#  A Revisit to Optimal Control of Forward-Backward Stochastic Differential
System with Observation Noise ††thanks: This work was supported by the Natural
Science Foundation of Zhejiang Province for Distinguished Young Scholar
(No.LR15A010001), and the National Natural Science Foundation of China
(No.11471079, 11301177)

Qingxin Meng Qiuhong Shi Maoning Tang
Department of Mathematics, Huzhou University, Zhejiang 313000, China
Corresponding author. E-mail<EMAIL_ADDRESS>
###### Abstract

This paper revisits the partial information optimal control problem considered
by Wang, Wu and Xiong [13], where the system is derived by a controlled
forward-backward stochastic differential equation with correlated noises
between the system and the observation. For this type of partial information
optimal control problem, one necessary and one suffcient (a verification
theorem) conditions of optimality are derived using a unified way. We improve
the $L^{p}-$ bounds on the control from $L^{8}$ in [13] to $L^{4}$ in this
paper.

Keywords Maximum Principle, Forward-Backward Stochastic Differential Equation,
Partial Information, Girsanov s Theorem

##  1  Introduction

It is well known that forward-backward stochastic differential equations
(FBSDEs in short) consists of a forward stochastic differential equation (SDE
in short) of Itô type and a backward stochastic differential equation (BSDE in
short) of Pardoux-Peng (for details see [11],[5]).

FBSDEs are not only encountered in stochastic optimal control problems when
applying the stochastic maximum principle but also used in mathematical
finance (see Antonelli [1], Duffie and Epstein [3], El Karoui, Peng and Quenez
[5] for example). It now becomes more clear that certain important problems in
mathematical economics and mathematical finance, especially in the
optimization problem, can be formulated to be FBSDEs.

There are two important approaches to the general stochastic optimal control
problem. One is the Bellman dynamic programming principle, which results in
the Hamilton-Jacobi-Bellman equation. The other is the maximum principle. Now
the maximum principle of forward-backward stochastic systems driven by
Brownian motion have been studied extensively in the literature. We refer to
[12, 14, 6, 15]and references therein.

In recent years, there have been growing interests on stochastic optimal
control problems under partial information, partly due to the applications in
mathematical finance. For the partial information optimal control problem, the
objective is to find an optimal control for which the controller has less
information than the complete information filtration. In particular, sometimes
an economic model in which there are information gaps among economic agents
can be formulated as a partial information optimal control problem (see
${\O}$ksendal [10], Kohlmann and Xiong [9]).

Recently, Baghery and ${\O}$ksendal [2] established a maximum principle of
forward systems with jumps under partial information. In 2009, Meng [7]
studied a partial information stochastic optimal control problem of continuous
fully coupled forward-backward stochastic systems driven by a Brownian motion.
As in [2], the author established one sufficient (a verification theorem) and
one necessary conditions of optimality. The main limitation of [2] and [7] is
that the stochastic systems do not contain observation noise and the partial
information filtration is too general to have more practical application. In
2013, Wang, Wu and Xiong [13] studied a partial information optimal control
problem derived by forward-backward stochastic systems with correlated noises
between the system and the observation. Utilizing a direct method, an
approximation method, and a Malliavin derivative method, they established
three versions of maximum principle (i.e., necessary condition) for optimal
control, where the following $L^{8}-$ bounds is imposed on the admissible
controls:

$\displaystyle\mathbb{E}\bigg{[}\sup_{0\leq t\leq T}|u(t)|^{8}\bigg{]}.$ (1)

The present paper revisits the partial information optimal control problem
considered by Wang, Wu and Xiong [13]. Its one of the two main contributions
is that we improve the $L^{p}-$ bounds on the control from $L^{8}-$ bounds (1)
to the following $L^{4}-$ bounds

$\displaystyle\mathbb{E}\bigg{[}\bigg{(}\int_{0}^{T}|u(t)|^{2}dt\bigg{)}^{2}\bigg{]}<\infty.$
(2)

Another contribution of this paper is that we will establish necessary and
sufficient conditions for an optimal control in a unified way. The main idea
is to get directly a variation formula in terms of the Hamiltonian and the
associated adjoint system which is a linear forward-backward stochastic
differential equation and neither the variational systems nor the
corresponding Taylor type expansions of the state process and the cost
functional will be considered.

The paper is organized as follows. In section 2, we formulate the problem and
give various assumptions used throughout the paper. Section 3 is devoted to
derive necessary as well as sufficient optimality conditions in the form of
stochastic maximum principles in a unified way.

##  2 Formulation of Problem

In this section, we introduce some basic notations which will be used in this
paper. Let ${\cal T}:=[0,T]$ denote a finite time index, where $0<T<\infty$.
We consider a complete probability space $(\Omega,{\mathscr{F}},{\mathbb{P}})$
equipped with two one-dimensional standard Brownian motions $\\{W(t),t\in{\cal
T}\\}$ and $\\{Y(t),t\in{\cal T}\\},$ respectively. Let
$\\{\mathscr{F}^{W}_{t}\\}_{t\in{\cal T}}$ and
$\\{\mathscr{F}^{Y}_{t}\\}_{t\in{\cal T}}$ be $\mathbb{P}$-completed natural
filtration generated by $\\{W(t),t\in{\cal T}\\}$ and $\\{Y(t),t\in{\cal
T}\\},$ respectively. Set $\\{\mathscr{F}_{t}\\}_{t\in{\cal
T}}:=\\{\mathscr{F}^{W}_{t}\\}_{t\in{\cal
T}}\bigvee\\{\mathscr{F}^{Y}_{t}\\}_{t\in{\cal
T}},\mathscr{F}=\mathscr{F}_{T}.$ Denote by $\mathbb{E}[\cdot]$ the
expectation under the probablity $\mathbb{P}.$ Let $E$ be a Euclidean space.
The inner product in $E$ is denoted by $\langle\cdot,\cdot\rangle,$ and the
norm in $E$ is denoted by $|\cdot|.$ Let $A^{\top}$ denote the transpose of
the matrix or vector $A.$ For a function
$\psi:\mathbb{R}^{n}\longrightarrow\mathbb{R},$ denote by $\psi_{x}$ its
gradient. If $\psi:\mathbb{R}^{n}\longrightarrow\mathbb{R}^{k}$ (with $k\geq
2),$ then $\psi_{x}=(\frac{\partial\phi_{i}}{\partial x_{j}})$ is the
corresponding $k\times n$-Jacobian matrix. By $\mathscr{P}$ we denote the
predictable $\sigma$ field on $\Omega\times[0,T]$ and by
$\mathscr{B}(\Lambda)$ the Borel $\sigma$-algebra of any topological space
$\Lambda.$ In the follows, $K$ represents a generic constant, which can be
different from line to line. Next we introduce some spaces of random variable
and stochastic processes. For any $\alpha,\beta\in[1,\infty),$ denote by
$M_{\mathscr{F}}^{\beta}(0,T;E)$ the space of all $E$-valued and
${\mathscr{F}}_{t}$-adapted processes $f=\\{f(t,\omega),\ (t,\omega)\in\cal
T\times\Omega\\}$ satisfying
$\|f\|_{M_{\mathscr{F}}^{\beta}(0,T;E)}\triangleq{\left(\mathbb{E}\bigg{[}\displaystyle\int_{0}^{T}|f(t)|^{\beta}dt\bigg{]}\right)^{\frac{1}{\beta}}}<\infty,$
by $S_{\mathscr{F}}^{\beta}(0,T;E)$ the space of all $E$-valued and
${\mathscr{F}}_{t}$-adapted càdlàg processes $f=\\{f(t,\omega),\
(t,\omega)\in{\cal T}\times\Omega\\}$ satisfying
$\|f\|_{S_{\mathscr{F}}^{\beta}(0,T;E)}\triangleq{\left(\mathbb{E}\bigg{[}\displaystyle\sup_{t\in{\cal
T}}|f(t)|^{\beta}\bigg{]}\right)^{\frac{1}{\beta}}}<+\infty,$ by
$L^{\beta}(\Omega,{\mathscr{F}},P;E)$ the space of all $E$-valued random
variables $\xi$ on $(\Omega,{\mathscr{F}},P)$ satisfying
$\|\xi\|_{L^{\beta}(\Omega,{\mathscr{F}},P;E)}\triangleq\sqrt{\mathbb{E}|\xi|^{\beta}}<\infty,$
by $M_{\mathscr{F}}^{\beta}(0,T;L^{\alpha}(0,T;E))$ the space of all
$L^{\alpha}(0,T;E)$-valued and ${\mathscr{F}}_{t}$-adapted processes
$f=\\{f(t,\omega),\ (t,\omega)\in[0,T]\times\Omega\\}$ satisfying
$\|f\|_{\alpha,\beta}\triangleq{\left\\{\mathbb{E}\bigg{[}\left(\displaystyle\int_{0}^{T}|f(t)|^{\alpha}dt\right)^{\frac{\beta}{\alpha}}\bigg{]}\right\\}^{\frac{1}{\beta}}}<\infty.$

Consider the following forward-backward stochastic differential equation

$\displaystyle\left\\{\begin{aligned}
dx(t)=&b(t,x(t),u(t))dt+\sigma_{1}(t,x(t),u(t))dW(t)+\sigma_{2}(t,x(t),u(t))dW^{u}(t),\\\
dy(t)=&f(t,x(t),y(t),z_{1}(t),z_{2}(t),u(t))dt+z_{1}(t)dW(t)+z_{2}(t)dW^{u}(t),\\\
x(0)=&x,\\\ y(T)=&\phi(x(T))\end{aligned}\right.$ (3)

with one observation processes $Y(\cdot)$ driven by the following stochastic
differential equation

