤+ΔΥ^𝛿(𝑡,𝑒)𝑋₁^𝛿,𝜀(𝑡−)^⊤+Υ^𝛿_𝑋(𝑡,𝑒)𝑋₁^𝛿,𝜀(𝑡−)ΔΥ^𝛿(𝑡,𝑒)^⊤+ΔΥ^𝛿(𝑡,𝑒)𝑋₁^𝛿,𝜀(𝑡−)Υ^𝛿_𝑋(𝑡,𝑒)^⊤+ΔΥ^𝛿(𝑡,𝑒)ΔΥ^𝛿(𝑡,𝑒)^⊤𝐼_𝑆𝜀(𝑡)𝜋(𝑑𝑒)𝑑𝑡=𝔼^𝑇₀tr𝑃^𝛿,𝜀(𝑡)ΔΣ^𝛿(𝑡,⋅)ΔΣ^𝛿(𝑡,⋅)^⊤+_𝐄ΔΥ^𝛿(𝑡,𝑒)ΔΥ^𝛿(𝑡,𝑒)^⊤𝜋(𝑑𝑒)𝐼_𝑆𝜀(𝑡)+_𝐄𝐾^𝛿,𝜀(𝑡,𝑒)ΔΥ^𝛿(𝑡,𝑒)ΔΥ^𝛿(𝑡,𝑒)^⊤𝐼_𝑆𝜀(𝑡)𝜋(𝑑𝑒)−𝐻^{𝛿,𝜀,Φ0𝛿,𝜀}_𝑋𝑋(𝑡,⋅)𝑌^𝛿,𝜀(𝑡)𝑑𝑡+𝑂𝜀^3/2=𝔼^𝑇₀trΔΣ^𝛿(𝑡,⋅)^⊤𝑃^𝛿,𝜀(𝑡)ΔΣ^𝛿(𝑡,⋅)𝐼_𝑆𝜀(𝑡)+_𝐄ΔΥ^𝛿(𝑡,𝑒)^⊤𝑃^𝛿,𝜀(𝑡)+𝐾^𝛿,𝜀(𝑡,𝑒)ΔΥ^𝛿(𝑡,𝑒)𝐼_𝑆𝜀(𝑡)𝜋()−𝐻𝑑𝑒_{𝛿,𝜀,Φ0𝛿,𝜀𝑋𝑋}(𝑡,⋅)𝑋₁^𝛿,𝜀(𝑡),𝑋₁^𝛿,𝜀(𝑡)𝑑𝑡+𝑂𝜀^3/2. (3.69) Consequently, from (3.64), we have −√𝛿√𝜀𝑦₀+𝜀(𝑇+𝐾)≤𝔼^𝑇₀Φ₀^𝛿,𝜀𝑙𝑡,𝑋^𝛿(𝑡),𝑣^𝛿,𝜀(𝑡,⋅)−𝑙𝑡,𝑋^𝛿(𝑡),𝑣^𝛿()−Φ𝑡,⋅_𝛿,𝜀(𝑡),Δ𝐵^𝛿(𝑡,⋅)−Ψ^𝛿,𝜀(𝑡),ΔΣ^𝛿(−𝑡,⋅)_𝐄Ξ^𝛿,𝜀(𝑡,𝑒),ΔΥ^𝛿(𝜋()−1𝑡,𝑒)𝑑𝑒2ΔΣ^𝛿(𝑡,⋅)^⊤𝑃^𝛿,𝜀(𝑡)ΔΣ^𝛿()−1𝑡,⋅2_𝐄ΔΥ^𝛿(𝑡,𝑒)^⊤𝑃^𝛿,𝜀(𝑡)+𝐾^𝛿,𝜀(𝑡,𝑒)ΔΥ^𝛿(𝑡,𝑒)𝜋(𝑑𝑒)𝐼_𝑆𝜀(𝑡)√𝑑𝑡+𝔼𝜀Φ₀^𝛿,𝜀Γ_𝑋0𝑋^𝛿₀,𝑋^𝛿(𝑇)−Φ^𝛿,𝜀(0),0𝑦₀+𝜀2Φ₀^𝛿,𝜀Γ_𝑋0𝑋0𝑋^𝛿₀,𝑋^𝛿(𝑇)−𝑃^𝛿,𝜀(00𝑦)₀,0𝑦₀+𝑂𝜀^3/2. (3.70)

Step 6 (variational inequality). In (3.70), dividing √𝜀 and then sending 𝜀→0 followed by sending 𝛿→0, we get 0≤𝔼Φ₀Γ_𝑋0𝑋₀(𝑇)−Φ(0),0𝑦,𝑋₀=𝔼Φ₀0𝛾_𝑦𝑦₀−Φ(0),0𝑦₀,∀𝑦₀∈𝐑^𝑚, (3.71) which implies Φ(0)=Φ₀0𝔼𝛾_𝑦𝑦₀. (3.72)

