#  Stochastic Evolution Equation Driven by Teugels Martingale and Its Optimal
Control ††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

The paper is concerned with a class of stochastic evolution equations in
Hilbert space with random coefficients driven by Teugel’s martingales and an
independent multi-dimensional Brownian motion and its optimal control problem.
Here Teugels martingales are a family of pairwise strongly orthonormal
martingales associated with Lévy processes (see Nualart and Schoutens [16]).
There are three major ingredients. The first is to prove the existence and
uniqueness of the solutions by continuous dependence theorem of solutions
combining with the parameter extension method. The second is to establish the
stochastic maximum principle and verification theorem for our optimal control
problem by the classic convex variation method and dual technique. The third
is to represent an example of a Cauchy problem for a controlled stochastic
partial differential equation driven by Teugels martingales which our
theoretical results can solve.

Keywords: Stochastic Evolution Equation; Teugels Martingales; Optimal Control;
Stochastic Maximum Principle; Verification Theorem

##  1  Introduction

This paper is concerned with the following stochastic evolution equation in
Hilbert space

$\displaystyle\left\\{\begin{aligned} dX(t)=&\
[A(t)X(t)+b(t,X(t))]dt+[B(t)X(t)+g(t,X(t))]dW(t)+\sum_{i=1}^{\infty}\sigma^{i}(t,X(t-))dH^{i}(t),\\\
X(0)=&\ x,\quad t\in[0,T]\end{aligned}\right.$ (1.1)

in the framework of a Gelfand triple $V\subset H=H^{*}\subset V^{*},$ where
$H$ and $V$ are two given Hilbert spaces. Here on a given filtrated
probability space $(\Omega,\mathscr{F},\\{{\mathscr{F}}_{t}\\}_{0\leq t\leq
T},P),$ $W$ is a one-dimensional Brownian motion and $\\{H^{i}(t),0\leq t\leq
T\\}_{i=1}^{\infty}$ is is Teugels martingale associated with a one-
dimensional Lévy process $\\{L(t),0\leq t\leq T\\}$,
$A:[0,T]\times\Omega\longrightarrow{\mathscr{L}}(V,V^{*})$,
$B:[0,T]\times\Omega\longrightarrow{\mathscr{L}}(V,H),$
$b:[0,T]\times\Omega\times H\longrightarrow H$, $g:[0,T]\times\Omega\times
H\longrightarrow H$ and $\sigma^{i}:[0,T]\times\Omega\times E\times
H\longrightarrow H$ are given random mappings. Here we denote by
$\mathscr{L}(V,V^{*})$ the space of bounded linear transformations of V into
$V^{*}$, by ${\mathscr{L}}(V,H)$ the space of bounded linear transformations
of $H$ into $V.$ An adapted solution of (1.1) is a $V$-valued,
$\\{{\mathscr{F}}_{t}\\}_{0\leq t\leq T}$-adapted process $X(\cdot)$ which
satisfies (1.1) under some appropriate sense. Such a model as (1.1) represents
a large classes of stochastic partial differential equations, for instance the
nonlinear filtering equation and other stochastic parabolic PDEs (cf.[5]), but
it is by no means the largest one. Partial differential equation are too
diverse to be covered by a single model, like ordinary equations.

In 2000, Nualart and Schoutens [16] got a martingale representation theorem
for a type of Lévy processes through Teugels martingales which are a family of
pairwise strongly orthonormal martingales associated with Lévy processes.
Later, they proved in [17] the existence and uniqueness theory of BSDE driven
by Teugels martingales. The above results are further extended to the one-
dimensional BSDE driven by Teugels martingales and an independent multi-
dimensional Brownian motion by Bahlali et al [2]. One can refer to [18, 19,
25, 24] for more results on such kind of BSDEs.

In the mean time, the stochastic optimal control problems related to Teugels
martingales were studied. In 2008, a stochastic linear-quadratic problem with
Lévy processes was considered by Mitsui and Tabata [15], in which they
established the closeness property of a multi-dimensional backward stochastic
Riccati differential equation (BSRDE) with Teugels martingales and proved the
existence and uniqueness of the 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 [15], Meng
and Tang [14] studied the general stochastic optimal control problem for the
forward stochastic systems driven by Teugels 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. In 2012, Tang and Zhang [31] studied the
optimal control problem of backward stochastic systems driven by Teugels
martingales and an independent multi-dimensional Brownian motion and obtained
the corresponding stochastic maximum principle.

Due to the interesting analytical contents and wide applications in various
sciences such as physics, mechanical engineering, control theory and
economics, the theory of SPDEs driven by Wiener processes or Gaussian random
processes now has been investigated extensively and has already achieved
fruitful results on the existence uniqueness, stability, invariant measure
other quantitative and qualitative properties of solution and so on. There are
a great amount of literature on this topic, for example [5][6][26] and
references therein. On the one hand, non-Gaussian random processes play an
increasing role in modeling stochastic dynamical systems. Typical examples of
non-Gaussian stochastic processes are Lévy processes and processes arising by
Poisson random measures. In neurophysiology the driving noise of the cable
equation is basically impulsiv e.g., of a Poisson type (see [33] ) or, on the
other hand, Woyczyński describes in [34] a number of phenomena from fluid
mechanics, solid state physics, polymer chemistry, economic science etc., for
which non-Gaussian Lévy processes can be used as their mathematical model in
describing the related stochastic behavior. Thus, from the point of view of
applications one might feel that the restriction to Wiener processes or
Gaussian noise is unsatisfactory; to handle such cases one can replace Wiener
processes or Gaussian noise by a Poisson random measure. Most recently, thanks
to comprehensive practical applications, many attentions have been paid to
SPDEs driven by jump processes, (cf., for exampl [1][27][23][26]
[28][32][35][36] and the references therein). It is worth mentioning that
Röcker and Zhang [26] established the existence and uniqueness theory for
solutions of stochastic evolution equations of type (1.1) by a successive
approximations, in which case the operator $B$ does not exist.

One of the purposes of this paper is to establish the continuous dependence of
the solution on the coefficients and the existence and uniqueness of solutions
to the stochastic evolution equation (1.1). It is well known that there are
two different methods to analyzing SPDEs: the semigroup (or mild solution )
approach (cf. [6]) and the variational approach (cf.[22]). For (1.1), since
its coefficients are allowed to be random, we need to use the variational
approach in the weak solution framework (in the PDE sense) of the Gelfand
triple and can not use the mild solution approach to study it. In fact, when
the coefficients are deterministic, we always study the stochastic evolution
equation in the mild solution framework. However, due to the randomness of
coefficients, it seems very difficult or even impossible to tackle the problem
in the mild solution sense. Indeed, if we define the mild solution as usual,
the adaptability of the integrand in the stochastic integral may not be
satisfied due to the randomness of the operator $A$. The advantage of the
variational approach is that a version of Itô’s formula exists in the context
of the Gelfand triple of Hilbert spaces (see [10] for details). Such a formula
will play an important role in proving the main results throughout this paper.

Another purpose of this paper is to establish the maximum principle and
verification theorem for the optimal control problem where the state process
is driven by a controlled stochastic evolution equation (1.1). A classical
approach for optimal control problems is to derive necessary conditions
satisfied by an optimal control, such as Pongtryagin s maximum principle.
Since the 1970s, the maximum principle has been extensively studied for
stochastic control systems: in the finite-dimensional case it has been solved
by Peng [21] in a general setting where the control was allowed to take values
in a nonconvex set and enter into the diffusion, while in the infinte-
dimensional case the existing literatur e.g.,[3], [11] [12][37], required at
least one of the following three assumptions: (1) the control domain was
convex, (2) the diffusion did not depend on the control, (3) the state
equation and cost functional were both linear in the state variable. So far
the general maximum principle for infinite-dimensional stochastic control
systems, the counterpart of Peng’s result, remained open for a long time.
Until recently, Du and Meng attempt to fill this gap in [7] where they
developed a new procedure to perform the second-order duality analysis: by
virtue of the Lebesgue differentiation theorem and an approximation argument
to establish the corresponding maximum principle. Meanwhile other very
important works concerned with the general stochastic maximum principle in
infinite dimensions were given in [9] [13] besides [7]. From the above
references, works on optimal control problems of infinite dimension stochastic
evolution equation or stochastic partial differential equation are mainly
concerned with systems driven only by Wiener processes. In contrast, there
have not been a number of results on the optimal control for stochastic
partial differential equations driven by jump processes. In 2005, Øksendal,
Prosk and Zhang [20] studied the optimal control problem of quasilinear
semielliptic SPDEs driven by Poisson random measure and gave sufficient
maximum principle results, not necessary ones. In 2017, Tang and Meng
[30]studied the optimal control problem of more general stochastic evolution
equations driven by Poisson random measure with random coefficients and gave
necessary and sufficient maximum principle results.In this paper, for a
controlled stochastic evolution equation (1.1), we suppose that the control
domain is convex. We adopt the convex variation method and the first adjoint
duality analysis to show a necessary maximum principle where the continuous
dependence theorem (see Theorem 3.2) plays a key role in proving the variation
inequality for the cost functional (see Lemma 6.2). Under the convexity
assumption of the Hamiltonian and the terminal cost, we provide a sufficient
maximum principle for this optimal problem which is the so-called verification
theorem. It is worth mentioning that if the admissible control set is non-
convex and the diffusion terms of the state equation is independent of the
control variable we can use the first-order spike variation method to obtain
the maximum principle in the global form by establishing some subtle
$L^{2}$-estimate for the state equation. All the details shall be given in our
forthcoming paper. But for the general setting, it seems very difficult or
even impossible to obtain the corresponding maximum principle because it seems
impossible to establish some $L^{p}$ $(p>2)$ estimates as in [7] for the state
process which play a key role in the second variation analysis. Finally, to
illustrate our results, we apply the stochastic maximum principles to solve an
optimal control of a Cauchy problem for a controlled stochastic linear partial
differential equation.

The rest of this paper is structured as follows. In Section 2, we provide the
basic notations and recall Itô formula for Teugels martingales in Hilbert
space used frequently in this paper. Section 3 establishes the continuous
dependence and the existence and uniqueness of solutions to the stochastic
evolution equation (1.1). Section 4 formulates the optimal control problem
specifying the hypotheses. In section 5, adjoint equation is introduced which
turns out to be a backward stochastic evolution equation driven by Teugels
martingales. In Section 6, we establish the stochastic maximum principle by
the classical convex variation method. In Sections 7, the verification theorem
for optimal controls is obtained by dual technique. In section 8, we present
an application of our results. The final section concludes the paper.

##  2 Notations and Itô Formula for Teugels Martingales in Hilbert Space

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 one-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

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

where $a\in\mathbb{R}^{1}$, $\sigma>0$ and $v$ is a measure on
$\mathbb{R}^{1}$ satisfying (i)$\displaystyle\int_{-T}^{T}(1\wedge
x^{2})v(dx)<\infty$ 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.

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

$\bullet$ $(\cdot,\cdot)_{X}:$ the inner product in Hilbert space $X.$

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

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

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

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

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

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

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

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

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

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

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

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

We denote by $\\{H^{i}(t),0\leq t\leq T\\}_{i=1}^{\infty}$ the Teugels
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 $c_{ij}$ corresponding 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 Teugels 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 Teugels martingales, we invite the reader to consult
Nualart and Schoutens [16].

Let $V$ and $H$ be two separable (real) Hilbert spaces such that $V$ is
densely embedded in $H$. We identify $H$ with its dual space by the Riesz
mapping. Then we can take $H$ as a pivot space and get a Gelfand triple
$V\subset H=H^{*}\subset V^{*},$ where $H^{*}$ and $V^{*}$ denote the dual
spaces of $H$ and $V$, respectively. Denote by $\|\cdot\|_{V},\|\cdot\|_{H}$
and $\|\cdot\|_{V^{*}}$ the norms of $V,H$ and $V^{*}$, respectively, by
$(\cdot,\cdot)_{H}$ the inner product in $H$, by $\langle\cdot,\cdot\rangle$
the duality product between $V$ and $V^{*}$. Moreover we write
$\mathscr{L}(V,V^{*})$ the space of bounded linear transformations of V into
$V^{*}$. Throughout this paper, we let $C$ and $K$ be two generic positive
constants, which may be different from line to line.

Now we present an Itô’s formula in Hilbert space which will be frequently used
in this paper.

