# Optimal Variational Principle for Backward Stochastic Control Systems
Associated with Lévy Processes ††thanks: This work is partially supported by
the National Basic Research Program of China (973 Program) (Grant
No.2007CB814904), the National Natural Science Foundation of China (Grants
No.10325101, 11071069), the Specialized Research Fund for the Doctoral Program
of Higher Education of China (Grant No.20090071120002) and the Innovation Team
Foundation of the Department of Education of Zhejiang Province (Grant
No.T200924).

Maoning Tanga Qi Zhangb
aDepartment of Mathematical Sciences, Huzhou University, Zhejiang 313000,
China
Email<EMAIL_ADDRESS>
bSchool of Mathematical Sciences, Fudan University, Shanghai 200433, China
Email<EMAIL_ADDRESS>
###### Abstract

The paper is concerned with optimal control of backward stochastic
differential equation (BSDE) driven by Teugel’s martingales and an independent
multi-dimensional Brownian motion, where Teugel’s martingales are a family of
pairwise strongly orthonormal martingales associated with Lévy processes (see
Nualart and Schoutens [14]). We derive the necessary and sufficient conditions
for the existence of the optimal control by means of convex variation methods
and duality techniques. As an application, the optimal control problem of
linear backward stochastic differential equation with a quadratic cost
criteria (called backward linear-quadratic problem, or BLQ problem for short)
is discussed and characterized by stochastic Hamilton system.

Keywords: stochastic control, stochastic maximum principle, Lévy processes,
Teugel’s martingales, backward stochastic differential equations

##  1  Introduction

It is well known that the maximum principle for a stochastic optimal control
problem involves the so-called adjoint processes which solve the corresponding
adjoint equation. In fact, the adjoint equation is in general a linear
backward stochastic differential equation (BSDE) with a specified a random
terminal condition on the state. Unlike a forward stochastic differential
equation, the solution of a BSDE is a pair of adapted solutions. Thus, in
order to obtain the maximum principle, we need first obtain the existence and
uniqueness theorem for the pair of adapted solutions of adjoint equation.

The linear BSDE was first proposed by Bismut [4] in 1973. This research field
developed fast after the pioneer work of Pardoux and Peng [16] in 1990 got the
existence and uniqueness theorem for the solution of nonlinear BSDE driven by
Brownian motion under Lipschitz condition. Now BSDE theory has been playing a
key role not only in dealing with stochastic optimal control problems, but in
mathematical finance, particularly in hedging and nonlinear pricing theory for
imperfect market (see e.g. [7]).

As for BSDE driven by the non-continuous martingale, Tang and Li [20] first
discussed the existence and uniqueness theorem of the solution of BSDE driven
by Poisson point process and consequently proved the maximum principle for
optimal control of stochastic systems with random jumps. In 2000, Nualart and
Schoutens [14] got a martingale representation theorem for a type of Lévy
processes through Teugel’s martingales, where Teugel’s martingales are a
family of pairwise strongly orthonormal martingales associated with Lévy
processes. Later, they proved in [15] the existence and uniqueness theory of
BSDE driven by Teugel’s martingales. The above results are further extended to
the one-dimensional BSDE driven by Teugel’s martingales and an independent
multi-dimensional Brownian motion by Bahlali et al [1]. One can refer to [8,
9, 17, 18] for more results on such kind of BSDEs.

In the mean time, the stochastic optimal control problems related to Teugel’s
martingales were studied. In 2008, a stochastic linear-quadratic problem with
Lévy processes was considered by Mitsui and Tabata [13], in which they
established the closeness property of multi-dimensional backward stochastic
Riccati differential equation(BSRDE) with Teugel’s martingales and proved the
existence and uniqueness of solution to such kind of one-dimensional BSRDE,
moreover, in their paper an application of BSDE to a financial problem with
full and partial observations was demonstrated. Motivated by [13], Meng and
Tang [12] studied the general stochastic optimal control problem for the
forward stochastic systems driven by Teugel’s martingales and an independent
multi-dimensional Brownian motion, of which the necessary and sufficient
optimality conditions in the form of stochastic maximum principle with the
convex control domain are obtained.

However, [12] and [13] are only concerned with the optimal control problem of
the forward controlled stochastic system. Since a BSDE is a well-defined
dynamic system itself and has important applications in mathematical finance,
it is necessary and natural to consider the optimal control problem of BSDE.
Actually, there has been much literature on BSDE control system driven by
Brownian motion (see e.g. [2, 3, 5, 11, 10]). But to our best knowledge, there
is no discussion on the optimal control problem of BSDE driven by Teugel
martingales and an independent Brownian motion, which motives us to write this
paper.

In this paper, by means of convex variation methods and duality techniques, we
will give the necessary and sufficient conditions for the existence of the
optimal control for BSDE system driven by Teugel martingales and an
independent multi-dimensional Brownian motion. As an application, the optimal
control for linear backward stochastic differential equation with a quadratic
cost criteria or called backward linear-quadratic (BLQ) problem is discussed
in details. The optimal control of BLQ problem will be characterized by
stochastic Hamilton systems. In this case, the stochastic Hamilton system is a
linear forward-backward stochastic differential equation driven by Teugel’s
martingales and an independent multi-dimensional Brownian motion, consisting
of the state equation, the adjoint equation and the dual presentation of the
optimal control.

The rest of this paper is organized as follows. In section 2, we introduce
useful notation and some existing results on stochastic differential equations
(SDEs) and BSDEs driven by Teugel’s martingales. In section 3, we state the
optimal control problem we study, give needed assumptions and prove some
preliminary results on variational equation and variational inequality. In
section 4, we prove the necessary and sufficient optimality conditions for the
optimal control problem put forward in section 3. As an application, the
optimal control for BLQ problem is discussed in section 5.

##  2 Notation and preliminaries

Let $(\Omega,\mathscr{F},\\{\mathscr{F}_{t}\\}_{0\leq t\leq T},P)$ be a
complete probability space. The filtration $\\{\mathscr{F}_{t}\\}_{0\leq t\leq
T}$ is right-continuous and generated by a $d$-dimensional standard Brownian
motion $\\{W(t),0\leq t\leq T\\}$ and a one-dimensional Lévy process
$\\{L(t),0\leq t\leq T\\}$. It is known that $L(t)$ has a characteristic
function of the form

$Ee^{i\theta L(t)}=\exp\bigg{[}ia\theta t-{1\over
2}\sigma^{2}\theta^{2}t+t\int_{\mathbb{R}^{1}}(e^{i\theta x}-1-i\theta
xI_{\\{|x|<1\\}})v(dx)\bigg{]},$

where $a\in\mathbb{R}^{1}$, $\sigma>0$ and $v$ is a measure on
$\mathbb{R}^{1}$ satisfying (i)$\displaystyle\int_{0}^{T}(1\wedge
x^{2})v(dx)<\infty$ and (ii) there exists $\varepsilon>0$ and $\lambda>0$,
s.t.
$\displaystyle\int_{\\{-\varepsilon,\varepsilon\\}^{c}}e^{\lambda|x|}v(dx)<\infty$.
These settings imply that the random variables $L(t)$ have moments of all
orders. Denote by $\mathscr{P}$ the predictable sub-$\sigma$ field of
$\mathscr{B}([0,T])\times\mathscr{F}$, then we introduce the following
notation used throughout this paper.

$\bullet$ $H$: a Hilbert space with norm $\|\cdot\|_{H}$.

$\bullet$ $\langle\alpha,\beta\rangle:$ the inner product in
$\mathbb{R}^{n},\forall\alpha,\beta\in\mathbb{R}^{n}.$

$\bullet$ $|\alpha|=\sqrt{\langle\alpha,\alpha\rangle}:$ the norm of
$\mathbb{R}^{n},\forall\alpha\in\mathbb{R}^{n}.$

$\bullet$ $\langle A,B\rangle=tr(AB^{T}):$ the inner product in
$\mathbb{R}^{n\times m},\forall A,B\in\mathbb{R}^{n\times m}.$

$\bullet$ $|A|=\sqrt{tr(AA^{T})}:$ the norm of $\mathbb{R}^{n\times m},\forall
A\in\mathbb{R}^{n\times m}$.

