# A local maximum principle for robust optimal control problems of quadratic
BSDEs††thanks: This paper is supported by National Key R&D Program of China
(Grant Nos.2022YFA1006101, 2018YFA0703900), National Natural Science
Foundation of China (Grant Nos.12171279, 72171133, 12101291, 12371445),
Natural Science Foundation of Shandong Province (Grant Nos.ZR2020MA032,
ZR2022MA029), Natural Science Foundation of Guangdong Province of China (Grant
No.2214050003543), and Science and Technology Commission of Shanghai
Municipality (Grant No.22ZR1407600).

Tao Hao , Jiaqiang Wen , Qi Zhang , School of Statistics and Mathematics,
Shandong University of Finance and Economics, Jinan 250014, China (Email:
taohao@sdufe.edu.cn). Department of Mathematics, Southern University of
Science and Technology, Shenzhen 518055, China (Email: wenjq@sustech.edu.cn).
School of Mathematical Sciences, Fudan University, Shanghai 200433, China &
Laboratory of Mathematics for Nonlinear Science, Fudan University, Shanghai
200433, China (Email:qzh@fudan.edu.cn).

Abstract: The paper concerns the necessary maximum principle for robust
optimal control problems of quadratic BSDEs. The coefficient of the systems
depends on the parameter $\theta$, and the generator of BSDEs is of quadratic
growth in $z$. Since the model is uncertain, the variational inequality is
proved by weak convergence technique. In addition, due to the generator being
quadratic with respect to $z$, the forward adjoint equations are SDEs with
unbounded coefficient involving mean oscillation martingales. Using reverse
Hölder inequality and John-Nirenberg inequality, we show that its solutions
are continuous with respect to the parameter $\theta$. The necessary and
sufficient conditions for robust optimal control are proved by linearization
method.

Key words: quadratic BSDE, model uncertainty, maximum principle, robust
optimal control

AMS subject classifications. 93E20, 60H10, 35K15

##  1 Introduction

The stochastic maximum principle, namely, the necessary condition for
optimality, is an important approach to study stochastic optimal control
problems. In 1990, Peng [27] first proved a global maximum principle for
classical stochastic control systems. From then on, this topic has been
extensively explored for various stochastic systems. We refer to Hu and
Tessitore [10] for infinite-dimensional dynamics, Buckdahn, Li and Ma [4] for
mean-field control systems, Fuhrman, Hu, Ji and Xue [18] for fully coupled
forward-backward stochastic control systems, to name but a few. In addition,
many works on maximum principle for the stochastic recursive optimal control
problems have been published. For example, Peng [28] considered a local
maximum principle in the case that the control domain is convex and the
diffusion coefficient depends on control. Ji and Zhou [22] investigated a
local maximum principle for stochastic optimal control with terminal state
constraints. Xu [31] established a stochastic maximum principle for the
nonconvex control domain in the case that the diffusion coefficient is
independent of control. Hu and Ji [15] studied stochastic maximum principle
for stochastic recursive optimal control problems under volatility ambiguity.
Hu [13] obtained a global stochastic maximum principle by introducing two new
adjoint equations, and solved Peng’s open problem completely.

In many scenarios, we meet the situation that the model is uncertain and the
generator of BSDEs is quadratic growth. Our motivation of the present paper is
to study the control problem of such kind of model. We give two examples for
the concerned model which come from risk-sensitive control problems and
optimal strategy problems for large investor in section 2.

Inspired by the two examples in section 2, we consider the following robust
optimal control problem with quadratic generator. Let $\theta\in\Theta$
represent different market conditions, where $\Theta$ is a locally compact
Polish space with distance $d$. Assume that $V$ is a given nonempty convex
subset of $\mathbb{R}^{k}$, and consider the following forward-backward
control system

$\left\\{\begin{aligned}
X^{v}_{\theta}(t)=&x+\int_{0}^{t}b_{\theta}(s,X^{v}_{\theta}(s),\mathbb{E}[X^{v}_{\theta}(s)],v(s))ds\\\
&+\int_{0}^{t}\sigma_{\theta}(s,X^{v}_{\theta}(s),v(s))dW(s),\ t\in[0,T],\\\
Y^{v}_{\theta}(t)=&\Phi_{\theta}(X^{v}_{\theta}(T),\mathbb{E}[X^{v}_{\theta}(T)])+\int_{t}^{T}f_{\theta}(s,X^{v}_{\theta}(s),\mathbb{E}[X^{v}_{\theta}(s)],Y^{v}_{\theta}(s),Z^{v}_{\theta}(s),v(s))ds\\\
&-\int_{t}^{T}(Z_{\theta}^{v}(s))^{\top}dW_{s},\\\ \end{aligned}\right.$ (1.1)

where $b_{\theta}:[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times
V\rightarrow\mathbb{R}^{n},\sigma_{\theta}:[0,T]\times\mathbb{R}^{n}\times
V\rightarrow\mathbb{R}^{n\times d},$
$f_{\theta}:[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}\times\mathbb{R}^{d}\times
V\rightarrow\mathbb{R},\
\Phi_{\theta}:\mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow\mathbb{R}$;
$f_{\theta}$ is quadratic growth with respect to $z$; $v(\cdot)$ is a control
process; the precise assumptions on
$b_{\theta},\sigma_{\theta},f_{\theta},\Phi_{\theta}$ refer to 1. In order to
obtain a general result, we assume that the forward stochastic differential
equation (SDE, for short) is mean-field type, and consider the following cost
functional

$J(v(\cdot))=\sup\limits_{Q\in{\cal
Q}}\int_{\Theta}\mathbb{E}\Big{[}\phi_{\theta}(X_{\theta}^{v}(T))+\gamma_{\theta}(Y^{v}_{\theta}(0))\Big{]}Q(d\theta).$

The assumptions on $\phi_{\theta}$ and $\gamma_{\theta}$ lie in 2. The robust
optimal control problem with quadratic generator is to minimize the cost
function $J(v(\cdot))$ over ${\cal V}_{ad}$ (see 3.1). The target of the
present paper is to give the necessary and sufficient conditions of the
optimal control.

Note that the solvability of BSDEs with quadratic generators has been
intensively investigated over the past twenty years. In 2000, Kobylanski [24]
investigated the existence and uniqueness of one-dimensional BSDEs in the case
that the generator $g$ is of quadratic growth in $z$ and the terminal value
$\xi$ is bounded. Briand and Hu [2, 3] obtained the existence and uniqueness
of one-dimensional BSDEs with quadratic growth and unbounded terminal value.
The multi-dimensional situation with bounded terminal value was studied by Hu
and Tang [20] and Xing and Zitkovic [30]; and the multi-dimensional situation
with unbounded terminal value was investigated by Fan, Hu and Tang [8]. Some
other recent developments concerning the quadratic BSDEs can be found in Fan,
Hu and Tang [9], Hu, Li and Wen [19], Hao, Hu, Tang and Wen [12] and so on. In
addition, much progress has been made on stochastic optimal control problems
involving quadratic BSDEs. Let us cite [25] for its applications to the risk-
sensitive control problems with the risk-sensitive parameter; and [29, 6, 14,
26] for the exponential utility maximization problems.

Since the generator is quadratic and the model is uncertain, there are three
essential difficulties in the present work. (i) The fact that $f$ is of
quadratic growth in $z$ causes that the variational equations are linear BSDEs
with stochastic unbounded Lipschitz coefficients involving BMO-martingales.
Those results for BSDEs with bounded Lipschitz coefficients (see e.g. [21,
Lemma 3.2]) do not work in this setting. In order to obtain the maximum
principle, we prove a new estimate (4.4), i.e., for $p\in(1\vee
2p_{\partial_{z}f_{\theta}}^{-1},2)$,

$\displaystyle\lim\limits_{\lambda\rightarrow
0}\sup\limits_{\theta\in\Theta}\mathbb{E}\bigg{[}\sup\limits_{0\leqslant
t\leqslant T}\Gamma_{\theta}(t)|\delta
Y^{\lambda}_{\theta}(t)|^{p}+\Big{(}\int_{0}^{T}(\Gamma_{\theta}(t))^{\frac{2}{p}}|\delta
Z^{\lambda}_{\theta}(t)|^{2}dt\Big{)}^{\frac{p}{2}}\bigg{]}=0,$

where $\Gamma_{\theta}(t):={\cal
E}\Big{(}\int_{0}^{t}\partial_{z}f_{\theta}(s)^{\top}dW(s)\Big{)},\
t\in[0,T].$ (ii) The forward adjoint equations are linear SDEs with stochastic
unbounded coefficients. We obtain the wellposedness of this kinds of
equations, which develops the work of Gal’chuk [11]. The continuity of its
solutions with respect to $\theta$, i.e., for
$1<p<(p_{\partial_{z}f_{\theta}}\wedge
p_{(\partial_{z}f_{\tilde{\theta}}-\partial_{z}f_{\theta})})$,

$\displaystyle\lim_{\epsilon\rightarrow
0}\sup_{d(\theta,\tilde{\theta})\leqslant\epsilon}\mathbb{E}\bigg{[}\sup_{t\in[0,T]}|p_{\theta}(t)-p_{\tilde{\theta}}(t)|^{p}\bigg{]}=0,$

is proved by the theory of BMO-martingales (5.2), which is necessary to prove
that $\Lambda_{\theta}(\cdot)$ defined by (5.11) is ${\cal F}$-progressively
measurable. (iii) Since we consider the robust optimal control problems, i.e.,
the cost functional is a supremum over a set of probability measures, the
classical convex variational method is not suitable for this situation. We
adopt the weak convergence technique to prove the variational inequality.

Compared with the existing literatures, our paper has the following
contributions. First, our model is uncertain and the generator $f$ is of
quadratic growth in $z$. In this case, the robust optimal control problems
rather than classical optimal control problems are considered. We prove a new
and nontrivial estimate (4.4) and develop the work of [21] from the Lipschitz
case to quadratic case. Second, we give the existence and uniqueness result of
$L^{p}$-solution to linear SDEs with stochastic unbounded coefficients, and
prove the continuity of its solutions with respect to $\theta$ by reverse
Hölder inequality and John-Nirenberg inequality. In addition, the continuity
of solution to linear BSDEs with stochastic unbounded coefficients with
respect to $\theta$ is obtained. Third, since the model is uncertain, the
necessary maximum principe is proved by linearization technique and weak
convergence method. Moreover, under certain convexity assumptions, with the
help of Sion’s minimax theorem, we prove the sufficient maximum principle.

The paper is arranged as follows. In section 3, we formulate the problem.
Variational inequality is proved in section 4. The section 5 is devoted to the
necessary and sufficient maximum principles.

##  2 Two examples

In this section, we present two examples which demonstrate the applications of
robust optimal control problems of quadratic BSDEs.

######  Example 2.1.

(risk-sensitive control) Assume that a market is roughly divided into two
states $A$ and $B$ (for example, a share market is divided into a bull market
or a bear market) which usually leads to the different coefficients in two
states. Let $\theta=1$ and 2 denote the state $A$ and the state $B$,
respectively, which can be characterized as $\theta\in\Theta=\\{1,2\\}$. Let
${\cal Q}=\\{Q^{\lambda}:\lambda\in[0,1]\\},$ where $Q^{\lambda}$ is the
probability such that $Q^{\lambda}(\\{1\\})=\lambda,\quad
Q^{\lambda}(\\{2\\})=1-\lambda.$ For simplicity, we only consider the
1-dimensional case. Suppose that there exist $N$-individual agents in a
system. The state process of the $i$-th agent is described by

$\left\\{\begin{aligned}
dX_{\theta}^{i,v}(t)&=b_{\theta}^{i}(t,X_{\theta}^{i,v}(t),\frac{1}{N}\sum_{j=1}^{N}X_{\theta}^{j,v}(t),v(t))dt+\sigma_{\theta}^{i}(t,X_{\theta}^{i,v}(t),v(t))dW^{i}(t),\\\
X_{\theta}^{i,v}(t)&=x_{0},\end{aligned}\right.$ (2.1)

where $W^{i},i=1,2,\cdots,N$ are independent copies of a 1-dimensional
standard Brownian motion $W$; $v(\cdot)$ is the control process. By ${\cal
V}_{ad}$ we denote the set of admissible controls (see 3.1). The robust
objective functional of the $i$-th agent is

$\displaystyle
J^{i}(v(\cdot))=\max\limits_{\lambda\in\\{0,1\\}}\bigg{\\{}\lambda
Y^{i,v}_{1}+(1-\lambda)Y^{i,v}_{2}\bigg{\\}}=\sup_{Q^{\lambda}\in{\cal
Q}}\int_{\Theta}Y^{i,v}_{\theta}Q^{\lambda}(d\theta),$ (2.2)

where $(Y_{\theta}^{i,v},Z_{\theta}^{i,v})$ is the solution of the following
BSDE with quadratic generator (see El Karoui and Hamadenè [7])

$\left\\{\begin{aligned}
dY_{\theta}^{i,v}(t)&=-\frac{\kappa}{2}|Z_{\theta}^{i,v}(t)|^{2}dt+Z_{\theta}^{i,v}(t)dW^{i}(t),\\\
Y_{\theta}^{i,v}(T)&=\Phi_{\theta}^{i}(X_{\theta}^{i,v}(T),\frac{1}{N}\sum_{j=1}^{N}X_{\theta}^{j,v}(T)).\end{aligned}\right.$
(2.3)

Note that the parameter $\kappa$ is called risk-sensitive parameter. The
$i$-th agent wants to minimize her/his objective functional. The above problem
is regarded as a risk-sensitive robust optimal control problem for particle
systems.

Now assume that the game is symmetric. In other words, let
$b^{i}_{\theta}=b_{\theta},\sigma^{i}_{\theta}=\sigma_{\theta},\Phi^{i}_{\theta}=\Phi_{\theta}.$
As $N\rightarrow\infty$ in (2.1)-(2.3), from strong law of large number, we
obtain the following FBSDE:

$\left\\{\begin{aligned}
dX_{\theta}^{v}(t)&=b_{\theta}(t,X_{\theta}^{v}(t),\mathbb{E}[X_{\theta}^{v}(t)],v(t))dt+\sigma_{\theta}(t,X_{\theta}^{v}(t),v(t))dW(t),\\\
dY_{\theta}^{v}(t)&=-\frac{\kappa}{2}|Z_{\theta}^{v}(t)|^{2}dt+Z_{\theta}^{v}(t)dW(t),\\\
X_{\theta}^{v}(t)&=x_{0},\
Y_{\theta}^{v}(T)=\Phi_{\theta}(X_{\theta}^{v}(T),\mathbb{E}[X_{\theta}^{v}(T)]),\end{aligned}\right.$

and, meanwhile, the cost functional becomes
$J(v(\cdot))=\sup\limits_{Q^{\lambda}\in{\cal
Q}}\int_{\Theta}Y_{\theta}^{v}(0)Q^{\lambda}(d\theta).$ Consequently, the
control problem can be written as below.

Problem R Find an optimal control $\bar{v}(\cdot)$ such that

$J(\bar{v}(\cdot))=\inf_{v(\cdot)\in{\cal
V}_{ad}}J(v(\cdot))=\inf_{v(\cdot)\in{\cal V}_{ad}}\sup_{Q^{\lambda}\in{\cal
Q}}\int_{\Theta}Y_{\theta}^{v}(0)Q^{\lambda}(d\theta).$

######  Example 2.2.

(optimal strategy for large investor) Let $K\subset\mathbb{R}$ be a closed
convex cone. Assume that there are $N$ investors in a market and the market
consists of a bond and a stock whose prices denoted by $S_{j}(t),j=0,1$ are
governed by the following ODE and SDE:

$\left\\{\begin{aligned}
dS_{0}(t)&=S_{0}(t)\Big{[}r(t)+f^{i}_{0}(t,x^{i,\pi}(t),\frac{1}{N}\sum_{j=1}^{N}x^{j,\pi}(t),\pi(t))\Big{]}dt,\\\
dS_{1}(t)&=S_{1}(t)\Big{[}(\mu(t)-f^{i}_{1}(t,x^{i,\pi}(t),\frac{1}{N}\sum_{j=1}^{N}x^{j,\pi}(t),\pi(t))dt+\sigma
dW(t)\Big{]},\end{aligned}\right.$

where $(r,\mu):\Omega\times[0,T]\rightarrow\mathbb{R}\times\mathbb{R}$ are the
interest rate and return rate, respectively; $\sigma$ is the volatility of
stock; $(r,\mu)$ are $\\{{\cal F}_{t}\\}_{t\geqslant 0}$-adapted and uniformly
bounded processes and $\sigma\neq 0$; $W$ is a 1-dimensional Brownian motion;
for each $i=1,2,\cdots,N$,
$f^{i}_{\ell},\ell=0,1:[0,T]\times\mathbb{R}^{+}\times\mathbb{R}^{+}\times\mathbb{R}\rightarrow\mathbb{R}$
are two given functions representing the effect of the strategies chosen by
the investors on the prices.

The $i$-th investor intends to invest $\pi^{i}(t)$ in stock at time $t$ and
the remaining asset in bond. Under a self-financed portfolio, the wealth
process of the $i$-th investor, starting with an initial wealth $x^{i}_{0}$,
satisfies the following wealth equation:

$\left\\{\begin{aligned}
dx^{i,\pi}(t)&=\Big{(}r(t)x^{i,\pi}(t)+(x^{i,\pi}(t)-\pi^{i}(t))f^{i}_{0}(t,x^{i,\pi}(t),\frac{1}{N}\sum_{j=1}^{N}x^{j,\pi}(t),\pi(t))\\\
&\quad\
+\pi^{i}(t)[\mu(t)-r(t)+f^{i}_{1}(t,x^{i,\pi}(t),\frac{1}{N}\sum_{j=1}^{N}x^{j,\pi}(t),\pi(t))]\Big{)}dt+\pi^{i}(t)\sigma
dW(t),\quad t\in[0,T],\\\ x^{i,\pi}(0)&=x^{i}_{0}.\end{aligned}\right.$

For $i=1,2,\cdots,N$, let $\gamma^{i}>0$ be a given parameter and $C^{i}$ be
the consumption at time T for the $i$-investor. Consider the following cost
functional

$J^{i}(\pi)=\frac{1}{\gamma^{i}}\ln\mathbb{E}\Big{[}\exp[-\gamma^{i}(x^{i,\pi}(T)-C^{i})]\Big{]}.$

A strategy $\pi$ taking values in $K$ is called admissible if for any
$t\in[0,T]$, $\mathbb{E}\Big{[}\int_{0}^{T}|\pi(t)|^{2}dt\Big{]}<\infty.$ By
${\cal K}$ we denote the set of all admissible strategies. The $i$-th investor
intends to minimize her/his cost functional over ${\cal K}$. Next, let
$N\rightarrow\infty$ and consider the asymptotic behavior of $N$ investors.
For simplicity, we set
$\pi=\pi^{i},x_{0}=x_{0}^{i},\gamma=\gamma^{i},f_{0}=f^{i}_{0},f_{1}=f^{i}_{1}$
and $C=C^{i}$. This optimal strategy problem can be summarized as

Problem Find an optimal strategy $\bar{\pi}(\cdot)$ such that

$J(\bar{\pi}(\cdot))=\inf\limits_{\pi(\cdot)\in{\cal
K}}\frac{1}{\gamma}\ln\mathbb{E}\Big{[}\exp[-\gamma(x^{\pi}(T)-C)]\Big{]},$

subject to a mean-field SDE

$\left\\{\begin{aligned}
dx^{\pi}(t)&=\Big{(}r(t)x^{\pi}(t)+(x^{\pi}(t)-\pi(t))f_{0}(t,x^{\pi}(t),\mathbb{E}[x^{\pi}(t)],\pi(t))\\\
&\quad\ \
+\pi(t)[\mu(t)-r(t)+f_{1}(t,x^{\pi}(t),\mathbb{E}[x^{\pi}(t)],\pi(t))]\Big{)}dt+\pi(t)\sigma
dW(t),\quad t\in[0,T],\\\ x^{\pi}(0)&=x_{0}.\end{aligned}\right.$

To illustrate our problem in a concise way, set

$f_{0}(t,x^{\pi}(t),\mathbb{E}[x^{\pi}(t)],\pi(t)):=\alpha(t),\quad
f_{1}(t,x^{\pi}(t),\mathbb{E}[x^{\pi}(t)],\pi(t)):=\beta(t).$

Here $\alpha(\cdot)$ and $\beta(\cdot)$ are two bounded functions. In order to
compute the forward value of a portfolio, let us consider a zero-coupon bond
$\Lambda$ with a maturity of $T$, which is a financial asset generating a
cash-flow of $1$ at time $T$. In other words, there is an $\mathbb{R}$-valued
progressively measurable process $\Gamma$ such that the zero-coupon price
$\Lambda$ satisfies BSDE:

$\left\\{\begin{aligned}
d\Lambda(t)&=\Lambda(t)\Big{[}\Big{(}r(t)+\Gamma(t)(\mu(t)-r(t)+\beta(t)-\alpha(t))\Big{)}dt+\Gamma(t)\sigma
dW(t)\Big{]},\quad t\in[0,T],\\\ \Lambda(T)&=1.\end{aligned}\right.$

Here, for simplicity, we assume that $\Gamma$ is $\\{{\cal
F}_{t}\\}_{t\geqslant 0}$-adapted and uniformly bounded for any
$(t,\omega)\in[0,T]\times\Omega.$ According to [29, Theorem 2.1], one can see

$J(\bar{\pi})=\inf\limits_{\pi(\cdot)\in{\cal
K}}\frac{1}{\gamma}\ln\mathbb{E}\Big{[}\exp[-\gamma(x^{\pi}(T)-C)]\Big{]}=-\frac{x_{0}}{\Lambda(0)}+\sup_{v(\cdot)\in{\cal
V}}Y^{v}(0),$ (2.4)

where $(Y^{v},Z^{v})$ satisfies BSDE

$\left\\{\begin{aligned}
dY^{v}(t)&=-\Big{(}-\frac{1}{2\gamma}|\delta(t)+\Pi_{\sigma^{-1}\tilde{K}}(-\delta(t)-\gamma
Z^{v}(t))|^{2}-(\delta(t)+\Pi_{\sigma^{-1}\tilde{K}}(-\delta(t)-\gamma
Z^{v}(t)))Z^{v}(t)\Big{)}dt\\\ &\quad+Z^{v}(t)dW(t),\ \ \ t\in[0,T],\\\
Y^{v}(T)&=C,\end{aligned}\right.$ (2.5)

where
$\delta(t)=\sigma^{-1}(\mu(t)-r(t)+\beta(t)-\alpha(t)-\sigma^{2}\Gamma(t))$;
$\widetilde{K}=\\{x\in\mathbb{R}|\sup\limits_{\pi\in K}(-\pi x)<\infty\\}$;
$\Pi_{\sigma^{-1}\tilde{K}}(u)$ is the projection of $u\in\mathbb{R}$ on
$\sigma^{-1}\tilde{K}$; ${\cal
V}:=\\{v:\Omega\times[0,T]\rightarrow\sigma^{-1}\widetilde{K}|\ v\ \text{is}\
\\{{\cal F}_{t}\\}_{t\geqslant 0}\text{-adapted}\ \text{and}$
$\text{bounded}.$ Since $|\Pi_{\sigma^{-1}\tilde{K}}(u)|\leqslant|u|,$ for
$\forall u\in\mathbb{R}$, there exists a constant $L>0$ such that the
generator of (2.5) is dominated by $L(1+|Z^{v}(t)|^{2})$.