Using a standard argument of Tang and Li [24] in (3.70), we have the following variational inequality:Φ₀𝑙𝑡,𝑋(𝑡),𝑣(𝑡,⋅)−𝑙𝑡,𝑋(𝑡),𝑣(𝑡,⋅)−Φ(𝑡),𝐵𝑡,𝑋(𝑡),𝑣(𝑡,⋅)−𝐵𝑡,𝑋(𝑡),𝑣(−Ψ(𝑡)𝑡,⋅),Σ𝑡,𝑋(𝑡),𝑣(𝑡,⋅)−Σ𝑡,𝑋(𝑡),𝑣(𝑡,⋅)−_𝐄Ξ(𝑡,𝑒),Υ𝑡,𝑋(𝑡),𝑣(𝑡,𝑒)−Υ𝑡,𝑋(𝑡),𝑣()𝜋()−1𝑡,𝑒𝑑𝑒2Σ𝑡,𝑋(𝑡),𝑣(𝑡,⋅)−Σ𝑡,𝑋(𝑡),𝑣(𝑡,⋅)^⊤𝑃(𝑡)Σ𝑡,𝑋(𝑡),𝑣(𝑡,⋅)−Σ𝑡,𝑋(𝑡),𝑣()−1𝑡,⋅2_𝐄Υ𝑡,𝑋(𝑡),𝑣(𝑡,𝑒)−Υ𝑡,𝑋(𝑡),𝑣(𝑡,𝑒)^⊤[𝑃(𝑡)(×Υ+𝐾𝑡,𝑒)]𝑡,𝑋(𝑡),𝑣(𝑡,𝑒)−Υ𝑡,𝑋(𝑡),𝑣(𝑡,𝑒)𝜋(𝑑𝑒)≥0,∀𝑣(⋅),a.e.,a.s. (3.73)

Step 7 (finalizing the proof). Since Φ₀≠0, by rescaling, we let Φ₀=1. Then the first-order adjoint equation (3.61) reduces to −𝑑Φ(𝑡)=𝐵_𝑋(𝑡,⋅)^⊤Φ(𝑡)+Σ_𝑋(𝑡,⋅)^⊤Ψ(𝑡)+_𝐄Υ_𝑋(𝑡,𝑒)^⊤Ξ(𝑡,𝑒)𝜋(𝑑𝑒)−𝑙_𝑋(−Ψ(𝑡)(𝑡)−𝑡,⋅)𝑑𝑡𝑑𝑊_𝐄Ξ()𝑁(),Φ(𝑇)Γ𝑡,𝑒𝑑𝑒𝑑𝑡=−_𝑋𝑇𝑋₀,𝑋(𝑇)+Π_𝑋𝑇0,𝑋(𝑇0Φ)_𝑇. (3.74) Let (𝛿,𝜀)→0 in (3.66), then (𝑃^𝛿,𝜀(⋅),𝑄^𝛿,𝜀(⋅),𝐾^𝛿,𝜀(⋅,⋅))→(𝑃(⋅),𝑄(⋅),𝐾(⋅,⋅)) which is the adapted solution to the following second-order adjoint equation:

−𝑑𝑃(𝑡)=𝐵_𝑋(𝑡,⋅)^⊤𝑃(𝑡)+𝑃(𝑡)𝐵_𝑋(𝑡,⋅)+Σ_𝑋(𝑡,⋅)^⊤𝑃(𝑡)Σ_𝑋(𝑡,⋅)+Σ_𝑋(𝑡,⋅)^⊤𝑄(𝑡)+𝑄(𝑡)Σ_𝑋()+𝑡,⋅_𝐄Υ_𝑋(𝑡,𝑒)^⊤𝑃(𝑡)Υ_𝑋(𝑡,𝑒)+Υ_𝑋(𝑡,𝑒)^⊤𝐾(𝑡,𝑒)Υ_𝑋()+𝑡,𝑒Υ_𝑋(𝑡,𝑒)^⊤𝐾(𝑡,𝑒)+𝐾(𝑡,𝑒)Υ_𝑋(𝑡,𝑒)𝜋(𝑑𝑒)+𝐻_𝑋𝑋((𝑡)(𝑡)−𝑡,⋅)𝑑𝑡−𝑄𝑑𝑊_𝐄𝐾()𝑁(),𝑃(𝑇)Γ𝑡,𝑒𝑑𝑒𝑑𝑡=−_{𝑋𝑇𝑋𝑇}𝑋₀,𝑋(𝑇)+Π_{𝑋𝑇𝑋𝑇}0,𝑋(𝑇0Φ)_𝑇, (3.75) where𝐻_𝑋𝑋(𝑡,⋅)≐𝐻_𝑋𝑋𝑡,𝑋(𝑡),𝑣(𝑡,⋅),Φ(𝑡),Ψ(𝑡),Ξ(𝑡,⋅), (3.76) with 𝐻_𝑋𝑋(𝑡,𝑋,𝑣(⋅),Φ,Ψ,Ξ(⋅)) being the Hessian of the following𝐻(⋅),Ξ(⋅)≐((⋅+((⋅+𝑡,𝑋,𝑣Φ,Ψ,Φ,𝐵𝑡,𝑋,𝑣))Ψ,Σ𝑡,𝑋,𝑣))_𝐄Ξ(𝑒),Υ(𝑡,𝑋,𝑣(𝑒))𝜋(𝑑𝑒)−𝑙(𝑡,𝑋,𝑣(⋅)). (3.77) Also, from (3.72), we haveΦ(0)=0𝔼𝛾_𝑦𝑦₀. (3.78)