$\displaystyle\left\\{\begin{aligned} dY(t)=&h(t,x(t),u(t))dt+dW^{u}(t),\\\
Y(t)=&0,\end{aligned}\right.$ (4)

where $b:{\cal T}\times\Omega\times{\mathbb{R}}^{n}\times
U\rightarrow{\mathbb{R}}^{n}$, $\sigma_{1}:{\cal
T}\times\Omega\times{\mathbb{R}}^{n}\times U\rightarrow{\mathbb{R}}^{n}$,
$\sigma_{2}:{\cal T}\times\Omega\times{\mathbb{R}}^{n}\times
U\rightarrow{\mathbb{R}}^{n}$, $f:{\cal
T}\times\Omega\times{\mathbb{R}}^{n}\times{\mathbb{R}}^{m}\times{\mathbb{R}}^{m}\times{\mathbb{R}}^{m}\times
U\rightarrow{\mathbb{R}}^{m},\phi:\Omega\times{\mathbb{R}}^{n}\rightarrow{\mathbb{R}^{m}}$
and $h:{\cal T}\times\Omega\times{\mathbb{R}}^{n}\times
U\rightarrow{\mathbb{R}}$ are given random mapping with $U$ being a nonempty
convex subset of $\mathbb{R}^{k}.$ In the above equations, $u(\cdot)$ is our
admissible control process defined as follows.

######  Definition 2.1.

An admissible control process is defined as an ${\mathscr{F}}^{Y}_{t}$-adapted
process valued in an nonempty convex subset $U$ in $\mathbb{R}^{K}$ such that

$\mathbb{E}\bigg{[}\bigg{(}\int_{0}^{T}|u(t)|^{2}dt\bigg{)}^{2}\bigg{]}<\infty.$

The set of all admissible controls is denoted by $\cal A.$

Now we make the following standard assumptions on the coefficients of the
equations (3) and (4).

######  Assumption 2.1.

(i)The coefficients $b$, $\sigma_{1},\sigma_{2}$ and $h$ are
${\mathscr{P}}\otimes{\mathscr{B}}({\mathbb{R}}^{n})\otimes{\mathscr{B}}(U)$-measurable.
For each $(x,u)\in\mathbb{R}^{n}\times U$,
$b(\cdot,x,u),\sigma_{1}(\cdot,x,u)$, $\sigma_{2}(\cdot,x,u)$ and
$h(\cdot,x,u)$ are all $\\{\mathscr{F}_{t}\\}_{t\in\cal T}$-adapted processes.
For almost all $(t,\omega)\in{\cal T}\times\Omega$, the mapping

$\displaystyle(x,u)\rightarrow\psi(t,\omega,x,u)$

is continuous differentiable with respect to $(x,u)$ with appropriate growths,
where $\psi=b,\sigma_{1},\sigma_{2}$ and $h.$ More precisely, there exists a
constant $C>0$ such that for all $x\in\mathbb{R}^{n},u\in U$ and a.e.
$(t,\omega)\in{\cal T}\times\Omega,$

$\displaystyle\left\\{\begin{aligned}
&(1+|x|+|u|)^{-1}|\alpha(t,x,u)|+|\alpha_{x}(t,x,u)|+|\alpha_{u}(t,x,u)|\leq
C,\alpha=b,\sigma_{1},\\\
&|\beta(t,x,u)|+|\beta_{x}(t,x,u)|+|\beta_{u}(t,x,u)|\leq
C,\beta=h,\sigma_{2},.\end{aligned}\right.$

(ii)The coefficient $f$ is
${\mathscr{P}}\otimes{\mathscr{B}}({\mathbb{R}}^{n})\otimes{\mathscr{B}}({\mathbb{R}}^{m})\otimes{\mathscr{B}}({\mathbb{R}}^{m})\otimes{\mathscr{B}}({\mathbb{R}}^{m})\otimes{\mathscr{B}}(U)$-measurable.
For each
$(x,y,z_{1},z_{2},u)\in\mathbb{R}^{n}\times\mathbb{R}^{m}\times\mathbb{R}^{m}\times\mathbb{R}^{m}\times
U$, $f(\cdot,x,y,z_{1},z_{2},u)$ is $\\{\mathscr{F}_{t}\\}_{t\in\cal
T}$-adapted processes. For almost all $(t,\omega)\in{\cal T}\times\Omega$, the
mapping

$\displaystyle(x,y,z_{1},z_{2},u)\rightarrow f(t,\omega,x,y,z_{1},z_{2},u)$

is continuous differentiable with respect to $(x,y,z_{1},z_{2},u)$ with
appropriate growths. More precisely, there exists a constant $C>0$ such that
for all
$(x,y,z_{1},z_{2},u)\in\mathbb{R}^{n}\times\mathbb{R}^{m}\times\mathbb{R}^{m}\times\mathbb{R}^{m}\times
U$ and a.e. $(t,\omega)\in{\cal T}\times\Omega,$

$\displaystyle\begin{split}&(1+|x|+|y|+|z_{1}|+|z_{2}|+|u|)^{-1}|f(t,x,y,z_{1},z_{2},u)|+|f_{x}(t,x,y,z_{1},z_{2},u)|+|f_{y}(t,x,y,z_{1},z_{2},u)|\\\
&+|f_{z_{1}}(t,x,y,z_{1},z_{2},u)|+|f_{z_{2}}(t,x,y,z_{1},z_{2},u)|+|f_{u}(t,x,y,z_{1},z_{2},u)|\leq
C.\end{split}$ (5)

(iii) The coefficient $\phi$ is
${\mathscr{F}}_{T}\otimes{\mathscr{B}}({\mathbb{R}}^{n})$-measurable. For
almost all $(t,\omega)\in[0,T]\times\Omega$, the mapping

$\displaystyle x\rightarrow\phi(\omega,x)$

is continuous differentiable with respect to $x$ with appropriate growths,
respectively. More precisely, there exists a constant $C>0$ such that for all
$x\in\mathbb{R}^{n}$ and a.e. $\omega\in\Omega,$

$\displaystyle\begin{split}&(1+|x|)^{-1}|\phi(x)|+|\phi_{x}(x)|\leq
C.\end{split}$ (6)

Now we begin to discuss the well- posedness of (3) and (4). Indeed, putting
(4) into the state equation (3), we get that

$\displaystyle\left\\{\begin{aligned}
dx(t)=&(b-\sigma_{2}h)(t,x(t),u(t))dt+\sigma_{1}(t,x(t),u(t))dW(t)+\sigma_{2}(t,x(t),u(t))dY(t),\\\
dy(t)=&(f(t,x(t),y(t),z_{1}(t),z_{2}(t),u(t))-z_{2}(t)h(t,x(t),u(t)))dt+z_{1}(t)dW(t)+z_{2}(t)dY(t),\\\
x(0)=&x,\\\ y(T)=&\Phi(x(T)).\end{aligned}\right.$ (7)

Under Assumption 2.1, for any admissible control $u(\cdot)\in\cal A,$ we have
the following basic result.

######  Lemma 2.1.

Let Assumption 2.1 be satisfied. Then for any admissible control
$u(\cdot)\in\cal A,$ the equation (7) admits a unique strong solution
$(x(\cdot),y(\cdot),z_{1}(\cdot),z_{2}(\cdot))\in
S_{\mathscr{F}}^{4}(0,T;\mathbb{R}^{n})\times
S_{\mathscr{F}}^{4}(0,T;\mathbb{R}^{m})\times
M_{\mathscr{F}}^{2}(0,T;L^{2}(0,T;\mathbb{R}^{m}))\times
M_{\mathscr{F}}^{2}(0,T;L^{2}(0,T;\mathbb{R}^{m})).$ Moreover, we have the
following estimate:

$\displaystyle\begin{split}{\mathbb{E}}\bigg{[}\sup_{t\in\cal
T}|x(t)|^{4}\bigg{]}+{\mathbb{E}}\bigg{[}\sup_{t\in\cal
T}|y(t)|^{4}\bigg{]}+{\mathbb{E}}\bigg{[}\bigg{(}\int_{0}^{T}|z_{1}(t)|^{2}dt\bigg{)}^{2}+{\mathbb{E}}\bigg{[}\bigg{(}\int_{0}^{T}|z_{2}(t)|^{2}dt\bigg{)}^{2}\bigg{]}&\leq&K\bigg{\\{}1+|x|^{4}+\mathbb{E}\bigg{[}\Big{(}\int_{0}^{T}|u(t)|^{2}dt\Big{)}^{2}\bigg{]}\bigg{\\}}.\end{split}$
(8)

Further, if
$(\bar{x}(\cdot),\bar{y}(\cdot),\bar{z}_{1}(\cdot),\bar{z}_{2}(\cdot))$ is the
unique strong solution corresponding to another admissible control
$\bar{u}(\cdot)\in\cal A,$ then the following estimate holds:

