---
abstract: 'This paper investigates the near-optimal control for a kind of linear stochastic control systems governed by the forward-backward stochastic differential equations, where both the drift and diffusion terms are allowed to depend on controls and the control domain is non-convex. In the previous works of the second and third authors (see \[*Automatica* **46** (2010) 397-404\]), some open problem of near optimal control with the control-dependent diffusion is addressed and our current paper can be viewed as some direct response to it. The necessary and sufficient conditions of the near-optimality are established within the framework of optimality variational principle developed by Yong \[SIAM J. Control Optim. **48** (2010) 4119–4156\] and obtained by the convergence technique to treat the optimal control of FBSDEs in unbounded control domains by Wu \[*Automatica* **49** (2013) 1473–1480\]. Some new estimates are given here to handle the near optimality. In addition, two illustrating examples are discussed as well.'
author:
- |
    Liangquan Zhang$^{1,2}$[^1], Jianhui Huang$^{3}$[^2], Xun Li$^{3}$ [^3]\
    <span>1. School of mathematics, Shandong University, China.</span>\
    <span>2. Laboratoire de Math$\acute{\text{e}}$matiques</span>\
    <span>  Universit$\acute{\text{e}}$ de Bretagne Occidentale, 29285 Brest C$\acute{\text{e}}$dex, France.</span>\
    <span>3. Department of Applied Mathematics</span>\
    <span>  The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China.</span>
title: 'Stochastic Maximum Principle for Near-Optimal Control of FBSDEs'
---

**AMS subject classifications:** 93E20, 49L20.

**Key words:** Near-optimal control, Forward-Backward stochastic differential equations, Adjoint equations, Ekeland's principle.

Introduction {#sect:1}
============

Due to the nature of uncertainty, solutions to a forward stochastic system governed by Itô-based *stochastic differential equations* (SDE, for short) need to be non-anticipative. The equation for a conventional Itô SDE can be naturally solved in a forward-looking way by starting with the initial state. In some financial engineering problems, however, it is inherent that some terminal states are specified and one must consider a stochastic dynamic system in a backward fashion. For example, one needs to determine the option price with a given terminal payoff (as a random variable on the underlying asset). This results in a *backward stochastic differential equation* (BSDE, for short) with a terminal condition. This theory can be traced back to Bismut @B who studied linear BSDEs motivated by stochastic control problems, and Pardoux and Peng @PP who proved the well-posedness for nonlinear BSDEs. Since then, BSDEs have been extensively studied and used in the areas of applied probability and optimal stochastic controls, particularly in financial engineering. Moreover, initiated by Antonelli @A, *forward-backward stochastic differential equations* (FBSDEs, for short) have also been investigated systematically. For instance, Ma, Protter and Yong @MPY established the four-step-scheme, Hu and Peng @HP and Yong @Y1997 developed the method of continuation, Huang, Li and Wang <span>HLW,HLW1</span> analyzed partial information control problems using FBSDEs, Lim and Zhou @LZ formulated and solved backward linear-quadratic controls, and Yong @Y [@Y2010] further considered coupled FBSDEs with mixed initial-terminal conditions.

Near-optimization has been investigated by many literatures for both theory and applications. On the one hand, near-optimal controls are more available than optimal ones. Indeed, optimal controls may not exist in lots of situations, while near-optimal controls always exist, and it is much easier to derive the near-optimal controls than optimal ones, both analytically and numerically. On the other hand, since the there are many candidates for near-optimal controls, it is possible to select among them appropriates ones that are easier for analysis and implementation (see @SZ [@ZS] reference therein).

As a matter of fact, the near optimal control for the forward deterministic and stochastic systems have been extensively studied. We refer the reader to the monographs @Y [@Z1995; @Z1996; @Z1998; @ZS] for deterministic and stochastic cases. In @HLW, Huang, Li and Wang investigated the near optimal control of linear FBSDEs. Based on Ekeland's principle and spike variation, a necessary and sufficient condition of near-optimality for the near optimal control are established. However, in their work, the diffusion coefficient is independent of the control variable. The similar hypothesis was put in the work of Bahlali, Khelfallah and Mezerdi @BKB. Besides, Hui, Huang, Li and Wang @HHLW also considered the near optimal control for general form of FBSDEs, with the assumption that the control domain is convex. It is remarkable that some open problem to near optimal control is addressed in previous work of second and third authors (see @HLW) in which the diffusion terms depends on the control. The difficulties to this problem when the controlled system is FBSDE is also discussed. Our paper aims to fulfill this research gap by removing this assumptions, that is, *the diffusion coefficient is independent of the control variable and control domain is convex*. Our methods are mainly based on the Ekeland's principle, spike variation and reduction technique developed recently by Yong @Y and the methodology recently introduced by Wu @W to consider the optimal control problem for FBSDEs in the general case of control domains including Lagrange multipliers.

Let us make it more precise. First of all, we introduce the controlled initial value problem for a system of SDEs, where the pair $\left( x\left(
\cdot \right) ,y\left( \cdot \right) \right) $ is regarded as the state process and $\left( z\left( \cdot \right) ,u\left( \cdot \right) \right) $ is regarded as the control process in bounded control domains. At the same time, we regard the original terminal condition $y\left( T\right) =Mx\left(
T\right) $ as the terminal state constraint. Next it is possible to translate the near optimal control Problem $(\tilde{C}^{\varepsilon })$ into a high-dimensional reduced near optimal control problem driven by the standard SDEs with state constraint (for more information see Problem $(C^{\varepsilon })$ in Section \[sect:3\]). We mention that the advantage of this reduced near optimal control problem is that one needs not much regularity/integrability of processes $z\left( \cdot \right) $ since it is treated as a control process. Hence, it is possible to apply the Ekeland's variational principle to handle this high-dimensional reduced near optimal control problem with state constraint. Afterwards, the necessary conditions for the near optimal control of Problem $(C)$ are derived by Problem $(\tilde{C}^{\varepsilon })$. Finally, by convergence technique we obtain the general case of control domains and complete our proof.

The paper is organized as follows. The notations and preliminaries and some basic definitions are given in Section \[sect:2\]. In Section \[sect:3\], under some suitable assumptions, we state the main result of this paper, together some discussions on special cases. The application of our theoretical results will be show in the Section \[sect:4\]. Some conclusion is given in Section \[sect:5\]. Finally, we present some technique proofs in Appendix.

Notations and preliminaries {#sect:2}
===========================

Throughout this paper, we denoted by $\mathbb{R}$ the space of one-dimensional Euclidean space, by $\mathbb{R}^{n\times d}$ the space the matrices with order $n\times d$, by $\mathbf{S}^{n}$ the space of symmetric matrices with order $n\times n$. $\left\langle \cdot ,\cdot \right\rangle $ and $\left\vert \cdot \right\vert $ denote the scalar product and norm in the Euclidean space, respectively. $\top $ is the transpose of a matrix.

Let $\mathbb{U}$ be a given closed set in some Euclidean space $\mathbb{R}$. Let $(\Omega ,\mathcal{F},\{\mathcal{F}_{t}\}_{t\geq 0},P)$ be a complete filtered probability space on which a one-dimensional standard Brownian motion $W(\cdot )$ is defined, with $\{\mathcal{F}_{t}\}_{t\geq 0}$ being its natural filtration, augmented by all the $P$-null sets. We now introduce the following spaces of process: $$\begin{aligned}
\mathcal{S}^{2}(0,1;\mathbb{R})\triangleq & \left\{ \mathbb{R}\text{-valued }\mathcal{F}_{t}\text{-adapted process }\phi (t)\text{; }\mathbb{E}\left[
\sup\limits_{0\leq t\leq 1}\left\vert \phi (t)\right\vert ^{2}\right]
<\infty \right\} , \\
\mathcal{M}^{2}(0,1;\mathbb{R})\triangleq & \left\{ \mathbb{R}\text{-valued }\mathcal{F}_{t}\text{-adapted process }\varphi (t)\text{; }\mathbb{E}\left[
\int_{0}^{1}\left\vert \varphi (t)\right\vert ^{2}\mbox{\rm d}t\right]
<\infty \right\} ,\end{aligned}$$and denote $\mathcal{N}^{2}\left[ 0,1\right] =\mathcal{S}^{2}(0,1;\mathbb{R})\times \mathcal{S}^{2}(0,1;\mathbb{R})\times \mathcal{M}^{2}(0,1;\mathbb{R}).$ Clearly, $\mathcal{N}^{2}\left[ 0,1\right] $ forms a Banach space. Any process in $\mathcal{N}^{2}\left[ 0,1\right] $ is denoted by $\Theta (\cdot
)=(x(\cdot ),y(\cdot ),z(\cdot ))$, whose norm is given by $$\left\Vert \Theta (\cdot )\right\Vert _{\mathcal{N}^{2}\left[ 0,1\right] }=\mathbb{E}\left[ \sup\limits_{t\in \left[ 0,1\right] }\left\vert
x(t)\right\vert ^{2}+\sup\limits_{t\in \left[ 0,1\right] }\left\vert
y(t)\right\vert ^{2}+\int_{0}^{1}\left\vert z(t)\right\vert ^{2}\mbox{\rm d}t\right] .$$

Formulation of the Optimal Control Problem and Basic Assumptions
----------------------------------------------------------------

We study the stochastic control systems which are described by a linear forward-backward stochastic differential equations of the type: $$\left\{ 
\begin{array}{rcl}
\mbox{\rm d}x(t) & \3n=\3n & \left[ A(t)x(t)+B(t)u(t)\right] \mbox{\rm d}t+\left[ C(t)x(t)+D(t)u(t)\right] \mbox{\rm d}W(t), \\[2mm] 
\mbox{\rm d}y(t) & \3n=\3n & -\left[ a(t)x(t)+b(t)y(t)+c(t)u(t)\right] \mbox{\rm d}t+z(t)\mbox{\rm d}W(t), \\[2mm] 
x(0) & \3n=\3n & x_{0},\qquad y(1)=Mx(1),\end{array}\right.  \label{2.1}$$where $A(\cdot ),B(\cdot ),C(\cdot ),D(\cdot ),a(\cdot ),b(\cdot )$ and $c(\cdot )$ are bounded deterministic functions with values in $\mathbb{R}$, $M$ is a constant, and $u(\cdot )$ is a control process.

Let $\mathbb{U}\subseteq \mathbb{R}$ be a compact set. The control process $u(\cdot ):\left[ 0,1\right] \times \Omega \rightarrow \mathbb{U}$ is called admissible, if it is an $\mathcal{F}_{t}$-adapted process with values in $\mathbb{U}$. The set of all admissible controls is denoted by $\mathcal{U}_{ad}[0,1]$.

Under the above assumptions, for any $u( \cdot ) \in \mathcal{U} _{ad}\left[
0,1\right]$, it is easy to check that FBSDEs (\[2.1\]) admits a unique $\mathcal{F}_{t}$-adapted solution denoted by the triple $( x( \cdot ) ,y(
\cdot ) ,z( \cdot ) ) \in \mathcal{S}^{2}( 0,1;\mathbb{R}) \times \mathcal{S}^{2}( 0,1;\mathbb{R}) \times \mathcal{M}^{2}( 0,1;\mathbb{R} )$.

The cost functional is given by $$J(u(\cdot ))=\mathbb{E}\left[ \int_{0}^{1}l(t,x(t),y(t),u(t))\mbox{\rm d}t+\phi (x(1))+\gamma (y(0))\right] ,  \label{2.2}$$where $$\begin{array}{rl}
\phi : & \mathbb{R}\rightarrow \mathbb{R}, \\ 
\gamma : & \mathbb{R}\rightarrow \mathbb{R}, \\ 
l: & \left[ 0,1\right] \times \mathbb{R\times R}\times \mathbb{U}\rightarrow 
\mathbb{R}.\end{array}$$ **Problem $(C)$**. Find $\bar{u}( \cdot ) \in \mathcal{U}_{ad}[0,1]$, such that $$\label{2.3}
J( \bar{u}( \cdot ) ) =\inf\limits_{u( \cdot ) \in \mathcal{U}_{ad}\left[ 0,1\right] }J( u( \cdot ) ).$$ Any $\bar{u}( \cdot ) \in \mathcal{U}_{ad}\left[ 0,1\right] $ satisfying (\[2.3\]) is called an optimal control process of Problem $(C)$, and the corresponding state process, denoted by $\Phi ( \cdot ) =( \bar{x}( \cdot ) ,\bar{y}( \cdot ) ,\bar{z}( \cdot ) ) $, is called optimal state process. We also refer to $( \bar{x}( \cdot ) ,\bar{y}( \cdot ) ,\bar{z}( \cdot ) ,\bar{u}( \cdot ) ) $ as an optimal 4-tuple of Problem $(C)$.

However, the control problem under consideration in this paper is to find the a control in $\mathcal{U}_{ad}\left[ 0,1\right] $, which minimizes or "nearly" minimizes $J( \bar{u}( \cdot ) ) $ over $\mathcal{U}_{ad}[0,1]$. From this point, we need the following definitions.

\[**Optimal Control**\] Any admissible control $\bar{u}( \cdot ) \in \mathcal{U}_{ad} \left[ 0,1\right] ,$ is called optimal$,$ if $\bar{u}( \cdot ) $ attains the minimum of $J( u( \cdot ) ) .$

\[$\protect\varepsilon $**-Optimal Control**\] For a given $\varepsilon >0,$ an admissible control $u^{\varepsilon }( \cdot
) $ is called $\varepsilon $-optimal if $$\left\vert J( u^{\varepsilon }( \cdot ) ) -J( \bar{u}( \cdot ) ) \right\vert
\leq \varepsilon .$$

Both a family of admissible controls $\left\{ u^{\varepsilon }(\cdot
)\right\} $ parameterized by $\varepsilon >0$ and any element $u^{\varepsilon }(\cdot ),$ in the family, are called near-optimal if $$\left\vert J(u^{\varepsilon }(\cdot ))-J(\bar{u}(\cdot ))\right\vert \leq
r(\varepsilon )$$holds for sufficient small $\varepsilon ,$ where $r$ is a function of $\varepsilon $ satisfying $r(\varepsilon )\rightarrow 0$ as $\varepsilon
\rightarrow 0.$ The estimate $r(\varepsilon )$ is called an error bound. If $r(\varepsilon )=C\varepsilon ^{\delta }$ for some $\delta >0$ independent of the constant $C$, then $u^{\varepsilon }(\cdot )$ is called near-optimal with order $\varepsilon ^{\delta }$.

**Problem $(C^{\varepsilon })$**. Find $\bar{u}^{\varepsilon
}(\cdot )\in \mathcal{U}_{ad}[0,1]$, such that $$J(\bar{u}(\cdot ))=\inf\limits_{u(\cdot )\in \mathcal{U}_{ad}\left[ 0,1\right] }J(u(\cdot ))+\varepsilon .  \label{2.31}$$

Any $\bar{u}^{\varepsilon }(\cdot )\in \mathcal{U}_{ad}\left[ 0,1\right] $ satisfying (\[2.31\]) is called a near optimal control process of Problem $(C)$, and the corresponding state process, denoted by $\Phi
^{\varepsilon }(\cdot )=(\bar{x}^{\varepsilon }(\cdot ),\bar{y}^{\varepsilon
}(\cdot ),\bar{z}^{\varepsilon }(\cdot ))$, is called optimal state process. We also refer to $(\bar{x}^{\varepsilon }(\cdot ),\bar{y}^{\varepsilon
}(\cdot ),\bar{z}^{\varepsilon }(\cdot ),\bar{u}^{\varepsilon }(\cdot ))$ as an optimal 4-tuple of Problem $(C)$.

For any $u(\cdot )\in \mathcal{U}_{ad}\left[ 0,1\right] $ and $x_{0}\in \mathbb{R},$ FBSDEs (\[2.1\]) admit a unique adapted solution denoted by $$\Theta (\cdot )=(x(\cdot ),y(\cdot ),z(\cdot ))\in \mathcal{S}^{2}(0,1;\mathbb{\ R})\times \mathcal{S}^{2}(0,1;\mathbb{R})\times \mathcal{\ M}^{2}(0,1;\mathbb{R}).$$Moreover, the following estimate holds:$$\begin{aligned}
\left\Vert \Theta (\cdot )\right\Vert _{\mathcal{N}^{2}\left[ 0,1\right] }=&
\;\mathbb{E}\left[ \sup\limits_{t\in \left[ 0,1\right] }\left\vert
x(t)\right\vert ^{2}+\sup\limits_{t\in \lbrack 0,1]}\left\vert
y(t)\right\vert ^{2}+\int_{0}^{1}\left\vert z(t)\right\vert ^{2}\mbox{\rm d}t\right]  \notag  \label{2.4} \\
\leq & C\Bigg[\left\vert x_{0}\right\vert ^{2}+M^{2}+(\int_{0}^{1}\left\vert
B\left( t\right) u(t)\right\vert \mbox{\rm d}t)^{2}  \notag \\
& +\int_{0}^{1}\left\vert D(t)u(t)\right\vert ^{2}\text{d}t+(\int_{0}^{1}\left\vert c(t)u(t)\right\vert \mbox{\rm d}t)^{2}\Bigg].\end{aligned}$$Hereafter, $C>0$ stands for a generic constant which can be different at different places. Further, if $\tilde{\Theta}(\cdot )=(\tilde{x}(\cdot ),\tilde{y}(\cdot ),\tilde{z}(\cdot ))\in \mathcal{S}^{2}(0,1;\mathbb{R})\times \mathcal{S}^{2}(0,1;\mathbb{R})\times \mathcal{M}^{2}(0,1;\mathbb{R}) $ is the unique adapted solution of (2.1) with $u(\cdot )$ replaced by $\tilde{u}(\cdot )\in \mathcal{U}_{ad}\left[ 0,1\right] ,$ respectively, then the following estimate holds: $$\begin{aligned}
\left\Vert \Theta (\cdot )-\tilde{\Theta}(\cdot )\right\Vert _{\mathcal{N}^{2}\left[ 0,1\right] }=& \;\mathbb{E}\left[ \sup\limits_{t\in \left[ 0,1\right] }\left\vert x(t)-\tilde{x}(t)\right\vert ^{2}+\sup\limits_{t\in \left[ 0,1\right] }\left\vert y(t)-\tilde{y}(t)\right\vert
^{2}+\int_{0}^{1}\left\vert z(t)-\tilde{z}(t)\right\vert ^{2}\mbox{\rm d}t\right]  \notag  \label{2.5} \\
\leq & C\mathbb{E}\Bigg[\left\vert M(x(1)-\tilde{x}(1))\right\vert
^{2}+(\int_{0}^{1}\left\vert A(t)(x(t)-\tilde{x}(t))+B(t)(u(t)-\tilde{u}(t))\right\vert \mbox{\rm d}t)^{2}  \notag \\
& +(\int_{0}^{1}\left\vert a(t)(x(t)-\tilde{x}(t))+b(t)(y(t)-\tilde{y}(t))+c(t)(u(t)-\tilde{u}(t))\right\vert \mbox{\rm d}t)^{2}  \notag \\
& +\int_{0}^{1}\left\vert C(t)(x(t)-\tilde{x}(t))+D(t)(u(t)-\tilde{u}(t))\right\vert ^{2}\mbox{\rm d}t\Bigg].\end{aligned}$$The proof is standard, therefore we omit it.

Main Result {#sect:3}
===========

Necessary Condition of Near-Optimality
--------------------------------------

In this section, we first present our necessary conditions for the near-optimal control of Problem $(C)$ under some suitable assumptions. Then, some interesting special cases will be discussed.

Due to the assumptions introduced in Section \[sect:2\]. For any $(\Theta
,u)\in \mathbb{R\times R\times R}\times \mathbb{U},$ there exists a constant $L>0$ such that$$\begin{aligned}
\left\vert A(t)(x-x^{\prime })\right\vert ^{2}+\left\vert C(t)(x-x^{\prime
})\right\vert ^{2}+\left\vert a(t)(x-x^{\prime })+b(t)(y-y^{\prime
})\right\vert ^{2}& \leq L(\left\vert x-x^{\prime }\right\vert
^{2}+\left\vert y-y^{\prime }\right\vert ^{2}), \\
\forall (x,y),(x^{\prime },y^{\prime })& \in \mathbb{R}\times \mathbb{R},\end{aligned}$$and $$\begin{aligned}
\left\vert A(t)x+B(t)u\right\vert ^{2}+\left\vert C(t)x+D(t)u\right\vert
^{2}+\left\vert a(t)x+b(t)y+c(t)u\right\vert ^{2}& \leq L(1+\left\vert
x\right\vert ^{2}+\left\vert y\right\vert ^{2}), \\
\forall (t,z,u)& \in \left[ 0,1\right] \times \mathbb{R}\times \mathbb{U},~(x,y)\in \mathbb{R}\times \mathbb{R}.\end{aligned}$$

To establish the necessary condition, we need the following assumptions

1.  The maps $\phi ,$ $\gamma $ are twice continuously differentiable with respect to $\left( x,y\right) $. $l_{x},$ $l_{y},$ $\phi
    _{x}$ and $\gamma _{y}$ grow linearly about $\left( x,y,u\right) $ and is continuous in $\left( t,u\right) .$ Moreover, $l_{xx},$ $l_{yy},$ $l_{xy},$ $\phi _{xx}$ and $\gamma _{yy}$ are bounded.