Note thatΓ_𝑋𝑇𝑋₀,𝑋(𝑇)=𝜙_𝑥(𝑥(𝑇0)),Γ_{𝑋𝑇𝑋𝑇}𝑋₀,𝑋(𝑇)=𝜙_𝑥𝑥(𝑥(𝑇))000. (3.79) On the other hand, sinceΠ0,𝑋(𝑇),0Φ_𝑇=𝑦(𝑇)−ℎ𝑋(𝑇)Φ_𝑇, (3.80) we haveΠ_𝑋𝑇0,𝑋(𝑇)0Φ_𝑇=−ℎ_𝑥(𝑥(𝑇))^⊤Φ_𝑇Φ_𝑇,Π_{𝑋𝑇𝑋𝑇}0,𝑋(𝑇)0Φ_𝑇=−ℎ_𝑥𝑥(𝑥(𝑇))^⊤Φ_𝑇0.00 (3.81)

LetΦ(⋅)≡𝑝(⋅)𝑞(⋅),Ψ(⋅)≡𝑘(⋅)𝑘(⋅),Ξ()≡𝑟()()⋅,⋅⋅,⋅𝑟̃⋅,⋅. (3.82) Then it follows from (3.74), (3.78), and (3.82) that𝑝(0)𝑞(0)=0𝔼𝛾_𝑦𝑦₀,𝑝(𝑇)𝑞(𝑇)=−𝜙_𝑥(𝑥(𝑇))+ℎ_𝑥(𝑥(𝑇))^⊤Φ_𝑇−Φ_𝑇. (3.83) By (3.82), we can rewrite the equations in (3.74) as𝑝(𝑡)𝑞(𝑡)−𝑑=𝑏_𝑥(𝑡,⋅)^⊤−𝑓_𝑥(𝑡,⋅)^⊤𝑏_𝑦(𝑡,⋅)^⊤−𝑓_𝑦(𝑡,⋅)^⊤𝑝(𝑡)𝑞(𝑡)+𝜎_𝑥(𝑡,⋅)^⊤0𝜎_𝑦(𝑡,⋅)^⊤0𝑘(𝑡)𝑘(𝑡)+_𝐄𝑔_𝑥(𝑡,𝑒)^⊤0𝑔_𝑦(𝑡,𝑒)^⊤0𝑟()()𝜋()−𝑡,𝑒𝑟̃𝑡,𝑒𝑑𝑒𝑙_𝑥(𝑡,⋅)𝑙_𝑦()−𝑘(𝑡)𝑘(𝑡)(𝑡)−𝑡,⋅𝑑𝑡𝑑𝑊_𝐄𝑟()()𝑁(),𝑡,𝑒𝑟̃𝑡,𝑒𝑑𝑒𝑑𝑡 (3.84) and the variational inequality (3.73) now takes the form: 𝑙𝑡,𝑥(𝑡),𝑦(𝑡),𝑧,𝑐(⋅),𝑢−𝑙𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)−𝑝(𝑡),𝑏𝑡,𝑥(𝑡),𝑦(𝑡),𝑧,𝑐(⋅),𝑢−𝑏𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)+𝑞(𝑡),𝑓𝑡,𝑥(𝑡),𝑦(𝑡),𝑧,𝑐(⋅),𝑢−𝑓𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)−𝑘(𝑡),𝜎𝑡,𝑥(𝑡),𝑦(𝑡),𝑧,𝑐(⋅),𝑢−𝜎𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)−_𝐄𝑟(𝑡,𝑒),𝑔𝑡,𝑥(𝑡),𝑦(𝑡),z,𝑐(𝑒),𝑢−𝑔𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,𝑒),𝑢(𝑡)𝜋()−𝑘(𝑡)𝑑𝑒,𝑧−𝑧(𝑡)−_𝐄⟨𝑟̃(𝑡,𝑒),𝑐(𝑒)−𝑐(𝜋()−1𝑡,𝑒)⟩𝑑𝑒2𝜎𝑡,𝑥(𝑡),𝑦(𝑡),𝑧,𝑐(⋅),𝑢−𝜎𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)𝑧−𝑧(𝑡)^⊤𝑃(𝑡)×𝜎𝑡,𝑥(𝑡),𝑦(𝑡),𝑧,𝑐(⋅),𝑢−𝜎𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)𝑧−𝑧(𝑡)−12_𝐄𝑔𝑡,𝑥(𝑡),𝑦(𝑡),𝑧,𝑐(𝑒),𝑢,𝑒−𝑔𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,𝑒),𝑢(𝑡)𝑐(𝑒)−,𝑒𝑐(𝑡,𝑒)^⊤×[𝑃(𝑡)(×𝑔+𝐾𝑡,𝑒)]𝑡,𝑥(𝑡),𝑦(𝑡),𝑧,𝑐(𝑒),𝑢,𝑒−𝑔𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,𝑒),𝑢(𝑡)𝑐(𝑒)−,𝑒𝑐()𝜋()(⋅)𝑡,𝑒𝑑𝑒≥0,∀𝑧,𝑐,𝑢,a.e.,a.s. (3.85) Thus, taking 𝑢=𝑢(⋅), 𝑧=𝑧(𝑡)+𝜀𝑧₀ and 𝑐(⋅)=𝑐(𝑡,⋅)+𝜀𝑐₀(⋅), then dividing by 𝜀 and