######  Lemma 2.1.

Let $\varphi\in L^{2}(\Omega,\mathscr{F}_{0},P;H)$. Let $Y,Z,\Gamma$ and
$R\equiv(R^{i})_{i=1}^{\infty}$ be three progressively measurable stochastic
processes defined on $[0,T]\times\Omega$ with values in $V,H$ and $V^{*}$ such
that $Y\in{M}_{\mathscr{F}}^{2}(0,T;V),Z\in{M}_{\mathscr{F}}^{2}(0,T;H),$
$\Gamma\in{M}_{\mathscr{F}}^{2}(0,T;V^{*})$ and $R\in
l_{\mathscr{F}}^{2}(0,T;H)$, respectively. Suppose that for every $\eta\in V$
and almost every $(\omega,t)\in\Omega\times[0,T]$, it holds that

$(\eta,Y(t))_{H}=(\eta,\varphi)_{H}+\int_{0}^{t}\langle\eta,\Gamma(s)\rangle
ds+\int_{0}^{t}(\eta,Z(s))_{H}dW(s)+\sum_{i=1}^{\infty}\int_{0}^{t}(\eta,R^{i}(s))_{H}dH^{i}(s).$

Then $Y$ is a $H$-valued strongly càdlàg $\mathscr{F}_{t}$-adapted process
such that the following Itô formula holds

$\displaystyle\begin{split}||Y(t)||_{H}^{2}=&||\varphi||^{2}+2\int_{0}^{t}\langle\Gamma(s),Y(s)\rangle
ds+2\int_{0}^{t}(Y(s),Z(s))_{H}dW(s)+\int_{0}^{t}||Z(s)||_{H}^{2}ds\\\
&+2\sum_{i=1}^{\infty}\int_{0}^{t}(Y(s),R^{i}(s))_{H}dH^{i}(s)+\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}\int_{0}^{t}(R^{i}(s),R^{j}(s))_{H}d[H^{i}(s),H^{j}(s)].\end{split}$
(2.1)

###### Proof.

The proof follows that of Theorem 1 in Gyöngy and Krylov [13]. ∎

##  3 Stochastic Evolution Equation driven by Teugels Martingales

In this section, we present some preliminary results of the following
stochastic evolution equation (SEE for short) driven by Brownian motion
$\\{W(t),0\leq t\leq T\\}$ and Teugels Martingales $\\{H^{i}(t),0\leq t\leq
T\\}_{i=1}^{\infty}$:

$\displaystyle\left\\{\begin{aligned} dX(t)=&\
[A(t)X(t)+b(t,X(t))]dt+[B(t)X(t)+g(t,{X(t)})]dW(t)+\sum_{i=1}^{\infty}\sigma^{i}(s,X(s-))dH^{i}(s),\\\
X(0)=&\ x\in H,\quad t\in[0,T],\end{aligned}\right.$ (3.1)

where $A,B,b,g$ and $\sigma\equiv{(\sigma^{i})}_{i=1}^{\infty}$ are given
random mappings which satisfy the following standard assumptions.

######  Assumption 3.1.

The operator processes
$A:[0,T]\times\Omega\longrightarrow{\mathscr{L}}(V,V^{*})$ and
$B:[0,T]\times\Omega\longrightarrow{\mathscr{L}}(V,H)$ are weakly predictable;
i.e., $\langle A(\cdot)x,y\rangle$ and $(B(\cdot)x,y)_{H}$ are both
predictable process for every $x,y\in V,$ and satisfy the coercive condition,
i.e., there exist some constants $C,\alpha>0$ and $\lambda$ such that for any
$x\in V$ and each $(t,\omega)\in[0,T]\times\Omega,$

$\displaystyle\begin{split}-\langle
A(t)x,x\rangle+\lambda||x||_{H}^{2}\geq\alpha||x||_{V}^{2}+||Bx||_{H}^{2},\end{split}$
(3.2)

and

$\displaystyle\sup_{(t,\omega)\in[0,T]\times\Omega}\|A(t,\omega)\|_{{\mathscr{L}}(V,V^{*})}+\sup_{(t,\omega)\in[0,T]\times\Omega}\|B(t,\omega)\|_{{\mathscr{L}}(V,H)}\leq
C\ .$ (3.3)

######  Assumption 3.2.

The mappings $b:[0,T]\times\Omega\times H\longrightarrow H$ and
$g:[0,T]\times\Omega\times H\longrightarrow H$ are both
$\mathscr{P}\times\mathscr{B}(H)/\mathscr{B}(H)$-measurable such that
$b(\cdot,0),g(\cdot,0)\in M_{\mathscr{F}}^{2}(0,T;H)$; the mapping
$\sigma:[0,T]\times\Omega\times H\longrightarrow l^{2}(H)$ is
$\mathscr{P}\times\mathscr{B}(H)/\mathscr{B}(l^{2}(H))$-measurable such that
$\sigma(\cdot,0)\in l_{\mathscr{F}}^{2}(0,T,H)$. And there exists a constant
$C$ such that for all $x,\bar{x}\in V$ and
a.s.$(t,\omega)\in[0,T]\times\Omega,$

$\displaystyle\begin{split}||b(t,x)-b(t,x)||_{H}+||g(t,x)-g(t,x)||_{H}+||\sigma(t,x)-\sigma(t,x)||_{l^{2}(H)}\leq
C||x-\bar{x}||_{H}.\end{split}$ (3.4)

######  Definition 3.1.

A $V$-valued, $\\{{\mathscr{F}}_{t}\\}_{0\leq t\leq T}$-adapted process
$X(\cdot)$ is said to be a solution to the SEE (3.1), if
$X(\cdot)\in{M}_{\mathscr{F}}^{2}(0,T;V)$ such that for every $\phi\in V$ and
a.e. $(t,\omega)\in[0,T]\times\Omega$, it holds that

$\displaystyle\left\\{\begin{aligned} (X(t),\phi)_{H}=&\
(x,\phi)_{H}+\int_{0}^{t}\left<A(s)X(s),\phi\right>ds+\int_{0}^{t}(b(s,X(s)),\phi)_{H}ds+\int_{0}^{t}(B(s)X(s)+g(s,X(s)),\phi)_{H}dW(s)\\\
&+\sum_{i=1}^{\infty}\int_{0}^{t}(\sigma^{i}(s,X(s-)),\phi)_{H}dH^{i}(s),\quad
t\in[0,T],\\\ X(0)=&\ x\in H,\end{aligned}\right.$ (3.5)

or alternatively, $X(\cdot)$ satisfies the following Itô’s equation in
$V^{*}$:

$\displaystyle\left\\{\begin{aligned} X(t)=&\
x+\int_{0}^{t}A(s)X(s)ds+\int_{0}^{t}b(s,X(s))ds+\int_{0}^{t}[B(s)X(s)+g(s,X(s))]dW(s)+\sum_{i=1}^{\infty}\int_{0}^{t}\sigma^{i}(s,X(s-))dH^{i}(s),\\\
X(t)=&\ x\in H.\end{aligned}\right.$ (3.6)

Now we state our main result.

######  Theorem 3.1.

Let Assumptions 3.1-3.2 be satisfied by any given coefficients
$(A,B,b,g,\sigma)$ of the SEE (3.1). Then for any initial value $X(0)=x,$ the
SEE (3.1) has a unique solution $X(\cdot)\in M_{\mathscr{F}}^{2}(0,T;V).$

To prove this theorem, we first show the following result on the continuous
dependence of the solution to the SEE (3.1).

######  Theorem 3.2.

Let $X(\cdot)$ be a solution to the SEE (3.1) with the initial value $X(0)=x$
and the coefficients $(A,B,b,g,\sigma)$ which satisfy Assumptions 3.1-3.2.
Then the following estimate holds:

$\displaystyle\begin{split}\sup_{0\leq t\leq
T}\mathbb{E}[||\hat{X}(t)||_{H}^{2}]+{\mathbb{E}}\bigg{[}\int_{0}^{T}\|X(t)\|_{V}^{2}dt\bigg{]}\leq
K&\bigg{\\{}||x||_{H}^{2}+{\mathbb{E}}\bigg{[}\int_{0}^{T}\|b(t,0)\|_{H}^{2}dt\bigg{]}+{\mathbb{E}}\bigg{[}\int_{0}^{T}\|g(t,0)\|_{H}^{2}dt\bigg{]}\\\
&+{\mathbb{E}}\bigg{[}\int_{0}^{T}\|\sigma(t,0)\|^{2}_{l^{2}(H)}dt\bigg{]}\bigg{\\}}.\end{split}$
(3.7)

Furthermore, suppose that $\bar{X}(\cdot)$ is a solution to the SEE (3.1) with
the initial value $\bar{X}(0)=\bar{x}\in H$ and the coefficients
$(A,B,\bar{b},\bar{g},\bar{\sigma})$ satisfying Assumptions 3.1-3.2, then we
have

$\displaystyle\sup_{0\leq t\leq
T}\mathbb{E}[||X(t)-\bar{X}(t)||_{H}^{2}]+{\mathbb{E}}\bigg{[}\int_{0}^{T}\|X(t)-{\bar{X}}(t)\|_{V}^{2}dt\bigg{]}$
$\displaystyle\leq
K\bigg{\\{}\|x-\bar{x}\|^{2}_{H}+{\mathbb{E}}\bigg{[}\int_{0}^{T}\|b(t,{\bar{X}}(t))-{\bar{b}}(t,{\bar{X}}(t))\|_{H}^{2}dt\bigg{]}+{\mathbb{E}}\bigg{[}\int_{0}^{T}\|g(t,{\bar{X}}(t))-{\bar{g}}(t,{\bar{X}}(t))\|_{H}^{2}dt\bigg{]}$
$\displaystyle~{}~{}~{}~{}~{}~{}+{\mathbb{E}}\bigg{[}\int_{0}^{T}\|\sigma(t,{\bar{X}}(t))-{\bar{\sigma}}(t,{\bar{X}}(t))\|_{l^{2}(H)}^{2}dt\bigg{]}\bigg{\\}}.$

###### Proof.

The estimate (3.7) can be directly obtained by the estimate (3.2) by taking
the initial value $\bar{X}(0)=0$ and the coefficients
$(A,B,\bar{b},\bar{g},\bar{\sigma})=(A,B,0,0,0)$ which imply that
$\bar{X}(\cdot)\equiv 0.$ Therefore, it suffices to prove the estimate (3.2).
For the sake of simplicity, in the following discussion, we will use the
following shorthand notation:

$\displaystyle\begin{split}&{\hat{X}}(t)\triangleq
X(t)-{\bar{X}}(t),\quad\hat{x}\triangleq x-\bar{x},\\\
&\Lambda\triangleq\|x-\bar{x}\|^{2}_{H}+{\mathbb{E}}\bigg{[}\int_{0}^{T}\|b(t,{\bar{X}}(t))-{\bar{b}}(t,{\bar{X}}(t))\|_{H}^{2}dt\bigg{]}+{\mathbb{E}}\bigg{[}\int_{0}^{T}\|g(t,{\bar{X}}(t))-{\bar{g}}(t,{\bar{X}}(t))\|_{H}^{2}dt\bigg{]}\\\
&\quad\quad\quad+{\mathbb{E}}\bigg{[}\int_{0}^{T}\|\sigma(t,{\bar{X}}(t))-{\bar{\sigma}}(t,{\bar{X}}(t))\|_{l^{2}(H)}^{2}dt\bigg{]},\end{split}$
(3.9)

and for $\phi=b,g,\sigma$

$\displaystyle\begin{split}&{\tilde{\phi}}(t)\triangleq\phi(t,{X}(t))-{\bar{\phi}}(t,{\bar{X}}(t)),{\hat{\phi}}(t)\triangleq\phi(t,{\bar{X}}(t))-{\bar{\phi}}(t,{\bar{X}}(t)),{\Delta\phi}(t)\triangleq\phi(t,{X}(t))-{\phi}(t,{\bar{X}}(t)),~{}~{}t\in[0,T],\end{split}$
(3.10)

where when $\phi=\sigma,$ the terms $X(t)$ and $\bar{X}(t)$ will be replaced
by $X(t-)$ and $\bar{X}(t-),$ respectively.

Applying Itô formula in Lemma 2.1 to $||\hat{X}(t)||_{H}^{2}$ and using
Assumptions 3.1-3.2 and the elementary inequalities $|a+b|^{2}\leq
2a^{2}+2b^{2}$ and $2ab\leq a^{2}+b^{2}$, $\forall a,b>0$, we get that

$\displaystyle||\hat{X}(t)||_{H}^{2}=$
$\displaystyle||\hat{x}||^{2}_{H}+2\int_{0}^{t}\langle
A(s)\hat{X}(s),\hat{X}(s)\rangle
ds+2\int_{0}^{t}(\hat{X}(s),\tilde{b}(s))_{H}ds+\int_{0}^{t}||B(s)\hat{X}(s)+\tilde{g}(s)||_{H}^{2}ds$
$\displaystyle+\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}\int_{0}^{t}(\tilde{\sigma}^{i}(s),\tilde{\sigma}^{j}(s))_{H}d[H^{i}(s),H^{j}(s)]+2\int_{0}^{t}(\hat{X}(s),B(s)\hat{X}(s)+\tilde{g}(s))_{H}dW(s)$
$\displaystyle+2\sum_{i=1}^{\infty}\int_{0}^{t}(\hat{X}(s),\tilde{\sigma}^{i}(s))dH^{i}(s)$
$\displaystyle=$
$\displaystyle||\hat{x}||^{2}_{H}+2\int_{0}^{t}\bigg{[}\langle
A(s)\hat{X}(s),\hat{X}(s)\rangle+||B(s)\hat{X}(s)||_{H}^{2}\bigg{]}ds+\int_{0}^{t}||\Delta
g(s)+\hat{g}(s)||_{H}^{2}ds$
$\displaystyle+2\int_{0}^{t}(B(s)\hat{X}(s),\Delta
g(s)+\hat{g}(s))_{H}ds+2\int_{0}^{t}(\hat{X}(s),\Delta b(s)+\hat{b}(s))_{H}ds$
$\displaystyle+\int_{0}^{t}||\Delta\sigma(s)+\hat{\sigma}(s)||_{l^{2}(H)}ds+\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}\int_{0}^{t}(\tilde{\sigma}^{i}(s),\tilde{\sigma}^{j}(s))_{H}d\\{[H^{i}(s),H^{j}(s)]-\langle
H^{i}(s),H^{j}(s)\rangle\\}$
$\displaystyle+2\int_{0}^{t}(\hat{X}(s),B(s)\hat{X}(s)+\tilde{g}(s))_{H}dW(s)+2\sum_{i=1}^{\infty}\int_{0}^{t}(\hat{X}(s),\tilde{\sigma}^{i}(s))dH^{i}(s)$
$\displaystyle\leq$ $\displaystyle
K\Lambda-\alpha\int_{0}^{t}\|\hat{X}(s)\|_{V}^{2}ds+K\int_{0}^{t}\|\hat{X}(s)\|_{H}^{2}dt$
$\displaystyle+\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}\int_{0}^{t}(\tilde{\sigma}^{i}(s),\tilde{\sigma}^{j}(s))_{H}d\\{[H^{i}(s),H^{j}(s)]-\langle
H^{i}(s),H^{j}(s)\rangle\\}$
$\displaystyle+2\int_{0}^{t}(\hat{X}(s),B(s)\hat{X}(s)+\tilde{g}(s))_{H}dW(s)+2\sum_{i=1}^{\infty}\int_{0}^{t}(\hat{X}(s),\tilde{\sigma}^{i}(s))dH^{i}(s)$