2.  For any $0\leq t\leq 1,$ $\left( x,y,v\right) \in 
    \mathbb{R\times R\times U}$ and $\left( x^{\prime },y^{\prime }\right) \in 
    \mathbb{R\times R},$ there is a positive constant $C$ such that$$\begin{aligned}
    \left\vert \phi _{x}(x)-\phi _{x}(x^{\prime })\right\vert &\leq &C\left\vert
    x-x^{\prime }\right\vert , \\
    \left\vert \gamma _{y}(y)-\gamma _{y}(y^{\prime })\right\vert &\leq
    &C\left\vert y-y^{\prime }\right\vert ,\end{aligned}$$and$$\left\vert l_{x}\left( t,x,y,v\right) -l_{x}\left( t,x^{\prime },y^{\prime
    },v\right) \right\vert +\left\vert l_{y}\left( t,x,y,v\right) -l_{y}\left(
    t,x^{\prime },y^{\prime },v\right) \right\vert \leq C\left( \left\vert
    x-x^{\prime }\right\vert +\left\vert y-y^{\prime }\right\vert \right) .$$

Now, let $(x^{\varepsilon }(\cdot ),y^{\varepsilon }(\cdot
),z^{\varepsilon }(\cdot ),u^{\varepsilon }(\cdot ))$ be a near optimal 4-tuple of Problem $(C^{\varepsilon })$. We introduce $$\begin{aligned}
\mathcal{B}_{X}(t,\cdot )& \triangleq \left( 
\begin{array}{cc}
A(t) & 0 \\ 
-a(t) & -b(t)\end{array}\right) , \\
\Sigma _{X}(t,\cdot )& \triangleq \left( 
\begin{array}{cc}
C(t) & 0 \\ 
0 & 0\end{array}\right) .\end{aligned}$$Our first main result of this paper is following:

\[thm:4\] Suppose (H1)-(H2) hold. Then, for any $\beta \in \lbrack 0,\frac{1}{3})$, there exist a constant $C_{1}=C_{1}(\beta )$ such that for any fixed $\varepsilon >0$ and any $\varepsilon $-optimal $(x^{\varepsilon
}(\cdot ),y^{\varepsilon }(\cdot ),z^{\varepsilon }(\cdot ),u^{\varepsilon
}(\cdot ))$ of the problem $(C)$, there exist two parameters $\theta
_{0}^{\varepsilon }$ and $\theta _{1}^{\varepsilon }$ ($\mathcal{F}_{1}$-measurable random variable) with $\left\vert \theta _{0}^{\varepsilon
}\right\vert ^{2}+\mathbb{E}\left\vert \theta _{1}^{\varepsilon }\right\vert
^{2}=1$, $\theta _{0}^{\varepsilon }\geq 0$ holds that $$\begin{aligned}
& \int_{0}^{1}\bigg[\left\langle p^{\varepsilon }(t),B(t)(u-u^{\varepsilon
}(t))\right\rangle +\left\langle k^{\varepsilon }(t),D(t)(u-u^{\varepsilon
}(t))\right\rangle -\left\langle q^{\varepsilon }(t),c(t)(u-u^{\varepsilon
}(t))\right\rangle  \label{3.3} \\
& +\theta _{0}^{\varepsilon }\left[ l(t,x^{\varepsilon }(t),y^{\varepsilon
}(t),u)-l(t,x^{\varepsilon }(t),y^{\varepsilon }(t),u^{\varepsilon }(t))\right] +\frac{1}{2}D^{\varepsilon }(t)(u-u^{\varepsilon }(t))^{2}P_{1}(t)\bigg]\mbox{\rm d}t\geq -C_{1}\theta _{0}^{\varepsilon }\varepsilon ^{\beta
},  \notag\end{aligned}$$where $$\left\{ 
\begin{array}{rcl}
-\mbox{\rm d}p^{\varepsilon }(t) & \3n=\3n & \left[ A(t)p^{\varepsilon
}(t)-a(t)q^{\varepsilon }(t)+C(t)k^{\varepsilon }(t)+\theta
_{0}^{\varepsilon }l_{x}^{\varepsilon }(t,\cdot )\right] \text{d}t-k^{\varepsilon }(t)\mbox{\rm d}W(t), \\[2mm] 
\text{ \ }\mbox{\rm d}q^{\varepsilon }(t) & \3n=\3n & \left[
-b(t)q^{\varepsilon }(t)-\theta _{0}^{\varepsilon }l_{y}^{\varepsilon
}(t,\cdot )\right] \mbox{\rm d}t, \\[2mm] 
\text{ \ }p^{\varepsilon }(1) & \3n=\3n & \theta _{0}^{\varepsilon }\phi
_{x}(x^{\varepsilon }(1))-M\theta _{1}^{\varepsilon },\qquad q^{\varepsilon
}(0)=-\theta _{0}^{\varepsilon }\mathbb{\gamma }_{y}(y^{\varepsilon }(0)),\end{array}\right.  \label{3.4}$$and $$\left\{ 
\begin{array}{rcl}
-\mbox{\rm d}P^{\varepsilon }(t) & \3n=\3n & \big[\mathcal{B}_{X}(t,\cdot
)^{\top }P^{\varepsilon }(t)+P^{\varepsilon }(t)\mathcal{B}_{X}(t,\cdot
)+\Sigma _{X}(t,\cdot )^{\top }P^{\varepsilon }(t)\Sigma _{X}(t,\cdot ) \\[2mm] 
&  & +\Sigma _{X}(t,\cdot )^{\top }Q^{\varepsilon }(t)+Q^{\varepsilon
}(t)\Sigma _{X}(t,\cdot )+H_{XX}^{\varepsilon }(t,\cdot )\big]\mbox{\rm d}t-Q^{\varepsilon }(t)\mbox{\rm d}W(t), \\[2mm] 
\text{ \ }P^{\varepsilon }(1) & \3n=\3n & \left( 
\begin{array}{cc}
\theta _{0}^{\varepsilon }\phi _{xx}(x^{\varepsilon }(1)) & 0 \\ 
0 & 0\end{array}\right) ,\end{array}\right.  \label{3.5}$$where$$\left\{ 
\begin{array}{lll}
\text{ \ \ }l_{x}^{\varepsilon }(t,\cdot ) & \3n=\3n & l_{x}\left(
t,x^{\varepsilon }(t),y^{\varepsilon }(t),u^{\varepsilon }(t)\right) , \\ 
H_{XX}^{\varepsilon }(t,\cdot ) & \3n=\3n & H_{XX}(t,x^{\varepsilon
}(t),y^{\varepsilon }(t),u^{\varepsilon }(t),p^{\varepsilon
}(t),q^{\varepsilon }(t),k^{\varepsilon }(t),\theta _{0}^{\varepsilon }),\end{array}\right.$$and the Hamiltonian function $H:\left[ 0,T\right] \times \mathbb{R}\times 
\mathbb{R}\times \mathbb{U}\times \mathbb{R}\times \mathbb{R}\times \mathbb{R\times R\rightarrow R}$ is defined as follows:$$\begin{aligned}
H(t,x,y,u,p,q,k,\theta )\triangleq & \left\langle p,A(t)x+B(t)u\right\rangle
-\left\langle q,a(t)x+b(t)y+c(t)u\right\rangle \\
& +\left\langle k,C(t)x+D(t)u\right\rangle +\theta l(t,x,y,u).\end{aligned}$$

The proof can be seen in Appendix. Some remarks are in order.

*Actually, the second order adjoint equations (\[3.5\]) can be rewritten as the following three BSDEs if we introduce that $$P^{\varepsilon }\left( \cdot \right) \triangleq \left( 
\begin{array}{cc}
P_{1}^{\varepsilon }(\cdot ) & P_{2}^{\varepsilon }(\cdot ) \\ 
P_{2}^{\varepsilon }(\cdot ) & P_{3}^{\varepsilon }(\cdot )\end{array}\right) ,\text{ }Q^{\varepsilon }(\cdot )\triangleq \left( 
\begin{array}{cc}
Q_{1}^{\varepsilon }(\cdot ) & Q_{2}^{\varepsilon }(\cdot ) \\ 
Q_{2}^{\varepsilon }(\cdot ) & Q_{3}^{\varepsilon }(\cdot )\end{array}\right) ,$$then we have $$\left\{ \3n\begin{array}{rcl}
-\mbox{\rm d}P_{1}^{\varepsilon }(t) & \3n=\3n & \left[ 2A(t)P_{1}^{\varepsilon }(t)+C(t)^{2}P_{1}^{\varepsilon }(t)+2C(t)Q_{1}^{\varepsilon
}(t)-2a(t)P_{2}^{\varepsilon }(t)-\theta _{0}^{\varepsilon
}l_{xx}^{\varepsilon }(t,\cdot )\right] \mbox{\rm d}t-Q_{1}^{\varepsilon }(t)\mbox{\rm d}W(t), \\[2mm] 
P_{1}^{\varepsilon }(1) & \3n=\3n & \theta _{0}^{\varepsilon }\phi
_{xx}(x^{\varepsilon }(1)),\end{array}\right.$$$$\left\{ \3n\begin{array}{rcl}
-\mbox{\rm d}P_{2}^{\varepsilon }(t) & \3n=\3n & \left[ A(t)P_{2}^{\varepsilon }(t)+C(t)Q_{2}^{\varepsilon }(t)-a(t)P_{3}^{\varepsilon
}(t)-P_{2}^{\varepsilon }(t)b(t)-\theta _{0}^{\varepsilon
}l_{xy}^{\varepsilon }(t,\cdot )\right] \mbox{\rm d}t-Q_{2}^{\varepsilon }(t)\mbox{\rm d}W(t), \\[2mm] 
P_{2}^{\varepsilon }(1) & \3n=\3n & 0,\end{array}\right.$$and $$\left\{ \3n\begin{array}{rcl}
-\mbox{\rm d}P_{3}^{\varepsilon }(t) & \3n=\3n & \left[ -2b(t)P_{3}^{\varepsilon }(t)-\theta _{0}^{\varepsilon }l_{yy}^{\varepsilon }(t,\cdot )\right] \mbox{\rm
d}t-Q_{3}^{\varepsilon }(t)\mbox{\rm d}W(t), \\[2mm] 
P_{3}^{\varepsilon }(1) & \3n=\3n & 0.\end{array}\right.$$*

Let us now look at some special cases. It can be seen that our theorem recovers the known results.

*If let* $\theta _{0}^{\varepsilon }=1,$ and*  $$\left\{ 
\begin{array}{ll}
M=0, & \gamma (y)=0, \\ 
l(t,x,y,u)=l(t,x,u), & a(t)=b(t)=c(t)=0,\end{array}\right.$$we get maximum principle for near optimal control problem considered in Zhou @Z1998 for the linear case. If* $\mathbb{U}$* is convex, our result essentially covers the results developed in Hui et al @HHLW, and if $\sigma $ is independent of $u(\cdot )$ and* $\mathbb{U}$* is not supposed convex, our result can be reduced to those of Huang et al in @HLW, and Khaled et al in @BKB for the linear case.*

*The necessary condition of near-optimal controls are derived in terms of the near-maximum condition in an integral form. It is well known that, for exact optimality, the integral form and the pointwise form of the maximum condition are equivalent, however it is certainly not the case for near-optimality.*

*If we set $\varepsilon =0$ in Theorem \[thm:4\], then it is just the necessary condition of exact optimality for linear SDEs and FBSDEs considered by Peng @P, Yong @Y and Wu @W, respectively.*

*It is necessary to point out that, the approach developed in this paper can also handle nonlinear form coupled FBSDEs for the near optimal control problems. This issue will be carried out as our future publications.*

Sufficient Condition of Near-Optimality
---------------------------------------

Theorem \[thm:4\]* *gives some necessary conditions for the near optimal control. In this subsection, we establish a sufficient condition for near optimality. To this end, let us first consider the following FBSDEs$$\left\{ 
\begin{array}{rcl}
-\text{d}p(t) & \3n=\3n & \left[ A(t)p(t)-a(t)q(t)+C(t)k(t)+l_{x}(t,\cdot )\right] \text{d}t-k(t)\mbox{\rm d}W(t), \\[2mm] 
\text{ \ d}q(t) & \3n=\3n & \left[ -b(t)q(t)-l_{y}(t,\cdot )\right] \mbox{\rm d}t, \\[2mm] 
\text{ \ }p(1) & \3n=\3n & \phi _{x}(x(1))-Mq(1),\qquad q(0)=-\mathbb{\gamma 
}_{y}(y(0)),\end{array}\right.  \label{3.6}$$$$\left\{ \3n\begin{array}{rcl}
-\text{d}P_{1}(t) & \3n=\3n & \left[
2A(t)P_{1}(t)+C(t)^{2}P_{1}(t)+2C(t)Q_{1}(t)-2a(t)P_{2}(t)-l_{xx}(t,\cdot )\right] \mbox{\rm d}t-Q_{1}(t)\mbox{\rm d}W(t), \\[2mm] 
P_{1}(1) & \3n=\3n & \phi _{xx}(x(1)),\end{array}\right.  \label{3.7}$$$$\left\{ \3n\begin{array}{rcl}
-\text{d}P_{2}(t) & \3n=\3n & \left[
A(t)P_{2}(t)+C(t)Q_{2}(t)-a(t)P_{3}(t)-P_{2}(t)b(t)-l_{xy}(t,\cdot )\right] \mbox{\rm d}t-Q_{2}(t)\mbox{\rm d}W(t), \\[2mm] 
P_{2}(1) & \3n=\3n & 0,\end{array}\right.  \label{3.8}$$and $$\left\{ \3n\begin{array}{rcl}
-\text{d}P_{3}(t) & \3n=\3n & \left[ -2b(t)P_{3}(t)-l_{yy}(t,\cdot )\right] 
\mbox{\rm
d}t-Q_{3}(t)\mbox{\rm d}W(t), \\[2mm] 
P_{3}(1) & \3n=\3n & 0.\end{array}\right.  \label{3.9}$$for some admissible triple $(x(\cdot ),y(\cdot ),u(\cdot ))$. Now define the usual Hamiltonian $$\begin{aligned}
H^{\prime }(t,x,y,u,p,q,k)\triangleq & -p\left[ A(t)x+B(t)\right] u+q\left[
a(t)x+b(t)y+c(t)u\right] \\
& -k\left[ C(t)x+D(t)u\right] -l(t,x,y,u),\end{aligned}$$and $$\begin{aligned}
\mathcal{H}^{(x(\cdot ),y(\cdot ),u(\cdot ))}(t,x,y,u)\triangleq &
\;H^{\prime }(t,x,y,u,p(t),q(t),k(t)-P_{1}(t)(C(t)x(t)+D(t)u(t))) \\
& -\frac{1}{2}(C(t)x(t)+D(t)u(t))^{2}P_{1}(t).\end{aligned}$$We need the following assumptions.

1.  \(i) The control set $\mathbb{U}$ is convex; (ii) For any $0\leq t\leq 1,$ $(x,y,z)\in \mathbb{R}\times \mathbb{R}\times \mathbb{U}
    $ and $u\in \mathbb{U},$ the function $l$ is differentiable in $u$, and there is a constant $C$ such that $$\left\vert l(t,x,y,u)-l(t,x,y,u^{\prime })\right\vert +\left\vert
    l_{u}(t,x,y,u)-l_{u}(t,x,y,u^{\prime })\right\vert \leq C\left\vert
    u-u^{\prime }\right\vert .$$

\[thm:10\] Suppose that (H1)-(H3) hold. Let $(x^{\varepsilon }(\cdot
),y^{\varepsilon }(\cdot ),u^{\varepsilon }(\cdot ))$ be an admissible triple, and $(p^{\varepsilon }(\cdot ),q^{\varepsilon }(\cdot
),k^{\varepsilon }(\cdot ))$ be the solutions to (\[2.1\]) and (\[3.6\]), respectively corresponding to $(x^{\varepsilon }(\cdot ),y^{\varepsilon
}(\cdot ),u^{\varepsilon }(\cdot ))$. Assume that $$H^{\prime }(t,\cdot ,\cdot ,\cdot ,p^{\varepsilon }(t),q^{\varepsilon
}(t),k^{\varepsilon }(t))$$is concave, a.e., a.s.. Moreover, both $\phi $ and $\gamma $ are convex. If for some $\varepsilon >0,$ $$\begin{aligned}
& \mathbb{E}\left[ \int_{0}^{1}\mathcal{H}^{(x^{\varepsilon }(\cdot
),y^{\varepsilon }(\cdot ),u^{\varepsilon }(\cdot ))}(t,x^{\varepsilon
}(t),y^{\varepsilon }(t),u^{\varepsilon }(t))\mbox{\rm d}t\right]  \notag \\
\geq & \sup\limits_{u(\cdot )\in \mathcal{U}_{ad}\left[ 0,1\right] }\mathbb{E}\left[ \int_{0}^{1}\mathcal{H}^{(x^{\varepsilon }(\cdot ),y^{\varepsilon
}(\cdot ),u^{\varepsilon }(\cdot ))}(t,x^{\varepsilon }(t),y^{\varepsilon
}(t),u(t))\mbox{\rm d}t\right] -\varepsilon ,  \label{3.91}\end{aligned}$$then $$J(u^{\varepsilon }(\cdot ))\leq \inf\limits_{u(\cdot )\in \mathcal{U}_{ad}\left[ 0,1\right] }J(u(\cdot ))+C\varepsilon ^{\frac{1}{2}}.  \label{3.10}$$

Let us fix an $\varepsilon >0.$ Define a new metric $\tilde{d}$ on $\mathcal{U}_{ad}\left[ 0,1\right] $ as follows: $$\tilde{d}(u(\cdot ),u^{\prime }(\cdot ))=\mathbb{E}\left[ \int_{0}^{1}v^{\varepsilon }(t)\left\vert u(t)-u^{\prime }(t)\right\vert \mbox{\rm d}t\right] ,$$where $$v^{\varepsilon }(t)=1+\left\vert p^{\varepsilon }(t)\right\vert +\left\vert
q^{\varepsilon }(t)\right\vert +\left\vert k^{\varepsilon }(t)\right\vert
+\left\vert P_{1}^{\varepsilon }(t)\right\vert +\left\vert
P_{1}^{\varepsilon }(t)\right\vert \left\vert x^{\varepsilon }(t)\right\vert
,$$It is easy to see that $\tilde{d}$ is a complete metric space as a weighted $L^{1}$ norm.

Define a functional $F(\cdot ):\mathcal{U}_{ad}\left[ 0,1\right]
\rightarrow \mathbb{R}$ $$F(u(\cdot ))=\mathbb{E}\left[ \int_{0}^{1}\mathcal{H}^{(x^{\varepsilon
}(\cdot ),y^{\varepsilon }(\cdot ),u^{\varepsilon }(\cdot
))}(t,x^{\varepsilon }(t),y^{\varepsilon }(t),u(t))\mbox{\rm d}t\right] .$$By virtue of (H2), we check that $$F(u(\cdot ))-F(u^{\prime }(\cdot ))=C\mathbb{E}\left[ \int_{0}^{1}v^{\varepsilon }(t)\left\vert u(\cdot )-u^{\prime }(\cdot )\right\vert \mbox{\rm d}t\right] .$$Hence, $F$ is continuous on $\mathcal{U}_{ad}\left[ 0,1\right] $ with respect to $\tilde{d}$. By (\[3.10\]) and Lemma \[lem:17\], there exists a $\hat{u}^{\varepsilon }(\cdot )\in \mathcal{U}_{ad}\left[ 0,1\right] $ such that $$\tilde{d}(u^{\varepsilon }(\cdot ),\hat{u}^{\varepsilon }(\cdot ))\leq
\varepsilon ^{\frac{1}{2}}$$and $$\mathbb{E}\left[ \int_{0}^{1}\mathcal{\tilde{H}}(t,x^{\varepsilon
}(t),y^{\varepsilon }(t),\hat{u}^{\varepsilon }(\cdot ))\text{d}t\right]
=\max\limits_{u(\cdot )\in \mathcal{U}_{ad}\left[ 0,1\right] }\mathbb{E}\left[ \int_{0}^{1}\mathcal{\tilde{H}}(t,x^{\varepsilon }(t),y^{\varepsilon
}(t),u(\cdot ))\mbox{\rm d}t\right] ,$$where $$\mathcal{\tilde{H}}(t,x,y,u)\triangleq \mathcal{H}^{(x^{\varepsilon }(\cdot
),y^{\varepsilon }(\cdot ),u^{\varepsilon }(\cdot ))}(t,x,y,u)-\varepsilon ^{\frac{1}{2}}v^{\varepsilon }(t)\left\vert u-\hat{u}^{\varepsilon
}(t)\right\vert ,$$which implies that $$\mathcal{\tilde{H}}(t,x^{\varepsilon }(t),y^{\varepsilon }(t),\hat{u}^{\varepsilon }(\cdot ))=\max\limits_{u\in U}\mathcal{\tilde{H}}(t,x^{\varepsilon }(t),y^{\varepsilon }(t),u).  \label{3.11}$$Applying Lemma 2.3 in Yong and Zhou @YZ to (\[3.11\]), we have $$\begin{aligned}
0\in & \;\partial _{u}\mathcal{\tilde{H}}(t,x^{\varepsilon
}(t),y^{\varepsilon }(t),\hat{u}^{\varepsilon }(t))  \notag  \label{3.12} \\
\subset & \;\partial _{u}\mathcal{\tilde{H}}(t,x^{\varepsilon
}(t),y^{\varepsilon }(t),\hat{u}^{\varepsilon }(t))+\left[ -\varepsilon ^{\frac{1}{2}}v^{\varepsilon }(t),\varepsilon ^{\frac{1}{2}}v^{\varepsilon }(t)\right]  \notag \\
=& \;\partial _{u}H^{\prime }(t,x^{\varepsilon }(t),y^{\varepsilon
}(t),u^{\varepsilon }(t),p^{\varepsilon }(t),q^{\varepsilon
}(t),k^{\varepsilon }(t))+\left[ -\varepsilon ^{\frac{1}{2}}v^{\varepsilon
}(t),\varepsilon ^{\frac{1}{2}}v^{\varepsilon }(t)\right]  \notag \\
& +(D(t))^{2}P_{1}^{\varepsilon }(t)(u^{\varepsilon }(t)-\hat{u}^{\varepsilon }(t)).\end{aligned}$$By (H2), the inclusion (\[3.12\]) implies that there is a $\lambda
^{\varepsilon }(t)\in \left[ -\varepsilon ^{\frac{1}{2}}v^{\varepsilon
}(t),\varepsilon ^{\frac{1}{2}}v^{\varepsilon }(t)\right] ,$ such that $$H_{u}^{\prime }(t,x^{\varepsilon }(t),y^{\varepsilon }(t),\hat{u}^{\varepsilon }(t),p^{\varepsilon }(t),q^{\varepsilon }(t),k^{\varepsilon
}(t))=-\lambda ^{\varepsilon }(t)-D(t)^{2}P_{1}^{\varepsilon
}(t)(u^{\varepsilon }(t)-\hat{u}^{\varepsilon }(t)).$$So by (H2)$,$ we have $$\begin{aligned}
& \;\left\vert H_{u}^{\prime }(t,x^{\varepsilon }(t),y^{\varepsilon
}(t),u^{\varepsilon }(t),p^{\varepsilon }(t),q^{\varepsilon
}(t),k^{\varepsilon }(t))\right\vert \\
\leq & \;\left\vert H_{u}^{\prime }(t,x^{\varepsilon }(t),y^{\varepsilon
}(t),u^{\varepsilon }(t),p^{\varepsilon }(t),q^{\varepsilon
}(t),k^{\varepsilon }(t))-H_{u}^{\prime }(t,x^{\varepsilon
}(t),y^{\varepsilon }(t),\hat{u}^{\varepsilon }(t),p^{\varepsilon
}(t),q^{\varepsilon }(t),k^{\varepsilon }(t))\right\vert \\
& +\left\vert H_{u}^{\prime }(t,x^{\varepsilon }(t),y^{\varepsilon }(t),\hat{u}^{\varepsilon }(t),p^{\varepsilon }(t),q^{\varepsilon }(t),k^{\varepsilon
}(t))\right\vert \\
\leq & \;Cv^{\varepsilon }(t)\left\vert u^{\varepsilon }(t)-\hat{u}^{\varepsilon }(t)\right\vert +\left\vert \lambda ^{\varepsilon
}(t)\right\vert +\left\vert (D(t))^{2}P_{1}^{\varepsilon }(t)(u^{\varepsilon
}(t)-\hat{u}^{\varepsilon }(t))\right\vert \\
\leq & \;C_{2}v^{\varepsilon }(t)\left\vert u^{\varepsilon }(t)-\hat{u}^{\varepsilon }(t)\right\vert +\varepsilon ^{\frac{1}{2}}v^{\varepsilon }(t).\end{aligned}$$By the concavity of $H^{\prime }(t,\cdot ,\cdot ,\cdot ,p^{\varepsilon
}(\cdot ),q^{\varepsilon }(\cdot ),k^{\varepsilon }(\cdot ))$, we have $$\begin{array}{rl}
& H^{\prime }(t,x(t),y(t),u(t),p^{\varepsilon }(t),q^{\varepsilon
}(t),k^{\varepsilon }(t))-H^{\prime }(t,x^{\varepsilon }(t),y^{\varepsilon
}(t),u^{\varepsilon }(t),p^{\varepsilon }(t),q^{\varepsilon
}(t),k^{\varepsilon }(t)) \\[2mm] 
\leq \3n & H_{x}^{\prime }(t,x^{\varepsilon }(t),y^{\varepsilon
}(t),u^{\varepsilon }(t),p^{\varepsilon }(t),q^{\varepsilon
}(t),k^{\varepsilon }(t))(x(t)-x^{\varepsilon }(t)) \\[2mm] 
& +H_{y}^{\prime }(t,x^{\varepsilon }(t),y^{\varepsilon }(t),u^{\varepsilon
}(t),p^{\varepsilon }(t),q^{\varepsilon }(t),k^{\varepsilon
}(t))(y(t)-y^{\varepsilon }(t)) \\[2mm] 
& +H_{u}^{\prime }(t,x^{\varepsilon }(t),y^{\varepsilon }(t),u^{\varepsilon
}(t),p^{\varepsilon }(t),q^{\varepsilon }(t),k^{\varepsilon
}(t))(u(t)-u^{\varepsilon }(t)),\end{array}
\label{3.13}$$for any admissible $(x(\cdot ),y(\cdot ),u(\cdot ))$.