sending 𝜀→0, we get 0≤𝑙_𝑧𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡),𝑧₀−𝑝(𝑡),𝑏_𝑧𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)𝑧₀+𝑞(𝑡),𝑓_𝑧𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)𝑧₀−𝑘(𝑡),𝜎_𝑧𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)𝑧₀−_𝐄𝑟(𝑡,𝑒),𝑔_𝑧𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,𝑒),𝑢(𝑡)𝑧₀−𝑘(𝑡),𝑧₀+𝑙_𝑐𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡),𝑐₀(⋅)−𝑝(𝑡),𝑏_𝑐𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)𝑐₀(⋅+𝑞(𝑡)),𝑓_𝑐𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)𝑐₀(⋅)−𝑘(𝑡),𝜎_𝑐𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)𝑐_0𝐄𝑟(𝑡,𝑒),𝑔_𝑐𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,𝑒),𝑢(𝑡)𝑐₀(𝑒−_𝐄)𝑟̃(𝑡,𝑒),𝑐₀(𝑒)𝜋(𝑑𝑒),∀𝑧₀∈𝐑^𝑚,𝑐₀(⋅)∈ℒ²𝐄,ℬ(𝐄),𝜋;𝐑^𝑚. (⋅−)(3.86) Hence,𝑘(𝑡)=𝑙_𝑧𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)−𝑏_𝑧𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)^⊤𝑝(𝑡)+𝑓_𝑧𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)^⊤𝑞(𝑡)−𝜎_𝑧𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)^⊤𝑘(𝑡)−_𝐄𝑔_𝑧𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,𝑒),𝑢(𝑡)^⊤𝑟()𝜋(),()𝑡,𝑒𝑑𝑒𝑟̃𝑡,⋅=𝑙_𝑐𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)−𝑏_𝑐𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)^⊤𝑝(𝑡)+𝑓_𝑐𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)^⊤𝑞(𝑡)−𝜎_𝑐𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)^⊤𝑘(𝑡)−_𝐄𝑔_𝑐𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,𝑒),𝑢(𝑡)^⊤𝑟(𝑡,𝑒)𝜋(𝑑𝑒). (3.87) Combining (3.83), (3.84) with (3.87), we arrive at (2.10).

Next, we have𝑃(𝑇)=−Γ_{𝑋𝑇𝑋𝑇}𝑋₀,𝑋(𝑇)+Π_{𝑋𝑇𝑋𝑇}0,𝑋(𝑇)0Φ_𝑇=−𝜙_𝑥𝑥(𝑥(𝑇))+ℎ_𝑥𝑥(𝑥(𝑇))^⊤Φ_𝑇0=00−𝜙_𝑥𝑥(𝑥(𝑇))−ℎ_𝑥𝑥(𝑥(𝑇))^⊤𝑞(𝑇)0.00 (3.88) Then (2.11) follows from (3.75).