Taking expectation on both sides of the above inequality, we get that

$\displaystyle\begin{split}\mathbb{E}[||\hat{X}(t)||_{H}^{2}]+\alpha\mathbb{E}\bigg{[}\int_{0}^{t}||\hat{X}(s)||_{V}^{2}ds\bigg{]}\leq
K\Lambda+K{\mathbb{E}}\bigg{[}\int_{0}^{t}\|\hat{X}(s)\|_{H}^{2}dt\bigg{]}.\end{split}$
(3.12)

Then by virtue of Grönwall’s inequality to $\mathbb{E}[||X(t)||_{H}^{2}],$ we
obtain

$\displaystyle\begin{split}\sup_{0\leq t\leq
T}\mathbb{E}[||\hat{X}(t)||_{H}^{2}]+\mathbb{E}\bigg{[}\int_{0}^{T}||\hat{X}(s)||_{V}^{2}ds\bigg{]}\leq&K\Lambda.\end{split}$
(3.13)

The proof is complete. ∎

In the following, we give the existence and uniqueness result for the solution
of the SEE (3.1) for a simple case where the coefficients $(b,g,\sigma)$ is
independent of the variable $x.$

######  Lemma 3.3.

Given three stochastic processes $b,g$ and $\sigma$ such that $b(\cdot)\in
M_{\mathscr{F}}^{2}(0,T;H),g(\cdot)\in M_{\mathscr{F}}^{2}(0,T;H)$ and
$\sigma(\cdot)\in M_{\mathscr{F}}^{2}(0,T;l^{2}(H)).$ Suppose that the
operators $A$ and $B$ satisfy Assumption 3.1. Then there exists a unique
solution $X(\cdot)\in M_{\mathscr{F}}^{2}(0,T;V)$ to the following SEE:

$\displaystyle\left\\{\begin{aligned} dX(t)=&\
[A(t)X(t)+b(t)]dt+[B(t)X(t)+g(t)]dW(t)+\sum_{i=1}^{\infty}\sigma^{i}(t)dH^{i}(t),\\\
X(0)=&\ x,\quad t\in[0,T].\end{aligned}\right.$ (3.14)

###### Proof.

The proof can be obtained by Galerkin approximations in the same way as the
proof of Theorem 3.2 in [4] with minor change. Now we begin our proof. First
of all, we fix a standard complete orthogonal basis
$\\{e_{i}|i=1,2,3,\dots\\}$ in the space $H$ which is dense in the space $V$.
For any $n,$ consider the following finite-dimensional stochastic differential
equation in $\mathbb{R}^{n}:$

$\displaystyle\left\\{\begin{aligned} x_{1}^{n}(t)=&\
(x,e_{1})_{H}+\int_{0}^{t}\bigg{[}\sum_{j=1}^{n}x_{j}^{n}(s)\langle
A(s)e_{j},e_{1}\rangle+(b(s),e_{1})_{H}\bigg{]}ds+\int_{0}^{t}\bigg{[}\sum_{j=1}^{n}x_{j}^{n}(s)(B(s)e_{j},e_{1})_{H}+(g(s),e_{1})_{H}\bigg{]}dW(s)\\\
&+\sum_{i=1}^{\infty}\int_{0}^{t}(\sigma^{i}(s),e_{1})_{H}dH^{i}(s),\\\
x_{2}^{n}(t)=&\
(x,e_{2})_{H}+\int_{0}^{t}\bigg{[}\sum_{j=1}^{n}x_{j}^{n}(s)\langle
A(s)e_{j},e_{1}\rangle+(b(s),e_{2})_{H}\bigg{]}ds+\int_{0}^{t}\bigg{[}\sum_{j=1}^{n}x_{j}^{n}(s)(B(s)e_{j},e_{2})_{H}+(g(s),e_{2})_{H}\bigg{]}dW(s)\\\
&+\sum_{i=1}^{\infty}\int_{0}^{t}(\sigma^{i}(s),e_{2})_{H}dH^{i}(s),\\\
&\vdots\\\ x_{n}^{n}(t)=&\
(x,e_{n})_{H}+\int_{0}^{t}\bigg{[}\sum_{j=1}^{n}x_{j}^{n}(s)\langle
A(s)e_{j},e_{n}\rangle+(b(s),e_{n})_{H}\bigg{]}ds+\int_{0}^{t}\bigg{[}\sum_{j=1}^{n}x_{j}^{n}(s)(B(s)e_{j},e_{n})_{H}+(g(s),e_{n})_{H}\bigg{]}dW(s)\\\
&+\sum_{i=1}^{\infty}\int_{0}^{t}(\sigma^{i}(s),e_{n})_{H}dH^{i}(s),\end{aligned}\right.$
(3.15)

Under Assumptions 3.1 and 3.2, from the existence and uniqueness theory for
the finite dimensional SDE driven by Teugels martingale , the above equation
admits a unique strong solution
$x^{n}(\cdot)\in{M}_{\mathscr{F}}^{2}(0,T;\mathbb{R}^{n})$ , where
$x^{n}(\cdot)=(x_{1}^{n}(\cdot),\cdots,x_{n}^{n}(\cdot)).$

Now we can define an approximation solution to (3.14) as follows:

$\displaystyle X^{n}(t):=\sum_{i=1}^{n}x_{i}^{n}(t)e_{i},$

where

$\displaystyle X^{n}(0):=\sum_{i=1}^{n}(x,e_{i})_{H}e_{i}.$

Then the equation (3.15) can be written as

$\displaystyle(X^{n}(t),e_{i})_{H}$ $\displaystyle=$
$\displaystyle\left(X^{n}(0),e_{i}\right)_{H}+\int_{0}^{t}\bigg{[}\left\langle
A(s)X^{n}(s),e_{i}\right\rangle+(b(s),e_{i})_{H}\bigg{]}ds$ (3.16)
$\displaystyle+\int_{0}^{t}\bigg{[}(B(s)X^{n}(s),e_{i})+\left(g(s),e_{i}\right)_{H}\bigg{]}dW(s)+\sum_{i=1}^{\infty}\int_{0}^{t}\int_{0}^{t}(\sigma^{i}(s),e_{n})_{H}dH^{i}(s),\quad
i=1,\cdots,n.$

Now applying Itô formula to $||X^{n}(t)||^{2}_{H}$, we get that

$\displaystyle||X^{n}(t)||_{H}^{2}=$
$\displaystyle||\hat{X}^{n}(0)||^{2}_{H}+2\int_{0}^{t}\langle
A(s)X^{n}(s),X^{n}(s)\rangle
ds+2\int_{0}^{t}(X^{n}(s),b(s))_{H}ds+\int_{0}^{t}||B(s)X^{n}(s)+g(s)||_{H}^{2}ds$
$\displaystyle+\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}\int_{0}^{t}(\sigma^{i}(s),\sigma^{j}(s))_{H}d[H^{i}(s),H^{j}(s)]+2\int_{0}^{t}(X^{n}(s),B(s)X^{n}(s)+g(s))_{H}dW(s)$
$\displaystyle+2\sum_{i=1}^{\infty}\int_{0}^{t}(X^{n}(s),\sigma^{i}(s))dH^{i}(s).$

Therefore, under Assumptions 3.1 and 3.2, similar to the proof of the estimate
(3.7), using Grönwall’s inequality, we can easily get the following estimate:

$\displaystyle\begin{split}&{\mathbb{E}}\bigg{[}\int_{0}^{T}\|X^{n}(t)\|_{V}^{2}dt\bigg{]}\leq
K\bigg{\\{}||x||_{H}^{2}+{\mathbb{E}}\bigg{[}\int_{0}^{T}\|b(t)\|_{H}^{2}dt\bigg{]}+{\mathbb{E}}\bigg{[}\int_{0}^{T}\|g(t)\|_{H}^{2}dt\bigg{]}+{\mathbb{E}}\bigg{[}\int_{0}^{T}\|\sigma(t)\|^{2}_{l^{2}(H)}dt\bigg{]}\bigg{\\}}.\end{split}$
(3.17)

This inequality implies that there is a subsequence $\\{n^{\prime}\\}$ of
$\left\\{n\right\\}$ and a triple $X(\cdot)\in{M}_{\mathscr{F}}^{2}(0,T;V)$
such that

$\displaystyle X^{n^{\prime}}\rightarrow X\text{ weakly in
}{M}_{\mathscr{F}}^{2}(0,T;V).$ (3.18)

Let $\Pi$ be an arbitrary bounded random variable on $(\Omega,\mathscr{F})$
and $\psi$ be an arbitrary bounded measurable function on $[0,T]$. From the
equality (3.16), for $n^{\prime}\in\mathbb{N}^{*}$ and basis $e_{i}$, where
$i\leq n^{\prime}$, we have

$\displaystyle\mathbb{E}\bigg{[}\int_{0}^{T}\Pi\psi(t)(X^{n^{\prime}}(t),e_{i})_{H}dt\bigg{]}$
$\displaystyle=\mathbb{E}\bigg{[}\int_{0}^{T}\Pi\psi(t)\bigg{\\{}\left(X^{n^{\prime}}(0),e_{i}\right)_{H}+\int_{0}^{t}\bigg{[}\left\langle
A(s)X^{n^{\prime}}(s),e_{i}\right\rangle+(b(s),e_{i})_{H}\bigg{]}ds$
$\displaystyle+\int_{0}^{t}\bigg{[}(B(s)X^{n^{\prime}}(s),e_{i})+\left(g(s),e_{i}\right)_{H}\bigg{]}dW(s)+\sum_{i=1}^{\infty}\int_{0}^{t}\int_{0}^{t}(\sigma^{i}(s),e_{i})_{H}dH^{i}(s)\bigg{\\}}dt\bigg{]}.$
(3.19)

Now letting $n^{\prime}\longrightarrow\infty$ on the both sides of the above
equation to get its limit. Firstly, from the weak convergence property of
$\\{X^{n}\\}_{n=1}^{\infty}$ in ${\cal M}_{\cal F}^{2}(0,T;V)$, we have

$\displaystyle\lim_{n^{\prime}\rightarrow\infty}\mathbb{E}\bigg{[}\int_{0}^{T}\Pi\psi(t)(X^{n^{\prime}}(t),e_{i})_{H}dt\bigg{]}$
$\displaystyle=$
$\displaystyle\lim_{n^{\prime}\rightarrow\infty}\mathbb{E}\bigg{[}\int_{0}^{T}\mathbb{E}[\Pi|\mathscr{F}_{t}]\psi(t)(X^{n^{\prime}}(t),e_{i})_{H}dt\bigg{]}$
(3.20) $\displaystyle=$
$\displaystyle\lim_{n^{\prime}\rightarrow\infty}\mathbb{E}\bigg{[}\int_{0}^{T}(X^{n^{\prime}}(t),\mathbb{E}[\Pi|\mathscr{F}_{t}]\psi(t)e_{i})_{H}dt\bigg{]}$
$\displaystyle=$
$\displaystyle\mathbb{E}\bigg{[}\int_{0}^{T}(X(t),\mathbb{E}[\Pi|\mathscr{F}_{t}]\psi(t)e_{i})dt\bigg{]}$
$\displaystyle=$
$\displaystyle\mathbb{E}\bigg{[}\int_{0}^{T}\Pi\psi(t)(X(t),e_{i})dt\bigg{]},$

and

$\displaystyle\lim_{n^{\prime}\rightarrow\infty}\mathbb{E}\bigg{[}\int_{0}^{t}\Pi\langle
A(s)X^{n^{\prime}}(s),e_{i}\rangle ds\bigg{]}$ $\displaystyle=$
$\displaystyle\lim_{n^{\prime}\rightarrow\infty}\mathbb{E}\bigg{[}\int_{0}^{t}\mathbb{E}[\Pi|\mathscr{F}_{s}]\langle
A(s)X^{n^{\prime}}(s),e_{i}\rangle ds\bigg{]}$ (3.21) $\displaystyle=$
$\displaystyle\lim_{n^{\prime}\rightarrow\infty}\mathbb{E}\bigg{[}\int_{0}^{t}\langle
A(s)X^{n^{\prime}}(s),\mathbb{E}[\Pi|\mathscr{F}_{s}]e_{i}\rangle ds\bigg{]}$
$\displaystyle=$
$\displaystyle\lim_{n^{\prime}\rightarrow\infty}\mathbb{E}\bigg{[}\int_{0}^{t}\langle
X^{n^{\prime}}(s),A^{*}(s)\mathbb{E}[\Pi|\mathscr{F}_{s}]e_{i}\rangle
ds\bigg{]}$ $\displaystyle=$
$\displaystyle\mathbb{E}\bigg{[}\int_{0}^{t}\langle
X(s),A^{*}(s)\mathbb{E}[\Pi|\mathscr{F}_{s}]e_{i}\rangle ds\bigg{]}$
$\displaystyle=$ $\displaystyle\mathbb{E}\bigg{[}\int_{0}^{t}\Pi\langle
A(s)X(s),e_{i}\rangle ds\bigg{]}.$