Taking expectation on both side of (\[3.13\]), it yields that $$\begin{aligned}
& \;\mathbb{E}\bigg[\int_{0}^{1}H^{\prime }(t,x(t),y(t),u(t),p^{\varepsilon
}(t),q^{\varepsilon }(t),k^{\varepsilon }(t))-H^{\prime }(t,x^{\varepsilon
}(t),y^{\varepsilon }(t),u^{\varepsilon }(t),p^{\varepsilon
}(t),q^{\varepsilon }(t),k^{\varepsilon }(t))\mbox{\rm d}t\bigg] \\
\leq & \;\mathbb{E}\bigg[\int_{0}^{1}H_{x}^{\prime }(t,x^{\varepsilon
}(t),y^{\varepsilon }(t),u^{\varepsilon }(t),p^{\varepsilon
}(t),q^{\varepsilon }(t),k^{\varepsilon }(t))(x(t)-x^{\varepsilon }(t))\mbox{\rm d}t \\
& +\int_{0}^{1}H_{y}^{\prime }(t,x^{\varepsilon }(t),y^{\varepsilon
}(t),u^{\varepsilon }(t),p^{\varepsilon }(t),q^{\varepsilon
}(t),k^{\varepsilon }(t))(y(t)-y^{\varepsilon }(t))\mbox{\rm d}t\bigg] \\
& +C_{3}\left[ \varepsilon ^{\frac{1}{2}}+\tilde{d}(u^{\varepsilon }(\cdot ),\hat{u}^{\varepsilon }(\cdot ))\right] \\
\leq & \;\mathbb{E}\bigg[\int_{0}^{1}H_{x}^{\prime }(t,x^{\varepsilon
}(t),y^{\varepsilon }(t),u^{\varepsilon }(t),p^{\varepsilon
}(t),q^{\varepsilon }(t),k^{\varepsilon }(t))(x(t)-x^{\varepsilon }(t))\mbox{\rm d}t \\
& +\mathbb{E}\int_{0}^{1}H_{y}^{\prime }(t,x^{\varepsilon
}(t),y^{\varepsilon }(t),u^{\varepsilon }(t),p^{\varepsilon
}(t),q^{\varepsilon }(t),k^{\varepsilon }(t))(y(t)-y^{\varepsilon }(t))\mbox{\rm d}t\bigg]+(C_{3}+1)\varepsilon ^{\frac{1}{2}}.\end{aligned}$$On the other hand, applying Itô's formula to $p^{\varepsilon }(\cdot
)(x^{\varepsilon }(\cdot )-x(\cdot ))$ and $q^{\varepsilon }(\cdot
)(y^{\varepsilon }(\cdot )-y(\cdot ))$, we derive that $$\begin{aligned}
\mathbb{E}\left[ \phi _{x}(x^{\varepsilon }(1))(x^{\varepsilon }(1)-x(1))\right] =& \;\mathbb{E}\bigg[\int_{0}^{1}H_{x}^{\prime }(t,x^{\varepsilon
}(t),y^{\varepsilon }(t),u^{\varepsilon }(t),p^{\varepsilon
}(t),q^{\varepsilon }(t),k^{\varepsilon }(t))(x^{\varepsilon }(t)-x(t))\mbox{\rm d}t  \notag \\
& +\int_{0}^{1}\left[ A(t)(x^{\varepsilon }(t)-x(t))+B(t)(u^{\varepsilon
}(t)-u(t))\right] p^{\varepsilon }(t)\mbox{\rm d}t  \notag \\
& +\int_{0}^{1}\left[ C(t)(x^{\varepsilon }(t)-x(t))+D(t)(u^{\varepsilon
}(t)-u(t))\right] q^{\varepsilon }(t)\mbox{\rm d}t\bigg]  \notag \\
& +\mathbb{E}\left[ Mq^{\varepsilon }\left( 1\right) \left( x^{\varepsilon
}\left( 1\right) -x\left( 1\right) \right) \right]  \label{3.14}\end{aligned}$$and $$\begin{aligned}
\mathbb{E}[\gamma _{y}(y^{\varepsilon }(0))(y^{\varepsilon }(0)-y(0))]=& \;\mathbb{E}\bigg[\int_{0}^{1}H_{y}^{\prime }(t,x^{\varepsilon
}(t),y^{\varepsilon }(t),u^{\varepsilon }(t),p^{\varepsilon
}(t),q^{\varepsilon }(t),k^{\varepsilon }(t))(y^{\varepsilon }(t)-y(t))\mbox{\rm d}t  \notag \\
& +\int_{0}^{1}\left[ a(t)((x(t)-x^{\varepsilon }(t))+b\left( t\right)
(y(t)-y^{\varepsilon }(t))+c(t)(u(t)-u^{\varepsilon }(t)))\right]
q^{\varepsilon }(t)\mbox{\rm d}t  \notag \\
& +\mathbb{E}\left[ q^{\varepsilon }\left( 1\right) \left( y\left( 1\right)
-y^{\varepsilon }\left( 1\right) \right) \right] \bigg].  \label{3.15}\end{aligned}$$Noting the definition of $H^{\prime }$, we have $$\begin{aligned}
& \;\mathbb{E}\left\{ \int_{0}^{1}\left[ l(t,x^{\varepsilon
}(t),y^{\varepsilon }(t),u^{\varepsilon }(t))-l(t,x(t),y(t),u(t))\right] \mbox{\rm d}t\right\}  \notag  \label{3.16} \\
=& \;\mathbb{E}\bigg\{\int_{0}^{1}H^{\prime
}[(t,x(t),y(t),u(t),p^{\varepsilon }(t),q^{\varepsilon }(t),k^{\varepsilon
}(t))-H^{\prime }(t,x^{\varepsilon }(t),y^{\varepsilon }(t),u^{\varepsilon
}(t),p^{\varepsilon }(t),q^{\varepsilon }(t),k^{\varepsilon }(t))]\mbox{\rm
d}t\bigg\}  \notag \\
=& \;\mathbb{E}\left\{ \int_{0}^{1}\left[ A(t)(x(t)-x^{\varepsilon
}(t))+B(t)(y(t)-y^{\varepsilon }(t))\right] p^{\varepsilon }(t)\mbox{\rm d}t\right\}  \notag \\
& +\mathbb{E}\left\{ \int_{0}^{1}\left[ a(t)((x^{\varepsilon
}(t)-x(t))+b\left( t\right) (y^{\varepsilon }(t)-y(t))+c(t)(u^{\varepsilon
}(t)-u(t)))\right] q^{\varepsilon }(t)\mbox{\rm d}t\right\}  \notag \\
& +\mathbb{E}\left\{ \int_{0}^{1}\left[ C(t)(x(t)-x^{\varepsilon
}(t))+D(t)(u(t)-u^{\varepsilon }(t))\right] q^{\varepsilon }(t)\mbox{\rm d}t\right\} .\end{aligned}$$Note that $$\begin{aligned}
J(u^{\varepsilon }(\cdot ))-J(u(\cdot ))=& \;\mathbb{E}\left\{ \int_{0}^{1}\left[ l(t,x^{\varepsilon }(t),y^{\varepsilon }(t),u^{\varepsilon
}(t))-l(t,x(t),y(t),u(t))\right] \mbox{\rm d}t\right\} \\
& +\mathbb{E}\left[ \phi (x^{\varepsilon }(1)-\phi (x(1)))\right] +\mathbb{E}\left[ \gamma (y^{\varepsilon }(0)-\gamma (y(0)))\right] .\end{aligned}$$From Lemma \[lem:14\] in Section 6, (\[3.13\])-(\[3.16\]), we derive that $$\begin{aligned}
J(u^{\varepsilon }(\cdot ))-J(u(\cdot ))\leq & \;\mathbb{E}\left[
\int_{0}^{1}H_{u}^{\prime }(t,x^{\varepsilon }(t),y^{\varepsilon
}(t),u^{\varepsilon }(t),p^{\varepsilon }(t),q^{\varepsilon
}(t),k^{\varepsilon }(t))\mbox{\rm d}t\right] \\
\leq & \;C\varepsilon ^{\frac{1}{2}}.\end{aligned}$$The proof is complete.

\[col:16\] Under the same assumptions in Theorem \[thm:10\], a sufficient condition for an admissible pair $(x^{\varepsilon
}(t),y^{\varepsilon }(t),u^{\varepsilon }(t))$ to be $\varepsilon $-optimal is $$\begin{aligned}
& \;\mathbb{E}\left[ \int_{0}^{1}\mathcal{H}^{(x^{\varepsilon }(\cdot
),y^{\varepsilon }(\cdot ),u^{\varepsilon }(\cdot ))}(t,x^{\varepsilon
}(t),y^{\varepsilon }(t),u^{\varepsilon }(t))\mbox{\rm d}t\right] \\
\geq & \sup\limits_{u(\cdot )\in \mathcal{U}_{ad}\left[ 0,1\right] }\mathbb{E}\left[ \int_{0}^{1}\mathcal{H}^{(x^{\varepsilon }(\cdot ),y^{\varepsilon
}(\cdot ),u^{\varepsilon }(\cdot ))}(t,x^{\varepsilon }(t),y^{\varepsilon
}(t),u(t))\mbox{\rm d}t\right] -\Big(\frac{\varepsilon }{C_{1}}\Big)^{2},\end{aligned}$$for some positive constant $C_{1}$.

Examples {#sect:4}
========

We now illustrate the results of Section \[sect:3\] by looking an example. Observe that the FBSDEs considered in this paper are linear, it is possible to implement our principles directly.

Next, we are devote to showing the application of Theorem \[thm:4\] and Theorem \[thm:10\], by which we obtain a near-optimal control of the following example which is modified from Zhou @Z1998. For convenience, we still adopt the notations introduced in Section \[sect:3\].

\[**Necessary condition**\] \[e1\]*Let the admissible control domain $\Gamma =\left[ 0,1\right] .$ Consider the following $\varepsilon $-optimal control problem$$\min\limits_{u(\cdot )\in \mathcal{U}_{ad}\left[ 0,1\right] }J(u(\cdot )),$$where $$J(u(\cdot ))=\mathbb{E}\left[ \int_{0}^{1}u(t)\mbox{\rm d}t+\frac{\sqrt{2}}{2}x^{2}(1)+x(1)-y(0)\right] ,  \label{4.2}$$with $$\left\{ 
\begin{array}{rcl}
\mbox{\rm d}x(t) & \3n=\3n & u(t)\mbox{\rm d}W(t), \\ 
\mbox{\rm d}y(t) & \3n=\3n & -\left( 1+\sqrt{2}\right) u(t)\mbox{\rm d}t+z(t)\mbox{\rm d}W(t), \\ 
x(0) & \3n=\3n & 0,\qquad y(1)=x(1).\end{array}\right.  \label{4.3}$$Set* $\theta _{0}^{\varepsilon }=\frac{\sqrt{2}}{2}.$ *For a given admissible triple $(x^{\varepsilon }(\cdot ),y^{\varepsilon }(\cdot
),u^{\varepsilon }(\cdot )),$ the corresponding first and second adjoint equations are$$\left\{ 
\begin{array}{rcl}
\mbox{\rm d}p^{\varepsilon }(t) & \3n=\3n & k^{\varepsilon }(t)\mbox{\rm d}W(t), \\ 
\mbox{\rm d}q^{\varepsilon }(t) & \3n=\3n & 0, \\ 
p^{\varepsilon }(1) & \3n=\3n & x^{\varepsilon }(1),\qquad q^{\varepsilon
}(0)=\frac{\sqrt{2}}{2},\end{array}\right.  \label{4.4}$$and $$\left\{ 
\begin{array}{rcl}
\mbox{\rm d}P_{1}^{\varepsilon }(t) & \3n=\3n & Q_{1}^{\varepsilon }(t)\mbox{\rm d}W(t), \\ 
P_{1}^{\varepsilon }(1) & \3n=\3n & 1,\end{array}\right.  \label{4.5}$$respectively. Obviously, by the uniqueness of equations (\[4.4\]) and (<span>4.5</span>), we derive $$\left\{ 
\begin{array}{lll}
p^{\varepsilon }(t) & \3n=\3n & u^{\varepsilon }(t)W(t), \\ 
k^{\varepsilon }(t) & \3n=\3n & u^{\varepsilon }(t), \\ 
q^{\varepsilon }(t) & \3n=\3n & \frac{\sqrt{2}}{2},~t\in \left[ 0,1\right] ,\end{array}\right.$$and $$\left\{ 
\begin{array}{rcl}
P_{1}^{\varepsilon }(t) & \3n=\3n & 1, \\ 
Q_{1}^{\varepsilon }(t) & \3n=\3n & 0,~t\in \left[ 0,1\right] ,\end{array}\right.$$respectively. On the other hand, Theorem \[thm:4\] gives $$\mathbb{E}\left\{ \int_{0}^{1}\left[ \frac{1}{2}(u(t))^{2}+u(t)(k^{\varepsilon }(t)-u^{\varepsilon }(t)-1)+\frac{1}{2}(u^{\varepsilon
}(t))^{2}-k^{\varepsilon }(t)u^{\varepsilon }(t)+u^{\varepsilon }(t)\right] \mbox{\rm d}t\right\} \geq -C\varepsilon ^{\beta }.$$Hence a simple calculation shows that if $$u^{\varepsilon }(t)+1-k^{\varepsilon }(t)\in \Gamma ,  \label{4.7}$$then, we get $$\mathbb{E}\left[ \int_{0}^{1}(k^{\varepsilon }(t)-1)^{2}\mbox{\rm d}t\right]
\leq C\varepsilon ^{\beta }.  \label{4.8}$$The above condition reveals the minimumqualification for the pair $(x^{\varepsilon }(\cdot ),u^{\varepsilon }(\cdot
))$ to be $\varepsilon $-optimal. Actually, $u^{\varepsilon
}(t)=1-\varepsilon ^{\frac{1}{2}}$ is one of the candidates for $\varepsilon 
$-optimal. Indeed, if we choose $u^{\varepsilon }(t)=1-\varepsilon ^{\frac{1}{2}}$ with the corresponding state $$x^{\varepsilon }(t)=(1-\varepsilon ^{\frac{1}{2}})W(t),~t\in \left[ 0,1\right] ,$$then the solutions of first order adjoint equations are $$(p^{\varepsilon }(t),k^{\varepsilon }(t))=\Big(\Big(1-\varepsilon ^{\frac{1}{2}}\Big)W(t),1-\varepsilon ^{\frac{1}{2}}\Big).$$Obviously, (\[4.7\]) and (\[4.8\]) are fulfilled.*

\[**Sufficient condition**\] \[e2\]*Modify (\[4.2\]) and (\[4.3\]) as follows:$$J(u(\cdot ))=\mathbb{E}\left[ \int_{0}^{1}u(t)\mbox{\rm d}t+\frac{1}{2}x^{2}(1)+x(1)-y(0)\right] ,$$and* $$\left\{ 
\begin{array}{rcl}
\mbox{\rm d}x(t) & \3n=\3n & u(t)\mbox{\rm d}W(t), \\ 
\mbox{\rm d}y(t) & \3n=\3n & -2u(t)\mbox{\rm d}t+z(t)\mbox{\rm d}W(t), \\ 
x(0) & \3n=\3n & 0,\qquad y(1)=x(1).\end{array}\right.$$*For a given admissible triple* $(x^{\varepsilon }(\cdot
),y^{\varepsilon }(\cdot ),u^{\varepsilon }(\cdot )),$* the corresponding first and second adjoint equations (\[3.6\])-(\[3.9\] ) are*$$\left\{ 
\begin{array}{rcl}
\mbox{\rm d}p^{\varepsilon }(t) & \3n=\3n & k^{\varepsilon }(t)\mbox{\rm d}W(t), \\ 
\mbox{\rm d}q^{\varepsilon }(t) & \3n=\3n & 0, \\ 
p^{\varepsilon }(1) & \3n=\3n & x(1),\qquad q^{\varepsilon }(0)=1,\end{array}\right.  \label{4.9}$$*and* $$\left\{ 
\begin{array}{rcl}
\mbox{\rm d}P_{1}^{\varepsilon }(t) & \3n=\3n & Q_{1}^{\varepsilon }(t)\mbox{\rm d}W(t), \\ 
P_{1}^{\varepsilon }(1) & \3n=\3n & 1,\end{array}\right.  \label{4.10}$$*respectively. Obviously, by the uniqueness of equations (\[4.9\]) and (\[4.10\]), we derive* $$\left\{ 
\begin{array}{lll}
p^{\varepsilon }(t) & \3n=\3n & u^{\varepsilon }(t)W(t), \\ 
k^{\varepsilon }(t) & \3n=\3n & u^{\varepsilon }(t), \\ 
q^{\varepsilon }(t) & \3n=\3n & 1,~t\in \left[ 0,1\right] ,\end{array}\right.$$*and* $$\left\{ 
\begin{array}{rcl}
P_{1}^{\varepsilon }(t) & \3n=\3n & 1, \\ 
Q_{1}^{\varepsilon }(t) & \3n=\3n & 0,~t\in \left[ 0,1\right] ,\end{array}\right.$$*respectively. It is easy to check that when* $u^{\varepsilon
}(t)=1-\varepsilon ^{2},$ we have* *$$\mathbb{E}\left\{ \int_{0}^{1}\left[ \frac{1}{2}(u(t))^{2}+u(t)(k^{\varepsilon }(t)-u^{\varepsilon }(t)-1)+\frac{1}{2}(u^{\varepsilon
}(t))^{2}-k^{\varepsilon }(t)u^{\varepsilon }(t)+u^{\varepsilon }(t)\right] \mbox{\rm d}t\right\} \geq -C\varepsilon ^{2}.$$Moreover, $H^{\prime }(t,x,y,u,p,q,k)\overset{\vartriangle }{=}u-ku,$* is concave in* $(x,y,u).$* By Corollary \[col:16\], we conclude that* $u^{\varepsilon }(t)=1-\varepsilon ^{2}$* is an* $\varepsilon $*-optimal control.*

*In Example \[e2\], the control process* $u^{\varepsilon
}(t)=1-\varepsilon ^{\frac{1}{2}}$ actually fulfils the condition (\[3.91\]) (see (\[4.8\])). However, the underlying cost functional$$\begin{aligned}
J(u(\cdot )) &=&\mathbb{E}\left[ \int_{0}^{1}\left( \frac{1}{2}u^{2}(t)-u(t)\right) \mbox{\rm d}t\right] \\
&=&\mathbb{E}\left[ \int_{0}^{1}\frac{1}{2}\left( u\left( t\right) -1\right)
^{2}\mbox{\rm d}t-\frac{1}{2}\right] .\end{aligned}$$*Hence, the minimum value of the cost functional is* $-\frac{1}{2}.$ *Moreover, it is easy to check that* $u^{\varepsilon
}(t)=1-\varepsilon ^{\frac{1}{2}}$* is actually a near optimal control with order* $\varepsilon .$ *The main reason comes from the Ekeland principle @E*, *where a small value* $\lambda >0$* ensures a better position of* $u^{\lambda }$* but pays the price of a low sharp estimate of value of* $\rho (u^{\lambda }).$

Concluding Remarks {#sect:5}
==================

In this article, by Ekeland's principle, a spike variation, some dedicated estimates and reduction method, we established necessary and sufficient conditions for near-optimal controls to stochastic recursive optimization problems in terms of a small parameter $\varepsilon >0$. In particular, we solve the problems posed in @HLW (Huang, Li and Wang *Near-optimal control problems for linear forward-backward stochastic systems*, Automatica 46 (2010), 397-404) Page 402 for non-convex domain and diffusion term containing control variable. This result is partially based on the work from @E [@HLW; @Y; @Z1998; @W], etc. Our results extends that of Zhou's @Z1998 with second order adjoint equations in the setup of FBSDEs. Hopefully, the theoretical result obtained in this paper may inspire some real applications in finance and economics.

Appendix {#appendix .unnumbered}
========

The Proof of Theorem \[thm:4\] {#sect:6}
==============================

To establish the necessary condition, we need the following results mainly from Lemma 2.1, Lemma 2.2, Lemma 3.1 and Lemma 3.2 in @HLW (note that $\left\vert \theta _{0}^{\varepsilon }\right\vert ^{2}+\mathbb{E}\left\vert
\theta _{1}^{\varepsilon }\right\vert ^{2}=1$, $1\geq \theta
_{0}^{\varepsilon }\geq 0$ for any fixed $\varepsilon >0$ which don't change these results$)$. For simplicity, we omit the superscript $\varepsilon $.

\[lem:13\] There exists a constant $C>0$ such that for any $\alpha \geq
0 $ and any $u( \cdot ) \in \mathcal{U}_{ad}\left[ 0,1\right] ,$$$\begin{array}{c}
\mathbb{E}\left[ \sup\limits_{0\leq t\leq 1}\left\vert x( t) \right\vert
^{\alpha }\right] \leq C, \quad \mathbb{E}\left[ \sup\limits_{0\leq t\leq
1}\left\vert y( t) \right\vert ^{\alpha }\right] \leq C.\end{array}$$

\[lem:14\] There exists a constant $C>0$ such that $$\mathbb{E}\left[ \sup\limits_{0\leq t\leq 1}\left\vert q(t)\right\vert
^{2}+\sup\limits_{0\leq t\leq 1}\left\vert p(t)\right\vert
^{2}+\int_{0}^{1}\left\vert k(t)\right\vert ^{2}\mbox{\rm d}t\right] \leq C,$$where $C$ is independent of $(x(\cdot ),y(\cdot ),z(\cdot )).$

There exists a constant $C>0$ such that $$\mathbb{E}\left[ \sup\limits_{0\leq t\leq 1}\left\vert P_{1}(t)\right\vert
^{2}+\int_{0}^{1}\left\vert Q_{1}(t)\right\vert ^{2}\mbox{\rm d}t\right]
\leq C,$$where $C$ is independent of $(x(\cdot ),y(\cdot ),z(\cdot )).$

Applying Itô's formula to $\left\vert P_{3}(t)\right\vert ^{2},$ we have $$\left\vert P_{3}\left( t\right) \right\vert ^{2}+\mathbb{E}^{\mathcal{F}_{t}}\left[ \int_{t}^{1}\left\vert Q_{3}\left( s\right) \right\vert ^{2}\mbox{\rm
d}s\right] \leq C^{\prime }\mathbb{E}\int_{t}^{1}\left\vert P_{3}\left(
s\right) \right\vert ^{2}\mbox{\rm d}s+2\int_{t}^{1}\left\vert
l_{yy}\right\vert ^{2}\mbox{\rm d}s.$$By Burkholder-Davis-Gundy's inequality and Gronwall inequality, there is a constant $C$ such that, $$\mathbb{E}\left[ \sup\limits_{0\leq t\leq 1}\left\vert P_{3}\left( t\right)
\right\vert ^{2}+\int_{t}^{1}\left\vert Q_{3}\left( s\right) \right\vert ^{2}\mbox{\rm d}s\right] \leq C.$$The same method to deal with $P_{2}\left( t\right) ,$ and $P_{1}\left(
t\right) ,$ we get the desired result.

\[lem:15\] For any $\tau \geq 0$ and $0<\beta <1$ satisfying $\tau \beta
<1,$ there is a positive constant $C>0$ such that for any $u(\cdot )$ and $u^{\prime }(\cdot )\in U_{ad}\left[ 0,1\right] $ along with the corresponding trajectories $(x(\cdot ),y(\cdot ),z(\cdot ))$ and $(x^{\prime
}(\cdot ),y^{\prime }(\cdot ),z^{\prime }(\cdot ))$, it follows that $$\left\{ 
\begin{array}{c}
\mathbb{E}\left[ \sup\limits_{0\leq t\leq 1}\left\vert x(t)-x^{\prime
}(t)\right\vert ^{2\tau }\right] \leq Cd(u(\cdot ),u^{\prime }(\cdot
))^{\tau \beta }, \\ 
\mathbb{E}\left[ \sup\limits_{0\leq t\leq 1}\left\vert y(t)-y^{\prime
}(t)\right\vert ^{2\tau }\right] \leq Cd(u(\cdot ),u^{\prime }(\cdot
))^{\tau \beta }.\end{array}\right.$$

\[lem:16\] Assume (H1)-(H2) hold. For any $1<\tau <2$ and $0<\beta <1$ satisfying $(1+\beta )\tau <2,$ there is a constant $C$ such that for any $u(\cdot )$ and $u^{\prime }(\cdot )\in U_{ad}\left[ 0,1\right] $ along with the corresponding trajectories $(x(\cdot ),y(\cdot ),z(\cdot ))$, $(x^{\prime }(\cdot ),y^{\prime }(\cdot ),z^{\prime }(\cdot ))$, and solutions $(p(\cdot ),q(\cdot ),k(\cdot )),(p^{\prime }(\cdot ),q^{\prime
}(\cdot ),k^{\prime }(\cdot ))$ of the corresponding adjoint equations, it holds that $$\left\{ 
\begin{array}{l}
\mathbb{E}\left[ \sup\limits_{0\leq t\leq 1}\left\vert q(t)-q^{\prime
}(t)\right\vert ^{\tau }\right] \leq Cd(u(\cdot ),u^{\prime }(\cdot ))^{\frac{\tau \beta }{2}}, \\[2mm] 
\mathbb{E}\left[ \int_{0}^{1}\left\vert p(t)-p^{\prime }(t)\right\vert
^{\tau }+\left\vert k(t)-k^{\prime }(t)\right\vert ^{\tau }\mbox{\rm d}t\right] \leq Cd(u(\cdot ),u^{\prime }(\cdot ))^{\frac{\tau \beta }{2}}, \\[2mm] 
\mathbb{E}\left[ \int_{0}^{1}\left\vert P_{i}(t)-P_{i}^{\prime
}(t)\right\vert ^{\tau }+\left\vert Q_{i}(t)-Q_{i}^{\prime }(t)\right\vert
^{\tau }\mbox{\rm d}t\right] \leq Cd(u(\cdot ),u^{\prime }(\cdot ))^{\frac{\tau \beta }{2}},~i=1,2,3.\end{array}\right.$$

We are going to prove the third assertion. Note that $(\bar{P}_{3}(t),\bar{Q}_{3}(t))=(P_{3}(t)-P_{3}^{\prime }(t),Q_{3}(t)-Q_{3}^{\prime }(t))$ satisfies the following BSDEs$$\left\{ 
\begin{array}{rcl}
-\text{d}\bar{P}_{3}(t) & \3n=\3n & \left[ -2b(t)\bar{P}_{3}(t)-\theta
_{0}(l_{yy}(t,x(t),y(t),u(t))-l_{yy}(t,x^{\prime }(t),y^{\prime
}(t)),u^{\prime }(t))\right] \mbox{\rm d}t \\[2mm] 
&  & -Q_{3}(t)\mbox{\rm d}W(t), \\[2mm] 
\bar{P}_{3}(1) & \3n=\3n & 0.\end{array}\right.$$Set $\rho _{3}(\cdot )$ to be the following linear SDEs:$$\left\{ 
\begin{array}{rcl}
\text{d}\rho _{3}(t) & \3n=\3n & \left[ 2b(t)\rho _{3}(t)+\left\vert \bar{P}_{3}(t)\right\vert ^{\tau -1}\text{sgn}(\bar{P}_{3}(t))\right] \mbox{\rm d}t+\left\vert \bar{Q}_{3}(t)\right\vert ^{\tau -1}\text{sgn}(\bar{Q}_{3}(t))\mbox{\rm d}W(t), \\ 
\rho _{3}(0) & \3n=\3n & 0,\end{array}\right.  \label{6.6}$$It is easy to check that (\[6.6\]) admit a unique solution, and the following estimate can be obtained by Cauchy-Schwartz's inequality $$\mathbb{E}\left[ \sup\limits_{0\leq t\leq 1}\left\vert \rho
_{3}(t)\right\vert ^{\gamma }\right] \leq C\mathbb{E}\left[ \int_{0}^{1}\left[ \left\vert \bar{P}_{3}(t)\right\vert ^{\tau }+\left\vert \bar{Q}_{3}(t)\right\vert ^{\tau }\right] \mbox{\rm d}t\right] ,  \label{6.7}$$where $\gamma >2$ and $\frac{1}{\gamma }+\frac{1}{\tau }=1$.