Further, by taking 𝑧=𝑧(⋅),𝑐(⋅)=𝑐(⋅,⋅) in the variational inequality (3.85), we obtain𝑙𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢−𝑙𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)−𝑝(𝑡),𝑏𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢−𝑏𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)+𝑞(𝑡),𝑓𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢−𝑓𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)−𝑘(𝑡),𝜎𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢−𝜎𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)−_𝐄𝑟(𝑡,𝑒),𝑔𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,𝑒),𝑢,𝑒−𝑔𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,𝑒),𝑢(𝑡)𝜋()−1,𝑒𝑑𝑒2𝜎𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢−𝜎𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)0^⊤𝑃(𝑡)×𝜎𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢−𝜎𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢(𝑡)0−12_𝐄𝑔𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢,𝑒−𝑔𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,𝑒),𝑢(𝑡)0,𝑒^⊤×[𝑃(𝑡)(×𝑔+𝐾𝑡,𝑒)]𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,⋅),𝑢,𝑒−𝑔𝑡,𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑐(𝑡,𝑒),𝑢(𝑡)0𝜋(),𝑒𝑑𝑒≥0,∀𝑢∈𝐔,a.e.,a.s., (3.89) 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:𝑑𝑥(𝑡)=𝑏₁𝑥(𝑡)+𝑏₂𝑦(𝑡)+𝑏₃𝑧(𝑡)+𝑏₄𝑐(𝑡,⋅)+𝑏₅𝑢(𝑡+𝜎)𝑑𝑡₁𝑥(𝑡)+𝜎₂𝑦(𝑡)+𝜎₃𝑧(𝑡)+𝜎₄𝑐(𝑡,⋅)+𝜎₅𝑢(𝑡(𝑡)+)𝑑𝑊_𝐄𝑔₁𝑥(𝑡−)+𝑔₂𝑦(𝑡−)+𝑔₃𝑧(𝑡)+𝑔₄𝑐(𝑡,𝑒)+𝑔₅𝑢(𝑡𝑁(),(𝑡)=𝑓)𝑑𝑒𝑑𝑡−𝑑𝑦₁𝑥(𝑡)+𝑓₂𝑦(𝑡)+𝑓₃𝑧(𝑡)+𝑓₄𝑐(𝑡,⋅)+𝑓₅𝑢(𝑡(𝑡)(𝑡)−)𝑑𝑡−𝑧𝑑𝑊_𝐄𝑐()𝑁(),𝑥(0)(𝑇)𝑡,𝑒𝑑𝑒𝑑𝑡=𝑎,𝑦=ℎ₁𝑥(𝑇), (4.1) with the quadratic cost functional given by𝐽(𝑢(⋅=1))2𝔼^𝑇₀𝑙₁𝑥²(𝑡)+𝑙₂𝑦²(𝑡)+𝑙₃𝑧²(𝑡)+𝑙₄𝑐²(𝑡,⋅)+𝑙₅𝑢²(𝑡)𝑑𝑡+𝜙₁𝑥²(𝑇)+𝛾₁𝑦²(0). (4.2) In the above, 𝑏_𝑖,𝜎_𝑖,𝑔_𝑖,𝑓_𝑖,𝑙_𝑖, 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 ℱ_𝑡-predictable processes 𝑢∶[0,𝑇]×Ω→𝐔 such that sup_{0≤𝑡≤𝑇}𝔼|𝑢(𝑡)|^𝑖<+∞,∀𝑖=1,2,….