The cost functional (2.4) tells us that solving an optimal strategy problem is
equivalent to solving an optimal control problem of BSDE, whose generator is
dominated by a quadratic generator. Following this line, a natural question is
if $\gamma$ and $C$ depend on a parameter $\theta$ and are continuous in
$\theta$, how to describe the equivalent optimal control problem, where
$\theta\in\Theta$ represents the different market states and $\Theta$ is a
closed set of $\mathbb{R}$. Actually, in this case, we need to consider the
following cost functional

$\widehat{J}(v(\cdot)):=\sup_{Q\in{\cal
Q}}\int_{\Theta}Y^{v}_{\theta}(0)Q(d\theta),$

where ${\cal Q}$ is a weakly compact and convex set of probability measures on
$(\Theta,{\cal B}(\Theta))$; $(Y^{v}_{\theta},Z^{v}_{\theta})$ satisfies the
following BSDE

$\left\\{\begin{aligned}
dY^{v}_{\theta}(t)&=-\Big{(}-\frac{1}{2\gamma_{\theta}}|\delta(t)+\Pi_{\sigma^{-1}\tilde{K}}(-\delta(t)-\gamma_{\theta}Z^{v}(t))|^{2}-(\delta(t)+\Pi_{\sigma^{-1}\tilde{K}}(-\delta(t)-\gamma_{\theta}Z^{v}(t)))Z^{v}(t)\Big{)}dt\\\
&\quad\ +Z^{v}_{\theta}(t)dW(t),\ t\in[0,T],\\\
Y^{v}_{\theta}(T)&=C_{\theta}.\end{aligned}\right.$ (2.6)

Similar to (2.5), the generator of (2.6) is dominated by
$L(1+|Z^{v}_{\theta}(t)|^{2})$, where $L$ is a constant independent of
$\theta$. Consequently, the corresponding optimal control problem is
formulated as follow.

Problem S Find an optimal control $\bar{v}(\cdot)$ such that

$\widehat{J}(\bar{v}(\cdot))=\sup_{v(\cdot)\in{\cal V}}\sup_{Q\in{\cal
Q}}\int_{\Theta}Y^{v}_{\theta}(0)Q(d\theta).$

##  3 Problem formulation

###  3.1 Some properties for BMO-martingales

Denote by $\mathbb{N}$ the set of all natural numbers and by $\mathbb{R}^{+}$
the set of positive real numbers, respectively. Let the superscript $\top$
denote the transpose of vectors or matrices. Let $M=(M_{t},\mathscr{F}_{t})$
be a uniformly integrable martingale with $M_{0}=0$, and we set, for
$p\geqslant 1$,

$\|M\|_{BMO_{p}(\mathbb{P})}:=\sup_{\tau}\bigg{\|}\mathbb{E}_{\tau}\Big{[}\Big{(}\langle
M\rangle_{\infty}-\langle
M\rangle_{\tau}\Big{)}^{\frac{p}{2}}\Big{]}^{\frac{1}{p}}\bigg{\|}_{\infty},$

where $\tau$ is a stopping time on $[0,T]$. The class
$\big{\\{}M:\|M\|_{BMO_{p}(\mathbb{P})}<\infty\big{\\}}$ is denoted by
$BMO_{p}(\mathbb{P})$. Note that $BMO_{p}(\mathbb{P})$ is a Banach space under
the norm $\|\cdot\|_{BMO_{p}(\mathbb{P})}$. For simplicity,
$BMO_{2}(\mathbb{P})$ is written as $BMO$.

Next, we list some properties for $BMO$-martingales. The reader can refer to
Kazamaki [23] for more details.

$\bullet$ Denote by ${\cal E}(M)$ the Doléans–Dade exponential of a continuous
local martingale $M$, i.e., ${\cal E}(M_{t})=\exp\\{M_{t}-\frac{1}{2}\langle
M\rangle_{t}\\}$, for any $t\in[0,T]$. If $M\in BMO$, then ${\cal E}(M)$ is a
uniformly integrable martingale.

$\bullet$ Let $\Psi$ be the monotonically decreasing function defined on
$(1,\infty)$ by

$\displaystyle\Psi(p)=\bigg{(}1+p^{-2}\ln\frac{2p-1}{2(p-1)}\bigg{)}^{\frac{1}{2}}-1$

and then $\Psi(+\infty):=\lim_{x\rightarrow+\infty}\Psi(x)=0$. For $M\in BMO$,
we can find a positive constant $p_{M}$ which satisfies
$\Psi(p_{M})=\|M\|_{BMO}$. In particular, set $p_{M}=+\infty$ if
$\|M\|_{BMO}=0$. Then $p_{M}$ is uniquely determined.

$\bullet$ The reverse Hölder inequality: If $p\in(1,p_{M})$, for any stopping
time $\tau\in[0,T]$,

$\mathbb{E}\Big{[}({\cal E}(M_{T}))^{p}/({\cal E}(M_{\tau}))^{p}|{\cal
F}_{\tau}\Big{]}\leqslant K(p,||M||_{BMO}),\ \text{a.s.},$

where

$K(p,||M||_{BMO})=2\bigg{(}1-\frac{2p-2}{2p-1}\exp\bigg{\\{}p^{2}\Big{[}||M||^{2}_{BMO}+2||M||_{BMO}\Big{]}\bigg{\\}}\bigg{)}^{-1}.$

$\bullet$ The John-Nirenberg inequality: For $M\in BMO$, if
$\theta\in(0,\|M\|^{-2}_{BMO}),$ for any stopping time $\tau\in[0,T]$,

$\mathbb{E}\Big{[}\exp\Big{\\{}\theta(\langle M\rangle_{T}-\langle
M\rangle_{\tau})\Big{\\}}\Big{|}{\cal
F}_{\tau}\Big{]}\leqslant\Big{(}1-\theta\|M\|_{BMO}^{2}\Big{)}^{-1},\
\text{a.s.}$

$\bullet$ By $p^{*}_{M}$ we denote the conjugate exponent of $p_{M}$, i.e.,
$(p_{M})^{-1}+(p^{*}_{M})^{-1}=1$.

For any $p\geqslant 1$, $t\in[0,T]$ and a filtration $\mathbb{F}$, we
introduce some useful spaces:

$\displaystyle\displaystyle{\cal
S}_{\mathbb{F}}^{p}(t,T;\mathbb{R}^{m})=\Big{\\{}$
$\displaystyle\varphi:\Omega\times[t,T]\to\mathbb{R}^{m}\bigm{|}\varphi\hbox{
is $\mathbb{F}$-adapted, continuous, }$
$\displaystyle\qquad\qquad\qquad\|\varphi\|_{{\cal
S}_{\mathbb{F}}^{p}(t,T)}\triangleq\Big{\\{}\mathbb{E}\Big{(}\sup_{s\in[t,T]}|\varphi_{s}|^{p}\Big{)}\Big{\\}}^{\frac{1}{p}}<\infty\Big{\\}},$
$\displaystyle\displaystyle{\cal
S}_{\mathbb{F}}^{\infty}(t,T;\mathbb{R}^{m})=\Big{\\{}$
$\displaystyle\varphi:\Omega\times[t,T]\to\mathbb{R}^{m}\bigm{|}\varphi\hbox{
is $\mathbb{F}$-adapted, continuous, }$
$\displaystyle\qquad\qquad\qquad\|\varphi\|_{{\cal
S}_{\mathbb{F}}^{\infty}(t,T)}\triangleq\mathop{\rm
esssup}_{(s,\omega)\in[t,T]\times\Omega}|\varphi_{s}(\omega)|<\infty\Big{\\}},$
$\displaystyle\displaystyle{\cal
H}_{\mathbb{F}}^{2,p}(t,T;\mathbb{R}^{m})=\Big{\\{}$
$\displaystyle\varphi:\Omega\times[t,T]\to\mathbb{R}^{m}\bigm{|}\varphi\hbox{
is $\mathbb{F}$-progressively measurable, }$
$\displaystyle\qquad\qquad\qquad\|\varphi\|_{{\cal
H}^{2,p}_{\mathbb{F}}(t,T)}\triangleq\mathbb{E}\Big{[}\Big{(}\int^{T}_{t}|\varphi_{s}|^{2}ds\Big{)}^{p\over
2}\Big{]}^{\frac{1}{p}}<\infty\Big{\\}}.$

Next, we formulate the optimal control problem. For this end, we first
introduce the set of admissible controls.

######  Definition 3.1.

An $\mathbb{F}$-adapted process $v$ taking values in $V\subset\mathbb{R}^{K}$
is called an admissible control, if for any $\ell>0$,
$\mathbb{E}[\int_{0}^{T}|v(t)|^{\ell}dt]<\infty.$ By ${\cal V}_{ad}$ we denote
the set of all admissible controls.

The following assumptions on coefficients
$b_{\theta},\sigma_{\theta},f_{\theta},\Phi_{\theta}$ are forced.

######  Assumption 1.

  1. (i)

There exists a constant $C_{0}>0$ independent of $\theta$ such that, for
$t\in[0,T]$,
$x_{1},x_{2},x^{\prime}_{1},x^{\prime}_{2}\in\mathbb{R}^{n},v,v^{\prime}\in
V$,

$\displaystyle|b_{\theta}(t,0,0,v)|+|\sigma_{\theta}(t,0,v)|\leqslant
C_{0}(1+|v|),$
$\displaystyle|b_{\theta}(t,x_{1},x^{\prime}_{1},v)-b_{\theta}(t,x_{2},x^{\prime}_{2},v^{\prime})|\leqslant
C_{0}(|x_{1}-x_{2}|+|x^{\prime}_{1}-x^{\prime}_{2}|+|v-v^{\prime}|),$
$\displaystyle|\sigma_{\theta}(t,x_{1},v)-\sigma_{\theta}(t,x_{2},v^{\prime})|\leqslant
C_{0}(|x_{1}-x_{2}|+|v-v^{\prime}|);$

$b_{\theta},\sigma_{\theta}$ is continuously differentiable in
$(x,x^{\prime},v)$ and $(x,v)$, respectively;
$\partial_{x}b_{\theta},\partial_{x^{\prime}}b_{\theta},\partial_{v}b_{\theta},$
$\partial_{x}\sigma_{\theta},\partial_{v}\sigma_{\theta}$ are Lipschitz
continuous in $(x,x^{\prime},v)$.

  2. (ii)

$\Phi_{\theta},\partial_{x}\Phi_{\theta},\partial_{x^{\prime}}\Phi_{\theta}$
are continuous in $(x,x^{\prime})$ and bounded.

  3. (iii)

$f$ is continuously differentiable in $(x,x^{\prime},y,z,v)$;
$\partial_{x}f_{\theta},\partial_{x^{\prime}}f_{\theta},\partial_{y}f_{\theta},\partial_{z}f_{\theta},\partial_{v}f_{\theta}$
are Lipschitz continuous in $(x,x^{\prime},y,z,v)$; there exists a constant
$C_{1}>0$ independent of $\theta$ such that, for $t\in[0,T]$,
$x_{1},x_{2},x^{\prime}_{1},x^{\prime}_{2}\in\mathbb{R}^{n},y_{1},y_{2}\in\mathbb{R},z_{1},z_{2}\in\mathbb{R}^{d},v\in
V$,

$\displaystyle|f_{\theta}(t,x_{1},x^{\prime}_{1},0,0,v)|\leqslant C_{1},$
$\displaystyle|f_{\theta}(t,x_{1},x^{\prime}_{1},y_{1},z_{1},v)-f_{\theta}(t,x_{2},x^{\prime}_{2},y_{2},z_{2},v)|$
$\displaystyle\leqslant
C_{1}(|x_{1}-x_{2}|+|x^{\prime}_{1}-x^{\prime}_{2}|+|y_{1}-y_{2}|)+C_{1}(1+|z_{1}|+|z_{2}|)|z_{1}-z_{2}|,$
$\displaystyle|\partial_{v}f_{\theta}(t,x_{1},x^{\prime}_{1},y_{1},z_{1},v)|\leqslant
C_{1}.$

  4. (iv)

There exists a constant $C_{2}>0$ independent of $\theta$ such that, for any
$t\in[0,T]$, $\theta,\bar{\theta}\in\Theta$,
$x_{1},x^{\prime}_{1}\in\mathbb{R}^{n},y_{1}\in\mathbb{R},z_{1}\in\mathbb{R}^{d},v\in
V$,

$|\varphi_{\theta}(t,x_{1},x^{\prime}_{1},y_{1},z_{1},v)-\varphi_{\bar{\theta}}(t,x_{1},x^{\prime}_{1},y_{1},z_{1},v)|\leqslant
C_{2}d(\theta,\bar{\theta}),$

where $\varphi_{\theta}$ is
$b_{\theta},\sigma_{\theta},f_{\theta},\Phi_{\theta}$ and their derivatives
w.r.t. their respective variables.

  5. (v)

${\cal Q}$ is a weakly compact and convex set of probability measures on
$(\Theta,{\cal B}(\Theta))$.

According [3, Proposition 3] and [20, Lemma 2.1], we know the existence and
uniqueness of $(Y^{v}_{\theta},Z^{v}_{\theta}).$

######  Theorem 3.2.

Under 1, for any $v(\cdot)\in{\cal V}_{ad}$ and $p>1$, the equation (1.1) has
a unique solution $(X^{v}_{\theta},Y^{v}_{\theta},Z^{v}_{\theta})\in{\cal
S}^{p}_{\mathbb{F}}(0,T;\mathbb{R}^{n})\times{\cal
S}^{\infty}_{\mathbb{F}}(0,T;\mathbb{R})\times{\cal
H}^{2,p}_{\mathbb{F}}(0,T;\mathbb{R}^{d})$. Moreover, the following estimates
hold:

$\displaystyle\mathrm{(i)}\ \|X^{v}_{\theta}\|^{p}_{{\cal
S}^{p}_{\mathbb{F}}(0,T)}\leqslant
C\bigg{\\{}|x|^{p}+\mathbb{E}\Big{[}\Big{(}\int_{0}^{T}|b_{\theta}(s,0,0,v(s))|ds\Big{)}^{p}\Big{]}+\mathbb{E}\Big{[}\Big{(}\int_{0}^{T}|\sigma_{\theta}(s,0,v(s))|^{2}ds\Big{)}^{\frac{p}{2}}\Big{]}\bigg{\\}},$
(3.1) $\displaystyle\mathrm{(ii)}\ \|Y^{v}_{\theta}\|_{{\cal
S}_{\mathbb{F}}^{\infty}(0,T)}\leqslant
M_{1}\quad~{}\text{and}\quad~{}\|Z^{v}_{\theta}\cdot W\|_{BMO}\leqslant
M_{2},$

where the constant $C$ depends on $C_{0},p,T$ and the constants $M_{1},M_{2}$
depends on $C_{0},C_{1},T,\|\Phi\|_{\infty}$.

In the rest of this paper we use $C$ to represent a generic constant which
only depends on given parameters and could be different from line to line.

The continuity of $(X_{\theta}^{v},Y_{\theta}^{v},Z_{\theta}^{v})$ with
respect to $\theta$ is proved in the following proposition.

######  Proposition 3.3.

Under 1, for any $v(\cdot)\in{\cal V}_{ad}$, $p>1$ and
$q>p_{\partial_{z}f_{\theta}}^{*}$,

$\displaystyle\mathrm{(i)}\quad\lim_{\epsilon\rightarrow
0}\sup_{d(\theta,\tilde{\theta})\leqslant\epsilon}\mathbb{E}\bigg{[}\sup_{0\leqslant
t\leqslant T}|X_{\theta}^{v}(t)-X_{\tilde{\theta}}^{v}(t)|^{p}\bigg{]}=0,$
$\displaystyle\mathrm{(ii)}\quad\lim_{\epsilon\rightarrow
0}\sup_{d(\theta,\tilde{\theta})\leqslant\epsilon}\mathbb{E}\bigg{[}\sup_{0\leqslant
t\leqslant
T}|Y_{\theta}^{v}(t)-Y_{\tilde{\theta}}^{v}(t)|^{q}+\Big{(}\int_{0}^{T}|Z_{\theta}^{v}(t)-Z_{\tilde{\theta}}^{v}(t)|^{2}dt\Big{)}^{\frac{q}{2}}\bigg{]}=0.$

###### Proof.

As for $\mathrm{(i)}$, denote $\Delta
X(s)=X_{\theta}^{v}(s)-X_{\tilde{\theta}}^{v}(s),$ and for $h=b,\sigma,$

$\Delta
h(s):=h_{\theta}(s,X_{\theta}^{v}(s),\mathbb{E}[X_{\theta}^{v}(s)],v(s))-h_{\tilde{\theta}}(s,X_{\theta}^{v}(s),\mathbb{E}[X_{\theta}^{v}(s)],v(s)).$

Then it follows that

$\displaystyle\Delta X(t)$ $\displaystyle=\int_{0}^{t}\Big{(}\Delta
b(s)+b_{\tilde{\theta}}(s,X_{\tilde{\theta}}^{v}(s)+\Delta
X(s),\mathbb{E}[X_{\tilde{\theta}}^{v}(s)]+\mathbb{E}[\Delta X(s)],v(s))$
$\displaystyle\quad\ \ \ \ \ \ \
-b_{\tilde{\theta}}(s,X_{\tilde{\theta}}^{v}(s),\mathbb{E}[X_{\tilde{\theta}}^{v}(s)],v(s))\Big{)}ds$
$\displaystyle\quad+\int_{0}^{t}\Big{(}\Delta\sigma(s)+\sigma_{\tilde{\theta}}(s,X_{\tilde{\theta}}^{v}(s)+\Delta
X(s),v(s))-\sigma_{\tilde{\theta}}(s,X_{\tilde{\theta}}^{v}(s),v(s))\Big{)}dW(s).$

Thanks to Theorem 3.2, it yields, for $p>1$,

$\displaystyle\mathbb{E}\Big{[}\sup\limits_{t\in[0,T]}|\Delta
X(t)|^{p}\Big{]}\leqslant C\mathbb{E}\Big{[}\Big{(}\int_{0}^{T}|\Delta
b(s)|ds\Big{)}^{p}+\Big{(}\int_{0}^{T}|\Delta\sigma(s)|^{2}ds\Big{)}^{\frac{p}{2}}\Big{]}\leqslant
Cd(\theta,\tilde{\theta})^{p}.$

Then we turn to $\mathrm{(ii)}$. For simplicity, we denote

$\displaystyle\Delta Y(s)$
$\displaystyle=Y_{\theta}^{v}(s)-Y_{\tilde{\theta}}^{v}(s),\quad\Delta
Z(s)=Z_{\theta}^{v}(s)-Z_{\tilde{\theta}}^{v}(s),$
$\displaystyle\Delta\Phi(T)$
$\displaystyle=\Phi_{\theta}(X_{\tilde{\theta}}^{v}(T),\mathbb{E}[X_{\tilde{\theta}}^{v}(T)])-\Phi_{\tilde{\theta}}(X_{\tilde{\theta}}^{v}(T),\mathbb{E}[X_{\tilde{\theta}}^{v}(T)]),$
$\displaystyle\Delta f(s)$
$\displaystyle=f_{\theta}(s,X_{\tilde{\theta}}^{v}(s),\mathbb{E}[X_{\tilde{\theta}}^{v}(s)],Y_{\tilde{\theta}}^{v}(s),Z_{\tilde{\theta}}^{v}(s),v(s))-f_{\tilde{\theta}}(s,X_{\tilde{\theta}}^{v}(s),\mathbb{E}[X_{\tilde{\theta}}^{v}(s)],Y_{\tilde{\theta}}^{v}(s),Z_{\tilde{\theta}}^{v}(s),v(s)).$

Then

$\displaystyle\Delta Y(t)$
$\displaystyle=\Delta\Phi(T)+\Phi_{\theta}(X_{\theta}^{v}(T),\mathbb{E}[X_{\theta}^{v}(T)])-\Phi_{\theta}(X_{\tilde{\theta}}^{v}(T),\mathbb{E}[X_{\tilde{\theta}}^{v}(T)])$
$\displaystyle\quad+\int_{t}^{T}\Big{(}\Delta
f(s)+f_{\theta}(s,X_{\theta}^{v}(s),\mathbb{E}[X_{\theta}^{v}(s)],Y_{\tilde{\theta}}^{v}(s)+\Delta
Y(s),Z_{\tilde{\theta}}^{v}(s)+\Delta Z(s),v(s))$ $\displaystyle\quad\ \ \ \ \
\ \ \ \ \
-f_{\theta}(s,X_{\tilde{\theta}}^{v}(s),\mathbb{E}[X_{\tilde{\theta}}^{v}(s)],Y_{\tilde{\theta}}^{v}(s),Z_{\tilde{\theta}}^{v}(s),v(s))\Big{)}ds$
$\displaystyle\quad-\int_{t}^{T}\Delta Z(s)dW(s),t\in[0,T].$

According to [1, Corollary 9], we know, for any
$q>p_{\partial_{z}f_{\theta}}^{*}$,

$\displaystyle\mathbb{E}\Big{[}\sup\limits_{t\in[0,T]}|\Delta
Y(t)|^{q}+\Big{(}\int_{0}^{T}|\Delta Z(t)|^{2}dt\Big{)}^{\frac{q}{2}}\Big{]}$
$\displaystyle\leqslant
C\bigg{(}\mathbb{E}\Big{[}|\Delta\Phi(T)|^{q+1}+|\Delta
X(T)|^{q+1}+|\mathbb{E}[\Delta X(T)]|^{q+1}\Big{]}$ $\displaystyle\quad\
+\mathbb{E}\Big{[}\int_{0}^{T}\Big{(}|\Delta
f(t)|+|f_{\theta}(t,X_{\theta}^{v}(t),\mathbb{E}[X_{\theta}^{v}(t)],Y_{\tilde{\theta}}^{v}(t),Z_{\tilde{\theta}}^{v}(t),v(t))$
$\displaystyle\quad\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-f_{\theta}(t,X_{\tilde{\theta}}^{v}(t),\mathbb{E}[X_{\tilde{\theta}}^{v}(t)],Y_{\tilde{\theta}}^{v}(t),Z_{\tilde{\theta}}^{v}(t),v(t))|\Big{)}^{q+1}dt\Big{]}\bigg{)}^{\frac{q}{q+1}}.$

Hence, from (i), we have, for $q>p_{\partial_{z}f_{\theta}}^{*}$,

$\displaystyle\mathbb{E}\Big{[}\sup\limits_{t\in[0,T]}|\Delta
Y(t)|^{q}+\Big{(}\int_{0}^{T}|\Delta
Z(t)|^{2}dt\Big{)}^{\frac{q}{2}}\Big{]}\leqslant
Cd(\theta,\tilde{\theta})^{q}.$

∎

In order to introduce the cost functional, we need the following assumption
for the mappings $\phi_{\theta}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ and
$\gamma_{\theta}:\mathbb{R}\rightarrow\mathbb{R}$.

