# The Global Maximum Principle for Progressive Optimal Control of Partially
Observed Forward- Backward Stochastic Systems with Random Jumps ††thanks: This
work is supported by National Natural Science Foundations of China (Grant Nos.
11971266, 11831010, 11571205), and Shandong Provincial Natural Science
Foundations (Grant Nos. ZR2020ZD24, ZR2019ZD42).

Yueyang Zheng , Jingtao Shi School of Mathematics, Shandong University, Jinan
250100, P.R.China, E-mail<EMAIL_ADDRESS>author.
School of Mathematics, Shandong University, Jinan 250100, P.R.China, E-mail:
<EMAIL_ADDRESS>
Abstract: In this paper, we study a partially observed progressive optimal
control problem of forward-backward stochastic differential equations with
random jumps, where the control domain is not necessarily convex, and the
control variable enter into all the coefficients. In our model, the
observation equation is not only driven by a Brownian motion but also a
Poisson random measure, which also have correlated noises with the state
equation. For preparation, we first derive the existence and uniqueness of the
solutions to the fully coupled forward-backward stochastic system with random
jumps and its estimation in $L^{\beta}(\beta\geq 2)$-space under some
assumptions, and the non-linear filtering equation of partially observed
stochastic system with random jumps. Then we derive the partially observed
global maximum principle with random jumps. To show its applications, a
partially observed linear quadratic progressive optimal control problem with
random jumps is investigated, by the maximum principle and stochastic
filtering. State estimate feedback representation of the optimal control is
given in a more explicit form by introducing some ordinary differential
equations.

Keywords: Global maximum principle, forward-backward stochastic differential
equation, partial observation, random jumps, stochastic filtering,
$L^{\beta}$-estimate

Mathematics Subject Classification: 93E20, 49K45, 49N10, 49N70, 60H10

##  1 Introduction

In modern stochastic control theory, it is acknowledged that the main way to
obtain the global maximum principle is the spike variational technique when
the control domain is not necessarily convex. But different from the classical
stochastic control problem, when the diffusion term contains the control
variable, first-order expansion is not enough to formulate the variational
equation for the state process $x^{u}$ which generally is the adapted solution
to the following controlled stochastic differential equation (SDE for short):

$\left\\{\begin{aligned}
dx_{t}^{u}&=b(t,x_{t}^{u},u_{t})dt+\sigma(t,x_{t}^{u},u_{t})dW_{t},\quad
t\in[0,T],\\\ x_{0}^{u}&=x,\end{aligned}\right.$

where $W$ is a Brownian motion. The main reason is the Itô’s integral
$\int_{t}^{t+\epsilon}\sigma(s,x_{s}^{u},u_{s})dW_{s}$ is only of order
$O(\epsilon^{\frac{1}{2}})$. In 1990, Peng [24] first introduced the second-
order term in the Taylor’s expansion of the variation and completely obtained
the global maximum principle for the stochastic control problem.

Consider the following controlled backward stochastic differential equation
(BSDE for short), which is coupled with the above controlled SDE:

$\left\\{\begin{aligned}
-dy_{t}^{u}&=g(t,x_{t}^{u},y_{t}^{u},z_{t}^{u},u_{t})dt-z_{t}^{u}dW_{t},\quad
t\in[0,T],\\\ y_{T}^{u}&=\phi(x_{T}^{u}).\end{aligned}\right.$ (1.1)

Pardoux and Peng [23] first obtained its existence and uniqueness result under
the assumption that the generator $g$ is linear growth and Lipschitz
continuous in $(y,z)$. Independently, Duffie and Epstein [5] introduced the
notion of recursive utilities in continuous time which is a type of BSDE where
$g$ is independent of $z$. In El Karoui et al. [13], a general stochastic
control problem with recursive utilities by the solutions to BSDEs, where the
generator $g$ contains $z$, is studied. When the control domain is convex, the
global maximum principle reduces to the so-called local maximum principle
which is derived by the classical convex variation instead of the spike
variation. Peng [25] first established a local maximum principle for the
stochastic recursive optimal control problem, where the state equation of the
problem is a controlled forward-backward stochastic differential equation
(FBSDE for short). Wu [53] proved a local maximum principle for the optimal
control problem of fully coupled FBSDEs. Wu [49] and Yong [66] obtained global
maximum principles for optimal control problems of decoupled and fully coupled
FBSDEs, respectively.

Next, consider the following backward stochastic differential equation with
random jumps (BSDEP for short):

$\left\\{\begin{aligned}
-dy_{t}^{u}&=g(t,x_{t}^{u},y_{t}^{u},z_{t}^{u},\tilde{z}^{u}_{(t,\cdot)},u_{t})dt-
z_{t}^{u}dW_{t}-\int_{\mathcal{E}}\tilde{z}_{(t,e)}\tilde{N}(de,dt),\quad
t\in[0,T],\\\ y_{T}^{u}&=\phi(x_{T}^{u}),\end{aligned}\right.$

where $\tilde{N}(\cdot,\cdot)$ is a Poisson random martingale measure. Under
standard assumptions, Tang and Li [41] first proved their existence and
uniqueness results for general BSDEPs, which are derived as adjoint equations
when they studied the global maximum principle for the controlled stochastic
differential equation with random jumps (SDEP for short). Wu [51] studied the
existence and uniqueness result of the solutions to fully coupled forward-
backward SDEP (FBSDEP for short). Situ [35] first obtained the global maximum
principle for the controlled SDEP where the control variable is not included
in the jump coefficient. Tang and Li [41] completely proved the global maximum
principle where the control variable enters into both diffusion and jump
coefficients. Shi and Wu [33] extended the result of Peng [25] to
discontinuous case, and proved a local maximum principle for a kind of
controlled FBSDEPs. Recently, Song et al. [38] found that, in the work of
[41], the third estimate in (2.10) is not correct, which was also put forward
by Situ [37] in the early, and the subsequent variational calculus then
underestimated the influence of jumps. In [38], they overcome the deficiencies
with a new spike variation technique and obtained a new global maximum
principle for controlled SDEPs.

An inspiring thought appearing in the nonlinear expectation theory motivates
us to study the optimal control problem of FBSDEP. We consider an SDEP as
follows:

$\left\\{\begin{aligned}
dx_{t}^{u}&=b(t,x_{t}^{u},u_{t})dt+\sigma(t,x_{t}^{u},u_{t})dW_{t}+\int_{\mathcal{E}}f(t,x_{t-}^{u},u_{t},e)\tilde{N}(de,dt),\quad
t\in[0,T],\\\ x_{0}^{u}&=x_{0},\end{aligned}\right.$ (1.2)

and the cost functional can be formulated as:

$J_{g^{p}}(u)=\mathcal{E}_{g^{p}}\bigg{[}\int_{0}^{T}l(t,x_{t}^{u},u_{t})dt+\Phi(x_{T}^{u})\bigg{]},$
(1.3)

where $\mathcal{E}_{g^{p}}[\cdot]$ is a nonlinear expectation called
$g^{p}$-expectation, which is related to the BSDEP:

$\left\\{\begin{aligned}
-d\eta_{t}&=g(t,\zeta_{t},\lambda_{(t,\cdot)})dt-\zeta_{t}dW_{t}-\int_{\mathcal{E}}\lambda_{(t,e)}\tilde{N}(de,dt),\quad
t\in[0,T],\\\ \eta_{T}&=\xi.\end{aligned}\right.$ (1.4)

If we set $g(t,0,0)\equiv 0$, then we can define a $g^{p}$-expectation in the
following.

######  Definition 1.1.

For any $\xi\in L^{2}_{\mathcal{F}_{T}}$, we set
$\mathcal{E}_{g^{p}}[\xi]=\eta_{0}$.

It is known that in 1997, Peng [26] introduced the notion of $g$-expectation,
for a random variable $\xi$, it is defined as the initial value of a classical
BSDE driven by a generator $g$ and with a terminal value $\xi$. More
precisely, when we consider the following BSDE without jumps:

$\left\\{\begin{aligned} -dy_{t}&=g(t,y_{t},z_{t})dt-z_{t}dW_{t},\quad
t\in[0,T],\\\ y_{T}&=\xi,\end{aligned}\right.$ (1.5)

then we denote $g$-expectation by the operator $\mathcal{E}_{g}[\cdot]$ and
define $\mathcal{E}_{g}[\xi]=y_{0}$. More properties about $g$-expectation can
be seen in [27]. In 2006, Royer [31] generalized the notion of $g$-expectation
and introduced $f$-expectation related to some BSDEP driven by a generator $f$
and with terminal value $\xi$, under which the filtration is generated by both
Brownian motion and Poisson random measure. Under some conditions,
$f$-expectation, like $g$-expectation, preserve most properties of the
classical expectation except the linearity. Here, in (1.3) of this paper, we
put a superscript $p$ on $g$, i.e. $g^{p}$, to represent some $f$-expectation
with jumps which is distinct form the $g$-expectation without jumps. Moreover,
$\mathcal{E}_{g^{p}}[\cdot]$ and $\mathcal{E}_{g}[\cdot]$ can be reduced to
classical expectation $E[\cdot]$ provided that $g=0$.

Then the aim is to find an optimal control $\bar{u}\in\mathcal{U}_{ad}$ such
that

$J_{g^{p}}(\bar{u})=\inf_{u\in\,\mathcal{U}_{ad}}J_{g^{p}}(u),$ (1.6)

where $\mathcal{U}_{ad}$ is a certain admissible control set. Next, we
consider that $\xi=\int_{0}^{T}l(t,x_{t}^{u},u_{t})dt+\Phi(x_{T}^{u})$, then
we get

$\eta_{T}=\int_{0}^{T}l(t,x_{t}^{u},u_{t})dt+\Phi(x_{T}^{u}),$ (1.7)

and define

$y^{u}_{t}=\eta_{t}-\int_{0}^{t}l(s,x_{s}^{u},u_{s})ds,\quad
z^{u}_{t}=\zeta^{u}_{t},\quad\tilde{z}^{u}_{(t,\cdot)}=\lambda^{u}_{(t,\cdot)},$
(1.8)

where $(y^{u},z^{u},\tilde{z}^{u})$ satisfies the following BSDEP coupled with
(1.2):

$\left\\{\begin{aligned}
-dy^{u}_{t}&=\big{[}g(t,z_{t}^{u},\tilde{z}_{(t,\cdot)}^{u})+l(t,x_{t}^{u},u_{t})\big{]}dt-z^{u}_{t}dW_{t}-\int_{\mathcal{E}}\tilde{z}^{u}_{(t,e)}\tilde{N}(de,dt),\quad
t\in[0,T],\\\ y^{u}_{T}&=\Phi(x_{T}^{u}).\end{aligned}\right.$ (1.9)

According to the discussion above, we can transform the original optimal
control problem with $g^{p}$-expectation into a problem of FBSDEP with
classical expectation. In this case, the state equation becomes

$\left\\{\begin{aligned}
dx_{t}^{u}&=b(t,x_{t}^{u},u_{t})dt+\sigma(t,x_{t}^{u},u_{t})dW_{t}+\int_{\mathcal{E}}f(t,x_{t-}^{u},u_{t},e)\tilde{N}(de,dt),\\\
-dy^{u}_{t}&=\big{[}g(t,z_{t}^{u},\tilde{z}_{(t,\cdot)}^{u})+l(t,x_{t}^{u},u_{t})\big{]}dt-z^{u}_{t}dW_{t}-\int_{\mathcal{E}}\tilde{z}^{u}_{(t,e)}\tilde{N}(de,dt),\quad
t\in[0,T],\\\ x_{0}^{u}&=x_{0},\
y^{u}_{T}=\Phi(x_{T}^{u}),\end{aligned}\right.$ (1.10)

with the cost functional

$J_{g^{p}}(u)=y_{0}^{u},$ (1.11)

and the aim is transformed to minimize the cost functional (1.11) subject to
(1.10), which can be regarded as a special recursive optimal control problem.