$\displaystyle\begin{split}&{\mathbb{E}}\bigg{[}\sup_{t\in\cal
T}|x(t)-\bar{x}(t)|^{4}\bigg{]}+{\mathbb{E}}\bigg{[}\sup_{t\in\cal
T}|y(t)-\bar{y}(t)|^{4}\bigg{]}+{\mathbb{E}}\bigg{[}\bigg{(}\int_{0}^{T}|z_{1}(t)-\bar{z}_{1}(t)|^{2}dt\bigg{)}^{2}\bigg{]}+{\mathbb{E}}\bigg{[}\bigg{(}\int_{0}^{T}|z_{2}(t)-\bar{z}_{2}(t)|^{2}dt\bigg{)}^{2}\bigg{]}\\\
&\leq
K{\mathbb{E}}\bigg{[}\int_{0}^{T}|u(t)-\bar{u}(t)|^{2}dt\bigg{]}^{2}.\end{split}$
(9)

###### Proof.

The proof can be directly obtained by combining Proposition 2.1 in [8] and
Lemma 2 in [6]. ∎

For the strong solution
$(x^{u}(\cdot),y^{u}(\cdot),z_{1}^{u}(\cdot),z_{2}^{u}(\cdot))$ of the
equation (7) associated with any given admissible control $u(\cdot)\in\cal A,$
we introduce a process

$\displaystyle\begin{split}\rho^{u}(t)=\displaystyle\exp^{\bigg{\\{}\displaystyle\int_{0}^{t}h(s,x^{u}(s),u(s))dY(s)-\frac{1}{2}h^{2}(s,x^{u}(s),u(s))ds\bigg{\\}}},\end{split}$
(10)

which is abviously the solution to the following SDE

$\displaystyle\left\\{\begin{aligned}
d\rho^{u}(t)=&\rho^{u}(t)h(s,x^{u}(s),u(s))dY(s)\\\
\rho^{u}(0)=&1.\end{aligned}\right.$ (11)

For the stochastic process $\rho^{u}(\cdot),$ we have the following basic
result.

######  Lemma 2.2.

Let Assumption 2.1 holds. Then for any $u(\cdot)\in\cal A,$ we have for any
$\alpha\geq 2,$

$\displaystyle\begin{split}{\mathbb{E}}\bigg{[}\sup_{{t\in\cal
T}}|\rho^{u}(t)|^{\alpha}\bigg{]}\leq K.\end{split}$ (12)

Further, if $\bar{\rho}(\cdot)$ is the process defined by (10) or (11)
corresponding to another admissible control $\bar{u}(\cdot)\in\cal A,$ then
the following estimate holds

$\displaystyle{\mathbb{E}}\bigg{[}\sup_{t\in\cal
T}|\rho^{u}(t)-\bar{\rho}(t)|^{2}\bigg{]}\leq
K\bigg{\\{}{\mathbb{E}}\bigg{[}\int_{0}^{T}|u(t)-\bar{u}(t)|^{2}dt\bigg{]}^{{2}}\bigg{\\}}^{\frac{1}{2}}.$
(13)

###### Proof.

The proof can be directly obtained by combining Proposition 2.1 in [8]. ∎

Under Assumption 2.1, $\rho^{u}(\cdot)$ is an
$(\Omega,{\mathscr{F}},\\{\mathscr{F}_{t}\\}_{t\in{\cal T}},{\mathbb{P}})-$
martingale. Define a new probability measure $\mathbb{P}^{u}$ on
$(\Omega,\mathscr{F})$ by

$\displaystyle d\mathbb{P}^{u}=\rho^{u}(1)d\mathbb{P}.$ (14)

Then from Girsanov’s theorem and (4), $(W(\cdot),W^{u}(\cdot))$ is an
$\mathbb{R}^{2}$-valued standard Brownian motion defined in the new
probability space $(\Omega,\mathscr{F},\\{\mathscr{F}_{t}\\}_{0\leq t\leq
T},\mathbb{P}^{u}).$ So
$(\mathbb{P}^{u},x^{u}(\cdot),y^{u}(\cdot),z^{u}_{1}(\cdot),z^{u}_{2}(\cdot),\rho^{u}(\cdot),W(\cdot),W^{u}(\cdot))$
is a weak solution on $(\Omega,\mathscr{F},\\{\mathscr{F}_{t}\\}_{t\in\cal
T})$ of (3) and (4).

The cost functional is given by

$\displaystyle\begin{split}J(u(\cdot)=\mathbb{E}^{u}\bigg{[}\int_{0}^{T}l(t,x(t),y(t),z_{1}(t),z_{2}(t),u(t))dt+\Phi(x(T))+\gamma(y(0))\bigg{]}.\end{split}$
(15)

where $\mathbb{E}^{u}$ denotes the expectation with respect to the probability
space $(\Omega,\mathscr{F},\\{\mathscr{F}_{t}\\}_{0\leq t\leq
T},\mathbb{P}^{u})$ and $l:{\cal
T}\times\Omega\times{\mathbb{R}}^{n}\times{\mathbb{R}}^{m}\times{\mathbb{R}}^{m}\times{\mathbb{R}}^{m}\times
U\rightarrow{\mathbb{R}},$
$\Phi:\Omega\times{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}$ and
$\gamma:\Omega\times{\mathbb{R}}^{m}\rightarrow{\mathbb{R}}$ are given random
mappings satisfying the following assumption:

######  Assumption 2.2.

$l$ is
${\mathscr{P}}\otimes{\mathscr{B}}({\mathbb{R}}^{n})\otimes{\mathscr{B}}({\mathbb{R}}^{m})\otimes{\mathscr{B}}({\mathbb{R}}^{m})\otimes{\mathscr{B}}({\mathbb{R}}^{m})\otimes{\mathscr{B}}(U)$-measurable,
$\Phi$ is
${\mathscr{F}}_{T}\otimes{\mathscr{B}}({\mathbb{R}}^{n})$-measurable, and
$\gamma$ is
${\mathscr{F}}_{0}\otimes{\mathscr{B}}({\mathbb{R}}^{n})$-measurable. For each
$(x,y,z_{1},z_{2},u)\in\mathbb{R}^{n}\times\mathbb{R}^{m}\times\mathbb{R}^{m}\times\mathbb{R}^{m}\times
U$, $f(\cdot,x,y,z_{1},z_{2},u)$ is an ${\mathbb{F}}$-adapted process,
$\Phi(x)$ is an ${\mathscr{F}}_{T}$-measurable random variable, and
$\gamma(y)$ is an ${\mathscr{F}}_{0}$-measurable random variable. For almost
all $(t,\omega)\in[0,T]\times\Omega$, the mappings

$\displaystyle(x,y,z_{1},z_{2},u)\rightarrow l(t,\omega,x,y,z_{1},z_{2},u),$
$\displaystyle x\rightarrow\Phi(\omega,x)$

and

$\displaystyle y\rightarrow\gamma(\omega,y)$

are continuous differentiable with respect to $(x,y,z_{1},z_{2},u)$ with
appropriate growths, respectively. More precisely, there exists a constant
$C>0$ such that for all
$(x,y,z_{1},z_{2},u)\in\mathbb{R}^{n}\times\mathbb{R}^{m}\times\mathbb{R}^{m}\times\mathbb{R}^{m}\times
U$ and a.e. $(t,\omega)\in[0,T]\times\Omega,$

$\displaystyle\left\\{\begin{aligned}
&(1+|x|+|y|+|z_{1}|+|z_{2}|+|u|)^{-1}(|l_{x}(t,x,y,z_{1},z_{2},u)|+|l_{y}(t,x,y,z_{1},z_{2},u)|+|l_{z_{1}}(t,x,y,z_{1},z_{2},u)|\\\
&\quad\quad+|l_{z_{2}}(t,x,y,z_{1},z_{2},u)|+|l_{u}(t,x,y,z_{1},z_{2},u)|)+(1+|x|^{2}+|y|^{2}+|z_{1}|^{2}+|z_{2}|^{2}+|u|^{2})^{-1}|l(t,x,y,z_{1},z_{2},u)|\leq
C;\\\ &(1+|x|^{2})^{-1}|\Phi(x)|+(1+|x|)^{-1}|\Phi_{x}(x)|\leq C,\\\
&(1+|y|^{2})^{-1}|\gamma(y)|+(1+|y|)^{-1}|\gamma_{y}(y)|\leq
C.\end{aligned}\right.$

Under Assumption 2.1 and 2.2, by the estimates (8) and (12), we get that

$\displaystyle\begin{split}|J(u(\cdot))|\leq&K\bigg{\\{}\mathbb{E}\bigg{[}\sup_{t\in{\cal
T}}|\rho^{u}(t)|^{2}\bigg{]}\bigg{\\}}^{\frac{1}{2}}\bigg{\\{}\mathbb{E}\bigg{[}\sup_{{t\in\cal
T}}|x(t)|^{4}\bigg{]}+\mathbb{E}\bigg{[}\sup_{{t\in\cal
T}}|y(t)|^{4}\bigg{]}+\mathbb{E}\bigg{[}\bigg{(}\int_{0}^{T}|z_{1}(t)|^{2}dt\bigg{)}^{2}\bigg{]}\\\
&\quad\quad+\mathbb{E}\bigg{[}\bigg{(}\int_{0}^{T}|z_{2}(t)|^{2}dt\bigg{)}^{2}\bigg{]}+\mathbb{E}\bigg{[}\bigg{(}\int_{0}^{T}|u(t)|^{2}dt\bigg{)}^{2}\bigg{]}+1\bigg{\\}}^{\frac{1}{2}}\\\
<&\infty,\end{split}$ (16)

which implies that the cost functional is well-defined.