######  Assumption 2.

  1. (i)

$\phi_{\theta}$ and $\gamma_{\theta}$ are continuously differentiable in their
respective variables and bounded.

  2. (ii)

There exists a constant $L_{1}>0$ such that, for $x,x^{\prime}\in\mathbb{R}$,
$\theta\in\Theta$,

$\displaystyle\begin{aligned}
|\phi_{\theta}(x)-\phi_{\theta}(x^{\prime})|\leqslant
L_{1}(1+|x|+|x^{\prime}|)|x-x^{\prime}|,\quad|\partial_{x}\phi_{\theta}(x)-\partial_{x}\phi_{\theta}(x^{\prime})|\leqslant
L_{1}|x-x^{\prime}|.\end{aligned}$

  3. (iii)

$\gamma_{\theta}$ and its derivative $\partial_{y}\gamma_{\theta}$ are
Lipschitz continuous with respect to $y$ uniformly in $\theta$ and bounded.

  4. (iv)

There exists a constant $L_{2}>0$ such that, for $x\in\mathbb{R}^{n}$,
$\theta,\theta^{\prime}\in\Theta$,

$\displaystyle|h_{\theta}(x)-h_{\theta^{\prime}}(x)|\leqslant
L_{2}d(\theta,\theta^{\prime}),$

where $h$ denotes $\phi,\gamma$ and their derivatives w.r.t. their respective
variables.

Since the control system (1.1) is model uncertainty, we consider the robust
cost functional

$J(v(\cdot))=\sup\limits_{Q\in{\cal
Q}}\int_{\Theta}\mathbb{E}\Big{[}\phi_{\theta}(X_{\theta}^{v}(T))+\gamma_{\theta}(Y^{v}_{\theta}(0))\Big{]}Q(d\theta).$
(3.2)

The optimal control problem is

Problem (PQU) Find an optimal control $\bar{v}(\cdot)$ such that

$J(\bar{v}(\cdot))=\inf_{v(\cdot)\in{\cal V}_{ad}}J(v(\cdot)),$ (3.3)

subject to (1.1) and (3.2).

The above control problem is what we concern in the rest of the paper, for
which we will give the necessary and sufficient conditions of the optimal
control.

##  4 Variational inequality

In this section, we investigate the variational equations and variational
inequality, which are essential materials to study the stochastic maximum
principle.

Let $\bar{v}(\cdot)$ be an optimal control and
$(\bar{X}_{\theta},\bar{Y}_{\theta},\bar{Z}_{\theta})$ be the corresponding
state processes of (1.1). Notice that ${\cal V}_{ad}$ is convex, and hence for
any $v(\cdot)\in{\cal V}_{ad}$ and $0<\lambda<1$,
$v^{\lambda}(\cdot):=\bar{v}(\cdot)+\lambda(v(\cdot)-\bar{v}(\cdot))\in{\cal
V}_{ad}$. By
$(X_{\theta}^{\lambda},Y_{\theta}^{\lambda},Z_{\theta}^{\lambda})$ we denote
the corresponding trajectories of (1.1) with $v^{\lambda}(\cdot)$ for each
$\theta\in\Theta$. To avoid heavy notation, set

$\displaystyle b_{\theta}(t)$
$\displaystyle=b_{\theta}(t,\bar{X}_{\theta}(t),\mathbb{E}[\bar{X}_{\theta}(t)],\bar{v}(t)),$
$\displaystyle f_{\theta}(t)$
$\displaystyle=f_{\theta}(t,\bar{X}_{\theta}(t),\mathbb{E}[\bar{X}_{\theta}(t)],\bar{Y}_{\theta}(t),\bar{Z}_{\theta}(t),\bar{v}(t)),$

$\sigma_{\theta}(t),\partial_{x}b_{\theta}(t),\partial_{x^{\prime}}b_{\theta}(t),\partial_{v}b_{\theta}(t),\partial_{x}\sigma_{\theta}(t),\partial_{x^{\prime}}\sigma_{\theta}(t),\partial_{v}\sigma_{\theta}(t),\partial_{x}f_{\theta}(t),\partial_{x^{\prime}}f_{\theta}(t),\partial_{y}f_{\theta}(t),\partial_{z}f_{\theta}(t),\partial_{v}f_{\theta}(t)$
can be set similarly.

###  4.1 Variational equation

Consider the following variational SDE on $[0,T]$: for each $\theta\in\Theta$,

$\left\\{\begin{aligned}
dX^{1}_{\theta}(t)&=\Big{(}\partial_{x}b_{\theta}(t)X^{1}_{\theta}(t)+\partial_{x^{\prime}}b_{\theta}(t)\mathbb{E}[X^{1}_{\theta}(t)]+\partial_{v}b_{\theta}(t)(v(t)-\bar{v}(t))\Big{)}dt\\\
&\quad+\sum_{i=1}^{d}\Big{(}\partial_{x}\sigma^{i}_{\theta}(t)X^{1}_{\theta}(t)+\partial_{v}\sigma^{i}_{\theta}(t)(v(t)-\bar{v}(t))\Big{)}dW^{i}(t),\\\
X^{1}_{\theta}(0)&=0.\end{aligned}\right.$ (4.1)

The equation (4.1) is a linear mean-field SDE. Since the coefficients
$\partial_{x}b_{\theta}(t),\partial_{x^{\prime}}b_{\theta}(t),\partial_{u}b_{\theta}(t),$
$\partial_{x}\sigma_{\theta}(t),\partial_{x^{\prime}}\sigma_{\theta}(t),\partial_{u}\sigma_{\theta}(t)$
are bounded, according to Theorem 3.2, SDE (4.1) has a unique solution
$X^{1}_{\theta}\in{\cal S}^{p}_{\mathbb{F}}(0,T;\mathbb{R}^{n})$ for $p>1.$
Furthermore, we have, for $p>1$,

$\mathbb{E}\bigg{[}\sup_{0\leqslant t\leqslant
T}|X_{\theta}^{1}(t)|^{p}\bigg{]}\leqslant
C\bigg{\\{}\mathbb{E}\Big{[}\int_{0}^{T}(|v(t)|^{p}+|\bar{v}(t)|^{p})dt\Big{]}+\Big{(}\mathbb{E}\int_{0}^{T}|v(t)|^{2}dt\Big{)}^{\frac{p}{2}}+\Big{(}\mathbb{E}\int_{0}^{T}|\bar{v}(t)|^{2}dt\Big{)}^{\frac{p}{2}}\bigg{\\}}.$
(4.2)

The following lemma shows that $X^{1}_{\theta}$ is continuous in $\theta$.

######  Proposition 4.1.

Under 1, we have for $p>1$,

$\lim_{\epsilon\rightarrow
0}\sup_{d(\theta,\tilde{\theta})\leqslant\epsilon}\mathbb{E}\bigg{[}\sup_{0\leqslant
t\leqslant T}|X_{\theta}^{1}(t)-X_{\tilde{\theta}}^{1}(t)|^{p}\bigg{]}=0.$

###### Proof.

Notice

$\left\\{\begin{aligned}
d\Big{(}X^{1}_{\theta}(t)-X^{1}_{\widetilde{\theta}}(t)\Big{)}&=\partial_{x}b_{\theta}(t)(X^{1}_{\theta}(t)-X^{1}_{\widetilde{\theta}}(t))+\partial_{x^{\prime}}b_{\theta}(t)\mathbb{E}[X^{1}_{\theta}(t)-X^{1}_{\widetilde{\theta}}(t)]+I_{1,\theta,\widetilde{\theta}}(t)dt\\\
&\ \ \
+\sum\limits_{i=1}^{d}\Big{(}\partial_{x}\sigma^{i}_{\theta}(t)(X^{1}_{\theta}(t)-X^{1}_{\widetilde{\theta}}(t))+I^{i}_{2,\theta,\widetilde{\theta}}(t)\Big{)}dW^{i}(t),\\\
X^{1}_{\theta}(t)-X^{1}_{\widetilde{\theta}}(t)&=0,\end{aligned}\right.$

where

$\displaystyle I_{1,\theta,\widetilde{\theta}}(t)$
$\displaystyle=\Big{(}\partial_{x}b_{\theta}(t)-\partial_{x}b_{\widetilde{\theta}}(t)\Big{)}X^{1}_{\widetilde{\theta}}(t)+\Big{(}\partial_{x^{\prime}}b_{\theta}(t)-\partial_{x^{\prime}}b_{\widetilde{\theta}}(t)\Big{)}\mathbb{E}[X^{1}_{\widetilde{\theta}}(t)]$
$\displaystyle\quad\
+\Big{(}\partial_{v}b_{\theta}(t)-\partial_{v}b_{\widetilde{\theta}}(t)\Big{)}\Big{(}v(t)-\bar{v}(t)\Big{)},$
$\displaystyle I^{i}_{2,\theta,\widetilde{\theta}}(t)$
$\displaystyle=\Big{(}\partial_{x}\sigma^{i}_{\theta}(t)-\partial_{x}\sigma^{i}_{\widetilde{\theta}}(t)\Big{)}X^{1}_{\widetilde{\theta}}(t)+\Big{(}\partial_{v}\sigma^{i}_{\theta}(t)-\partial_{v}\sigma^{i}_{\widetilde{\theta}}(t)\Big{)}\Big{(}v(t)-\bar{v}(t)\Big{)}.$

Then we can prove 4.1 by Theorem 3.2 and Buckholder-Davis-Gundy inequality,
following a standard proof. ∎

Set $\delta
X^{\lambda}_{\theta}(t)=\frac{1}{\lambda}(X_{\theta}^{\lambda}(t)-\bar{X}_{\theta}(t))-X^{1}_{\theta}(t)$,
and we have the following estimate on it.

######  Lemma 4.2.

Under 1, there exists a constant $C>0$ depending on $C_{0}$ and $T$ such that,
for each $\theta\in\Theta$ and $p>1$,

$\displaystyle\mathrm{(i)}\ \mathbb{E}\bigg{[}\sup\limits_{0\leqslant
t\leqslant T}|\delta X^{\lambda}_{\theta}(t)|^{p}\bigg{]}\leqslant
C\bigg{\\{}\mathbb{E}\Big{[}\int_{0}^{T}(|v(t)|^{p}+|\bar{v}(t)|^{p})dt\Big{]}+\Big{(}\mathbb{E}\int_{0}^{T}|v(t)|^{2}dt\Big{)}^{\frac{p}{2}}+\Big{(}\mathbb{E}\int_{0}^{T}|\bar{v}(t)|^{2}dt\Big{)}^{\frac{p}{2}}\bigg{\\}}.$
$\displaystyle\mathrm{(ii)}\ \lim\limits_{\lambda\rightarrow
0}\sup\limits_{\theta\in\Theta}\mathbb{E}\bigg{[}\sup\limits_{0\leqslant
t\leqslant T}|\delta X^{\lambda}_{\theta}(t)|^{p}\bigg{]}=0.$

###### Proof.

Set
$b^{\lambda}_{\theta}(s)=b_{\theta}(s,X^{\lambda}_{\theta}(s),\mathbb{E}[X^{\lambda}_{\theta}(s)],v^{\lambda}(s)),\sigma^{i,\lambda}_{\theta}(s)=\sigma^{i}_{\theta}(s,X^{\lambda}_{\theta}(s),v^{\lambda}(s)).$
According to the definition of $\delta X^{\lambda}_{\theta}$, we have

$\displaystyle\delta X^{\lambda}_{\theta}(t)$
$\displaystyle=\int_{0}^{t}\bigg{\\{}\frac{1}{\lambda}\Big{(}b_{\theta}^{\lambda}(s)-b_{\theta}(s)\Big{)}-\Big{[}\partial_{x}b_{\theta}(s)X^{1}_{\theta}(s)+\partial_{x^{\prime}}b_{\theta}(s)\mathbb{E}[X^{1}_{\theta}(s)]+\partial_{v}b_{\theta}(s)(v(s)-\bar{v}(s))\Big{]}\bigg{\\}}ds$
(4.3)
$\displaystyle+\int_{0}^{t}\sum\limits_{i=1}^{d}\bigg{\\{}\frac{1}{\lambda}\Big{(}\sigma^{\lambda,i}_{\theta}(s)-\sigma^{i}_{\theta}(s)\Big{)}-\Big{[}\partial_{x}\sigma^{i}_{\theta}(s)X^{1}_{\theta}(s)+\partial_{v}\sigma^{i}_{\theta}(s)(v(s)-\bar{v}(s))\Big{]}\bigg{\\}}dW^{i}(s).$

Denote

$\displaystyle\pi_{\theta}^{\rho\lambda}(t)$
$\displaystyle=(\bar{X}_{\theta}(t)+\rho\lambda(\delta
X^{\lambda}_{\theta}(t)+X^{1}_{\theta}(t)),\mathbb{E}[\bar{X}_{\theta}(t)]+\rho\lambda(\mathbb{E}[\delta
X^{\lambda}_{\theta}(t)]+\mathbb{E}[X^{1}_{\theta}(t)]),\bar{v}(t)+\rho\lambda(v(t)-\bar{v}(t))),$
$\displaystyle\tilde{\pi}_{\theta}^{\rho\lambda}(t)$
$\displaystyle=(\bar{X}_{\theta}(t)+\rho\lambda(\delta
X^{\lambda}_{\theta}(t)+X^{1}_{\theta}(t)),\bar{v}(t)+\rho\lambda(v(t)-\bar{v}(t))),$
$\displaystyle A_{\theta}^{\lambda}(t)$
$\displaystyle=\int_{0}^{1}\partial_{x}b_{\theta}(t,\pi_{\theta}^{\rho\lambda}(t))d\rho,\qquad
B_{\theta}^{\lambda}(t)=\int_{0}^{1}\partial_{x^{\prime}}b_{\theta}(t,\pi_{\theta}^{\rho\lambda}(t))d\rho,$
$\displaystyle C_{\theta}^{\lambda}(t)$
$\displaystyle=\int_{0}^{1}[\partial_{v}b_{\theta}(t,\pi_{\theta}^{\rho\lambda}(t))-\partial_{v}b_{\theta}(t)](v(t)-\bar{v}(t))d\rho$
$\displaystyle\quad+[A_{\theta}^{\lambda}(t)-\partial_{x}b_{\theta}(t)]X_{\theta}^{1}(t)+[B_{\theta}^{\lambda}(t)-\partial_{x^{\prime}}b_{\theta}(t)]\mathbb{E}[X_{\theta}^{1}(t)],$
$\displaystyle D_{\theta}^{\lambda,i}(t)$
$\displaystyle=\int_{0}^{1}\partial_{x}\sigma^{i}_{\theta}(t,\tilde{\pi}_{\theta}^{\rho\lambda}(t))d\rho,$
$\displaystyle F_{\theta}^{\lambda,i}(t)$
$\displaystyle=\int_{0}^{1}[\partial_{v}\sigma^{i}_{\theta}(t,\tilde{\pi}_{\theta}^{\rho\lambda}(t))-\partial_{v}\sigma^{i}_{\theta}(t)](v(t)-\bar{v}(t))d\rho+[D_{\theta}^{\lambda,i}(t)-\partial_{x}\sigma^{i}_{\theta}(t)]X_{\theta}^{1}(t).$

Then the equation (4.3) can be written as

$\displaystyle\delta X^{\lambda}_{\theta}(t)$
$\displaystyle=\int_{0}^{t}\Big{(}A_{\theta}^{\lambda}(s)\delta
X^{\lambda}_{\theta}(s)+B_{\theta}^{\lambda}(s)\mathbb{E}[\delta
X^{\lambda}_{\theta}(s)]+C_{\theta}^{\lambda}(s)\Big{)}ds$
$\displaystyle\quad+\sum_{i=1}^{d}\int_{0}^{t}\Big{(}D_{\theta}^{\lambda,i}(s)\delta
X^{\lambda}_{\theta}(s)+F_{\theta}^{\lambda,i}(s)\Big{)}dW^{i}(s).$

Notice that
$\partial_{x}b_{\theta}(t),\partial_{x^{\prime}}b_{\theta}(t),\partial_{v}b_{\theta}(t),\partial_{x}\sigma^{i}_{\theta}(t),\partial_{v}\sigma^{i}_{\theta}(t)$
are bounded. Thanks to Theorem 3.2 and (4.2), it yields, for $p>1$,

$\displaystyle\mathbb{E}\Big{[}\sup\limits_{0\leqslant t\leqslant T}|\delta
X^{\lambda}_{\theta}(t)|^{p}\Big{]}$ $\displaystyle\leqslant
C\mathbb{E}\bigg{(}\int_{0}^{T}|C_{\theta}^{\lambda}(s)|ds\bigg{)}^{p}+C\mathbb{E}\bigg{(}\int_{0}^{T}\sum\limits_{i=1}^{d}|F_{\theta}^{\lambda,i}(s)|^{2}ds\bigg{)}^{\frac{p}{2}}$
$\displaystyle\leqslant
C\bigg{\\{}\mathbb{E}\Big{[}\int_{0}^{T}(|v(t)|^{p}+|\bar{v}(t)|^{p})dt\Big{]}+\Big{(}\mathbb{E}\int_{0}^{T}|v(t)|^{2}dt\Big{)}^{\frac{p}{2}}+\Big{(}\mathbb{E}\int_{0}^{T}|\bar{v}(t)|^{2}dt\Big{)}^{\frac{p}{2}}\bigg{\\}}.$

Next, we turn to prove $\mathrm{(ii)}$. It is enough to prove

$\left\\{\begin{aligned} &\lim\limits_{\lambda\rightarrow
0}\sup\limits_{\theta\in\Theta}\mathbb{E}\bigg{(}\int_{0}^{T}|C_{\theta}^{\lambda}(s)|ds\bigg{)}^{p}=0,\\\
&\lim\limits_{\lambda\rightarrow
0}\sup\limits_{\theta\in\Theta}\mathbb{E}\bigg{(}\int_{0}^{T}\sum\limits_{i=1}^{d}|F_{\theta}^{\lambda,i}(s)|^{2}ds\bigg{)}^{\frac{p}{2}}=0.\end{aligned}\right.$

On one hand, we have

$\displaystyle|C_{\theta}^{\lambda}(s)|^{p}$
$\displaystyle\leqslant\Big{(}2^{p-1}\vee
1\Big{)}\bigg{\\{}\int_{0}^{1}|\partial_{v}b_{\theta}(s,\pi_{\theta}^{\rho\lambda}(s))-\partial_{v}b_{\theta}(s)|^{p}|v(s)-\bar{v}(s)|^{p}d\rho$
$\displaystyle\qquad\qquad\qquad\quad+\int_{0}^{1}|\partial_{x}b_{\theta}(s,\pi_{\theta}^{\rho\lambda}(s))-\partial_{x}b_{\theta}(s)|^{p}|X_{\theta}^{1}(s)|^{p}d\rho$
$\displaystyle\qquad\qquad\qquad\quad+\int_{0}^{1}|\partial_{x^{\prime}}b_{\theta}(s,\pi_{\theta}^{\rho\lambda}(s))-\partial_{x^{\prime}}b_{\theta}(s)|^{p}|\mathbb{E}[X_{\theta}^{1}(s)]|^{p}d\rho\bigg{\\}}.$

From $ab\leqslant\frac{1}{2}a^{2}+\frac{1}{2}b^{2},a,b\in\mathbb{R}^{+}$, we
have

$\displaystyle|C_{\theta}^{\lambda}(s)|^{p}$ $\displaystyle\leqslant
C_{p}\lambda^{p}\bigg{\\{}|\delta
X^{\lambda}_{\theta}(s)+X^{1}_{\theta}(s)|^{2p}+|\mathbb{E}[\delta
X^{\lambda}_{\theta}(s)+X^{1}_{\theta}(s)]|^{2p}$
$\displaystyle\quad+|v(s)-\bar{v}(s)|^{2p}+|X_{\theta}^{1}(s)|^{2p}+|\mathbb{E}[X_{\theta}^{1}(s)]|^{2p}\bigg{\\}},$

where the constant $C_{p}$ depends on $p$. By (4.2) and $\mathrm{(i)}$ in 4.2
$\lim\limits_{\lambda\rightarrow
0}\sup\limits_{\theta\in\Theta}\mathbb{E}(\int_{0}^{T}|C_{\theta}^{\lambda}(s)|ds)^{p}=0$
follows immediately. On the other hand, based on the fact that

$\displaystyle|F_{\theta}^{\lambda,i}(s)|^{2}$ $\displaystyle\leqslant
C\lambda^{2}\bigg{\\{}|\delta
X^{\lambda}_{\theta}(s)+X^{1}_{\theta}(s)|^{4}+|\mathbb{E}[\delta
X^{\lambda}_{\theta}(s)+X^{1}_{\theta}(s)]|^{4}$
$\displaystyle\quad+|v(s)-\bar{v}(s)|^{4}+|X_{\theta}^{1}(s)|^{4}+|\mathbb{E}[X_{\theta}^{1}(s)]|^{4}\bigg{\\}},$

we obtain

$\displaystyle\mathbb{E}\bigg{(}\int_{0}^{T}\sum\limits_{i=1}^{d}|F_{\theta}^{\rho\lambda,i}(s)|^{2}ds\bigg{)}^{\frac{p}{2}}$
$\displaystyle\leqslant
C_{p,d}\lambda^{p}\bigg{\\{}\mathbb{E}\Big{(}\int_{0}^{T}|v(s)-\bar{v}(s)|^{4}ds\Big{)}^{\frac{p}{2}}$
$\displaystyle\quad+\mathbb{E}\Big{[}\sup_{0\leqslant s\leqslant T}|\delta
X^{\lambda}_{\theta}(s)|^{2p}\Big{]}+\mathbb{E}\Big{[}\sup_{0\leqslant
s\leqslant T}|X^{1}_{\theta}(s)|^{2p}\Big{]}\bigg{\\}},$

where the constant $C_{p,d}$ depends on $p$ and $d$. Notice that by Hölder
inequality we have