$\bullet$ $l^{2}$: the space of all real-valued sequences $x=(x_{n})_{n\geq
0}$ satisfying

$\|x\|_{l^{2}}\triangleq\sqrt{\displaystyle\sum_{i=1}^{\infty}x_{i}^{2}}<+\infty.$

$\bullet$ $l^{2}(H):$ the space of all H-valued sequence $f=\\{f^{i}\\}_{i\geq
1}$ satisfying

$\|f\|_{l^{2}(H)}\triangleq\sqrt{\displaystyle\sum_{i=1}^{\infty}||f^{i}||_{H}^{2}}<+\infty.$

$\bullet$ $l_{\mathscr{F}}^{2}(0,T,H):$ the space of all $l^{2}(H)$-valued and
${\mathscr{F}}_{t}$-predictable processes $f=\\{f^{i}(t,\omega),\
(t,\omega)\in[0,T]\times\Omega\\}_{i\geq 1}$ satisfying

$\|f\|_{l_{\mathscr{F}}^{2}(0,T,H)}\triangleq\sqrt{E\displaystyle\int_{0}^{T}\sum_{i=1}^{\infty}||f^{i}(t)||_{H}^{2}dt}<\infty.$

$\bullet$ $M_{\mathscr{F}}^{2}(0,T;H):$ the space of all $H$-valued and
${\mathscr{F}}_{t}$-adapted processes $f=\\{f(t,\omega),\
(t,\omega)\in[0,T]\times\Omega\\}$ satisfying

$\|f\|_{M_{\mathscr{F}}^{2}(0,T;H)}\triangleq\sqrt{E\displaystyle\int_{0}^{T}\|f(t)\|_{H}^{2}dt}<\infty.$

$\bullet$ $S_{\mathscr{F}}^{2}(0,T;H):$ the space of all $H$-valued and
${\mathscr{F}}_{t}$-adapted càdlàg processes $f=\\{f(t,\omega),\
(t,\omega)\in[0,T]\times\Omega\\}$ satisfying

$\|f\|_{S_{\mathscr{F}}^{2}(0,T;H)}\triangleq\sqrt{E\displaystyle\sup_{0\leq
t\leq T}\|f(t)\|_{H}^{2}dt}<+\infty.$

$\bullet$ $L^{2}(\Omega,{\mathscr{F}},P;H):$ the space of all $H$-valued
random variables $\xi$ on $(\Omega,{\mathscr{F}},P)$ satisfying

$\|\xi\|_{L^{2}(\Omega,{\mathscr{F}},P;H)}\triangleq E\|\xi\|_{H}^{2}<\infty.$

We denote by $\\{H^{i}(t),0\leq t\leq T\\}_{i=1}^{\infty}$ the Teugel’s
martingales associated with the Lévy process $\\{L(t),0\leq t\leq T\\}$.
$H^{i}(t)$ is given by

$H^{i}(t)=c_{i,i}Y^{(i)}(t)+c_{i,i-1}Y^{(i-1)}(t)+\cdots+c_{i,1}Y^{(1)}(t),$

where $Y^{(i)}(t)=L^{(i)}(t)-E[L^{(i)}(t)]$ for all $i\geq 1$, $L^{(i)}(t)$
are so called power-jump processes with $L^{(1)}(t)=L(t)$,
$L^{(i)}(t)=\displaystyle\sum_{0<s\leq t}(\Delta L(s))^{i}$ for $i\geq 2$ and
the coefficients $c_{ij}$ correspond to the orthonormalization of polynomials
$1,x,x^{2},\cdots$ w.r.t. the measure
$\mu(dx)=x^{2}v(dx)+\sigma^{2}\delta_{0}(dx)$. The Teugel’s martingales
$\\{H^{i}(t)\\}_{i=1}^{\infty}$ are pathwise strongly orthogonal and their
predictable quadratic variation processes are given by

$\langle H^{(i)}(t),H^{(j)}(t)\rangle=\delta_{ij}t$

For more details of Teugel’s martingales, we invite the reader to consult
Nualart and Schoutens [14, 15].

In what follows, we will state some basic results on SDE and BSDE driven by
Teugel’s martingales $\\{H^{i}(t),0\leq t\leq T\\}_{i=1}^{\infty}$ and the
$d$-dimensional Brownian motion $\\{W(t),0\leq t\leq T\\}.$

Consider SDE:

$\begin{array}[]{ll}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X(t)=&a+\displaystyle\int_{0}^{t}b(s,X(s))ds+\sum_{i=1}^{d}\int_{0}^{t}g^{i}(s,X(s))dW^{i}(s)\\\
&+\displaystyle\sum_{i=1}^{\infty}\int_{0}^{t}\sigma^{i}(s,X(s-))dH^{i}(s),\ \
t\in[0,T],\end{array}$ (2.1)

where $(a,b,g,\sigma)$ are given mappings satisfying the assumptions below.

######  Assumption 2.1.

Random variable $a$ is ${\mathscr{F}}_{0}$-measurable and $(b,g,\sigma)$ are
three random mappings

$b:[0,T]\times\Omega\times\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n},$
$g\equiv(g^{1},g^{2},\cdots,g^{d}):[0,T]\times\Omega\times\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n\times
d},$
$\sigma\equiv{(\sigma^{i})}_{i=1}^{\infty}:[0,T]\times\Omega\times\mathbb{R}^{n}\longrightarrow
l^{2}(\mathbb{R}^{n})$

satisfying
(i) $b,g$ and $\sigma$ are
${\mathscr{P}}\bigotimes{\mathscr{B}}(\mathbb{R}^{n})$ measurable with
$b(\cdot,0)\in M_{\mathscr{F}}^{2}(0,T;\mathbb{R}^{n})$, $g(\cdot,0)\in
M_{\mathscr{F}}^{2}(0,T;\mathbb{R}^{n\times d})$ and $\sigma(\cdot,0)\in
l_{\mathscr{F}}^{2}(0,T;\mathbb{R}^{n}).$
(ii) $b,g$ and $\sigma$ are uniformly Lipschitz continuous w.r.t. $x$, i.e.
there exists a constant $C>0$ s.t. for all
$(t,x,\bar{x})\in[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}$ and a.s.
$\omega\in\Omega$,

$\begin{array}[]{ll}|b(t,x)-b(t,\bar{x})|+|g(t,x)-g(t,\bar{x})|+||\sigma(t,x)-\sigma(t,\bar{x})||_{l^{2}(\mathbb{R}^{n})}\leq
C|x-\bar{x}|.\end{array}$

######  Lemma 2.1 ([19], Existence and Uniqueness Theorem of SDE).

If coefficients $(a,b,g,\sigma)$ satisfy Assumption 2.1, then SDE (2.1) has a
unique solution $x(\cdot)\in S_{\mathscr{F}}^{2}(0,T;\mathbb{R}^{n})$.

######  Lemma 2.2 ([12], Continuous Dependence Theorem of SDE).

Assume coefficients $(a,b,g,\sigma)$ and
$(\bar{a},\bar{b},\bar{g},\bar{\sigma})$ satisfy Assumption 2.1. If $x(\cdot)$
and $\bar{x}(\cdot)$ are the solutions to SDE (2.1) corresponding to
$(a,b,g,\sigma)$ and $(\bar{a},\bar{b},\bar{g},\bar{\sigma})$, respectively,
then we have

$\begin{array}[]{ll}~{}E\displaystyle\sup_{0\leq t\leq
T}|x(t)-\bar{x}(t)|^{2}\leq&K\bigg{[}|a-\bar{a}|^{2}+E\displaystyle\int_{0}^{T}|b(t,\bar{x}(t))-\bar{b}(t,\bar{x}(t))|^{2}dt\\\
&+E\displaystyle\int_{0}^{T}|g(t,\bar{x}(t))-\bar{g}(t,\bar{x}(t))|^{2}dt\\\
&+E\displaystyle\int_{0}^{T}||\sigma(t,\bar{x}(t))-\bar{\sigma}(t,\bar{x}(t))||_{l^{2}(\mathbb{R}^{n})}^{2}dt\bigg{]},\end{array}$

where $K$ is a positive constant depending only on $T$ and the Lipschitz
constant $C$.

In particular, for $(\bar{a},\bar{b},\bar{g},\bar{\sigma})=(0,0,0,0),$ we have

$\begin{array}[]{ll}&E\displaystyle\sup_{0\leq t\leq T}|x(t)|^{2}\\\
\leq&K\bigg{[}|a|^{2}+E\displaystyle\int_{0}^{T}|b(t,0)|^{2}dt+E\displaystyle\int_{0}^{T}|g(t,0)|^{2}dt+E\displaystyle\int_{0}^{T}||\sigma(t,0)||_{l^{2}(\mathbb{R}^{n})}^{2}dt\bigg{]}<+\infty.\end{array}$

Now we consider BSDE:

$\displaystyle\begin{split}y(t)=&\xi+\displaystyle\int_{t}^{T}f(s,y{(s)},q(s),z(s))ds-\displaystyle\sum_{i=1}^{d}\int_{t}^{T}q^{i}(s)dW^{i}(s)\\\
&-\displaystyle\sum_{i=1}^{\infty}\int_{t}^{T}z^{i}(s)dH^{i}(s),\ \
t\in[0,T],\end{split}$ (2.2)

where coefficients $(\xi,f)$ are given mappings satisfying the assumptions
below.

######  Assumption 2.2.

The terminal value $\xi\in L^{2}(\Omega,{\mathscr{F}}_{T},P;\mathbb{R}^{n})$
and $f$ is a random mapping

$f:[0,T]\times\Omega\times\mathbb{R}^{n}\times\mathbb{R}^{n\times d}\times
l^{2}(\mathbb{R}^{n})\longrightarrow\mathbb{R}^{n}$

satisfying
(i) $f$ is
${\mathscr{P}}\bigotimes{\mathscr{B}}(\mathbb{R}^{n})\bigotimes{\mathscr{B}}(\mathbb{R}^{n\times
d})\bigotimes{\mathscr{B}}(l^{2}(\mathbb{R}^{n}))$ measurable with
$f(\cdot,0,0,0)\in M_{\mathscr{F}}^{2}(0,T;\mathbb{R}^{n})$.
(ii) $f$ is uniformly Lipschitz continuous w.r.t. $(y,q,z)$, i.e. there exists
a constant $C>0$ s.t. for all
$(t,y,q,z,\bar{y},\bar{q},\bar{z})\in[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n\times
d}\times l^{2}(\mathbb{R}^{n})\times\mathbb{R}^{n}\times\mathbb{R}^{n\times
d}\times l^{2}(\mathbb{R}^{n})$ and a.s. $\omega\in\Omega$,

$\begin{array}[]{ll}&|f(t,y,q,z)-f(t,\bar{y},\bar{q},\bar{z})|\leq
C\bigg{[}|y-\bar{y}|+|q-\bar{q}|+\|z-\bar{z}\|_{l^{2}({\mathbb{R}^{n}})}\bigg{]}.\end{array}$