Applying Itô's formula to $\bar{P}_{3}(\cdot )\rho _{3}(\cdot )$ on $\left[
0,1\right] ,$ we have $$\begin{aligned}
& \;\mathbb{E}\left[ \int_{0}^{1}\left\vert \bar{P}_{3}(t)\right\vert ^{\tau
}+\left\vert Q_{3}(t)\right\vert ^{\tau }\mbox{\rm d}t\right]  \notag
\label{6.8} \\
=& \;\mathbb{E}\left[ \int_{0}^{1}\rho _{3}(t)\theta
_{0}(l_{yy}(t,x(t),y(t),u(t))-l_{yy}(t,x^{\prime }(t),y^{\prime
}(t),u^{\prime }(t)))\mbox{\rm d}t\right]  \notag \\
\leq & \;C\left( \mathbb{E}\int_{0}^{1}\left\vert
(l_{yy}(t,x(t),y(t),u(t))-l_{yy}(t,x^{\prime }(t),y^{\prime }(t),u^{\prime
}(t)))\right\vert ^{\tau }\mbox{\rm d}t\right) ^{\frac{1}{\tau }}\left( 
\mathbb{E}\int_{0}^{1}\left\vert \rho _{3}(t)\right\vert ^{\gamma }\mbox{\rm
d}t\right) ^{\frac{1}{\gamma }}.\end{aligned}$$Substituting (\[6.7\]) into (\[6.8\]), we get $$\mathbb{E}\left[ \int_{0}^{1}\left\vert \bar{P}_{3}(t)\right\vert ^{\tau
}+\left\vert Q_{3}(t)\right\vert ^{\tau }\mbox{\rm d}t\right] \leq C\mathbb{E}\left[ \int_{0}^{1}\theta _{0}\left\vert
(l_{yy}(t,x(t),y(t),u(t))-l_{yy}(t,x^{\prime }(t),y^{\prime }(t),u^{\prime
}(t)))\right\vert ^{\tau }\mbox{\rm d}t\right] .$$From (H1), it follows that $$\begin{aligned}
& \left[ \int_{0}^{1}\theta _{0}\left\vert
l_{yy}(t,x(t),y(t),u(t))-l_{yy}(t,x^{\prime }(t),y^{\prime }(t),u^{\prime
}(t))\right\vert ^{\tau }\mbox{\rm d}t\right] \\
\leq & \;\mathbb{E}\left[ \int_{0}^{1}\left\vert
l_{yy}(t,x(t),y(t),u(t))-l_{yy}(t,x(t),y(t),u^{\prime }(t))\right\vert
^{\tau }\mathcal{X}_{u(t)=u^{^{\prime }}(t)}\mbox{\rm d}t\right] \\
& +\mathbb{E}\left[ \int_{0}^{1}\left\vert l_{yy}(t,x(t),y(t),u^{\prime
}(t))-l_{yy}(t,x^{\prime }(t),y^{\prime }(t),u^{\prime }(t))\right\vert
^{\tau }\mbox{\rm d}t\right] \\
\leq & \;C\mathbb{E}\left[ \left( \int_{0}^{1}\left\vert
l_{yy}(t,x(t),y(t),u(t))-l_{yy}(t,x(t),y(t),u^{\prime }(t))\right\vert ^{2}\mbox{\rm d}t\right) ^{\frac{\tau }{2}}d(u(t),u^{\prime }(t))^{1-\frac{\tau 
}{2}}\mbox{\rm d}t\right] \\
& +C\mathbb{E}\left[ \int_{0}^{1}\left[ \left\vert x(t)-x^{\prime
}(t)\right\vert ^{\tau }+\left\vert y(t)-y^{\prime }(t)\right\vert ^{\tau }\right] \mbox{\rm d}t\right] \\
\leq & \;Cd(u(t),u^{\prime }(t))^{\frac{\tau \beta }{2}}.\end{aligned}$$Combining (\[6.8\]) with the above inequality, the result for $i=3$ holds immediately.

We proceed to estimate the case, $i=2$. Similarly, we define the following SDEs: $$\left\{ 
\begin{array}{rcl}
\text{d}\rho _{2}(t) & \3n=\3n & \left[ (b(t)-A(t))\rho _{2}(t)+\left\vert 
\bar{P}_{2}(t)\right\vert ^{\tau -1}\text{sgn}(\bar{P}_{2}(t))\right] \mbox{\rm d}t \\ 
&  & +\left[ c(t)\rho _{2}(t)+\left\vert \bar{Q}_{2}(t)\right\vert ^{\tau -1}\text{sgn}(\bar{Q}_{2}(t))\right] \mbox{\rm d}W(t), \\ 
\rho _{2}(0) & \3n=\3n & 0.\end{array}\right.$$Applying Itô's formula to $\bar{P}_{2}(\cdot )\rho _{2}(\cdot )$ on $\left[
0,1\right] ,$ we have $$\begin{aligned}
& \;\mathbb{E}\left[ \int_{0}^{1}\left\vert \bar{P}_{2}(t)\right\vert ^{\tau
}+\left\vert Q_{2}(t)\right\vert ^{\tau }\mbox{\rm d}t\right] \\
=& \;\mathbb{E}\left\{ \int_{0}^{1}\rho _{2}(t)\left[ \theta
_{0}(l_{xy}(t,x(t),y(t),u(t))-l_{xy}(t,x^{\prime }(t),y^{\prime
}(t),u^{\prime }(t)))+a(t)\bar{P}_{3}(t)\right] \mbox{\rm d}t\right\} .\end{aligned}$$By Cauchy-Schwartz's inequality, we obtain $$\begin{aligned}
& \;\mathbb{E}\left[ \int_{0}^{1}\left\vert \bar{P}_{2}(t)\right\vert ^{\tau
}+\left\vert Q_{2}(t)\right\vert ^{\tau }\mbox{\rm d}t\right] \\
\leq & \;C\mathbb{E}\left[ \int_{0}^{1}\left\vert a(t)\bar{P}_{3}(t)\right\vert ^{\tau }\mbox{\rm d}t\right] +C\mathbb{E}\left[
\int_{0}^{1}\theta _{0}\left\vert
l_{xy}(t,x(t),y(t),u(t))-l_{xy}(t,x^{\prime }(t),y^{\prime }(t),u^{\prime
}(t))\right\vert ^{\tau }\mbox{\rm d}t\right] \\
\leq & \;Cd(u(\cdot ),u^{\prime }(\cdot ))^{\frac{\tau \beta }{2}}.\end{aligned}$$Analogously, we define the following SDEs: $$\left\{ 
\begin{array}{lll}
\text{d}\rho _{1}(t) & \3n=\3n & \left[ (2A(t)+C(t)^{2})\rho
_{1}(t)+\left\vert \bar{P}_{1}(t)\right\vert ^{\tau -1}\text{sgn}(\bar{P}_{1}(t))\right] \mbox{\rm d}t \\ 
&  & +\left[ 2C(t)\rho _{1}(t)+\left\vert \bar{Q}_{1}(t)\right\vert ^{\tau
-1}\text{sgn}(\bar{Q}_{1}(t))\right] \mbox{\rm d}W(t), \\ 
\rho _{1}(0) & \3n=\3n & 0.\end{array}\right.$$Repeating the method used above, we have $$\mathbb{E}\left[ \int_{0}^{1}\left\vert \bar{P}_{1}(t)\right\vert ^{\tau
}+\left\vert Q_{1}(t)\right\vert ^{\tau }\mbox{\rm d}t\right] \leq
Cd(u(t),u^{\prime }(t))^{\frac{\tau \beta }{2}}.$$The proof is complete. The proof of Theorem \[thm:4\] will be accomplished step by step. The idea is mainly borrowed from @Y and @W

**Step 1** **(The bounded control domains)**.

When $(x(\cdot ),y(\cdot ))$ is regarded as the state process and $(z(\cdot ),u(\cdot ))$ as the control process. Then, we consider the following initial value problem for a control system of SDEs:$$\left\{ 
\begin{array}{rcl}
\mbox{\rm d}x(t) & \3n=\3n & \left[ A(t)x(t)+B(t)u(t)\right] \mbox{\rm d}t+\left[ C(t)x(t)+D(t)u(t)\right] \mbox{\rm d}W(t), \\[2mm]
-\mbox{\rm d}y(t) & \3n=\3n & \left[ a(t)x(t)+b(t)y(t)+c(t)u(t)\right] \text{d}t-z(t)\mbox{\rm d}W(t), \\[2mm]
x(0) & \3n=\3n & x_{0},\qquad y(0)=y_{0},\end{array}\right.   \label{6.16}$$

Clearly, it is easy to check that, for any $(z(\cdot ),u(\cdot
))\in \mathcal{M}^{2}(0,1;\mathbb{R})\times \mathcal{U}_{ad}\left[ 0,1\right]
,$ $y_{0}\in \mathbb{R}$, there exists a unique strong solution $$(x(\cdot ),y(\cdot ))\equiv (x(\cdot ,z(\cdot ),u(\cdot )),y(\cdot ,z(\cdot
),u(\cdot )))\in \mathcal{S}^{2}(0,1;\mathbb{R})\times \mathcal{S}^{2}(0,1;\mathbb{R})$$to (\[6.16\]) depending on $(z(\cdot ),u(\cdot ))$. Next, we regard the original terminal condition as the terminal state constraint: $$y(1)=Mx(1).  \label{6.17}$$Since $\mathbb{R},$ $\mathcal{M}^{2}(0,1;\mathbb{R})$ are all unbounded, Thus, we adopt a convergence technique developed by Tang and Li (1994) and Wu (2013).

Let $y_{0}$, $z\left( \cdot \right) ,$ take value in $\mathbb{M},$ $\mathbb{N\subset R},$ and $\mathbb{M}$ be convex. Moreover, $\mathbb{M},$ $\mathbb{N}
$ are all bounded. Let $\mathcal{A}$ be the set of all 3-triples $(y_{0},z(\cdot ),u(\cdot ))\in \mathbb{M}\times \mathcal{M}^{2}(0,1;\mathbb{N})\times \mathcal{U}_{ad}\left[ 0,1\right] $ such that the unique corresponding state process $(x(\cdot ),y(\cdot ))$ satisfies the constraint (\[6.17\]). Note that, for any $u(\cdot )\in \mathcal{U}_{ad}[0,1]$, there exists a unique $(y_{0},z(\cdot ))\in \mathbb{R}\times \mathcal{M}^{2}(0,1;\mathbb{R})$ such that state equation (\[2.1\]) admits a unique state process $(x(\cdot ),y(\cdot ))\in \mathcal{S}^{2}(0,1;\mathbb{R})\times 
\mathcal{M}^{2}(0,1;\mathbb{R})$ satisfying the state constraint (\[6.17\]). Hence, (H1) implies $\mathcal{A\neq \phi }.$ The cost functional is given by $$J(y_{0},z(\cdot ),u(\cdot ))=\mathbb{E}\left[ \int_{0}^{1}l(t,x(t),y(t),u(t))\mbox{\rm d}t+\phi (x(1))+\gamma (y(0))\right] .$$We state the following problem.

**Problem $(\tilde{C}^{\varepsilon })$**. Find $(
y_{0}^{\varepsilon },z^{\varepsilon }( \cdot ) ,u^{\varepsilon }( \cdot ) )
\in \mathcal{A}$, such that $$J( y_{0}^{\varepsilon },z^{\varepsilon }( \cdot ) ,u^{\varepsilon }( \cdot )
) =\inf\limits_{( y_{0},z( \cdot ) ,u( \cdot ) ) \in \mathcal{A}}J( y_{0},z(
\cdot ) ,u( \cdot ) ) +\varepsilon .$$ We, respectively, refer to $( y_{0}^{\varepsilon },z^{\varepsilon }( \cdot )
,u^{\varepsilon }( \cdot ) ) $ as an optimal control process, to $(
x^{\varepsilon }( \cdot ) ,y^{\varepsilon }( \cdot ) ) $ as the corresponding optimal state process, and to $( y_{0}^{\varepsilon
},z^{\varepsilon }( \cdot ) ,u^{\varepsilon }( \cdot ) ) $ as an optimal 3-tuple of Problem $(\tilde{C}^{\varepsilon })$.

Problems $(C^{\varepsilon })$ is embedded into $(\tilde{C}^{\varepsilon })$. Suppose that $\left( \tilde{y}_{0}^{\varepsilon },\tilde{z}^{\varepsilon
}(\cdot ),\tilde{u}^{\varepsilon }(\cdot )\right) $ is the near optimal control of Problem $(\tilde{C}^{\varepsilon })$, clearly, we know that $\tilde{u}^{\varepsilon }(\cdot )$ is the near optimal control of Problem $(C^{\varepsilon }).$ The advantage of Problem $(\tilde{C}^{\varepsilon })$ is that one does not need much regularity/integrability on $z(\cdot )$ since it is treated as part of a control process; the disadvantage is that one has to treat terminal constraint (\[6.17\]).

\[**Ekeland Principle @E**\] \[lem:17\] Let $(S,d)$ be a complete metric space and $\rho
:S\rightarrow R\cup \left\{ +\infty \right\} $ be a lower semicontinuous function, bounded from below. If for each $\varepsilon >0,$ there exists $u^{\varepsilon }\in S$ such that $\rho (u^{\varepsilon })\leq
\inf\limits_{u\in S}\rho (u)+\varepsilon .$ Then for any $\lambda >0,$ there exists $u^{\lambda }\in S$ such that $$\left\{ 
\begin{array}{ll}
\mbox{(i)} & \rho (u^{\lambda })\leq \rho (u^{\varepsilon }), \\ 
\mbox{(ii)} & d(u^{\lambda },u^{\varepsilon })\leq \lambda , \\ 
\mbox{(iii)} & \rho (u^{\lambda })\leq \rho (u)+\frac{\varepsilon }{\lambda }d(u,u^{\lambda }),\quad \text{for all }u\in S.\end{array}\right.$$

For $u,v$ in $\mathcal{U}_{ad}\left[ 0,1\right] $ or in $\mathcal{M}^{2}(0,1;\mathbb{R}),$ we define $$d(u,v)=\text{d}t\otimes P\left\{ (t,\omega )\in \left[ 0,1\right] \times
\Omega :u(t,\omega )\neq v(t,\omega )\right\} ,$$where d$t\otimes P$ is the product measure of the Lebesgue measure d$t$ with the probability measure $P$. It is well known that $(\mathcal{U}_{ad}\left[
0,1\right] ,d)$ is a complete metric space (see @YZ). Then $\mathbb{R\times }\mathcal{M}^{2}(0,1;\mathbb{R})\times \mathcal{U}_{ad}\left[ 0,1\right] $ is a complete metric space under the following metric: for any $(y_{0},z(\cdot ),u(\cdot )),(\tilde{y}_{0},\tilde{z}(\cdot ),\tilde{u}(\cdot
))\in \mathcal{A},$ $$d_{\mathcal{A}}(\theta (\cdot ),\tilde{\theta}(\cdot ))=\left[ \left\vert
y_{0}-\tilde{y}_{0}\right\vert ^{2}+d\left( z(\cdot ),\tilde{z}(\cdot
)\right) ^{2}+d(u(\cdot ),\tilde{u}(\cdot ))^{2}\right] ^{\frac{1}{2}},$$where $\theta (\cdot )=(y_{0},z(\cdot ),u(\cdot ))$ and $\tilde{\theta}(\cdot )=(\tilde{y}_{0},\tilde{z}(\cdot ),\tilde{u}(\cdot )),$ respectively.

By assumption (H1), it is easy to see that $J(y_{0},z(\cdot ),u(\cdot ))$ is lower semicontinuous on $\mathcal{A}$. By virtue of Ekeland principle (Lemma \[lem:17\]) with $\lambda =\varepsilon ^{\frac{2}{3}}$ (fixed $\varepsilon
>0$) there is an admissible 3-triple $(\tilde{y}_{0}^{\varepsilon },\tilde{z}^{\varepsilon }(\cdot ),\tilde{u}^{\varepsilon }(\cdot ))\in \mathcal{A}$ such that $$d_{\mathcal{A}}((y_{0}^{\varepsilon },z^{\varepsilon }(\cdot
),u^{\varepsilon }(\cdot )),(\tilde{y}_{0}^{\varepsilon },\tilde{z}^{\varepsilon }(\cdot ),\tilde{u}^{\varepsilon }(\cdot )))\leq \varepsilon ^{\frac{2}{3}}  \label{6.18}$$and $$\tilde{J}^{\varepsilon }((\tilde{y}_{0}^{\varepsilon },\tilde{z}^{\varepsilon }(\cdot ),\tilde{u}^{\varepsilon }(\cdot )))\leq \tilde{J}^{\varepsilon }(v(\cdot )),\text{ for any }v(\cdot )\in \mathcal{A},$$where $$\tilde{J}^{\varepsilon }(v(\cdot ))=J(v(\cdot ))+\varepsilon ^{\frac{1}{3}}d_{\mathcal{A}}(v(\cdot ),\tilde{\theta}^{\varepsilon }(\cdot )),
\label{6.22}$$which means that $(\tilde{y}_{0},\tilde{z}^{\varepsilon }(\cdot ),\tilde{u}^{\varepsilon }(\cdot ))$ is an optimal pair for the system (\[6.16\]) with a new cost functional $\tilde{J}^{\varepsilon }$.

Let $(\tilde{y}_{0}^{\varepsilon },\tilde{z}^{\varepsilon }(\cdot ),\tilde{u}^{\varepsilon }(\cdot ))$ be an optimal 3-triple of Problem $(\tilde{C}^{\varepsilon })$ with new functional (\[6.22\]), with the corresponding optimal state process $(\tilde{x}^{\varepsilon }(\cdot ),\tilde{y}^{\varepsilon }(\cdot )).$ For any $\delta >0,$ we define, for any $\forall
(y_{0},z(\cdot ),u(\cdot ))\in \mathbb{M}\times \mathcal{M}^{2}(0,1;\mathbb{N})\times \mathcal{U}_{ad}\left[ 0,1\right] $, $$J^{\delta ,\varepsilon }(y_{0},z(\cdot ),u(\cdot ))=\left\{ \left[ (\tilde{J}^{\varepsilon }(y_{0},z(\cdot ),u(\cdot ))-\tilde{J}^{\varepsilon }(\tilde{y}_{0}^{\varepsilon },\tilde{z}^{\varepsilon }(\cdot ),\tilde{u}^{\varepsilon
}(\cdot ))+\delta )^{+}\right] ^{2}+\mathbb{E}\left\vert
y(1)-Mx(1)\right\vert ^{2}\right\} ^{\frac{1}{2}},$$where $(x(\cdot ),y(\cdot ))$ is the unique solution of (\[6.16\]). Also, it is clear that $$\begin{aligned}
& J^{\delta ,\varepsilon }(y_{0},z(\cdot ),u(\cdot ))>0,~\forall
(y_{0},z(\cdot ),u(\cdot ))\in \mathbb{M}\times \mathcal{M}^{2}(0,1;\mathbb{N})\times \mathcal{U}_{ad}\left[ 0,1\right] , \\
& J^{\delta ,\varepsilon }(\tilde{y}_{0}^{\varepsilon },\tilde{z}^{\varepsilon }(\cdot ),\tilde{u}^{\varepsilon }(\cdot ))=\delta \leq
\inf\limits_{(y_{0},z(\cdot ),u(\cdot ))\in \mathbb{M}\times \mathcal{M}^{2}(0,1;\mathbb{N})\times \mathcal{U}_{ad}\left[ 0,1\right] }J^{\delta
,\varepsilon }(y_{0},z(\cdot ),u(\cdot ))+\delta .\end{aligned}$$Hence, by Lemma \[lem:17\], there exists a 3-triple $(y_{0}^{\delta
,\varepsilon },z^{\delta ,\varepsilon }(\cdot ),u^{\delta ,\varepsilon
}(\cdot ))\in \mathbb{M}\times \mathcal{M}^{2}(0,1;\mathbb{N})\times 
\mathcal{U}_{ad}\left[ 0,1\right] $ such that $$\left\{ 
\begin{array}{l}
\text{(1) }J^{\delta ,\varepsilon }(y_{0}^{\delta ,\varepsilon },z^{\delta
,\varepsilon }(\cdot ),u^{\delta ,\varepsilon }(\cdot ))\leq J^{\delta
,\varepsilon }(\tilde{y}_{0}^{\varepsilon },\tilde{z}^{\varepsilon }(\cdot ),\tilde{u}^{\varepsilon }(\cdot ))=\delta , \\[2mm] 
\text{(2) }\left\vert y_{0}^{\delta ,\varepsilon }-\tilde{y}_{0}^{\varepsilon }\right\vert ^{2}+d\left( z^{\delta ,\varepsilon }(\cdot
)-z(\cdot )\right) ^{2}+d(u^{\delta ,\varepsilon }(\cdot ),\tilde{u}^{\varepsilon }(\cdot ))^{2}\leq \delta , \\[2mm] 
\text{(3) }-\sqrt{\delta }\left[ \left\vert y_{0}^{\delta ,\varepsilon
}-y_{0}\right\vert ^{2}+d\left( z^{\delta ,\varepsilon }(\cdot )-z(\cdot
)\right) ^{2}+d(u^{\delta ,\varepsilon }(\cdot ),u(\cdot ))^{2}\right] ^{\frac{1}{2}} \\[2mm] 
\qquad \leq J^{\delta ,\varepsilon }(y_{0},z(\cdot ),u(\cdot ))-J^{\delta
,\varepsilon }(y_{0}^{\delta ,\varepsilon },z^{\delta ,\varepsilon }(\cdot
),u^{\delta ,\varepsilon }(\cdot )), \\[2mm] 
\hfill \forall (y_{0},z(\cdot ),u(\cdot ))\in \mathbb{M}\times \mathcal{M}^{2}(0,1;\mathbb{N})\times \mathcal{U}_{ad}\left[ 0,1\right] .\end{array}\right.  \label{6.26}$$Hence, $(y_{0}^{\delta ,\varepsilon },z^{\delta ,\varepsilon }(\cdot
),u^{\delta ,\varepsilon }(\cdot ))$ is a global minimum point of the following penalized cost functional$$J^{\delta ,\varepsilon }(y_{0},z(\cdot ),u(\cdot ))+\sqrt{\delta }\left[
\left\vert y_{0}-y_{0}^{\delta ,\varepsilon }\right\vert ^{2}+d\left(
z^{\delta ,\varepsilon }(\cdot )-z(\cdot )\right) ^{2}+d(u^{\delta
,\varepsilon }(\cdot ),u(\cdot ))^{2}\right] ^{\frac{1}{2}}.  \label{6.27}$$In other words, fix $\varepsilon >0,$ if we pose a penalized optimal control problem with the state constraint (\[6.17\]) and the cost functional (<span>6.27</span>) , then $(y_{0}^{\delta ,\varepsilon },z^{\delta ,\varepsilon }(\cdot
),u^{\delta ,\varepsilon }(\cdot ))$ is an optimal 3-triple of the problem. Note that this problem does not have state constraints, and the optimal 3-triple $(y_{0}^{\delta ,\varepsilon },z^{\delta ,\varepsilon }(\cdot
),u^{\delta ,\varepsilon }(\cdot ))$ approaches $(\tilde{y}_{0}^{\varepsilon
},\tilde{z}^{\varepsilon }(\cdot ),\tilde{u}^{\varepsilon }(\cdot ))$ as $\delta \rightarrow 0$. Let us turn back to the new cost functional $$\begin{aligned}
\mathcal{J}^{\delta ,\varepsilon }(y_{0},z(\cdot ),u(\cdot )) &=&J^{\delta
,\varepsilon }(y_{0},z(\cdot ),u(\cdot ))  \notag \\
&&+\sqrt{\delta }\left[ \left\vert y_{0}-y_{0}^{\delta ,\varepsilon
}\right\vert ^{2}+d\left( z^{\delta ,\varepsilon }(\cdot )-z(\cdot )\right)
^{2}+d(u^{\delta ,\varepsilon }(\cdot ),u(\cdot ))^{2}\right] ^{\frac{1}{2}}.
\label{6.28}\end{aligned}$$