For any 𝑢(⋅)∈𝒰[0,𝑇], we denote𝑥𝑦𝑧Ξ=,𝒜(𝑡,Ξ,𝑐(⋅),𝑢)=−𝑓₁𝑥−𝑓₂𝑦−𝑓₃𝑧−𝑓₄𝑐(⋅)−𝑓₅𝑢𝑏₁𝑥+𝑏₂𝑦+𝑏₃𝑧+𝑏₄𝑐(⋅)+𝑏₅𝑢𝜎₁𝑥+𝜎₂𝑦+𝜎₃𝑧+𝜎₄𝑐(⋅)+𝜎₅𝑢, (4.3) and assume that (H4.1)𝒜(𝑡,Ξ,𝑐(⋅),𝑢)−𝒜𝑡,Ξ^′,𝑐^′(⋅),𝑢,Ξ−Ξ^′+_𝐄𝑔₁𝑥−𝑥^′+𝑔₂𝑦−𝑦^′+𝑔₃𝑧−𝑧^′+𝑔₄c(𝑒)−𝑐^′(𝑒),𝑐(𝑒)−𝑐^′(𝑒)𝜋(𝑑𝑒)≤−𝛽₁𝑥−𝑥^′2−𝛽₂𝑦−𝑦^′2+𝑧−𝑧^′2+_𝐄𝑐(𝑒)−𝑐^′(𝑒)²𝜋(),ℎ𝑑𝑒₁≥𝜇₁>0, (4.4) 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−𝑑𝑝(𝑡)=𝑏₁𝑝(𝑡)−𝑓₁𝑞(𝑡)+𝜎₁𝑘(𝑡)+_𝐄𝑔₁𝑟(𝑡,𝑒)𝜋(𝑑𝑒)−𝑙₁𝑥(𝑡(𝑡)(𝑡)−)𝑑𝑡−𝑘𝑑𝑊_𝐄𝑟()𝑁(),(𝑡)=𝑡,𝑒𝑑𝑒𝑑𝑡𝑑𝑞−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(𝑡)+)𝑑𝑊_𝐄−𝑏₄𝑝(𝑡−)+𝑓₄𝑞(𝑡−)−𝜎₄𝑘(𝑡)−𝑔₄𝑟(𝑡,𝑒)+𝑙₄𝑐(𝑁(),𝑝(𝑇)𝑡,𝑒)𝑑𝑒𝑑𝑡=−𝜙₁𝑥(𝑇)−ℎ₁𝑞(𝑇),𝑞(0)=𝛾₁𝑦₀. (4.5) Similarly, we denote𝑞𝑝𝑘𝜗=,ℬ(𝑡,𝜗,𝑟(⋅))=−𝑏₁𝑝+𝑓₁𝑞−𝜎₁𝑘−𝑔₁𝑟(⋅)+𝑙₁𝑥−𝑏₂𝑝+𝑓₂𝑞−𝜎₂𝑘−𝑔₂𝑟(⋅)+𝑙₂𝑦−𝑏₂𝑝+𝑓₃𝑞−𝜎₃𝑘−𝑔₃𝑟(⋅)+𝑙₃𝑧. (4.6) Since controlled FBSDEJ satisfies (H4.1), we can easily check that the above FBSDEJ (4.4) satisfies the following condition (H4.2): ℬ(𝑡,𝜗,𝑟(⋅),𝑢)−ℬ𝑡,𝜗^′,𝑟^′(⋅),𝑢,𝜗−𝜗^′+_𝐄−𝑏₄𝑝−𝑝^′+𝑓₄𝑞−𝑞^′−𝜎₄𝑘−𝑘^′−𝑔₄𝑟(𝑒)−𝑟^′(𝑒)+𝑙₄𝑐(𝑒),𝑟(𝑒)−𝑟^′(𝑒)𝜋(𝑑𝑒)≥𝛽₁𝑞−𝑞^′2+𝛽₂𝑝−𝑞^′2+𝑘−𝑘^′2+_𝐄𝑟(𝑒)−𝑟^′(𝑒)²𝜋(𝑑𝑒),−ℎ₁≤−𝜇₁, (4.7) 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(⋅,⋅)).

The 2×2 matrix-valued second-order adjoint equation (2.11) becomes−𝑑𝑃(𝑡)=𝐵^⊤𝑃(𝑡)+𝑃(𝑡)𝐵+Σ^⊤𝑃(𝑡)Σ+Σ^⊤𝑄(𝑡)+𝑄(𝑡)Σ+Υ^⊤𝑃(𝑡)Υ+_𝐄Υ^⊤𝐾(𝑡,𝑒)Υ+Υ^⊤𝐾()()Υ𝜋()(𝑡)(𝑡)−𝑡,𝑒+𝐾𝑡,𝑒𝑑𝑒−𝐿𝑑𝑡−𝑄𝑑𝑊_𝐄𝐾()𝑁(),𝑃(𝑇)=𝑡,𝑒𝑑𝑒𝑑𝑡−𝜙₁0,00 (4.8) where𝑏𝐵=¹𝑏₂−𝑓₁−𝑓₂𝜎,Σ=¹𝜎₂𝑔00,Υ=¹𝑔₂𝑙00,𝐿=¹00𝑙₂. (4.9)

As before, (4.5) can be split into the following three scalar BSDEJs: −𝑑𝑃₁(𝑡)=2𝑏₁𝑃₁(𝑡)+𝜎²₁𝑃₁(𝑡)+2𝜎₁𝑄₁(𝑡)−2𝑓₁𝑃₂(𝑡)+𝑔²₁𝑃₁(𝑡)+_𝐄𝑔²₁𝐾₁(𝑡,𝑒)+2𝑔₁𝐾₁(𝑡,𝑒)𝜋(𝑑𝑒)−𝑙₁𝑑𝑡−𝑄₁(𝑡)𝑑𝑊(𝑡)−_𝐄𝐾₁()𝑁(),𝑃𝑡,𝑒𝑑𝑒𝑑𝑡₁(𝑇)=−𝜙₁,−𝑑𝑃₂(𝑡)=𝑏₂𝑃₁(𝑡)+𝑏₁𝑃₂(𝑡)+𝜎₁𝜎₂𝑃₁(𝑡)+𝜎₁𝑄₂(𝑡)+𝜎₂𝑄₁(𝑡)−𝑓₁𝑃₃(𝑡)−𝑓₂𝑃₂(𝑡)+𝑔₁𝑔₂𝑃₁(𝑡)+_𝐄𝑔₁𝑔₂𝐾₁(𝑡,𝑒)+𝑔₁𝐾₂(𝑡,𝑒)+𝑔₂𝐾₁(𝑡,𝑒)𝜋(𝑑𝑒)𝑑𝑡−𝑄₂(𝑡)𝑑𝑊(𝑡)−_𝐄𝐾₂()𝑁(),𝑃𝑡,𝑒𝑑𝑒𝑑𝑡₂(𝑇)=0,−𝑑𝑃₃(𝑡)=2𝑏₂𝑃₂(𝑡)+𝜎²₂𝑃₁(𝑡)+2𝜎₂𝑄₂(𝑡)−2𝑓₂𝑃₃(𝑡)+𝑔²₂𝑃₁(𝑡)+_𝐄𝑔²₂𝐾₁(𝑡,𝑒)+2𝑔₂𝐾₂(𝑡,𝑒)𝜋(𝑑𝑒)−𝑙₂𝑑𝑡−𝑄₃(𝑡)𝑑𝑊(𝑡)−_𝐄𝐾₃()𝑁(),𝑃𝑡,𝑒𝑑𝑒𝑑𝑡₃(𝑇)=0. (4.10)(4.11)(4.12)