######  Lemma 2.3 ([1], Existence and Uniqueness of BSDE).

If coefficients $(\xi,f)$ satisfy Assumption 2.2, then BSDE (2.2) has a unique
solution

$(y(\cdot),q(\cdot),z(\cdot))\in S_{\mathscr{F}}^{2}(0,T;\mathbb{R}^{n})\times
M_{\mathscr{F}}^{2}(0,T;\mathbb{R}^{n\times d})\times
l_{\mathscr{F}}^{2}(0,T;\mathbb{R}^{n}).$

######  Lemma 2.4 ([1], Continuous Dependence Theorem of BSDE).

Assume that coefficients $(\xi,f)$ and $(\bar{\xi},\bar{f})$ satisfy
Assumption 2.2. If $(y(\cdot),q(\cdot),z(\cdot))$ and
$(\bar{y}(\cdot),\bar{q}(\cdot),\bar{z}(\cdot))$ are the solutions to BSDE
(2.2) corresponding to $(\xi,f)$ and $(\bar{\xi},\bar{f})$, respectively, then
we have

$\begin{array}[]{ll}{}&E\displaystyle\sup_{0\leq t\leq
T}|y(t)-\bar{y}(t)|^{2}+E\int_{0}^{T}|q(t)-\bar{q}(t)|^{2}dt+E\int_{0}^{T}||z(t)-\bar{z}(t)||^{2}_{l^{2}({\mathbb{R}^{n}})}dt\\\
\leq&K\bigg{[}E|\xi-\bar{\xi}|^{2}+E\displaystyle\int_{0}^{T}|f(t,\bar{y}{(t)},\bar{q}(t),\bar{z}(t))-\bar{f}(t,\bar{y}{(t)},\bar{q}(t),\bar{z}(t))|^{2}dt\bigg{]},\end{array}$

where $K$ is a positive constant depending only on $T$ and the Lipschitz
constant $C$.

In particular, if $(\bar{\xi},\bar{f})=(0,0)$, we have

$\begin{array}[]{ll}&E\displaystyle\sup_{0\leq t\leq
T}|y(t)|^{2}+E\int_{0}^{T}|q(t)|^{2}dt+E\int_{0}^{T}||z(t)||^{2}_{l^{2}({\mathbb{R}^{n}})}dt\\\
\leq&K\bigg{[}E|\xi|^{2}+E\displaystyle\int_{0}^{T}|f(t,0,0,0)|^{2}dt\bigg{]}.\end{array}$
(2.3)

In view of Assumptions 2.1-2.2, Lemmas 2.1-2.4 follow from an application of
Itô’s formula, Gronwall’s inequality and Burkholder-Davis-Gundy inequality.
One can refer to [1], [12] and [19] for details.

##  3 Formulation of the problem and preliminary lemmas

Let the admissible control set $U$ be a nonempty convex subset of
$\mathbb{R}^{m}$. An admissible control process $u(\cdot)$ is defined as a
${\mathscr{F}}_{t}$-predictable process with values in $U$ s.t.
$E\displaystyle\int_{0}^{T}|u(t)|^{2}dt<+\infty$. We denote by ${\mathcal{A}}$
the set including all admissible control processes.

For any given admissible control $u(\cdot)\in{\mathcal{A}}$, we consider the
following controlled nonlinear BSDE driven by multi-dimensional Brownian
motion $W$ and Teugel’s martingales $\\{H^{i}\\}_{i=1}^{\infty}$:

$\displaystyle\begin{split}y(t)=&\xi+\displaystyle\int_{t}^{T}f(s,y{(s)},q(s),z(s),u(s))ds\\\
&-\displaystyle\sum_{i=1}^{d}\int_{t}^{T}q^{i}(s)dW^{i}(s)-\displaystyle\sum_{i=1}^{\infty}\int_{t}^{T}z^{i}(s)dH^{i}(s),\
\ t\in[0,T]\end{split}$ (3.1)

with the cost functional

$J(u(\cdot))=E\displaystyle\bigg{[}\int_{0}^{T}l(t,y(t),q(t),z(t),u(t))dt+\Phi(y(0))\bigg{]},$
(3.2)

where

$\xi:\Omega\longrightarrow\mathbb{R}^{n},$
$f:[0,T]\times\Omega\times\mathbb{R}^{n}\times\mathbb{R}^{n\times d}\times
l^{2}(\mathbb{R}^{n})\times U\longrightarrow\mathbb{R}^{n},$
$l:[0,T]\times\Omega\times\mathbb{R}^{n}\times\mathbb{R}^{n\times d}\times
l^{2}(\mathbb{R}^{n})\times U\longrightarrow\mathbb{R}^{1}$

and

$\phi:\Omega\times\mathbb{R}^{n}\longrightarrow\mathbb{R}^{1}$

are given coefficients.

Throughout this paper, we introduce the following basic assumptions on
coefficients $(\xi,f,l,\phi)$.

######  Assumption 3.1.

The terminal value $\xi\in L^{2}(\Omega,{\mathscr{F}}_{T},P;\mathbb{R}^{n})$
and the random mapping $f$ is
${\mathscr{P}}\bigotimes{\mathscr{B}}(\mathbb{R}^{n})\bigotimes{\mathscr{B}}(\mathbb{R}^{n\times
d})\bigotimes{\mathscr{B}}(l^{2}(\mathbb{R}^{n}))\bigotimes{\mathscr{B}}(U)$
measurable with $f(\cdot,0,0,0,0)\in M^{2}(0,T;\mathbb{R}^{n})$. For almost
all $(t,\omega)\in[0,T]\times\Omega$, $f(t,\omega,y,p,z,u)$ is Fréchet
differentiable w.r.t. $(y,p,z,u)$ and the corresponding Fréchet derivatives
$f_{y},f_{p},f_{z},f_{u}$ are continuous and uniformly bounded.

######  Assumption 3.2.

The random mapping $l$ is
${\mathscr{P}}\bigotimes{\mathscr{B}}(\mathbb{R}^{n})\bigotimes{\mathscr{B}}(\mathbb{R}^{n\times
d})\bigotimes{\mathscr{B}}(l^{2}(\mathbb{R}^{n}))\bigotimes{\mathscr{B}}(U)$
measurable and for almost all $(t,\omega)\in[0,T]\times\Omega$, $l$ is Fréchet
differentiable w.r.t. $(y,p,z,u)$ with continuous Fréchet derivatives
$l_{y},l_{q},l_{z},l_{u}$. The random mapping $\phi$ is
${\mathscr{F}}_{T}\bigotimes{\mathscr{B}}(\mathbb{R}^{n})$ measurable and for
almost all $(t,\omega)\in[0,T]\times\Omega$, $\phi$ is Fréchet differentiable
w.r.t. $y$ with continuous Fréchet derivative $\phi_{y}$. Moreover, for almost
all $(t,\omega)\in[0,T]\times\Omega$, there exists a constant $C$ s.t. for all
$(p,q,z,u)\in\mathbb{R}^{n}\times\mathbb{R}^{n\times d}\times
l^{2}(\mathbb{R}^{n})\times U$,

$|l|\leq C(1+|y|^{2}+|q|^{2}+|z|^{2}+|u|^{2}),\ \ |\phi|\leq C(1+|y|^{2}),$
$|l_{y}|+|l_{q}|+|l_{z}|+|l_{u}|\leq C(1+|y|+|q|+|z|+|u|)\ and\ |\phi_{y}|\leq
C(1+|y|).$

Under Assumption 3.1, we can get from Lemma 2.3 that for each
$u(\cdot)\in{\mathcal{A}}$, the system (3.1) admits a unique strong solution.
We denote the strong solution of (3.1) by
$(y^{u}(\cdot),q^{u}(\cdot),z^{u}(\cdot))$, or $(y(\cdot),q(\cdot),z(\cdot))$
if its dependence on admissible control $u(\cdot)$ is clear from context. Then
we call $(y(\cdot),q(\cdot),z(\cdot))$ the state processes corresponding to
the control process $u(\cdot)$ and call
$(u(\cdot);y(\cdot),q(\cdot),z(\cdot))$ the admissible pair. Furthermore, by
Assumption 3.2 and a priori estimate (2.3), it is easy to check that

$|J(u(\cdot))|<\infty.$

Then we put forward the optimal control problem we study.

######  Problem 3.1.

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

$J(\bar{u}(\cdot))=\displaystyle\inf_{u(\cdot)\in{\mathcal{A}}}J(u(\cdot)).$

Any $\bar{u}(\cdot)\in{\mathcal{A}}$ satisfying above is called an optimal
control process of Problem 3.1 and the corresponding state processes
$(\bar{y}(\cdot),\bar{q}(\cdot),\bar{z}(\cdot))$ are called the optimal state
processes. Correspondingly
$(\bar{u}(\cdot);\bar{y}(\cdot),\bar{q}(\cdot),\bar{z}(\cdot))$ is called an
optimal pair of Problem 3.1.