Denote $$\begin{array}{cccc}
X\triangleq \left( 
\begin{array}{c}
x \\ 
y\end{array}\right) , & v(\cdot )\triangleq \left( 
\begin{array}{c}
z \\ 
u\end{array}\right) , & X_{0}\triangleq \left( 
\begin{array}{c}
x_{0} \\ 
y_{0}\end{array}\right) , & X(1)\triangleq \left( 
\begin{array}{c}
x(1) \\ 
y(1)\end{array}\right) ,\end{array}$$$$\begin{aligned}
\mathcal{B}(t,X,v(\cdot ))\triangleq & \left( 
\begin{array}{c}
A(t)x(t)+B(t)u(t) \\ 
-a(t)x(t)-b(t)y(t)-c(t)u(t)\end{array}\right) , \\
\Sigma (t,X,v(\cdot ))\triangleq & \left( 
\begin{array}{c}
C(t)x(t)+D(t)u(t) \\ 
z\end{array}\right) , \\
\Xi (X(0),X(1))\triangleq & \;\phi (x(1))+\gamma (y(0)), \\
\Pi (X(0),X(1))\triangleq & \left( 
\begin{array}{c}
0 \\ 
y(1)-Mx(1)\end{array}\right) ,\end{aligned}$$and $$\begin{aligned}
\mathcal{H}& \triangleq \;\mathbb{R}^{2}\times L_{\mathcal{F}_{1}}^{2}(\Omega ;\mathbb{R}^{2})\equiv \mathbb{R}^{2}\times \mathcal{X}_{2}^{2}, \\
\mathcal{H}_{0}& \triangleq \;\mathbb{R}\times L_{\mathcal{F}_{1}}^{2}(\Omega ;\mathbb{R})\equiv \mathbb{R}\times \mathcal{X}_{1}^{2}.\end{aligned}$$Consequently, $$\begin{aligned}
\tilde{J}^{\varepsilon }(y_{0},z(\cdot ),u(\cdot ))=& \;\tilde{J}^{\varepsilon }(y_{0},v(\cdot )), \\
J^{\delta ,\varepsilon }(y_{0},z(\cdot ),u(\cdot ))=& \;J^{\delta
,\varepsilon }(y_{0},v(\cdot )).\end{aligned}$$Note that $\mathcal{H}$ and $\mathcal{H}_{0}$ are Hilbert spaces. We identify $\mathcal{H}^{\ast }=\mathcal{H}$ and $\mathcal{H}_{0}^{\ast }=\mathcal{H}_{0}.$ Also $$\Xi :\mathcal{H}\rightarrow \mathbb{R},\qquad \Pi :\mathcal{H}\rightarrow 
\mathcal{H}_{0}.$$The gradient of $D\Xi $ and the Hessian $D^{2}\Xi $ of $\Xi $ are defined as follows: $$\begin{aligned}
D\Xi (X(0),X(1))=& (D_{X_{0}}\Xi (X(0),X(1)),D_{X_{1}}\Xi (X(0),X(1)))\in 
\mathcal{L}(\mathcal{H};\mathbb{R})\equiv \mathcal{H}^{\ast }=\mathcal{H}, \\
D^{2}\Xi (X(0),X(1))=& \left( 
\begin{array}{cc}
D_{X_{0}X_{0}}\Xi (X(0),X(1)) & D_{X_{0}X_{1}}\Xi (X(0),X(1)) \\ 
D_{X_{1}X_{0}}\Xi (X(0),X(1)) & D_{X_{1}X_{1}}\Xi (X(0),X(1))\end{array}\right) \in \mathcal{L}_{s}(\mathcal{H};\mathcal{H}),\end{aligned}$$where $\mathcal{L}(\mathcal{H}_{1};\mathcal{H}_{2})$ is the set of all linear bounded operator from $\mathcal{H}_{1}$ to $\mathcal{H}_{2}$, and $\mathcal{L}_{s}(\mathcal{H};\mathcal{H})$ is the set of all linear bounded self-adjoint operators from $\mathcal{H}$ to itself. We have$$\begin{aligned}
\Xi _{X_{0}}(X(0),X(1)) &=&(0,\gamma _{y}(y_{0}))^{\top }\in \mathbb{R}^{2},
\\
\Xi _{X_{1}}(X(0),X(1)) &=&(\phi _{x}(x(1)),0)^{\top }\in \mathcal{X}_{1}^{2}, \\
\Xi _{X_{0}X_{0}}(X(0),X(1)) &=&\left( 
\begin{array}{cc}
0 & 0 \\ 
0 & \gamma _{yy}(y_{0})\end{array}\right) \in \mathcal{S}^{2}, \\
\Xi _{X_{0}X_{1}}(X(0),X(1)) &=&\left( 
\begin{array}{cc}
0 & 0 \\ 
0 & 0\end{array}\right) \in \mathcal{L}(\mathcal{X}_{1}^{2};\mathbb{R}^{2}), \\
\Xi _{X_{1}X_{0}}(X(0),X(1)) &=&\left( 
\begin{array}{cc}
0 & 0 \\ 
0 & 0\end{array}\right) \in \mathcal{L}(\mathbb{R}^{2};\mathcal{X}_{1}^{2}), \\
\Xi _{X_{1}X_{1}}(X(0),X(1)) &=&\left( 
\begin{array}{cc}
\phi _{xx}(x(1)) & 0 \\ 
0 & 0\end{array}\right) \in \mathcal{L}(\mathcal{X}_{2}^{2};\mathcal{X}_{2}^{2}).\end{aligned}$$ For $$\begin{aligned}
D\Pi (X(0),X(1))=& (D_{X_{0}}\Pi (X(0),X(1)),D_{X_{1}}\Pi (X(0),X(1)))\in 
\mathcal{L}(\mathcal{H};\mathcal{H}_{0}), \\
D^{2}\Pi (X(0),X(1))=& (\begin{array}{cc}
D_{X_{0}X_{0}}\Pi (X(0),X(1)) & D_{X_{0}X_{1}}\Pi (X(0),X(1)) \\ 
D_{X_{1}X_{0}}\Pi (X(0),X(1)) & D_{X_{1}X_{1}}\Pi (X(0),X(1))\end{array})\in \mathcal{L}(\mathcal{H};\mathcal{L}(\mathcal{H};\mathcal{H}_{0})).\end{aligned}$$Take any $\hat{\Phi}=(\hat{\Phi}_{0},\hat{\Phi}_{1})\in \mathcal{H}_{0}.$ Then, $$\left\langle \Pi (X(0),X(1)),\hat{\Phi}\right\rangle =\left\langle
y(1)-M(1)x(1),\hat{\Phi}\right\rangle .$$Thus, $$\begin{aligned}
D\Pi (X(0),X(1))\hat{\Phi}=& \;D\left[ \left\langle \Pi (X(0),X(1)),\hat{\Phi}\right\rangle \right] \\
=& \left( \left\langle \Pi (X(0),X(1)),\hat{\Phi}\right\rangle
_{X_{0}},\left\langle \Pi (X(0),X(1)),\hat{\Phi}\right\rangle _{X_{1}}\right)
\\
=& (\Pi _{X_{0}}(X(0),X(1))\hat{\Phi},\Pi _{X_{1}}(X(0),X(1))\hat{\Phi}),\end{aligned}$$with $$\begin{aligned}
\Pi _{X_{0}}(X(0),X(1))\hat{\Phi}=& (0,0), \\
\Pi _{X_{1}}(X(0),X(1))\hat{\Phi}=& (-M\hat{\Phi}_{1},\hat{\Phi}_{1}),\end{aligned}$$$$\begin{aligned}
D^{2}\Pi (X(0),X(1))\hat{\Phi}=& \;D^{2}\left[ \left\langle \Pi (X(0),X(1)),\hat{\Phi}\right\rangle \right] \\
=& \left( 
\begin{array}{cc}
D_{X_{0}X_{0}}\Pi (X(0),X(1))\hat{\Phi} & D_{X_{0}X_{1}}\Pi (X(0),X(1))\hat{\Phi} \\ 
D_{X_{1}X_{0}}\Pi (X(0),X(1))\hat{\Phi} & D_{X_{1}X_{1}}\Pi (X(0),X(1))\hat{\Phi}\end{array}\right) \in \mathcal{L}(\mathcal{H};\mathcal{H}),\end{aligned}$$and $$\begin{aligned}
D_{X_{0}X_{0}}\Pi (X(0),X(1))\hat{\Phi}=& \left( 
\begin{array}{cc}
0 & 0 \\ 
0 & 0\end{array}\right) , \\
D_{X_{1}X_{0}}\Pi (X(0),X(1))\hat{\Phi}=& \left( 
\begin{array}{cc}
0 & 0 \\ 
0 & 0\end{array}\right) , \\
D_{X_{0}X_{1}}\Pi (X(0),X(1))\hat{\Phi}=& \left( 
\begin{array}{cc}
0 & 0 \\ 
0 & 0\end{array}\right) , \\
D_{X_{1}X_{1}}\Pi (X(0),X(1))\hat{\Phi}=& \left( 
\begin{array}{cc}
0 & 0 \\ 
0 & 0\end{array}\right) .\end{aligned}$$We now construct spike variation. For$,$ $u(\cdot )\in \mathcal{U}_{ad}\left[
0,1\right] $, and any $0<\alpha <1,$ let $y_{0}\in \mathbb{M}$  such that $y_{0}^{\delta ,\varepsilon }+y_{0}\in \mathbb{M}$ . Define $$y_{0}^{\delta ,\varepsilon ,\alpha }=\alpha y_{0}+y_{0}^{\delta ,\varepsilon
},\text{\qquad }(z^{\delta ,\varepsilon ,\alpha }(t),u^{\delta ,\varepsilon
,\alpha }(t))=\left\{ 
\begin{array}{ll}
(z,,u), & t\in \left[ \tau ,\tau +\alpha \right] , \\ 
(z^{\delta ,\varepsilon }(t),u^{\delta ,\varepsilon }(t)), & \text{otherwise,}\end{array}\right.$$where $z\in \mathbb{N}$, $u\in \mathbb{U}$ are $\mathcal{F}_{\tau }$-measurable random variables, such that $\sup\limits_{\omega \in \Omega
}\left\vert u\left( \omega \right) \right\vert <+\infty $ and $\sup\limits_{\omega \in \Omega }\left\vert z\left( \omega \right)
\right\vert <+\infty $. Note that $y_{0}$ is a control independent of time variable, so convex perturbation can be applied here.

Let $X^{\delta ,\varepsilon ,\alpha }(\cdot )$ be the state process (<span>6.16</span>) corresponding to $(X_{0}^{\delta ,\varepsilon ,\alpha }\cdot
,v^{\delta ,\varepsilon ,\alpha }(\cdot ))$. Let $X_{1}^{\delta ,\varepsilon
,\alpha }(\cdot )$ and $X_{2}^{\delta ,\varepsilon ,\alpha }(\cdot )$ be, respectively, the solutions to the following SDEs: $$\left\{ 
\begin{array}{lll}
\mbox{\rm d}X_{1}^{\delta ,\varepsilon ,\alpha }(t) & \3n=\3n & \mathcal{B}_{X}^{\delta ,\varepsilon }(t,\cdot )X_{1}^{\delta ,\varepsilon ,\alpha }(t)\mbox{\rm d}t+\left[ \Sigma _{X}^{\delta ,\varepsilon }(t,\cdot
)X_{1}^{\delta ,\varepsilon ,\alpha }(t)+\bigtriangleup \Sigma ^{\delta
,\varepsilon }(t,\cdot )I_{S_{\alpha }}\right] \mbox{\rm d}W(t), \\ 
X_{1}^{\delta ,\varepsilon ,\alpha }(t) & \3n=\3n & \sqrt{\alpha }y_{0},\end{array}\right.$$and $$\left\{ 
\begin{array}{lll}
\mbox{\rm d}X_{2}^{\delta ,\varepsilon ,\alpha }(t) & \3n=\3n & \mathcal{B}_{X}^{\delta ,\varepsilon }(t,\cdot )X_{2}^{\delta ,\varepsilon ,\alpha
}(t)+\bigtriangleup \mathcal{B}^{\delta ,\varepsilon }(t,\cdot )I_{S_{\alpha
}}+\frac{1}{2}\mathcal{B}_{XX}^{\delta ,\varepsilon }(t,\cdot )X_{1}^{\delta
,\varepsilon ,\alpha }(t)^{2}\mbox{\rm d}t \\[2mm] 
&  & +\left[ \Sigma _{X}^{\delta ,\varepsilon }(t,\cdot )X_{2}^{\delta
,\varepsilon ,\alpha }(t)+\bigtriangleup \Sigma _{X}^{\delta ,\varepsilon
}(t,\cdot )X_{1}^{\delta ,\varepsilon ,\alpha }(t)I_{S_{\alpha }}+\frac{1}{2}\Sigma _{XX}^{\delta ,\varepsilon }(t,\cdot )X_{1}^{\delta ,\varepsilon
,\alpha }(t)^{2}\right] \mbox{\rm d}W(t), \\[2mm] 
X_{2}^{\delta ,\varepsilon ,\alpha }(0) & \3n=\3n & 0,\end{array}\right.$$where $I_{S_{\alpha }}$ denotes the indicator function of the set $S_{\alpha
}$ and for any $X\in \mathbb{R}^{2}$.