Note that since all the coefficients in (4.5) are constants and the terminal condition is deterministic, we must have𝑄(⋅)=0,𝐾(⋅,⋅)=0, (4.13) and 𝑃(⋅) is deterministic. Consequently,𝑃̇(𝑡)=−𝐵^⊤𝑃(𝑡)+𝑃(𝑡)𝐵+Σ^⊤𝑃(𝑡)Σ+Υ^⊤𝑃(𝑡)Υ−𝐿,𝑃(𝑇)=−𝜙₁000. (4.14) And at the same time (4.7)–(4.10) reduce to𝑃̇₁(𝑡)=−2𝑏₁+𝜎²₁+𝑔²₁𝑃₁(𝑡)+2𝑓₂𝑃₂(𝑡)+𝑙₁,𝑃₁(𝑇)=−𝜙₁,𝑃̇₂(𝑡)=−𝑏₂+𝜎₁𝜎₂−𝑔₁𝑔₂𝑃₁(𝑡)+𝑏₁−𝑓₂𝑃₂(𝑡)−𝑓₁𝑃₃(𝑡),𝑃₂(𝑇)𝑃̇=0,₃(𝑡)=−𝜎²₂+𝑔²₂𝑃₁(𝑡)−𝜎²₂𝑃₂(𝑡)+2𝑓₂𝑃₃(𝑡)−𝑙₂,𝑃₃(𝑇)=0, (4.15) and 𝑄_𝑖(⋅)=0,𝐾_𝑖(⋅,⋅)=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) reads𝐻(⋅)(⋅)=𝑏𝑥,𝑦,𝑧,𝑐,𝑢,𝑝,𝑞,𝑘,𝑟₁𝑥+𝑏₂𝑦+𝑏₃𝑧+𝑏₄𝑐(⋅)+𝑏₅𝑢𝑝−𝑓₁𝑥+𝑓₂𝑦+𝑓₃𝑧+𝑓₄𝑐(⋅)+𝑓₅𝑢𝑞+𝜎₁𝑥+𝜎₂𝑦+𝜎₃𝑧+𝜎₄𝑐(⋅)+𝜎₅𝑢𝑘+_𝐄𝑔₁𝑥+𝑔₂𝑦+𝑔₃𝑧+𝑔₄𝑐(𝑒)+𝑔₅𝑢𝑟(𝑒)𝜋()−1𝑑𝑒2𝑙₁𝑥²+𝑙₂𝑦²+𝑙₃𝑧²+𝑙₄𝑐²(⋅)+𝑙₅𝑢², (4.16) the maximum condition (2.13) now takes the form (noting that 𝐾₁(⋅,⋅)=0)𝑏₅𝑝(𝑡)−𝑓₅𝑞(𝑡)+𝜎₅𝑘(𝑡)+_𝐄𝑔₅𝑟(𝑡,𝑒)𝜋(𝑑𝑒)[𝑢(𝑡)]−1−𝑢2𝑙₅𝑢²(𝑡)−𝑢²−12𝜎²₅+𝑔²₅[𝑢(𝑡)−𝑢]²𝑃₁(𝑡)≥0,∀𝑢∈𝐔,𝑡∈[0,𝑇]. (4.17) This implies that𝑏₅𝑝(𝑡)−𝑓₅𝑞(𝑡)+𝜎₅𝑘(𝑡)+_𝐄𝑔₅𝑟()𝜋()≥1𝑡,𝑒𝑑𝑒2𝑙₅[𝑢(𝑡)]+1+𝑢2𝜎²₅+𝑔²₅[𝑢(𝑡)−𝑢]𝑃₁(𝑡),∀𝑢∈𝐔,𝑢>𝑢(𝑡)[];𝑏,𝑡∈0,𝑇₅𝑝(𝑡)−𝑓₅𝑞(𝑡)+𝜎₅𝑘(𝑡)+_𝐄𝑔₅𝑟()𝜋()≤1𝑡,𝑒𝑑𝑒2𝑙₅[𝑢(𝑡)]+1+𝑢2𝜎²₅+𝑔²₅[𝑢(𝑡)−𝑢]𝑃₁(𝑡),∀𝑢∈𝐔,𝑢≤𝑢(𝑡),𝑡∈[0,𝑇]. (4.18)