In view of (3.3) and (3.17), we conclude that

$\displaystyle\mathbb{E}\bigg{[}\bigg{|}\int_{0}^{t}\Pi\langle
A(s)X^{n^{\prime}}(s),e_{i}\rangle ds\bigg{|}\bigg{]}\leq
C\bigg{\\{}\mathbb{E}\bigg{[}\int_{0}^{T}||X^{n^{\prime}}(s)||_{V}^{2}ds\bigg{]}\bigg{\\}}^{\frac{1}{2}}<C<\infty,$

where the constant $C$ is independent of $n^{\prime}$. Hence from Fubini’s
Theorem and Lebesgue’s Dominated Convergence Theorem, we have

$\displaystyle\lim_{n^{\prime}\longrightarrow\infty}\mathbb{E}\bigg{[}\int_{0}^{T}\Pi\psi(t)\int_{0}^{t}\langle
A(s)X^{n^{\prime}}(s),e_{i}\rangle dsdt\bigg{]}=$
$\displaystyle\lim_{n^{\prime}\longrightarrow\infty}\int_{0}^{T}\psi(t)\mathbb{E}\bigg{[}\int_{0}^{t}\Pi\langle
A(s)X^{n^{\prime}}(s),e_{i}\rangle ds\bigg{]}dt$ $\displaystyle=$
$\displaystyle\int_{0}^{T}\psi(t)\mathbb{E}\bigg{[}\int_{0}^{t}\Pi\langle
A(s)X(s),e_{i}\rangle ds\bigg{]}dt$ $\displaystyle=$
$\displaystyle\mathbb{E}\bigg{[}\int_{0}^{T}\psi(t)\int_{0}^{t}\Pi\langle
A(s)X(s),e_{i}\rangle ds\bigg{]}dt.$ (3.22)

Similarly, in view of (3.3), from (3.17) and (3.18), it is easy to check that

$(BX^{n^{\prime}},e_{i})_{H}\rightarrow(BX,e_{i})_{H}\text{ weakly in
}{M}_{\mathscr{F}}^{2}(0,t;\mathbb{R}).$

Since the stochastic integral with respect to the Brownian motion $W$ are
linear and strong continuous mappings from
${M}_{\mathscr{F}}^{2}(0,t;\mathbb{R})$ to
$L^{2}(\Omega,{\mathscr{F}}_{T},P;\mathbb{R})$, it is weakly continuous.
Therefore,

$\displaystyle\lim_{n^{\prime}\rightarrow\infty}\mathbb{E}\bigg{[}\Pi\int_{0}^{t}\left(B(s)X^{n^{\prime}}(s),e_{i}\right)_{H}dW(s)\bigg{]}=\mathbb{E}\bigg{[}\Pi\int_{0}^{t}\left(B(s)X(s),e_{i}\right)_{H}dW(s)\bigg{]}.$
(3.23)

Moreover, from (3.17), we get

$\displaystyle\psi(t)\mathbb{E}\bigg{[}\Pi\bigg{(}\int_{0}^{t}\left(B(s)X^{n^{\prime}}(s),e_{i}\right)_{H}dW(s)\bigg{]}\leq\frac{1}{2}\psi^{2}(t)\mathbb{E}|\Pi|^{2}+C\bigg{\\{}\mathbb{E}\bigg{[}\int_{0}^{T}||B(s)X^{n^{\prime}}(s)||^{2}_{H}dt\bigg{]}\bigg{\\}}\leq
C.$

Hence, by Fubini’s Theorem and Lebesgue’s Dominated Convergence Theorem, we
have

$\displaystyle\lim_{n^{\prime}\rightarrow\infty}\mathbb{E}\bigg{[}\int_{0}^{T}\psi(t)\Pi\bigg{(}\int_{0}^{t}\left(B(s)X^{n^{\prime}}(s),e_{i}\right)_{H}dW(s)\bigg{]}dt$
$\displaystyle=\lim_{n^{\prime}\rightarrow\infty}\int_{0}^{T}\psi(t)\mathbb{E}\bigg{[}\Pi\bigg{(}\int_{0}^{t}\left(B(s)X^{n^{\prime}}(s),e_{i}\right)_{H}dW(s)\bigg{]}dt$
$\displaystyle=\int_{t}^{T}\psi(t)\mathbb{E}\bigg{[}\Pi\bigg{(}\int_{0}^{t}\left(B(s)X(s),e_{i}\right)_{H}dW(s)\bigg{)}\bigg{]}dt$
(3.24)
$\displaystyle=\mathbb{E}\int_{t}^{T}\bigg{[}\psi(t)\Pi\bigg{(}\int_{0}^{t}\left(B(s)X(s),e_{i}\right)_{H}dW(s)\bigg{)}\bigg{]}dt.$
(3.25)

Therefore, combining (3.20),(3), (3.23) and (3), and letting
$n^{\prime}\rightarrow\infty$ in (3), we can conclude that

$\displaystyle\mathbb{E}\bigg{[}\int_{0}^{T}\Pi\psi(t)(X(t),e_{i})_{H}dt\bigg{]}$
$\displaystyle=\mathbb{E}\bigg{[}\int_{0}^{T}\Pi\psi(t)\bigg{\\{}\left(X(0),e_{i}\right)_{H}+\int_{0}^{t}\bigg{[}\left\langle
A(s)X(s),e_{i}\right\rangle+(b(s),e_{i})_{H}\bigg{]}ds$
$\displaystyle+\int_{0}^{t}\bigg{[}(B(s)X(s),e_{i})+\left(g(s),e_{i}\right)_{H}\bigg{]}dW(s)+\sum_{i=1}^{\infty}\int_{0}^{t}\int_{0}^{t}(\sigma^{i}(s),e_{i})_{H}dH^{i}(s)\bigg{\\}}dt\bigg{]}.$
(3.26)

This implies that for a.s. $\left(t,\omega\right)\in[0,T]\times\Omega$,

$\displaystyle(X(t),e_{i})_{H}$ $\displaystyle=$
$\displaystyle\left(X(0),e_{i}\right)_{H}+\int_{0}^{t}\bigg{[}\left\langle
A(s)X(s),e_{i}\right\rangle+(b(s),e_{i})_{H}\bigg{]}ds$ (3.27)
$\displaystyle+\int_{0}^{t}\bigg{[}(B(s)X(s),e_{i})+\left(g(s),e_{i}\right)_{H}\bigg{]}dW(s)+\sum_{i=1}^{\infty}\int_{0}^{t}\int_{0}^{t}(\sigma^{j}(s),e_{i})_{H}dH^{j}(s),$

thanks to the arbitrariness of $\Pi$ and $\psi(\cdot)$. Since the standard
complete orthogonal basis $\\{e_{i}|i=1,2,3,\dots\\}$ in $H$ is dense in the
space $V$, for every $\phi\in V$ and a.e. $(t,\omega)\in[0,T]\times\Omega$, it
holds that

$\displaystyle(X(t),\phi)_{H}$ $\displaystyle=$
$\displaystyle\left(X(0),\phi\right)_{H}+\int_{0}^{t}\bigg{[}\left\langle
A(s)X(s),\phi\right\rangle+(b(s),\phi)_{H}\bigg{]}ds$ (3.28)
$\displaystyle+\int_{0}^{t}\bigg{[}(B(s)X(s),\phi)+\left(g(s),\phi\right)_{H}\bigg{]}dW(s)+\sum_{i=1}^{\infty}\int_{0}^{t}\int_{0}^{t}(\sigma^{j}(s),\phi)_{H}dH^{j}(s).$

Therefore, from the Definition 3.1, we conclude that the triple $X(\cdot)$ is
the solution to the SEE (3.14). Thus the existence is proved. For the
uniqueness, the proof can be directly by the priori estimate (3.2). The proof
is complete. ∎

_Proof of Theorem 3.1._ The uniqueness of the solution to the SEE (3.1) can be
got by the a priori estimate (3.2) directly. For $\rho\in[0,1]$ and any three
given stochastic processes $b_{0}\in{M}_{\mathscr{F}}^{2}(0,T;H),$
$g_{0}\in{M}_{\mathscr{F}}^{2}(0,T;H),$ and
$\sigma_{0}\in{M}_{\mathscr{F}}^{2}(0,T;l^{2}(H))$ we introduce a family of
parameterized SEEs as follows:

$\displaystyle\begin{split}X(t)=&\
x+\int_{0}^{t}A(s)X(s)ds+\int_{0}^{t}\Big{[}\rho
b(s,X(s))]+b_{0}(s)\Big{]}ds+\int_{0}^{t}\Big{[}B(s)X(s)+\rho
g(s,X(s))+g_{0}(s)\Big{]}dW(s)\\\
&+\sum_{i=1}^{\infty}\int_{0}^{t}\Big{[}\rho\sigma^{i}(s,X(s))+\sigma_{0}^{i}(s)\Big{]}dH^{i}(s).\end{split}$
(3.29)

It is easy to see that when we take the parameter $\rho=1$ and $b_{0}\equiv
0,g_{0}\equiv 0,\sigma_{0}\equiv 0,$ the SEE (3.29) is reduced to the original
SEE (3.1). Obviously, the coefficients of the SEE (3.29) satisfy Assumption
3.1 and 3.2 with $(A,B,b,g,\sigma)$ replaced by $(A,B,\rho b+b_{0},\rho
g+g_{0},\rho\sigma+\sigma_{0})$. Suppose for any
$b_{0}\in{M}_{\mathscr{F}}^{2}(0,T;H),$
$g_{0}\in{M}_{\mathscr{F}}^{2}(0,T;H),$
$\sigma_{0}\in{M}_{\mathscr{F}}^{2}(0,T;l^{2}(H))$ and some parameter
$\rho=\rho_{0}$, there exists a unique solution
$X(\cdot)\in{M}_{\mathscr{F}}^{2}(0,T;V)$ to the SEE (3.29). For any parameter
$\rho$, the SEE (3.29) can be rewritten as

$\displaystyle\begin{split}X(t)=&\
x+\int_{0}^{t}A(s)X(s)ds+\int_{0}^{t}\Big{[}\rho_{0}b(s,X(s))+b_{0}(s)+(\rho-\rho_{0})b(s,X(s))\Big{]}ds\\\
&+\int_{0}^{t}\Big{[}B(s)X(s)+\rho_{0}g(s,X(s))+g_{0}(s)+(\rho-\rho_{0})g(s,X(s))\Big{]}dW(s)\\\
&+\sum_{i=1}^{\infty}\int_{0}^{t}\Big{[}\rho_{0}\sigma^{i}(s,X(s))+\sigma^{i}_{0}(t)+(\rho-\rho_{0})\sigma^{i}(s,X(s))\Big{]}dH^{i}(s).\end{split}$
(3.30)

Therefor by the above assumption, for any
$x(\cdot)\in{M}_{\mathscr{F}}^{2}(0,T;V),$ the following SEE

$\displaystyle\begin{split}X(t)=&\
x+\int_{0}^{t}A(s)X(s)ds+\int_{0}^{t}\Big{[}\rho_{0}b(s,X(s))+b_{0}(s)+(\rho-\rho_{0})b(s,x(s))\Big{]}ds\\\
&+\int_{0}^{t}\Big{[}B(s)X(s)+\rho_{0}g(s,X(s))+g_{0}(s)+(\rho-\rho_{0})g(s,x(s))\Big{]}dW(s)\\\
&+\sum_{i=1}^{\infty}\int_{0}^{t}\Big{[}\rho_{0}\sigma^{i}(s,X(s))+\sigma^{i}_{0}(s)+(\rho-\rho_{0})\sigma^{i}(s,x(s))\Big{]}dH^{i}(s)\end{split}$
(3.31)

admits a unique solution $X(\cdot)\in{M}_{\mathscr{F}}^{2}(0,T;V)$. Now define
a mapping from ${M}_{\mathscr{F}}^{2}(0,T;V)$ onto itself denoted by

$X(\cdot)=\Gamma(x(\cdot)).$

Then for any $x_{i}(\cdot)\in{M}_{\mathscr{F}}^{2}(0,T;V)$, $i=1,2$, from the
Lipschitz continuity of $b$, $g$, $\sigma$ and a priori estimate (3.2), it
follows that

$\displaystyle||\Gamma(x_{1}(\cdot))-\Gamma(x_{2}(\cdot))||^{2}_{{M}_{\mathscr{F}}^{2}(0,T;V)}$
$\displaystyle=$
$\displaystyle||X_{1}(\cdot)-X_{2}(\cdot)||^{2}_{{M}_{\mathscr{F}}^{2}(0,T;V)}$
$\displaystyle\leq$ $\displaystyle
K|\rho-\rho_{0}|^{2}\cdot||x_{1}(\cdot)-x_{2}(\cdot)||^{2}_{{M}_{\mathscr{F}}^{2}(0,T;V)}.$

Here $K$ is a positive constant independent of $\rho$. If
$|\rho-\rho_{0}|<\frac{1}{2\sqrt{K}}$, the mapping $\Gamma$ is strictly
contractive in ${M}_{\mathscr{F}}^{2}(0,T;V)$. Hence it implies that the SEE
(3.14) with the coefficients $(A,B,\rho b+b_{0},\rho
g+g_{0},\rho\sigma+\sigma_{0})$ admits a unique solution $X(\cdot)\in{M}_{\cal
F}^{2}(0,T;V)$. From Lemma 3.3, the uniqueness and existence of the solution
to the SEE (3.14) is true for $\rho=0$. Then starting from $\rho=0$, one can
reach $\rho=1$ in finite steps and this finishes the proof of solvability of
the SEE (3.1). This completes the proof.