Set $$\begin{aligned}
\mathcal{B}_{X}^{\delta ,\varepsilon }(t,\cdot )=& \;\mathcal{B}_{X}(t,X^{\delta ,\varepsilon }(t),v^{\delta ,\varepsilon }(t)), \\
\Sigma _{X}^{\delta ,\varepsilon }(t,\cdot )=& \;\Sigma _{X}(t,X^{\delta
,\varepsilon }(t),v^{\delta ,\varepsilon }(t)), \\
\bigtriangleup \mathcal{B}^{\delta ,\varepsilon }(t,\cdot )=& \;\mathcal{B}(t,X^{\delta ,\varepsilon }(t),v(t))-\mathcal{B}(t,X^{\delta ,\varepsilon
}(t),v^{\delta ,\varepsilon }(t)), \\
\bigtriangleup \Sigma ^{\delta ,\varepsilon }(t,\cdot )=& \;\Sigma
(t,X^{\delta ,\varepsilon }(t),v(t))-\Sigma (t,X^{\delta ,\varepsilon
}(t),v^{\delta ,\varepsilon }(t)), \\
\bigtriangleup \Sigma _{X}^{\delta ,\varepsilon }(t,\cdot )=& \;\Sigma
_{X}(t,X^{\delta ,\varepsilon }(t),v(t))-\Sigma _{X}(t,X^{\delta
,\varepsilon }(t),v^{\delta ,\varepsilon }(t)),\end{aligned}$$$$\begin{aligned}
\mathcal{B}_{XX}^{\delta ,\varepsilon }(t,\cdot )X^{2}=& \;\left( 
\begin{array}{c}
\left\langle \mathcal{B}_{XX}^{1,\delta ,\varepsilon }(t,\cdot
)X,X\right\rangle \\[2mm] 
\left\langle \mathcal{B}_{XX}^{2,\delta ,\varepsilon }(t,\cdot
)X,X\right\rangle\end{array}\right) ,~i=1,2, \\
\mathcal{B}_{XX}^{i,\delta ,\varepsilon }(t,\cdot )=& \;\mathcal{B}_{XX}^{i}(t,X^{\delta ,\varepsilon }(t),v^{\delta ,\varepsilon }(t)),~i=1,2,
\\
\Sigma _{XX}^{\delta ,\varepsilon }(t,\cdot )=& \left( 
\begin{array}{c}
\left\langle \Sigma _{XX}^{1,\delta ,\varepsilon }(t,\cdot )X,X\right\rangle
\\[2mm] 
\left\langle \Sigma _{XX}^{2,\delta ,\varepsilon }(t,\cdot )X,X\right\rangle\end{array}\right) ,~i=1,2, \\
\Sigma _{XX}^{i,\delta ,\varepsilon }(t,\cdot )=& \;\Sigma
_{XX}^{i}(t,X^{\delta ,\varepsilon }(t),v^{\delta ,\varepsilon }(t)),~i=1,2.\end{aligned}$$The following results can be seen in @W or @YZ: $$\begin{aligned}
& \sup\limits_{0\leq t\leq 1}\mathbb{E}\left\vert X_{1}^{\delta ,\varepsilon
,\alpha }(t)\right\vert ^{2k}+\sup\limits_{0\leq t\leq 1}\mathbb{E}\left\vert X^{\delta ,\varepsilon ,\alpha }(t)-X^{\delta ,\varepsilon
}(t)\right\vert ^{2k}\leq C\alpha ^{k}, \\
& \sup\limits_{0\leq t\leq 1}\mathbb{E}\left\vert X_{2}^{\delta ,\varepsilon
,\alpha }(t)\right\vert ^{2k}+\sup\limits_{0\leq t\leq 1}\mathbb{E}\left\vert X^{\delta ,\varepsilon ,\alpha }(t)-X^{\delta ,\varepsilon
}(t)-X_{1}^{\delta ,\varepsilon ,\alpha }(t)\right\vert ^{2k}\leq C\alpha
^{2k}, \\
& \sup\limits_{0\leq t\leq 1}\mathbb{E}\left\vert X^{\delta ,\varepsilon
,\alpha }(t)-X^{\delta ,\varepsilon }(t)-X_{1}^{\delta ,\varepsilon ,\alpha
}(t)-X_{2}^{\delta ,\varepsilon ,\alpha }(t)\right\vert ^{2k}=o(\alpha
^{2k}).\end{aligned}$$Now from the last relation in (\[6.26\]), we derive$$\begin{aligned}
-\sqrt{\delta }\alpha \sqrt{2+\left\vert y_{0}\right\vert ^{2}}\leq &
\;J^{\delta ,\varepsilon }(y_{0}^{\delta ,\varepsilon ,\alpha },z^{\delta
,\varepsilon ,\alpha }(\cdot ),u^{\delta ,\varepsilon ,\alpha }(\cdot
))-J^{\delta ,\varepsilon }(y_{0}^{\delta ,\varepsilon },z^{\delta
,\varepsilon }(\cdot ),u^{\delta ,\varepsilon }(\cdot )) \\
=& \frac{J^{\delta ,\varepsilon }(y_{0}^{\delta ,\varepsilon ,\alpha
},z^{\delta ,\varepsilon ,\alpha }(\cdot ),u^{\delta ,\varepsilon ,\alpha
}(\cdot ))^{2}}{J^{\delta ,\varepsilon }(y_{0}^{\delta ,\varepsilon ,\alpha
},z^{\delta ,\varepsilon ,\alpha }(\cdot ),u^{\delta ,\varepsilon ,\alpha
}(\cdot ))-J^{\delta ,\varepsilon }(y_{0}^{\delta ,\varepsilon },z^{\delta
,\varepsilon }(\cdot ),u^{\delta ,\varepsilon }(\cdot ))} \\
& -\frac{J^{\delta ,\varepsilon }(y_{0}^{\delta ,\varepsilon },z^{\delta
,\varepsilon }(\cdot ),u^{\delta ,\varepsilon }(\cdot ))^{2}}{J^{\delta
,\varepsilon }(y_{0}^{\delta ,\varepsilon ,\alpha },z^{\delta ,\varepsilon
,\alpha }(\cdot ),u^{\delta ,\varepsilon ,\alpha }(\cdot ))-J^{\delta
,\varepsilon }(y_{0}^{\delta ,\varepsilon },z^{\delta ,\varepsilon }(\cdot
),u^{\delta ,\varepsilon }(\cdot ))} \\
=& \frac{\left[ (\tilde{J}^{\varepsilon }(y_{0}^{\delta ,\varepsilon ,\alpha
},z^{\delta ,\varepsilon ,\alpha }(\cdot ),u^{\delta ,\varepsilon ,\alpha
}(\cdot ))-\tilde{J}^{\varepsilon }(\tilde{y}_{0}^{\varepsilon },\tilde{z}^{\varepsilon }(\cdot ),\tilde{u}^{\varepsilon }(\cdot ))+\delta )^{+}\right]
^{2}}{J^{\delta ,\varepsilon }(y_{0}^{\delta ,\varepsilon ,\alpha
},z^{\delta ,\varepsilon ,\alpha }(\cdot ),u^{\delta ,\varepsilon ,\alpha
}(\cdot ))-J^{\delta ,\varepsilon }(y_{0}^{\delta ,\varepsilon },z^{\delta
,\varepsilon }(\cdot ),u^{\delta ,\varepsilon }(\cdot ))} \\
& -\frac{\left[ (\tilde{J}^{\varepsilon }(y_{0}^{\delta ,\varepsilon
},z^{\delta ,\varepsilon }(\cdot ),u^{\delta ,\varepsilon }(\cdot ))-\tilde{J}^{\varepsilon }(\tilde{y}_{0}^{\varepsilon },\tilde{z}^{\varepsilon }(\cdot
),\tilde{u}^{\varepsilon }(\cdot ))+\delta )^{+}\right] ^{2}}{J^{\delta
,\varepsilon }(y_{0}^{\delta ,\varepsilon ,\alpha },z^{\delta ,\varepsilon
,\alpha }(\cdot ),u^{\delta ,\varepsilon ,\alpha }(\cdot ))-J^{\delta
,\varepsilon }(y_{0}^{\delta ,\varepsilon },z^{\delta ,\varepsilon }(\cdot
),u^{\delta ,\varepsilon }(\cdot ))} \\
& +\frac{\mathbb{E}\left[ \Pi (X^{\delta ,\varepsilon ,\alpha }(0),X^{\delta
,\varepsilon ,\alpha }(1))^{2}-\Pi (X^{\delta ,\varepsilon }(0),X^{\delta
,\varepsilon }(1))^{2}\right] }{J^{\delta ,\varepsilon }(y_{0}^{\delta
,\varepsilon ,\alpha },z^{\delta ,\varepsilon ,\alpha }(\cdot ),u^{\delta
,\varepsilon ,\alpha }(\cdot ))-J^{\delta ,\varepsilon }(y_{0}^{\delta
,\varepsilon },z^{\delta ,\varepsilon }(\cdot ),u^{\delta ,\varepsilon
}(\cdot ))} \\
& =\;\theta _{0}^{\delta ,\varepsilon ,\alpha }\left[ \tilde{J}^{\varepsilon
}(y_{0}^{\delta ,\varepsilon ,\alpha },z^{\delta ,\varepsilon ,\alpha
}(\cdot ),u^{\delta ,\varepsilon ,\alpha }(\cdot ))-\tilde{J}^{\varepsilon
}(y_{0}^{\delta ,\varepsilon },z^{\delta ,\varepsilon }(\cdot ),u^{\delta
,\varepsilon }(\cdot ))\right] \\
& +\mathbb{E}\left\langle \binom{0}{\theta _{1}^{\delta ,\varepsilon ,\alpha
}},\Pi (X^{\delta ,\varepsilon ,\alpha }(0),X^{\delta ,\varepsilon ,\alpha
}(1))-\Pi (X^{\delta ,\varepsilon }(0),X^{\delta ,\varepsilon
}(1))\right\rangle \\
& =(\theta _{0}^{\delta ,\varepsilon }+o(1))\left[ \tilde{J}^{\varepsilon
}(y_{0}^{\delta ,\varepsilon ,\alpha },z^{\delta ,\varepsilon ,\alpha
}(\cdot ),u^{\delta ,\varepsilon ,\alpha }(\cdot ))-\tilde{J}^{\varepsilon
}(y_{0}^{\delta ,\varepsilon },z^{\delta ,\varepsilon }(\cdot ),u^{\delta
,\varepsilon }(\cdot ))\right] \\
& +\mathbb{E}\left\langle \left( 
\begin{array}{c}
0 \\ 
\theta _{1}^{\delta ,\varepsilon }+o(1)\end{array}\right) ,\Pi (X^{\delta ,\varepsilon ,\alpha }(0),X^{\delta ,\varepsilon
,\alpha }(1))-\Pi (X^{\delta ,\varepsilon }(0),X^{\delta ,\varepsilon
}(1))\right\rangle ,\end{aligned}$$where$$\begin{aligned}
\theta _{0}^{\delta ,\varepsilon ,\alpha }=& \frac{2}{J^{\delta ,\varepsilon
}(y_{0}^{\delta ,\varepsilon ,\alpha },z^{\delta ,\varepsilon ,\alpha
}(\cdot ),u^{\delta ,\varepsilon ,\alpha }(\cdot ))+J^{\delta ,\varepsilon
}(y_{0}^{\delta ,\varepsilon },z^{\delta ,\varepsilon }(\cdot ),u^{\delta
,\varepsilon }(\cdot ))}  \notag  \label{6.35} \\
& \times \Bigg \{\int_{0}^{1}\bigg[\beta (\tilde{J}(y_{0}^{\delta
,\varepsilon ,\alpha },z^{\delta ,\varepsilon ,\alpha }(\cdot ),u^{\delta
,\varepsilon ,\alpha }(\cdot ))-\tilde{J}(\tilde{y}_{0}^{\varepsilon },\tilde{z}^{\varepsilon }(\cdot ),\tilde{u}^{\varepsilon }(\cdot )))  \notag
\\
& +(1-\beta )(\tilde{J}(y_{0}^{\delta ,\varepsilon },z^{\delta ,\varepsilon
}(\cdot ),u^{\delta ,\varepsilon }(\cdot ))-\tilde{J}(\tilde{y}_{0}^{\varepsilon },\tilde{z}^{\varepsilon }(\cdot ),\tilde{u}^{\varepsilon
}(\cdot )))\bigg]\mbox{\rm d}\beta +\delta \Bigg \}^{+},\end{aligned}$$$$\begin{aligned}
\theta _{1}^{\delta ,\varepsilon ,\alpha }=& \frac{y^{\delta ,\varepsilon
,\alpha }(1)-Mx^{\delta ,\varepsilon ,\alpha }(1)+y^{\delta ,\varepsilon
}(1)-Mx^{\delta ,\varepsilon }(1)}{J^{\delta ,\varepsilon }(y_{0}^{\delta
,\varepsilon ,\alpha },z^{\delta ,\varepsilon ,\alpha }(\cdot ),u^{\delta
,\varepsilon ,\alpha }(\cdot ))+J^{\delta ,\varepsilon }(y_{0}^{\delta
,\varepsilon },z^{\delta ,\varepsilon }(\cdot ),u^{\delta ,\varepsilon
}(\cdot ))},  \notag  \label{6.38} \\
\theta _{0}^{\delta ,\varepsilon }=& \frac{(\tilde{J}^{\varepsilon
}(y_{0}^{\delta ,\varepsilon },z^{\delta ,\varepsilon }(\cdot ),u^{\delta
,\varepsilon }(\cdot ))-\tilde{J}^{\varepsilon }(\tilde{y}_{0}^{\varepsilon
},\tilde{z}^{\varepsilon }(\cdot ),\tilde{u}^{\varepsilon }(\cdot ))+\delta
)^{+}}{J^{\delta ,\varepsilon }(y^{\delta ,\varepsilon }(0),z^{\delta
,\varepsilon }(\cdot ),u^{\delta ,\varepsilon }(\cdot ))}\in \left[ 0,1\right] ,  \notag \\
\theta _{1}^{\delta ,\varepsilon }=& \frac{y^{\delta ,\varepsilon
}(1)-Mx^{\delta ,\varepsilon }(1)}{J^{\delta ,\varepsilon }(y^{\delta
,\varepsilon }(0),z^{\delta ,\varepsilon }(\cdot ),u^{\delta ,\varepsilon
}(\cdot ))}\in L_{\mathcal{F}_{1}}^{2}(\Omega ;\mathbb{R}).\end{aligned}$$On the other hand, $$\begin{aligned}
& \;\theta _{0}^{\delta ,\varepsilon ,\alpha }\left[ \tilde{J}^{\varepsilon
}(y_{0}^{\delta ,\varepsilon ,\alpha },z^{\delta ,\varepsilon ,\alpha
}(\cdot ),u^{\delta ,\varepsilon ,\alpha }(\cdot ))-\tilde{J}^{\varepsilon
}(y_{0}^{\delta ,\varepsilon },z^{\delta ,\varepsilon }(\cdot ),u^{\delta
,\varepsilon }(\cdot ))\right] \\
=& \;\theta _{0}^{\delta ,\varepsilon ,\alpha }\left[ J(y_{0}^{\delta
,\varepsilon ,\alpha },z^{\delta ,\varepsilon ,\alpha }(\cdot ),u^{\delta
,\varepsilon ,\alpha }(\cdot ))+\varepsilon ^{\frac{1}{3}}d_{\mathcal{A}}(y_{0}^{\delta ,\varepsilon ,\alpha },z^{\delta ,\varepsilon }(\cdot
),u^{\delta ,\varepsilon ,\alpha }(\cdot ),\tilde{\theta}^{\varepsilon
}(\cdot ))\right. \\
& \left. -J(y_{0}^{\delta ,\varepsilon },z^{\delta ,\varepsilon }(\cdot
),u^{\delta ,\varepsilon }(\cdot ))-\varepsilon ^{\frac{1}{3}}d_{\mathcal{A}}(y_{0}^{\delta ,\varepsilon },z^{\delta ,\varepsilon }(\cdot ),u^{\delta
,\varepsilon }(\cdot ),\tilde{\theta}^{\varepsilon }(\cdot ))\right] \\
=& \;\theta _{0}^{\delta ,\varepsilon ,\alpha }\left[ J(y_{0}^{\delta
,\varepsilon ,\alpha },z^{\delta ,\varepsilon ,\alpha }(\cdot ),u^{\delta
,\varepsilon ,\alpha }(\cdot ))-J(y_{0}^{\delta ,\varepsilon },z^{\delta
,\varepsilon }(\cdot ),u^{\delta ,\varepsilon }(\cdot ))\right] \\
& +\varepsilon ^{\frac{1}{3}}\theta _{0}^{\delta ,\varepsilon ,\alpha }\left[
d_{\mathcal{A}}(y_{0}^{\delta ,\varepsilon ,\alpha },z^{\delta ,\varepsilon
,\alpha }(\cdot ),u^{\delta ,\varepsilon ,\alpha }(\cdot ),\tilde{\theta}^{\varepsilon }(\cdot ))-d_{\mathcal{A}}(y_{0}^{\delta ,\varepsilon
},z^{\delta ,\varepsilon }(\cdot ),u^{\delta ,\varepsilon }(\cdot ),\tilde{\theta}^{\varepsilon }(\cdot ))\right] \\
\leq & \;\theta _{0}^{\delta ,\varepsilon ,\alpha }\left[ J(y_{0}^{\delta
,\varepsilon ,\alpha },z^{\delta ,\varepsilon ,\alpha }(\cdot ),u^{\delta
,\varepsilon ,\alpha }(\cdot ))-J(y_{0}^{\delta ,\varepsilon },z^{\delta
,\varepsilon }(\cdot ),u^{\delta ,\varepsilon }(\cdot ))\right] +\alpha
\varepsilon ^{\frac{1}{3}}\theta _{0}^{\delta ,\varepsilon ,\alpha }\sqrt{\left\vert y_{0}\right\vert ^{2}+2},\end{aligned}$$since the triangle inequality $$\begin{aligned}
&&d_{\mathcal{A}}(y_{0}^{\delta ,\varepsilon ,\alpha },z^{\delta
,\varepsilon ,\alpha }(\cdot ),u^{\delta ,\varepsilon ,\alpha }(\cdot ),\tilde{\theta}^{\varepsilon }(\cdot ))-d_{\mathcal{A}}(y_{0}^{\delta
,\varepsilon },z^{\delta ,\varepsilon }(\cdot ),u^{\delta ,\varepsilon
}(\cdot ),\tilde{\theta}^{\varepsilon }(\cdot )) \\
&\leq &d_{\mathcal{A}}\left( \left( y_{0}^{\delta ,\varepsilon ,\alpha
},z^{\delta ,\varepsilon ,\alpha }(\cdot ),u^{\delta ,\varepsilon ,\alpha
}(\cdot )\right) ,\left( y_{0}^{\delta ,\varepsilon },z^{\delta ,\varepsilon
}(\cdot ),u^{\delta ,\varepsilon }(\cdot )\right) \right) \leq \alpha \sqrt{\left\vert y_{0}\right\vert ^{2}+2}.\end{aligned}$$Note that $$\begin{aligned}
& \;J(y_{0}^{\delta ,\varepsilon ,\alpha },z^{\delta ,\varepsilon ,\alpha
}(\cdot ),u^{\delta ,\varepsilon ,\alpha }(\cdot ))-J(y_{0}^{\delta
,\varepsilon },z^{\delta ,\varepsilon }(\cdot ),u^{\delta ,\varepsilon
}(\cdot )) \\
=& \;\mathbb{E}\left[ \int_{0}^{1}l(t,X^{\delta ,\varepsilon ,\alpha
}(t),u^{\delta ,\varepsilon ,\alpha }(t))-l(t,X^{\delta ,\varepsilon
}(t),u^{\delta ,\varepsilon }(t))\mbox{\rm d}t\right] \\
& +\mathbb{E}\left[ \Xi (X^{\delta ,\varepsilon ,\alpha }(0),X^{\delta
,\varepsilon ,\alpha }(1))-\Xi (X^{\delta ,\varepsilon }(0),X^{\delta
,\varepsilon }(1))\right] \\
=& \;I_{1}+I_{2}.\end{aligned}$$We deal with $I_{1},$ $I_{2},$ respectively. By Theorem 4.4 in @YZ, we have $$\begin{aligned}
I_{1}=& \;\mathbb{E}\bigg[\int_{0}^{1}\bigg(l(t,X^{\delta ,\varepsilon
,\alpha }(t),u^{\delta ,\varepsilon ,\alpha }(t))-l(t,X^{\delta ,\varepsilon
}(t),u^{\delta ,\varepsilon }(t))\bigg)\mbox{\rm d}t \\
& +\int_{0}^{1}l_{X}(t,X^{\delta ,\varepsilon }(t),u^{\delta ,\varepsilon
}(t))(X_{1}^{\delta ,\varepsilon ,\alpha }(t)+X_{2}^{\delta ,\varepsilon
,\alpha }(t))\mbox{\rm d}t \\
& +\int_{0}^{1}\frac{1}{2}l_{XX}(t,X^{\delta ,\varepsilon }(t),u^{\delta
,\varepsilon }(t))(X_{1}^{\delta ,\varepsilon ,\alpha }(t))^{2}\mbox{\rm d}t
\\
& +\int_{0}^{1}(l_{X}(t,X^{\delta ,\varepsilon ,\alpha }(t),u^{\delta
,\varepsilon ,\alpha }(t))-l_{X}(t,X^{\delta ,\varepsilon }(t),u^{\delta
,\varepsilon }(t)))(X^{\delta ,\varepsilon ,\alpha }(t)-X^{\delta
,\varepsilon }(t))\mbox{\rm d}t \\
& +\int_{0}^{1}(l_{X}(t,X^{\delta ,\varepsilon }(t),u^{\delta ,\varepsilon
}(t)))(X^{\delta ,\varepsilon ,\alpha }(t)-X^{\delta ,\varepsilon
}(t)-X_{1}^{\delta ,\varepsilon ,\alpha }(t)-X_{2}^{\delta ,\varepsilon
,\alpha }(t))\mbox{\rm d}t\bigg] \\
& +\mathbb{E}\bigg[\int_{0}^{1}\beta \left[ l_{XX}(t,\beta X^{\delta
,\varepsilon }(t)+(1-\beta )X^{\delta ,\varepsilon ,\alpha }(t),u^{\delta
,\varepsilon ,\alpha }(t))\right. \\
& \left. -l_{XX}(t,X^{\delta ,\varepsilon }(t),u^{\delta ,\varepsilon
,\alpha }(t))\right] (X^{\delta ,\varepsilon ,\alpha }(t)-X^{\delta
,\varepsilon }(t))^{2}\mbox{\rm d}\beta \bigg] \\
& +\frac{1}{2}\mathbb{E}\left[ \int_{0}^{1}(l_{XX}(t,X^{\delta ,\varepsilon
,\alpha }(t),u^{\delta ,\varepsilon ,\alpha }(t))-l_{XX}(t,X^{\delta
,\varepsilon }(t),u^{\delta ,\varepsilon }(t)))(X^{\delta ,\varepsilon
,\alpha }(t)-X^{\delta ,\varepsilon }(t))^{2}\right] \mbox{\rm d}t \\
& +\frac{1}{2}\mathbb{E}\bigg[\int_{0}^{1}l_{XX}(t,X^{\delta ,\varepsilon
}(t),u^{\delta ,\varepsilon }(t))(X^{\delta ,\varepsilon ,\alpha
}(t)-X^{\delta ,\varepsilon }(t)-X_{1}^{\delta ,\varepsilon ,\alpha }(t)) \\
& \times (X^{\delta ,\varepsilon ,\alpha }(t)-X^{\delta ,\varepsilon
}(t)+X_{1}^{\delta ,\varepsilon ,\alpha }(t))\mbox{\rm d}t\bigg],\end{aligned}$$and $$\begin{aligned}
I_{2}=& \;\mathbb{E}\left[ \Xi (X^{\delta ,\varepsilon ,\alpha
}(0),X^{\delta ,\varepsilon ,\alpha }(1))-\Xi (X^{\delta ,\varepsilon
}(0),X^{\delta ,\varepsilon }(1))\right] \\
=& \;\mathbb{E}\left[ \Xi _{X(0)}(X^{\delta ,\varepsilon }(0),X^{\delta
,\varepsilon }(1))(X^{\delta ,\varepsilon ,\alpha }(0)-X^{\delta
,\varepsilon }(0))\right] \\
& +\mathbb{E}\left[ \Xi _{X(1)}(X^{\delta ,\varepsilon }(0),X^{\delta
,\varepsilon }(1))(X^{\delta ,\varepsilon ,\alpha }(1)-X^{\delta
,\varepsilon }(1))\right] \\
& +\mathbb{E}\left[ \frac{1}{2}\Xi _{X(0)X(0)}(X^{\delta ,\varepsilon
,\alpha }(0)-X^{\delta ,\varepsilon }(0))^{2}\right] +\mathbb{E}\left[ \frac{1}{2}\Xi _{X(1)X(1)}(X^{\delta ,\varepsilon ,\alpha }(1)-X^{\delta
,\varepsilon }(1))^{2}\right] \\
& +\mathbb{E}\left\langle D^{2}\Xi ^{\delta ,\varepsilon ,\alpha }\left( 
\begin{array}{c}
X^{\delta ,\varepsilon ,\alpha }(0)-X^{\delta ,\varepsilon }(0) \\ 
X^{\delta ,\varepsilon ,\alpha }(1)-X^{\delta ,\varepsilon }(1)\end{array}\right) ,\left( 
\begin{array}{c}
X^{\delta ,\varepsilon ,\alpha }(0)-X^{\delta ,\varepsilon }(0) \\ 
X^{\delta ,\varepsilon ,\alpha }(1)-X^{\delta ,\varepsilon }(1)\end{array}\right) \right\rangle ,\end{aligned}$$where $$\begin{aligned}
D^{2}\Xi ^{\delta ,\varepsilon ,\alpha }=& \int_{0}^{1}[\beta D^{2}\Xi
(\beta X^{\delta ,\varepsilon }(0)+(1-\beta )X^{\delta ,\varepsilon ,\alpha
}(0),\beta X^{\delta ,\varepsilon }(1)+(1-\beta X^{\delta ,\varepsilon
,\alpha }(1))) \\
& -D^{2}\Xi (X^{\delta ,\varepsilon }(0),X^{\delta ,\varepsilon }(1))]\mbox{\rm d}\beta .\end{aligned}$$Besides, $$\begin{aligned}
& \;\mathbb{E}\left\langle \binom{0}{\theta _{1}^{\delta ,\varepsilon
,\alpha }},\Pi (X^{\delta ,\varepsilon ,\alpha }(0),X^{\delta ,\varepsilon
,\alpha }(1))-\Pi (X^{\delta ,\varepsilon }(0),X^{\delta ,\varepsilon
}(1))\right\rangle \\
=& \;\mathbb{E}\left\langle \binom{0}{\theta _{1}^{\delta ,\varepsilon
,\alpha }},\Pi (0,X^{\delta ,\varepsilon ,\alpha }(1))-\Pi (0,X^{\delta
,\varepsilon }(1))\right\rangle \\
=& \;\mathbb{E}\left\langle \Pi _{X(1)}(0,X^{\delta ,\varepsilon }(1))\binom{0}{\theta _{1}^{\delta ,\varepsilon ,\alpha }}(X^{\delta ,\varepsilon
,\alpha }(1)-X^{\delta ,\varepsilon }(1))\right\rangle \\
& +\frac{1}{2}\mathbb{E}\left[ \left\langle \Pi _{X(1)X(1)}(0,X^{\delta
,\varepsilon }(1))\binom{0}{\theta _{1}^{\delta ,\varepsilon ,\alpha }}(X^{\delta ,\varepsilon ,\alpha }(1)-X^{\delta ,\varepsilon
}(1))^{2}\right\rangle \right] \\
& +\mathbb{E}\left[ D^{2}\Pi ^{\delta ,\varepsilon ,\alpha }\binom{0}{\theta
_{1}^{\delta ,\varepsilon ,\alpha }}(X^{\delta ,\varepsilon ,\alpha
}(1)-X^{\delta ,\varepsilon }(1))^{2}\right] ,\end{aligned}$$where $$D^{2}\Pi ^{\delta ,\varepsilon ,\alpha }=\int_{0}^{1}\beta \left[ \Pi
_{X(1)X(1)}(0,\beta X^{\delta ,\varepsilon }(1)+(1-\beta )X^{\delta
,\varepsilon ,\alpha }(1))-\Pi _{X(1)X(1)}(0,X^{\delta ,\varepsilon }(1))\right] \mbox{\rm d}\beta .$$Clearly, under assumptions (H1), we have$$\begin{aligned}
& -\alpha \sqrt{\left\vert y_{0}\right\vert ^{2}+2}\left( \sqrt{\delta }+\varepsilon ^{\frac{1}{3}}\theta _{0}^{\delta ,\varepsilon ,\alpha }\right)
\\
\leq & \;\theta _{0}^{\delta ,\varepsilon ,\alpha }\left[ J(y_{0}^{\delta
,\varepsilon ,\alpha },z^{\delta ,\varepsilon ,\alpha }(\cdot ),u^{\delta
,\varepsilon ,\alpha }(\cdot ))-J(y_{0}^{\delta ,\varepsilon },z^{\delta
,\varepsilon }(\cdot ),u^{\delta ,\varepsilon }(\cdot ))\right] \\
& +\mathbb{E}\left\langle \binom{0}{\theta _{1}^{\delta ,\varepsilon ,\alpha
}},\Pi (X^{\delta ,\varepsilon ,\alpha }(0),X^{\delta ,\varepsilon ,\alpha
}(1))-\Pi (X^{\delta ,\varepsilon }(0),X^{\delta ,\varepsilon
}(1))\right\rangle \\
& =\;\theta _{0}^{\delta ,\varepsilon ,\alpha }\mathbb{E}\bigg[\int_{0}^{1}\bigg(l_{X}(t,X^{\delta ,\varepsilon }(t),u^{\delta ,\varepsilon
}(t))(X_{1}^{\delta ,\varepsilon ,\alpha }(t)+X_{2}^{\delta ,\varepsilon
,\alpha }(t)) \\
& +l(t,X^{\delta ,\varepsilon ,\alpha }(t),u^{\delta ,\varepsilon ,\alpha
}(t))-l(t,X^{\delta ,\varepsilon }(t),u^{\delta ,\varepsilon }(t))+\frac{1}{2}l_{XX}(t,X^{\delta ,\varepsilon }(t),u^{\delta ,\varepsilon
}(t))(X_{1}^{\delta ,\varepsilon ,\alpha }(t))^{2}\bigg)\mbox{\rm d}t\bigg]
\\
& +\mathbb{E}\left[ \Xi _{X(0)}(X^{\delta ,\varepsilon ,\alpha
}(0)-X^{\delta ,\varepsilon }(0))+\Xi _{X(1)}(X^{\delta ,\varepsilon ,\alpha
}(1)-X^{\delta ,\varepsilon }(1))\right] \\
& +\mathbb{E}\left[ \frac{1}{2}\Xi _{X(0)X(0)}(X^{\delta ,\varepsilon
}(t),u^{\delta ,\varepsilon }(\cdot ))(X^{\delta ,\varepsilon ,\alpha
}(0)-X^{\delta ,\varepsilon }(0))^{2}\right] \\
& +\mathbb{E}\left[ \frac{1}{2}\Xi _{X(1)X(1)}(X^{\delta ,\varepsilon
}(t),u^{\delta ,\varepsilon }(\cdot ))(X^{\delta ,\varepsilon ,\alpha
}(1)-X^{\delta ,\varepsilon }(1))^{2}\right] \\
& +\mathbb{E}\left\langle \Pi _{X(1)}(0,X^{\delta ,\varepsilon }(1))\binom{0}{\theta _{1}^{\delta ,\varepsilon ,\alpha }}(X^{\delta ,\varepsilon ,\alpha
}(1)-X^{\delta ,\varepsilon }(1))\right\rangle \\
& +\mathbb{E}\left[ \frac{1}{2}\left\langle \Pi _{X(1)X(1)}(0,X^{\delta
,\varepsilon }(1))\binom{0}{\theta _{1}^{\delta ,\varepsilon ,\alpha }}(X^{\delta ,\varepsilon ,\alpha }(1)-X^{\delta ,\varepsilon
}(1))^{2}\right\rangle \right] +o(\alpha ) \\
& =\;\mathbb{E}\Bigg \{\int_{0}^{1}\bigg \{\theta _{0}^{\delta ,\varepsilon
,\alpha }\left[ l(t,X^{\delta ,\varepsilon ,\alpha }(t),u^{\delta
,\varepsilon ,\alpha }(t))-l(t,X^{\delta ,\varepsilon }(t),u^{\delta
,\varepsilon }(t))\right] \\
& +\theta _{0}^{\delta ,\varepsilon ,\alpha }\left[ l_{X}(t,X^{\delta
,\varepsilon }(t),u^{\delta ,\varepsilon }(t))(X_{1}^{\delta ,\varepsilon
,\alpha }(t)+X_{2}^{\delta ,\varepsilon ,\alpha }(t))\right] \\
& +\frac{1}{2}\theta _{0}^{\delta ,\varepsilon ,\alpha }l_{XX}(t,X^{\delta
,\varepsilon }(t),u^{\delta ,\varepsilon }(t))\left( X_{1}^{\delta
,\varepsilon ,\alpha }\left( t\right) \right) ^{2}\bigg \}\mbox{\rm d}t\end{aligned}$$$$\begin{aligned}
&&+\sqrt{\alpha }\left\langle \theta _{0}^{\delta ,\varepsilon ,\alpha }\Xi
_{X(0)}\left( X^{\delta ,\varepsilon }\left( 0\right) ,X^{\delta
,\varepsilon }\left( 1\right) \right) ,\binom{0}{y_{0}}\right\rangle \\
&&+\frac{\alpha }{2}\left\langle \theta _{0}^{\delta ,\varepsilon ,\alpha
}\Xi _{X(0)X(0}\left( X^{\delta ,\varepsilon }\left( 0\right) ,X^{\delta
,\varepsilon }\left( 1\right) \right) \binom{0}{y_{0}},\binom{0}{y_{0}}\right\rangle \\
&&+\left( \theta _{0}^{\delta ,\varepsilon ,\alpha }\Xi _{X(1)}(X^{\delta
,\varepsilon }(0),X^{\delta ,\varepsilon }(1))+\Pi _{X(1)}(0,X^{\delta
,\varepsilon }(1))\binom{0}{\theta _{1}^{\delta ,\varepsilon ,\alpha }}\right) (X_{1}^{\delta ,\varepsilon ,\alpha }(1)+X_{2}^{\delta ,\varepsilon
,\alpha }(1)) \\
&&+\frac{1}{2}\Bigg[\theta _{0}^{\delta ,\varepsilon ,\alpha }\Xi
_{X(1)X(1)}(X^{\delta ,\varepsilon }(1),u^{\delta ,\varepsilon }(\cdot
))+\Pi _{X(1)X(1)}(0,X^{\delta ,\varepsilon }(1))\binom{0}{\theta
_{1}^{\delta ,\varepsilon ,\alpha }}\Bigg](X_{1}^{\delta ,\varepsilon
,\alpha }(1))^{2}\Bigg \}+o(\alpha ).\end{aligned}$$$$\label{6.45}$$

Let us introduce the following the first order BSDEs: $$\left\{ 
\begin{array}{rcl}
-\mbox{\rm d}\tilde{\Phi}^{\delta ,\varepsilon ,\alpha }(t) & \3n=\3n & 
\left[ \mathcal{B}_{X}^{\delta ,\varepsilon }(t,\cdot )\tilde{\Phi}^{\delta
,\varepsilon ,\alpha }(t)+\Sigma _{X}^{\delta ,\varepsilon }(t,\cdot )\tilde{\Psi}^{\delta ,\varepsilon ,\alpha }(t)+\theta _{0}^{\delta ,\varepsilon
,\alpha }l_{X}^{\delta ,\varepsilon }(t,\cdot )\right] \mbox{\rm d}t-\tilde{\Psi}^{\delta ,\varepsilon ,\alpha }(t)\mbox{\rm d}W(t), \\[2mm] 
\tilde{\Phi}^{\delta ,\varepsilon ,\alpha }(1) & \3n=\3n & \displaystyle\left[ \theta _{0}^{\delta ,\varepsilon ,\alpha }\Xi _{X(1)}(X^{\delta
,\varepsilon }(0),X^{\delta ,\varepsilon }(1))+\Pi _{X(1)}(0,X^{\delta
,\varepsilon }(1))\binom{0}{\theta _{1}^{\delta ,\varepsilon ,\alpha }}\right] ,\end{array}\right.$$where $l_{X}^{\delta ,\varepsilon }(t,\cdot )=l_{X}(t,X^{\delta ,\varepsilon
}(t),u^{\delta ,\varepsilon }(t))$.