Before we deduce the necessary and sufficient conditions for the optimal
control of Problem 3.1, we need do some preparations. Since the control domain
$U$ is convex, the classical method to get necessary conditions for optimal
control processes is the so-called convex perturbation method. More precisely,
assuming that $(\bar{u}(\cdot);\bar{y}(\cdot),\bar{q}(\cdot),\bar{z}(\cdot))$
is an optimal pair of Problem 3.1, for any given admissible control
${u}(\cdot)$, we define an admissible control in the form of convex variation

$u^{\varepsilon}(\cdot)=\bar{u}(\cdot)+\varepsilon(u(\cdot)-\bar{u}(\cdot)),$

where $\varepsilon>0$ can be chosen sufficiently small. Denoting by
$(y^{\varepsilon}(\cdot),q^{\varepsilon}(\cdot),z^{\varepsilon}(\cdot))$ the
state processes of the control system (3.1) corresponding to the control
process $u^{\varepsilon}(\cdot)$, we obtain the variational inequality

$J(u^{\varepsilon}(\cdot))-J(\bar{u}(\cdot))\geq 0.$

In what follows, we do some estimates on the optimal pair and the convex
variable pair.

######  Lemma 3.2.

Under Assumptions 3.1-3.2, we have

$\displaystyle\begin{split}E\sup_{0\leq t\leq
T}|y^{\varepsilon}(t)-\bar{y}(t)|^{2}+E\int_{0}^{T}|q^{\varepsilon}(t)-\bar{q}(t)|^{2}dt+E\int_{0}^{T}||z^{\varepsilon}(t)-\bar{z}(t)||^{2}_{l^{2}({\mathbb{R}^{n}})}dt=O(\varepsilon^{2}).\end{split}$

###### Proof.

By continuous dependence theorem of BSDE (Lemma 2.4) and the uniformly bounded
property of Fréchet derivative $f_{u}$, we have

$\displaystyle\begin{split}&E\sup_{0\leq t\leq
T}|y^{\varepsilon}(t)-\bar{y}(t)|^{2}+E\int_{0}^{T}|q^{\varepsilon}(t)-\bar{q}(t)|^{2}dt+E\int_{0}^{T}||z^{\varepsilon}(t)-\bar{z}(t)||^{2}_{l^{2}({\mathbb{R}^{n}})}dt\\\
\leq&KE\displaystyle\int_{0}^{T}|f(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),{u}^{\varepsilon}(t))-f(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t))\big{|}^{2}dt\\\
\leq&KE\displaystyle\int_{0}^{T}|u^{\varepsilon}(t)-\bar{u}(t)|^{2}dt\\\
=&KE\displaystyle\int_{0}^{T}|(\bar{u}(t)+\varepsilon(u(t)-\bar{u}(t))-\bar{u}(t))|^{2}dt\\\
=&K\varepsilon^{2}E\displaystyle\int_{0}^{T}|u(t)-\bar{u}(t)|^{2}dt=O(\varepsilon^{2}).\end{split}$

Here and in the rest of this paper, $K$ is a generic positive constant and
might change from line to line. ∎

Then we consider the following linear BSDE served as a variational equation:

$\displaystyle
dY_{t}=-\bigg{[}f_{y}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}_{t})Y(t)+f_{q}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}_{t})Q_{t}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+f_{z}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}_{t})Z(t)+f_{u}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}_{t})(u(t)-\bar{u}(t))\bigg{]}dt$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}\displaystyle\sum_{i=1}^{d}\int_{t}^{T}Q^{i}(s)dW^{i}(s)+\displaystyle\sum_{i=1}^{\infty}Z^{i}(t)dH^{i}(t)$
(3.3) $\displaystyle Y(T)=0.$

Under Assumption 3.1, by Lemma 2.3 we know that BSDE (3) has a unique solution

$(Y,Q,Z)\in S_{\mathscr{F}}^{2}(0,T;\mathbb{R}^{n})\times
M_{\mathscr{F}}^{2}(0,T;\mathbb{R}^{n\times d})\times
l_{\mathscr{F}}^{2}(0,T;\mathbb{R}^{n}).$

######  Lemma 3.3.

Under Assumptions 3.1-3.2, it follows that

$\displaystyle\begin{split}&E\displaystyle\sup_{0\leq t\leq
T}|y^{\varepsilon}(t)-\bar{y}(t)-\varepsilon
Y(t)|^{2}+E\int_{0}^{T}|q^{\varepsilon}(t)-\bar{q}(t)-\varepsilon
Q(t)|^{2}dt\\\ &+E\int_{0}^{T}||z^{\varepsilon}(t)-\bar{z}(t)-\varepsilon
Z(t)||^{2}_{l^{2}({\mathbb{R}^{n}})}dt=o(\varepsilon^{2}).\end{split}$

###### Proof.

Firstly, one can check that

$\begin{array}[]{ll}&y^{\varepsilon}(t)-\bar{y}(t)\\\
=&\displaystyle\int_{t}^{T}\bigg{[}{f}_{y}^{\varepsilon}(s)(y^{\varepsilon}(s)-\bar{y}(s))+{f}_{q}^{\varepsilon}(s)(q^{\varepsilon}(s)-\bar{q}(s))\\\
&\ \ \ \ \ \ \ \
+{f}_{z}^{\varepsilon}(s)(z^{\varepsilon}(s)-\bar{z}(s))+{f}_{u}^{\varepsilon}(s)(u^{\varepsilon}(s)-\bar{u}(s))\bigg{]}ds\\\
&\ \ \ \ \
-\displaystyle\sum_{i=1}^{d}\int_{t}^{T}\big{(}q^{i\varepsilon}(s)-\bar{q}^{i}(s)\big{)}dW^{i}(s)-\displaystyle\sum_{i=1}^{\infty}\int_{t}^{T}\big{(}z^{i\varepsilon}(s)-\bar{z}^{i}(s)\big{)}dH^{i}(s)\end{array}$

and

$\displaystyle\begin{split}\varepsilon
Y(t)=&\displaystyle\displaystyle\int_{t}^{T}\bigg{[}{f}_{y}(s)\varepsilon{Y}(s)++{f}_{q}(s)\varepsilon
Q(s)+{f}_{z}(s)\varepsilon
Z(s)+{f}_{u}(s)\varepsilon(u(s)-\bar{u}(s))\bigg{]}ds\\\ &\ \ \ \ \
-\displaystyle\sum_{i=1}^{d}\int_{t}^{T}\varepsilon
Q^{i}(s)dW^{i}(s)-\displaystyle\sum_{i=1}^{\infty}\int_{t}^{T}\varepsilon
Z^{i}(s)dH^{i}(s),\end{split}$

where we have used the abbreviations for $\varphi=f,l$ as follows:

$\displaystyle\varphi_{y}(t)=\varphi_{y}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}_{t}),$
$\displaystyle\varphi_{z}(t)=\varphi_{z}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}_{t}),$
$\displaystyle\varphi_{q}(t)=\varphi_{q}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}_{t}),$
$\displaystyle\varphi_{u}(t)=\varphi_{u}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}_{t}),$
(3.4)
$\displaystyle\tilde{\varphi}_{y}^{\varepsilon}(t)=\displaystyle\int_{0}^{1}\varphi_{y}(t,\bar{y}(t)+\lambda(y^{\varepsilon}(t)-\bar{y}(t)),\bar{z}(t)+\lambda(z^{\varepsilon}(t)-\bar{z}(t)),\bar{u}(t)+\lambda(u^{\varepsilon}(t)-u(t)))d\lambda,$
$\displaystyle\tilde{\varphi}_{z}^{\varepsilon}(t)=\displaystyle\int_{0}^{1}\varphi_{z}(t,\bar{y}(t)+\lambda(y^{\varepsilon}(t)-\bar{y}(t)),\bar{z}(t)+\lambda(z^{\varepsilon}(t)-\bar{z}(t)),\bar{u}(t)+\lambda(u^{\varepsilon}(t)-u(t)))d\lambda,$
$\displaystyle\tilde{\varphi}_{q}^{\varepsilon}(t)=\displaystyle\int_{0}^{1}\varphi_{q}(t,\bar{y}(t)+\lambda(q^{\varepsilon}(t)-\bar{y}(t)),\bar{z}(t)+\lambda(q^{\varepsilon}(t)-\bar{z}(t)),\bar{u}(t)+\lambda(u^{\varepsilon}(t)-u(t)))d\lambda,$
$\displaystyle\tilde{\varphi}_{u}^{\varepsilon}(t)=\displaystyle\int_{0}^{1}\varphi_{u}(t,\bar{y}(t)+\lambda(y^{\varepsilon}(t)-\bar{y}(t)),\bar{z}(t)+\lambda(z^{\varepsilon}(t)-\bar{z}(t)),\bar{u}(t)+\lambda(u^{\varepsilon}(t)-u(t)))d\lambda.$

Thus by Lemma 2.4 again, we get

$\displaystyle\begin{array}[]{ll}&E\displaystyle\sup_{0\leq t\leq
T}|y^{\varepsilon}(t)-\bar{y}(t)-\varepsilon
Y(t)|^{2}+E\int_{0}^{T}|q^{\varepsilon}(t)-\bar{q}(t)-\varepsilon
Q(t)|^{2}dt\\\
&~{}+E\displaystyle\int_{0}^{T}||z^{\varepsilon}(t)-\bar{z}(t)-\varepsilon
Z(t)||^{2}_{l^{2}({\mathbb{R}^{n}})}dt\\\
\leq&K\varepsilon^{2}\bigg{[}E\displaystyle\int_{0}^{T}\bigg{|}(\tilde{f}^{\varepsilon}_{y}(t)-f_{y}(t))Y(t)+(\tilde{f}^{\varepsilon}_{q}(t)-f_{q}(t))Q(t)+(\tilde{f}^{\varepsilon}_{z}(t)-f_{z}(t))Z(t)\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+(\tilde{f}^{\varepsilon}_{u}(t)-f_{u}(t))(u(t)-\bar{u}(t))\bigg{|}^{2}dt\bigg{]}\\\
=&K\varepsilon^{2}\cdot\alpha(\varepsilon),\end{array}$ (3.10)

where

$\displaystyle\begin{split}\alpha(\varepsilon)=E\displaystyle&\int_{0}^{T}\bigg{|}(\tilde{f}^{\varepsilon}_{y}(t)-f_{y}(t))Y(t)+(\tilde{f}^{\varepsilon}_{q}(t)-f_{q}(t))Q(t)\\\
&\ \ \ \ \ \ \
+(\tilde{f}^{\varepsilon}_{z}(t)-f_{z}(t))Z(t)+(\tilde{f}^{\varepsilon}_{u}(t)-f_{u}(t))(u(t)-\bar{u}(t))\bigg{|}^{2}dt.\end{split}$