##  4  Formulation of Optimal Control Problem

Let $U$ be a real-valued Hilbert space standing for the control space. Let
${\mathscr{U}}$ be a nonempty convex closed subset of $U$. An admissible
control process $u(\cdot)\triangleq\\{u(t),0\leq t\leq T\\}$ is defined as
follows.

######  Definition 4.1.

A stochastic process $u(\cdot)$ defined on $[0,T]\times\Omega$ is called an
admissible control process if it is a predictable process such that
$u(\cdot)\in{M}^{2}_{\mathscr{F}}(0,T;U)$ and $u(t)\in{\mathscr{U}}$, a.e.
$t\in[0,T]$, ${\mathbb{P}}$-a.s.. Write ${\cal A}$ for the set of all
admissible control processes.

In the Gelfand triple $(V,H,V^{*}),$ for any admissible control
$u(\cdot)\in\cal A,$ we consider the following controlled SEE driven by
Teugels martingales

$\displaystyle\left\\{\begin{aligned} dX(t)=&\
[A(t)X(t)+b(t,X(t),u(t))]dt+[B(t)X(t)+g(t,X(t),u(t))]dW(t)\\\
&\quad+\sum_{i=1}^{\infty}\sigma^{i}(t,X(t-),u(t))dH^{i}(t),\\\ X(0)=&\
x,\quad t\in[0,T]\end{aligned}\right.$ (4.1)

with the cost functional

$\displaystyle
J(u(\cdot))={\mathbb{E}}\bigg{[}\int_{0}^{T}l(t,x(t),u(t))dt+\Phi(x(T))\bigg{]}.$
(4.2)

where the coefficients satisfy the following basic assumptions:

######  Assumption 4.1.

  1.   2. (i)

$A:[0,T]\times\Omega\longrightarrow{\mathscr{L}}(V,V^{*})$ and
$B:[0,T]\times\Omega\longrightarrow{\mathscr{L}}(V,H)$ are operator-valued
stochastic processes satisfying (i) in Assumption 3.1;

  3. (iii)

$b,g:[0,T]\times\Omega\times H\times{\mathscr{U}}\rightarrow H$ are
$\mathscr{P}\times\mathscr{B}(H)\times\mathscr{B}(\mathscr{U})/\mathscr{B}(H)$
measurable mappings and
$\sigma=(\sigma)_{i=1}^{\infty}:[0,T]\times\Omega\times
H\times\mathscr{U}\longrightarrow l^{2}(H)$ is a
$\mathscr{P}\times\mathscr{B}(H)\times\mathscr{B}(U)/\mathscr{B}(l^{2}(H))$-measurable
mapping such that
$b(\cdot,0,0),g(\cdot,0,0)\in{M}^{2}_{\mathscr{F}}(0,T;H),\sigma(\cdot,0,0)\in{M}_{\mathscr{F}}^{2}(0,T;l^{2}(H)).$
Moreover, for almost all $(t,\omega)\in[0,T]\times\Omega$, $b$, $g$ and
$\sigma$ are Gâteaux differentiable in $(x,u)$ with continuous bounded Gâteaux
derivatives $b_{x},g_{x},\sigma_{x},b_{u},g_{u}$ and $\sigma_{u}$;

  4. (iv)

$l:[0,T]\times\Omega\times H\times{\mathscr{U}}\rightarrow\mathbb{R}$ is a
${\mathscr{P}}\otimes{\mathscr{B}}(H)\otimes{\mathscr{B}}({\mathscr{U}})/{\mathscr{B}}({\mathbb{R}})$-measurable
mapping and $\Phi:\Omega\times H\rightarrow{\mathbb{R}}$ is a
${\mathscr{F}}_{T}\otimes{\mathscr{B}}(H)/{\mathscr{B}}({\mathbb{R}})$-measurable
mapping. For almost all $(t,\omega)\in[0,T]\times\Omega$, $l$ is continuous
Gâteaux differentiable in $(x,u)$ with continuous Gâteaux derivatives $l_{x}$
and $l_{u}$, and $\Phi$ is Gâteaux differentiable in $x$ with continuous
Gâteaux derivative $\Phi_{x}$. Moreover, for almost all
$(t,\omega)\in[0,T]\times\Omega$, there exists a constant $C>0$ such that for
all $(x,u)\in H\times{\mathscr{U}}$

$\displaystyle|l(t,x,u)|\leq C(1+\|x\|^{2}_{H}++\|u\|_{U}^{2}),$
$\displaystyle\|l_{x}(t,x,u)\|_{H}++\|l_{u}(t,x,u)\|_{U}\leq
C(1+\|x\|_{H}+\|u\|_{U}),$

and

$\displaystyle|\Phi(x)|\leq C(1+\|x\|^{2}_{H}),$
$\displaystyle\|\Phi_{x}(x)\|_{H}\leq C(1+\|x\|_{H}).$

For any admissible control $u(\cdot),$ the solution of the system (4.1),
denoted by $X^{u}(\cdot)$ or $X(\cdot),$ if its dependence on $u(\cdot)$ is
clear from the context, is called the state process corresponding to the
control process $u(\cdot)$, and $(u(\cdot);X(\cdot))$ is called an admissible
pair. The following result gives the well-posedness of the state equation as
well as some useful estimates.

######  Lemma 4.1.

Let Assumption 4.1 be satisfied. Then for any admissible control $u(\cdot)$,
the state equation (4.1) has a unique solution $X^{u}(\cdot)\in
M_{\mathscr{F}}^{2}(0,T;V).$ Moreover, the following estimate holds

$\displaystyle\begin{split}&\sup_{0\leq t\leq
T}\mathbb{E}[||\hat{X}^{u}(t)||_{H}^{2}]+{\mathbb{E}}\bigg{[}\int_{0}^{T}\|X^{u}(t)\|_{V}^{2}dt\bigg{]}\leq
K\bigg{\\{}1+||x||_{H}^{2}+{\mathbb{E}}\bigg{[}\int_{0}^{T}\|u(t)\|_{U}^{2}dt\bigg{]}\bigg{\\}}\end{split}$
(4.3)

and

$\displaystyle\begin{split}|J(u(\cdot))|<\infty.\end{split}$ (4.4)

Furthermore, let $X^{v}(\cdot)$ be the state process corresponding to another
admissible control $v(\cdot),$ then

$\displaystyle\sup_{0\leq t\leq
T}\mathbb{E}[||\hat{X}^{u}(t)-X^{v}(t)||_{H}^{2}]+{\mathbb{E}}\bigg{[}\int_{0}^{T}\|X^{u}(t)-{X}^{v}(t)\|_{V}^{2}dt\bigg{]}\leq
K{\mathbb{E}}\bigg{[}\int_{0}^{T}\|u(t)-v(t)\|_{U}^{2}dt\bigg{]}.$ (4.5)

###### Proof.

Under Assumption 4.1, by Theorem 3.1, we can get directly the existence and
uniqueness of the solution of the state equation (3.1). And the estimates
(4.3) and (4.5) can be obtained by the estimates (3.7) and (3.2),
respectively. Furthermor from Assumption 4.1 and the estimate (4.3), it
follows that

$\displaystyle\begin{split}|J(u(\cdot))|\leq K\bigg{\\{}\sup_{0\leq t\leq
T}\mathbb{E}\bigg{[}||X(t)||_{H}^{2}\bigg{]}+\mathbb{E}\bigg{[}\int_{0}^{T}||u(t)||^{2}_{U}dt\bigg{]}+1\bigg{\\}}\leq
K\bigg{\\{}1+||x||_{H}^{2}+\mathbb{E}\bigg{[}\int_{0}^{T}||u(t)||_{U}^{2}dt\bigg{]}\bigg{\\}}<\infty.\end{split}$
(4.6)

The proof is complete. ∎

Therefor by Lemma 4.1, we claim that the cost functional (4.2) is well-
defined. Our optimal control problem can be stated as follows.

######  Problem 4.2.

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

$\displaystyle J({\bar{u}}(\cdot))=\inf_{u(\cdot)\in{\cal A}}J(u(\cdot)).$
(4.7)

The admissible control ${\bar{u}}(\cdot)$ satisfying (4.7) is called an
optimal control process of Problem 4.2. Correspondingly, the state process
${\bar{X}}(\cdot)$ associated with ${\bar{u}}(\cdot)$ is called an optimal
state process. Then $({\bar{u}}(\cdot);{\bar{X}}(\cdot))$ is called an optimal
pair of Problem 4.2.

##  5 Regularity Result for the Adjoint Equation

For any admissible pair $(\bar{u}(\cdot);\bar{X}(\cdot))$, the corresponding
adjoint processes is defined as a triple
$(\bar{p}(\cdot),\bar{q}(\cdot),\bar{r}(\cdot))$ of stochastic processes,
which is a solution to the following backward stochastic evolution equation
(BSEE for short) driven by Teugels martingales, called the adjoint equation,

$\displaystyle\begin{split}\left\\{\begin{array}[]{ll}d\bar{p}(t)=&-\bigg{[}A^{*}(t)\bar{p}(t)+b_{x}^{*}(t,\bar{X}(t),\bar{u}(t))\bar{p}(t)+B^{*}(t)\bar{q}(t)+g_{x}^{*}(t,\bar{X}(t),\bar{u}(t))\bar{q}(t)\\\
&\quad\quad+\displaystyle\sum_{i=1}^{\infty}\sigma{{}^{i}}_{x}^{*}(t,\bar{X}(t),\bar{u}(t))\bar{r}^{i}(t)+l_{x}(t,\bar{X}(t),\bar{u}(t))\bigg{]}dt\\\
&\quad\quad+\bar{q}(t)dW(t)+\displaystyle\sum_{i=1}^{\infty}\bar{r}^{i}(t)dH^{i}(t),~{}~{}~{}~{}0\leqslant
t\leqslant T,\\\
\bar{p}(T)=&\Phi_{x}(\bar{X}(T)).\end{array}\right.\end{split}$ (5.1)

Here $A^{*}$ denotes the adjoint operator of the operator $A.$ Similarly, we
can define the corresponding adjoint operator for other coefficients.
Under Assumptions 4.1, we have the following basic result for the adjoint
process.