In the literature, when the control domain in nonconvex, it is difficult to
derive the first-order and second-order Taylor’s expansions for BSDE (1.1),
which is proposed as an open problem in Peng [28]. Recently, Hu [8] solved
this open problem and obtained a novel global maximum principle. In [8], the
relationship among the terms of the first-order Taylor expansions, i.e.,

$\displaystyle y^{1}_{t}$ $\displaystyle=p_{t}x^{1}_{t},\ z^{1}_{t}$
$\displaystyle=p_{t}\delta\sigma(t)\mathbb{I}_{E_{\epsilon}}(t)+[\sigma_{x}(t)p_{t}+q_{t}]x^{1}_{t},$

is built, where $(p,q)$ is the solution to the first-order adjoint equation
which possesses a linear generator. More importantly, a novel feature is that
the variation of $z_{t}$ contains an additional term
$p_{t}\delta\sigma(t)\mathbb{I}_{E_{\epsilon}}(t)$. Another novel idea is to
apply Taylor’s expansion to
$\bar{z}_{t}+p_{t}\delta\sigma(t)\mathbb{I}_{E_{\epsilon}}(t)$ and then to
derive the global maximum principle. Motivated by these ideas, Hu et al. [9]
further extended the global maximum principle to the fully coupled controlled
FBSDE. In [9], they decoupled the fully coupled linear first-order and second-
order forward-backward variation equations by establishing the following
relationships:

$\displaystyle y^{1}_{t}$ $\displaystyle=p_{t}x^{1}_{t},\ z^{1}_{t}$
$\displaystyle=\Delta(t)\mathbb{I}_{E_{\epsilon}}(t)+k_{1}(t)x^{1}_{t},$

where
$k_{1}(t)=(1-p_{t}\sigma_{z}(t))^{-1}[\sigma_{x}(t)p_{t}+\sigma_{y}(t)p_{t}^{2}+q_{t}]$
and $\Delta(t)$ satisfies an algebra equation

$\displaystyle\Delta(t)=p_{t}\big{[}\sigma(t,\bar{x}_{t},\bar{y}_{t},\bar{z}_{t}+\Delta(t),u_{t})-\sigma(t,\bar{x}_{t},\bar{y}_{t},\bar{z}_{t},\bar{u}_{t})\big{]},$

with $(p,q)$ is the solution to the adjoint equation with a quadratic
generator. They also applied Taylor’s expansion to
$\bar{z}_{t}+\Delta(t)\mathbb{I}_{E_{\epsilon}}(t)$ and derived the global
maximum principle by assuming $q$ is bounded and unbounded, respectively.

However, in the above, the controlled systems are all considered under
complete information. In practice, the state equation usually could not be
observed directly. There are two kinds of incomplete information forms. The
first one is that one could only observe a related stochastic process $Y$
which usually satisfies an SDE:

$\left\\{\begin{aligned} dY_{t}&=h(t,x_{t},u_{t})dt+\sigma(t)dW_{t},\quad
t\in[0,T],\\\ Y_{0}&=0.\end{aligned}\right.$

The second one is that the known information is generated by a standard
Brownian noise, which, in fact, can be regarded as a special case of the first
situation.

For stochastic control problems with partial information, there has been
considerable literature about forward-backward stochastic controlled systems.
Wang and Wu [43] obtained a global maximum principle for partially observed
stochastic recursive optimal control problem, under the assumption that
control domain is not necessarily convex and the forward diffusion coefficient
does not contain control variable. Wu [50] studied a partially observed
stochastic control problem of FBSDEs and a local maximum principle was proved.
Shi and Wu [34] obtained a partially observed maximum principle of controlled
fully coupled FBSDEs on the assumption that the forward diffusion coefficient
does not contain the control variable and the control domain is not
necessarily convex. Wang et al. [44] studied a partial observed stochastic
control problem of FBSDEs with correlated noises between the system and the
observation. Three versions of maximum principle were established by using a
direct method, an approximation method and a Malliavin derivative method,
respectively. By introducing a state decomposition and backward separation
approach, Wang et al. [45] studied an linear-quadratic (LQ for short)
stochastic control problem of FBSDEs, where the drift coefficient of the
observation equation is linear with respect to state $x$, and the observation
noise is correlated with the state noise. Some recent progress for partially
observed problems, has been made in mean-field type controls (see [20], [47],
[7]), differential games (see [48], [11], [54], [61], [68]), random jumps (see
[22], [40], [55], [56], [57], [67]), time delay (see [17]) and applications in
finance (see [60], [62]). The partially observed stochastic control problem is
usually associated with the state filtering and estimation technique, and we
only list some main references such as [19], [2], [59], [42], [57], [46],
[37].

The motivation of our work in this paper comes from two aspects. The first one
is Hu [8], in which a global maximum principle for optimal control problem
with recursive utilities has been obtained completely. The second one is Song
et al. [38], in which a new global maximum principle for stochastic control
problem of SDEPs, where the control variable enters into both the coefficients
of diffusion and jump, has been derived by using a novel spike variation
technique, which modifies a critical incorrectness in Tang and Li [41], which
has been pointed out in the book of Situ [37] for over sixteen years.
Combining the two aspects above with the partially observed feature, in this
paper, we aim to study a global maximum principle for partially observed
controlled forward-backward stochastic system with random jumps, where all the
coefficients contain the control variable. We note that, very recently, Hao
and Meng [7] obtained a global maximum principle for optimal control of the
general mean-field forward-backward stochastic system with jumps. However, the
jump term in the SDEP of their paper does not contain the control variable and
the cost functional is a special recursive case.

To our best knowledge, there are rare literature on the partially observed
forward-backward stochastic optimal control problem with random jumps. In this
paper, we initiate to prove the partially observed global maximum principle
with random jumps, where the state noise is correlated with the observation
noise. Our work distinguishes itself from the existing literatures in the
following aspects:

(1) We extend the work in [8] into the partially observed case with random
jumps. There exist correlated noises between the state equation and the
observation equation, which make the original decoupled forward-backward
stochastic system become a kind of fully-coupled one. We first prove the
existence and uniqueness of the solution to fully coupled FBSDEPs in
$L^{2}$-space on small interval. However, due to some technical difficulties,
we only obtain the existence and uniqueness of the solution to the fully
coupled FBSDEPs and give its estimate under (A4), (A6) in
$L^{\beta}(\beta>2)$-space, which, motivated by the recent work of Meng and
Yang [21] in which they proved that an adapted $L^{2}$-solution is an
$L^{\beta}$-solution for some $\beta>2$, extends to a general case for any
given terminal time $T$. For further resolving the difficulties, we relax the
Assumption (A6), and obtain the $L^{\beta}(\beta>2)$-solution to fully coupled
FBSDEPs and give its estimate under (A4), (A5) by defined a new space
$\tilde{F}^{2,\beta}[0,T]$ for any given $T$.

(2) Then we derive the partially observed global maximum principle. Compared
to [41] and [40], our result is both a correction and an extension. To our
best knowledge, in the optimal control problems, we initiate to consider that
the observation equation is not only driven by the Brownian motion, but also
by the Poisson random measure. Moreover, the correlated noises between state
and the observation equation not only come from the Brownian motion, but from
the Poisson random measure. It is an extension of the case in [57], in which
an LQ problem with special Poisson process and convex control domain is
considered.

(3) Similar to [8], we also guess that
$z_{t}^{1,1},z_{t}^{2,1},\tilde{z}_{(t,e)}^{1,1},\tilde{z}_{(t,e)}^{2,1}$ have
the following form, respectively:

$\displaystyle z_{t}^{1,1}$
$\displaystyle=\Delta^{1}(t)\mathbbm{1}_{[\bar{t},\bar{t}+\epsilon]}+z_{t}^{1,1^{\prime}},\
z_{t}^{2,1}=\Delta^{2}(t)\mathbbm{1}_{[\bar{t},\bar{t}+\epsilon]}+z_{t}^{2,1^{\prime}},$
(1.12) $\displaystyle\tilde{z}_{(t,e)}^{1,1}$
$\displaystyle=\tilde{\Delta}^{1}(t)\mathbbm{1}_{\mathcal{O}}+\tilde{z}_{(t,e)}^{1,1^{\prime}},\
\tilde{z}_{(t,e)}^{2,1}=\tilde{\Delta}^{2}(t)\mathbbm{1}_{\mathcal{O}}+\tilde{z}_{(t,e)}^{2,1^{\prime}},$

where
$\Delta^{1}(\cdot),\Delta^{2}(\cdot),\tilde{\Delta}^{1}(\cdot),\tilde{\Delta}^{2}(\cdot)$
are four $\mathcal{F}_{t}$-adapted processes, and
$z^{1,1^{\prime}},z^{2,1^{\prime}},\tilde{z}^{1,1^{\prime}},\tilde{z}^{2,1^{\prime}}$
have good estimates similarly as $x^{1}$. However, different from the results
in [8] or [9], we find
$\tilde{\Delta}^{1}(\cdot)=\tilde{\Delta}^{2}(\cdot)=0$. The main reason is
that we subtract the jump part in the random interval $\mathcal{O}$. There is
no need for such a term
$\tilde{\Delta}^{1}(t)\mathbbm{1}_{\mathcal{O}},\tilde{\Delta}^{2}(t)\mathbbm{1}_{\mathcal{O}}$
in the third and fourth equality in (1.12), which is in accordance with the
variational equation of $x$ in Section 3.1. Therefore, we mainly do Taylor
expansions at
$\bar{x}_{t},\bar{y}_{t},\bar{z}^{1}_{t}+p_{t}\delta\sigma_{1}(t)\mathbb{I}_{E_{\epsilon}}(t),\bar{z}^{2}_{t}+p_{t}\delta\sigma_{2}(t)\mathbb{I}_{E_{\epsilon}}(t)$
and $\bar{\tilde{z}}_{t}^{1},\bar{\tilde{z}}_{t}^{2}$ for
$\tilde{b}_{1},\sigma_{1},\sigma_{2},f_{1},f_{2},l$, respectively.