Consequently, using Lemma 3.2 and Assumption 3.1, by the dominated convergence
theorem we can deduce

$\displaystyle\lim_{\varepsilon\rightarrow 0}\alpha(\varepsilon)=0.$

Then the lemma follows from above and (3.10). ∎

######  Lemma 3.4.

Under Assumptions 3.1-3.2, using the abbreviations (3) we have

$\begin{array}[]{ll}J(u^{\varepsilon}(\cdot))-J(\bar{u}(\cdot))=&\varepsilon
E\phi_{y}(\bar{y}(0))Y(0)+\varepsilon
E\displaystyle\int_{0}^{T}l_{y}(t)Y(t)dt+\varepsilon
E\displaystyle\int_{0}^{T}l_{q}(t)Q(t)dt\\\ &+\varepsilon
E\displaystyle\int_{0}^{T}l_{z}(t)Z(t)dt+\varepsilon
E\displaystyle\int_{0}^{T}l_{u}(t)(u(t)-\bar{u}(t))dt+o(\varepsilon).\end{array}$

###### Proof.

After a first order development, we have

$\displaystyle J(u^{\varepsilon}(\cdot))-J(\bar{u}(\cdot))$ $\displaystyle=$
$\displaystyle
E\displaystyle\int_{0}^{1}\phi_{y}(\bar{y}(0)+\lambda(y^{\varepsilon}(0)-\bar{y}(0)))(y^{\varepsilon}(0)-\bar{y}(0))d\lambda$
$\displaystyle+E\displaystyle\int_{0}^{T}\tilde{l}_{y}^{\varepsilon}(t)(y^{\varepsilon}(t)-\bar{y}(t))dt+E\displaystyle\int_{0}^{T}\tilde{l}_{q}^{\varepsilon}(t)(q^{\varepsilon}(t)-\bar{q}(t))dt$
$\displaystyle+E\displaystyle\int_{0}^{T}\tilde{l}_{z}^{\varepsilon}(t)(z^{\varepsilon}(t)-\bar{z}(t))dt+E\displaystyle\int_{0}^{T}\tilde{l}_{u}^{\varepsilon}(t)(u^{\varepsilon}(t)-\bar{u}(t))dt$
$\displaystyle=$ $\displaystyle\varepsilon
E\phi_{y}(\bar{y}(0))Y(0)+E\phi_{y}(\bar{y}(0))(y^{\varepsilon}(0)-\bar{y}(0)-\varepsilon
Y(0))$
$\displaystyle+E\displaystyle\int_{0}^{1}\bigg{[}\phi_{y}(\bar{y}(0)+\lambda(y^{\varepsilon}(0)-\bar{y}(0)))-\phi_{y}(\bar{y}(0))\bigg{]}(y^{\varepsilon}(0)-\bar{y}(0))d\lambda$
$\displaystyle+\varepsilon
E\displaystyle\int_{0}^{T}l_{y}(t)Y(t)dt+E\displaystyle\int_{0}^{T}l_{y}(t)(y^{\varepsilon}(t)-\bar{y}(t)-\varepsilon
Y(t))dt$
$\displaystyle+E\displaystyle\int_{0}^{T}(\tilde{l}_{y}^{\varepsilon}(t)-l_{y}(t))(y^{\varepsilon}(t)-\bar{y}(t))dt$
$\displaystyle+\varepsilon
E\displaystyle\int_{0}^{T}l_{q}(t)q(t)dt+E\displaystyle\int_{0}^{T}l_{q}(t)(q^{\varepsilon}(t)-\bar{q}(t)-\varepsilon
Q(t))dt$
$\displaystyle+E\displaystyle\int_{0}^{T}(\tilde{l}_{q}^{\varepsilon}(t)-l_{q}(t))(q^{\varepsilon}(t)-\bar{q}(t))dt$
$\displaystyle+\varepsilon
E\displaystyle\int_{0}^{T}l_{z}(t)Z(t)dt+E\displaystyle\int_{0}^{T}l_{z}(t)(z^{\varepsilon}(t)-\bar{z}(t)-\varepsilon
Z(t))dt$
$\displaystyle+E\displaystyle\int_{0}^{T}(\tilde{l}_{z}^{\varepsilon}(t)-l_{z}(t))(z^{\varepsilon}(t)-\bar{z}(t))dt$
$\displaystyle+E\displaystyle\int_{0}^{T}l_{u}(t)\varepsilon(u(t)-\bar{u}(t))dt+E\displaystyle\int_{0}^{T}(\tilde{l}_{u}^{\varepsilon}(t)-l_{u}(t))\varepsilon(u(t)-\bar{u}(t))dt$
$\displaystyle=$ $\displaystyle\varepsilon
E\phi_{y}(\bar{y}(0))Y(0)+\varepsilon
E\displaystyle\int_{0}^{1}l_{y}(t)Y(t)dt+\varepsilon
E\displaystyle\int_{0}^{1}l_{q}(t)Q(t)dt$ $\displaystyle+\varepsilon
E\displaystyle\int_{0}^{1}l_{z}(t)Z(t)dt+\varepsilon
E\displaystyle\int_{0}^{1}l_{u}(t)(u(t)-\bar{u}(t))dt+\beta(\varepsilon),$

where $\beta(\varepsilon)$ is given by

$\begin{array}[]{ll}\beta(\varepsilon)=&E\phi_{y}(\bar{y}(0))(y^{\varepsilon}(0)-\bar{y}(0)-\varepsilon
Y(0))\\\
&+E\displaystyle\int_{0}^{1}\bigg{[}\phi_{y}(\bar{y}(0)+\lambda(y^{\varepsilon}(0)-\bar{y}(0)))-\phi_{y}(\bar{y}(0))\bigg{]}(y^{\varepsilon}(0)-\bar{y}(0))d\lambda\\\
&+E\displaystyle\int_{0}^{T}l_{y}(t)(y^{\varepsilon}(t)-\bar{y}(t)-\varepsilon
Y(t))dt+E\displaystyle\int_{0}^{T}(\tilde{l}_{y}^{\varepsilon}(t)-l_{y}(t))(y^{\varepsilon}(t)-\bar{y}(t))dt\\\
&+E\displaystyle\int_{0}^{T}l_{q}(t)(q^{\varepsilon}(t)-\bar{q}(t)-\varepsilon
Q(t))dt+E\displaystyle\int_{0}^{T}(\tilde{l}_{q}^{\varepsilon}(t)-l_{q}(t))(q^{\varepsilon}(t)-\bar{q}(t))dt\\\
&+E\displaystyle\int_{0}^{T}l_{z}(t)(z^{\varepsilon}(t)-\bar{z}(t)-\varepsilon
Z(t))dt+E\displaystyle\int_{0}^{T}(\tilde{l}_{z}^{\varepsilon}(t)-l_{z}(t))(z^{\varepsilon}(t)-\bar{z}(t))dt\\\
&+E\displaystyle\int_{0}^{T}(\tilde{l}_{u}^{\varepsilon}(t)-l_{u}(t))\varepsilon(u(t)-\bar{u}(t))dt.\end{array}$

Thus combining Lemma 3.2, Lemma 3.4 and Assumption 3.2, by the dominated
convergence theorem we conclude that $\beta(\varepsilon)=o(\varepsilon)$. ∎

By Lemma 3.4 and the fact that $\displaystyle\lim_{\varepsilon\rightarrow
0^{+}}\frac{J(u^{\varepsilon})-J(\bar{u})}{\varepsilon}\geq 0$, we can further
deduce

######  Corollary 3.5.

Under Assumptions 3.1-3.2, we have the variation inequality below

$\begin{array}[]{ll}&E\phi_{y}(\bar{y}(0))Y(0)+E\displaystyle\int_{0}^{T}l_{y}(t)Y(t)dt+E\displaystyle\int_{0}^{T}l_{y}(t)Y(t)dt\\\
&+E\displaystyle\int_{0}^{T}l_{z}(t)Z(t)dt+E\displaystyle\int_{0}^{T}l_{u}(t)(u(t)-\bar{u}(t))dt\geq
0.\end{array}$ (3.11)

##  4 Necessary and sufficient optimality conditions

We first introduce the adjoint equation corresponding to the variational
equation (3):

$\displaystyle
dk(t)=-\bigg{[}-f_{y}^{*}(t)k(t)+l_{y}(t)\bigg{]}dt-\displaystyle\sum_{i=1}^{d}\bigg{[}-f_{q^{i}}^{*}(t)k(t)+l_{q^{i}}(t)\bigg{]}dW^{i}(t)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \
-\displaystyle\sum_{i=1}^{\infty}\bigg{[}-f_{z^{i}}^{*}(t)k(t)+l_{z^{i}}(t)\bigg{]}dH^{i}(t)$
(4.1) $\displaystyle k(0)=-\phi_{y}(\bar{y}(0)),~{}~{}~{}~{}0\leq t\leq T,$

where $f_{y}^{*},f_{q^{i}}^{*}$and $f_{z^{i}}^{*}$ are the dual operators of
$f_{y},f_{q^{i}}$ and $f_{z^{i}}$, respectively.