######  Lemma 5.1.

Let Assumptions 4.1 be satisfied. Then for any admissible pair
$(\bar{u}(\cdot);\bar{X}(\cdot)),$ there exists a unique adjoint process
$(\bar{p}(\cdot),\bar{q}(\cdot),\bar{r}(\cdot))\in
M_{\mathscr{F}}^{2}(0,T;V)\times M_{\mathscr{F}}^{2}(0,T;H)\times
M_{\mathscr{F}}^{2}(0,T;l^{2}(H)).$ Moreover, the following estimate holds:

$\displaystyle\begin{split}&{\mathbb{E}}\bigg{[}\int_{0}^{T}\|\bar{p}(t)\|_{V}^{2}dt\bigg{]}+{\mathbb{E}}\bigg{[}\int_{0}^{T}\|\bar{q}(t)\|_{H}^{2}dt\bigg{]}+{\mathbb{E}}\bigg{[}\int_{0}^{T}\|r(t)\|_{l^{2}(H)}^{2}dt\bigg{]}\leq
K\bigg{\\{}{\mathbb{E}}\bigg{[}\int_{0}^{T}\|l_{x}(t,\bar{X}(t),\bar{u}(t))\|_{H}^{2}dt\bigg{]}+\mathbb{E}[||\Phi_{x}(\bar{X}(T))||_{H}^{2}]\bigg{\\}}.\end{split}$
(5.2)

###### Proof.

From the property of adjoint operator, the adjoint operator $A^{*}$ of $A$ and
the adjoint operator $B^{*}$ of $B$ also satisfies (i) in Assumption 3.1.
Therefore, similarly to Theorem 3.1, the existence and uniqueness of the
solution can be proved by Galerkin approximations and parameter extension
method. ∎

Define the Hamiltonian ${\cal H}:[0,T]\times\Omega\times
H\times{\mathscr{U}}\times H\times H\times l^{2}(H)\rightarrow{\mathbb{R}}$ by

$\displaystyle{\cal
H}(t,x,u,p,q,r):=\left(b(t,x,u),p\right)_{H}+\left(g(t,x,u),q\right)_{H}+\left(\sigma(t,x,u),r(t)\right)_{l^{2}(H)}+l(t,x,u).$
(5.3)

Using Hamiltonian ${\cal H}$, the adjoint equation (5.1) can be written in the
following form:

$\displaystyle\begin{split}\left\\{\begin{array}[]{ll}d\bar{p}(t)=&-\bigg{[}A^{*}(t)\bar{p}(t)+B(t)^{*}\bar{q}(t)+\bar{\cal
H}_{x}(t)\bigg{]}dt+\bar{q}(t)dW(t)+\displaystyle\sum_{i=1}^{\infty}\bar{r}^{i}(t)dH^{i}(t),~{}~{}~{}~{}0\leqslant
t\leqslant T,\\\
\bar{p}(T)=&\Phi_{x}(\bar{X}(T)),\end{array}\right.\end{split}$ (5.4)

where we denote

$\displaystyle\bar{\cal H}(t)\triangleq{\cal
H}(t,\bar{x}(t),\bar{u}(t),\bar{p}(t),\bar{q}(t),\bar{r}(t)).$ (5.5)

##  6  Stochastic Maximum Principle

###  6.1 Variation of the State Process and Cost Functional

Let$({\bar{u}}(\cdot);{\bar{X}}(\cdot))$ be an optimal pair of Problem 4.2.
Define a convex perturbation of $\bar{u}(\cdot)$ as follows:

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

where $v(\cdot)$ is an arbitrarily admissible control. Since the control
domain $\mathscr{U}$ is convex, $u^{\varepsilon}(\cdot)$ is also an element of
$\cal A.$ We denote by $X^{\varepsilon}(\cdot)$ the state process
corresponding to the control $u^{\varepsilon}(\cdot).$ Now we introduce the
following first order variation equation:

$\displaystyle\left\\{\begin{aligned} dY(t)=&\
[A(t)Y(t)+b_{x}(t,\bar{X}(t),\bar{u}(t))Y(t)+b_{u}(t,\bar{X}(t),\bar{u}(t))(v(t)-\bar{u}(t))]dt\\\
&+[B(t)Y(t)+g_{x}(t,\bar{X}(t),\bar{u}(t))Y(t)+g_{u}(t,\bar{X}(t),\bar{u}(t))(v(t)-\bar{u}(t))]dW(t)\\\
&+\sum_{i=1}^{\infty}\bigg{[}\sigma^{i}_{x}(t,\bar{X}(t-),\bar{u}(t))Y(t-)+\sigma^{i}_{u}(t,\bar{X}(t-),\bar{u}(t))(v(t)-\bar{u}(t))\bigg{]}dH^{i}(t),\\\
Y(0)=&0.\end{aligned}\right.$ (6.1)

Under Assumption 4.1, by Theorem 3.1, we see that the variation equation (6.1)
has a unique solution $Y(\cdot)\in M_{\mathscr{F}}^{2}(0,T;V).$

######  Lemma 6.1.

Let Assumption 4.1 be satisfied. Then we have the following estimates:

$\displaystyle\sup_{0\leq t\leq
T}{\mathbb{E}}\bigg{[}\|X^{\epsilon}(t)-{\bar{X}}(t)\|^{2}_{H}\bigg{]}+{\mathbb{E}}\bigg{[}\int_{0}^{T}\|X^{\epsilon}(t)-{\bar{X}}(t)\|^{2}_{V}dt\bigg{]}=O(\epsilon^{2}),$
(6.2)

$\displaystyle\sup_{0\leq t\leq
T}{\mathbb{E}}\bigg{[}\|X^{\epsilon}(t)-{\bar{X}}(t)-\varepsilon
Y(t)\|^{2}_{H}\bigg{]}+{\mathbb{E}}\bigg{[}\int_{0}^{T}\|X^{\epsilon}(t)-{\bar{X}}(t)-\varepsilon
Y(t)\|^{2}_{V}dt\bigg{]}=o(\epsilon^{2})\ .$ (6.3)

###### Proof.

From the estimate (4.5), we have

$\displaystyle\begin{split}\sup_{0\leq t\leq
T}{\mathbb{E}}\bigg{[}\|X^{\epsilon}(t)-{\bar{X}}(t))\|^{2}_{H}\bigg{]}+{\mathbb{E}}\bigg{[}\int_{0}^{T}\|X^{\varepsilon}(t)-{\bar{X}}(t)\|_{V}^{2}dt\bigg{]}&\leq
K{\mathbb{E}}\bigg{[}\int_{0}^{T}\|u^{\varepsilon}(t)-\bar{u}(t)\|_{U}^{2}dt\bigg{]}\\\
&=K\varepsilon^{2}{\mathbb{E}}\bigg{[}\int_{0}^{T}\|v(t)-\bar{u}(t)\|_{U}^{2}dt\bigg{]}\\\
&=O(\varepsilon^{2}).\end{split}.$ (6.4)

Denote

$\displaystyle\Xi^{\varepsilon}(t):=X^{\varepsilon}(t)-\bar{X}(t)-\varepsilon
Y(t).$ (6.5)

From Taylor expanding, we have

$\displaystyle\left\\{\begin{aligned} d\Xi^{\varepsilon}(t)=&\
[A(t)\Xi^{\varepsilon}(t)+b_{x}(t,\bar{X}(t),\bar{u}(t))\Xi^{\varepsilon}(t)+\alpha^{\varepsilon}(t)]dt+[B(t)\Xi^{\varepsilon}(t)+g_{x}(t,\bar{X}(t),\bar{u}(t))\Xi^{\varepsilon}(t)+\beta^{\varepsilon}(t)]dW(t)\\\
&+\sum_{i=1}^{\infty}\bigg{[}\sigma^{i}_{x}(t,\bar{X}(t-),\bar{u}(t))\Xi^{\varepsilon}(t)+\gamma^{i\varepsilon}(t)\bigg{]}dH^{i}(t),\\\
X(0)=&\ x,\quad t\in[0,T],\end{aligned}\right.$ (6.6)

where

$\displaystyle\left\\{\begin{aligned}
\alpha^{\varepsilon}(t)=&\int_{0}^{1}\bigg{[}\big{(}b_{x}(t,\bar{X}(t)+\lambda(X^{\varepsilon}(t)-\bar{X}(t)),\bar{u}(t)+\lambda(u^{\varepsilon}(t)-\bar{u}(t)))-b_{x}(t,\bar{X}(t),\bar{u}(t))\big{)}(X^{\varepsilon}(t)-\bar{X}(t))\\\
&+\big{(}b_{u}(t,\bar{X}(t)+\lambda(X^{\varepsilon}(t)-\bar{X}(t)),\bar{u}(t)+\lambda(u^{\varepsilon}(t)-\bar{u}(t)))-b_{u}(t,\bar{X}(t),\bar{u}(t))\big{)}(u^{\varepsilon}(t)-\bar{u}(t))\bigg{]}d\lambda,\\\
\beta^{\varepsilon}(t)=&\int_{0}^{1}\bigg{[}\big{(}g_{x}(t,\bar{X}(t)+\lambda(X^{\varepsilon}(t)-\bar{X}(t)),\bar{u}(t)+\lambda(u^{\varepsilon}(t)-\bar{u}(t)))-g_{x}(t,\bar{X}(t),\bar{u}(t))\big{)}(X^{\varepsilon}(t)-\bar{X}(t))\\\
&+\big{(}g_{u}(t,\bar{X}(t)+\lambda(X^{\varepsilon}(t)-\bar{X}(t)),\bar{u}(t)+\lambda(u^{\varepsilon}(t)-\bar{u}(t)))-g_{u}(t,\bar{X}(t),\bar{u}(t))\big{)}(u^{\varepsilon}(t)-\bar{u}(t))\bigg{]}d\lambda,\\\
\gamma^{i\varepsilon}(t)=&\int_{0}^{1}\bigg{[}\big{(}\sigma^{i}_{x}(t,\bar{X}(t-)+\lambda(X^{\varepsilon}(t-)-\bar{X}(t-)),\bar{u}(t)+\lambda(u^{\varepsilon}(t)-\bar{u}(t))-\sigma^{i}_{x}(t,\bar{X}(t-),\bar{u}(t))\big{)}(X^{\varepsilon}(t-)-\bar{X}(t-))\\\
&+\big{(}\sigma^{i}_{u}(t,\bar{X}(t-)+\lambda(X^{\varepsilon}(t-)-\bar{X}(t-)),\bar{u}(t)+\lambda(u^{\varepsilon}(t)-\bar{u}(t)))-\sigma^{i}_{x}(t,\bar{X}(t-),\bar{u}(t))\big{)}(u^{\varepsilon}(t)-\bar{u}(t))\bigg{]}d\lambda.\end{aligned}\right.$
(6.7)

From the estimates (3.7), (6.2) and Lebesgue dominated convergence theorem, we
get that

$\displaystyle\sup_{0\leq t\leq
T}{\mathbb{E}}\bigg{[}\|\Xi(t)\|^{2}_{H}\bigg{]}+{\mathbb{E}}\bigg{[}\int_{0}^{T}\|\Xi(t)\|^{2}_{V}dt\bigg{]}\leq
K\bigg{\\{}\mathbb{E}\bigg{[}\int_{0}^{T}||\alpha^{\varepsilon}(t)||^{2}_{H}dt\bigg{]}+\mathbb{E}\bigg{[}\int_{0}^{T}||\beta^{\varepsilon}(t)||^{2}_{H}dt\bigg{]}+\mathbb{E}\bigg{[}\int_{0}^{T}||\gamma^{\varepsilon}(t)||^{2}_{l^{2}(H)}dt\bigg{]}\bigg{\\}}=o(\varepsilon).$

The proof is complete. ∎

######  Lemma 6.2.

Let Assumption 4.1 be satisfied. Let $({\bar{u}}(\cdot);{\bar{X}}(\cdot))$ be
an optimal pair of Problem 4.2 associated with the first order variation
process $Y(\cdot)$ (see (6.1)). Then,

$\displaystyle\begin{split}J(u^{\varepsilon}(\cdot))-J(\bar{u}(\cdot))=&\varepsilon\mathbb{E}\bigg{[}(\Phi_{x}(\bar{X}(T)),Y(T))_{H}\bigg{]}+\varepsilon\mathbb{E}\bigg{[}\int_{0}^{T}(l_{x}(t,\bar{X}(t),\bar{u}(t)),Y(t))_{H}dt\bigg{]}\\\
&+\varepsilon\mathbb{E}\bigg{[}\int_{0}^{T}(l_{u}(t,\bar{X}(t),\bar{u}(t)),v(t)-u(t))_{U}dt\bigg{]}+o(\varepsilon).\end{split}$
(6.8)

###### Proof.

From the definition of the cost functional (see (4.2)), we have

$\displaystyle
J(u^{\varepsilon}(\cdot))-J(\bar{u}(\cdot))={\mathbb{E}}\bigg{[}\int_{0}^{T}\Big{(}l(t,X^{\varepsilon}(t),u^{\varepsilon}(t))-l(t,\bar{X}(t),\bar{u}(t)\Big{)}dt\bigg{]}+\mathbb{E}\bigg{[}\Phi(X^{\varepsilon}(T))-\Phi(\bar{X}(T))\bigg{]}=:I_{1}+I_{2}{\color[rgb]{0,0,1},}$
(6.9)

where

$\displaystyle\begin{split}I_{1}=&{\mathbb{E}}\bigg{[}\int_{0}^{T}\Big{(}l(t,X^{\varepsilon}(t),u^{\varepsilon}(t))-l(t,\bar{X}(t),\bar{u}(t)\Big{)}dt\bigg{]},\\\
I_{2}=&\mathbb{E}\bigg{[}\Phi(X^{\varepsilon}(T))-\Phi(\bar{X}(T))\bigg{]}.\end{split}$
(6.10)

Let us concentrate on $I_{1}.$ In terms of Taylor expanding, Lemma 6.1 and the
control convergence theorem, we have

$\displaystyle\begin{split}I_{1}=&\mathbb{E}\bigg{[}\int_{0}^{T}\int_{0}^{1}\big{(}l_{x}(t,\bar{X}(t)+\lambda(X^{\varepsilon}(t)-\bar{X}(t)),\bar{u}(t)+\lambda(u^{\varepsilon}(t)-\bar{u}(t)))-l_{x}(t,\bar{X}(t),\bar{u}(t))\big{)}(X^{\varepsilon}(t)-\bar{X}(t))d\lambda
dt\bigg{]}\\\
&+E\bigg{[}\int_{0}^{T}\int_{0}^{1}\big{(}l_{u}(t,\bar{X}(t)+\lambda(X^{\varepsilon}(t)-\bar{X}(t)),\bar{u}(t)+\lambda(u^{\varepsilon}(t)-\bar{u}(t)))-l_{u}(t,\bar{X}(t),\bar{u}(t))\big{)}(u^{\varepsilon}(t)-\bar{u}(t))d\lambda
dt\bigg{]}\\\
&+\mathbb{E}\bigg{[}\int_{0}^{T}l_{x}(t,\bar{X}(t),\bar{u}(t))\big{)}\Xi^{\varepsilon}(t)dt\bigg{]}+\varepsilon\mathbb{E}\bigg{[}\int_{0}^{T}l_{x}(t,\bar{X}(t),\bar{u}(t))\big{)}Y(t)dt\bigg{]}\\\
&+\varepsilon
E\bigg{[}\int_{0}^{T}l_{u}(t,\bar{X}(t),\bar{u}(t))\big{)}(u(t)-\bar{u}(t))dt\bigg{]}\\\
=&\varepsilon\mathbb{E}\bigg{[}\int_{0}^{T}l_{x}(t,\bar{X}(t),\bar{u}(t))Y(t)dt\bigg{]}+\varepsilon
E\bigg{[}\int_{0}^{T}l_{u}(t,\bar{X}(t),\bar{u}(t))(u(t)-\bar{u}(t))dt\bigg{]}\\\
&+o(\varepsilon).\end{split}$ (6.11)

Similarly, we have

$\displaystyle\begin{split}I_{2}=\varepsilon\mathbb{E}\bigg{[}\Phi_{x}(\bar{X}(T))Y(T)\bigg{]}+o(\varepsilon).\end{split}$
(6.12)

Then putting (6.11) and (6.12) into (6.9), we get (6.8). The proof is
complete.