Then we can put forward the following partially observed optimal control
problem in its weak formulation, i.e., with changing the reference probability
space $(\Omega,\mathscr{F},\\{\mathscr{F}_{t}\\}_{0\leq t\leq
T},\mathbb{P}^{u}),$ as follows.

######  Problem 2.1.

Find an admissible control $\bar{u}(\cdot)\in\cal$ such that

$J(\bar{u}(\cdot))=\displaystyle\inf_{u(\cdot)\in\cal A}J(u(\cdot)),$

subject to the state equation (3), the observation equation (4) and the cost
functional (15).

Obviously, according to Bayes’ formula, the cost functional (15) can be
rewritten as

$\begin{split}J(u(\cdot))=&\mathbb{E}\displaystyle\bigg{[}\int_{0}^{T}\rho^{u}(t)l(t,x(t),y(t),z_{1}(t),z_{2}(t),u(t))dt+\rho^{u}(T)\Phi(x(T))+\gamma(y(0))\bigg{]}.\end{split}$
(17)

Therefore, we can translate Problem 2.1 into the following equivalent optimal
control problem in its strong formulation, i.e., without changing the
reference probability space $(\Omega,\mathscr{F},\\{\mathscr{F}_{t}\\}_{0\leq
t\leq T},\mathbb{P}),$ where $\rho^{u}(\cdot)$ will be regarded as an
additional state process besides the state process
$(x^{u}(\cdot),y^{u}(\cdot),z^{u}_{1}(\cdot),z^{u}_{2}(\cdot)).$

######  Problem 2.2.

Find an admissible control $\bar{u}(\cdot)$ such that

$J(\bar{u}(\cdot))=\displaystyle\inf_{u(\cdot)\in\cal A}J(u(\cdot)),$

subject to the cost functional (17) and the following state equation

$\displaystyle\left\\{\begin{array}[]{lll}dx(t)=&(b-\sigma_{2}h)(t,x(t),u(t))dt+\sigma_{1}(t,x(t),u(t))dW(t)+\sigma_{2}(t,x(t),u(t))dY(t),\\\
dy(t)=&(f(t,x(t),y(t),z_{1}(t),z_{2}(t),u(t))-z_{2}(t)h(t,x(t),u(t)))dt+z_{1}(t)dW(t)+z_{2}(t)dY(t),\\\
d\rho^{u}(t)=&\rho^{u}(t)h(s,x^{u}(s),u(s))dY(s),\\\ \rho^{u}(0)=&1,\\\
x(0)=&x,\\\ y(T)=&\Phi(x(T)).\end{array}\right.$ (18)

Any $\bar{u}(\cdot)\in\cal A$ satisfying above is called an optimal control
process of Problem 2.2 and the corresponding state process
$(\bar{x}(\cdot),\bar{y}(\cdot),\bar{z}_{1}(\cdot),\bar{z}_{2}(\cdot),\bar{\rho}(\cdot))$
is called the optimal state process. Correspondingly
$(\bar{u}(\cdot);\bar{x}(\cdot),\bar{y}(\cdot),\bar{z}_{1}(\cdot),\bar{z}_{2}(\cdot),\bar{\rho}(\cdot))$
is called an optimal pair of Problem 2.2.

######  Remark 2.1.

The present formulation of the partially observed optimal control problem is
quite similar to a completely observed optimal control problem; the only
difference lies in the admissible class $\cal A$ of controls.

##  3 Stochastic Maximum Principle

This section is devoted to establishing the stochastic maximum principle of
Problem 2.1 or Problem 2.2, i.e., establishing the necessary and sufficient
optimality conditions of Pontryagin’s type for an admissible control to be
optimal. To this end, for the state equation (18), we first introduce the
corresponding adjoint equation.

Define the Hamiltonian ${\cal H}:\Omega\times{\cal
T}\times\mathbb{R}^{n}\times\mathbb{R}^{m}\times\mathbb{R}^{m}\times\mathbb{R}^{m}\times
U\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^{m}\times\mathbb{R}\rightarrow\mathbb{R}$
as follows:

$\displaystyle{\cal H}(t,x,y,z_{1},z_{2},u,p,q_{1},q_{2},k,R_{2})$
$\displaystyle=l(t,x,y,z_{1},z_{2},u)+\langle
b(t,x,u),p\rangle+\langle\sigma_{1}(t,x,u),q_{1}\rangle+\langle\sigma_{2}(t,x,u),q_{2}\rangle+\langle
f(t,x,y,z_{1},z_{2},u),k\rangle+\langle R_{2},h(t,x,u)\rangle\ .$ (19)

For any given admissible control pair
$(\bar{u}(\cdot);\bar{x}(\cdot),\bar{y}(\cdot),\bar{z}_{1}(\cdot),\bar{z}_{2}(\cdot)),$
the corresponding adjoint equation is defined as follows.

$\displaystyle\begin{split}d\bar{r}(t)&=-l(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t))dt+\bar{R}_{1}\left(t\right)dW\left(t\right)+{\bar{R}}_{2}\left(t\right)dW^{\bar{u}}\left(t\right),\\\
d\bar{p}\left(t\right)&=-{\cal{H}}_{x}\left(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t)\right)dt+\bar{q}_{1}\left(t\right)dW\left(t\right)+{\bar{q}}_{2}\left(t\right)dW^{\bar{u}}\left(t\right),\\\
d\bar{k}\left(t\right)&=-{\cal{H}}_{y}\left(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t)\right)dt-{\cal{H}}_{z_{1}}\left(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t)\right)dW\left(t\right)\\\
&\quad\quad-{\cal{H}}_{z_{2}}\left(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t)\right)dW^{u}\left(t\right),\\\
\bar{p}(T)&=\Phi_{x}(\bar{x}(T))-\phi_{x}^{*}(\bar{x}(T))\bar{k}(T),\\\
\bar{r}(T)&=\Phi(\bar{x}(T)),\\\
\bar{k}(0)&=-\gamma_{y}(\bar{y}(0)).\end{split}$ (20)

Here we have used the following notation

$\displaystyle\begin{split}&{\cal
H}_{a}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t))\\\
=&{\cal{H}}_{a}\left(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t),\bar{p}(t),\bar{q}_{1}(t),\bar{q}_{2}(t),\bar{k}(t),\bar{R}_{2}(t)-\bar{\sigma}_{2}^{*}(t,x(t),u(t))\bar{p}(t)-\bar{z}_{2}^{*}(t)\bar{k}(t)\right)\end{split}$
(21)

where $a=x,y,z_{1},z_{2},u.$

Note the adjoint equation (20) is a forward-backward stochastic differential
equation whose solution consists of an 7-tuple process
$(\bar{p}(\cdot),\bar{q}_{1}(\cdot),\bar{q}_{2}(\cdot),\bar{k}(\cdot),\bar{r}(\cdot),\bar{R}_{1}(\cdot),\bar{R}_{2}(\cdot)).$
Under Assumptions 2.1 and 2.2, by Proposition 2.1 in [8] and Lemma 2 in [6] ,
it is easily to see that the adjoint equation (20) admits a unique solution
$(\bar{p}(\cdot),\bar{q}_{1}(\cdot),\bar{q}_{2}(\cdot),\bar{k}(\cdot),\bar{r}(\cdot),\bar{R}_{1}(\cdot),\bar{R}_{2}(\cdot))\in
S_{\mathscr{F}}^{4}(0,T;\mathbb{R}^{n})\times
M_{\mathscr{F}}^{2}(0,T;L^{2}(0,T;\mathbb{R}^{n}))\times
M_{\mathscr{F}}^{2}(0,T;L^{2}(0,T;\mathbb{R}^{n}))\times
S_{\mathscr{F}}^{4}(0,T;\mathbb{R}^{m})\times
S_{\mathscr{F}}^{4}(0,T;\mathbb{R})\times
M_{\mathscr{F}}^{2}(0,T;L^{2}(0,T;\mathbb{R}))\times
M_{\mathscr{F}}^{2}(0,T;L^{2}(0,T;\mathbb{R}^{n})),$ also called the adjoint
process corresponding the admissible pair
$(\bar{u}(\cdot);\bar{x}(\cdot),\bar{y}(\cdot),\bar{z}_{1}(\cdot),\bar{z}_{2}(\cdot),\bar{\rho}(\cdot))$.

Let
$({u}(\cdot);x^{u}(\cdot),y^{u}(\cdot),z^{u}_{1}(\cdot),z^{u}_{2}(\cdot),\rho^{u}(\cdot))$
and
$(\bar{u}(\cdot);\bar{x}(\cdot),\bar{y}(\cdot),\bar{z}_{1}(\cdot),\bar{z}_{2}(\cdot),\bar{\rho}(\cdot))$
be two admissible pairs. Next we represent the difference
$J(u(\cdot))-J(\bar{u}(\cdot))$ in terms of the Hamiltonian ${\cal H}$ and the
adjoint process
$(\bar{p}(\cdot),\bar{q}_{1}(\cdot),\bar{q}_{2}(\cdot),\bar{k}(\cdot),\bar{r}(\cdot),\bar{R}_{1}(\cdot),\bar{R}_{2}(\cdot))$
as well as other relevant expressions.