$\text{if}\
1<p<2,\quad\mathbb{E}\Big{(}\int_{0}^{T}|v(s)-\bar{v}(s)|^{4}ds\Big{)}^{\frac{p}{2}}\leqslant\Big{(}\mathbb{E}\int_{0}^{T}|v(s)-\bar{v}(s)|^{4}ds\Big{)}^{\frac{p}{2}},$

and

$\text{if}\
p>2,\quad\mathbb{E}\Big{(}\int_{0}^{T}|v(s)-\bar{v}(s)|^{4}ds\Big{)}^{\frac{p}{2}}\leqslant
T^{\frac{p}{p-2}}\mathbb{E}\int_{0}^{T}|v(s)-\bar{v}(s)|^{2p}ds.$

Finally, by (4.2) and $\mathrm{(i)}$ in 4.2 again, we derive

$\lim\limits_{\lambda\rightarrow
0}\sup\limits_{\theta\in\Theta}\mathbb{E}\bigg{(}\int_{0}^{T}\sum\limits_{i=1}^{d}|F_{\theta}^{\rho\lambda,i}(s)|^{2}ds\bigg{)}^{\frac{p}{2}}=0.$

∎

We next introduce the variational BSDE on $[0,T]$: for each $\theta\in\Theta$,

$\left\\{\begin{aligned}
-dY^{1}_{\theta}(t)&=\bigg{(}\partial_{x}f_{\theta}(t)X^{1}_{\theta}(t)+\partial_{x^{\prime}}f_{\theta}(t)\mathbb{E}[X^{1}_{\theta}(t)]+\partial_{y}f_{\theta}(t)Y^{1}_{\theta}(t)+\partial_{z}f_{\theta}(t)Z^{1}_{\theta}(t)\\\
&\quad\ \ \
+\partial_{v}f_{\theta}(t)(v(t)-\bar{v}(t))\bigg{)}dt-Z^{1}_{\theta}(t)dW(t),\\\
Y^{1}_{\theta}(T)&=\partial_{x}\Phi_{\theta}(T)X_{\theta}^{1}(T)+\partial_{x^{\prime}}\Phi_{\theta}(T)\mathbb{E}[X_{\theta}^{1}(T)].\end{aligned}\right.$
(4.4)

Since $|\partial_{z}f_{\theta}(t)|\leqslant C_{2}(1+|\bar{Z}_{\theta}(t)|)$,
we know from (ii) in (3.1) that $\partial_{z}f_{\theta}\cdot W$ is a BMO
martingale and the equation (4.4) is a linear BSDE with unbounded coefficient.
Due to $|\partial_{x}\Phi_{\theta}(T)|\leqslant C$ and
$|\partial_{x^{\prime}}\Phi_{\theta}(T)|\leqslant C$, by [16, Proposition
3.1], it has a unique solution $(Y,Z)\in\bigcap_{p>1}({\cal
S}^{p}_{\mathbb{F}}(0,T;\mathbb{R})\times{\cal
H}^{2,\frac{p}{2}}_{\mathbb{F}}(0,T;\mathbb{R}^{d}))$, and we further have for
all $\bar{p}>p_{\partial_{z}f_{\theta}}^{*}$ and
$p_{\partial_{z}f_{\theta}}^{*}<p<\bar{p}$, there exists a constant $C>0$
depending on
$p,\bar{p},T,\|\partial_{x}f_{\theta}\|_{\infty},\|\partial_{x^{\prime}}f_{\theta}\|_{\infty},\|\partial_{y}f_{\theta}\|_{\infty}$
and $\|\partial_{z}f_{\theta}\cdot W\|_{BMO}$ such that,

$\displaystyle\mathbb{E}\bigg{[}\sup_{0\leqslant t\leqslant
T}|Y_{\theta}^{1}(t)|^{p}+\bigg{(}\int_{0}^{T}|Z_{\theta}^{1}(t)|^{2}dt\bigg{)}^{\frac{p}{2}}\bigg{]}$
(4.5) $\displaystyle\leqslant
C\bigg{\\{}\mathbb{E}\Big{[}\int_{0}^{T}(|v(t)|^{\bar{p}}+|\bar{v}(t)|^{\bar{p}})dt\Big{]}+\Big{(}\mathbb{E}\int_{0}^{T}|v(t)|^{2}dt\Big{)}^{\frac{\bar{p}}{2}}+\Big{(}\mathbb{E}\int_{0}^{T}|\bar{v}(t)|^{2}dt\Big{)}^{\frac{{}^{\bar{p}}}{2}}\bigg{\\}}^{\frac{p}{\bar{p}}}.$

Then we consider the continuity of the pair $(Y^{1}_{\theta},Z^{1}_{\theta})$
with respect to $\theta$.

######  Proposition 4.3.

Under 1, for $q>p_{\partial_{z}f_{\theta}}^{*}$,

$\lim_{\epsilon\rightarrow
0}\sup_{d(\theta,\tilde{\theta})\leqslant\epsilon}\mathbb{E}\bigg{[}\sup_{0\leqslant
t\leqslant
T}|Y_{\theta}^{1}(t)-Y_{\tilde{\theta}}^{1}(t)|^{q}+\Big{(}\int_{0}^{T}|Z_{\theta}^{1}(t)-Z_{\tilde{\theta}}^{1}(t)|^{2}\Big{)}^{\frac{q}{2}}\bigg{]}=0.$

The proof is similar to that of 3.3. Hence we omit it to save space.

Set

$\displaystyle\delta Y^{\lambda}_{\theta}(t)$
$\displaystyle=\frac{1}{\lambda}(Y_{\theta}^{\lambda}(t)-\bar{Y}_{\theta}(t))-Y^{1}_{\theta}(t),\quad\delta
Z^{\lambda}_{\theta}(t)=\frac{1}{\lambda}(Z_{\theta}^{\lambda}(t)-\bar{Z}_{\theta}(t))-Z^{1}_{\theta}(t),$
(4.6) $\displaystyle\Gamma_{\theta}(t)$ $\displaystyle:={\cal
E}\Big{(}\int_{0}^{t}\partial_{z}f_{\theta}(s)^{\top}dW(s)\Big{)},\ \ \
t\in[0,T].$

The following proposition demonstrates the continuity of $(\delta
Y^{\lambda}_{\theta},\delta Z^{\lambda}_{\theta})$ with respect to $\theta$.

######  Proposition 4.4.

Under 1, for $p\in(1\vee 2p_{\partial_{z}f_{\theta}}^{-1},2)$,

$\displaystyle\lim\limits_{\lambda\rightarrow
0}\sup\limits_{\theta\in\Theta}\mathbb{E}\bigg{[}\sup\limits_{0\leqslant
t\leqslant T}\Gamma_{\theta}(t)|\delta
Y^{\lambda}_{\theta}(t)|^{p}+\Big{(}\int_{0}^{T}(\Gamma_{\theta}(t))^{\frac{2}{p}}|\delta
Z^{\lambda}_{\theta}(t)|^{2}dt\Big{)}^{\frac{p}{2}}\bigg{]}=0.$

###### Proof.

Set

$\displaystyle\kappa^{\lambda\rho}(T)$
$\displaystyle:=\Big{(}\bar{X}_{\theta}(T)+\lambda\rho(X^{1}_{\theta}(T)+\delta
X_{\theta}^{\lambda}(T)),\mathbb{E}[\bar{X}_{\theta}(T)]+\lambda\rho(\mathbb{E}[X^{1}_{\theta}(T)]+\mathbb{E}[\delta
X_{\theta}^{\lambda}(T)])\Big{)},$
$\displaystyle\pi_{\theta}^{\lambda\rho}(t)$
$\displaystyle:=\Big{(}\bar{X}_{\theta}(t)+\lambda\rho(X^{1}_{\theta}(t)+\delta
X_{\theta}^{\lambda}(t)),\mathbb{E}[\bar{X}_{\theta}(t)]+\lambda\rho(\mathbb{E}[X^{1}_{\theta}(t)]+\mathbb{E}[\delta
X_{\theta}^{\lambda}(t)])$ $\displaystyle\quad\ \ \
\bar{Y}_{\theta}(t)+\lambda\rho(Y^{1}_{\theta}(t)+\delta
Y_{\theta}^{\lambda}(t)),\bar{Z}_{\theta}(t)+\lambda\rho(Z^{1}_{\theta}(t)+\delta
Z_{\theta}^{\lambda}(t)),\bar{v}(t)+\lambda\rho(v(t)-\bar{v}(t))\Big{)},$
$\displaystyle A^{\lambda}_{\theta}(t)$
$\displaystyle:=\int_{0}^{1}\partial_{x}f_{\theta}(t,\pi_{\theta}^{\lambda\rho}(t))d\rho,\quad\bar{A}^{\lambda}_{\theta}(t):=\int_{0}^{1}\partial_{x^{\prime}}f_{\theta}(t,\pi_{\theta}^{\lambda\rho}(t))d\rho,$
$\displaystyle B^{\lambda}_{\theta}(t)$
$\displaystyle:=\int_{0}^{1}\partial_{y}f_{\theta}(t,\pi_{\theta}^{\lambda\rho}(t))d\rho,\quad
C^{\lambda}_{\theta}(t):=\int_{0}^{1}\partial_{z}f_{\theta}(t,\pi_{\theta}^{\lambda\rho}(t))d\rho.$

By the definitions of $\delta Y^{\lambda}_{\theta},\delta
Z^{\lambda}_{\theta}$, we arrive at

$\displaystyle\delta Y^{\lambda}_{\theta}(t)$
$\displaystyle=I^{1,\lambda}_{\theta}+I^{2,\lambda}_{\theta}$
$\displaystyle\quad+\int_{t}^{T}\Big{(}\partial_{x}f(s)\delta
X_{\theta}^{\lambda}(s)+\partial_{x^{\prime}}f(s)\mathbb{E}[\delta
X_{\theta}^{\lambda}(s)]+\partial_{y}f(s)\delta
Y_{\theta}^{\lambda}(s)+\partial_{z}f(s)\delta
Z_{\theta}^{\lambda}(s)+D^{\lambda}_{\theta}(s)\Big{)}ds$
$\displaystyle\quad-\int_{t}^{T}\delta Z_{\theta}^{\lambda}(s)dW(s),\
t\in[0,T],$

where

$\displaystyle I^{1,\lambda}_{\theta}$
$\displaystyle:=\int_{0}^{1}\partial_{x}\Phi_{\theta}(\kappa^{\lambda\rho}(T))d\rho\cdot\delta
X^{\lambda}_{\theta}(T)+\int_{0}^{1}\partial_{x^{\prime}}\Phi_{\theta}(\kappa^{\lambda\rho}(T))d\rho\cdot\mathbb{E}[\delta
X^{\lambda}_{\theta}(T)],$ $\displaystyle I^{2,\lambda}_{\theta}$
$\displaystyle:=\Big{(}\int_{0}^{1}\partial_{x}\Phi_{\theta}(\kappa^{\lambda\rho}(T))d\rho-\partial_{x}\Phi_{\theta}(T)\Big{)}X_{\theta}^{1}(T)+\Big{(}\int_{0}^{1}\partial_{x^{\prime}}\Phi_{\theta}(\kappa^{\lambda\rho}(T))d\rho-\partial_{x^{\prime}}\Phi_{\theta}(T)\Big{)}\mathbb{E}[X_{\theta}^{1}(T)],$
$\displaystyle D^{\lambda}_{\theta}(t)$
$\displaystyle:=\int_{0}^{1}\Big{(}\partial_{v}f_{\theta}(t,\pi_{\theta}^{\lambda\rho}(t))-\partial_{v}f_{\theta}(t)\Big{)}(v(t)-\bar{v}(t))d\rho+\Big{(}A^{\lambda}_{\theta}(t)-\partial_{x}f_{\theta}(t)\Big{)}(X^{\lambda}_{\theta}(t)-\bar{X}_{\theta}(t))$
$\displaystyle\quad\ \
+\Big{(}\bar{A}^{\lambda}_{\theta}(t)-\partial_{x^{\prime}}f_{\theta}(t)\Big{)}\mathbb{E}[X^{\lambda}_{\theta}(t)-\bar{X}_{\theta}(t)]+\Big{(}B^{\lambda}_{\theta}(t)-\partial_{y}f_{\theta}(t)\Big{)}(Y^{\lambda}_{\theta}(t)-\bar{Y}_{\theta}(t))$
$\displaystyle\quad\ \
+\Big{(}C^{\lambda}_{\theta}(t)-\partial_{z}f_{\theta}(t)\Big{)}(Z^{\lambda}_{\theta}(t)-\bar{Z}_{\theta}(t)).$

By [16, Proposition 3.2], it yields, for any $p\in(1\vee
2p_{\partial_{z}f_{\theta}}^{-1},2)$ and $p<\bar{p}<2$,

$\displaystyle\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\Gamma_{\theta}(t)|\delta
Y^{\lambda}_{\theta}(t)|^{p}+\Big{(}\int_{0}^{T}(\Gamma_{\theta}(t))^{\frac{2}{p}}|\delta
Z^{\lambda}_{\theta}(t)|^{2}dt\Big{)}^{\frac{p}{2}}\bigg{]}$ (4.7)
$\displaystyle\leqslant
C_{0}\bigg{(}\mathbb{E}\Big{[}(\Gamma_{\theta}(T))^{\frac{\bar{p}}{p}}|I^{1,\lambda}_{\theta}+I^{2,\lambda}_{\theta}|^{\bar{p}}+\Big{(}\int_{0}^{T}(\Gamma_{\theta}(t))^{\frac{1}{p}}|\partial_{x}f(t)\delta
X_{\theta}^{\lambda}(t)+\partial_{x^{\prime}}f(t)\mathbb{E}[\delta
X_{\theta}^{\lambda}(t)]+D^{\lambda}_{\theta}(t)|dt\Big{)}^{\bar{p}}\Big{]}\bigg{)}^{\frac{p}{\bar{p}}}.$

First, reverse Hölder inequality leads to that, for $p\in(1\vee
2p_{\partial_{z}f_{\theta}}^{-1},2)$,

$\mathbb{E}[(\Gamma_{\theta}(T))^{\frac{2}{p}}]\leqslant
K(\frac{2}{p},\|\partial_{z}f_{\theta}\cdot W\|).\mathcal{}$

Consequently, from 4.2, we know, for all $p\in(1\vee
2p_{\partial_{z}f_{\theta}}^{-1},2)$ and $p<\bar{p}<2$,

$\displaystyle\mathbb{E}\Big{[}(\Gamma_{\theta}(T))^{\frac{\bar{p}}{p}}|I^{1,\lambda}_{\theta}|^{\bar{p}}\Big{]}$
$\displaystyle\leqslant
C\mathbb{E}\Big{[}(\Gamma_{\theta}(T))^{\frac{\bar{p}}{p}}\cdot(|\delta
X_{\theta}^{\lambda}(T)|^{\bar{p}}+|\mathbb{E}[\delta
X_{\theta}^{\lambda}(T)]|^{\bar{p}})\Big{]}$ $\displaystyle\leqslant
C\bigg{\\{}\mathbb{E}\Big{[}(\Gamma_{\theta}(T))^{\frac{2}{p}}\Big{]}\bigg{\\}}^{\frac{\bar{p}}{2}}\cdot\bigg{\\{}\mathbb{E}\Big{[}\sup_{t\in[0,T]}|\delta
X_{\theta}^{\lambda}(t)|^{\frac{2\bar{p}}{2-\bar{p}}}\Big{]}\bigg{\\}}^{\frac{(2-\bar{p})}{2}}$
$\displaystyle\leqslant C\Big{(}K(\frac{2}{p},\|\partial_{z}f_{\theta}\cdot
W\|)\Big{)}^{\frac{\bar{p}}{2}}\bigg{\\{}\mathbb{E}\Big{[}\sup_{t\in[0,T]}|\delta
X_{\theta}^{\lambda}(t)|^{\frac{2\bar{p}}{2-\bar{p}}}\Big{]}\bigg{\\}}^{\frac{(2-\bar{p})}{2}}\rightarrow
0,\quad\text{as}\ \lambda\rightarrow 0.$

We can similarly deduce that, for all $p\in(1\vee
2p_{\partial_{z}f_{\theta}}^{-1},2)$ and $p<\bar{p}<2$,

$\displaystyle\lim_{\lambda\rightarrow
0}\mathbb{E}\Big{[}(\Gamma_{\theta}(T))^{\frac{\bar{p}}{p}}|I^{2,\lambda}_{\theta}|^{\bar{p}}+\Big{(}\int_{0}^{T}(\Gamma_{\theta}(t))^{\frac{1}{p}}|\partial_{x}f(t)\delta
X_{\theta}^{\lambda}(t)+\partial_{x^{\prime}}f(t)\mathbb{E}[\delta
X_{\theta}^{\lambda}(t)]|dt\Big{)}^{\bar{p}}\Big{]}=0.$

Next, we focus on
$\mathbb{E}\Big{[}\Big{(}\int_{0}^{T}(\Gamma_{\theta}(t))^{\frac{1}{p}}|D^{\lambda}_{\theta}(t)|dt\Big{)}^{\bar{p}}\Big{]}$.
Notice

$\displaystyle|C^{\lambda}_{\theta}(t)-\partial_{z}f_{\theta}(t)|$
$\displaystyle\leqslant C\lambda\Big{(}|\delta
X_{\theta}^{\lambda}(t)+X_{\theta}^{1}(t)|+|\mathbb{E}[\delta
X_{\theta}^{\lambda}(t)+X_{\theta}^{1}(t)]|$ $\displaystyle\quad+|\delta
Y_{\theta}^{\lambda}(t)+Y_{\theta}^{1}(t)|+|\delta
Z_{\theta}^{\lambda}(t)+Z_{\theta}^{1}(t)|+|v(t)-\bar{v}(t)|\Big{)}.$

We only deal with the most difficult term
$\mathbb{E}\Big{[}\Big{(}\int_{0}^{T}(\Gamma_{\theta}(t))^{\frac{1}{p}}\cdot|\bar{Z}_{\theta}(t)|\cdot|Z_{\theta}^{1}(t)|dt\Big{)}^{\bar{p}}\Big{]}$.
The other terms can be estimated similarly. Actually, by Hölder inequality, we
have

$\displaystyle\mathbb{E}\Big{[}\Big{(}\int_{0}^{T}(\Gamma_{\theta}(t))^{\frac{1}{p}}\cdot|\bar{Z}_{\theta}(t)|\cdot|Z_{\theta}^{1}(t)|dt\Big{)}^{\bar{p}}\Big{]}$
$\displaystyle\leqslant\bigg{\\{}\mathbb{E}\Big{[}\int_{0}^{T}(\Gamma_{\theta}(t))^{\frac{2}{p}}\cdot|\bar{Z}_{\theta}(t)|^{2}dt\Big{]}\bigg{\\}}^{\frac{\bar{p}}{2}}\bigg{\\{}\mathbb{E}\Big{[}\Big{(}\int_{0}^{T}|Z_{\theta}^{1}(t)|^{2}dt\Big{)}^{\frac{{\bar{p}}}{2-\bar{p}}}\bigg{]}\bigg{\\}}^{\frac{2-\bar{p}}{2}}.$

By Hölder inequality again, together with Doob’s inequality and reverse Höler
inequality, we further have for fixed $p_{1}\in(1\vee
2p_{\partial_{z}f_{\theta}}^{-1},2)$ and any $p>p_{1}$,

$\displaystyle\mathbb{E}\Big{[}\int_{0}^{T}(\Gamma_{\theta}(t))^{\frac{2}{p}}\cdot|\bar{Z}_{\theta}(t)|^{2}dt\Big{]}$
$\displaystyle\leqslant\bigg{\\{}\mathbb{E}\Big{[}\sup_{t\in[0,T]}(\Gamma_{\theta}(t))^{\frac{2}{p_{1}}}\Big{]}\bigg{\\}}^{\frac{p_{1}}{p}}\cdot\bigg{\\{}\mathbb{E}\Big{[}\Big{(}\int_{0}^{T}|\bar{Z}_{\theta}(t)|^{2}dt\Big{)}^{\frac{p}{p-p_{1}}}\Big{]}\bigg{\\}}^{\frac{p-p_{1}}{p}}$
$\displaystyle\leqslant\bigg{\\{}\mathbb{E}\Big{[}(\Gamma_{\theta}(T))^{\frac{2}{p_{1}}}\Big{]}\bigg{\\}}^{\frac{p_{1}}{p}}\cdot\bigg{\\{}\mathbb{E}\Big{[}\Big{(}\int_{0}^{T}|\bar{Z}_{\theta}(t)|^{2}dt\Big{)}^{\frac{p}{p-p_{1}}}\Big{]}\bigg{\\}}^{\frac{p-p_{1}}{p}}<\infty.$

The above and (4.5) implies

$\displaystyle\mathbb{E}\Big{[}\Big{(}\int_{0}^{T}(\Gamma^{\lambda}_{\theta}(t))^{\frac{1}{p}}\cdot|\bar{Z}_{\theta}(t)|\cdot|Z_{\theta}^{1}(t)|dt\Big{)}^{\bar{p}}\Big{]}<\infty.$

The proof is complete. ∎

######  Remark 4.5.

$\mathrm{(i)}$ From (4.7), we also have

$\displaystyle\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\Gamma^{\lambda}_{\theta}(t)|\delta
Y^{\lambda}_{\theta}(t)|^{p}+\Big{(}\int_{0}^{T}(\Gamma^{\lambda}_{\theta}(t))^{\frac{2}{p}}|\delta
Z^{\lambda}_{\theta}(t)|^{2}dt\Big{)}^{\frac{p}{2}}\bigg{]}<\infty,$

which implies $|\delta Y^{\lambda}_{\theta}(0)|<\infty$.

$\mathrm{(ii)}$ Taking $t=0$ it follows from 4.4 that, for $p\in(1\vee
2p_{\partial_{z}f_{\theta}}^{-1},2)$,

$\lim\limits_{\lambda\rightarrow 0}\sup\limits_{\theta\in\Theta}|\delta
Y^{\lambda}_{\theta}(0)|^{p}=0.$

######  Remark 4.6.

If the coefficient $f$ depends on the mean-field terms $\mathbb{E}[Y_{s}]$ and
$\mathbb{E}[Z_{s}]$, there is an essential difficulty in the present method to
prove the above 4.4. We leave it for further work.

###  4.2 Variational inequality

This subsection is devoted to studying variational inequality. For given
$v(\cdot)\in{\cal V}_{ad}$, define

$\displaystyle{\cal Q}^{v}$ $\displaystyle=\bigg{\\{}Q\in{\cal
Q}\bigg{|}J(v(\cdot))=\int_{\Theta}\mathbb{E}\Big{[}\phi_{\theta}(X^{v}_{\theta}(T))+\gamma_{\theta}(Y_{\theta}^{v}(0))\Big{]}Q(d\theta)\bigg{\\}}.$

######  Lemma 4.7.

Under 2, ${\cal Q}^{v}$ is nonempty.