∎

###  6.2 Main Results

Now we are in position to state and prove the maximum principle for Problem
4.2.

######  Theorem 6.3 (Maximum Principle).

Let Assumption 4.1 be satisfied. Let $({\bar{u}}(\cdot);{\bar{X}}(\cdot))$ be
an optimal pair of Problem 4.2 associated with the adjoint processes
$({\bar{p}}(\cdot),{\bar{q}}(\cdot),\bar{r}(\cdot)).$ Then the following
minimum condition holds:

$\displaystyle\big{(}{\cal
H}_{u}(t,\bar{X}(t-),\bar{u}(t),\bar{p}(t-),\bar{q}(t),\bar{r}(t)),v-{\bar{u}}(t)\big{)}_{U}\geq
0,\quad\quad\forall
v\in{\mathscr{U}},for~{}a.e.~{}t\in[0,T],{\mathbb{P}}-a.s.$ (6.13)

###### Proof.

Recalling the adjoint equation (5.4) and the first order variational equation
(6.1), and then applying Itô formula to $(\bar{p}(t),Y(t))_{H},$ we have

$\displaystyle\begin{split}&\mathbb{E}[(\Phi_{x}(\bar{X}(T)),Y(T))_{H}]+\mathbb{E}\bigg{[}\int_{0}^{T}(l_{x}(t,\bar{X}(t),\bar{u}(t)),Y(t))_{H}dt\bigg{]}\\\
=&\mathbb{E}\bigg{[}\int_{0}^{T}\bigg{(}v(t)-\bar{u}(t),b_{u}^{*}(t,\bar{X}(t),\bar{u}(t))\bar{p}(t)+g_{u}^{*}(t,\bar{X}(t),\bar{u}(t))\bar{q}(t)+\sum_{i=1}^{\infty}\sigma{{}^{i}}_{u}^{*}(t,\bar{X}(t),\bar{u}(t))\bar{r}^{i}(t)\bigg{)}_{U}dt\bigg{]}.\end{split}$
(6.14)

Since $\bar{u}(\cdot)$ is the optimal control, from (6.8), the duality
relation (6.14) and the definition of the Hamiltonian ${\cal H}$ (see (5.3)) ,
we have

$\displaystyle\begin{split}0\leq&\lim_{\varepsilon\longrightarrow
0^{+}}\frac{J(u^{\varepsilon}(\cdot))-J(\bar{u}(\cdot))}{\varepsilon}\\\
=&\mathbb{E}[\big{(}\Phi_{x}(\bar{X}(T)),Y(T)\big{)}_{H}]+\mathbb{E}\bigg{[}\int_{0}^{T}(l_{x}(t,\bar{X}(t),\bar{u}(t)),Y(t))_{H}dt\bigg{]}+\mathbb{E}\bigg{[}\int_{0}^{T}(l_{u}(t,\bar{X}(t),\bar{u}(t)),v(t)-u(t))_{U}dt\bigg{]}\\\
=&\mathbb{E}\bigg{[}\int_{0}^{T}\bigg{(}v(t)-\bar{u}(t),b_{u}^{*}(t,\bar{X}(t),\bar{u}(t))\bar{p}(t)+g_{u}^{*}(t,\bar{X}(t),\bar{u}(t))\bar{q}(t)+\sum_{i=1}^{\infty}\sigma{{}^{i}}_{u}^{*}(t,\bar{X}(t),\bar{u}(t))\bar{r}^{i}(t)\bigg{)}_{U}dt\bigg{]}\\\
&+\mathbb{E}\bigg{[}\int_{0}^{T}(l_{u}(t,\bar{X}(t),\bar{u}(t)),v(t)-\bar{u}(t))_{U}dt\bigg{]}\\\
=&E\bigg{[}\int_{0}^{T}\bigg{(}v(t)-\bar{u}(t),{\cal
H}_{u}(t,\bar{X}(t),\bar{u}(t),\bar{p}(t),\bar{q}(t),\bar{r}(t)))_{U}dt\bigg{]}.\end{split}$
(6.15)

This implies the minimum condition (6.13) holds since $v(\cdot)$ is any given
admissible control. ∎

##  7 Verification Theorem

In the following, we give a sufficient condition of optimality for the
existence of an optimal control of Problem 4.2, which is the so-called
verification theorem.

######  Theorem 7.1 (Verification Theorem).

Let Assumption 4.1 be satisfied. Let $({\bar{u}}(\cdot);{\bar{X}}(\cdot))$ be
an admissible pair of Problem 4.2 associated with the adjoint processes
$(\bar{p}(\cdot),\bar{q}(\cdot),\bar{r}(\cdot)).$ Suppose that ${\cal
H}(t,x,u,{\bar{p}}(t),{\bar{q}}(t),\bar{r}(t))$ is convex in $(x,u)$, and
$\Phi(x)$ is convex in $x$, moreover assume that the following optimality
condition holds for almost all $(t,\omega)\in[0,T]\times\Omega$:

$\displaystyle{\cal
H}(t,{\bar{X}}(t),\bar{u}(t),{\bar{p}}(t),{\bar{q}}(t),\bar{r}(t))=\min_{u\in{\mathscr{U}}}{\cal
H}(t,{\bar{X}}(t),u,{\bar{p}}(t),{\bar{q}}(t),\bar{r}(t)).$ (7.1)

Then $({\bar{u}}(\cdot);{\bar{X}}(\cdot))$ is an optimal pair of Problem 4.2.

###### Proof.

Let $({u}(\cdot);{X}(\cdot))$ be an any given admissible pair. To simplify our
notations, we define

$\displaystyle\begin{split}&b(t)\triangleq
b(t,X(t),u(t)),{\bar{b}}(t)\triangleq b(t,{\bar{X}}(t),{\bar{u}}(t)),\\\
&g(t)\triangleq g(t,X(t),u(t)),{\bar{g}}(t)\triangleq
g(t,{\bar{X}}(t),{\bar{u}}(t)),\\\
&\sigma{{}^{i}}(t)\triangleq\sigma{{}^{i}}(t,X(t-),u(t)),{\bar{\sigma}{{}^{i}}}(t)\triangleq\sigma{{}^{i}}(t,{\bar{X}}(t-),{\bar{u}}(t)),\\\
&{\cal H}(t)\triangleq{\cal
H}(t,X(t),u(t),{\bar{p}}(t),{\bar{q}}(t),\bar{r}(t)),\\\ &{\bar{\cal
H}}(t)\triangleq{\cal
H}(t,\bar{X}(t),\bar{u}(t),{\bar{p}}(t),{\bar{q}}(t),\bar{r}(t)).\end{split}$
(7.2)

From the definitions of the cost functional $J(u(\cdot))$ and the Hamiltonian
${\cal H}$ (see (4.2) and (5.3)), we can represent
$J(u(\cdot))-J(\bar{u}(\cdot))$ as follows:

$\displaystyle J(u(\cdot))-J({\bar{u}}(\cdot))$ $\displaystyle=$
$\displaystyle{\mathbb{E}}\bigg{[}\int_{0}^{T}\bigg{(}{\cal H}(t)-{\bar{\cal
H}}(t)-({\bar{p}}(t),b(t)-{\bar{b}}(t))_{H}-({\bar{q}}(t),g(t)-{\bar{g}}(t))_{H}$
(7.3)
$\displaystyle-\sum_{i=1}^{\infty}({\bar{r}}^{i}(t),\sigma{{}^{i}}(t)-{\bar{\sigma}{{}^{i}}}(t))_{H}\bigg{)}dt\bigg{]}+{\mathbb{E}}\bigg{[}\Phi(X(T))-\Phi({\bar{X}}(T))\bigg{]}.$

Then recalling the adjoint equation (5.1) and applying Itô’s formula to
$({\bar{p}}(t),X(t)-{\bar{X}}(t))_{H}$, we get that

$\displaystyle{\mathbb{E}}\bigg{[}\int_{0}^{T}\bigg{(}({\bar{p}}(t),b(t)-{\bar{b}}(t))_{H}+({\bar{q}}(t),g(t)-{\bar{g}}(t))_{H}+\sum_{i=1}^{\infty}({\bar{r}}^{i}(t),\sigma{{}^{i}}(t)-{\bar{\sigma}{{}^{i}}}(t))_{H}\bigg{)}dt\bigg{]}$
$\displaystyle=$ $\displaystyle{\mathbb{E}}\bigg{[}\int_{0}^{T}({\bar{\cal
H}}_{x}(t),X(t)-{\bar{X}}(t))_{H}dt\bigg{]}+{\mathbb{E}}\Big{[}(\Phi_{x}(\bar{X}(T)),X(T)-{\bar{X}}(T))_{H}\Big{]}.$
(7.4)

Then substituting (7) into (7.3) leads to

$\displaystyle J(u(\cdot))-J({\bar{u}}(\cdot))$ $\displaystyle=$
$\displaystyle{\mathbb{E}}\bigg{[}\int_{0}^{T}\bigg{(}{\cal H}(t)-{\bar{\cal
H}}(t)-({\bar{\cal H}}_{x}(t),X(t)-{\bar{X}}(t))_{H}\bigg{)}dt\bigg{]}$ (7.5)
$\displaystyle+{\mathbb{E}}[\Phi(X(T))-\Phi({\bar{X}}(T))-(\Phi_{x}({\bar{X}}(T)),X(T)-{\bar{x}}(T))_{H}].$

On the other hand, the convexity of ${\cal H}(t)$ and $\Phi(x)$ yields

$\displaystyle{\cal H}(t)-{\bar{\cal H}}(t)$ $\displaystyle\geq$
$\displaystyle({\bar{\cal H}}_{x}(t),X(t)-{\bar{X}}(t))_{H}+({\bar{\cal
H}}_{u}(t),u(t)-{\bar{u}}(t))_{U},$ (7.6)

and

$\displaystyle\Phi(X(T))-\Phi({\bar{X}}(T))\geq(\Phi_{x}({\bar{X}}(T)),x(T)-{\bar{x}}(T))_{H}.$
(7.7)

In addition, the optimality condition (7.1) and the convex optimization
principle (see Proposition 2.21 of [8] ) yield that for almost all
$(t,\omega)\in[0,T]\times\Omega$,

$\displaystyle({\bar{\cal H}}_{u}(t),u(t)-{\bar{u}}(t))_{U}\geq 0.$ (7.8)

Then putting (7.6), (7.7) and (7.8) into (7.5), we get that

$\displaystyle J(u(\cdot))-J({\bar{u}}(\cdot))\geq 0.$ (7.9)

Therefore, since $u(\cdot)$ is arbitrary, ${\bar{u}}(\cdot)$ is an optimal
control process and $({\bar{u}}(\cdot);{\bar{X}}(\cdot))$ is an optimal pair.
The proof is complete. ∎

##  8 Application

In this section, we will apply our theoretical results to solve a specific
exampl i.e., an optimal control problem for a controlled Cauchy problem driven
by Teugels martingales.

First of all, let us recall some preliminaries of Sobolev spaces. For $m=0,1$,
we define the space $H^{m}\triangleq\\{\phi:\partial_{z}^{\alpha}\phi\in
L^{2}({\mathbb{R}}^{d}),\ \mbox{for any}\
\alpha:=(\alpha_{1},\cdots,\alpha_{d})\ \mbox{with}\
|\alpha|:=|\alpha_{1}|+\cdots+|\alpha_{d}|\leq m\\}$ with the norm

$\displaystyle\|\phi\|_{m}\triangleq\left\\{\sum_{|\alpha|\leq
m}\int_{{\mathbb{R}}^{d}}|\partial_{z}^{\alpha}\phi(z)|^{2}dz\right\\}^{\frac{1}{2}}.$

We denote by $H^{-1}$ the dual space of $H^{1}$. We set $V=H^{1}$, $H=H^{0}$,
$V^{*}=H^{-1}$. Then $(V,H,V^{*})$ is a Gelfand triple.