To unburden our notation, we will use the following abbreviations:

$\displaystyle\left\\{\begin{aligned} &\gamma^{u}(0)=\gamma(y^{u}(0))\
,\bar{\gamma}(0)=\gamma(\bar{y}(0)),\ \\\
&\phi^{u}(T)=\phi(x^{u}(T)),\bar{\phi}(T)=\phi(\bar{x}(T))\ ,\\\
&\Phi^{u}(T)=\Phi(x^{u}(T)),\bar{\Phi}(T)=\Phi(\bar{x}(T)),\\\
&\alpha^{u}(t)=\alpha(t,x^{u}(t),u(t)),\\\
&\bar{\alpha}(t)=\alpha(t,\bar{x}(t),\bar{u}(t)),\quad\alpha=b,\sigma_{1},\sigma_{2},h,\\\
&\beta^{u}(t)=\alpha(t,x^{u}(t),y^{u}(t),z^{u}_{1}(t),z^{u}_{2}(t),u(t)),\\\
&\bar{\beta}(t)=\alpha(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t)),\quad\beta=f,l.\end{aligned}\right.$
(22)

######  Lemma 3.1.

Let Assumptions 2.1 and 2.2 be satisfied. Using the notations (21) and (22),
we have

$\displaystyle J(u(\cdot))-J(\bar{u}(\cdot))$ (23) $\displaystyle=$
$\displaystyle{\mathbb{E}}^{\bar{u}}\bigg{[}\int_{0}^{T}\bigg{[}{\cal
H}(t,x^{u}(t),y^{u}(t),z^{u}_{1}(t),z^{u}_{2}(t),u(t))-{\cal
H}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t))$
$\displaystyle-\big{<}{\cal
H}_{x}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t)),x^{u}(t)-\bar{x}(t)\big{>}$
$\displaystyle-\big{<}{\cal
H}_{y}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t)),y^{u}(t)-\bar{y}(t)\big{>}$
$\displaystyle-\big{<}{\cal
H}_{z_{1}}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t)),z^{u}_{1}(t)-\bar{z}_{1}(t)\big{>}$
$\displaystyle-\big{<}{\cal
H}_{z_{2}}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t)),z^{u}_{2}(t)-\bar{z}_{2}(t)\big{>}$
$\displaystyle-\langle(\sigma_{2}^{u}(t)-\bar{\sigma}_{2}(t))(h^{u}(t)-\bar{h}(t)),\bar{p}(t)\rangle$
$\displaystyle-\langle(z_{2}^{u}(t)-\bar{z}_{2}(t))(h^{u}(t)-\bar{h}(t)),\bar{k}(t)\rangle
dt\bigg{]}$
$\displaystyle+{\mathbb{E}}^{\bar{u}}\big{[}\Phi^{u}(T)-\bar{\Phi}(T)-\left<x^{u}(T)-\bar{x}(T),\bar{\Phi}_{x}(T)\right>\big{]}$
$\displaystyle-{\mathbb{E}}^{\bar{u}}\big{[}\langle\phi^{u}(T)-\bar{\phi}(T),\bar{k}(T)\rangle-\left<\bar{\phi}_{x}^{*}(T)\bar{k}(T),x^{u}(T)-\bar{x}(T)\right>\big{]}$
$\displaystyle+{\mathbb{E}}\big{[}\gamma^{u}(0)-\bar{\gamma}(0)-\left<y^{u}(0)-\bar{y}(0),\bar{\gamma}_{y}(0)\right>\big{]}$
$\displaystyle+\mathbb{E}\bigg{[}\int_{0}^{T}\bar{R}_{2}(t)(\rho^{u}(t)-\rho^{\bar{u}}(t))(h^{u}(t)-\bar{h}(t))dt\bigg{]}$
$\displaystyle+\mathbb{E}\bigg{[}\int_{0}^{T}(l^{u}(t)-\bar{l}(t))(\rho^{u}(t)-\bar{\rho}(t))dt\bigg{]}$
$\displaystyle+\mathbb{E}\bigg{[}(\rho^{u}(T)-\bar{\rho}(T))(\Phi^{u}(T)-\bar{\Phi}(T))\bigg{]}.$

###### Proof.

From (4), it is easy to check that the
$(x^{u}(\cdot),y^{u}(\cdot),z^{u}(\cdot))$ satisfies the following FBSDE:

$\displaystyle\left\\{\begin{aligned}
dx(t)=&\big{[}b^{u}(t)+\sigma_{2}^{u}(t)(\bar{h}(t)-h^{u}(t))\big{]}dt+\sigma^{u}_{1}(t)dW(t)+\sigma_{2}^{u}(t)dW^{\bar{u}}(t)\\\
dy(t)=&\big{[}f^{u}(t)+z_{2}(t)(\bar{h}(t)-h^{u}(t))\big{]}dt+z_{1}(t)dW(t)+z_{2}(t)dW^{\bar{u}}(t)\\\
x(0)=&x,\\\ y(T)=&\phi(x(T))\end{aligned}\right.$ (24)

Therefore $(x^{u}(t)-\bar{x}(t),y^{u}(t)-\bar{y}(t),z^{u}(t)-\bar{z}(t))$
satisfies the following FBSDE:

$\displaystyle\left\\{\begin{aligned}
dx(t)-\bar{x}(t)=&\big{[}b^{u}(t)-\bar{b}(t)+\sigma_{2}^{u}(t)(h^{\bar{u}}(t)-h^{u}(t))\big{]}dt+[\sigma^{u}_{1}(t)-\bar{\sigma}_{1}(t)]dW(t)+[\sigma_{2}^{u}(t)-\bar{\sigma}_{2}(t)]dW^{\bar{u}}(t)\\\
dy(t)-\bar{y}(t)=&\big{[}f^{u}(t)-\bar{f}(t)+z_{2}(t)(\bar{h}(t)-h(t))\big{]}dt+[z_{1}(t)-\bar{z}_{1}(t)]dW(t)+[z_{2}(t)-\bar{z}_{2}(t)]dW^{\bar{u}}(t)\\\
x(0)-\bar{x}(0)=&0,\\\
y(T)-\bar{y}(T)=&\phi(x(T))-\phi(\bar{x}(T)).\end{aligned}\right.$ (25)

From (20), we know that
$(\bar{p}(\cdot),\bar{q}_{1}(\cdot),\bar{q}_{2}(\cdot),\bar{k}(\cdot))$
satisfies the following FBSDE

$\displaystyle\begin{split}d\bar{p}\left(t\right)&=-{\cal{H}}_{x}\left(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t)\right)dt+\bar{q}_{1}\left(t\right)dW\left(t\right)+{\bar{q}}_{2}\left(t\right)dW^{\bar{u}}\left(t\right),\\\
d\bar{k}\left(t\right)&=-{\cal{H}}_{y}\left(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t)\right)dt-{\cal{H}}_{z_{1}}\left(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t)\right)dW\left(t\right)\\\
&\quad\quad-{\cal{H}}_{z_{2}}\left(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t)\right)dW^{\bar{u}}\left(t\right),\\\
\bar{p}(T)&=\bar{\Phi}_{x}(T)-\bar{\phi}_{x}^{*}(T)\bar{k}(T),\\\
\bar{k}(0)&=-\bar{\gamma}_{y}(0),\end{split}$ (26)

It is easy to check that
$(\bar{r}(\cdot),\bar{R}_{1}(\cdot),\bar{R}_{2}(\cdot))$ satisfying the
following BSDE

$\displaystyle\begin{split}d\bar{r}(t)&=-[\bar{l}(t)+\bar{R}_{2}(t)\bar{h}(t)]dt+\bar{R}_{1}\left(t\right)dW\left(t\right)+{\bar{R}}_{2}\left(t\right)dY(t),\\\
\bar{r}(T)&=\bar{\Phi}(T),\end{split}$ (27)

From the definition of the cost function $J(u(\cdot)),$ we have

$\displaystyle\begin{split}J(u(\cdot))-J(\bar{u}(\cdot))=&\mathbb{E}^{u}\bigg{[}\int_{0}^{T}l^{u}(t)dt+\Phi^{u}(T)+\gamma^{u}(0)\bigg{]}-\mathbb{E}^{\bar{u}}\bigg{[}\int_{0}^{T}\bar{l}(t)dt+\bar{\Phi}(T)+\bar{\gamma}(0)\bigg{]}\\\
=&\mathbb{E}\bigg{[}\int_{0}^{T}(\rho^{u}(t)l^{u}(t)-\bar{\rho}(t)\bar{l}(t))dt\bigg{]}+\mathbb{E}[\rho^{u}(T)\Phi^{u}(T)-\bar{\rho}(t)\bar{\Phi}(T)]+\mathbb{E}[\gamma^{u}(0)-\bar{\gamma}(0)]\\\
=&\mathbb{E}^{\bar{u}}\bigg{[}\int_{0}^{T}[l^{u}(t)-\bar{l}(t)]dt\bigg{]}+\mathbb{E}^{\bar{u}}[\Phi^{u}(T)-\bar{\Phi}(T)]+\mathbb{E}\bigg{[}\int_{0}^{T}(\rho^{u}(t)-\bar{\rho}(t))l^{u}(t)dt\bigg{]}\\\
&+\mathbb{E}[(\rho^{u}(T)-\bar{\rho}(T))\Phi^{u}(T)]+\mathbb{E}[\gamma^{u}(0)-\bar{\gamma}(0)]\end{split}$
(28)