###### Proof.

From the definition of $J(v(\cdot))$, there exists a sequence $Q^{n}\in{\cal
Q}$ such that, for all $v(\cdot)\in{\cal V}_{ad}$,

$\int_{\Theta}\mathbb{E}\Big{[}\phi_{\theta}(X^{v}_{\theta}(T))+\gamma_{\theta}(Y_{\theta}^{v}(0))\Big{]}Q^{n}(d\theta)\geqslant
J(v(\cdot))-\frac{1}{n}.$

Since ${\cal Q}$ is weakly compact, there is some $Q^{v}\in{\cal Q}$ such that
a subsequence of $Q^{n}$ (if necessary) converges weakly to $Q^{v}$. Thanks to
Theorem 3.2, 3.3 and 2, the mapping
$\theta\mapsto\mathbb{E}[\phi_{\theta}(X^{v}_{\theta}(T))+\gamma_{\theta}(Y_{\theta}^{v}(0))]$
is continuous and bounded. Hence, we obtain

$\displaystyle J(v(\cdot))$
$\displaystyle\geqslant\int_{\Theta}\mathbb{E}\Big{[}\phi_{\theta}(X^{v}_{\theta}(T))+\gamma_{\theta}(Y_{\theta}^{v}(0))\Big{]}Q^{v}(d\theta)$
$\displaystyle=\lim_{n\rightarrow\infty}\int_{\Theta}\mathbb{E}\Big{[}\phi_{\theta}(X^{v}_{\theta}(T))+\gamma_{\theta}(Y_{\theta}^{v}(0))\Big{]}Q^{n}(d\theta)\geqslant
J(v(\cdot)).$

Thereby, ${\cal Q}^{v}$ is nonempty. ∎

We now give the variational inequality.

######  Theorem 4.8.

Under 1 and 2, we have

$\displaystyle\lim_{\lambda\rightarrow
0}\frac{1}{\lambda}\Big{(}J(v^{\lambda}(\cdot))-J(\bar{v}(\cdot))\Big{)}=\sup_{Q\in{\cal
Q}^{\bar{v}}}\int_{\Theta}\mathbb{E}\Big{[}\partial_{x}\phi_{\theta}(\bar{X}_{\theta}(T))X^{1}_{\theta}(T)+\partial_{y}\gamma_{\theta}(\bar{Y}_{\theta}(0))Y^{1}_{\theta}(0)\Big{]}Q(d\theta).$

###### Proof.

The proof is split into three steps.

Step 1. We show

$\displaystyle\liminf_{\lambda\rightarrow
0}\frac{1}{\lambda}\Big{(}J(v^{\lambda}(\cdot))-J(\bar{v}(\cdot))\Big{)}\geqslant\sup_{Q\in{\cal
Q}^{\bar{v}}}\int_{\Theta}\mathbb{E}\Big{[}\partial_{x}\phi_{\theta}(\bar{X}_{\theta}(T))X^{1}_{\theta}(T)+\partial_{y}\gamma_{\theta}(\bar{Y}_{\theta}(0))Y^{1}_{\theta}(0)\Big{]}Q(d\theta).$
(4.8)

From the definition of ${\cal Q}^{\bar{v}}$, it yields that, for any
$Q\in{\cal Q}^{\bar{v}}$,

$\displaystyle J(v^{\lambda}(\cdot))$
$\displaystyle\geqslant\int_{\Theta}\mathbb{E}\Big{[}\phi_{\theta}(X^{\lambda}_{\theta}(T))+\gamma_{\theta}(Y_{\theta}^{\lambda}(0))\Big{]}Q(d\theta),$
$\displaystyle J(\bar{v}(\cdot))$
$\displaystyle=\int_{\Theta}\mathbb{E}\Big{[}\phi_{\theta}(\bar{X}_{\theta}(T))+\gamma_{\theta}(\bar{Y}_{\theta}(0))\Big{]}Q(d\theta).$

Consequently, for any $Q\in{\cal Q}^{\bar{v}}$,

$\displaystyle\frac{1}{\lambda}\Big{(}J(v^{\lambda}(\cdot))-J(\bar{v}(\cdot))\Big{)}\geqslant\int_{\Theta}\mathbb{E}\Big{[}\partial_{x}\phi_{\theta}(\bar{X}_{\theta}(T))X^{1}_{\theta}(T)+\partial_{y}\gamma_{\theta}(\bar{Y}_{\theta}(0))Y^{1}_{\theta}(0)\Big{]}Q(d\theta)+J_{1}^{\lambda}+J_{2}^{\lambda},$

where

$\displaystyle
J_{1}^{\lambda}=\int_{\Theta}\mathbb{E}\Big{[}\int_{0}^{1}\partial_{x}\phi_{\theta}^{\lambda\rho}(T)d\rho\cdot\
\delta
X_{\theta}^{\lambda}(T)+\int_{0}^{1}\partial_{y}\gamma_{\theta}^{\lambda\rho}(0)d\rho\cdot\
\delta Y_{\theta}^{\lambda}(0)\Big{]}Q(d\theta),$ (4.9) $\displaystyle
J_{2}^{\lambda}=\int_{\Theta}\mathbb{E}\Big{[}\int_{0}^{1}\Big{(}\partial_{x}\phi_{\theta}^{\lambda\rho}(T)-\partial_{x}\phi_{\theta}(\bar{X}_{\theta}(T))\Big{)}d\rho\cdot\
X_{\theta}^{1}(T)$ $\displaystyle\qquad\
+\int_{0}^{1}\Big{(}\partial_{y}\gamma_{\theta}^{\lambda\rho}(0)-\partial_{y}\gamma_{\theta}(\bar{Y}_{\theta}(0)\Big{)}d\rho\cdot\
Y_{\theta}^{1}(0)\Big{]}Q(d\theta),$
$\displaystyle\partial_{x}\phi_{\theta}^{\lambda\rho}(T)=\partial_{x}\phi_{\theta}(\bar{X}_{\theta}(T)+\lambda\rho(X^{1}_{\theta}(T)+\delta
X^{\lambda}_{\theta}(T))),$
$\displaystyle\partial_{y}\gamma_{\theta}^{\lambda\rho}(0)=\partial_{y}\gamma_{\theta}(\bar{Y}_{\theta}(0)+\lambda\rho(Y^{1}_{\theta}(0)+\delta
Y^{\lambda}_{\theta}(0))).$

On the one hand, since
$|\int_{0}^{1}\partial_{x}\phi_{\theta}^{\lambda\rho}(T)d\rho|\leqslant
L_{1}(1+|\bar{X}_{\theta}(T)|+|X^{1}_{\theta}(T)|+|\delta
X^{\lambda}_{\theta}(T)|)$ and
$|\partial_{y}\gamma_{\theta}^{\lambda\rho}(0)|\leqslant C$, it follows that,
for any $\theta\in\Theta$,

$\displaystyle\mathbb{E}\Big{[}\int_{0}^{1}\partial_{x}\phi_{\theta}^{\lambda\rho}(T)d\rho\cdot\
\delta
X_{\theta}^{\lambda}(T)+\int_{0}^{1}\partial_{y}\gamma_{\theta}^{\lambda\rho}(0)d\rho\cdot\
\delta Y_{\theta}^{\lambda}(0)\Big{]}$ $\displaystyle\leqslant
C\bigg{\\{}\mathbb{E}\Big{[}\sup_{t\in[0,T]}(1+|\bar{X}_{\theta}(t)|^{2}+|X^{1}_{\theta}(t)|^{2}+|\delta
X^{\lambda}_{\theta}(t)|^{2})\Big{]}\bigg{\\}}^{\frac{1}{2}}\cdot\
\bigg{\\{}\mathbb{E}\Big{[}\sup_{t\in[0,T]}|\delta
X_{\theta}^{\lambda}(t)|^{2}\Big{]}\bigg{\\}}^{\frac{1}{2}}+|\delta
Y_{\theta}^{\lambda}(0)|.$

Making use of (3.1), (4.2), 4.2 and 4.4 (or 4.5) we have

$\displaystyle 0\leqslant\lim_{\lambda\rightarrow
0}|J_{1}^{\lambda}|\leqslant\lim_{\lambda\rightarrow
0}\sup_{\theta\in\Theta}\mathbb{E}\Big{[}\int_{0}^{1}\partial_{x}\phi_{\theta}^{\lambda\rho}(T)d\rho\cdot\
\delta
X_{\theta}^{\lambda}(T)+\int_{0}^{1}\partial_{y}\gamma_{\theta}^{\lambda\rho}(0)d\rho\cdot\
\delta Y_{\theta}^{\lambda}(0)\Big{]}=0.$

Moreover, based on (4.2), 4.2, (4.5), 4.5 and the assumption that
$\partial_{y}\gamma_{\theta}$ is Lipschitz continuous with respect to $y$
uniformly in $\theta$, we have

$\displaystyle|J_{2}^{\lambda}|$
$\displaystyle\leqslant\sup_{\theta\in\Theta}\mathbb{E}\Big{[}|\int_{0}^{1}(\partial_{x}\phi_{\theta}^{\lambda\rho}(T)-\partial_{x}\phi_{\theta}(\bar{X}_{\theta}(T)))d\rho|\cdot\
|X_{\theta}^{1}(T)|\Big{]}$
$\displaystyle\qquad+\int_{0}^{1}\sup_{\theta\in\Theta}|\partial_{y}\gamma_{\theta}^{\lambda\rho}(0)-\partial_{y}\gamma_{\theta}(\bar{Y}_{\theta}(0))|d\rho\cdot\
\sup_{\theta\in\Theta}|Y_{\theta}^{1}(0)|$ $\displaystyle\leqslant
C\lambda\mathbb{E}\Big{[}\sup_{t\in[0,T]}|X_{\theta}^{1}(t)|^{2}+\sup_{t\in[0,T]}|\delta
X_{\theta}^{\lambda}(t)|^{2}\Big{]}$
$\displaystyle\quad+\int_{0}^{1}\sup_{\theta\in\Theta}|\partial_{y}\gamma_{\theta}^{\lambda\rho}(0)-\partial_{y}\gamma_{\theta}(\bar{Y}_{\theta}(0))|d\rho\cdot\
\sup_{\theta\in\Theta}|Y_{\theta}^{1}(0)|\rightarrow 0,\quad\text{as}\
\lambda\rightarrow 0.$

From the above estimate, (4.8) follows immediately.

Step 2. Find a subsequence $\lambda_{n}\rightarrow 0$ such that

$\displaystyle\limsup_{\lambda\rightarrow
0}\frac{J(v^{\lambda}(\cdot))-J(\bar{v}(\cdot))}{\lambda}=\lim_{n\rightarrow\infty}\frac{J(v^{\lambda_{n}}(\cdot))-J(\bar{v}(\cdot))}{\lambda_{n}}.$

For each $n\in\mathbb{N}$, due to ${\cal Q}^{v^{\lambda_{n}}}$ being nonempty,
there exists a probability measure $Q^{\lambda_{n}}\in{\cal
Q}^{v^{\lambda_{n}}}$ such that

$\displaystyle
J(v^{\lambda_{n}}(\cdot))=\int_{\Theta}\mathbb{E}\Big{[}\phi_{\theta}(X^{\lambda_{n}}_{\theta}(T))+\gamma_{\theta}(Y_{\theta}^{\lambda_{n}}(0))\Big{]}Q^{\lambda_{n}}(d\theta),$
$\displaystyle
J(\bar{v}(\cdot))\geqslant\int_{\Theta}\mathbb{E}\Big{[}\phi_{\theta}(\bar{X}_{\theta}(T))+\gamma_{\theta}(\bar{Y}_{\theta}(0))\Big{]}Q^{\lambda_{n}}(d\theta).$

To see this, first notice

$\displaystyle\frac{J(v^{\lambda_{n}}(\cdot))-J(\bar{v}(\cdot))}{\lambda_{n}}$
$\displaystyle\leqslant\int_{\Theta}\frac{\mathbb{E}[\phi_{\theta}(X^{\lambda_{n}}_{\theta}(T))-\phi_{\theta}(\bar{X}_{\theta}(T))]+\gamma_{\theta}(Y_{\theta}^{\lambda_{n}}(0))-\gamma_{\theta}(\bar{Y}_{\theta}(0))}{\lambda_{n}}Q^{\lambda_{n}}(d\theta)$
(4.10)
$\displaystyle=\int_{\Theta}\mathbb{E}\Big{[}\partial_{x}\phi_{\theta}(\bar{X}_{\theta}(T))X^{1}_{\theta}(T)+\partial_{y}\gamma_{\theta}(\bar{Y}_{\theta}(0))Y^{1}_{\theta}(0)\Big{]}Q^{\lambda_{n}}(d\theta)+J_{1}^{\lambda_{n}}+J_{2}^{\lambda_{n}},$

where $J_{1}^{\lambda_{n}},J_{2}^{\lambda_{n}}$ are given in (4.9). Similar to
Step 1, we can show

$\lim_{n\rightarrow\infty}\Big{(}|J_{1}^{\lambda_{n}}|+|J_{2}^{\lambda_{n}}|\Big{)}=0.$

Since ${\cal Q}$ is weakly compact, there exists a probability measure
$\widehat{Q}\in{\cal Q}$ such that the sequence
$(Q^{\lambda_{n}})_{n\in\mathbb{N}}$ (choosing its subsequence if necessary)
converges weakly to $\widehat{Q}$. In addition, according to Theorem 3.2, 3.3,
4.1, 4.3, (4.2), (4.5) and 2, we have that the mapping
$\theta\mapsto\mathbb{E}[\partial_{x}\phi_{\theta}(\bar{X}_{\theta}(T))X^{1}_{\theta}(T)+\partial_{y}\gamma_{\theta}(\bar{Y}_{\theta}(0))Y^{1}_{\theta}(0)]$
is continuous and bounded. As $n\rightarrow\infty$ in (4.10), it yields

$\displaystyle\limsup_{\lambda\rightarrow
0}\frac{J(v^{\lambda}(\cdot))-J(\bar{v}(\cdot))}{\lambda}=\lim_{n\rightarrow\infty}\frac{J(v^{\lambda_{n}}(\cdot))-J(\bar{v}(\cdot))}{\lambda_{n}}$
$\displaystyle\leqslant\int_{\Theta}\mathbb{E}\Big{[}\partial_{x}\phi_{\theta}(\bar{X}_{\theta}(T))X^{1}_{\theta}(T)+\partial_{y}\gamma_{\theta}(\bar{Y}_{\theta}(0))Y^{1}_{\theta}(0)\Big{]}\widehat{Q}(d\theta).$

Step 3. We prove $\widehat{Q}\in{\cal Q}^{\bar{v}}$. For this, we need to show

$\displaystyle
J(\bar{v}(\cdot))=\int_{\Theta}\mathbb{E}[\phi_{\theta}(\bar{X}_{\theta}(T))+\gamma_{\theta}(\bar{Y}_{\theta}(0))]\widehat{Q}(d\theta).$

In fact, since $|\partial_{x}\phi_{\theta}^{\lambda_{n}\rho}(T)|\leqslant
L_{1}(1+|\bar{X}_{\theta}(T)|+|X^{1}_{\theta}(T)|+|\delta
X^{\lambda}_{\theta}(T)|)$ and
$|\partial_{y}\gamma_{\theta}^{\lambda_{n}\rho}(0)|\leqslant C$, it follows
from (3.1), (4.2), 4.2, (4.5) and 4.5 that, for all $\theta\in\Theta$,

$\displaystyle\Big{|}\mathbb{E}[\phi_{\theta}(X^{\lambda_{n}}_{\theta}(T))-\phi_{\theta}(\bar{X}_{\theta}(T))]+\gamma_{\theta}(Y_{\theta}^{\lambda_{n}}(0))-\gamma_{\theta}(\bar{Y}_{\theta}(0))\Big{|}$
$\displaystyle=\lambda_{n}\Big{|}\mathbb{E}\Big{[}\int_{0}^{1}\partial_{x}\phi_{\theta}^{\lambda_{n}\rho}(T)d\rho\cdot(X_{\theta}^{1}(T)+\delta
X_{\theta}^{\lambda_{n}}(T))\Big{]}\Big{|}+\lambda_{n}\Big{|}\int_{0}^{1}\partial_{y}\gamma_{\theta}^{\lambda_{n}\rho}(0)d\rho\cdot(Y_{\theta}^{1}(0)+\delta
Y_{\theta}^{\lambda_{n}}(0))\Big{|}$
$\displaystyle\leqslant\lambda_{n}\bigg{\\{}\mathbb{E}\Big{[}1+|\bar{X}_{\theta}(T)|^{2}+|X^{1}_{\theta}(T)|^{2}+|\delta
X^{\lambda_{n}}_{\theta}(T)|^{2}\Big{]}+|Y_{\theta}^{1}(0)+\delta
Y_{\theta}^{\lambda_{n}}(0)|\bigg{\\}}\rightarrow
0,\quad\text{as}\quad\lambda_{n}\rightarrow 0.$

Above inequality leads to

$\displaystyle\lim_{n\rightarrow\infty}|J(v^{\lambda_{n}}(\cdot))-J(\bar{v}(\cdot))|$
$\displaystyle\leqslant\lim_{n\rightarrow\infty}\int_{\Theta}\Big{|}\mathbb{E}[\phi_{\theta}(X^{\lambda_{n}}_{\theta}(T))-\phi_{\theta}(\bar{X}_{\theta}(T))]\Big{|}+\Big{|}\gamma_{\theta}(Y_{\theta}^{\lambda_{n}}(0))-\gamma_{\theta}(\bar{Y}_{\theta}(0))\Big{|}Q^{\lambda_{n}}(d\theta)$
$\displaystyle\leqslant\lim_{n\rightarrow\infty}\sup_{\theta\in\Theta}\mathbb{E}\Big{[}\Big{|}\phi_{\theta}(X^{\lambda_{n}}_{\theta}(T))-\phi_{\theta}(\bar{X}_{\theta}(T))\Big{|}\Big{]}+\Big{|}\gamma_{\theta}(Y_{\theta}^{\lambda_{n}}(0))-\gamma_{\theta}(\bar{Y}_{\theta}(0))\Big{|}=0$

and

$\displaystyle\lim_{n\rightarrow\infty}\Big{|}\int_{\Theta}\Big{(}\mathbb{E}[\phi_{\theta}(X^{\lambda_{n}}_{\theta}(T))+\gamma_{\theta}(Y_{\theta}^{\lambda_{n}}(0))]-\mathbb{E}[\phi_{\theta}(\bar{X}_{\theta}(T))+\gamma_{\theta}(\bar{Y}_{\theta}(0))]\Big{)}Q^{\lambda_{n}}(d\theta)\Big{|}$
$\displaystyle\leqslant\lim_{n\rightarrow\infty}\int_{\Theta}\Big{|}\mathbb{E}[\phi_{\theta}(X^{\lambda_{n}}_{\theta}(T))-\phi_{\theta}(\bar{X}_{\theta}(T))]\Big{|}+\Big{|}\gamma_{\theta}(Y_{\theta}^{\lambda_{n}}(0))-\gamma_{\theta}(\bar{Y}_{\theta}(0))\Big{|}Q^{\lambda_{n}}(d\theta)$
$\displaystyle\leqslant\lim_{n\rightarrow\infty}\sup_{\theta\in\Theta}\bigg{\\{}\mathbb{E}\Big{[}\Big{|}\phi_{\theta}(X^{\lambda_{n}}_{\theta}(T))-\phi_{\theta}(\bar{X}_{\theta}(T))\Big{|}\Big{]}+\Big{|}\gamma_{\theta}(Y_{\theta}^{\lambda_{n}}(0))-\gamma_{\theta}(\bar{Y}_{\theta}(0))\Big{|}\bigg{\\}}=0.$

Consequently, it yields

$\displaystyle J(\bar{v}(\cdot))$
$\displaystyle=\lim_{n\rightarrow\infty}J(v^{\lambda_{n}}(\cdot))=\lim_{n\rightarrow\infty}\int_{\Theta}\mathbb{E}[\phi_{\theta}(X^{\lambda_{n}}_{\theta}(T))+\gamma_{\theta}(Y_{\theta}^{\lambda_{n}}(0))]Q^{\lambda_{n}}(d\theta)$
$\displaystyle=\lim_{n\rightarrow\infty}\int_{\Theta}\mathbb{E}[\phi_{\theta}(\bar{X}_{\theta}(T))+\gamma_{\theta}(\bar{Y}_{\theta}(0))]Q^{\lambda_{n}}(d\theta)=\int_{\Theta}\mathbb{E}[\phi_{\theta}(\bar{X}_{\theta}(T))+\gamma_{\theta}(\bar{Y}_{\theta}(0))]\widehat{Q}(d\theta).$

∎

From Theorem 4.8, we have the main theorem in this section.