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 that𝑓₁>0,𝑏₂≤0,𝜎₃≤0,𝑔₄𝑏≤0,₁=𝑓₂,𝜎₁=𝑓₃,𝑔₁=𝑓₄,𝜎₂=−𝑏₃,𝑔₂=−𝑏₄,𝑔₃=−𝜎₄. (4.19) It is very easy to check that the assumptions (H4.1) and (H4.2) hold. Now the first-order adjoint equation (4.4) reads−𝑑𝑝(𝑡)=𝑏₁𝑝(𝑡)−𝑓₁𝑞(𝑡)+𝜎₁𝑘(𝑡)+_𝐄𝑔₁𝑟(𝑡,𝑒)𝜋(𝑑𝑒)−𝑙₁𝑥(𝑡(𝑡)(𝑡)−)𝑑𝑡−𝑘𝑑𝑊_𝐄𝑟()𝑁(),(𝑡)=𝑡,𝑒𝑑𝑒𝑑𝑡𝑑𝑞−𝑏₂𝑝(𝑡)+𝑏₁𝑞(𝑡)−𝜎₂𝑘(𝑡)−_𝐄𝑔₂𝑟(𝑡,𝑒)𝜋(𝑑𝑒)+𝑙₂𝑦(𝑡+𝜎)𝑑𝑡₂𝑝(𝑡)+𝜎₁𝑞(𝑡)−𝜎₃𝑘(𝑡)+_𝐄𝜎₄𝑟(𝑡,𝑒)𝜋(𝑑𝑒)+𝑙₃𝑧(𝑡(𝑡)+)𝑑𝑊_𝐄𝑔₂𝑝(𝑡−)+𝑔₁𝑞(𝑡−)−𝜎₄𝑘(𝑡)−𝑔₄𝑟(𝑡,𝑒)+𝑙₄𝑐(𝑁(),𝑝(𝑇)𝑡,𝑒)𝑑𝑒𝑑𝑡=−𝜙₁𝑥(𝑇)−ℎ₁𝑞(𝑇),𝑞(0)=𝛾₁𝑦₀, (4.20) and the second-order adjoint ODEs (4.12) reduce to𝑃̇₁(𝑡)=−2𝑏₁+𝜎²₁+𝑔²₁𝑃₁(𝑡)+2𝑏₁𝑃₂(𝑡)+𝑙₁,𝑃₁(𝑇)=−𝜙₁,𝑃̇₂(𝑡)=−𝑏₂+𝜎₁𝜎₂−𝑔₁𝑔₂𝑃₁(𝑡)−𝑓₁𝑃₃(𝑡),𝑃₂(𝑇)𝑃̇=0,₃(𝑡)=−𝜎²₂+𝑔²₂𝑃₁(𝑡)−𝜎²₂𝑃₂(𝑡)+2𝑏₁𝑃₃(𝑡)−𝑙₂,𝑃₃(𝑇)=0. (4.21)

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 to𝑃̇₁(𝑡)=−𝜎²₁+𝑔²₁𝑃₁(𝑡)+𝑙₁,𝑃₁(𝑇)=−𝜙₁,𝑃̇₂(𝑡)=−𝑏₂−𝑔₁𝑔₂𝑃₁(𝑡)−𝑓₁𝑃₃(𝑡),𝑃₂(𝑇)𝑃̇=0,₃(𝑡)=−𝑔²₂𝑃₁(𝑡)−𝑙₂,𝑃₃(𝑇)=0. (4.22)(4.23)(4.24) We can first solve 𝑃₁(⋅) explicitly from (4.22):𝑃₁(𝑡)=−𝑒^{(𝜎21+𝑔21)(𝑇−𝑡)𝑇}_𝑡𝑙₁𝑒^{−(𝜎21+𝑔21)(𝑇−𝑠)}𝑑𝑠+𝜙₁, (4.25) 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.

Acknowledgments

This work was supported by China Postdoctoral Science Foundation Funded Project (no. 20100481278), Postdoctoral Innovation Foundation Funded Project of Shandong Province (no. 201002026), National Natural Sciences Foundations of China (no. 11126209) and Shandong Province (no. ZR2011AQ012), and Independent Innovation Foundation of Shandong University (IIFSDU, no. 2010TS060).