The second order BSDEs: $$\left\{ 
\begin{array}{rcl}
-\mbox{\rm d}\tilde{P}^{\delta ,\varepsilon ,\alpha }(t) & \3n=\3n & \left[ 
\mathcal{B}_{X}^{\delta ,\varepsilon }(t,\cdot )^{\top }\tilde{P}^{\delta
,\varepsilon ,\alpha }(t)+\tilde{P}^{\delta ,\varepsilon ,\alpha }(t)\mathcal{B}_{X}^{\delta ,\varepsilon }(t,\cdot )+\Sigma _{X}^{\delta
,\varepsilon }(t,\cdot )^{\top }\tilde{P}^{\delta ,\varepsilon ,\alpha
}(t)\Sigma _{X}^{\delta ,\varepsilon }(t,\cdot )\right. \\[2mm] 
&  & \left. \Sigma _{X}^{\delta ,\varepsilon }(t,\cdot )^{\top }\tilde{Q}^{\delta ,\varepsilon ,\alpha }(t)+\tilde{Q}^{\delta ,\varepsilon ,\alpha
}(t)\Sigma _{X}^{\delta ,\varepsilon }(t,\cdot )+H_{XX}^{\delta ,\varepsilon
,\alpha }\right] \mbox{\rm d}t-\tilde{Q}^{\delta ,\varepsilon ,\alpha }(t)\text{d}W(t), \\[2mm] 
\tilde{P}^{\delta ,\varepsilon ,\alpha }(1) & \3n=\3n & \displaystyle\left[
\theta _{0}^{\delta ,\varepsilon ,\alpha }\Xi _{X(1)X(1)}(X^{\delta
,\varepsilon }(0),X^{\delta ,\varepsilon }(1))+\Pi _{X(1)X(1)}(0,X^{\delta
,\varepsilon }(1))\binom{0}{\theta _{1}^{\delta ,\varepsilon ,\alpha }}\right] ,\end{array}\right.$$where $$\tilde{H}_{XX}^{\delta ,\varepsilon ,\alpha }(t)=\tilde{H}_{XX}(t,\theta
_{0}^{\delta ,\varepsilon ,\alpha },X^{\delta ,\varepsilon ,\alpha
}(t),u^{\delta ,\varepsilon ,\alpha }(t),\tilde{\Phi}^{\delta ,\varepsilon
,\alpha }(t),\tilde{\Psi}^{\delta ,\varepsilon ,\alpha }(t)),$$with $\tilde{H}_{XX}(t,\theta _{0}^{\delta ,\varepsilon ,\alpha },X^{\delta
,\varepsilon }(t),u^{\delta ,\varepsilon }(t),\tilde{\Phi}^{\delta
,\varepsilon ,\alpha }(t),\tilde{\Psi}^{\delta ,\varepsilon ,\alpha }(t))$ is defined as follows: $$\tilde{H}(t,\theta ,X,v,p,k)=\left\langle p,\mathcal{B}(t,X,v)\right\rangle
+\left\langle k,\Sigma (t,X,v)\right\rangle +\theta l(t,X,v).$$Set $\mathcal{Y}^{\delta ,\varepsilon ,\alpha }(\cdot )=X_{1}^{\delta
,\varepsilon ,\alpha }(\cdot )X_{1}^{\delta ,\varepsilon ,\alpha }(\cdot ).$ Then, $$\left\{ 
\begin{array}{rcl}
\mbox{\rm d}\mathcal{Y}^{\delta ,\varepsilon ,\alpha }(t) & \3n=\3n & \Bigg
\{\mathcal{B}_{X}^{\delta }(t,\cdot )\mathcal{Y}^{\delta ,\varepsilon
,\alpha }(t)+\mathcal{Y}^{\delta ,\varepsilon ,\alpha }(t)\mathcal{B}_{X}^{\delta }(t,\cdot )^{\top }+\Sigma _{X}^{\delta }(t,\cdot )\mathcal{Y}^{\delta ,\varepsilon ,\alpha }(t)\Sigma _{X}^{\delta }(t,\cdot )^{\top } \\[2mm] 
&  & +\bigg[\triangle \Sigma ^{\delta }(t,\cdot )\triangle \Sigma ^{\delta
}(t,\cdot )^{\top }+\Sigma _{X}^{\delta }(t,\cdot )X_{1}^{\delta
,\varepsilon ,\alpha }(t)\triangle \Sigma ^{\delta }(t,\cdot )^{\top } \\[2mm] 
&  & +\triangle \Sigma ^{\delta }(t,\cdot )X_{1}^{\delta ,\varepsilon
,\alpha }(t)\Sigma ^{\delta }(t,\cdot )^{\top }\bigg]I_{S_{\alpha }}(t)\Bigg
\}\mbox{\rm d}t \\[2mm] 
&  & +\bigg \{\Sigma _{X}^{\delta }(t,\cdot )\mathcal{Y}^{\delta
,\varepsilon ,\alpha }(t)+\mathcal{Y}^{\delta ,\varepsilon ,\alpha
}(t)\Sigma _{X}^{\delta }(t,\cdot )^{\top } \\[2mm] 
&  & +\left[ X_{1}^{\delta ,\varepsilon ,\alpha }(t)\triangle \Sigma
^{\delta }(t,\cdot )^{\top }+\triangle \Sigma ^{\delta }(t,\cdot
)X_{1}^{\delta ,\varepsilon ,\alpha }(t)^{\top }\right] I_{S_{\alpha }}(t)\bigg \}\mbox{\rm d}W(t), \\[2mm] 
\mathcal{Y}^{\delta ,\varepsilon ,\alpha }(0) & \3n=\3n & \left( 
\begin{array}{cc}
0 & 0 \\ 
0 & \alpha y_{0}^{2}\end{array}\right) ,\end{array}\right.$$Applying Itô's formula to $\left\langle \tilde{\Phi}^{\delta ,\varepsilon
,\alpha }(\cdot ),X_{1}^{\delta ,\varepsilon ,\alpha }(\cdot )+X_{2}^{\delta
,\varepsilon ,\alpha }(\cdot )\right\rangle $ and $P^{\delta ,\varepsilon
,\alpha }(\cdot )\mathcal{Y}^{\delta ,\varepsilon ,\alpha }(\cdot )$ respectively, we have $$\begin{aligned}
& \;\mathbb{E}\left[ \left\langle \tilde{\Phi}^{\delta ,\varepsilon ,\alpha
}(1),X_{1}^{\delta ,\varepsilon ,\alpha }(1)+X_{2}^{\delta ,\varepsilon
,\alpha }(1)\right\rangle \right] -\mathbb{E}\left[ \left\langle \tilde{\Phi}^{\delta ,\varepsilon ,\alpha }(0),\binom{0}{\sqrt{\alpha }y_{0}}\right\rangle \right]  \notag  \label{6.49} \\
=& \;\mathbb{E}\bigg[\int_{0}^{1}-\left\langle \theta _{0}^{\delta
,\varepsilon ,\alpha }l_{X}^{\delta ,\alpha }(t,\cdot ),(X_{2}^{\delta
,\varepsilon ,\alpha }(t)+X_{1}^{\delta ,\varepsilon ,\alpha
}(t))\right\rangle  \notag \\
& +\left\langle \tilde{\Phi}^{\delta ,\varepsilon ,\alpha }(t),\triangle 
\mathcal{B}^{\delta ,\varepsilon }(t,\cdot )I_{S_{\alpha }}(t)+\frac{1}{2}\mathcal{B}_{XX}^{\delta ,\varepsilon }(t,\cdot )(X_{1}^{\delta ,\varepsilon
,\alpha }(t))^{2}\right\rangle  \notag \\
& +\left\langle \tilde{\Psi}^{\delta ,\varepsilon ,\alpha }(t),\triangle
\Sigma ^{\delta ,\varepsilon }(t,\cdot )X_{1}^{\delta ,\varepsilon ,\alpha
}(t)I_{S_{\alpha }}(t)+\frac{1}{2}\Sigma _{XX}^{\delta ,\varepsilon
}(t,\cdot )(X_{1}^{\delta ,\varepsilon ,\alpha }(t))^{2}\right\rangle \text{d}t\bigg]+o(\alpha ).\end{aligned}$$and $$\begin{aligned}
& \;\mathbb{E}\left[ \text{tr}\left[ P^{\delta ,\varepsilon ,\alpha }(1)\mathcal{Y}^{\delta ,\varepsilon ,\alpha }(1)\right] -\left\langle P^{\delta
,\varepsilon ,\alpha }(1)\binom{0}{\sqrt{\varepsilon }y_{0}},\binom{0}{\sqrt{\varepsilon }y_{0}}\right\rangle \right]  \notag  \label{6.50} \\
=& \;\mathbb{E}\left\{ \int_{0}^{1}\text{tr}\left[ \triangle \Sigma ^{\delta
,\varepsilon }(t,\cdot )^{\top }P^{\delta ,\varepsilon ,\alpha }(t)\triangle
\Sigma ^{\delta ,\varepsilon }(t,\cdot )-\left\langle H_{XX}^{\delta
,\varepsilon ,\alpha }(t)X_{1}^{\delta ,\varepsilon ,\alpha
}(t),X_{1}^{\delta ,\varepsilon ,\alpha }(t)\right\rangle \right] 
\mbox{\rm
d}t\right\} +o(\alpha ).\end{aligned}$$Then, from (\[6.45\]), (\[6.49\]) and (\[6.50\]), we obtain$$\begin{aligned}
&&-\alpha \sqrt{\left\vert y_{0}\right\vert ^{2}+2}\left( \sqrt{\delta }+\varepsilon ^{\frac{1}{3}}\theta _{0}^{\delta ,\varepsilon ,\alpha }\right)
\\
&\leq &\;\mathbb{E}\bigg[\int_{0}^{1}\theta _{0}^{\delta ,\varepsilon
,\alpha }\left[ l(t,X^{\delta ,\varepsilon ,\alpha }(t),u^{\delta
,\varepsilon ,\alpha }(\cdot ))-l(t,X^{\delta ,\varepsilon }(t),u^{\delta
,\varepsilon }(\cdot ))\right] \\
&&+\left\langle \tilde{\Phi}^{\delta ,\varepsilon ,\alpha }(t),\triangle 
\mathcal{B}^{\delta ,\varepsilon }(t,\cdot )\right\rangle +\left\langle 
\tilde{\Psi}^{\delta ,\varepsilon ,\alpha }(t),\triangle \Sigma ^{\delta
,\varepsilon }(t,\cdot )\right\rangle \\
&&+\frac{1}{2}\triangle \Sigma ^{\delta ,\varepsilon }(t,\cdot )^{\top
}P^{\delta ,\varepsilon ,\alpha }(t)\triangle \Sigma ^{\delta ,\varepsilon
}(t,\cdot )\mbox{\rm d}t\bigg] \\
&&+\mathbb{E}\left[ \sqrt{\alpha }\left\langle \theta _{0}^{\delta
,\varepsilon ,\alpha }\Xi _{X(0)}\left( X^{\delta ,\varepsilon }\left(
0\right) ,X^{\delta ,\varepsilon }\left( 1\right) \right) +\tilde{\Phi}^{\delta ,\varepsilon ,\alpha }(0),\binom{0}{y_{0}}\right\rangle \right] \\
&&+\mathbb{E}\left[ \frac{\alpha }{2}\left\langle \left( \theta _{0}^{\delta
,\varepsilon ,\alpha }\Xi _{X(0)X(0}\left( X^{\delta ,\varepsilon }\left(
0\right) ,X^{\delta ,\varepsilon }\left( 1\right) \right) +P^{\delta
,\varepsilon ,\alpha }(0)\right) \binom{0}{y_{0}},\binom{0}{y_{0}}\right\rangle \right] +o(\alpha ).\end{aligned}$$$$\label{6.51}$$To derive the adjoint equations, in (6.51)$,$ dividing $\sqrt{\alpha }$ and then sending $\alpha \rightarrow 0,$ followed by sending $\delta \rightarrow
0,$ we get

$$\begin{aligned}
0\leq & \;\mathbb{E}\left\langle \theta _{0}^{\varepsilon }\Xi _{X(0)}(\tilde{X}^{\varepsilon }(0),\tilde{X}^{\varepsilon }(1))+\tilde{\Phi}^{\varepsilon }(0),\binom{0}{y_{0}}\right\rangle  \notag \\
=& \;\mathbb{E}\left\langle \theta _{0}^{\varepsilon }\binom{0}{\gamma _{y}(\tilde{y}_{0}^{\varepsilon })}+\tilde{\Phi}^{\varepsilon }(0),\binom{0}{y_{0}}\right\rangle .  \label{6.52}\end{aligned}$$