(4) Compared to the work in [8], we extend the cost functional into a general
case, which is similar to the LQ part in [9]. However, two new adjoint
equations are introduced. The variational inequality is firstly derived.
Compared to the work in [8], besides the vital first-order adjoint equation
with Poisson measure, we can obtain four new adjoint equations, however, the
second-order adjoint equation is a new BSDEP. Then we derive a new partially
observed global maximum principle. More importantly, the maximum principle can
reduce to the work [18] when the forward-backward system with random jumps
degenerate into the forward one without random jumps. Different from the
framework in Li and Tang [18] (or Tang [39]) where the observed process $Y$ is
viewed as a Brownian motion at the beginning, we not only consider a more
natural framework(Hu et al. [10], Zhang et al. [67]), but also prove the
equivalence between the two frameworks. More details can be seen in Remark
2.2, Remark 3.2, Remark 3.3.

(5) We consider a special partially observed LQ forward-backward stochastic
optimal control problem with random jumps. By the method of [37], we first
deduce a non-linear filtering equation for partially observed stochastic
system with random jumps, which extends the case with Poisson process into the
case with Poisson martingale measure. And we obtain the state estimate
feedback of the optimal control in a closed-loop form. Moreover, an explicit
representation for the optimal control is given by some ordinary differential
equations (ODEs for short).

The rest of this paper is organized as follows. In Section 2, we formulate a
general partially observed progressive optimal control problem of forward-
backward stochastic system with random jumps, and prove the existence and
uniqueness of the solution to the fully coupled FBSDEPs and its
$L^{\beta}$-estimate under some assumptions and different spaces, for any
$\beta\geq 2$. In Section 3, we derive the partially observed global maximum
principle of FBSDEPs with general cost functional. In Section 4, as an
application, our theoretical results are applied to study a partially observed
LQ optimal control problem of FBSDEPs. In a special case, we first give the
maximum condition for the optimal control and verify its sufficiency. We then
represent the optimal control as a state estimate feedback form. Moreover, we
obtain a more explicit optimal control with some ODEs. Section 5 gives some
concluding remarks. In the Appendix, we give some results about non-linear
filtering equation of signal-observation system with Poisson martingale
measure that will be used in the LQ problem in Section 4.

##  2 Problem Formulation and Preliminaries

Let $T>0$ be fixed. Consider a complete filtered probability space
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{0\leq t\leq T},\bar{P})$ and two one-
dimensional independent standard Brownian motions $W^{1}$ and $\tilde{W}^{2}$
defined in $\mathbb{R}^{2}$ with $W^{1}_{0}=\tilde{W}^{2}_{0}=0$. Let
$(\mathcal{E},\mathcal{B}(\mathcal{E}))$ be a Polish space with the
$\sigma$-finite measure $\nu_{1}$ on $\mathcal{E}_{1}$, $\nu_{2}$ on
$\mathcal{E}_{2}$ and
$\mathcal{E}_{1}\subset\mathcal{E},\mathcal{E}_{2}\subset\mathcal{E}$. Suppose
that $N_{1}(de,dt)$ is a Poisson random measure on
$(\mathbb{R}^{+}\times\mathcal{E}_{1},\mathcal{B}(\mathbb{R}^{+})\times\mathcal{B}(\mathcal{E}_{1}))$
under $\bar{P}$ and for any $E_{1}\in\mathcal{B}(\mathcal{E}_{1})$,
$\nu_{1}(E_{1})<\infty$, then the compensated Poisson random measure is given
by $\tilde{N}_{1}(de,dt)=N_{1}(de,dt)-\nu_{1}(de)dt$. Let $N_{2}(de,dt)$ be an
integer-valued random measure and its predictable compensator is given by
$\lambda(t,x_{t-},e)\nu_{2}(de)dt$, where the function
$\lambda(t,x,e)\in[l,1),0<l<1$, and for any
$E_{2}\in\mathcal{B}(\mathcal{E}_{2})$, $\nu_{2}(E_{2})<\infty$, then its
compensated random measure is given by
$\tilde{N}^{\prime}_{2}(de,dt)=N_{2}(de,dt)-\lambda(t,x_{t-},e)\nu_{2}(de)dt$.
Moreover, $W^{1},\tilde{W}^{2},N_{1},N_{2}$ are mutually independent under
$\bar{P}$, and let
$\mathcal{F}_{t}^{W^{1}},\mathcal{F}_{t}^{\tilde{W}^{2}},\mathcal{F}_{t}^{N_{1}},\mathcal{F}_{t}^{N_{2}}$
be the $\bar{P}$-completed natural filtrations generated by
$W^{1},\tilde{W}^{2},N_{1},N_{2}$, respectively. Set
$\mathcal{F}_{t}:=\mathcal{F}_{t}^{W^{1}}\vee\mathcal{F}_{t}^{\tilde{W}^{2}}\vee\mathcal{F}_{t}^{N_{1}}\vee\mathcal{F}_{t}^{N_{2}}\vee\mathcal{N}$
and $\mathcal{F}=\mathcal{F}_{T}$, where $\mathcal{N}$ denotes the totality of
$\bar{P}$-null sets. $\bar{E}$ denotes the expectation under the probability
measure $\bar{P}$.

Different from [41] and [40], the integrand of the stochastic integral in our
paper is $\mathcal{E}$-progressive measurable rather than
$\mathcal{E}$-predictable, which is referred to [38]. We first introduce some
preliminaries from [38].

######  Definition 2.1.

Suppose that $\mathcal{H}$ is an Euclidean space, and
$\mathcal{B}(\mathcal{H})$ is the Borel $\sigma$-field on $\mathcal{H}$. Given
$T>0$, a process $x:[0,T]\times\Omega\rightarrow\mathcal{H}$ is called
_progressive measurable (predictable)_ if $x$ is
$\mathcal{G}/\mathcal{B}(\mathcal{H})(\mathcal{P}/\mathcal{B}(\mathcal{H}))$
measurable, where $\mathcal{G}(\mathcal{P})$ is the corresponding progressive
measurable (predictable) $\sigma$-field on $[0,T]\times\Omega$, and a process
$x:[0,T]\times\Omega\times\mathcal{E}\rightarrow\mathcal{H}$ is called
_$\mathcal{E}$ -progressive measurable ($\mathcal{E}$-predictable)_ if $x$ is
$\mathcal{G}\times\mathcal{B}(\mathcal{E})/\mathcal{B}(\mathcal{H})(\mathcal{P}\times\mathcal{B}(\mathcal{E})/\mathcal{B}(\mathcal{H}))$
measurable.

Now, given a process $x$ which has RCLL paths, $x_{0-}:=0$ and $\Delta
x_{t}:=x_{t}-x_{t-},t\geq 0$, for $i=1,2$, let $m_{i}$ denote the measure on
$\mathcal{F}\otimes\mathcal{B}([0,T])\otimes\mathcal{B}(\mathcal{E}_{i})$
generated by $N_{i}$ that
$m_{i}(A)=\bar{E}\int_{0}^{T}\int_{\mathcal{E}_{i}}\mathbb{I}_{A}N_{i}(de,dt)$.
For any
$\mathcal{F}\otimes\mathcal{B}([0,T])\otimes\mathcal{B}(\mathcal{E}_{i})/\mathcal{B}(\mathbb{R})$
measurable integrable process $x$, we set $\mathbb{E}_{i}[x]:=\int xdm_{i}$
and denote by
$\mathbb{E}_{i}[x|\mathcal{P}\otimes\mathcal{B}(\mathcal{E}_{i})]$ the Radon-
Nikodym derivatives w.r.t. $\mathcal{P}\otimes\mathcal{B}(\mathcal{E}_{i})$.
In fact, $\mathbb{E}_{i}$ is not an expectation (for $m_{i}$ is not a
probability measure), but it owns similar properties to expectation. A more
general definition of stochastic integral of random measure has been
introduced by [38] where the theory of dual predictable projection is
utilized. Therefore, we omit the details here and give the following lemma
directly.

######  Lemma 2.1.

If $g$ is a positive $\mathcal{E}_{i}$-progressive measurable process that

$\bar{E}\int_{0}^{T}\int_{\mathcal{E}_{i}}gN_{i}(de,dt)<\infty,\ i=1,2,$

then we have the following results:

$\displaystyle(i)\qquad\bigg{(}\int_{0}^{\cdot}\int_{\mathcal{E}_{i}}gN_{i}(de,dt)\bigg{)}_{t}^{p}=\int_{0}^{t}\int_{\mathcal{E}_{i}}\mathbb{E}_{i}\big{[}g|\mathcal{P}\otimes\mathcal{B}(\mathcal{E}_{i})\big{]}\nu_{i}(de)dt,$
$\displaystyle(ii)\qquad\int_{0}^{T}\int_{\mathcal{E}_{i}}g\tilde{N}_{i}(de,dt)=\int_{0}^{T}\int_{\mathcal{E}_{i}}gN_{i}(de,dt)-\bigg{(}\int_{0}^{\cdot}\int_{\mathcal{E}_{i}}gN_{i}(de,dt)\bigg{)}_{T}^{p},$
$\displaystyle(iii)\qquad\int_{0}^{T}\int_{\mathcal{E}_{i}}g\tilde{N}_{i}(de,dt)=\int_{0}^{T}\int_{\mathcal{E}_{i}}gN_{i}(de,dt)-\int_{0}^{T}\int_{\mathcal{E}_{i}}\mathbb{E}_{i}\big{[}g|\mathcal{P}\otimes\mathcal{B}(\mathcal{E}_{i})\big{]}\nu_{i}(de)dt,$
$\displaystyle(iv)\qquad\bar{E}\int_{0}^{T}\int_{\mathcal{E}_{i}}gN_{i}(de,dt)=\bar{E}\int_{0}^{T}\int_{\mathcal{E}_{i}}\mathbb{E}_{i}\big{[}g|\mathcal{P}\otimes\mathcal{B}(\mathcal{E}_{i})\big{]}\nu_{i}(de)dt,$
$\displaystyle(v)\qquad\Delta(g_{\cdot}\tilde{N}_{i})_{t}=\int_{\mathcal{E}_{i}}gN_{i}(de,\\{t\\}),$
$\displaystyle(vi)\qquad\big{[}g_{\cdot}\tilde{N}_{i,t},g_{\cdot}\tilde{N}_{i,t}\big{]}=\int_{0}^{t}\int_{\mathcal{E}_{i}}g^{2}N_{i}(de,dt),$

where $x^{p}$ is the dual predictable projection of $x$.