From the definition of $\cal H,$ we get that

$\displaystyle\begin{split}\mathbb{E}^{\bar{u}}\bigg{[}\int_{0}^{T}(l^{u}(t)-\bar{l}(t))dt\bigg{]}=&{\mathbb{E}}^{\bar{u}}\bigg{[}\int_{0}^{T}\bigg{(}{\cal
H}(t,x^{u}(t),y^{u}(t),z^{u}_{1}(t),z^{u}_{2}(t),u(t))-{\cal
H}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t))\bigg{)}dt\bigg{]}\\\
&-\mathbb{E}^{\bar{u}}\bigg{[}\int_{0}^{T}\bigg{(}\langle\bar{p}(t),b^{u}(t)-\bar{b}(t)\rangle+\langle\bar{q}_{1}(t),\sigma_{1}^{u}(t)-\bar{\sigma}_{1}(t)\rangle+\langle\bar{q}_{2}(t),\sigma_{2}^{u}(t)-{\bar{\sigma}}_{2}(t)\rangle\\\
&+\langle\bar{k}(t),f^{u}(t)-\bar{f}(t)\rangle+\langle\bar{R}_{2}(t)-\bar{\sigma}_{2}^{*}(t)\bar{p}(t)-\bar{z}_{2}^{*}(t)\bar{k}(t),h^{u}(t)-\bar{h}(t)\rangle\bigg{)}dt\bigg{]}\end{split}$
(29)

Applying Itô formula to
$\langle\bar{p}(t),x^{u}(t)-\bar{x}(t)\rangle+\langle\bar{k}(t),y^{u}(t)-\bar{y}(t)\rangle$
and taking expectation under $P^{\bar{u}}$, we have

$\displaystyle\mathbb{E}^{\bar{u}}\big{[}\langle\bar{\Phi}_{x}(T)-\bar{\phi}_{x}^{*}(T)\bar{k}(T),x^{u}(T)-\bar{x}(T)\rangle\big{]}+\mathbb{E}^{\bar{u}}\big{[}\langle\bar{k}(T),\phi^{u}(T)-\bar{\phi}(T)\rangle\big{]}$
$\displaystyle=$
$\displaystyle\mathbb{E}^{\bar{u}}\bigg{[}\int_{0}^{T}\bigg{(}\langle\bar{p}(t),b^{u}(t)-\bar{b}(t)\rangle+\langle\bar{q}_{1}(t),\sigma_{1}^{u}(t)-\bar{\sigma}_{1}(t)\rangle+\langle\bar{q}_{2}(t),\sigma_{2}^{u}(t)-{\bar{\sigma}}_{2}(t)\rangle$
(30)
$\displaystyle+\langle\bar{k}(t),f^{u}(t)-\bar{f}(t)\rangle+\langle\bar{p}(t),\sigma_{2}^{u}(t)(\bar{h}(t)-h^{u}(t))\rangle+\langle\bar{k}(t),z_{2}^{u}(t)(\bar{h}(t)-h^{u}(t))\rangle\bigg{)}dt\bigg{]}$
$\displaystyle-\mathbb{E}^{\bar{u}}\bigg{[}\int_{0}^{T}\langle{\cal
H}_{x}(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}(t)),x^{u}(t)-\bar{x}(t)\rangle
dt\bigg{]}$
$\displaystyle-\mathbb{E}^{\bar{u}}\bigg{[}\int_{0}^{T}\langle{\cal
H}_{y}(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}(t)),y^{u}(t)-\bar{y}(t)\rangle
dt\bigg{]}$
$\displaystyle-\mathbb{E}^{\bar{u}}\bigg{[}\int_{0}^{T}\langle{\cal
H}_{z_{1}}(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}(t)),z_{1}^{u}(t)-\bar{z}_{1}(t)\rangle
dt\bigg{]}$
$\displaystyle-\mathbb{E}^{\bar{u}}\bigg{[}\int_{0}^{T}\langle{\cal
H}_{z_{2}}(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}(t)),z_{2}^{u}(t)-\bar{z}_{2}(t)\rangle
dt\bigg{]}$
$\displaystyle-{\mathbb{E}}\big{[}\left<y^{u}(0)-\bar{y}(0),\bar{\gamma}_{y}(0)\right>\big{]}\bigg{]}$

which implies that

$\displaystyle\mathbb{E}^{\bar{u}}\bigg{[}\int_{0}^{T}\bigg{(}\langle\bar{p}(t),b^{u}(t)-\bar{b}(t)\rangle+\langle\bar{q}_{1}(t),\sigma_{1}^{u}(t)-\bar{\sigma}_{1}(t)\rangle+\langle\bar{q}_{2}(t),\sigma_{2}^{u}(t)-{\bar{\sigma}}_{2}(t)\rangle+\langle\bar{k}(t),f^{u}(t)-\bar{f}(t)\rangle\bigg{)}dt\bigg{]}$
$\displaystyle=$
$\displaystyle\mathbb{E}^{\bar{u}}\big{[}\langle\bar{\Phi}_{x}(T),x^{u}(T)-\bar{x}(T)\rangle\big{]}+\mathbb{E}^{\bar{u}}\big{[}\langle\bar{k}(T),\phi^{u}(T)-\bar{\phi}(T)-\bar{\phi}_{x}(T)(x^{u}(T)-\bar{x}(T))\rangle\big{]}$
(31)
$\displaystyle+\mathbb{E}\bigg{[}\int_{0}^{T}\bigg{(}\langle\bar{p}(t),\sigma_{2}^{u}(t)(h^{u}(t)-\bar{h}(t))\rangle+\langle\bar{k}(t),z_{2}^{u}(t)(h^{u}(t)-\bar{h}(t))\rangle\bigg{)}dt\bigg{]}$
$\displaystyle+\mathbb{E}^{\bar{u}}\bigg{[}\int_{0}^{T}\langle{\cal
H}_{x}(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}(t)),x^{u}(t)-\bar{x}(t)\rangle
dt\bigg{]}$
$\displaystyle+\mathbb{E}^{\bar{u}}\bigg{[}\int_{0}^{T}\langle{\cal
H}_{y}(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}(t)),y^{u}(t)-\bar{y}(t)\rangle
dt\bigg{]}$
$\displaystyle+\mathbb{E}^{\bar{u}}\bigg{[}\int_{0}^{T}\langle{\cal
H}_{z_{1}}(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}(t)),z_{1}^{u}(t)-\bar{z}_{1}(t)\rangle
dt\bigg{]}$
$\displaystyle+\mathbb{E}^{\bar{u}}\bigg{[}\int_{0}^{T}\langle{\cal
H}_{z_{2}}(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}(t)),z_{2}^{u}(t)-\bar{z}_{2}(t)\rangle
dt\bigg{]}$
$\displaystyle+{\mathbb{E}}\big{[}\left<y^{u}(0)-\bar{y}(0),\bar{\gamma}_{y}(0)\right>\big{]}\bigg{]}$

Applying Itô formula to $(\rho^{u}(t)-\bar{\rho}(t))\bar{r}(t),$ we have

$\displaystyle\begin{split}\mathbb{E}[(\rho^{u}(T)-\bar{\rho}(T))\bar{\Phi}(T)]=&-\mathbb{E}\bigg{[}\int_{0}^{T}(\rho^{u}(t)-\bar{\rho}(t))(\bar{l}(t)+\bar{R}_{2}(t)\bar{h}(t))dt\bigg{]}+\mathbb{E}\bigg{[}\int_{0}^{T}\bar{R}_{2}(t)(\rho^{u}(t)h^{u}(t)-\bar{\rho}(t)\bar{h}(t))dt\bigg{]},\end{split}$
(32)

which implies that

$\displaystyle\begin{split}\mathbb{E}[(\rho^{u}(T)-\bar{\rho}(T))\bar{\Phi}(T)]+\mathbb{E}\bigg{[}\int_{0}^{T}(\rho^{u}(t)-\bar{\rho}(t))\bar{l}(t)dt\bigg{]}=\mathbb{E}\bigg{[}\int_{0}^{T}\bar{R}_{2}(t)\rho^{u}(t)(h^{u}(t)-\bar{h}(t))dt\bigg{]}\end{split}$
(33)

Putting (3) into (29), we have

$\displaystyle\begin{split}\mathbb{E}^{\bar{u}}\bigg{[}\int_{0}^{T}(l^{u}(t)-\bar{l}(t))dt\bigg{]}={\mathbb{E}}^{\bar{u}}\bigg{[}\int_{0}^{T}\bigg{(}&{\cal
H}(t,x^{u}(t),y^{u}(t),z^{u}_{1}(t),z^{u}_{2}(t),u(t))-{\cal
H}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t))\\\
&-\big{<}{\cal
H}_{x}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t)),x^{u}(t)-\bar{x}(t)\big{>}\\\
&-\big{<}{\cal
H}_{y}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t)),y^{u}(t)-\bar{y}(t)\big{>}\\\
&-\big{<}{\cal
H}_{z_{1}}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t)),z^{u}_{1}(t)-\bar{z}_{1}(t)\big{>}\\\
&-\big{<}{\cal
H}_{z_{2}}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t)),z^{u}_{2}(t)-\bar{z}_{2}(t)\big{>}\\\
&-\langle(\sigma_{2}^{u}(t)-\bar{\sigma}_{2}(t))(h^{u}(t)-\bar{h}(t)),\bar{p}(t)\rangle\bigg{)}dt\bigg{]}\\\
&-{\mathbb{E}}^{u}\big{[}\left<x^{u}(T)-\bar{x}(T),\bar{\Phi}_{x}(T)\right>\big{]}-{\mathbb{E}}\big{[}\left<y^{u}(0)-\bar{y}(0),\bar{\gamma}_{y}(0)\right>\big{]}\\\
&-{\mathbb{E}}^{u}\big{[}\langle\phi^{u}(T)-\bar{\phi}(T),k^{u}(T)\rangle-\left<\bar{\phi}_{x}^{*}(T)\bar{k}(T),x^{u}(T)-\bar{x}(T)\right>\big{]}\end{split}$
(34)