Under Assumptions 3.1-3.2, by Lemma 2.1 it is easy to see that the above
adjoint equation has a unique solution
$k(\cdot)\in{\mathcal{S}}^{2}_{\mathscr{F}}(0,T;\mathbb{R}^{n})$. Then we
define the Hamiltonian function
$H:[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n\times d}\times
l^{2}(\mathbb{R}^{n})\times U\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{1}$ by

$\begin{array}[]{ll}\displaystyle H(t,y,q,z,u,k)=\langle
k,-f(t,y,q,z,u)\rangle+l(t,y,q,z,u)\end{array}$ (4.2)

and rewrite the adjoint equation in the Hamiltonian system form:

$\left\\{\begin{array}[]{ll}dk(t)=-H_{y}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),k(t))dt\\\
\ \ \ \ \ \ \ \ \ \ \
-\displaystyle\sum_{i=1}^{d}H_{q}^{i}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),k(t))dW^{i}(t)\\\
\ \ \ \ \ \ \ \ \ \ \
-\displaystyle\sum_{i=1}^{\infty}H_{z^{i}}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),k(t))dH^{i}(t)\\\
k(0)=-\phi_{y}(\bar{y}(0)).\end{array}\right.$ (4.3)

Now we are ready to give the necessary conditions for an optimal control of
Problem 3.1.

######  Theorem 4.1.

Under Assumptions 3.1-3.2, if
$(\bar{u}(\cdot);\bar{y}(\cdot),\bar{q}(\cdot),\bar{z}(\cdot))$ is an optimal
pair of Problem 3.1, then we have

$H_{u}(t,\bar{y}(t-),\bar{q}(t),\bar{z}(t),\bar{u}(t),k(t-))(u-\bar{u}(t))\geq
0,~{}\forall u\in U,\ \ a.e.\ a.s.,$ (4.4)

where $k(\cdot)$ is the solution to the adjoint equation (4).

###### Proof.

By (3) and (4), applying Itô formula to $\langle Y(t),k(t)\rangle$ we have

$\begin{array}[]{ll}&E\phi_{y}(\bar{y}(0))Y(0)+E\displaystyle\int_{0}^{T}l_{y}(t)Y(t)dt+E\displaystyle\int_{0}^{T}l_{z}(t)Z(t)dt+E\displaystyle\int_{0}^{T}l_{u}(t)(u(t)-\bar{u}(t))dt\\\
=&-E\displaystyle\int_{0}^{T}\langle
k(t),f_{u}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}_{t})(u(t)-\bar{u}(t))\rangle
dt+E\displaystyle\int_{0}^{T}l_{u}(t)(u(t)-\bar{u}(t))dt.\end{array}$

Then noticing the definition of Hamilton function (4.2) and the variational
inequality (3.11), for any $u(\cdot)\in{\mathcal{A}}$, we have

$E\displaystyle\int_{0}^{T}H_{u}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),k(t))(u(t)-\bar{u}(t))dt\geq
0,$

which implies (4.4). ∎

We then consider the sufficient conditions for an optimal control of Problem
3.1.

######  Theorem 4.2.

Under Assumptions 3.1-3.2, let
$(\bar{u}(\cdot);\bar{y}(\cdot),\bar{q}(\cdot),\bar{z}(\cdot))$ be an
admissible pair and ${k}(\cdot)$ be the unique solution of the corresponding
adjoint equation (4.3). Assume that for almost all
$(t,\omega)\in[0,T]\times\Omega$ , $H(t,y,q,z,u,{k}(t))$ and $\phi(y)$ are
convex w.r.t. $(y,q,z,u)$ and $y$, respectively, and the optimality condition

$H(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),k(t))=\displaystyle\min_{u\in
U}H(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),u,k(t))$

holds, then $(\bar{u}(\cdot);\bar{y}(\cdot),\bar{q}(\cdot),\bar{z}(\cdot))$ is
an optimal pair of Problem 3.1.

###### Proof.

Let $(u(\cdot);y(\cdot),q(\cdot),z(\cdot))$ be an arbitrary admissible pair.
It follows from the form of the cost functional (3.2) that

$\displaystyle J(u(\cdot))-J(\bar{u}(\cdot))$ (4.5) $\displaystyle=$
$\displaystyle
E\displaystyle\int_{0}^{T}\bigg{[}l(t,y(t),q(t),z(t),u(t))-l(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t))\bigg{]}dt+E\bigg{[}\phi(y(0))-\phi(\bar{y}(0))\displaystyle\bigg{]}$
$\displaystyle=$ $\displaystyle I_{1}+I_{2},$

where

$\displaystyle
I_{1}=E\int_{0}^{T}\bigg{[}l(t,y(t),q(t),z(t),u(t))-l(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t))\bigg{]}dt$

and

$I_{2}=E\bigg{[}\phi(y(0))-\phi(\bar{y}(0))\bigg{]}.$

Due to the convexity of $\phi$, applying Itô formula to
$\langle{k}(t),y(t)-\bar{y}(t)\rangle$, we have

$\displaystyle\begin{split}I_{2}=&E[\phi(y(0))-\phi(\bar{y}(0))]\geq
E[\langle\phi_{y}(\bar{y}(0)),y(0)-\bar{y}(0)\rangle]=-E[\langle{k}(0),y(0)-\bar{y}(0)\rangle]\\\
=&-E\displaystyle\int_{0}^{T}\langle
H_{y}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),{k}(t)),y(t)-\bar{y}(t)\rangle
dt\\\ &-\sum_{i=1}^{d}E\displaystyle\int_{0}^{T}\langle
H_{q}^{i}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),{k}(t)),q^{i}(t)-\bar{q}^{i}(t)\rangle
dt\\\ &-\displaystyle\sum_{i=1}^{\infty}E\displaystyle\int_{0}^{T}\langle
H_{z}^{i}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),{k}(t)),z^{i}(t)-\bar{z}^{i}(t)\rangle
dt\\\ &-E\displaystyle\int_{0}^{T}\langle
f(t,{y}(t),{q}(t),{z}(t),{u}(t))-f(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t)),k(t)\rangle
dt\\\ =&-J_{1}+J_{2},\end{split}$ (4.6)

where

$\begin{array}[]{ll}J_{1}=&E\displaystyle\int_{0}^{T}\langle
H_{y}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),{k}(t)),y(t)-\bar{y}(t)\rangle
dt\\\ &+\displaystyle\sum_{i=1}^{d}E\displaystyle\int_{0}^{T}\langle
H_{q}^{i}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),{k}(t)),q^{i}(t)-\bar{q}^{i}(t)\rangle
dt\\\ &+\displaystyle\sum_{i=1}^{\infty}E\displaystyle\int_{0}^{T}\langle
H_{z}^{i}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),{k}(t)),z^{i}(t)-\bar{z}^{i}(t)\rangle
dt\end{array}$

and

$J_{2}=-E\displaystyle\int_{0}^{T}\langle
f(t,{y}(t),{q}(t),{z}(t),{u}(t))-f(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t)),k(t)\rangle
dt.$

Using the definition of the Hamiltonian function (4.2) again, we have

$\begin{array}[]{ll}I_{1}&=E\displaystyle\int_{0}^{T}\bigg{[}l(t,y(t),q(t),z(t),u(t))-l(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t))\bigg{]}dt\\\
&=E\displaystyle\int_{0}^{T}\bigg{[}H(t,y(t),q(t),z(t),u(t),k(t))-H(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),k(t))\bigg{]}dt\\\
&\ \ \ +E\displaystyle\int_{0}^{T}\langle
f(t,{y}(t),{q}(t),{z}(t),{u}(t))-f(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t)),k(t)\rangle
dt\\\ &=J_{3}-J_{2},\end{array}$ (4.7)

where

$\begin{array}[]{ll}J_{3}=E\displaystyle\int_{0}^{T}\bigg{[}H(t,y(t),q(t),z(t),u(t),k(t))-H(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),k(t))\bigg{]}dt.\end{array}$
(4.8)

Since $H(t,y,q,z,u,{k}(t))$ is convex w.r.t. $(y,q,z,u)$ for almost all
$(t,\omega)\in[0,T]\times\Omega$, it turns out that

$\displaystyle\begin{split}&H(t,y(t),q(t),z(t),u(t),{k}(t))-H(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),{k}(t))\\\
\geq&\langle
H_{y}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),{k}(t)),y(t)-\bar{y}(t)\rangle\\\
&+\sum_{i=1}^{d}\langle
H_{q}^{i}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),{k}(t)),q^{i}(t)-\bar{q}^{i}(t)\rangle\\\
&+\sum_{i=1}^{\infty}\langle
H_{z}^{i}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),{k}(t)),z^{i}(t)-\bar{z}^{i}(t)\rangle\\\
&+\langle
H_{u}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),{k}(t)),u(t)-\bar{u}(t)\rangle,\
\ a.s.\ a.e.\end{split}$ (4.9)

On the other hand, for almost all $(t,\omega)\in[0,T]\times\Omega$,
$u\rightarrow H(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),u,k(t))$ takes its minimal
value at $\bar{u}(t)$ in the domain $U$, thus