Consider the following state equation which is a controlled FBSDEP:

$\left\\{\begin{aligned}
dx_{t}^{u}&=b_{1}(t,x_{t}^{u},u_{t})dt+\sigma_{1}(t,x_{t}^{u},u_{t})dW^{1}_{t}+\sigma_{2}(t,x_{t}^{u},u_{t})d\tilde{W}^{2}_{t}\\\
&\quad+\int_{\mathcal{E}_{1}}f_{1}(t,x_{t-}^{u},u_{t},e)\tilde{N}_{1}(de,dt)+\int_{\mathcal{E}_{2}}f_{2}(t,x_{t-}^{u},u_{t},e)\tilde{N}^{\prime}_{2}(de,dt),\\\
-dy_{t}^{u}&=g\big{(}t,x_{t}^{u},y_{t}^{u},z^{1,u}_{t},z^{2,u}_{t},\int_{\mathcal{E}_{1}}\tilde{z}^{1,u}_{(t,e)}\nu_{1}(de),\int_{\mathcal{E}_{2}}\tilde{z}^{2,u}_{(t,e)}\nu_{2}(de),u_{t}\big{)}dt-z^{1,u}_{t}dW^{1}_{t}\\\
&\quad-z^{2,u}_{t}dW^{2}_{t}-\int_{\mathcal{E}_{1}}\tilde{z}^{1,u}_{(t,e)}\tilde{N}_{1}(de,dt)-\int_{\mathcal{E}_{2}}\tilde{z}^{2,u}_{(t,e)}\tilde{N}_{2}(de,dt),\quad
t\in[0,T],\\\ x_{0}^{u}&=x_{0},\ \
y_{T}^{u}=\phi(x_{T}^{u}),\end{aligned}\right.$ (2.1)

where, as we concluded in the following Remark 2.2, the first framework where
probability measure $P$ is given originally is equivalent to the second
framework (as presented in our paper) where probability measure $\bar{P}$ is
given originally. Therefore, by the following Girsanov’s theorem method,
$W^{1},W^{2}$ are mutually independent Brownian motions and
$\tilde{N}_{1},\tilde{N}_{2}$ are mutually independent Poisson martingale
measures under $P$ and $W^{1},\tilde{W}^{2}$ are mutually independent Brownian
motions and $\tilde{N}_{1},\tilde{N}^{\prime}_{2}$ are mutually independent
Poisson martingale measures under $\bar{P}$. Then we give the fact that the
stochastic process $W^{2}$ and Poisson random measure $\tilde{N}_{2}$ under
$\bar{P}$ is standard Brownian motion and Poisson martingale measure under
$P$, respectively. Equivalently, we also have the understanding that
stochastic process $\tilde{W}^{2}$ and Poisson random measure
$\tilde{N}^{\prime}_{2}$ under $P$ is standard Brownian motion and Poisson
martingale measure under ${\bar{P}}$ as presented exactly in our paper, which
is accordance with the non-jumps framework in [44]. Moreover, we have explicit
representations of $W^{2},\tilde{N}_{2}$ by
$\tilde{W}^{2},\tilde{N}^{\prime}_{2}$ as follows:

$\displaystyle dW^{2}_{t}$
$\displaystyle=d\tilde{W}^{2}_{t}+\sigma_{3}^{-1}(t)b_{2}(t,\Theta^{u}(t),u_{t})dt,$
(2.2) $\displaystyle\tilde{N}_{2}(de,dt)$
$\displaystyle=\tilde{N}^{\prime}_{2}(de,dt)+(\lambda(t,x_{t-}^{u},e)-1)\nu_{2}(de)dt.$

where $\sigma_{3}(t)$ and $b_{2}(t,\Theta^{u}(t),u_{t})$ are defined in the
following.

Suppose that the state process
$(x^{u},y^{u},z^{1,u},z^{2,u},\tilde{z}^{1,u},\tilde{z}^{2,u})$ cannot be
observed directly. Indeed, we can observe a related process $Y$ which is
governed by the following SDEP:

$\left\\{\begin{aligned}
dY_{t}^{u}&=b_{2}\big{(}t,x_{t}^{u},y_{t}^{u},z^{1,u}_{t},z^{2,u}_{t},\int_{\mathcal{E}_{1}}\tilde{z}^{1,u}_{(t,e)}\nu_{1}(de),\int_{\mathcal{E}_{2}}\tilde{z}^{2,u}_{(t,e)}\nu_{2}(de),u_{t}\big{)}dt\\\
&\quad+\sigma_{3}(t)d\tilde{W}^{2}_{t}+\int_{\mathcal{E}_{2}}f_{3}(t,e)\tilde{N}^{\prime}_{2}(de,dt),\quad
t\in[0,T],\\\ Y_{0}^{u}&=0,\end{aligned}\right.$ (2.3)

and consider the cost functional as follows:

$J(u)=\bar{E}\bigg{\\{}\int_{0}^{T}l\big{(}t,x_{t}^{u},y_{t}^{u},z_{t}^{1,u},z_{t}^{2,u},\int_{\mathcal{E}_{1}}\tilde{z}_{(t,e)}^{1,u}\nu_{1}(de),\int_{\mathcal{E}_{2}}\tilde{z}_{(t,e)}^{2,u}\nu_{2}(de),u_{t}\big{)}dt+\Phi(x_{T}^{u})+\Gamma(y_{0}^{u})\bigg{\\}},$
(2.4)

where we suppose that
$(b_{1},\sigma_{1},\sigma_{2})(t,x,u):[0,T]\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$,
$(f_{1},f_{2})(t,x,u,e):[0,T]\times\mathbb{R}\times\mathbb{R}\times\mathcal{E}_{1}(\mathcal{E}_{2})\rightarrow\mathbb{R}$,
and
$(g,b_{2},l)(t,x,y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2},u):[0,T]\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$,
$\phi(x):\mathbb{R}\rightarrow\mathbb{R}$,
$\sigma_{3}(t):[0,T]\rightarrow\mathbb{R}$,
$f_{3}(t,e):[0,T]\times\mathcal{E}_{2}\rightarrow\mathbb{R}$,
$\Phi(x):\mathbb{R}\rightarrow\mathbb{R}$,
$\Gamma(y):\mathbb{R}\rightarrow\mathbb{R}$ are suitable maps. The following
assumptions are assumed.

(A1) (1) $b_{1},\sigma_{1},\sigma_{2}$ are
$\mathcal{G}\otimes\mathcal{B}(\mathbb{R})\otimes\mathcal{B}(\mathbb{R})/\mathcal{B}(\mathbb{R})$
measurable, $f_{1},f_{2}$ are
$\mathcal{G}\otimes\mathcal{B}(\mathbb{R})\otimes\mathcal{B}(\mathbb{R})\otimes\mathcal{B}(\mathcal{E}_{1})(\mathcal{B}(\mathcal{E}_{2}))/\mathcal{B}(\mathbb{R})$
measurable, $\phi$ is $\mathcal{B}(\mathbb{R})/\mathcal{B}(\mathbb{R})$
measurable.

(2) $b_{1},\sigma_{1},\sigma_{2},f_{1}$ and $f_{2}$ are twice continuously
differentiable in $x$ with bounded first and second order derivatives,
$\sigma_{2}$ is also bounded and there is a constant $C$ such that

$\big{|}(b_{1},\sigma_{1},\sigma_{2},f_{1},f_{2})(t,x,u)\big{|}\leq
C(1+|x|+|u|).$

(3) $\phi$ are twice continuous differentiable in $x$ with bounded second
order derivatives and there is a constant $C$ such that

$|\phi(x)|\leq C(1+|x|).$

(4) For $\beta\geq 2$, the following hold:

$\displaystyle\bigg{(}\int_{0}^{T}|b_{1}(t,0,0)|dt\bigg{)}^{\beta}<\infty,\quad\bigg{(}\int_{0}^{T}|\sigma_{1}(t,0,0)|^{2}dt\bigg{)}^{\frac{\beta}{2}}<\infty,\quad\bigg{(}\int_{0}^{T}|\sigma_{2}(t,0,0)|^{2}dt\bigg{)}^{\frac{\beta}{2}}<\infty,$
$\displaystyle\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{1}}|f_{1}(t,0,0,e)|^{2}\nu_{1}(de)dt\bigg{)}^{\frac{\beta}{2}}<\infty,\quad\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{2}}|f_{2}(t,0,0,e)|^{2}\nu_{2}(de)dt\bigg{)}^{\frac{\beta}{2}}<\infty,$
$\displaystyle\bigg{(}\int_{0}^{T}|b_{2}(t,0,0,0,0,0,0,0)|dt\bigg{)}^{\beta}<\infty,\quad\bigg{(}\int_{0}^{T}|g(t,0,0,0,0,0,0,0)|dt\bigg{)}^{\beta}<\infty.$

(5) $b_{2},g$ are
$\mathcal{G}\otimes\mathcal{B}(\mathbb{R})\otimes\mathcal{B}(\mathbb{R})\otimes\mathcal{B}(\mathbb{R})\otimes\mathcal{B}(\mathbb{R})\otimes\mathcal{B}(\mathbb{R})\otimes\mathcal{B}(\mathbb{R})\otimes\mathcal{B}(\mathbb{R})/\mathcal{B}(\mathbb{R})$
measurable, and are twice continuously differentiable with respect to
$(x,y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})$;
$b_{2},g,Db_{2},Dg,D^{2}b_{2},D^{2}g$ are continuous in
$(x,y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2},u)$;
$b_{2},Db_{2},Dg,D^{2}b_{2},D^{2}g$ are bounded, and $|\sigma_{3}^{-1}(t)|$ is
bounded by some constant $C$, and

$\displaystyle\big{|}b_{2}(t,x,,y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2},u)\big{|}+\big{|}g(t,x,,y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2},u)\big{|}$
$\displaystyle\ \leq
C(1+|x|+|y|+|z^{1}|+|z^{2}|+||\tilde{z}^{1}||+||\tilde{z}^{2}||+|u|).$

(6) For any $(t,x,e)\in[0,T]\times\mathbb{R}\times\mathcal{E}_{2}$, there
exists a constant $C$ such that
$|\lambda_{x}(t,x,e)|+|\lambda_{xx}(t,x,e)|\leq C$.

(A2) $l,\Phi,\Gamma$ are twice continuous differentiable in
$(x,y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})$ with bounded second order
derivatives and there is a constant $C$ such that

$\displaystyle\big{|}l_{(x,y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})}(t,x,y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2},u)\big{|}\leq
C\big{(}1+|x|+|y|+|z^{1}|+|z^{2}|+||\tilde{z}^{1}||+||\tilde{z}^{2}||+|u|\big{)},$
$\displaystyle\big{|}l(t,x,y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2},u)\big{|}\leq
C\big{(}1+|x|^{2}+|y|^{2}+|z^{1}|^{2}+|z^{2}|^{2}+||\tilde{z}^{1}||^{2}+||\tilde{z}^{2}||^{2}+|u|^{2}\big{)},$
$\displaystyle|\Phi(x)|\leq C(1+|x|^{2}),\quad|\Phi_{x}(x)|\leq
C(1+|x|),\quad\Gamma(y)\leq C(1+|y|^{2}),\quad\Gamma_{y}(y)\leq C(1+|y|).$

######  Remark 2.1.

We consider the deterministic coefficient in our paper, that is, fixing
$(x,y,z^{1},z^{2},\\\ \tilde{z}^{1},\tilde{z}^{2},u)$,
$b_{1},b_{2},\sigma_{1},\sigma_{2},g,f_{1},f_{2},l$ are deterministic function
from $[0,T]\times\mathcal{E}_{1}(\mathcal{E}_{2})$ to $\mathbb{R}$. In fact,
we can regard $[0,T]$ as a kind of cylinder set $[0,T]\times\Omega$ in a
binary form, and here, $\mathcal{G}(\mathcal{P})$ is the corresponding
progressive measurable (predictable) $\sigma$-field on $[0,T]$ which can be
viewed as a special case on $[0,T]\times\Omega$. The only difference is that
the coefficients $b_{1},b_{2},\sigma_{1},\sigma_{2},g,f_{1},f_{2},l$ do not
contain $\omega$ exactly.