Then putting(33) and (34) into (28) , we get (23). The proof is complete. ∎

Since the control domain $U$ is convex, for any given admissible controls
$u(\cdot)\in{\cal A}$, the following perturbed control process
$u^{\epsilon}(\cdot)$:

$\displaystyle
u^{\epsilon}(\cdot):=\bar{u}(\cdot)+\epsilon(u(\cdot)-\bar{u}(\cdot)),\quad
0\leq\epsilon\leq 1,$

is also in ${\cal A}$. We denote by
$(\bar{x}(\cdot),\bar{y}(\cdot),\bar{z}_{1}(\cdot),\bar{z}_{2}(\cdot))$ and
$(x^{\epsilon}(\cdot),y^{\epsilon}(\cdot),z^{\epsilon}_{1}(\cdot),z^{\varepsilon}_{2}(\cdot))$
the corresponding state processes associated with $\bar{u}(\cdot)$ and
$u^{\epsilon}(\cdot)$, respectively. Denote by
$(\bar{p}(\cdot),\bar{q}_{1}(\cdot),\bar{q}_{2}(\cdot),\bar{k}(\cdot),\bar{r}(\cdot),\bar{R}_{1}(\cdot),\bar{R}_{2}(\cdot))$
the adjoint process associated with the admissible pair
$(\bar{u}(\cdot);\bar{x}(\cdot),\bar{y}(\cdot),\bar{z}_{1}(\cdot),\bar{z}_{2}(\cdot))$.

######  Lemma 3.2.

Let Assumptions 2.1 and 2.2 be satisfied. Then we have

$\displaystyle{\mathbb{E}}\bigg{[}\sup_{t\in\cal
T}|x^{\epsilon}(t)-\bar{x}(t)|^{4}\bigg{]}+{\mathbb{E}}\bigg{[}\sup_{t\in\cal
T}|y^{\epsilon}(t)-\bar{y}(t)|^{4}\bigg{]}+{\mathbb{E}}\bigg{[}\bigg{(}\int_{0}^{T}|z^{\epsilon}_{1}(t)-\bar{z}_{1}(t)|^{2}dt\bigg{)}^{2}\bigg{]}+{\mathbb{E}}\bigg{[}\bigg{(}\int_{0}^{T}|z^{\epsilon}_{2}(t)-\bar{z}_{2}(t)|^{2}dt\bigg{)}^{2}\bigg{]}=O(\epsilon^{4})\
.$

and

$\displaystyle{\mathbb{E}}\bigg{[}\sup_{t\in\cal
T}|\rho^{\epsilon}(t)-\bar{\rho}(t)|^{2}\bigg{]}=O(\epsilon^{2})\ .$

###### Proof.

The proof can be obtained directly by Lemmas 2.1 and 2.2. ∎

Now we are in the position to use Lemma 3.1 and Lemma 3.2 to derive the
variational formula for the cost functional $J(u(\cdot))$ in terms of the
Hamiltonian ${\cal H}$.

######  Theorem 3.1.

Let Assumptions 2.1 and 2.2 be satisfied. Then for any admissible control
$u(\cdot)\in{\cal A}$, the directional derivative of the cost functional
$J(u(\cdot))$ at $\bar{u}(\cdot)$ in the direction $u(\cdot)-\bar{u}(\cdot)$
exists and is given by

$\displaystyle\frac{d}{d\epsilon}J(\bar{u}(\cdot)+\epsilon(u(\cdot)-\bar{u}(\cdot)))|_{\epsilon=0}$
$\displaystyle:=\lim_{\epsilon\rightarrow
0^{+}}\frac{J(\bar{u}(\cdot)+\epsilon(u(\cdot)-\bar{u}(\cdot)))-J(\bar{u}(\cdot))}{\epsilon}$
$\displaystyle={\mathbb{E}}^{\bar{u}}\bigg{[}\int_{0}^{T}\left<{\cal
H}_{u}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t)),u(t)-\bar{u}(t)\right>dt\bigg{]}.$
(35)

###### Proof.

For notational simplicity, write

$\displaystyle\beta^{\epsilon}$ $\displaystyle:=$
$\displaystyle{\mathbb{E}}^{\bar{u}}\bigg{[}\int_{0}^{T}\bigg{[}{\cal
H}(t,x^{u^{\varepsilon}}(t),y^{u^{\varepsilon}}(t),z_{1}^{u^{\varepsilon}}(t),z_{2}^{u^{\varepsilon}}(t),u(t))-{\cal
H}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t))$ (36)
$\displaystyle-\big{<}{\cal
H}_{x}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t)),x^{u^{\varepsilon}}(t)-\bar{x}(t)\big{>}-\big{<}{\cal
H}_{y}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t)),y^{u^{\varepsilon}}(t)-\bar{y}(t)\big{>}$
$\displaystyle-\big{<}{\cal
H}_{z_{1}}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t)),z_{1}^{u^{\varepsilon}}(t)-\bar{z}_{1}(t)\big{>}-\big{<}{\cal
H}_{z_{2}}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t)),z_{2}^{u^{\varepsilon}}(t)-\bar{z}_{2}(t)\big{>}$
$\displaystyle-\big{<}{\cal
H}_{u}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t)),u^{\varepsilon}(t)-\bar{u}(t)\big{>}-\langle(\sigma_{2}^{u^{\varepsilon}}(t)-\bar{\sigma}_{2}(t))(h^{u^{\varepsilon}}(t)-\bar{h}(t)),\bar{p}(t)\rangle$
$\displaystyle-\langle(z_{2}^{u^{\varepsilon}}(t)-\bar{z}_{2}(t))(h^{u^{\varepsilon}}(t)-\bar{h}(t)),\bar{k}(t)\rangle
dt\bigg{]}$
$\displaystyle+{\mathbb{E}}^{\bar{u}}\big{[}\Phi^{u^{\varepsilon}}(T)-\bar{\Phi}(T)-\left<x^{u^{\varepsilon}}(T)-\bar{x}(T),\bar{\Phi}_{x}(T)\right>\big{]}$
$\displaystyle-{\mathbb{E}}^{\bar{u}}\big{[}\langle\phi^{u^{\varepsilon}}(T)-\bar{\phi}(T),\bar{k}(T)\rangle-\left<\bar{\phi}_{x}^{*}(T)\bar{k}(T),x^{u^{\varepsilon}}(T)-\bar{x}(T)\right>\big{]}$
$\displaystyle+{\mathbb{E}}\big{[}\gamma^{u^{\varepsilon}}(0)-\bar{\gamma}(0)-\left<y^{u^{\varepsilon}}(0)-\bar{y}(0),\bar{\gamma}_{y}(0)\right>\big{]}$
$\displaystyle+\mathbb{E}\bigg{[}\int_{0}^{T}\bar{R}_{2}(t)(\rho^{u^{\varepsilon}}(t)-\rho^{\bar{u}}(t))(h^{u^{\varepsilon}}(t)-\bar{h}(t))dt\bigg{]}$
$\displaystyle+\mathbb{E}\bigg{[}\int_{0}^{T}(l^{u^{\varepsilon}}(t)-\bar{l}(t))(\rho^{u^{\varepsilon}}(t)-\bar{\rho}(t))dt\bigg{]}$
$\displaystyle+\mathbb{E}\bigg{[}(\rho^{u^{\varepsilon}}(T)-\bar{\rho}(T))(\Phi^{u^{\varepsilon}}(T)-\bar{\Phi}(T))\bigg{]}$

By Lemma 3.1, we have

$\displaystyle
J(u^{\epsilon}(\cdot))-J(\bar{u}(\cdot))=\beta^{\epsilon}+\epsilon{\mathbb{E}}^{\bar{u}}\bigg{[}\int_{0}^{T}\left<{\cal
H}_{u}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t)),u(t)-\bar{u}(t)\right>dt\bigg{]}.$
(37)

Under Assumptions 2.1 and 2.2, combining the Taylor Expansions, Lemma 3.2, and
the dominated convergence theorem, we have

$\displaystyle\beta^{\epsilon}=o(\epsilon).$ (38)

Plugging (38) into (37) gives

$\displaystyle\lim_{\epsilon\rightarrow
0^{+}}\frac{J(u^{\epsilon}(\cdot))-J(\bar{u}(\cdot))}{\epsilon}={\mathbb{E}}^{\bar{u}}\bigg{[}\int_{0}^{T}\left<{\cal
H}_{u}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t)),u(t)-\bar{u}(t)\right>dt\bigg{]}.$

This completes the proof. ∎

Now we derive the necessary condition and sufficient maximum principles for
Problem 2.1 or 2.2. We first give the necessary condition of optimality for
the existence of an optimal control.