$\begin{array}[]{ll}\langle
H_{u}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),k(t)),u(t)-\bar{u}(t)\rangle\geq
0,\ \ a.s.\ a.e.\end{array}$ (4.10)

Therefore, by (4.8)–(4.10) we first have

$\begin{array}[]{ll}J_{3}\geq J_{1}.\end{array}$ (4.11)

By (4.11), together with (4.5)–(4.7), it follows that

$\begin{array}[]{ll}J(u(\cdot))-J(\bar{u}(\cdot))=I_{1}+I_{2}=(J_{3}-J_{2})+(-J_{1}+J_{2})\geq(J_{1}-J_{2})+(-J_{1}+J_{2})=0.\end{array}$

Due to the arbitrariness of $u(\cdot)$, we conclude that $\bar{u}(\cdot)$ is
an optimal control process and thus
$(\bar{u}(\cdot);\bar{y}(\cdot),\bar{q}(\cdot),\bar{z}(\cdot))$ is an optimal
pair. ∎

##  5 Applications in BLQ problems

In this section, we will apply our stochastic maximum principle to the so-
called BLQ problem, i.e. minimize the following quadratic cost functional over
$u(\cdot)\in\mathcal{A}$:

$\displaystyle\begin{split}J(u(\cdot)):=&E\langle
My(0),y(0)\rangle+E\displaystyle\int_{0}^{T}\langle E(s)y(s),y(s)\rangle
ds+\sum_{i=1}^{d}E\displaystyle\int_{0}^{T}\langle
F^{i}(s)q^{i}(s),q^{i}(s)\rangle ds\\\
&+\sum_{i=1}^{\infty}E\displaystyle\int_{0}^{T}\langle
G^{i}(s)z^{i}(s),z^{i}(s)\rangle ds+E\displaystyle\int_{0}^{T}\langle
N(s)u(s),u(s)\rangle ds,\end{split}$ (5.1)

where the state processes $(y(\cdot),q(\cdot),z(\cdot))$ are the solution to
the controlled linear backward stochastic system as follows:

$\displaystyle
dy(t)=-\bigg{[}A(t)y(t)+\displaystyle\sum_{i=1}^{d}B^{i}(t)q^{i}(t)+\displaystyle\sum_{i=1}^{\infty}C^{i}(t)z^{i}(t)+D(t)u(t)\bigg{]}dt$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \
+\displaystyle\sum_{i=1}^{d}q^{i}dW^{i}(t)+\displaystyle\sum_{i=1}^{\infty}z^{i}dH^{i}(t)$
(5.2) $\displaystyle y(T)=\xi.$

To study this problem, we need the assumptions on the coefficients below.

######  Assumption 5.1.

The $\\{{\mathscr{F}}_{t},0\leq t\leq T\\}$-predictable matrix processes
$A:[0,T]\times\Omega\rightarrow\mathbb{R}^{n\times
n},B^{i}:[0,T]\times\Omega\rightarrow\mathbb{R}^{n\times
n},i=1,2,\cdots,d,C^{i}:[0,T]\times\Omega\rightarrow\mathbb{R}^{n\times
n},i=1,2,\cdots,D:[0,T]\times\Omega\rightarrow\mathbb{R}^{n\times
m},E:[0,T]\times\Omega\rightarrow\mathbb{R}^{n\times
n},F^{i}:[0,T]\times\Omega\rightarrow\mathbb{R}^{n\times n},i=1,2,\cdots
d,G^{i}:[0,T]\times\Omega\rightarrow\mathbb{R}^{n\times
n},i=1,2,\cdots,N:[0,T]\times\Omega\rightarrow\mathbb{R}^{m\times m}$ and the
${\mathscr{F}}_{T}$-measurable random matrix
$M:\Omega\rightarrow\mathbb{R}^{n\times n}$ are uniformly bounded.

######  Assumption 5.2.

The state weighting matrix processes $E$, $F^{i}$, $G^{i}$, the control
weighting matrix process $N$ and the random matrix $M$ are a.e. a.s. symmetric
and nonnegative. Moreover, $N$ is a.e. a.s. uniformly positive, i.e.
$N\geq\delta I$ for some positive constant $\delta$ a.e. a.s.

######  Assumption 5.3.

There is no further constraint imposed on the control processes, i.e.

$\mathcal{A}=\bigg{\\{}u(\cdot)|u(\cdot)\ is\ \mathscr{F}_{t}-predictable\
with\ values\ in\ \mathbb{R}^{m}\ and\
E\displaystyle\int_{0}^{T}|u(t)|^{2}dt<\infty\\}.$

From Assumption 5.3, we know that $\mathcal{A}$ is a Hilbert space. If we
denote the norm of $\mathcal{A}$ by $\|\cdot\|_{\mathcal{A}}$, then for any
control process $u(\cdot)\in\mathcal{A}$,
$\|u(\cdot)\|_{\mathcal{A}}=E\displaystyle\sqrt{\int_{0}^{T}|u(t)|^{2}dt}$.

Under Assumptions 5.1, by Lemma 2.3 we first know that the linear BSDE (5) in
BLQ problem has a unique solution and thus the BLQ problem is well-defined.
Then, under Assumptions 5.1-5.3, we will demonstrate that BLQ problem has a
unique optimal control.

######  Lemma 5.1.

Under Assumptions 5.1-5.3, the cost functional $J$ is strictly convex over
$\mathcal{A}$ and
$\displaystyle\lim_{\|u(\cdot)\|_{\mathcal{A}}{\rightarrow\infty}}J(u(\cdot))=\infty.$

###### Proof.

The convexity of the cost functional $J$ over $\mathcal{A}$ is obvious.
Actually, since the weighting matrix process $N$ is uniformly positive, $J$ is
strictly convex. In view of the nonnegative property of $M,E,F^{i},G^{i}$ and
the strictly positive property of $N$, we have

$J(u(\cdot))\geq\delta
E\displaystyle\int_{0}^{T}|u(t)|^{2}dt=\delta\|u(\cdot)\|^{2}_{\mathcal{A}}.$

Therefore,
$\displaystyle\lim_{\|u(\cdot)\|_{\mathcal{A}}{\rightarrow\infty}}J(u(\cdot))=\infty.$
∎

######  Lemma 5.2.

Under Assumptions 5.1-5.3, the cost functional $J$ is Fréchet differentiable
over $\mathcal{A}$ and its Fréchet derivative $J^{\prime}$ at any admissible
control process $u(\cdot)\in{\mathcal{A}}$ is given by

$\displaystyle\begin{split}\langle
J^{\prime}(u(\cdot)),v(\cdot)\rangle=&2E\int_{0}^{T}\langle
E(t)y^{u}(t),Y^{v}(t)\rangle dt+2\sum_{i=1}^{d}E\int_{0}^{T}\langle
F^{i}(t)q^{iu}(t),Q^{iv}(t)\rangle dt\\\
&+2\sum_{i=1}^{\infty}E\int_{0}^{T}\langle G^{i}(t)z^{iu}(t),Z^{iv}(t)\rangle
dt+2E\int_{0}^{T}\langle N(t)u(t),v(t)\rangle dt\\\ &+2E\langle
My^{u}(0),Y^{v}(0)\rangle,\end{split}$ (5.3)

where $v(\cdot)\in\mathcal{A}$ is arbitrary, $(Y^{v},Q^{v},Z^{v})$ is the
solution of BSDE (5) corresponding to the control process
$v(\cdot)\in\mathcal{A}$ and the terminal value $0$, and
$(y^{u}(\cdot),q^{u}(\cdot),z^{u}(\cdot))$ are the state processes
corresponding to the control process $u(\cdot)$.

###### Proof.

For any $v(\cdot)\in\mathcal{A}$, we set

$\displaystyle\begin{split}\Delta
J=&J(u(\cdot)+v(\cdot))-J(u(\cdot))-2E\int_{0}^{T}\langle
E(t)y^{u}(t),Y^{v}(t)\rangle dt\\\ &-2\sum_{i=1}^{d}E\int_{0}^{T}\langle
F^{i}(t)q^{iu}(t),Q^{iv}(t)\rangle dt-2\sum_{i=1}^{\infty}E\int_{0}^{T}\langle
G^{i}(t)z^{iu}(t),Z^{iv}(t)\rangle dt\\\ &-2E\int_{0}^{T}\langle
N(t)u(t),v(t)\rangle dt-2E\langle My^{u}(0),Y^{v}(0)\rangle.\end{split}$

By the definition of cost functional (5.1), we have

$\displaystyle\begin{split}\Delta J=&E\langle
MY^{v}(0),Y^{v}(0)\rangle+E\displaystyle\int_{0}^{T}\langle
E(s)Y^{v}(s),Y^{v}(s)\rangle
ds+\sum_{i=1}^{d}E\displaystyle\int_{0}^{T}\langle
F^{i}(s)Q^{iv}(s),Q^{iv}(s)\rangle ds\\\
&+\sum_{i=1}^{\infty}E\displaystyle\int_{0}^{T}\langle
G^{i}(s)Z^{iv}(s),Z^{iv}(s)\rangle ds+E\displaystyle\int_{0}^{T}\langle
N(s)v(s),v(s)\rangle ds.\end{split}$

Then it follows from Assumption 5.1 and a priori estimate (2.3) that

$\displaystyle|\Delta J|$ $\displaystyle\leq$ $\displaystyle
K\bigg{[}E\sup_{0\leq t\leq
T}|Y^{v}(t)|^{2}+E\int_{0}^{T}|Q^{v}(t)|^{2}dt+E\int_{0}^{T}\|Z^{v}(t)\|^{2}_{l^{2}(\mathbb{R}^{N})}dt+E\int_{0}^{T}|v(t)|^{2}dt\bigg{]}$
$\displaystyle\leq$ $\displaystyle
KE\int_{0}^{T}|v(t)|^{2}dt=K\|v(\cdot)\|^{2}_{\mathcal{A}}.$

Consequently, we have

$\displaystyle\lim_{\|v(\cdot)\|_{\mathcal{A}}{\rightarrow 0}}\frac{|\Delta
J|}{\|v(\cdot)\|_{\mathcal{A}}}=0,$

which implies that $J$ is Fréchet differentiable and its Fréchet derivative
$J^{\prime}$ is given by (5.3). ∎

The strict convexity and the Fréchet differentiability of $J$ deduced from
Lemmas 5.1-5.2 lead to the lower semi-continuity of $J$, thus the following
lemma is applicable to $J$ and $\mathcal{A}$ in our BLQ problem.