######  Theorem 4.9.

Under 1 and 2, there exists a probability $\bar{Q}\in{\cal Q}^{\bar{v}}$ such
that, for any $v(\cdot)\in{\cal V}_{0,T}$,

$\displaystyle\int_{\Theta}\mathbb{E}\Big{[}\partial_{x}\phi_{\theta}(\bar{X}_{\theta}(T))X^{1}_{\theta}(T)+\partial_{y}\gamma_{\theta}(\bar{Y}_{\theta}(0))Y^{1}_{\theta}(0)\Big{]}\bar{Q}(d\theta)\geqslant
0.$

###### Proof.

For a more precise expression, we write $X_{\theta}^{1},Y_{\theta}^{1}$ as
$X_{\theta}^{1,v},Y_{\theta}^{1,v}$. It follows Theorem 4.8 that, for any
$v(\cdot)\in{\cal V}_{ad}$,

$\displaystyle 0\leqslant\lim_{\lambda\rightarrow
0}\frac{1}{\lambda}\Big{(}J(v^{\lambda}(\cdot))-J(\bar{v}(\cdot))\Big{)}=\sup_{Q\in{\cal
Q}^{\bar{v}}}\int_{\Theta}\mathbb{E}\Big{[}\partial_{x}\phi_{\theta}(\bar{X}_{\theta}(T))X^{1,v}_{\theta}(T)+\partial_{y}\gamma_{\theta}(\bar{Y}_{\theta}(0))Y^{1,v}_{\theta}(0)\Big{]}Q(d\theta),$

which implies

$\displaystyle\inf_{v(\cdot)\in{\cal V}_{ad}}\sup_{Q\in{\cal
Q}^{\bar{v}}}\int_{\Theta}\mathbb{E}\Big{[}\partial_{x}\phi_{\theta}(\bar{X}_{\theta}(T))X^{1,v}_{\theta}(T)+\partial_{y}\gamma_{\theta}(\bar{Y}_{\theta}(0))Y^{1,v}_{\theta}(0)\Big{]}Q(d\theta)\geqslant
0.$

Define
$\Upsilon_{\theta}^{v}:=\mathbb{E}\Big{[}\partial_{x}\phi_{\theta}(\bar{X}_{\theta}(T))X^{1,v}_{\theta}(T)+\partial_{y}\gamma_{\theta}(\bar{Y}_{\theta}(0))Y^{1,v}_{\theta}(0)\Big{]}$.
From the linearity of $X^{1,v},Y^{1,v}_{\theta}$ with respect to $v$, we know
that $v\mapsto\Upsilon_{\theta}^{v}$ is a linear mapping, i.e., for
$v(\cdot),v^{\prime}(\cdot)\in{\cal V}_{ad},0<l<1$ and $\theta\in\Theta$,

$\displaystyle\Upsilon_{\theta}^{lv+(1-l)v^{\prime}}=l\Upsilon_{\theta}^{v}+(1-l)\Upsilon_{\theta}^{v^{\prime}}.$

In addition, for $v(\cdot),v^{\prime}(\cdot)\in{\cal V}_{ad}$ and
$\theta\in\Theta,$

$\displaystyle|\Upsilon_{\theta}^{v}-\Upsilon_{\theta}^{v^{\prime}}|$
$\displaystyle\leqslant
C\mathbb{E}\Big{[}(1+|\bar{X}_{\theta}(T)|)|X_{\theta}^{1,v}(T)-X_{\theta}^{1,v^{\prime}}(T)|\Big{]}+|Y_{\theta}^{1,v}(0)-Y_{\theta}^{1,v^{\prime}}(0)|$
$\displaystyle\leqslant
C\Big{\\{}\mathbb{E}[1+|\bar{X}_{\theta}(T)|^{2}]\Big{\\}}^{\frac{1}{2}}\cdot\Big{\\{}\mathbb{E}[|X_{\theta}^{1,v}(T)-X_{\theta}^{1,v^{\prime}}(T)|^{2}]\Big{\\}}^{\frac{1}{2}}+|Y_{\theta}^{1,v}(0)-Y_{\theta}^{1,v^{\prime}}(0)|.$

This, together with (4.2) and (4.5), can show that $\Upsilon_{\theta}^{v}$ is
continuous with respect to $v$ uniformly in $\theta$. Making use of Sion’s
minimax theorem, we obtain

$\displaystyle\inf_{v(\cdot)\in{\cal V}_{ad}}\sup_{Q\in{\cal
Q}^{\bar{v}}}\int_{\Theta}\mathbb{E}\Big{[}\partial_{x}\phi_{\theta}(\bar{X}_{\theta}(T))X^{1,v}_{\theta}(T)+\partial_{y}\gamma_{\theta}(\bar{Y}_{\theta}(0))Y^{1,v}_{\theta}(0)\Big{]}Q(d\theta)$
$\displaystyle=\sup_{Q\in{\cal Q}^{\bar{v}}}\inf_{v(\cdot)\in{\cal
V}_{ad}}\int_{\Theta}\mathbb{E}\Big{[}\partial_{x}\phi_{\theta}(\bar{X}_{\theta}(T))X^{1,v}_{\theta}(T)+\partial_{y}\gamma_{\theta}(\bar{Y}_{\theta}(0))Y^{1,v}_{\theta}(0)\Big{]}Q(d\theta)\geqslant
0.$

Consequently, for any $\epsilon>0$, there exists a $Q^{\epsilon}\in{\cal
Q}^{\bar{v}}$ such that

$\displaystyle\inf_{v(\cdot)\in{\cal
V}_{ad}}\int_{\Theta}\mathbb{E}\Big{[}\partial_{x}\phi_{\theta}(\bar{X}_{\theta}(T))X^{1,v}_{\theta}(T)+\partial_{y}\gamma_{\theta}(\bar{Y}_{\theta}(0))Y^{1,v}_{\theta}(0)\Big{]}Q^{\epsilon}(d\theta)\geqslant-\epsilon.$

Finally, the compactness of ${\cal Q}^{\bar{v}}$ allows us to take a
subsequence $\epsilon_{n}\rightarrow 0$ such that $Q^{\epsilon_{n}}$ converges
weakly to a probability measure $\bar{Q}\in{\cal Q}^{\bar{v}}$ and, for all
$v(\cdot)\in{\cal V}_{ad}$,

$\displaystyle\int_{\Theta}\mathbb{E}\Big{[}\partial_{x}\phi_{\theta}(\bar{X}_{\theta}(T))X^{1,v}_{\theta}(T)+\partial_{y}\gamma_{\theta}(\bar{Y}_{\theta}(0))Y^{1,v}_{\theta}(0)\Big{]}\bar{Q}(d\theta)$
$\displaystyle=\lim_{\epsilon_{n}\rightarrow
0}\int_{\Theta}\mathbb{E}\Big{[}\partial_{x}\phi_{\theta}(\bar{X}_{\theta}(T))X^{1,v}_{\theta}(T)+\partial_{y}\gamma_{\theta}(\bar{Y}_{\theta}(0))Y^{1,v}_{\theta}(0)\Big{]}Q^{\epsilon_{n}}(d\theta)\geqslant
0.$

∎

##  5 Necessary and sufficient maximum principles

The necessary and sufficient maximum principles are discussed in this section.
For this, we introduce the adjoint equation:

$\left\\{\begin{aligned}
dp_{\theta}(t)&=\partial_{y}f_{\theta}(t)p_{\theta}(t)dt+\partial_{z}f_{\theta}(t)p_{\theta}(t)dW(t),\
\ \ t\in[0,T],\\\
-dq_{\theta}(t)&=\Big{(}-\partial_{x}f_{\theta}(t)p_{\theta}(t)+(\partial_{x}b_{\theta}(t))^{\top}q_{\theta}(t)+\sum_{i=1}^{d}(\partial^{i}_{x}\sigma_{\theta}(t))^{\top}r^{i}_{\theta}(t)-\mathbb{E}[\partial_{x^{\prime}}f_{\theta}(t)p_{\theta}(t)]\\\
&\quad+\mathbb{E}[(\partial_{x^{\prime}}b_{\theta}(t))^{\top}q_{\theta}(t)]\Big{)}dt-
r_{\theta}(t)dW(t),\ \ \ t\in[0,T],\\\
p_{\theta}(0)&=-\partial_{y}\gamma_{\theta}(\bar{Y}_{\theta}(0)),\\\
q_{\theta}(T)&=-(\partial_{x}\Phi_{\theta}(\bar{X}_{\theta}(T),\mathbb{E}[\bar{X}_{\theta}(T)]))^{\top}p_{\theta}(T)-\mathbb{E}\Big{[}\partial_{x^{\prime}}\Phi_{\theta}(\bar{X}_{\theta}(T),\mathbb{E}[\bar{X}_{\theta}(T)]))^{\top}p_{\theta}(T)\Big{]}\\\
&\quad+\partial_{x}\phi_{\theta}(\bar{X}_{\theta}(T)).\end{aligned}\right.$
(5.1)

The forward equation in (5.1) is a $1$-dimensional SDE with unbounded
coefficient.

The solvability of (5.1) comes from the following lemma which is an immediate
adaptation of [11, Lemma 3 and Lemma 4].

######  Lemma 5.1.

Assume
$\mathbb{E}\Big{[}\int_{0}^{T}|\varphi_{1}(t)|^{p}dt\Big{]}<\infty,\mathbb{E}\Big{(}\int_{0}^{T}|\varphi_{2}(t)|^{2}dt\Big{)}^{\frac{p}{2}}<\infty,$
and $a_{1}\cdot W$, $a_{2}\cdot W\in\text{BMO}$. Then for $p>1$, the
$1$-dimensional SDE with unbounded coefficient below

$\left\\{\begin{aligned}
dX(t)&=[a_{1}(t)X(t)+\varphi_{1}(t)]dt+[a_{2}(t)X(t)+\varphi_{2}(t)]dW(t),\
t\in[0,T],\\\ X(0)&=x_{0}\in\mathbb{R}.\end{aligned}\right.$ (5.2)

has a unique solution $X(\cdot)\in{\cal
S}_{\mathbb{F}}^{p}([0,T];\mathbb{R})$.

###### Proof.

Define

$\displaystyle\tau_{0}=0,\quad\tau_{n+1}$
$\displaystyle=\inf\bigg{\\{}t>\tau_{n}:\Big{(}\int_{\tau_{n}}^{t}|a_{2}(s)|^{2}ds\Big{)}^{\frac{p}{2}}\geqslant
2^{-p-1}C_{p}^{-1}\quad\text{or}\ \int_{\tau_{n}}^{t}|a_{1}(s)|^{p}ds\geqslant
2^{-p-1}\bigg{\\}}\wedge T,$

where $C_{p}$ only depending on $p$ is the constant in Buckholder-Davis-Gundy
inequality. Clearly, $\tau_{n}\uparrow T$, as $n\uparrow\infty$.

The proof is split into two steps.

Step 1. We show that the equation (5.2) possesses a unique solution
$X_{1}(\cdot)\in{\cal S}_{\mathbb{F}}^{p}([\tau_{0},\tau_{1}];\mathbb{R})$ on
the interval $[\tau_{0},\tau_{1}]$ with the fixed point theorem. Let
$X\in{\cal S}_{\mathbb{F}}^{p}([\tau_{0},\tau_{1}];\mathbb{R})$. Define

$\left\\{\begin{aligned}
\Gamma(X)(t)&:=x_{0}+\int_{\tau_{0}}^{t}\Big{[}a_{1}(s)X(s)+\varphi_{1}(s)\Big{]}ds+\int_{\tau_{0}}^{t}\Big{[}a_{2}(s)X(s)+\varphi_{2}(s)\Big{]}dW(s),\
t\in[\tau_{0},\tau_{1}],\\\ \Gamma(X)(\tau_{0})&:=x_{0}.\end{aligned}\right.$

It is easy to check $\Gamma(X)\in{\cal
S}_{\mathbb{F}}^{p}([\tau_{0},\tau_{1}];\mathbb{R})$. For
$X^{\prime},X^{\prime\prime}\in{\cal
S}_{\mathbb{F}}^{p}([\tau_{0},\tau_{1}];\mathbb{R})$, define

$\Delta X=X^{\prime}-X^{\prime\prime},\quad\Gamma(\Delta
X)=\Gamma(X^{\prime})-\Gamma(X^{\prime\prime}).$

Then it follows that

$\left\\{\begin{aligned} \Gamma(\Delta
X)(t)&=\int_{\tau_{0}}^{t}a_{1}(s)\Delta
X(s)ds+\int_{\tau_{0}}^{t}a_{2}(s)\Delta X(s)dW(s),\quad
t\in[\tau_{0},\tau_{1}],\\\ \Gamma(\Delta
X)(\tau_{0})&=0.\end{aligned}\right.$

By Buckholder-Davis-Gundy inequality and Hölder inequality, we have, for $p>1$

$\displaystyle\mathbb{E}\Big{[}\sup_{t\in[\tau_{0},\tau_{1}]}|\Gamma(\Delta
X)(t)|^{p}\Big{]}$ $\displaystyle\leqslant
2^{p-1}\mathbb{E}\Big{[}\int_{\tau_{0}}^{\tau_{1}}|a_{1}(s)|^{p}ds\cdot\sup_{t\in[\tau_{0},\tau_{1}]}|\Delta
X(t)|^{p}\Big{]}$ $\displaystyle\ \ \
+2^{p-1}C_{p}\mathbb{E}\Big{[}\Big{(}\int_{\tau_{0}}^{\tau_{1}}|a_{2}(s)|^{2}ds\Big{)}^{\frac{p}{2}}\cdot\sup_{t\in[\tau_{0},\tau_{1}]}|\Delta
X(t)|^{p}\Big{]}.$

Recalling the definition of $\tau_{1}$, we have

$\mathbb{E}\Big{[}\sup_{t\in[\tau_{0},\tau_{1}]}|\Gamma(\Delta
X)(t)|^{p}\Big{]}\leqslant\frac{1}{2}\mathbb{E}\Big{[}\sup_{t\in[\tau_{0},\tau_{1}]}|\Delta
X(t)|^{p}\Big{]}.$

By the fixed point theorem, it yields that SDE (5.2) has a unique solution
$X_{1}(\cdot)\in{\cal S}_{\mathbb{F}}^{p}([\tau_{0},\tau_{1}];\mathbb{R})$ on
the interval $[\tau_{0},\tau_{1}].$

Step 2. Assuming that a solution $X$ of (5.2) has been constructed on
$[\tau_{0},\tau_{n}]$, we construct it on $(\tau_{n},\tau_{n+1}]$. For this,
define

$\displaystyle\Gamma(X)(t):=X(\tau_{n})+\int_{\tau_{n}}^{t}\Big{[}a_{1}(s)X(s)+\varphi_{1}(s)\Big{]}ds+\int_{\tau_{n}}^{t}\Big{[}a_{2}(s)X(s)+\varphi_{2}(s)\Big{]}dW(s),\
t\in[\tau_{n},\tau_{n+1}].$ (5.3)

According to the definition of $\tau_{n+1}$, similar to Step 1, we know that
(5.3) has a unique solution $X_{n+1}\in{\cal
S}_{\mathbb{F}}^{p}([\tau_{n},\tau_{n+1}];\mathbb{R})$. Repeating this
procedure time and again, we get a solution of (5.2) on $[0,T]$. Moreover,
this solution is unique by the procedure of construction. ∎

According to 5.1, the forward equation in (5.1) has a unique strong solution
$p(\cdot)\in{\cal S}_{\mathbb{F}}^{p}([0,T];\mathbb{R})$ for $p>1$. The
backward equation in (5.1) is a $n$-dimensional mean-field BSDE with bounded
Lipschitz coefficient. According to [5, Theorem 3.4], for $p>1$, it has a
unique solution $(q(\cdot),r(\cdot))\in{\cal
S}_{\mathbb{F}}^{p}([0,T];\mathbb{R}^{n})\times{\cal
H}_{\mathbb{F}}^{2,p}([0,T];\mathbb{R}^{n\times d})$.

The following lemma show that $p_{\theta}$ is continuous with respect to
$\theta$.

######  Lemma 5.2.

Under 1, for $1<p<(p_{\partial_{z}f_{\theta}}\wedge
p_{(\partial_{z}f_{\tilde{\theta}}-\partial_{z}f_{\theta})})$,

$\displaystyle\lim_{\epsilon\rightarrow
0}\sup_{d(\theta,\tilde{\theta})\leqslant\epsilon}\mathbb{E}\bigg{[}\sup_{t\in[0,T]}|p_{\theta}(t)-p_{\tilde{\theta}}(t)|^{p}\bigg{]}=0.$

###### Proof.

Since $|p_{\theta}(t)|^{2}>0$ for $t\in[0,T]$, applying Itô’s formula to
$\ln|p_{\theta}(t)|^{2}$ on $[0,T]$, we have

$p_{\theta}(t)=\pm p_{\theta}(0)\cdot
e^{\int_{0}^{t}\partial_{y}f_{\theta}(s)ds}\cdot{\cal
E}(\int_{0}^{t}\partial_{z}f_{\theta}(s)^{\top}dW(s)).$

We only show $p_{\theta}(t)=-p_{\theta}(0)\cdot
e^{\int_{0}^{t}\partial_{y}f_{\theta}(s)ds}\cdot{\cal
E}(\int_{0}^{t}\partial_{z}f_{\theta}(s)^{\top}dW(s))$, i.e.,

$p_{\theta}(t)=\partial_{y}\gamma_{\theta}(\bar{Y}_{\theta}(0))\cdot
e^{\int_{0}^{t}\partial_{y}f_{\theta}(s)ds}\cdot{\cal
E}(\int_{0}^{t}\partial_{z}f_{\theta}(s)^{\top}dW(s)).$

Notice

$p_{\theta}(t)-p_{\tilde{\theta}}(t)=I_{1}(t)+I_{2}(t)+I_{3}(t),$

where

$\displaystyle I_{1}(t)$
$\displaystyle:=\Big{[}\partial_{y}\gamma_{\theta}(\bar{Y}_{\theta}(0))-\partial_{y}\gamma_{\tilde{\theta}}(\bar{Y}_{\tilde{\theta}}(0))\Big{]}\cdot
e^{\int_{0}^{t}\partial_{y}f_{\theta}(s)ds}\cdot{\cal
E}(\int_{0}^{t}\partial_{z}f_{\theta}(s)^{\top}dW(s)),$ $\displaystyle
I_{2}(t)$
$\displaystyle:=\partial_{y}\gamma_{\tilde{\theta}}(\bar{Y}_{\tilde{\theta}}(0))\cdot\Big{[}e^{\int_{0}^{t}\partial_{y}f_{\theta}(s)ds}-e^{\int_{0}^{t}\partial_{y}f_{\tilde{\theta}}(s)ds}\Big{]}\cdot{\cal
E}(\int_{0}^{t}\partial_{z}f_{\theta}(s)^{\top}dW(s)),$ $\displaystyle
I_{3}(t)$
$\displaystyle:=\partial_{y}\gamma_{\tilde{\theta}}(\bar{Y}_{\tilde{\theta}}(0))\cdot
e^{\int_{0}^{t}\partial_{y}f_{\tilde{\theta}}(s)ds}\cdot\Big{[}{\cal
E}(\int_{0}^{t}\partial_{z}f_{\theta}(s)^{\top}dW(s))-{\cal
E}(\int_{0}^{t}\partial_{z}f_{\tilde{\theta}}(s)^{\top}dW(s))\Big{]}.$

For $I_{1}(t)$, since

$\displaystyle|\partial_{y}\gamma_{\theta}(\bar{Y}_{\theta}(0))-\partial_{y}\gamma_{\tilde{\theta}}(\bar{Y}_{\tilde{\theta}}(0))|$
$\displaystyle\leqslant|\partial_{y}\gamma_{\theta}(\bar{Y}_{\theta}(0))-\partial_{y}\gamma_{\tilde{\theta}}(\bar{Y}_{\theta}(0))|+|\partial_{y}\gamma_{\tilde{\theta}}(\bar{Y}_{\theta}(0))-\partial_{y}\gamma_{\tilde{\theta}}(\bar{Y}_{\tilde{\theta}}(0))|$
$\displaystyle\leqslant
d(\theta,\tilde{\theta})+C|\bar{Y}_{\theta}(0)-\bar{Y}_{\tilde{\theta}}(0)|,$

it yields from Doob’s inequality and reverse Hölder’s inequality that

$\displaystyle\mathbb{E}\Big{[}\sup\limits_{t\in[0,T]}|I_{1}(t)|^{p}\Big{]}$
$\displaystyle\leqslant
C_{p}\mathbb{E}\Big{[}\sup\limits_{t\in[0,T]}\Big{\\{}(d^{p}(\theta,\tilde{\theta})+C|\bar{Y}_{\theta}(0)-\bar{Y}_{\tilde{\theta}}(0)|^{p})\cdot
e^{pTC_{1}}\cdot{\cal
E}(\int_{0}^{t}\partial_{z}f_{\theta}(s)^{\top}dW(s))^{p}\Big{\\}}\Big{]}$
$\displaystyle\leqslant
C_{p}d^{p}(\theta,\tilde{\theta})\mathbb{E}\Big{[}\sup\limits_{t\in[0,T]}{\cal
E}(\int_{0}^{t}\partial_{z}f_{\theta}(s)^{\top}dW(s))^{p}\Big{]}$
$\displaystyle\quad+C_{p}\Big{\\{}\mathbb{E}\Big{[}\sup\limits_{t\in[0,T]}|\bar{Y}_{\theta}(t)-\bar{Y}_{\tilde{\theta}}(t)|^{pq^{\prime}}\Big{]}\Big{\\}}^{\frac{1}{q^{\prime}}}\cdot\Big{\\{}\mathbb{E}\Big{[}\sup\limits_{t\in[0,T]}{\cal
E}(\int_{0}^{t}\partial_{z}f_{\theta}(s)^{\top}dW(s))^{pp^{\prime}}\Big{]}\Big{\\}}^{\frac{1}{p^{\prime}}}$
$\displaystyle\leqslant
C_{p}d^{p}(\theta,\tilde{\theta})\mathbb{E}\Big{[}{\cal
E}(\int_{0}^{T}\partial_{z}f_{\theta}(s)^{\top}dW(s))^{p}\Big{]}$
$\displaystyle\quad+C_{p}\Big{\\{}\mathbb{E}[\sup\limits_{t\in[0,T]}|\bar{Y}_{\theta}(t)-\bar{Y}_{\tilde{\theta}}(t)|^{pq^{\prime}}]\Big{\\}}^{\frac{1}{q^{\prime}}}\cdot\Big{\\{}\mathbb{E}[{\cal
E}(\int_{0}^{T}\partial_{z}f_{\theta}(s)^{\top}dW(s))^{pp^{\prime}}]\Big{\\}}^{\frac{1}{p^{\prime}}}$
$\displaystyle\leqslant C_{p}d^{p}(\theta,\tilde{\theta})\cdot
K(p,\|\partial_{z}f_{\theta}\cdot W\|_{BMO})$
$\displaystyle\quad+\Big{\\{}\mathbb{E}[\sup\limits_{t\in[0,T]}|\bar{Y}_{\theta}(t)-\bar{Y}_{\tilde{\theta}}(t)|^{pq^{\prime}}]\Big{\\}}^{\frac{1}{q^{\prime}}}\cdot
K(pp^{\prime},\|\partial_{z}f_{\theta}\cdot W\|_{BMO}),$

where $p^{\prime}=\frac{p+p_{\partial_{z}f_{\theta}}}{2p}$ and
$q^{\prime}=\frac{p^{\prime}}{p^{\prime}-1}$. Clearly
$pq^{\prime}>p^{*}_{\partial_{z}f_{\theta}}$, and by 3.3 we have, for
$1<p<p_{\partial_{z}f_{\theta}}$,

$\lim_{\epsilon\rightarrow
0}\sup_{d(\theta,\tilde{\theta})\leqslant\epsilon}\mathbb{E}\Big{[}\sup\limits_{t\in[0,T]}|I_{1}(t)|^{p}\Big{]}=0.$

For $I_{2}(t)$, since

$\Big{[}e^{\int_{0}^{t}\partial_{y}f_{\theta}(s)ds}-e^{\int_{0}^{t}\partial_{y}f_{\tilde{\theta}}(s)ds}\Big{]}\leqslant
e^{C_{1}T}\cdot\Big{[}e^{Td(\theta,\tilde{\theta})}-1\Big{]},$

we know, for $1<p<p_{\partial_{z}f_{\theta}}$,

$\displaystyle\mathbb{E}\Big{[}\sup\limits_{t\in[0,T]}|I_{2}(t)|^{p}\Big{]}$
$\displaystyle\leqslant
Le^{C_{1}T}\cdot\Big{[}e^{Td(\theta,\tilde{\theta})}-1\Big{]}\cdot\mathbb{E}\Big{[}\sup\limits_{t\in[0,T]}{\cal
E}(\int_{0}^{t}\partial_{z}f_{\theta}(s)^{\top}dW(s))^{p}\Big{]}$
$\displaystyle\leqslant
Le^{C_{1}T}\cdot\Big{[}e^{Td(\theta,\tilde{\theta})}-1\Big{]}\cdot
K(p,\|\partial_{z}f_{\theta}\cdot W\|_{BMO}).$