In our paper, for notational simplicity, we define
$\Theta^{u}(t):=(x_{t}^{u},y_{t}^{u},z^{1,u}_{t},z^{2,u}_{t},\tilde{z}^{1,u}_{(t,e)},\tilde{z}^{2,u}_{(t,e)})$
and

$\displaystyle\tilde{g}(t,\Theta^{u}(t),u_{t}):=\tilde{g}\big{(}t,x_{t}^{u},y_{t}^{u},z^{1,u}_{t},z^{2,u}_{t},\int_{\mathcal{E}_{1}}\tilde{z}^{1,u}_{(t,e)}\nu_{1}(de),\int_{\mathcal{E}_{2}}\tilde{z}^{2,u}_{(t,e)}\nu_{2}(de),u_{t}\big{)},$

for $\tilde{g}=g,b_{2},l$ and
$(t,x,y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})\in[0,T]\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\times
L^{2}(\mathcal{E}_{1},\mathcal{B}(\mathcal{E}_{1}),\nu_{1};\mathbb{R})\times
L^{2}(\mathcal{E}_{2},\mathcal{B}(\mathcal{E}_{2}),\nu_{2};\mathbb{R})$. Here,
the space
$L^{2}(\mathcal{E}_{i},\mathcal{B}(\mathcal{E}_{i}),\nu_{i};\mathbb{R})$,
$i=1,2$ are defined in the (A4), (3) in the following.

For $t\in[0,T]$, define ${\mathcal{F}}_{t}^{Y}:=\sigma\\{Y_{s};0\leq s\leq
t\\}$. For $U\subseteq\mathbb{R}$, the admissible control set is defined as
follows:

$\displaystyle\mathcal{U}_{ad}[0,T]:=$
$\displaystyle\Big{\\{}u\big{|}u_{t}\mbox{ is
}\mathcal{F}_{t}^{Y}\mbox{-progressive }U\mbox{-valued process, such that
}\sup_{0\leq t\leq T}\bar{E}\big{[}|u_{t}|^{\beta}\big{]}<\infty,$ (2.5)
$\displaystyle\mbox{ for any $\beta>1$ }\mbox{and
}\bar{E}\int_{0}^{T}|u_{t}|^{2}N_{i}(\mathcal{E}_{i},dt)<\infty,\mbox{\ for\
}i=1,2\Big{\\}}.$

Our aim in this paper, is to select an optimal control
$\bar{u}\in{\mathcal{U}_{ad}[0,T]}$ such that

$J(\bar{u})=\inf_{u\in\,\mathcal{U}_{ad}[0,T]}J(u).$

######  Remark 2.2.

In our setting, for overcoming the circular dependence between the controlled
filtration $\mathcal{F}_{t}^{Y}$ and the control $u$, we have adopted the
Girsanov measure transformation method. We point out that our framework is
different from that most used in ([18], [43], [50], [34], [44]), but similar
as ([10], [67]). The obvious difference between these two kinds of frameworks
depends on the original probability measure, that is, $P$ or $\bar{P}$, which
is given first. In the first framework, for a given original probability
measure $P$, the observation process $Y$ is given as a standard Brownian
motion. However, in the second framework which is used in our paper, under a
given original probability measure $\bar{P}$, the observation process $Y$ is a
stochastic process which becomes a standard Brownian motion under the new
probability measure $P$ by Girsanov’s theorem.

In fact, by analyzing, we could obtain an important conclusion that these two
kinds of frameworks are equivalent in some sense. First, no matter which kind
of framework is considered in the existing literature, as a goal, the
observation process $Y$ should be a control-independent standard Brownian
motion under the probability measure $P$. Indeed, this goal is the key to deal
with partially observed problems. More precisely, in the first framework, $W$
and observation process $Y$ are two uncontrolled Brownian motions, and the
cost functional is usually defined under the new controlled probability
measure $\bar{P}$. However, for obtaining necessary and sufficient maximum
principles, a controlled $\bar{P}$ would bring the obstacle when using
variation technique and Itô’s formula. Therefore, the given uncontrolled $P$
and uncontrolled Brownian motion $Y$ by the measure transformation can be
regarded as a “bridge” to fill in this gap. In the second framework, $P$ is a
new probability measure under which the original controlled stochastic process
$Y$ becomes a uncontrolled Brownian motion, which is more natural than the
given Brownian motion $Y$ at the beginning in the first framework. Moreover,
the nature mainly shows that, it is usual that the Brownian motion can not be
observed directly, but the role of Girsanov transformation shows that there
exists another probability measure $P$ such that the observation process $Y$
can be turned into certain Brownian motion. So we think that considering $Y$
as a Brownian motion at first, although it is unnatural, is a technical
demanding. Moreover, it means that the original $P$ (the new $\bar{P}$) in the
first framework and the new $P$ (the original $\bar{P}$) in the second
framework play the same roles and actually have the same effects. So these two
kinds of frameworks are equivalent in some sense. We will show its equivalence
in general mathematical sense in Remark 3.3.

As analyzed above, in order to solve the problem, we first set

$\displaystyle\Gamma^{u}_{t}:=$
$\displaystyle\exp\bigg{\\{}-\int_{0}^{T}\sigma_{3}^{-1}(t)b_{2}\big{(}t,\Theta^{u}(t),u_{t}\big{)}d\tilde{W}^{2}_{t}-\frac{1}{2}\int_{0}^{T}|\sigma_{3}^{-1}(t)b_{2}\big{(}t,\Theta^{u}(t),u_{t}\big{)}|^{2}dt$
(2.6)
$\displaystyle\qquad-\int_{0}^{T}\int_{\mathcal{E}_{2}}\log{\lambda(t,x_{t-}^{u},e)}N_{2}(de,dt)-\int_{0}^{T}\int_{\mathcal{E}_{2}}(1-\lambda(t,x_{t-}^{u},e))\nu_{2}(de)dt\bigg{\\}}.$

The following assumption is necessary to guarantee the Girsanov measure
transformation.

${\bf(A3)}\qquad\qquad\bar{E}\bigg{[}\exp\bigg{\\{}\int_{0}^{T}\int_{\mathcal{E}_{2}}\frac{(1-\lambda(t,x_{t-}^{u},e))^{2}}{\lambda(t,x_{t-}^{u},e)}\nu_{2}(de)dt\bigg{\\}}\bigg{]}<\infty.$

Under (A3), define

$\displaystyle M(t):=$
$\displaystyle-\int_{0}^{t}\sigma_{3}^{-1}(s)b_{2}\big{(}s,\Theta^{u}(s),u_{s}\big{)}d\tilde{W}^{2}_{s}+\int_{0}^{t}\int_{\mathcal{E}_{2}}\frac{1-\lambda(s,x_{s-}^{u},e)}{\lambda(s,x_{s-}^{u},e)}\tilde{N}^{\prime}_{2}(de,ds),$

which is a locally square integrable martingale. Moreover, $M_{t}-M_{t-}>-1,\
\bar{P}\mbox{-}a.s.$, and

$\displaystyle\bar{E}\bigg{[}\exp\bigg{\\{}\frac{1}{2}\langle
M^{c},M^{c}\rangle_{T}+\langle M^{d},M^{d}\rangle_{T}\bigg{\\}}\bigg{]}$
$\displaystyle=\bar{E}\bigg{[}\exp\bigg{\\{}\frac{1}{2}\int_{0}^{T}\big{|}\sigma_{3}^{-1}(t)b_{2}\big{(}t,\Theta^{u}(t),u_{t}\big{)}\big{|}^{2}dt+\int_{0}^{T}\int_{\mathcal{E}_{2}}\frac{(1-\lambda(t,x_{t}^{u},e))^{2}}{\lambda(t,x_{t}^{u},e)}\nu_{2}(de)dt\bigg{\\}}\bigg{]}<\infty,$

where $M^{c}$ and $M^{d}$ are continuous and purely discontinuous martingale
parts of $M$, respectively. Therefore, it follows that $\Gamma_{t}^{u}$, the
Doléans-Dade exponential of $M$, is a martingale (Protter and Shimbo [29]).
Then we can define a probability measure $P$ via

$\frac{d\bar{P}}{dP}:=\tilde{\Gamma}_{T}^{u}\equiv(\Gamma_{T}^{u})^{-1},$

where $\tilde{\Gamma}^{u}$ satisfies the following equation:

$\left\\{\begin{aligned}
d\tilde{\Gamma}_{t}^{u}&=\tilde{\Gamma}_{t}^{u}\sigma_{3}^{-1}(t)b_{2}\big{(}t,\Theta^{u}(t),u_{t}\big{)}dW^{2}_{t}+\int_{\mathcal{E}_{2}}\tilde{\Gamma}_{t-}^{u}(\lambda(t,x_{t-}^{u},e)-1)\tilde{N}_{2}(de,dt),\quad
t\in[0,T],\\\ \tilde{\Gamma}_{0}^{u}&=1.\end{aligned}\right.$ (2.7)

Moreover, under the new probability measure $P$ and denoting the expectation
with respect to $P$ by $E$, $W^{1},W^{2}$ are mutually independent Brownian
motions and $\tilde{N}_{1},\tilde{N}_{2}$ are mutually independent Poisson
martingale measures, where

$\displaystyle
dW^{2}_{t}=d\tilde{W}^{2}_{t}+\sigma_{3}^{-1}(t)b_{2}(t,\Theta^{u}(t),u_{t})dt,\quad\tilde{N}_{2}(de,dt)=N_{2}(de,dt)-\nu_{2}(de)dt.$
(2.8)

Observing $\tilde{N}_{2}$ and $\tilde{N}_{2}^{\prime}$, then their
relationship can be built as (2.2).

We introduce the following spaces:

$S^{\beta}[0,T]=\Big{\\{}y\big{|}y\mbox{ has }RCLL\mbox{ paths, adapted and
}E\Big{[}\sup_{0\leq t\leq T}|y_{t}|^{\beta}\Big{]}<\infty\Big{\\}},$

with norm $||y||_{\beta}^{\beta}:=E\big{[}\sup_{0\leq t\leq
T}|y_{t}|^{\beta}\big{]}$;

$M^{2,\beta}_{i}[0,T]=\Big{\\{}z^{i}\big{|}z^{i}\mbox{ is predictable and
}E\bigg{[}\Big{(}\int_{0}^{T}|z^{i}_{t}|^{2}dt\Big{)}^{\frac{\beta}{2}}\bigg{]}<\infty\Big{\\}},\
i=1,2,$

with norm
$||z^{i}||_{2,\beta}^{\beta}:=E\big{[}(\int_{0}^{T}|z^{i}_{t}|^{2}dt)^{\frac{\beta}{2}}\big{]}$;

$F^{2,\beta}_{i}[0,T]=\Big{\\{}\tilde{z}^{i}\big{|}\tilde{z}^{i}\mbox{ is
$\mathcal{E}_{i}$-predictable and
}E\bigg{[}\Big{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{z}^{i}_{(t,e)}|^{2}\nu_{i}(de)dt\Big{)}^{\frac{\beta}{2}}\bigg{]}<\infty\Big{\\}},\
i=1,2,$

with norm
$||\tilde{z}^{i}||_{2,\beta}^{\beta}:=E\big{[}\big{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{z}^{i}_{(t,e)}|^{2}\nu_{i}(de)dt\big{)}^{\frac{\beta}{2}}\big{]}$.
Specially, for $\beta=2$, we set $S^{2}[0,T]$, $M^{2}_{i}[0,T]\equiv
M^{2,2}_{i}[0,T]$ and $F^{2}_{i}[0,T]\equiv F^{2,2}_{i}[0,T]$ defined
similarly, for $i=1,2$, as the above.