######  Lemma 5.3.

(Proposition 1.2 of Chapter II in [6]) Let $\mathcal{A}$ be a reflexive Banach
space and $J:\mathcal{A}\mapsto\mathbb{R}^{1}$ be a convex function. Assume
that $J$ is lower semi-continuous and proper, and consider the minimization
problem

$\inf_{u\in\mathcal{A}}J(u).$

If the function $J$ is coercive over $\mathcal{A}$, i.e.

$\lim_{\|u\|_{\mathcal{A}}\to\infty}J(u)=\infty,$

then the minimization problem has at least one solution. Moreover, if $J$ is
strictly convex over $\mathcal{A}$, then the minimization problem has a unique
solution.

By Lemma 5.3 we can immediately conclude

######  Theorem 5.4.

Under Assumptions 5.1-5.3, BLQ problem has a unique optimal control.

In what follows, we will utilize the stochastic maximum principle to study the
dual representation of the optimal control to BLQ problem and construct its
stochastic Hamilton system. As in section 4, we first introduce the adjoint
forward equation corresponding to an admissible pair
$(u(\cdot);y(\cdot),q(\cdot),z(\cdot))$:

$\displaystyle
dk(t)=\bigg{(}A^{*}(t)k(t)-2E(t)y(t)\bigg{)}dt+\displaystyle\sum_{i=1}^{d}\bigg{(}B^{i*}(t)k^{i}(t)-2F^{i}(t)q^{i}(t)\bigg{)}dW^{i}(t)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \
+\displaystyle\sum_{i=1}^{\infty}\bigg{(}C^{i*}(t)k(t)-2G^{i}(t)z^{i}(t)\bigg{)}dH^{i}(t)$
(5.4) $\displaystyle k(0)={-2My(0)}.$

Also we define the Hamiltonian function
$H:[0,T]\times\Omega\times\mathbb{R}^{n}\times\mathbb{R}^{n\times d}\times
l^{2}(\mathbb{R}^{n})\times U\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{1}$ by

$\displaystyle\displaystyle H(t,y,q,z,u,k)$ $\displaystyle=$
$\displaystyle-\bigg{\langle}k,A(t)y+\displaystyle\sum_{i=1}^{d}B^{i}(t)q^{i}+\displaystyle\sum_{i=1}^{\infty}C^{i}(t)z^{i}+D(t)u\bigg{\rangle}$
$\displaystyle+\langle E(t)y,y\rangle+\displaystyle\sum_{i=1}^{d}\langle
F^{i}(t)q^{i},q^{i}\rangle+\displaystyle\sum_{i=1}^{\infty}\langle
G^{i}(t)z^{i},z^{i}\rangle+\langle N(t)u,u\rangle.$

Then the adjoint equation can be rewritten as a Hamiltonian form:

$\displaystyle
dk(t)=-H_{y}(t,{y}(t),{q}(t),{z}(t),{u}(t),k(t))dt-\displaystyle\sum_{i=1}^{d}H_{q}^{i}(t,{y}(t),{q}(t),{z}(t),{u}(t),k(t))dB^{i}(t)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \
-\displaystyle\sum_{i=1}^{\infty}H_{z^{i}}(t,{y}(t-),{q}(t),{z}(t),{u}(t),k(t))dH^{i}(t)$
(5.6) $\displaystyle k(0)=-2My(0).$

Under Assumption 5.1, for each admissible pair
$({u}(\cdot);{y}(\cdot),{q}(\cdot),{z}(\cdot))$, by Lemma 2.1 the adjoint
equation (5) has a unique solution $k(\cdot)$.

It is time to give the the dual characterization of the optimal control.

######  Theorem 5.5.

Under Assumptions 5.1-5.3, BLQ problem has a unique optimal control and the
optimal control is given by

$\displaystyle\begin{split}u(t)=-\frac{1}{2}N^{-1}(t)D^{*}(t)k(t-),\ \ a.e.\
a.s.,\end{split}$ (5.7)

where $k(\cdot)$ is the unique solution of the adjoint equation (5) (or
equivalently, (5)) corresponding to the optimal pair
$(u(\cdot);y(\cdot),q(\cdot),z(\cdot))$.

###### Proof.

By Theorem 5.4, we know the existence and uniqueness of optimal control to BLQ
problem and denote the optimal control by $u(\cdot)$. We only need to prove
$u$ has an expression as in (5.7). For this, let
$(y(\cdot),q(\cdot),z(\cdot))$ be the optimal state processes corresponding to
$u(\cdot)$ and $k(\cdot)$ be the unique solution of the adjoint equation (5)
corresponding to the optimal pair $(u(\cdot);y(\cdot),q(\cdot),z(\cdot))$. By
the necessary optimality condition (4.4) and Assumption 5.3, we have

$H_{u}(t,y(t-),q(t),z(t),u(t),k(t-))=0,\ \ a.e.\ a.s.$

Noticing the definition of $H$ in (5), we get

$2N(t)u(t)+D^{*}(t)k(t-)=0,\ \ a.e.\ a.s.$

Then the claim that the unique optimal control $u(\cdot)$ satisfies (5.7)
follows. ∎

Finally we introduce the so-called stochastic Hamilton system which consists
of the state equation (5), the adjoint equation (5) (or equivalently, (5)) and
the dual representation (5.7):

$\displaystyle
dy(t)=-\bigg{(}A(t)y(t)+\sum_{i=1}^{d}B^{i}(t)q^{i}(t)+\sum_{i=1}^{\infty}C^{i}(t)z^{i}(t)+D(t)u(t)\bigg{)}dt$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \
+\displaystyle\sum_{i=1}q^{i}dW^{i}(t)+\displaystyle\sum_{i=1}^{\infty}z^{i}dH^{i}(t)$
$\displaystyle y(T)=\xi,$ $\displaystyle
dk(t)=\bigg{(}A^{*}(t)k(t)-2E(t)y(t)\bigg{)}dt+\displaystyle\sum_{i=1}^{d}\bigg{(}B^{i*}(t)k^{i}(t)-2F^{i}(t)q^{i}(t)\bigg{)}dW^{i}(t)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \
+\displaystyle\sum_{i=1}^{\infty}\bigg{(}C^{i*}(t)k(t)-2G^{i}(t)z^{i}(t)\bigg{)}dH^{i}(t)$
(5.8) $\displaystyle k(0)=-2My(0),$ $\displaystyle
u_{t}=-\frac{1}{2}N^{-1}(t)D^{*}(t)k(t-).$

Clearly this is a fully coupled forward-backward stochastic differential
equation (FBSDE) driven by $d$-dimensional Brownian motion $W$ and Teugel’s
martingales $\\{H^{i}\\}_{i=1}^{\infty}$, and its solution is a stochastic
processes quaternary $(k(\cdot),y(\cdot),q(\cdot),z(\cdot))$.

######  Theorem 5.6.

Under Assumptions 5.1-5.3, the stochastic Hamilton system (5) has a unique
solution $(k(\cdot),y(\cdot),q(\cdot),z(\cdot))\in
S_{\mathscr{F}}^{2}(0,T;\mathbb{R}^{n})\times
S_{\mathscr{F}}^{2}(0,T;\mathbb{R}^{n})\times
M_{\mathscr{F}}^{2}(0,T;\mathbb{R}^{n\times d})\times
l_{\mathscr{F}}^{2}(0,T;\mathbb{R}^{n})$, where $u(\cdot)$ is the optimal
control of BLQ problem and $(y(\cdot),q(\cdot),z(\cdot))$ are its
corresponding optimal state. Moreover,

$\displaystyle\begin{split}&\displaystyle E\sup_{0\leqslant t\leqslant
T}|k(t)|^{2}+E\displaystyle\sup_{0\leq t\leq
T}|y(t)|^{2}+E\int_{0}^{T}|q(t)|^{2}dt+E\int_{0}^{T}||z(t)||^{2}_{l^{2}({\mathbb{R}^{n}})}dt\leqslant
KE{{|\xi|^{2}}}.\end{split}$ (5.9)

###### Proof.

The existence result follows from Theorem 5.5 and the uniqueness result is
obvious once a priori estimate (5.9) holds. But noticing Assumptions 5.1-5.3
and using Lemmas 2.2 and 2.4, we can deduce (5.9) immediately. ∎

In summary, the stochastic Hamilton system (5) completely characterize the
optimal control of BLQ problem in this section. Therefore, solving BLQ problem
is equivalent to solving the stochastic Hamilton system, moreover, the unique
optimal control of the stochastic Hamilton system can be given explicitly by
(5.7).

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