Finally, we prove, for $1<p<(p_{\partial_{z}f_{\theta}}\wedge
p_{(\partial_{z}f_{\tilde{\theta}}-\partial_{z}f_{\theta})})$,

$\lim_{\epsilon\rightarrow
0}\sup_{d(\theta,\tilde{\theta})\leqslant\epsilon}\mathbb{E}\Big{[}\sup\limits_{t\in[0,T]}|I_{3}(t)|^{p}\Big{]}=0.$

In fact, due to
$|\partial_{y}\gamma_{\tilde{\theta}}(\bar{Y}_{\tilde{\theta}}(0))\cdot
e^{\int_{0}^{t}\partial_{y}f_{\tilde{\theta}}(s)ds}|\leqslant C$, we only need
to prove, for $1<p<(p_{\partial_{z}f_{\theta}}\wedge
p_{(\partial_{z}f_{\tilde{\theta}}-\partial_{z}f_{\theta})})$,

$\lim_{\epsilon\rightarrow
0}\sup_{d(\theta,\tilde{\theta})\leqslant\epsilon}\mathbb{E}\Big{[}\sup\limits_{t\in[0,T]}|{\cal
E}(\int_{0}^{t}\partial_{z}f_{\theta}(s)^{\top}dW(s))-{\cal
E}(\int_{0}^{t}\partial_{z}f_{\tilde{\theta}}(s)^{\top}dW(s))|^{p}\Big{]}=0.$

For convenience, set $X_{\theta}(t)={\cal
E}(\int_{0}^{t}\partial_{z}f_{\theta}(s)^{\top}dW(s))$, and $X_{\theta}$
satisfies the following SDE

$X_{\theta}(t)=1+\int_{0}^{t}\partial_{z}f_{\theta}(s)X_{\theta}(s)dW(s),\
t\in[0,T].$

Set $\bar{X}(s)=X_{\theta}(s)-X_{\tilde{\theta}}(s)$ and
$\varphi(s)=(\partial_{z}f_{\theta}(s)-\partial_{z}f_{\tilde{\theta}}(s))X_{\tilde{\theta}}(s)$.
Then

$\bar{X}(t)=\int_{0}^{t}\Big{(}\partial_{z}f_{\theta}(s)\bar{X}(s)+\varphi(s)\Big{)}dW(s),\quad
t\in[0,T].$

Consider

$\left\\{\begin{aligned}
d\Pi_{\theta}(t)&=(\partial_{z}f_{\theta}(t))^{2}\Pi_{\theta}(t)dt-\partial_{z}f_{\theta}(t)\Pi_{\theta}(t)dW(t),\quad
t\in[0,T],\\\ \Pi_{\theta}(0)&=1.\end{aligned}\right.$

By 5.1, the above equation has a unique strong solution $\Pi_{\theta}\in{\cal
S}^{p}([0,T];\mathbb{R})$ for $p>1$. Then it follows from Itô’s formula to
$\bar{X}(s)\Pi_{\theta}(s)$ on $[0,t]$ that

$\bar{X}(t)\Pi_{\theta}(t)=\int_{0}^{t}(-\partial_{z}f_{\theta}(s)\Pi_{\theta}(s)\varphi(s))ds+\int_{0}^{t}\Pi_{\theta}(s)\varphi(s)dW(s),\quad
t\in[0,T].$ (5.4)

In addition, consider

$\left\\{\begin{aligned}
d\pi_{\theta}(t)&=\partial_{z}f_{\theta}(t)\pi_{\theta}(t)dW(t),\quad
t\in[0,T],\\\ \pi_{\theta}(0)&=1.\end{aligned}\right.$

By 5.1 again, the above equation also has a unique strong solution
$\pi_{\theta}\in{\cal S}^{p}([0,T];\mathbb{R})$ for $p>1$. Moreover, we know
from Itô’s formula that $\Pi_{\theta}(t)\pi_{\theta}(t)=1,\ t\in[0,T]$. Hence,

$\Pi_{\theta}(t)^{-1}=\pi_{\theta}(t)={\cal
E}\Big{(}\int_{0}^{t}\partial_{z}f_{\theta}(r)^{\top}dW(r)\Big{)}.$

From Hölder inequality we can see, for $1<p<(p_{\partial_{z}f_{\theta}}\wedge
p_{(\partial_{z}f_{\tilde{\theta}}-\partial_{z}f_{\theta})})$,

$\displaystyle\mathbb{E}\Big{[}\sup\limits_{t\in[0,T]}|\bar{X}(t)|^{p}\Big{]}=\mathbb{E}\Big{[}\sup\limits_{t\in[0,T]}|\bar{X}(t)\Pi_{\theta}(t)\pi_{\theta}(t)|^{p}\Big{]}$
$\displaystyle\leqslant\mathbb{E}\Big{[}\sup\limits_{t\in[0,T]}|\bar{X}(t)\Pi_{\theta}(t)|^{p}\cdot\sup\limits_{t\in[0,T]}|\pi_{\theta}(t)|^{p}\Big{]}$
$\displaystyle\leqslant\Big{\\{}\mathbb{E}\Big{[}\sup\limits_{t\in[0,T]}|\bar{X}(t)\Pi_{\theta}(t)|^{pq^{\prime}}\Big{]}\Big{\\}}^{\frac{1}{q^{\prime}}}\Big{\\{}\mathbb{E}\Big{[}\sup\limits_{t\in[0,T]}|\pi_{\theta}(t)|^{pp^{\prime}}\Big{]}\Big{\\}}^{\frac{1}{p^{\prime}}},$

where $p^{\prime}=\frac{p+(p_{\partial_{z}f_{\theta}}\wedge
p_{(\partial_{z}f_{\tilde{\theta}}-\partial_{z}f_{\theta})})}{2p}$ and
$q^{\prime}=\frac{p^{\prime}}{p^{\prime}-1}$. Since
$pp^{\prime}<p_{\partial_{z}f_{\theta}}$, it follows from Doob’s inequality
and reverse Hölder inequality that

$\mathbb{E}\Big{[}\sup\limits_{t\in[0,T]}\pi_{\theta}(t)^{pp^{\prime}}\Big{]}\leqslant
C\mathbb{E}\Big{[}\pi_{\theta}(T)^{pp^{\prime}}\Big{]}\leqslant
CK(pp^{\prime},\|\partial_{z}f_{\theta}\cdot W\|_{BMO}).$

Consequently, by Hölder inequality we have

$\displaystyle\mathbb{E}\Big{[}\sup\limits_{t\in[0,T]}|\bar{X}(t)|^{p}\Big{]}$
$\displaystyle\leqslant
C\Big{\\{}\mathbb{E}\Big{[}\sup\limits_{t\in[0,T]}|\bar{X}(t)\Pi_{\theta}(t)|^{pq^{\prime}}\Big{]}\Big{\\}}^{\frac{1}{q^{\prime}}}$
(5.5) $\displaystyle\leqslant
C\Big{\\{}\mathbb{E}\Big{[}\sup\limits_{t\in[0,T]}|\bar{X}(t)\Pi_{\theta}(t)|^{p\frac{p^{\prime}}{p^{\prime}-1}}\Big{]}\Big{\\}}^{\frac{1}{q^{\prime}}}$
$\displaystyle\leqslant
C\Big{\\{}\mathbb{E}\Big{[}\sup\limits_{t\in[0,T]}|\bar{X}(t)\Pi_{\theta}(t)|^{pp^{\prime}}\Big{]}\Big{\\}}^{\frac{1}{p^{\prime}}},$

where $C$ depends on $pp^{\prime}$ and $\|\partial_{z}f_{\theta}\cdot
W\|_{BMO}$. Besides, from (5.4) we arrive at

$\displaystyle\mathbb{E}\Big{[}\sup\limits_{t\in[0,T]}|\bar{X}(t)\Pi_{\theta}(t)|^{pp^{\prime}}\Big{]}$
$\displaystyle\leqslant
C\mathbb{E}\Big{[}\int_{0}^{T}|-\partial_{z}f_{\theta}(s)\Pi_{\theta}(s)\varphi(s))|^{pp^{\prime}}ds\Big{]}$
(5.6)
$\displaystyle\quad+C\mathbb{E}\Big{[}\sup\limits_{t\in[0,T]}|\int_{0}^{t}\Pi_{\theta}(s)\varphi(s)dW(s)|^{pp^{\prime}}\Big{]},$

where the constant $C>0$ depends on $pp^{\prime}$.

Then we estimate the $ds$-term and the $dW$-term in the above inequality one
by one. As for the $ds$-term, since

$\left\\{\begin{aligned} |\partial_{z}f_{\theta}(s)|&\leqslant
C(1+|\bar{Z}_{\theta}(s)|),\\\ |\varphi(s)|&\leqslant
d(\theta,\tilde{\theta})\cdot{\cal
E}\Big{(}\int_{0}^{s}\partial_{z}f_{\tilde{\theta}}(r)^{\top}dW(r)\Big{)},\\\
\Pi_{\theta}(s)&=\exp\Big{\\{}\int_{0}^{s}(-\partial_{z}f_{\theta}(r))^{\top}dW(r)+\frac{1}{2}\int_{0}^{s}|\partial_{z}f_{\theta}(r)|^{2}dr\Big{\\}},\end{aligned}\right.$
(5.7)

by Hölder inequality we have

$\displaystyle\mathbb{E}\Big{[}\int_{0}^{T}|-\partial_{z}f_{\theta}(s)\Pi_{\theta}(s)\varphi(s))|^{pp^{\prime}}ds\Big{]}$
(5.8) $\displaystyle\leqslant
Cd(\theta,\tilde{\theta})^{pp^{\prime}}\mathbb{E}\bigg{[}\Big{(}\int_{0}^{T}(1+|\bar{Z}_{\theta}(s)|)\cdot{\cal
E}\Big{(}\int_{0}^{s}(\partial_{z}f_{\tilde{\theta}}(r)-\partial_{z}f_{\theta}(r))^{\top}dW(r)\Big{)}\cdot$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad
e^{\int_{0}^{s}\partial_{z}f_{\theta}(r)^{\top}(\partial_{z}f_{\theta}(r)-\partial_{z}f_{\tilde{\theta}}(r))dr}ds\Big{)}^{pp^{\prime}}\bigg{]}$
$\displaystyle\leqslant
Cd(\theta,\tilde{\theta})^{pp^{\prime}}\mathbb{E}\Big{[}\sup\limits_{s\in[0,T]}{\cal
E}\Big{(}\int_{0}^{s}(\partial_{z}f_{\tilde{\theta}}(r)-\partial_{z}f_{\theta}(r))^{\top}dW(r)\Big{)}^{pp^{\prime}}\cdot$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
e^{Cd(\theta,\tilde{\theta})pp^{\prime}\int_{0}^{T}(1+|\bar{Z}_{\theta}(r)|)dr}\cdot\Big{(}\int_{0}^{T}(1+|\bar{Z}_{\theta}(r)|)dr\Big{)}^{pp^{\prime}}\Big{]}$
$\displaystyle\leqslant
Cd(\theta,\tilde{\theta})^{pp^{\prime}}\bigg{\\{}\mathbb{E}\Big{[}\sup\limits_{s\in[0,T]}{\cal
E}\Big{(}\int_{0}^{s}(\partial_{z}f_{\tilde{\theta}}(r)-\partial_{z}f_{\theta}(r))^{\top}dW(r)\Big{)}^{pp^{\prime}p^{\prime\prime}}\Big{]}\bigg{\\}}^{\frac{1}{p^{\prime\prime}}}\cdot$
$\displaystyle\qquad\qquad\qquad\bigg{\\{}\mathbb{E}\Big{[}e^{Cd(\theta,\tilde{\theta})pp^{\prime}q^{\prime\prime}\int_{0}^{T}(1+|\bar{Z}_{\theta}(r)|)dr}\cdot\Big{(}\int_{0}^{T}(1+|\bar{Z}_{\theta}(r)|)dr\Big{)}^{pp^{\prime}q^{\prime\prime}}\Big{]}\bigg{\\}}^{\frac{1}{q^{\prime\prime}}},$

where $p^{\prime\prime}=\frac{p^{\prime}+(p_{\partial_{z}f_{\theta}}\wedge
p_{(\partial_{z}f_{\tilde{\theta}}-\partial_{z}f_{\theta})})}{2pp^{\prime}}$
and $q^{\prime\prime}=\frac{p^{\prime\prime}}{p^{\prime\prime}-1}$. Since
$|\partial_{z}f_{\tilde{\theta}}(s)-\partial_{z}f_{\theta}(s)|\leqslant
C(1+|\bar{Z}_{\theta}(s)|+|\bar{Z}_{\tilde{\theta}}(s)|)$ we know
$(\partial_{z}f_{\tilde{\theta}}-\partial_{z}f_{\theta})\cdot W\in BMO$. Note
that
$1<p^{\prime}<pp^{\prime}p^{\prime\prime}<p_{(\partial_{z}f_{\tilde{\theta}}-\partial_{z}f_{\theta})}$.
It follows from Doob’s inequality and reverse Hölder inequality that

$\mathbb{E}\Big{[}\sup\limits_{s\in[0,T]}{\cal
E}\Big{(}\int_{0}^{s}(\partial_{z}f_{\tilde{\theta}}(r)-\partial_{z}f_{\theta}(r))^{\top}dW(r)\Big{)}^{pp^{\prime}p^{\prime\prime}}\Big{]}\leqslant
K(pp^{\prime}p^{\prime\prime},\|(\partial_{z}f_{\tilde{\theta}}-\partial_{z}f_{\theta})\cdot
W\|_{BMO}).$

On the other hand, by Hölder inequality and $\bar{Z}_{\theta}\cdot W\in BMO$,
we have

$\displaystyle\bigg{\\{}\mathbb{E}\Big{[}e^{Cd(\theta,\tilde{\theta})pp^{\prime}q^{\prime\prime}\int_{0}^{T}(1+|\bar{Z}_{\theta}(r)|)dr}\cdot\Big{(}\int_{0}^{T}(1+|\bar{Z}_{\theta}(r)|)dr\Big{)}^{pp^{\prime}q^{\prime\prime}}\Big{]}\bigg{\\}}^{\frac{1}{q^{\prime\prime}}}$
$\displaystyle\leqslant
C\bigg{\\{}\mathbb{E}\Big{[}e^{2Cd(\theta,\tilde{\theta})pp^{\prime}q^{\prime\prime}\int_{0}^{T}(1+|\bar{Z}_{\theta}(r)|)dr}\Big{]}\bigg{\\}}^{\frac{1}{2q^{\prime\prime}}}\cdot\bigg{\\{}\mathbb{E}\Big{[}\Big{(}\int_{0}^{T}(1+|\bar{Z}_{\theta}(r)|^{2})dr\Big{)}^{pp^{\prime}q^{\prime\prime}}\Big{]}\bigg{\\}}^{\frac{1}{2q^{\prime\prime}}}$
$\displaystyle\leqslant
C\bigg{\\{}\mathbb{E}\Big{[}e^{2Cd(\theta,\tilde{\theta})pp^{\prime}q^{\prime\prime}\int_{0}^{T}(1+|\bar{Z}_{\theta}(r)|)dr}\Big{]}\bigg{\\}}^{\frac{1}{2q^{\prime\prime}}}.$

Without loss of generality, assume $d(\theta,\tilde{\theta})<1$. Consequently,
it follows from John-Nirenberg inquality that

$\displaystyle\mathbb{E}\Big{[}e^{2Cd(\theta,\tilde{\theta})pp^{\prime}q^{\prime\prime}\int_{0}^{T}(1+|\bar{Z}_{\theta}(r)|)dr}\Big{]}<\mathbb{E}\Big{[}e^{2Cpp^{\prime}q^{\prime\prime}\int_{0}^{T}(1+|\bar{Z}_{\theta}(r)|)dr}\Big{]}$
$\displaystyle\leqslant
e^{2Cpp^{\prime}q^{\prime\prime}(T+\frac{T}{\delta})}\mathbb{E}\Big{[}e^{2Cpp^{\prime}q^{\prime\prime}\frac{\delta}{4}\int_{0}^{T}|\bar{Z}_{\theta}(r)|^{2}dr}\Big{]}$
$\displaystyle\leqslant
e^{2Cpp^{\prime}q^{\prime\prime}(T+\frac{T}{\delta})}\Big{(}1-2Cpp^{\prime}q^{\prime\prime}\frac{\delta}{4}\|\bar{Z}_{\theta}\cdot
W\|_{BMO}^{2}\Big{)}^{-1}$
$\displaystyle=\frac{4}{3}e^{2Cpp^{\prime}q^{\prime\prime}(T+\frac{T}{\delta})},$

where $\delta=\frac{\|\bar{Z}_{\theta}\cdot
W\|_{BMO}^{-2}}{2Cpp^{\prime}q^{\prime\prime}}$. Combining the above
inequalities, we have

$\mathbb{E}\Big{[}\int_{0}^{T}|-\partial_{z}f_{\theta}(s)\Pi_{\theta}(s)\varphi(s))|^{pp^{\prime}}ds\Big{]}\leqslant
Cd(\theta,\tilde{\theta})^{pp^{\prime}}.$ (5.9)

Next, we analyse the $dW$-term. Similar to the $ds$-term, based on Buckholder-
Davis-Gundy inequality, (5.7), Hölder inequality and John-Nirenberg
inequality, we have

$\displaystyle\mathbb{E}\Big{[}\sup\limits_{t\in[0,T]}|\int_{0}^{t}\Pi_{\theta}(s)\varphi(s)dW(s)|^{pp^{\prime}}\Big{]}$
(5.10) $\displaystyle\leqslant
Cd(\theta,\tilde{\theta})^{pp^{\prime}}\mathbb{E}\Big{[}\sup\limits_{s\in[0,T]}{\cal
E}\Big{(}\int_{0}^{s}(\partial_{z}f_{\tilde{\theta}}(r)-\partial_{z}f_{\theta}(r))^{\top}dW(r)\Big{)}^{pp^{\prime}}\cdot
e^{Cpp^{\prime}\int_{0}^{T}(1+|\bar{Z}_{\theta}(r)|)d(\theta,\tilde{\theta})dr}\Big{]}$
$\displaystyle\leqslant
Cd(\theta,\tilde{\theta})^{pp^{\prime}}\bigg{\\{}\mathbb{E}\Big{[}\sup\limits_{s\in[0,T]}{\cal
E}\Big{(}\int_{0}^{s}(\partial_{z}f_{\tilde{\theta}}(r)-\partial_{z}f_{\theta}(r))^{\top}dW(r)\Big{)}^{pp^{\prime}p^{\prime\prime}}\Big{]}\bigg{\\}}^{\frac{1}{p^{\prime\prime}}}\cdot$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\bigg{\\{}\mathbb{E}\Big{[}e^{Cpp^{\prime}q^{\prime\prime}\int_{0}^{T}(1+|\bar{Z}_{\theta}(r)|)d(\theta,\tilde{\theta})dr}\Big{]}\bigg{\\}}^{\frac{1}{q^{\prime\prime}}}$
$\displaystyle\leqslant Cd(\theta,\tilde{\theta})^{pp^{\prime}},$

where $C$ depends on
$\|(\partial_{z}f_{\tilde{\theta}}-\partial_{z}f_{\theta})\cdot W\|_{BMO}\|$
and $\partial_{z}f_{\theta}\cdot W\|_{BMO}$, and
$p^{\prime\prime},q^{\prime\prime}$ are given in (5.8).
Finally, taking into account (5.5), (5.6), (5.9) and (5.10), we have
$\mathbb{E}\Big{[}\sup\limits_{t\in[0,T]}|\bar{X}(t)|^{p}\Big{]}\leqslant
Cd(\theta,\tilde{\theta})^{p}.$ The proof is complete. ∎

Before we prove the stochastic maximum principle, we need to give the
continuity of $q_{\theta},r_{\theta}$ with respect to $\theta$.