For simplicity, we set $\mathcal{M}^{\beta}[0,T]:=S^{\beta}[0,T]\times
S^{\beta}[0,T]\times M_{1}^{2,\beta}[0,T]\times M_{2}^{2,\beta}[0,T]\times
F_{1}^{2,\beta}[0,T]\times F_{2}^{2,\beta}[0,T]$ and
$\mathcal{N}^{\beta}[0,T]:=S^{\beta}[0,T]\times M_{1}^{2,\beta}[0,T]\times
M_{2}^{2,\beta}[0,T]\times F_{1}^{2,\beta}[0,T]\times F_{2}^{2,\beta}[0,T]$,
for $\beta\geq 2$.

Then substituting (2.2) into the state equation (2.1), we get

$\left\\{\begin{aligned}
dx_{t}^{u}&=\tilde{b}_{1}\big{(}t,\Theta^{u}(t),u_{t}\big{)}dt+\sum_{i=1}^{2}\sigma_{i}(t,x_{t}^{u},u_{t})dW^{i}_{t}+\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}f_{i}(t,x_{t-}^{u},u_{t},e)\tilde{N}_{i}(de,dt),\\\
-dy_{t}^{u}&=g\big{(}t,\Theta^{u}(t),u_{t}\big{)}dt-\sum_{i=1}^{2}z^{i,u}_{t}dW^{i}_{t}-\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\tilde{z}^{i,u}_{(t,e)}\tilde{N}_{i}(de,dt),\quad
t\in[0,T],\\\ x_{0}^{u}&=x_{0},\ \
y_{T}^{u}=\phi(x_{T}^{u}),\end{aligned}\right.$ (2.9)

where

$\displaystyle\tilde{b}_{1}\big{(}t,\Theta^{u}(t),u\big{)}$
$\displaystyle:=b_{1}(t,x^{u},u)-\int_{\mathcal{E}_{2}}f_{2}(t,x^{u},u,e)(\lambda(t,x^{u},e)-1)\nu_{2}(de)$
$\displaystyle\qquad-\sigma_{3}^{-1}(t)\sigma_{2}(t,x^{u},u)b_{2}\big{(}t,\Theta^{u}(t),u\big{)}.$

######  Remark 2.3.

We note that, although we aim to study the decoupled forward-backward
stochastic systems originally, the system becomes a fully coupled one due to
the partially observed structure. Therefore, we need existence and uniqueness
of the solutions and the corresponding $L^{\beta}(\beta\geq 2)$-estimate for a
series of variations needed in the deduction of the global maximum principle.

For the $L^{p}$-estimate and existence and uniqueness of solutions to SDE(P)s,
BSDE(P)s and FBSDE(P)s, let us mention a few results. Tang and Li [41] studied
the existence and uniqueness of solution to BSDEP in $L^{2}$ space (see also
Barles et. al. [1]). Situ [36] derived an existence and uniqueness result for
the $L^{2}$-adapted solution to BSDEP, where he considered that its terminal
time is a bounded random stopping time and the coefficients are non-
Lipschitzian. Wu [51] obtained the existence and uniqueness results of
solution to fully coupled FBSDEP in $L^{2}$-space under the monotonicity
condition. Then Wu [52] gave the existence and uniqueness result and a
comparison theorem for such equation in stopping time (unbounded) duration.
Yin and Mao [64] dealt with a class of BSDEP with random terminal times and
proved the existence and uniqueness result of adapted solution under the
assumption of non-Lipschitzian coefficient. Quenez and Sulem [30] studied some
properties of linear BSDEP and also gave the existence and uniqueness of
solution to such a BSDEP with general driver in $L^{p}$-space $(p\geq 2)$ and
its $L^{2}$-estimate. Li and Wei [16] studied some $L^{p}$ $(p\geq 2)$
estimates for fully coupled FBSDEP which were proved under the monotonicity
assumption for arbitrary time intervals and in a small time intervals by
assuming that some Lipschitz constants are sufficiently small. Kruse and
Popier [14] gave the existence and uniqueness of solution to a BSDE with
Brownian and Poisson noises in a general filtration in $L^{p}$-spaces $(p>1)$,
especially for the case of $p\in(1,2)$ which has to be handled carefully, and
some mistake was pointed out when $p<2$, which was corrected by themselves in
[15]. Yao [63] studied $L^{p}$-solution $(p\in(1,2))$ of a multi-dimensional
BSDEP whose generator may not be Lipschitz continuous in $(y,z)$ and obtained
its existence and uniqueness of solution with $p$-integrable terminal data.
Geiss and Steinicke [6] proved existence and uniqueness of BSDEP allowing the
coefficients in the linear growth- and monotonicity-condition for the
generator to be random and time-dependent. In the $L^{2}$-case with linear
growth, this also generalized the results in [14]. Confortola [3] obtained the
existence and uniqueness in $L^{p}$ $(p>1)$ of solution to a BSDE driven by a
marked point process on a bounded interval. Moreover, it is also necessary to
mention the following recent works about $L^{p}$-theory of FBSDE. By firstly
establishing a measurable global implicit function theorem, then, under the
Lipschitz condition and monotonicity assumption, Xie and Yu [58] obtained an
$L^{p}(p>1)$-solution and its related $p$-th power estimates for coupled FBSDE
with random coefficients on small durations. It is worth noting that Yong [65]
investigated the problem of when an adapted $L^{2}$-solution of an FBSDE is an
adapted $L^{p}(p>2)$-solution. For giving an affirmative answer, several
important cases were explored, which is thought-provoking but is far from
having a satisfactory answer. Therefore, some open questions worthy of
studying are posed to construct a better theory of FBSDEs. For giving a
positive answer to the problem proposed by Yong [65], Meng and Yang [21]
proved that a unique $L^{2}$-solution of fully coupled FBSDEs is an
$L^{p}$-solution under standard conditions on the coefficients for any given
terminal time $T$, based on a key observation of the relation between $L^{2}$
and $L^{p}$ estimations of FBSDEs, by which we have developed a
$L^{p}(p>2)$-theory, including $L^{p}$-solution and its $p$-th power
estimation, of fully coupled FBSDEPs for any given terminal time $T$ as an
extension to the case with Poisson jumps.

###  2.1 The existence and uniqueness of the solution and $L^{\beta}$-estimate

Consider the following fully coupled FBSDEP:

$\left\\{\begin{aligned}
dx_{t}&=b(t,\Theta(t))dt+\sum_{i=1}^{2}\sigma_{i}(t,\Theta(t))dW^{i}_{t}+\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}f_{i}(t,\Theta(t-),e)\tilde{N}_{i}(de,dt),\\\
-dy_{t}&=g(t,\Theta(t))dt-\sum_{i=1}^{2}z^{i}_{t}dW^{i}_{t}-\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\tilde{z}^{i}_{(t,e)}\tilde{N}_{i}(de,dt),\quad
t\in[0,T],\\\ x_{0}&=x_{0},\ \ y_{T}=\phi(x_{T}),\end{aligned}\right.$ (2.10)

where
$\Theta(t):=(x_{t},y_{t},z^{1}_{t},z^{2}_{t},\tilde{z}^{1}_{(t,e)},\tilde{z}^{2}_{(t,e)})$,
and the coefficients $b,\sigma_{1},\sigma_{2},f_{1},f_{2},g,\phi$ could be
random.

(A4) (1) Let $(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{t\geq 0},P)$ be a
complete probability space, on which standrad Brownian motions
$\\{W^{1}_{t},W^{2}_{t}\\}_{t\geq 0}\in\mathbb{R}^{2}$ and Poisson random
measures $N_{i}$ with the compensator $EN_{i}(de,dt)=\nu_{i}(de)dt$, for
$i=1,2,$ are mutually independent. Here $\nu_{i}$ is assumed to be a
$\sigma$-finite Lévy measure on
$(\mathcal{E}_{i},\mathcal{B}(\mathcal{E}_{i}))$ with the property that
$\int_{\mathcal{E}_{i}}(1\land|e|^{2})\nu_{i}(de)<\infty$, for $i=1,2$.

(2) Define the space
$L^{2}(\mathcal{E}_{i},\mathcal{B}(\mathcal{E}_{i}),\nu_{i};\mathbb{R}):=\big{\\{}\tilde{z}^{i}_{(t,e)}\in\mathbb{R}\
\mbox{such that}\
\big{[}\int_{\mathcal{E}_{i}}|\tilde{z}^{i}_{(t,e)}|^{2}\nu_{i}(de)\big{]}^{\frac{1}{2}}<\infty\big{\\}}$,
for $i=1,2$.

(3) $b,\sigma_{1},\sigma_{2},g$ are
$\mathcal{G}\otimes\mathcal{B}(\mathbb{R})\otimes\mathcal{B}(\mathbb{R})\otimes\mathcal{B}(\mathbb{R})\otimes\mathcal{B}(\mathbb{R})\otimes\mathcal{B}(L^{2}(\mathcal{E}_{1},\mathcal{B}(\mathcal{E}_{1}),\nu_{1};\mathbb{R}))\\\
\otimes\mathcal{B}(L^{2}(\mathcal{E}_{2},\mathcal{B}(\mathcal{E}_{2}),\nu_{2};\mathbb{R}))/\mathcal{B}(\mathbb{R})$
measurable, $f_{1}(f_{2})$ is
$\mathcal{G}\otimes\mathcal{B}(\mathbb{R})\otimes\mathcal{B}(\mathbb{R})\otimes\mathcal{B}(\mathbb{R})\otimes\mathcal{B}(\mathbb{R})\otimes\mathcal{B}(\mathbb{R})\\\
\otimes\mathcal{B}(\mathbb{R})\otimes\mathcal{B}(\mathcal{E}_{1})(\otimes\mathcal{B}(\mathcal{E}_{2}))/\mathcal{B}(\mathbb{R})$
measurable, and $\phi$ is
$\mathcal{F}_{T}\otimes\mathcal{B}(\mathbb{R})/\mathcal{B}(\mathbb{R})$
measurable.

(4) $b,\sigma_{1},\sigma_{2},g$ are uniformly Lipschitz with respect to
$(x,y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})$, and $\phi(x)$ is uniformly
Lipschitz with respect to $x\in\mathbb{R}$.

(5) For $\beta\geq 2$, we have

$\displaystyle
E\bigg{(}\int_{0}^{T}|b(t,\omega,0,0,0,0,0,0)|dt\bigg{)}^{\beta}+\sum_{i=1}^{2}\bigg{(}E\bigg{(}\int_{0}^{T}|\sigma_{i}(t,\omega,0,0,0,0,0,0)|^{2}dt\bigg{)}^{\frac{\beta}{2}}$
$\displaystyle+E\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|f_{i}(t,\omega,0,0,0,0,0,0,e)|^{2}N_{i}(de,dt)\bigg{)}^{\frac{\beta}{2}}\bigg{)}+E\bigg{(}\int_{0}^{T}|g(t,\omega,0,0,0,0,0,0)|dt\bigg{)}^{\beta}<\infty,$

and $E[|\phi(0)|^{\beta}]<\infty$.