From continuous dependence of the solution of BSDEs on parameters $\left(
\theta _{0}^{\delta ,\varepsilon ,\alpha },\theta _{1}^{\delta ,\varepsilon
,\alpha }\right) $, we get $$\begin{aligned}
\left( \theta _{0}^{\delta ,\varepsilon ,\alpha },\theta _{1}^{\delta
,\varepsilon ,\alpha }\right) &\rightarrow &\left( \theta _{0}^{\varepsilon
},\theta _{1}^{\varepsilon }\right) \in \mathbb{R}\times L_{\mathcal{F}_{1}}^{2}(\Omega ;\mathbb{R}),\text{ weakly,} \\
\left( \tilde{\Phi}^{\delta ,\varepsilon ,\alpha }(\cdot ),\tilde{\Psi}^{\delta ,\varepsilon ,\alpha }(\cdot )\right) &\rightarrow &\left( \tilde{\Phi}^{\varepsilon }(\cdot ),\tilde{\Psi}^{\varepsilon }(\cdot )\right) ,\text{ in }\mathcal{M}^{2}(0,1;\mathbb{R}), \\
\left( \tilde{P}^{\delta ,\varepsilon ,\alpha }(\cdot ),\tilde{Q}^{\delta
,\varepsilon ,\alpha }(\cdot )\right) &\rightarrow &\left( \tilde{P}^{\varepsilon }(\cdot ),\tilde{Q}^{\varepsilon }(\cdot )\right) ,\text{ in }\mathcal{M}^{2}(0,1;\mathbb{R}),\text{ as }\delta \rightarrow 0,\alpha
\rightarrow 0.\end{aligned}$$Denote $$\tilde{\Phi}^{\varepsilon }(\cdot )=\binom{\tilde{p}^{\varepsilon }(\cdot )}{\tilde{q}^{\varepsilon }(\cdot )},\quad \tilde{\Psi}^{\varepsilon }(\cdot )=\binom{\tilde{k}^{\varepsilon }(\cdot )}{\tilde{h}^{\varepsilon }(\cdot )}.$$Then, from (\[6.52\]), we derive that $$\binom{\tilde{p}^{\varepsilon }(0)}{\tilde{q}^{\varepsilon }(0)}=\binom{0}{-\theta _{0}^{\varepsilon }\mathbb{E}\gamma _{y}(\tilde{y}^{\varepsilon }(1))}.\quad  \label{A18}$$Note that$$\left\{ 
\begin{array}{ccc}
\displaystyle\Xi _{X(1)}(\tilde{X}^{\varepsilon }(0),\tilde{X}^{\varepsilon
}(1)) & \3n=\3n & \displaystyle\binom{\phi _{x}(\tilde{x}^{\varepsilon }(1))}{0}, \\ 
\displaystyle\Xi _{X(1)X(1)}(\tilde{X}^{\varepsilon }(0),\tilde{X}^{\varepsilon }(1)) & \3n=\3n & \left( 
\begin{array}{cc}
\phi _{xx}(\tilde{x}^{\varepsilon }(1)) & 0 \\ 
0 & 0\end{array}\right) ,\end{array}\right.$$ and $$\left\{ 
\begin{array}{ccc}
\displaystyle\Pi _{X(1)}(0,\tilde{X}^{\varepsilon }(1))\binom{0}{\theta
_{1}^{\varepsilon }} & \3n=\3n & \displaystyle\binom{-M\theta
_{1}^{\varepsilon }}{\theta _{1}^{\varepsilon }}, \\ 
\displaystyle\Pi _{X(1)X(1)}(0,\tilde{X}^{\varepsilon }(1))\binom{0}{\theta
_{1}^{\varepsilon }} & \3n=\3n & \left( 
\begin{array}{cc}
0 & 0 \\ 
0 & 0\end{array}\right) .\end{array}\right.$$Next, for the first and second order BSDEs for Problem $(\tilde{C}^{\varepsilon })$, we have$$\binom{\tilde{p}^{\varepsilon }(1)}{\tilde{q}^{\varepsilon }(1)}=\binom{\theta _{0}^{\varepsilon }\phi _{x}(\tilde{x}^{\varepsilon }(1))-M\theta
_{1}^{\varepsilon }}{\theta _{1}^{\varepsilon }},  \label{A19}$$and $$\begin{aligned}
\tilde{P}^{\varepsilon }(1)=& \left[ \theta _{0}^{\varepsilon }\Xi
_{X(1)X(1)}(\tilde{X}^{\varepsilon }(0),\tilde{X}^{\varepsilon }(1))+\Pi
_{X(1)X(1)}(0,\tilde{X}^{\varepsilon }(1))\binom{0}{\theta _{1}^{\varepsilon
}}\right]  \notag \\
=& \left( 
\begin{array}{cc}
\theta _{0}^{\varepsilon }\phi _{xx}(\tilde{x}^{\varepsilon }(1)) & 0 \\ 
0 & 0\end{array}\right) .  \label{A20}\end{aligned}$$where the first and second order BSDEs are $$\left\{ 
\begin{array}{rcl}
-\mbox{\rm d}\tilde{\Phi}^{\varepsilon }(t) & \3n=\3n & \left[ \mathcal{B}_{X}(t,\cdot )\tilde{\Phi}^{\varepsilon }(t)+\Sigma _{X}(t,\cdot )\tilde{\Psi}^{\varepsilon }(t)+\theta _{0}^{\varepsilon }l_{X}(t,\cdot )\right] 
\mbox{\rm
d}t \\[2mm] 
&  & -\tilde{\Psi}^{\varepsilon }(t)\mbox{\rm d}W(t), \\[2mm] 
\tilde{\Phi}^{\varepsilon }(1) & \3n=\3n & \displaystyle\left[ \theta
_{0}^{\varepsilon }\Xi _{X(1)}(\tilde{X}^{\varepsilon }(0),\tilde{X}^{\varepsilon }(1))+\Pi _{X(1)}(0,\tilde{X}^{\varepsilon }(1))\binom{0}{\theta _{1}^{\varepsilon }}\right] ,\end{array}\right.$$and $$\left\{ 
\begin{array}{rcl}
-\mbox{\rm d}\tilde{P}^{\varepsilon }(t) & \3n=\3n & \left[ \mathcal{B}_{X}(t,\cdot )^{\top }\tilde{P}^{\varepsilon }(t)+\tilde{P}^{\varepsilon }(t)\mathcal{B}_{X}(t,\cdot )+\Sigma _{X}(t,\cdot )^{\top }\tilde{P}^{\varepsilon }(t)\Sigma _{X}(t,\cdot )\right. \\[2mm] 
&  & \left. \Sigma _{X}(t,\cdot )^{\top }\tilde{Q}^{\varepsilon }(t)+\tilde{Q}^{\varepsilon }(t)\Sigma _{X}(t,\cdot )+H_{XX}\right] \mbox{\rm d}t-\tilde{Q}^{\varepsilon }(t)\mbox{\rm d}W(t), \\[2mm] 
\tilde{P}^{\varepsilon }(1) & \3n=\3n & \displaystyle\left[ \theta
_{0}^{\varepsilon }\Xi _{X(1)X(1)}(\tilde{X}^{\varepsilon }(0),\tilde{X}^{\varepsilon }(1))+\Pi _{X(1)X(1)}(0,\tilde{X}^{\varepsilon }(1))\binom{0}{\theta _{1}^{\varepsilon }}\right] .\end{array}\right.$$Then using a standard argument of @YZ, taking $y_{0}=0,$ we have the following variational inequality: $$\begin{aligned}
-\sqrt{2}\varepsilon ^{\frac{1}{3}}\theta _{0}^{\varepsilon }& \leq \theta
_{0}^{\varepsilon }\left[ l(t,\tilde{x}^{\varepsilon }(t),\tilde{y}^{\varepsilon }(t),u)-l(t,\tilde{x}^{\varepsilon }(t),\tilde{y}^{\varepsilon
}(t),\tilde{u}^{\varepsilon }(t))\right]  \notag \\
& +\left\langle \tilde{\Phi}^{\varepsilon }(t),\mathcal{B}(t,\tilde{x}^{\varepsilon }(t),\tilde{y}^{\varepsilon }(t),u)-\mathcal{B}(t,\tilde{x}^{\varepsilon }(t),\tilde{y}^{\varepsilon }(t),\tilde{u}^{\varepsilon
}(t))\right\rangle  \notag \\
& +\left\langle \tilde{\Psi}^{\varepsilon }(t),\Sigma (t,\tilde{x}^{\varepsilon }(t),\tilde{y}^{\varepsilon }(t),u,z)-\Sigma (t,\tilde{x}^{\varepsilon }(t),\tilde{y}^{\varepsilon }(t),\tilde{u}^{\varepsilon }(t),\tilde{z}^{\varepsilon }\left( t\right) )\right\rangle  \notag \\
& +\frac{1}{2}(\Sigma (t,\tilde{x}^{\varepsilon }(t),\tilde{y}^{\varepsilon
}(t),u,z)-\Sigma (t,\tilde{x}^{\varepsilon }(t),\tilde{y}^{\varepsilon }(t),\tilde{u}^{\varepsilon }(t),\tilde{z}^{\varepsilon }\left( t\right) ))^{\top
}  \notag \\
& \times \tilde{P}^{\varepsilon }(t)(\Sigma (t,\tilde{x}^{\varepsilon }(t),\tilde{y}^{\varepsilon }(t),u)-\Sigma (t,\tilde{x}^{\varepsilon }(t),\tilde{y}^{\varepsilon }(t),\tilde{u}^{\varepsilon }(t)))+o(\alpha ),\text{ }  \notag
\\
\forall u& \in \mathbb{U},\text{ }\forall z\in \mathbb{N},\text{ }u\in 
\mathbb{U}\text{, .a.e, a.s..}  \label{6.53}\end{aligned}$$Then (\[6.53\]) can be rewrote as $$\begin{aligned}
-\sqrt{2}\varepsilon ^{\frac{1}{3}}\theta _{0}^{\varepsilon } &<&\theta
_{0}^{\varepsilon }\left[ l(t,\tilde{x}^{\varepsilon }(t),\tilde{y}^{\varepsilon }(t),u)-l(t,\tilde{x}^{\varepsilon }(t),\tilde{y}^{\varepsilon
}(t),\tilde{u}^{\varepsilon }(t))\right]  \notag \\
&&+\left\langle \tilde{p}^{\varepsilon }(t),B(t)(u-\tilde{u}^{\varepsilon
}(t))\right\rangle +\left\langle \tilde{k}^{\varepsilon }(t),D(t)(u-\tilde{u}^{\varepsilon }(t))\right\rangle  \notag \\
&&+\left\langle \tilde{h}^{\varepsilon }(t),z-\tilde{z}^{\varepsilon }\left(
t\right) \right\rangle -\left\langle \tilde{q}^{\varepsilon }(t),c(t)(u-\tilde{u}^{\varepsilon }(t))\right\rangle +\frac{1}{2}D^{2}(t)(u-\tilde{u}^{\varepsilon }(t))^{2}\tilde{P}_{1}^{\varepsilon }(t)  \notag \\
&&+\frac{1}{2}\binom{D(t)(u-\tilde{u}^{\varepsilon }(t)}{z-\tilde{z}^{\varepsilon }\left( t\right) }^{\top }\tilde{P}^{\varepsilon }\left(
t\right) \binom{D(t)(u-\tilde{u}^{\varepsilon }(t)}{z-\tilde{z}^{\varepsilon
}\left( t\right) },\text{ }  \notag \\
\forall z &\in &\mathbb{N},\text{ }u\in \mathbb{U}\text{, a.e, a.s..}
\label{A22}\end{aligned}$$Taking $u\left( t\right) =\tilde{u}^{\varepsilon }\left( t\right) ,$ $z\left( t\right) =\tilde{z}^{\varepsilon }\left( t\right) +\epsilon z_{0},$ $\forall z_{0}\in \mathbb{N}$, then dividing by sending $\epsilon \rightarrow
0,$ we have$$-\sqrt{2}\varepsilon ^{\frac{1}{3}}\theta _{0}^{\varepsilon }\leq
\left\langle \tilde{h}^{\varepsilon }(t),z_{0}\right\rangle .$$Hence, we derive that $\tilde{h}^{\varepsilon }\left( t\right) \equiv 0$ since $\theta _{0}^{\varepsilon }\geq 0.$ From (\[A18\])-(\[A20\]) we get $$\left\{ 
\begin{array}{rcl}
-\mbox{\rm d}\tilde{p}^{\varepsilon }(t) & \3n=\3n & \left[ A(t)\tilde{p}^{\varepsilon }(t)-a(t)\tilde{q}^{\varepsilon }(t)+C(t)\tilde{k}^{\varepsilon }(t)+\theta _{0}^{\varepsilon }l_{x}(t,\cdot )\right] \text{d}t-\tilde{k}^{\varepsilon }(t)\mbox{\rm d}W(t), \\[2mm] 
\text{ \ }\mbox{\rm d}\tilde{q}^{\varepsilon }(t) & \3n=\3n & \left[ -b(t)\tilde{q}^{\varepsilon }(t)-\theta _{0}^{\varepsilon }l_{y}(t,\cdot )\right] \mbox{\rm d}t, \\[2mm] 
\text{ \ }\tilde{p}^{\varepsilon }(1) & \3n=\3n & \theta _{0}^{\varepsilon
}\phi _{x}(x^{\varepsilon }(1))-M\theta _{1}^{\varepsilon },\text{ }\tilde{q}(0)=-\theta _{0}^{\varepsilon }\mathbb{\gamma }_{y}(\tilde{y}^{\varepsilon
}(0)),\text{ }\tilde{q}^{\varepsilon }(1)=\theta _{1}^{\varepsilon },\end{array}\right.  \label{6.531}$$and $$\left\{ 
\begin{array}{rcl}
-\mbox{\rm d}\tilde{P}^{\varepsilon }(t) & \3n=\3n & \left[ \mathcal{B}_{X}(t,\cdot )^{\top }\tilde{P}^{\varepsilon }(t)+\tilde{P}^{\varepsilon }(t)\mathcal{B}_{X}(t,\cdot )+\Sigma _{X}(t,\cdot )^{\top }\tilde{P}^{\varepsilon }(t)\Sigma _{X}(t,\cdot )\right. \\[2mm] 
&  & \left. +\Sigma _{X}(t,\cdot )^{\top }\tilde{Q}^{\varepsilon }(t)+\tilde{Q}^{\varepsilon }(t)\Sigma _{X}(t,\cdot )+H_{XX}(t,\cdot )\right] 
\mbox{\rm
d}t-\tilde{Q}^{\varepsilon }(t)\mbox{\rm d}W(t), \\[2mm] 
\text{ \ }\tilde{P}^{\varepsilon }(1) & \3n=\3n & \left( 
\begin{array}{cc}
\theta _{0}^{\varepsilon }\phi _{xx}(\tilde{x}^{\varepsilon }(1)) & 0 \\ 
0 & 0\end{array}\right) ,\end{array}\right.  \label{6.532}$$where $$H_{XX}(t,\cdot )=H_{XX}(t,\tilde{x}^{\varepsilon }(t),\tilde{y}^{\varepsilon
}(t),\tilde{u}^{\varepsilon }(t),\tilde{p}^{\varepsilon }(t),\tilde{q}^{\varepsilon }(t),\tilde{k}^{\varepsilon }(t),\theta _{0}^{\varepsilon }),$$and the Hamiltonian function $H:\left[ 0,T\right] \times \mathbb{R}\times 
\mathbb{R}\times \mathbb{U}\times \mathbb{R}\times \mathbb{R}\times \mathbb{R\times R\rightarrow R}$ is defined as follows:$$\begin{aligned}
H(t,x,y,u,p,q,k,\theta )\triangleq & \left\langle p,A(t)x+B(t)u\right\rangle
-\left\langle q,a(t)x+b(t)y+c(t)u\right\rangle \\
& +\left\langle k,C(t)x+D(t)u\right\rangle +\theta l(t,x,y,u).\end{aligned}$$Taking $y_{0}=0$ and $z\left( t\right) =\tilde{z}^{\varepsilon }(t)$ in (<span>A22</span>)$,$ we have the following variational inequality:$$\begin{aligned}
&&\left\langle \tilde{p}^{\varepsilon }(t),B(t)(u-\tilde{u}^{\varepsilon
}(t))\right\rangle +\left\langle \tilde{k}^{\varepsilon }(t),D(t)(u-\tilde{u}^{\varepsilon }(t))\right\rangle  \notag \\
&&-\left\langle \tilde{q}^{\varepsilon }(t),c(t)(u-\tilde{u}^{\varepsilon
}(t))\right\rangle  \notag \\
&&+\theta _{0}^{\varepsilon }\left[ l(t,\tilde{x}^{\varepsilon }(t),\tilde{y}^{\varepsilon }(t),u)-l(t,\tilde{x}^{\varepsilon }(t),\tilde{y}^{\varepsilon
}(t),\tilde{u}^{\varepsilon }(t))\right]  \notag \\
&&+\frac{1}{2}D^{2}(t)(u-\tilde{u}^{\varepsilon }(t))^{2}\tilde{P}_{1}^{\varepsilon }(t)  \notag \\
&\geq &-\sqrt{2}\varepsilon ^{\frac{1}{3}}\theta _{0}^{\varepsilon },\text{
a.e, a.s.}.  \label{6.54}\end{aligned}$$Now consider (\[6.531\])-(\[6.532\]) again but only $(\tilde{x}^{\varepsilon }(\cdot ),\tilde{y}^{\varepsilon }(\cdot ),\tilde{z}^{\varepsilon }(\cdot ),\tilde{u}^{\varepsilon }(\cdot ))$ replaced by $(x^{\varepsilon }(\cdot ),y^{\varepsilon }(\cdot ),z^{\varepsilon }(\cdot
),u^{\varepsilon }(\cdot )).$ We need to derive an estimate for the term similar to the right hand side of (\[6.54\]) with all $$(\tilde{x}^{\varepsilon }(\cdot ),\tilde{y}^{\varepsilon }(\cdot ),\tilde{z}^{\varepsilon }(\cdot ),\tilde{u}^{\varepsilon }(\cdot ))$$ replaced by $(x^{\varepsilon }(\cdot ),y^{\varepsilon }(\cdot
),z^{\varepsilon }(\cdot ),u^{\varepsilon }(\cdot )).$ To this end, we first estimate the following difference:$$\begin{aligned}
& \;\mathbb{E}\left[ \int_{0}^{1}D(t)\left[ (u-\tilde{u}^{\varepsilon }(t))\tilde{k}^{\varepsilon }(t)\right] \mbox{\rm d}t\right] -\mathbb{E}\left[
\int_{0}^{1}D(t)\left[ (u-u^{\varepsilon }(t))k^{\varepsilon }(t)\right] \mbox{\rm d}t\right] \\
=& \;\mathbb{E}\left[ \int_{0}^{1}D(t)(u-\tilde{u}^{\varepsilon }(t))(\tilde{k}^{\varepsilon }(t)-k^{\varepsilon }(t))\mbox{\rm d}t\right] +\mathbb{E}\left[ \int_{0}^{1}D(t)(u^{\varepsilon }(t)-\tilde{u}^{\varepsilon
}(t))k^{\varepsilon }(t)\mbox{\rm d}t\right]\end{aligned}$$with $$\begin{aligned}
I_{1}=& \;\mathbb{E}\left[ \int_{0}^{1}D(t)(u-\tilde{u}^{\varepsilon }(t))(\tilde{k}^{\varepsilon }(t)-k^{\varepsilon }(t))\mbox{\rm d}t\right] , \\
I_{2}=& \;\mathbb{E}\left[ \int_{0}^{1}D(t)(u^{\varepsilon }(t)-\tilde{u}^{\varepsilon }(t))k^{\varepsilon }(t)\mbox{\rm d}t\right] .\end{aligned}$$Due to Lemma \[lem:16\], for any $1<\tau <2$ and $0<\beta <1$ satisfying $(1+\beta )\tau <2,$ there is a constant $C>0$ such that $$\begin{aligned}
I_{1}\leq & \;\left( \mathbb{E}\int_{0}^{1}\left\vert \tilde{k}^{\varepsilon
}(t)-k^{\varepsilon }(t)\right\vert ^{\tau }\mbox{\rm d}t\right) ^{\frac{1}{\tau }}\times \left( \mathbb{E}\int_{0}^{1}\left\vert u-\tilde{u}^{\varepsilon }(t)\right\vert ^{\frac{\tau }{\tau -1}}\mbox{\rm
d}t\right) ^{\frac{\tau -1}{\tau }} \\
\leq & \;C\left( d(u^{\varepsilon }(t)-\tilde{u}^{\varepsilon }(t))^{\frac{\tau \beta }{2}}\right) ^{\frac{1}{\tau }}\times \left( \mathbb{E}\int_{0}^{1}(\left\vert u\right\vert ^{\frac{\tau }{\tau -1}}+\left\vert 
\tilde{u}^{\varepsilon }(t)\right\vert ^{\frac{\tau }{\tau -1}})\mbox{\rm
d}t\right) ^{\frac{\tau -1}{\tau }} \\
\leq & \;C\varepsilon ^{\frac{\beta }{3}},\end{aligned}$$and $$\begin{aligned}
I_{2}& \leq \;C\left( \mathbb{E}\int_{0}^{1}\left\vert k^{\varepsilon
}(t)\right\vert ^{2}\mbox{\rm d}t\right) ^{\frac{1}{2}}\left( \mathbb{E}\int_{0}^{1}\left\vert u^{\varepsilon }(t)-\tilde{u}^{\varepsilon
}(t)\right\vert ^{^{2}}I_{u^{\varepsilon }(t)\neq \tilde{u}^{\varepsilon
}(t)}(t)\mbox{\rm d}t\right) ^{\frac{1}{2}} \\
& \leq \;C\left( \mathbb{E}\int_{0}^{1}\left\vert u^{\varepsilon }(t)-\tilde{u}^{\varepsilon }(t)\right\vert ^{^{4}}\mbox{\rm d}t\right) ^{\frac{1}{4}}\left( \mathbb{E}\int_{0}^{1}I_{u^{\varepsilon }(t)\neq \tilde{u}^{\varepsilon }(t)}(t)\mbox{\rm d}t\right) ^{\frac{1}{4}} \\
& \leq \;C\left( \mathbb{E}\int_{0}^{1}\left\vert u^{\varepsilon
}(t)\right\vert ^{4}+\left\vert \tilde{u}^{\varepsilon }(t)\right\vert
^{^{4}}\mbox{\rm d}t\right) ^{\frac{1}{4}}\left( \mathbb{E}\int_{0}^{1}I_{u^{\varepsilon }(t)\neq \tilde{u}^{\varepsilon }(t)}(t)\mbox{\rm d}t\right) ^{\frac{1}{4}} \\
& \leq \;C\left( \mathbb{E}\int_{0}^{1}\left\vert u^{\varepsilon
}(t)\right\vert ^{4}+\left\vert \tilde{u}^{\varepsilon }(t)\right\vert
^{^{4}}\mbox{\rm d}t\right) ^{\frac{1}{4}}\left( \mathbb{E}\int_{0}^{1}I_{u^{\varepsilon }(t)\neq \tilde{u}^{\varepsilon }(t)}(t)\mbox{\rm d}t\right) ^{\frac{1}{4}} \\
& \leq \;C\left( \mathbb{E}\int_{0}^{1}\left\vert u^{\varepsilon
}(t)\right\vert ^{4}+\left\vert \tilde{u}^{\varepsilon }(t)\right\vert
^{^{4}}\mbox{\rm d}t\right) ^{\frac{1}{4}}\left( \mathbb{E}\int_{0}^{1}I_{u^{\varepsilon }(t)\neq \tilde{u}^{\varepsilon }(t)}(t)\mbox{\rm d}t\right) ^{\frac{1}{4}} \\
& \leq \;Cd(u^{\varepsilon }(\cdot ),\tilde{u}^{\varepsilon }(\cdot ))^{\frac{1}{4}} \\
& \leq \;C\varepsilon ^{\frac{1}{6}} \\
& \leq C\varepsilon ^{\frac{\beta }{3}}.\end{aligned}$$Similarly, $$\begin{aligned}
& \int_{0}^{1}\Big(\left\langle \tilde{p}^{\varepsilon }(t),B(t)(u-\tilde{u}^{\varepsilon }(t))\right\rangle -\left\langle p^{\varepsilon
}(t),B(t)(u-u^{\varepsilon }(t))\right\rangle \\
& +\left\langle q^{\varepsilon }(t),c(t)(u-u^{\varepsilon }(t))\right\rangle
-\left\langle \tilde{q}^{\varepsilon }(t),c(t)(u-\tilde{u}^{\varepsilon
}(t))\right\rangle \\
& +l(t,\tilde{x}^{\varepsilon }(t),\tilde{y}^{\varepsilon }(t),u)-l(t,\tilde{x}^{\varepsilon }(t),\tilde{y}^{\varepsilon }(t),\tilde{u}^{\varepsilon }(t))
\\
& -l(t,x^{\varepsilon }(t),y^{\varepsilon }(t),u)+l(t,x^{\varepsilon
}(t),y^{\varepsilon }(t),u^{\varepsilon }(t)) \\
& +\frac{1}{2}D^{2}(t)(u-\tilde{u}^{\varepsilon }(t))^{2}\tilde{P}_{1}^{\varepsilon }(t)-\frac{1}{2}D^{2}(t)(u-u^{\varepsilon
}(t))^{2}P_{1}^{\varepsilon }(t)\Big)\mbox{\rm d}t \\
\leq & \;C\varepsilon ^{\frac{\beta }{3}}.\end{aligned}$$Therefore, we get the first result on bounded control domains$$\begin{aligned}
&&\left\langle p^{\varepsilon }(t),B(t)(u-u^{\varepsilon }(t))\right\rangle
+\left\langle k^{\varepsilon }(t),D(t)(u-u^{\varepsilon }(t))\right\rangle \\
&&-\left\langle q^{\varepsilon }(t),c(t)(u-\tilde{u}^{\varepsilon
}(t))\right\rangle +\frac{1}{2}D^{2}(t)(u-u^{\varepsilon
}(t))^{2}P_{1}^{\varepsilon }(t) \\
&&+\theta _{0}^{\varepsilon }\left[ l(t,x^{\varepsilon }(t),y^{\varepsilon
}(t),u)-l(t,x^{\varepsilon }(t),y^{\varepsilon }(t),u^{\varepsilon }(t))\right] \\
&\geq &-C\varepsilon ^{\beta }\theta _{0}^{\varepsilon },\text{ }\forall
u\in \mathbb{U},\text{ a.e, a.s.}.\end{aligned}$$

**Step 2. (The general case of control domains)**

For every $K=1,2,\cdots ,$ set $$\begin{aligned}
\mathbb{M}^{K} &\triangleq &\left\{ \left. y_{0}\in \mathbb{R}\right\vert
\left\vert y_{0}\right\vert \leq \left\vert \tilde{y}_{0}^{\varepsilon
}\right\vert +K\right\} , \\
\mathbb{N}^{K} &\triangleq &\left\{ \left. z\left( t\right) \in \mathbb{R}\right\vert \left\vert z\left( t\right) \right\vert \leq \left\vert \tilde{z}^{\varepsilon }\left( t\right) \right\vert +K\right\} , \\
\mathcal{M}^{2}(0,1;\mathbb{N}^{K}) &\triangleq &\left\{ \left. z\left(
\cdot \right) \in \mathcal{M}^{2}(0,1;\mathbb{R})\right\vert z\left(
t\right) \in \mathbb{N}^{K}\right\} .\end{aligned}$$

Clearly, $\mathbb{M}^{K}$ is convex and $\tilde{y}_{0}^{\varepsilon }\in \mathbb{M}^{K}\subseteq \mathbb{M}^{K+1},$ $\mathbb{R}=\cup _{n=1}^{\infty }\mathbb{M}^{K}.$ $\tilde{z}^{\varepsilon }(\cdot )\in 
\mathcal{M}^{2}(0,1;\mathbb{N}^{K})\subseteq \mathcal{M}^{2}(0,1;\mathbb{N}^{K+1}),$ and $\mathcal{M}^{2}(0,1;\mathbb{R})=\cup _{K=1}^{\infty }\mathcal{M}^{2}(0,1;\mathbb{N}^{K}).$ Note that $(\tilde{y}_{0}^{\varepsilon },\tilde{z}^{\varepsilon }(\cdot ),\tilde{u}^{\varepsilon }(\cdot ))$ is still an optimal 3-triple of Problem $(\tilde{C}^{\varepsilon })$ with new functional (\[6.22\]) when the original admissible control set is replaced by $\mathbb{M}^{K}\times \mathcal{M}^{2}(0,1;\mathbb{N}^{K})\times \mathcal{U}_{ad}\left[ 0,1\right] ,$ $K=1,2,\cdots .$ Moreover, (\[6.18\]) also holds for fixed $\varepsilon >0$ on $\mathbb{M}^{K}\times \mathcal{M}^{2}(0,1;\mathbb{N}^{K})\times \mathcal{U}_{ad}\left[ 0,1\right] $ for every $K=1,2,\cdots .$ Then there exists a subsequence $$\left( \theta _{0}^{\varepsilon ,K},\theta _{1}^{\varepsilon
,K},p^{\varepsilon ,K}(\cdot ),q^{\varepsilon ,K}(\cdot ),k^{\varepsilon
,K}(\cdot ),P^{\varepsilon ,K}(\cdot ),Q^{\varepsilon ,K}\left( \cdot
\right) \right)$$satisfying $\left\vert \theta _{0}^{\varepsilon ,K}\right\vert ^{2}+\mathbb{E}\left\vert \theta _{1}^{\varepsilon ,K}\right\vert ^{2}=1$, $\theta
_{0}^{\varepsilon ,K}\geq 0,$ (\[6.531\])-(\[6.532\]) such that the following$$\begin{aligned}
&&\left\langle p^{\varepsilon ,K}(t),B(t)(u-u^{\varepsilon
}(t))\right\rangle +\left\langle k^{\varepsilon ,K}(t),D(t)(u-u^{\varepsilon
}(t))\right\rangle  \notag \\
&&-\left\langle q^{\varepsilon ,K}(t),c(t)(u-\tilde{u}^{\varepsilon
}(t))\right\rangle +\frac{1}{2}D^{2}(t)(u-u^{\varepsilon
}(t))^{2}P_{1}^{\varepsilon ,K}(t)  \notag \\
&&+\theta _{0}^{\varepsilon ,K}\left[ l(t,x^{\varepsilon
,K}(t),y^{\varepsilon ,K}(t),u)-l(t,x^{\varepsilon ,K}(t),y^{\varepsilon
,K}(t),u^{\varepsilon }(t))\right]  \notag \\
&\geq &-C\varepsilon ^{\beta }\theta _{0}^{\varepsilon ,K},\text{ }\forall
u\in \mathbb{U},\text{ a.e, a.s.}.  \label{6.55}\end{aligned}$$holds. Since $\left\vert \theta _{0}^{\varepsilon ,K}\right\vert ^{2}+\mathbb{E}\left\vert \theta _{1}^{\varepsilon ,K}\right\vert ^{2}=1,$ there is a subsequence also denoted by $\left( \theta _{0}^{\varepsilon ,K},\theta
_{1}^{\varepsilon ,K}\right) ,$ such that $\left( \theta _{0}^{\varepsilon
,K},\theta _{1}^{\varepsilon ,K}\right) \rightarrow \left( \theta
_{0}^{\varepsilon },\theta _{1}^{\varepsilon }\right) ,$ $\theta
_{0}^{\varepsilon }\geq 0.$ Hence, from continuous dependence of the solution of BSDEs on parameters (see Yong and Zhou @YZ), we have $$\left( p^{\varepsilon ,K}(\cdot ),q^{\varepsilon ,K}(\cdot ),k^{\varepsilon
,K}(\cdot ),P^{\varepsilon ,K}(\cdot ),Q^{\varepsilon ,K}\left( \cdot
\right) \right) \rightarrow \left( p^{\varepsilon }(\cdot ),q^{\varepsilon
}(\cdot ),k^{\varepsilon }(\cdot ),P^{\varepsilon }(\cdot ),Q^{\varepsilon
}\left( \cdot \right) \right)$$in $\mathcal{M}^{2}(0,1;\mathbb{R})$ as $K\rightarrow +\infty .$ Moreover, $\left( p^{\varepsilon }(\cdot ),q^{\varepsilon }(\cdot ),k^{\varepsilon
}(\cdot ),P^{\varepsilon }(\cdot ),Q^{\varepsilon }\left( \cdot \right)
\right) $ satisfies (\[6.531\])-(\[6.532\]). We have$$\begin{aligned}
&&\left\langle p^{\varepsilon }(t),B(t)(u-u^{\varepsilon }(t))\right\rangle
+\left\langle k^{\varepsilon }(t),D(t)(u-u^{\varepsilon }(t))\right\rangle 
\notag \\
&&-\left\langle q^{\varepsilon }(t),c(t)(u-u^{\varepsilon }(t))\right\rangle
+\frac{1}{2}D^{2}(t)(u-u^{\varepsilon }(t))^{2}P_{1}^{\varepsilon }(t) 
\notag \\
&&+\theta _{0}^{\varepsilon }\left[ l(t,x^{\varepsilon }(t),y^{\varepsilon
}(t),u)-l(t,x^{\varepsilon }(t),y^{\varepsilon }(t),u^{\varepsilon }(t))\right]  \notag \\
&\geq &-C\varepsilon ^{\beta }\theta _{0}^{\varepsilon },\text{ }\forall
u\in \mathbb{U},\text{ a.e, a.s.}.  \label{6.56}\end{aligned}$$

The proof is complete.

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[^1]: L. Zhang acknowledges the financial support partly by the National Nature Science Foundation of China (No. 11201263 and No. 11201264).) and the Nature Science Foundation of Shandong Province (No. ZR2012AQ004). E-mail: hobbscindy@example.org.

[^2]: J. Huang acknowledges the financial support partly by Hong Kong RGC grant 500909. E-mail: hobbscindy@example.org.

[^3]: X. Li acknowledges the financial support partly by Hong Kong RGC grant 524109. E-mail: hobbscindy@example.org.

{'source': 'arXiv_out/1203'}