######  Lemma 5.3.

Under 1, we have, for $1<p<(p_{\partial_{z}f_{\theta}}\wedge
p_{\partial_{z}f_{\tilde{\theta}}-\partial_{z}f_{\theta}})$,

$\displaystyle\lim_{\epsilon\rightarrow
0}\sup_{d(\theta,\tilde{\theta})\leqslant\epsilon}\mathbb{E}\bigg{[}\sup_{t\in[0,T]}|q_{\theta}(t)-q_{\tilde{\theta}}(t)|^{p}+\Big{(}\int_{0}^{T}|r_{\theta}(t)-r_{\tilde{\theta}}(t)|^{2}dt\Big{)}^{\frac{p}{2}}\bigg{]}=0.$

###### Proof.

Notice that

$\displaystyle q_{\theta}(t)-q_{\tilde{\theta}}(t)$
$\displaystyle=[q_{\theta}(T)-q_{\tilde{\theta}}(T)]+\int_{t}^{T}\Big{(}\partial_{x}b_{\theta}(s)^{\top}(q_{\theta}(s)-q_{\tilde{\theta}}(s))+\sum_{i=1}^{d}\partial_{x}\sigma^{i}_{\theta}(s)^{\top}(r^{i}_{\theta}(s)-r^{i}_{\tilde{\theta}}(s))$
$\displaystyle\quad+\mathbb{E}\Big{[}\partial_{x^{\prime}}b_{\theta}(s)^{\top}(q_{\theta}(s)-q_{\tilde{\theta}}(s))\Big{]}+I_{1,\theta,\tilde{\theta}}(s)+I_{2,\theta,\tilde{\theta}}(s)\Big{)}ds-\int_{t}^{T}(r_{\theta}(s)-r_{\tilde{\theta}}(s))dW(s),$

where

$\displaystyle I_{1,\theta,\tilde{\theta}}(s)$
$\displaystyle=\partial_{x}f_{\tilde{\theta}}(s)p_{\tilde{\theta}}(s)-\partial_{x}f_{\theta}(s)p_{\theta}(s)+\mathbb{E}[\partial_{x^{\prime}}f_{\tilde{\theta}}(s)p_{\tilde{\theta}}(s)-\partial_{x^{\prime}}f_{\theta}(s)p_{\theta}(s)],$
$\displaystyle I_{2,\theta,\tilde{\theta}}(s)$
$\displaystyle=(\partial_{x}b_{\theta}(t)-\partial_{x}b_{\tilde{\theta}}(t))^{\top}q_{\tilde{\theta}}(t)+\sum_{i=1}^{d}(\partial_{x}\sigma^{i}_{\theta}(t)-\partial_{x}\sigma^{i}_{\tilde{\theta}}(t))^{\top}r^{i}_{\tilde{\theta}}(t)+\mathbb{E}[(\partial_{x^{\prime}}b_{\theta}(t)-\partial_{x^{\prime}}b_{\tilde{\theta}}(t))^{\top}q_{\tilde{\theta}}(t)].$

Since
$\partial_{x}b_{\theta},\partial_{x}\sigma^{i}_{\theta},\partial_{x^{\prime}}b_{\theta}$
are bounded, the backward equation in (5.1) is a mean-field BSDE with
Lipschitz coefficient. By [5, Theorem 3.3], it yields that for $p>1$,

$\displaystyle\mathbb{E}\bigg{[}\sup_{t\in[0,T]}|q_{\theta}(t)-q_{\tilde{\theta}}(t)|^{p}+\Big{(}\int_{0}^{T}|r_{\theta}(t)-r_{\tilde{\theta}}(t)|^{2}dt\Big{)}^{\frac{p}{2}}\bigg{]}$
$\displaystyle\leqslant
C\mathbb{E}\bigg{[}|q_{\theta}(T)-q_{\tilde{\theta}}(T)|^{p}+\Big{(}\int_{0}^{T}|I_{1,\theta,\tilde{\theta}}(t)+I_{2,\theta,\tilde{\theta}}(t)|dt\Big{)}^{p}\bigg{]}.$

Since
$\partial_{x}\Phi_{\theta},\partial_{x^{\prime}}\Phi_{\theta},\partial_{x}b_{\theta},\partial_{x}\sigma_{\theta},\partial_{x^{\prime}}b_{\theta},\partial_{x}f_{\theta},\partial_{x^{\prime}}f_{\theta},\partial_{x}\phi_{\theta}$
are bounded and Lipschitz continuous in $\theta$, it follows from 5.2 and 3.3
that

$\displaystyle\lim_{\epsilon\rightarrow
0}\sup_{d(\theta,\tilde{\theta})\leqslant\epsilon}\mathbb{E}\bigg{[}\sup_{t\in[0,T]}|q_{\theta}(t)-q_{\tilde{\theta}}(t)|^{p}+\Big{(}\int_{0}^{T}|r_{\theta}(t)-r_{\tilde{\theta}}(t)|^{2}dt\Big{)}^{\frac{p}{2}}\bigg{]}=0.$

∎

Define the Hamiltonian: for
$t\in[0,T],x,x^{\prime}\in\mathbb{R}^{n},y\in\mathbb{R},z\in\mathbb{R}^{d},p\in\mathbb{R},q\in\mathbb{R}^{n},r\in\mathbb{R}^{n\times
d},v\in V,$

$\displaystyle H_{\theta}(t,x,x^{\prime},y,z,v,p,q,r)$
$\displaystyle:=q^{\top}b_{\theta}(t,x,x^{\prime},v)+\sum_{i=1}^{d}(r^{i})^{\top}\sigma_{\theta}^{i}(t,x,v)-pf_{\theta}(t,x,x^{\prime},y,z,v)$

and its partial derivative with respect to $v$:

$\displaystyle\Lambda_{\theta}(t,\omega):=\partial_{v}H_{\theta}(t,\bar{X}_{\theta}(t),\mathbb{E}[\bar{X}_{\theta}(t)],\bar{Y}_{\theta}(t),\bar{Z}_{\theta}(t),\bar{v}(t),p_{\theta}(t),q_{\theta}(t),r_{\theta}(t)).$
(5.11)

######  Lemma 5.4.

Under 1 and 2, the above $\Lambda$ is an ${\cal F}$-progressively measurable
process, i.e., for each $t\in[0,T]$, the function
$\Lambda_{\theta}(t,\omega):\Theta\times[0,t]\times\Omega\rightarrow\mathbb{R}$
is $\mathscr{B}(\Theta)\times\mathscr{B}([0,t])\times{\cal F}_{t}$-measurable.

###### Proof.

Since $\Theta$ is a Polish space, for each $M>1$, there exists a compact
subset $C^{M}\subset\Theta$ such that $\bar{Q}(\theta\notin
C^{M})\leqslant\frac{1}{M}$. Then we can find a subsequence of open
neighborhoods $\Big{(}B(\theta_{l},\frac{1}{2M})\Big{)}_{l=1}^{L_{M}}$ such
that
$C^{M}\subset\bigcup_{l=1}^{L_{M}}\Big{(}B(\theta_{l},\frac{1}{2M})\Big{)}$.
By the locally compact property of $\Theta$ and using partitions of unity,
there exists a sequence of continuous functions
$h_{l}:\Theta\rightarrow\mathbb{R}$ with values in $[0,1]$ such that

$h_{l}(\theta)=0,\ \text{if}\ \theta\notin
B\Big{(}\theta_{l},\frac{1}{2M}\Big{)},\ l=1,\cdots,L_{M},\ \text{and}\
\sum_{l=1}^{L_{M}}h_{l}(\theta)=1,\ \text{if}\ \theta\in C^{M}.$

We choose some $\theta_{l}^{*}$ satisfying $h_{l}(\theta_{l}^{*})>0$ and
define

$\Lambda^{M}_{\theta}(t):=\sum_{l=1}^{L_{M}}\Lambda_{\theta_{l}^{*}}(t)h_{l}(\theta)\mathbf{1}_{\\{\theta\in
C^{M}\\}}.$

Notice that
$\mathbb{E}\Big{[}\int_{0}^{T}|\Lambda_{\theta}(t)|dt\Big{]}\leqslant L.$ It
yields that

$\displaystyle\int_{\Theta}\mathbb{E}\Big{[}\int_{0}^{T}\Big{|}\Lambda_{\theta}^{M}(t)-\Lambda_{\theta}(t)\Big{|}dt\Big{]}\bar{Q}(d\theta)$
$\displaystyle\leqslant\int_{\Theta}\sum_{l=1}^{L_{M}}\mathbb{E}\bigg{[}\int_{0}^{T}|\Lambda_{\theta}^{M}(t)-\Lambda_{\theta}(t)|dt\bigg{]}h_{l}(\theta)\mathbf{1}_{\\{\theta\in
C^{M}\\}}+\mathbb{E}\bigg{[}\int_{0}^{T}|\Lambda_{\theta}(t)|dt\bigg{]}h_{l}(\theta)\mathbf{1}_{\\{\theta\notin
C^{M}\\}}\bar{Q}(d\theta)$
$\displaystyle\leqslant\sup_{d(\theta,\tilde{\theta})\leqslant\frac{1}{M}}\mathbb{E}\bigg{[}\int_{0}^{T}|\Lambda_{\tilde{\theta}}(t)-\Lambda_{\theta}(t)|dt\bigg{]}+L\bar{Q}(\theta\notin
C^{M})$
$\displaystyle\leqslant\sup_{d(\theta,\tilde{\theta})\leqslant\frac{1}{M}}\mathbb{E}\bigg{[}\int_{0}^{T}|\Lambda_{\tilde{\theta}}(t)-\Lambda_{\theta}(t)|dt\bigg{]}+\frac{L}{M}.$

Here we use the fact that $h_{l}(\theta)=0$ whenever
$d(\theta,\tilde{\theta})\geqslant\frac{1}{2N}$.

Then we prove
$\lim\limits_{M\rightarrow\infty}\sup\limits_{d(\theta,\tilde{\theta})\leqslant\frac{1}{M}}\mathbb{E}[\int_{0}^{T}|\Lambda_{\tilde{\theta}}(t)-\Lambda_{\theta}(t)|dt]=0.$
Recall that

$\displaystyle\Lambda_{\theta}(t,\omega)$
$\displaystyle=\partial_{v}b_{\theta}(t,\bar{X}_{\theta}(t),\mathbb{E}[\bar{X}_{\theta}(t)],\bar{v}(t))q_{\theta}(t)+\sum_{i=1}^{d}\partial_{v}\sigma^{i}_{\theta}(t,\bar{X}_{\theta}(t),\bar{v}(t))r^{i}_{\theta}(t)$
$\displaystyle\quad-\partial_{v}f_{\theta}(t,\bar{X}_{\theta}(t),\mathbb{E}[\bar{X}_{\theta}(t)],\bar{Y}_{\theta}(t),\bar{Z}_{\theta}(t),\bar{v}(t))p_{\theta}(t).$

Since
$\partial_{v}b_{\theta},\partial_{v}\sigma^{i}_{\theta},\partial_{v}f_{\theta}$
are bounded and Lipschitz continuous in $\theta$, the desired result comes
from 5.2 and 5.3.

∎

Now we are ready to get the necessary condition of the optimal control.

######  Theorem 5.5.

(Stochastic Maximum Principle) Let $\bar{v}(\cdot)$ be an optimal control to
the problem (3.3),
$(\bar{X}_{\theta}(\cdot),\bar{Y}_{\theta}(\cdot),\bar{Z}_{\theta}(\cdot))$ be
the solution to the state equation (1.1) with $\bar{v}(\cdot)$, and
$(p_{\theta}(\cdot),q_{\theta}(\cdot),r_{\theta}(\cdot))$ be the solution to
the adjoint equation (5.1) with $\bar{v}(\cdot)$. Under 1 and 2, there exists
a probability measure $\bar{Q}\in{\cal Q}^{\bar{v}}$ such that, for $v\in V$,
$dt\times d\mathbb{P}$-a.e.,

$\displaystyle\int_{\Theta}\langle\partial_{v}H_{\theta}(t,\bar{X}_{\theta}(t),\mathbb{E}[\bar{X}_{\theta}(t)],\bar{Y}_{\theta}(t),\bar{Z}_{\theta}(t),\bar{v}(t),p_{\theta}(t),q_{\theta}(t),r_{\theta}(t)),v-\bar{v}(t)\rangle\bar{Q}(d\theta)\geqslant
0.$

###### Proof.

Applying Itô’s formula to
$(X^{1}_{\theta}(\cdot))^{\top}q_{\theta}(\cdot)+Y^{1}_{\theta}(\cdot)p_{\theta}(\cdot)$,
we have

$\displaystyle\mathbb{E}\Big{[}(\partial_{x}\phi_{\theta}(\bar{X}_{\theta}(T)))^{\top}X_{\theta}^{1}(T)+\partial_{y}\gamma_{\theta}(\bar{Y}_{\theta}(0))Y_{\theta}^{1}(0)\Big{]}$
$\displaystyle=\mathbb{E}\Big{[}\int_{0}^{T}\langle(\partial_{v}b_{\theta}(t))^{\top}q_{\theta}(t)+\sum_{i=1}^{d}(\partial_{v}\sigma^{i}_{\theta}(t))^{\top}r^{i}_{\theta}(t)-\partial_{v}f_{\theta}(t)p_{\theta}(t),v(t)-\bar{v}(t)\rangle
dt\Big{]}$
$\displaystyle=\mathbb{E}\Big{[}\int_{0}^{T}\langle\partial_{v}H_{\theta}(t,\bar{X}_{\theta}(t),\mathbb{E}[\bar{X}_{\theta}(t)],\bar{Y}_{\theta}(t),\bar{Z}_{\theta}(t),\bar{v}(t),p_{\theta}(t),q_{\theta}(t),r_{\theta}(t)),v(t)-\bar{v}(t)\rangle
dt\Big{]}.$

Thanks to the above equality and Theorem 4.9, there exists some probability
$\bar{Q}\in{\cal Q}^{\bar{v}}$ such that, for all $v(\cdot)\in{\cal V}_{ad}$,

$\displaystyle\int_{\Theta}\mathbb{E}\Big{[}\int_{0}^{T}\langle\partial_{v}H_{\theta}(t,\bar{X}_{\theta}(t),\mathbb{E}[\bar{X}_{\theta}(t)],\bar{Y}_{\theta}(t),\bar{Z}_{\theta}(t),\bar{v}(t),p_{\theta}(t),q_{\theta}(t),r_{\theta}(t)),v(t)-\bar{v}(t)\rangle
dt\Big{]}\bar{Q}(d\theta)\geqslant 0.$

Thanks to 5.4, the process $\Lambda_{\theta}(\cdot)$ is ${\cal
F}$-progressively measurable. Then it follows from Fubini’s Theorem that

$\displaystyle\mathbb{E}\Big{[}\int_{0}^{T}\int_{\Theta}\langle\partial_{v}H_{\theta}(t,\bar{X}_{\theta}(t),\mathbb{E}[\bar{X}_{\theta}(t)],\bar{Y}_{\theta}(t),\bar{Z}_{\theta}(t),\bar{v}(t),p_{\theta}(t),q_{\theta}(t),r_{\theta}(t)),v(t)-\bar{v}(t)\rangle\bar{Q}(d\theta)dt\Big{]}\geqslant
0,$

which implies for $v\in V$, $dt\times d\mathbb{P}$-a.e.,

$\displaystyle\int_{\Theta}\langle\partial_{v}H_{\theta}(t,\bar{X}_{\theta}(t),\mathbb{E}[\bar{X}_{\theta}(t)],\bar{Y}_{\theta}(t),\bar{Z}_{\theta}(t),\bar{v}(t),p_{\theta}(t),q_{\theta}(t),r_{\theta}(t)),v(t)-\bar{v}(t)\rangle\bar{Q}(d\theta)\geqslant
0.$

∎

Then we turn to the sufficient condition of the optimal control.

######  Theorem 5.6.

Under the assumptions and the settings in Theorem 5.5, if $H_{\theta}$ is
convex in $(x,x^{\prime},y,z,v)$ and continuous in $t$, $\phi_{\theta}$ is
convex in $x$, $\gamma_{\theta}$ is convex in $y$ and $\Phi_{\theta}$ is
concave in $(x,x^{\prime})$, and there exists an admissible control
$\bar{v}(\cdot)$ and a probability measure $\bar{Q}\in{\cal Q}^{\bar{v}}$ such
that for $v\in V$, $dt\times d\mathbb{P}$-a.e.,

$\displaystyle\int_{\Theta}\langle\partial_{v}H_{\theta}(t,\bar{X}_{\theta}(t),\mathbb{E}[\bar{X}_{\theta}(t)],\bar{Y}_{\theta}(t),\bar{Z}_{\theta}(t),\bar{v}(t),p_{\theta}(t),q_{\theta}(t),r_{\theta}(t)),v-\bar{v}(t)\rangle\bar{Q}(d\theta)\geqslant
0,$

$\bar{v}(\cdot)$ is an optimal control.

###### Proof.

For $\theta\in\Theta$ and $v(\cdot)\in{\cal V}_{ad}$, by
$(X^{v}_{\theta},Y^{v}_{\theta},Z^{v}_{\theta})$ we denote the solution to the
equation (1.1). Set

$\displaystyle(\alpha_{\theta}^{\lambda},\beta_{\theta}^{\lambda},\zeta_{\theta}^{\lambda})=(X_{\theta}^{\lambda}-\bar{X}_{\theta},Y_{\theta}^{\lambda}-\bar{Y}_{\theta},Z_{\theta}^{\lambda}-\bar{Z}_{\theta}).$

Then

$\left\\{\begin{aligned}
\alpha_{\theta}^{\lambda}(t)&=\int_{0}^{t}(\partial_{x}b_{\theta}(s)\alpha_{\theta}^{\lambda}(s)+\partial_{x^{\prime}}b_{\theta}(s)\mathbb{E}[\alpha_{\theta}^{\lambda}(s)]+S_{\theta}(s))ds\\\
&\
+\sum_{i=1}^{d}\int_{0}^{t}(\partial_{x}\sigma^{i}_{\theta}(s)\alpha_{\theta}^{\lambda}(s)+J^{i}_{\theta}(s))dW^{i}(s),\\\
\beta_{\theta}^{\lambda}(t)&=K_{\theta}(T)+\int_{t}^{T}(\partial_{x}f_{\theta}(s)^{\top}\alpha_{\theta}^{\lambda}(s)+\partial_{x^{\prime}}f_{\theta}(s)^{\top}\mathbb{E}[\alpha_{\theta}^{\lambda}(s)]+\partial_{y}f_{\theta}(s)\beta_{\theta}^{\lambda}(s)\\\
&\quad+\partial_{z}f_{\theta}(s)\zeta_{\theta}^{\lambda}(s)+L_{\theta}(s))ds-\int_{t}^{T}\zeta_{\theta}(s)dW(s),\end{aligned}\right.$

where

$\displaystyle S_{\theta}(s)$
$\displaystyle=b_{\theta}(s,X_{\theta}^{v}(s),\mathbb{E}[X_{\theta}^{v}(s)],v(s))-b_{\theta}(s)-\partial_{x}b_{\theta}(s)\alpha_{\theta}^{\lambda}(s)-\partial_{x^{\prime}}b_{\theta}(s)\mathbb{E}[\alpha_{\theta}^{\lambda}(s)],$
$\displaystyle J^{i}_{\theta}(s)$
$\displaystyle=\sigma^{i}_{\theta}(s,X_{\theta}^{v}(s),v(s))-\sigma^{i}_{\theta}(s)-\partial_{x}\sigma^{i}_{\theta}(s)\alpha_{\theta}^{\lambda}(s),$
$\displaystyle K_{\theta}(T)$
$\displaystyle=\Phi_{\theta}(X_{\theta}^{v}(T),\mathbb{E}[X_{\theta}^{v}(T)])-\Phi_{\theta}(\bar{X}_{\theta}(T),\mathbb{E}[\bar{X}_{\theta}(T)]),$
$\displaystyle L_{\theta}(s)$
$\displaystyle=f_{\theta}(s,X_{\theta}^{v}(s),\mathbb{E}[X_{\theta}^{v}(s)],Y_{\theta}^{v}(s),Z_{\theta}^{v}(s),v(s))-f_{\theta}(s)-\partial_{x}f_{\theta}(s)\alpha_{\theta}^{\lambda}(s)$
$\displaystyle\quad-\partial_{x^{\prime}}f_{\theta}(s)\mathbb{E}[\alpha_{\theta}^{\lambda}(s)]-\partial_{y}f_{\theta}(s)\beta_{\theta}^{\lambda}(s)-\partial_{z}f_{\theta}(s)\zeta_{\theta}^{\lambda}(s).$

Applying Ito’s formula to
$\alpha_{\theta}^{\lambda}(\cdot)^{\top}q_{\theta}(\cdot)+\beta_{\theta}^{\lambda}(\cdot)p_{\theta}(\cdot)$
and then integrating from $0$ to $T$, we have

$\displaystyle\mathbb{E}[\partial_{x}\phi_{\theta}(\bar{X}_{\theta}(T))^{\top}\alpha_{\theta}^{\lambda}(T)+\partial_{y}\gamma_{\theta}(\bar{Y}_{\theta}(0))\beta_{\theta}^{\lambda}(0)]$
$\displaystyle=-\mathbb{E}\Big{[}p_{\theta}(T)(\Phi_{\theta}(X_{\theta}^{v}(T),\mathbb{E}[X_{\theta}^{v}(T)])-\Phi_{\theta}(\bar{X}_{\theta}(T),\mathbb{E}[\bar{X}_{\theta}(T)]))$
$\displaystyle\quad-
p_{\theta}(T)\partial_{x}\Phi(T)\alpha_{\theta}^{\lambda}(T)-\mathbb{E}[p_{\theta}(T)\partial_{x^{\prime}}\Phi(T)]\alpha_{\theta}^{\lambda}(T)\Big{]}$
$\displaystyle\quad+\mathbb{E}\Big{[}\int_{0}^{T}\Big{(}q_{\theta}(t)S_{\theta}(t)+\sum_{i=1}^{d}r^{i}_{\theta}(t)J^{i}_{\theta}(t)-p_{\theta}(t)L_{\theta}(t)\Big{)}dt\Big{]}.$

Since $\Phi_{\theta}$ is concave in $(x,x^{\prime})$ and the Hamiltonian
$H_{\theta}$ is convex in $(x,x^{\prime},y,z,v)$, it yields that

$\displaystyle\mathbb{E}[\partial_{x}\phi_{\theta}(\bar{X}_{\theta}(T))^{\top}\alpha_{\theta}^{\lambda}(T)+\partial_{y}\gamma_{\theta}(\bar{Y}_{\theta}(0))\beta_{\theta}^{\lambda}(0)]$
$\displaystyle\geqslant\mathbb{E}\Big{[}\int_{0}^{T}\langle\partial_{v}H_{\theta}(t,\bar{X}_{\theta}(t),\mathbb{E}[\bar{X}_{\theta}(t)],\bar{Y}_{\theta}(t),\bar{Z}_{\theta}(t),\bar{v}(t),p_{\theta}(t),q_{\theta}(t),r_{\theta}(t)),v^{\lambda}(t)-\bar{v}(t)\rangle
dt\Big{]}.$

Finally, recall the fact that $\bar{Q}\in{\cal Q}^{\bar{v}}$ and
$\phi_{\theta},\gamma_{\theta}$ are convex in $x,y$, respectively, it follows
from Sion’s minimax theorem that

$\displaystyle J(v^{\lambda}(\cdot))-J(v(\cdot))$
$\displaystyle=\int_{\Theta}\mathbb{E}\Big{[}\phi_{\theta}(X_{\theta}^{\lambda}(T))-\phi_{\theta}(\bar{X}_{\theta}(T))+\gamma_{\theta}(Y^{\lambda}_{\theta}(0))-\gamma_{\theta}(\bar{Y}_{\theta}(0))\Big{]}\bar{Q}(d\theta)$
$\displaystyle\geqslant\int_{\Theta}\mathbb{E}[\partial_{x}\phi_{\theta}(\bar{X}_{\theta}(T))^{\top}\alpha_{\theta}^{\lambda}(T)+\partial_{y}\gamma_{\theta}(\bar{Y}_{\theta}(0))\beta_{\theta}^{\lambda}(0)]\bar{Q}(d\theta)$
$\displaystyle\geqslant\int_{\Theta}\mathbb{E}\Big{[}\int_{0}^{T}\langle\partial_{v}H_{\theta}(t,\bar{X}_{\theta}(t),\mathbb{E}[\bar{X}_{\theta}(t)],\bar{Y}_{\theta}(t),\bar{Z}_{\theta}(t),\bar{v}(t),p_{\theta}(t),q_{\theta}(t),r_{\theta}(t)),v^{\lambda}(t)-\bar{v}(t)\rangle
dt\Big{]}\bar{Q}(d\theta)$
$\displaystyle\geqslant\mathbb{E}\Big{[}\int_{0}^{T}\int_{\Theta}\langle\partial_{v}H_{\theta}(t,\bar{X}_{\theta}(t),\mathbb{E}[\bar{X}_{\theta}(t)],\bar{Y}_{\theta}(t),\bar{Z}_{\theta}(t),\bar{v}(t),p_{\theta}(t),q_{\theta}(t),r_{\theta}(t)),v^{\lambda}(t)-\bar{v}(t)\rangle\bar{Q}(d\theta)dt\Big{]}\geqslant
0.$

The proof is complete.

∎

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