(6) For any $t\in[0,T]$,
$\Theta\in\mathbb{R}\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\times
L^{2}(\mathcal{E}_{1},\mathcal{B}(\mathcal{E}_{1}),\nu_{1};\mathbb{R})\times
L^{2}(\mathcal{E}_{2},\mathcal{B}(\mathcal{E}_{2}),\nu_{2};\mathbb{R})$,
$P\mbox{-}a.s.$,

$|b(t,\Theta)|+\sum_{i=1}^{2}|\sigma_{i}(t,\Theta)|+|g(t,\Theta)|+|\phi(x)|\leq
L(1+|x|+|y|+|z^{1}|+|z^{2}|+||\tilde{z}^{1}||+||\tilde{z}^{2}||),$

and there exists a measurable function
$\rho:\mathcal{E}_{i}(\subset\mathcal{E})\rightarrow\mathbb{R}^{+}$ with
$\int_{\mathcal{E}_{i}}\rho^{\beta}(e)\nu_{i}(de)<\infty(\beta\geq 2)$ such
that, for any $t\in[0,T]$, $\Theta\in\mathbb{R}^{6}$ and
$e\in\mathcal{E}_{i}$,
$|f_{i}(t,\Theta,e)|\leq\rho(e)(1+|x|+|y|+|z^{1}|+|z^{2}|+|\tilde{z}^{1}|+|\tilde{z}^{2}|)$,
i=1,2.

(7) For $i=1,2$, the Lipschitz constant $L_{\sigma_{i}}\geq 0$ of diffusion
$\sigma_{i}$ with respect to $(z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})$ is
sufficient small. That is, there exists some $L_{\sigma_{i}}\geq 0$ small
enough such that, for all $t\in[0,T]$,
$\Theta_{j}=(x_{j},y_{j},z^{1}_{j},z^{2}_{j},\tilde{z}^{1}_{j},\tilde{z}^{2}_{j})\in\mathbb{R}^{4}\times
L^{2}(\mathcal{E}_{1},\mathcal{B}(\mathcal{E}_{1}),\nu_{1};\mathbb{R})\times
L^{2}(\mathcal{E}_{2},\mathcal{B}(\mathcal{E}_{2}),\nu_{2};\mathbb{R})$,
$j=1,2$, $P\mbox{-}a.s.$,

$\big{|}\sigma_{i}(t,\Theta_{1})-\sigma_{i}(t,\Theta_{2})\big{|}\leq
K(|x_{1}-x_{2}|+|y_{1}-y_{2}|)+L_{\sigma_{i}}(|z^{1}_{1}-z^{1}_{2}|+|z^{2}_{1}-z^{2}_{2}|+||\tilde{z}^{1}_{1}-\tilde{z}^{1}_{2}||+||\tilde{z}^{2}_{1}-\tilde{z}^{2}_{2}||).$

Moreover, the Lipschitz coefficient $L_{f_{i}}$ of $f_{i}$ with respect to
$(z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})$ is sufficient small, i.e., there
exists a function $L_{f_{i}}:\mathcal{E}_{i}\rightarrow\mathbb{R}^{+}$ with
$\tilde{C}_{f_{i}}:=\max\big{\\{}\sup_{e\in\mathcal{E}_{i}}L^{\beta}_{f_{i}}(e),\int_{\mathcal{E}_{i}}L^{\beta}_{f_{i}}(e)\nu_{i}(de)\big{\\}}<\infty(\beta\geq
2)$ sufficient small, and for all $t\in[0,T]$,
$(\Theta_{1},\Theta_{2})\in\mathbb{R}^{12}$ and $e\in\mathcal{E}_{i}$,
$P\mbox{-}a.s.$,

$\big{|}f_{i}(t,\Theta_{1},e)-f_{i}(t,\Theta_{2},e)\big{|}\leq\rho(e)(|x_{1}-x_{2}|+|y_{1}-y_{2}|)+L_{f_{i}}(e)(|z^{1}_{1}-z^{1}_{2}|+|z^{2}_{1}-z^{2}_{2}|+|\tilde{z}^{1}_{1}-\tilde{z}^{1}_{2}|+|\tilde{z}^{2}_{1}-\tilde{z}^{2}_{2}|).$

Next, we first prove the existence and uniqueness of the solution to (2.10).

######  Theorem 2.1.

Suppose that (A4) holds, then there exists a constant $\tilde{T}>0$ only
depending on $K,L_{\sigma_{1}},L_{\sigma_{2}},L_{f_{1}},L_{f_{2}},\rho$ such
that, for every $0<T<\tilde{T}$, the fully coupled FBSDEP (2.10) has a unique
solution in $\mathcal{M}^{2}[0,T]$.

###### Proof.

For any
$(y^{\prime},z^{1,\prime},z^{2,\prime},\tilde{z}^{1,\prime},\tilde{z}^{2,\prime})\in\mathcal{N}^{2}[0,T]$,
it is classical that there exists a unique solution
$(y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})\in\mathcal{N}^{2}[0,T]$ to the
following decoupled FBSDEP:

$\left\\{\begin{aligned}
dx_{t}&=b(t,x_{t},y^{\prime}_{t},z^{1,\prime}_{t},z^{2,\prime}_{t},\tilde{z}^{1,\prime}_{(t,e)},\tilde{z}^{2,\prime}_{(t,e)})dt+\sum_{i=1}^{2}\sigma_{i}(t,x_{t},y^{\prime}_{t},z^{1,\prime}_{t},z^{2,\prime}_{t},\tilde{z}^{1,\prime}_{(t,e)},\tilde{z}^{2,\prime}_{(t,e)})dW^{i}_{t}\\\
&\quad+\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}f_{i}(t,x_{t},y^{\prime}_{t},z^{1,\prime}_{t},z^{2,\prime}_{t},\tilde{z}^{1,\prime}_{(t,e)},\tilde{z}^{2,\prime}_{(t,e)},e)\tilde{N}_{i}(de,dt),\\\
-dy_{t}&=g(t,\Theta(t))dt-\sum_{i=1}^{2}z^{i}_{t}dW^{i}_{t}-\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\tilde{z}^{i}_{(t,e)}\tilde{N}_{i}(de,dt),\quad
t\in[0,T],\\\ x_{0}&=x_{0},\ \ y_{T}=\phi(x_{T}).\end{aligned}\right.$ (2.11)

Define a map $\mathbb{I}_{1}$ from $\mathcal{N}^{2}[0,T]$ to itself. Let
$(y^{\prime}_{i},z^{1,\prime}_{i},z^{2,\prime}_{i},\tilde{z}^{1,\prime}_{i},\tilde{z}^{2,\prime}_{i})\in\mathcal{N}^{2}[0,T]$
and $(y_{i},z^{1}_{i},z^{2}_{i},\tilde{z}^{1}_{i},\tilde{z}^{2}_{i})\\\
=\mathbb{I}_{1}(y^{\prime}_{i},z^{1,\prime}_{i},z^{2,\prime}_{i},\tilde{z}^{1,\prime}_{i},\tilde{z}^{2,\prime}_{i}),i=1,2$
and define

$\displaystyle\hat{y}^{\prime}=y^{\prime}_{1}-y^{\prime}_{2},\
\hat{z}^{1,\prime}=z^{1,\prime}_{1}-z^{1,\prime}_{2},\
\hat{z}^{2,\prime}=z^{2,\prime}_{1}-z^{2,\prime}_{2},\
\hat{\tilde{z}}^{1,\prime}=\tilde{z}^{1,\prime}_{1}-\tilde{z}^{1,\prime}_{2},\
\hat{\tilde{z}}^{2,\prime}=\tilde{z}^{2,\prime}_{1}-\tilde{z}^{2,\prime}_{2}$
(2.12) $\displaystyle\hat{y}=y_{1}-y_{2},\ \hat{z}^{1}=z^{1}_{1}-z^{1}_{2},\
\hat{z}^{2}=z^{2}_{1}-z^{2}_{2},\
\hat{\tilde{z}}^{1}=\tilde{z}^{1}_{1}-\tilde{z}^{1}_{2},\
\hat{\tilde{z}}^{2}=\tilde{z}^{2}_{1}-\tilde{z}^{2}_{2},\
\hat{x}=x_{1}-x_{2}.$

By (A4), we get (We omit some time variables if there is no ambiguity.)

$\displaystyle|\hat{x}_{t}|^{2}\leq
C\bigg{[}\Big{|}\int_{0}^{t}L_{b}\big{(}|\hat{x}|+|\hat{y}^{\prime}|+|\hat{z}^{1,\prime}|+|\hat{z}^{2,\prime}|+||\hat{\tilde{z}}^{1,\prime}||+||\hat{\tilde{z}}^{2,\prime}||\big{)}dt\Big{|}^{2}$
(2.13)
$\displaystyle\quad+\sum_{i=1}^{2}\Big{|}\int_{0}^{t}\big{[}K(|\hat{x}|+|\hat{y}^{\prime}|)+L_{\sigma_{i}}(|\hat{z}^{1,\prime}|+|\hat{z}^{2,\prime}|+||\hat{\tilde{z}}^{1,\prime}||+||\hat{\tilde{z}}^{2,\prime}||)\big{]}dW^{i}_{t}\Big{|}^{2}$
$\displaystyle\quad+\sum_{i=1}^{2}\Big{|}\int_{0}^{t}\int_{\mathcal{E}_{i}}\big{[}\rho(e)(|\hat{x}|+|\hat{y}^{\prime}|)+L_{f_{i}}(e)(|\hat{z}^{1,\prime}|+|\hat{z}^{2,\prime}|+|\hat{\tilde{z}}^{1,\prime}|+|\hat{\tilde{z}}^{2,\prime}|)\big{]}\tilde{N}_{i}(de,dt)\Big{|}^{2}\bigg{]}.$