We choose control domain ${\mathscr{U}}=U=H$. The admissible control set $\cal
A$ is defined as ${M}^{2}_{\mathscr{F}}(0,T;U).$ For any admissible control
$u(\cdot,\cdot)\in{M}^{2}_{\mathscr{F}}(0,T;U),$ we consider a controlled
Cauchy problem, where the system is given by a stochastic partial differential
equation driven by Brownian motion $W$ and Poisson random martingale in the
following divergence form:

$\displaystyle\left\\{\begin{aligned} dy(t,z)=&\
\big{\\{}\partial_{z^{i}}[a^{ij}(t,z)\partial_{z^{j}}y(t,z)]+b^{i}(t,z)\partial_{z^{i}}y(t,z)+c(t,z)y(t,z)+u(t,z)\big{\\}}dt\\\
&+\\{\partial_{z^{i}}[\eta^{i}(t,z)y(t,z)]+\rho(t,z)y(t,z)+u(t,z)\\}dW(t)+\sum_{i=1}^{\infty}[\Gamma^{i}(t,z)y(t,z)+u(t,z)]dH^{i}(t),\\\
y(0,z)=&\
\xi(z)\in\mathbb{R}^{d}{\color[rgb]{0,0,1},}\quad(t,z)\in[0,T]\times{\mathbb{R}}^{d},\end{aligned}\right.$
(8.1)

where the coefficients
$a^{ij},b^{i},\eta^{i},c,\rho:[0,T]\times\Omega\times\mathbb{R}^{d}\rightarrow\mathbb{R}$
and $\Gamma^{i}:[0,T]\times\Omega\times\mathbb{R}^{d}$ are given random
mappings and satisfy the suitable measurability. Here we also use the Einstein
summation convention to
$\partial_{z^{i}}[a^{ij}(t,z)\partial_{z^{j}}y(t,z)],b^{i}(t,z)\partial_{z^{i}}y(t,z)$
and $\partial_{z^{i}}[\eta^{i}(t,z)y(t,z)].$

For any admissible control $u(\cdot,\cdot)\in\cal A$, the following definition
gives the generalized weak solution to (8.1).

######  Definition 8.1.

An $\mathbb{R}$-valued,
${\mathscr{P}}\times{\mathscr{B}}(\mathbb{R}^{d})$-measurable process
$y(\cdot,\cdot)$ is called a solution to (8.1), if $y(\cdot,\cdot)\in{\cal
M}_{\cal F}^{2}(0,T;H)$ such that for every $\phi\in H$ and a.e.
$(t,\omega)\in[0,T]\times\Omega$, it holds that

$\displaystyle\begin{split}\int_{\mathbb{R}^{d}}y(t,z)\phi(z)dz=&\
\int_{\mathbb{R}^{d}}\xi(z)\phi(z)dz-\int_{0}^{t}\int_{\mathbb{R}^{d}}a^{ij}(s,z)\partial_{z^{j}}y(s,z)\partial_{z^{i}}\phi(z)dzds+\int_{0}^{t}\int_{\mathbb{R}^{d}}\bigg{[}b^{i}(s,z)\partial_{z^{i}}y(s,z)\\\
&+c(s,z)y(s,z)+u(s,z)\bigg{]}\phi(z)dzds-\int_{0}^{t}\int_{\mathbb{R}^{d}}\eta^{i}(s,z)y(s,z)\partial_{i}\phi(z)dzdW(s)\\\
&+\int_{0}^{t}\int_{\mathbb{R}^{d}}\bigg{[}\rho(s,z)y(s,z)+u(s,z)\bigg{]}\phi(z)dzdW(s)+\sum_{i=1}^{\infty}\int_{0}^{t}\int_{\mathbb{R}^{d}}\bigg{[}\Gamma^{i}(s,z)y(s,z)+u(s,z)\bigg{]}\phi(z)dzdH^{i}(t).\end{split}$
(8.2)

For any admissible control process $u(\cdot,\cdot)$ and the solution
$y(\cdot,\cdot)$ of the corresponding state equation (8.1), the objective of
the control problem is to minimize the following cost functional

$\displaystyle
J(u(\cdot))={\mathbb{E}}\bigg{[}\int_{{\mathbb{R}}^{d}}y^{2}(T,z)dz+\iint_{[0,T]\times{{\mathbb{R}}^{d}}}y^{2}(s,z)dsdz+\iint_{[0,T]\times{{\mathbb{R}}^{d}}}u^{2}(s,z)dsdz\bigg{]}.$
(8.3)

To make the control problem well-defined, we make the following assumptions on
the coefficients $a$, $b$, $c$, $\eta$, $\rho$, $\Gamma$, for some fixed
constants $K\in(1,\infty)$ and $\kappa\in(0,1)$:

######  Assumption 8.1.

The functions $a$, $b$, $c$, $\eta$, and $\rho$ are
${\mathscr{P}}\times{\mathscr{B}}({\mathbb{R}}^{d})$-measurable with values in
the set of real symmetric $d\times d$ matrices, ${\mathbb{R}}^{d}$,
${\mathbb{R}}$, ${\mathbb{R}}^{d}$ and ${\mathbb{R}}$, respectively, and are
bounded by $K$. The function $\Gamma$ is
${\mathscr{P}}\times{\mathscr{B}}(E)\times{\mathscr{B}}({\mathbb{R}}^{d})$-measurable
with value ${l^{2}(\mathbb{R})}$ and is bounded by $K$. $\xi\in
L^{2}({\mathbb{R}}^{d}).$

######  Assumption 8.2.

The super-parabolic condition holds, i.e.,

$\displaystyle\kappa I+\eta(t,z)(\eta(t,z))^{*}\leq 2a(t,\omega,z)\leq
KI,\quad\forall(t,\omega,z)\in[0,T]\times\Omega\times{\mathbb{R}}^{d},$

where $I$ is the $(d\times d)$-identity matrix.

In order to apply our abstract theoretical results in Section 6 and 7 to our
optimal control problem, now we begin to transform (8.1) into a SEE driven by
Teugels martingales in the form of (3.1). Set

$\displaystyle X(t)\triangleq y(t,\cdot),$
$\displaystyle(A(t)\phi)(z)\triangleq\partial_{z^{i}}[a^{ij}(t,z)\partial_{z^{j}}\phi(z)]+b^{i}(t,z)\partial_{z^{i}}\phi(z)+c(t,z)\phi(z),\quad\forall\phi\in
V,$
$\displaystyle(B(t)\phi)(z)\triangleq\partial_{z^{i}}[\eta^{i}(t,z)\phi(z)]+\rho(t,z)\phi(z),\quad\forall\phi\in
V,$ $\displaystyle b(t,\phi,u)\triangleq u,\quad\forall\phi\in H,u\in{\cal
U},$ $\displaystyle g(t,\phi,u)\triangleq u,\quad\forall\phi\in H,u\in{\cal
U},$
$\displaystyle\sigma{{}^{i}}(t,\phi,u)\triangleq\Gamma^{i}(t)\phi+u,\quad\forall\phi\in
H,u\in\cal U,$ $\displaystyle
l(t,\phi,u)\triangleq(\phi,\phi)_{H}+(u,u)_{U},\quad\forall\phi\in H,u\in\cal
U,$ $\displaystyle\Phi(\phi)\triangleq(\phi,\phi)_{H},\quad\forall\phi\in H.$

In the Gelfand triple $(V,H,V^{*})$, using the above notations, we can rewrite
the state equation (8.1) as follows:

$\displaystyle\left\\{\begin{aligned} dX(t)=&\
[A(t)X(t)+b(t,X(t),u(t))]dt+[B(t)X(t)+g(t,X(t),u(t))]dW(t)\\\
&\quad+\sum_{i=1}^{\infty}\sigma{{}^{i}}(t,X(t),u(t))dH^{i}(t),\\\ X(0)=&\
x,\quad t\in[0,T],\end{aligned}\right.$ (8.4)

and the cost functional (8.3) can be rewritten as

$\displaystyle
J(u(\cdot))={\mathbb{E}}\bigg{[}\int_{0}^{T}l(t,x(t),u(t))dt+\Phi(x(T))\bigg{]}{\color[rgb]{0,0,1},}$
(8.5)

where we set

$\displaystyle\begin{split}&l(t,x,u)\triangleq(x,x)_{H}+(u,u)_{H},\forall x\in
H,u\in U,\\\ &\Phi(x)\triangleq(x,x)_{H},\forall x\in H.\end{split}$ (8.6)

Thus this optimal control problem is transformed into Problem 4.2 as a special
case. Under Assumptions 8.1-8.2, it is easy to check that the coefficients of
this optimal control problem satisfy Assumptions 4.1. So in this cas Theorem
6.3 and 7.1 hold. Moreover, from the a priori estimate (3.2), it is easy to
see that the cost functional $J(u(\cdot))$ is the strictly convex, coerciv
lower-semi continuous functional defined on the reflexive Banach space
${M}^{2}_{\mathscr{F}}(0,T;U).$ Therefor the uniqueness and existence of the
optimal control can be obtained by the convex optimality principle (see
Proposition 2.12 of [8]). Let $(\bar{u}(\cdot),\bar{X}(\cdot))$ be the optimal
pair. In the following, we will give the duality characterization of the
optimal control $\bar{u}(\cdot)$ by the maximum principle. More precisely, in
this case the corresponding Hamiltonian ${\cal H}$ becomes

$\displaystyle{\cal
H}(t,x,u,p,q,r):=\left(u,p\right)_{H}+\left(u,q\right)_{H}+\left(\Gamma(t)x+u,r(t)\right)_{l^{2}(H)}+(x,x)_{H}+(u,u)_{H}.$
(8.7)

Let $(\bar{u}(\cdot);\bar{X}(\cdot))$ be an optimal pair. Then the
corresponding adjoint equation becomes

$\displaystyle\begin{split}\left\\{\begin{array}[]{ll}d\bar{p}(t)=&-\bigg{[}A^{*}(t)\bar{p}(t)+B^{*}(t)\bar{q}(t)+\displaystyle\sum_{i=1}^{\infty}\Gamma^{i*}(t)\bar{r}^{i}(t)+2\bar{X}(t)\bigg{]}dt+\bar{q}(t)dW(t)+\displaystyle\sum_{i=1}^{\infty}\bar{r}^{i}(t)dH^{i}(t),0\leqslant
t\leqslant T,\\\ \bar{p}(T)=&2\bar{X}(T),\end{array}\right.\end{split}$ (8.8)

where

$\displaystyle
A^{*}(t)\phi(z)\triangleq-\partial_{z^{i}}[a^{ij}(t,z)\partial_{z^{j}}\phi(z)]+\partial_{z^{i}}[b^{i}(t,z)\phi(z)]+c(t,z)\phi(z),\quad\forall\phi\in
V,$ $\displaystyle
B^{*}(t)\phi(z)\triangleq-\eta^{i}(t,z)\partial_{z^{i}}\phi(z),\quad\forall\phi\in
H,$
$\displaystyle\Gamma^{i*}(t)\phi(z)\triangleq\Gamma^{i}(t,z)\phi(z),\quad\forall\phi\in
H.$

Since ${\mathscr{U}}=U$, there is no constraint on the control and therefore
the minimum condition (6.13) becomes

$\displaystyle{\cal
H}_{u}(t,\bar{X}(t-),\bar{u}(t),\bar{p}(t-),\bar{q}(t),\bar{r}(t))=0,$ (8.9)

which imply that

$\displaystyle 2{\bar{u}}(t)+\bar{p}(t-)+\bar{q}(t)+\bar{r}(t)=0,$ (8.10)

a.e. $t\in[0,T]$, ${\mathbb{P}}$-a.s.. Thus the optimal control
${\bar{u}}(\cdot)$ is given by

$\displaystyle{\bar{u}}(t)=-\frac{1}{2}\bigg{[}\bar{p}(t-)+\bar{q}(t)+\bar{r}(t)\bigg{]}.$

######  Remark 8.1.

The above example can be regarded as a special case of the infinite-
dimensional linear-quadratic control problem driven by Teugels martiangles
which can also be applied to some more practical problems such as the partial
observation optimal control driven by Teugels martingales and the optimal
harvesting problem associated with Lévy processes and so on. And we will give
detailed investigations on these applications in our future publication.

##  9 Conclusion

In this paper, we have developed an infinite-dimensional optimal control
problem of the stochastic evolution system driven by Teugels martingales. We
have considered the control variable enters the diffusion of the state
equation and the control domain is convex. We first provided the existence
uniqueness and continuous dependence theorems of solutions to SEE driven by
Teugels martingales. Then we established necessary and sufficient conditions
for optimal controls in the form of maximum principles by convex variational
technique. As an application, we considered an optimal control problem of a
Cauchy problem for a controlled stochastic partial differential equation and
obtained the dual characterization of the optimal control in terms of the
solution to the corresponding stochastic Hamiltonian system. And further
investigates will be carried out on the optimal control problem under convex
control domain assumption and more practical applications in our future
publications

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