######  Theorem 3.2 (Necessary Stochastic Maximum principle).

Let Assumptions 2.1 and 2.2 be satisfied. Let
$(\bar{u}(\cdot);\bar{x}(\cdot),\bar{y}(\cdot),\bar{z}_{1}(\cdot),\\\
\bar{z}_{2}(\cdot),\bar{\rho}(\cdot))$ be an optimal pair of Problem 2.2. Then

$\displaystyle\left<{\mathbb{E}}^{\bar{u}}[{\cal
H}_{u}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t))|{\mathscr{F}}_{t}^{Y}],v-\bar{u}(t)\right>\geq
0,\quad\forall v\in U,\ a.e.\ a.s..$ (39)

###### Proof.

Since all admissible controls are $\\{\mathscr{F}^{Y}_{t}\\}_{t\in\cal
T}$-adapted processes, from the property of conditional expectation, Theorem
3.1 and the optimality of $\bar{u}(\cdot)$, we deduce that

$\displaystyle{\mathbb{E}}\bigg{[}\int_{0}^{T}\langle{\mathbb{E}}[{\bar{\rho}(t)}{\bar{\cal
H}}_{u}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t))|{\mathscr{F}}_{t}^{Y}],u(t)-\bar{u}(t)\rangle
dt\bigg{]}$
$\displaystyle={\mathbb{E}}\bigg{[}\int_{0}^{T}\langle{\bar{\rho}(t)}{\bar{\cal
H}}_{u}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t)),u(t)-\bar{u}(t)\rangle\mathrm{d}t\bigg{]}$
$\displaystyle={\mathbb{E}}^{\bar{u}}\bigg{[}\int_{0}^{T}\langle{\bar{\cal
H}}_{u}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t)),u(t)-\bar{u}(t)\rangle\mathrm{d}t\bigg{]}$
$\displaystyle=\lim_{\epsilon\rightarrow
0^{+}}\frac{J(\bar{u}(\cdot)+\epsilon(u(\cdot)-\bar{u}(\cdot)))-J(\bar{u}(\cdot))}{\epsilon}\geq
0,$

which implies that

$\displaystyle\langle{\mathbb{E}}[{\bar{\rho}(t)}{\bar{\cal
H}}_{u}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t))|{\mathscr{F}}_{t}^{Y}],v-\bar{u}(t)\rangle\geq
0,\quad\forall v\in U,\ a.e.\ a.s..$ (40)

On the other hand, since $\bar{\rho}(t)>0,$

$\displaystyle\begin{split}&\left<{\mathbb{E}}^{\bar{u}}[{\cal
H}_{u}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t))|{\mathscr{F}}_{t}^{Y}],v-\bar{u}(t)\right>\\\
&=\frac{1}{\mathbb{E}[\bar{\rho}(t)|{\mathscr{F}}_{t}^{Y}]}\big{\langle}{\mathbb{E}}[{\bar{\rho}(t)}{\bar{\cal
H}}_{u}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t))|{\mathscr{F}}_{t}^{Y}],v-\bar{u}(t)\big{\rangle}\\\
&\geq 0.\end{split}$ (41)

The proof is complete.

∎

Next we give the sufficient condition of optimality for the existence of an
optimal control of Problem 2.2 in the case when the observation process is not
affected by the control process. Suppose that

$h(t,x,u)=h(t)$

is an $\mathscr{F}^{Y}_{t}-$ adapted bounded process. Define a new probability
measure $\mathbb{Q}$ on $(\Omega,\mathscr{F})$ by

$\displaystyle d\mathbb{Q}=\rho(1)d\mathbb{P},$ (42)

where

$\displaystyle\left\\{\begin{aligned} d\rho(t)=&\rho(t)h(s)dY(s)\\\
\rho(0)=&1.\end{aligned}\right.$ (43)

######  Theorem 3.3.

[Sufficient Maximum Principle]  Let Assumptions 2.1 and 2.2 be satisfied. Let
$(\bar{u}(\cdot);\bar{x}(\cdot),\bar{y}(\cdot),\bar{z}_{1}(\cdot),\bar{z}_{2}(\cdot))$
be an admissible pair with $\phi(x)=\phi x,$ where $\phi$ is
$\mathscr{F}_{T}-$measurable bounded random variable. If the following
conditions are satisfied,

  1. (i)

$\Phi$ and $\gamma$ is convex in $x$ and $y,$ respectively,

  2. (ii)

the Hamiltonian ${\cal H}$ is convex in $(x,y,z_{1},z_{2},u)$,

  3. (iii)

$\displaystyle\mathbb{E}\bigg{[}{\cal
H}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t))|\mathscr{F}^{Y}_{t}\bigg{]}$
$\displaystyle=\min_{u\in U}\mathbb{E}\bigg{[}{\cal
H}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),u)|\mathscr{F}^{Y}_{t}\bigg{]},\quad\mbox{a.e.\
a.s.},$

then
$(\bar{u}(\cdot),\bar{x}(\cdot),\bar{y}(\cdot),\bar{z}_{1}(\cdot),\bar{z}_{2}(\cdot))$
is an optimal pair of Problem 2.2.

###### Proof.

Let
$({u}(\cdot);x^{u}(\cdot),y^{u}(\cdot),z^{u}_{1}(\cdot),z^{u}_{2}(\cdot),\rho^{u}(\cdot))$
be an arbitrary admissible pair. By Lemma 3.1, we can represent the difference
$J(u(\cdot))-J(\bar{u}(\cdot))$ as follows

$\displaystyle J(u(\cdot))-J(\bar{u}(\cdot))$ (44) $\displaystyle=$
$\displaystyle{\mathbb{E}}^{Q}\bigg{[}\int_{0}^{T}\bigg{[}{\cal
H}(t,x^{u}(t),y^{u}(t),z^{u}(t),u(t))-{\cal
H}(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}(t))$ $\displaystyle-\big{<}{\cal
H}_{x}(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}(t)),x^{u}(t)-\bar{x}(t)\big{>}-\big{<}{\cal
H}_{y}(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}(t)),y^{u}(t)-\bar{y}(t)\big{>}$
$\displaystyle-\big{<}{\cal
H}_{z}(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}(t)),z^{u}(t)-\bar{z}(t)\big{>}dt\bigg{]}$
$\displaystyle+{\mathbb{E}}^{\bar{u}}\big{[}\Phi(T)-\bar{\Phi}^{u}(T)-\left<x^{u}(T)-\bar{x}(T),\bar{\Phi}_{x}(T)\right>\big{]}$
$\displaystyle+{\mathbb{E}}\big{[}\gamma^{u}(0)-\bar{\gamma}(0)-\left<y^{u}(0)-\bar{y}(0),\bar{\gamma}_{y}(0)\right>\big{]}.$

By the convexity of ${\cal H}$, $\Phi$ and $\gamma$ (i.e. Conditions (i) and
(ii)), we have

$\displaystyle{\cal
H}(t,x^{u}(t),y^{u}(t),z^{u}_{1}(t),z^{u}_{2}(t),u(t))-{\cal
H}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t))$ (45)
$\displaystyle\geq$ $\displaystyle\big{<}{\cal
H}_{x}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t)),x^{u}(t)-\bar{x}(t)\big{>}+\big{<}{\cal
H}_{y}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t)),y^{u}(t)-\bar{y}(t)\big{>}$
$\displaystyle+\big{<}{\cal
H}_{z_{1}}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t)),z^{u}_{1}(t)-\bar{z}_{1}(t)\big{>}+\big{<}{\cal
H}_{z_{2}}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t)),z^{u}_{2}(t)-\bar{z}_{2}(t)\big{>}$
$\displaystyle+\big{<}{\cal
H}_{u}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t)),u(t)-\bar{u}(t)\big{>},$

$\displaystyle\Phi^{u}(T)-\bar{\Phi}(T)\geq\left<X^{u}(T)-\bar{X}(T),\Phi_{x}(T)\right>$
(46)

and

$\displaystyle\gamma^{u}(0)-\bar{\gamma}(0)\geq\left<y^{u}(0)-\bar{y}(0),\gamma_{y}(0)\right>.$
(47)

Furthermore, from the optimality condition (iii) and the convex optimization
principle (see Proposition 2.21 of [4]), we have

$\displaystyle\left<u(t)-\bar{u}(t),\mathbb{E}\bigg{[}{\cal
H}_{u}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(t),\bar{u}(t))|\mathscr{F}^{Y}_{t}\bigg{]}\right>\geq
0.$ (48)

which imply that

$\displaystyle\mathbb{E}^{Q}\bigg{[}\left<u(t)-\bar{u}(t),{\cal
H}_{u}(t,\bar{x}(t),\bar{y}(t),\bar{z}_{1}(t),\bar{z}_{2}(\cdot),\bar{u}(t))\right>\bigg{]}\geq
0.$ (49)

Putting (45),(46),(47) and (49) into (44), we have

$\displaystyle J(u(\cdot))-J(\bar{u}(\cdot))\geq 0.$ (50)

Due to the arbitrariness of $u(\cdot)$, we can conclude that $\bar{u}(\cdot)$
is an optimal control process and thus
$(\bar{u}(\cdot),\bar{x}(\cdot),\bar{y}(\cdot),\bar{z}_{1}(\cdot),z_{2}(\cdot))$
is an optimal pair. The proof is completed. ∎

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