Applying B-D-G’s inequality and Hölder’s inequality, we obtain

$\displaystyle E\Big{[}\sup_{0\leq t\leq T}|\hat{x}_{t}|^{2}\Big{]}\leq
CTE\bigg{[}\int_{0}^{T}\big{(}|\hat{x}|^{2}+|\hat{y}^{\prime}|^{2}+|\hat{z}^{1,\prime}|^{2}+|\hat{z}^{2,\prime}|^{2}+||\hat{\tilde{z}}^{1,\prime}||^{2}+||\hat{\tilde{z}}^{2,\prime}||^{2}\big{)}dt\bigg{]}$
(2.14)
$\displaystyle\quad+CE\bigg{[}\int_{0}^{T}\big{[}K^{2}(|\hat{x}|^{2}+|\hat{y}^{\prime}|^{2})+(L_{\sigma_{1}}^{2}+L_{\sigma_{2}}^{2})(|\hat{z}^{1,\prime}|^{2}+|\hat{z}^{2,\prime}|^{2}+||\hat{\tilde{z}}^{1,\prime}||^{2}+||\hat{\tilde{z}}^{2,\prime}||^{2})\big{]}dt\bigg{]}$
$\displaystyle\quad+C\sum_{i=1}^{2}E\bigg{[}\int_{0}^{T}\int_{\mathcal{E}_{i}}\big{[}\rho^{2}(e)(|\hat{x}|^{2}+|\hat{y}^{\prime}|^{2})+L_{f_{i}}^{2}(e)(|\hat{z}^{1,\prime}|^{2}+|\hat{z}^{2,\prime}|^{2}+|\hat{\tilde{z}}^{1,\prime}|^{2}+|\hat{\tilde{z}}^{2,\prime}|^{2})\big{]}N_{i}(de,dt)\bigg{]}$
$\displaystyle\leq CTE\bigg{[}T\sup_{0\leq t\leq
T}(|\hat{x}|^{2}+|\hat{y}^{\prime}|^{2})+\sum_{i=1}^{2}\bigg{(}\int_{0}^{T}|\hat{z}^{i,\prime}|^{2}dt+\int_{0}^{T}\int_{\mathcal{E}_{i}}|\hat{\tilde{z}}^{i,\prime}|^{2}\nu_{i}(de)dt\bigg{)}\bigg{]}$
$\displaystyle\quad+CE\bigg{[}TK^{2}\sup_{0\leq t\leq
T}(|\hat{x}|^{2}+|\hat{y}^{\prime}|^{2})+(L_{\sigma_{1}}^{2}+L_{\sigma_{2}}^{2})\sum_{i=1}^{2}\bigg{(}\int_{0}^{T}|\hat{z}^{i,\prime}|^{2}dt+\int_{0}^{T}\int_{\mathcal{E}_{i}}|\hat{\tilde{z}}^{i,\prime}|^{2}\nu_{i}(de)dt\bigg{)}\bigg{]}$
$\displaystyle\quad+CE\bigg{[}T\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\rho^{2}(e)\nu_{i}(de)\sup_{0\leq
t\leq
T}(|\hat{x}|^{2}+|\hat{y}^{\prime}|^{2})+\bigg{(}\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}L_{f_{i}}^{2}(e)\nu_{i}(de)\bigg{)}\sum_{i=1}^{2}\int_{0}^{T}|\hat{z}^{i,\prime}|^{2}dt$
$\displaystyle\qquad\qquad+\bigg{(}\sum_{i=1}^{2}\sup_{e\in\mathcal{E}_{i}}L_{f_{i}}^{2}(e)\bigg{)}\sum_{i=1}^{2}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\hat{\tilde{z}}^{i,\prime}|^{2}\nu_{i}(de)dt\bigg{]}$
$\displaystyle\leq\bigg{[}CT^{2}+CTK^{2}+CT\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\rho^{2}(e)\nu_{i}(de)\bigg{]}E\Big{[}\sup_{0\leq
t\leq T}(|\hat{x}|^{2}+|\hat{y}^{\prime}|^{2})\Big{]}$
$\displaystyle\quad+\bigg{[}CT+C(L_{\sigma_{1}}^{2}+L_{\sigma_{2}}^{2})+C\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}L_{f_{i}}^{2}(e)\nu_{i}(de)\bigg{]}\sum_{i=1}^{2}E\int_{0}^{T}|\hat{z}^{i,\prime}|^{2}dt$
$\displaystyle\quad+\bigg{[}CT+C(L_{\sigma_{1}}^{2}+L_{\sigma_{2}}^{2})+C\sum_{i=1}^{2}\sup_{e\in\mathcal{E}_{i}}L_{f_{i}}^{2}(e)\bigg{]}\sum_{i=1}^{2}E\int_{0}^{T}\int_{\mathcal{E}_{i}}|\hat{\tilde{z}}^{i,\prime}|^{2}\nu_{i}(de)dt.$

Then we have

$\displaystyle\Big{\\{}1-\Big{[}CT^{2}+CTK^{2}+CT\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\rho^{2}(e)\nu_{i}(de)\Big{]}\Big{\\}}E\Big{[}\sup_{0\leq
t\leq T}|\hat{x}_{t}|^{2}\Big{]}$ (2.15)
$\displaystyle\leq\Big{[}CT^{2}+CTK^{2}+CT\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\rho^{2}(e)\nu_{i}(de)\Big{]}E\Big{[}\sup_{0\leq
t\leq
T}|\hat{y}^{\prime}|^{2}\Big{]}+\Big{[}CT+C(L_{\sigma_{1}}^{2}+L_{\sigma_{2}}^{2})$
$\displaystyle\quad+C\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}L_{f_{i}}^{2}(e)\nu_{i}(de)\Big{]}\sum_{i=1}^{2}E\int_{0}^{T}|\hat{z}^{i,\prime}|^{2}dt$
$\displaystyle\quad+\Big{[}CT+C(L_{\sigma_{1}}^{2}+L_{\sigma_{2}}^{2})+C\sum_{i=1}^{2}\sup_{e\in\mathcal{E}_{i}}L_{f_{i}}^{2}(e)\Big{]}\sum_{i=1}^{2}E\int_{0}^{T}\int_{\mathcal{E}_{i}}|\hat{\tilde{z}}^{i,\prime}|^{2}\nu_{i}(de)dt,$

and there exists $T_{0}$ such that
$CT_{0}^{2}+CT_{0}K^{2}+CT_{0}\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\rho^{2}(e)\nu_{i}(de)<1$,
we have, for $0<T<T_{0}$

$\displaystyle E\Big{[}\sup_{0\leq t\leq
T}|\hat{x}_{t}|^{2}\Big{]}\leq\frac{CT^{2}+CTK^{2}+CT\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\rho^{2}(e)\nu_{i}(de)}{1-\big{[}CT^{2}+CTK^{2}+CT\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\rho^{2}(e)\nu_{i}(de)\big{]}}E\Big{[}\sup_{0\leq
t\leq T}|\hat{y}^{\prime}|^{2}\Big{]}$ (2.16)
$\displaystyle\quad+\frac{\Big{[}CT+C(L_{\sigma_{1}}^{2}+L_{\sigma_{2}}^{2})+C(\tilde{C}_{f_{1}}+\tilde{C}_{f_{2}})\Big{]}}{1-\big{[}CT^{2}+CTK^{2}+CT\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\rho^{2}(e)\nu_{i}(de)\big{]}}\sum_{i=1}^{2}E\int_{0}^{T}|\hat{z}^{i,\prime}|^{2}dt$
$\displaystyle\quad+\frac{\Big{[}CT+C(L_{\sigma_{1}}^{2}+L_{\sigma_{2}}^{2})+C(\tilde{C}_{f_{1}}+\tilde{C}_{f_{2}})\Big{]}}{1-\big{[}CT^{2}+CTK^{2}+CT\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\rho^{2}(e)\nu_{i}(de)\big{]}}\sum_{i=1}^{2}E\int_{0}^{T}\int_{\mathcal{E}_{i}}|\hat{\tilde{z}}^{i,\prime}|^{2}\nu_{i}(de)dt.$

Noting that

$\displaystyle-d\hat{y}_{t}$
$\displaystyle=\big{[}g(t,\Theta_{1})-g(t,\Theta_{2})\big{]}dt-\sum_{i=1}^{2}(z^{i}_{1}-z^{i}_{2})dW^{i}_{t}-\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}(\tilde{z}^{i}_{1}-\tilde{z}^{i}_{2})\tilde{N}_{i}(de,dt)$
(2.17)
$\displaystyle=\big{[}g(t,\Theta_{1})-g(t,x_{2},y_{1}-(y_{1}-y_{2}),z^{1}_{1}-(z^{1}_{1}-z^{1}_{2}),z^{2}_{1}-(z^{2}_{1}-z^{2}_{2}),\tilde{z}^{1}_{1}-(\tilde{z}^{1}_{1}-\tilde{z}^{1}_{2}),$
$\displaystyle\quad\tilde{z}^{2}_{1}-(\tilde{z}^{2}_{1}-\tilde{z}^{2}_{2}))\big{]}dt-\sum_{i=1}^{2}(z^{i}_{1}-z^{i}_{2})dW^{i}_{t}-\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}(\tilde{z}^{i}_{1}-\tilde{z}^{i}_{2})\tilde{N}_{i}(de,dt)$
$\displaystyle=\big{[}g(t,\Theta_{1})-g(t,x_{2},y_{1}-\hat{y},z^{1}_{1}-\hat{z}^{1},z^{2}_{1}-\hat{z}^{2},\tilde{z}^{1}_{1}-\hat{\tilde{z}}^{1},\tilde{z}^{2}_{1}-\hat{\tilde{z}}^{2})\big{]}dt$
$\displaystyle\quad-\sum_{i=1}^{2}\hat{z}^{i}dW^{i}_{t}-\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\hat{\tilde{z}}^{i}\tilde{N}_{i}(de,dt),$

by standard estimates for solutions to BSDEPs (see [1], [14], [30] and its
derivative under our assumptions, which can be regarded as a special case of
Theorem 2.3 in the following), we have

$\displaystyle E\bigg{[}\sup_{0\leq t\leq
T}|\hat{y}_{t}|^{2}+\sum_{i=1}^{2}\bigg{(}\int_{0}^{T}|\hat{z}^{i}_{t}|^{2}dt+\int_{0}^{T}\int_{\mathcal{E}_{i}}|\hat{\tilde{z}}^{i}_{(t,e)}|^{2}\nu_{i}(de)dt\bigg{)}\bigg{]}$
(2.18) $\displaystyle\leq
CE\bigg{[}\big{|}\phi(x_{1,T})-\phi(x_{2,T})\big{|}^{2}+\int_{0}^{T}\big{|}g(t,\Theta_{1})-g(t,x_{2},y_{1},z^{1}_{1},z^{2}_{1},\tilde{z}^{1}_{1},\tilde{z}^{2}_{1})\big{|}^{2}dt\bigg{]}$
$\displaystyle\leq
CE\bigg{[}|C\hat{x}_{T}|^{2}+\int_{0}^{T}C^{2}|\hat{x}|^{2}dt\bigg{]}\leq
C(1+T)E\bigg{[}\sup_{0\leq t\leq T}|\hat{x}_{t}|^{2}\bigg{]}.$

Combined with (2.16), we have

$\displaystyle E\bigg{[}\sup_{0\leq t\leq
T}|\hat{y}_{t}|^{2}+\sum_{i=1}^{2}\bigg{(}\int_{0}^{T}|\hat{z}^{i}_{t}|^{2}dt+\int_{0}^{T}\int_{\mathcal{E}_{i}}|\hat{\tilde{z}}^{i}_{(t,e)}|^{2}\nu_{i}(de)dt\bigg{)}\bigg{]}$
(2.19) $\displaystyle\leq
C(1+T)\frac{C(T+T^{2})+TK^{2}+L_{\sigma_{1}}^{2}+L_{\sigma_{2}}^{2}+\tilde{C}_{f_{1}}+\tilde{C}_{f_{2}}+T\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\rho^{2}(e)\nu_{i}(de)}{1-\big{[}CT^{2}+CTK^{2}+CT\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\rho^{2}(e)\nu_{i}(de)\big{]}}$
$\displaystyle\quad\times E\bigg{[}\sup_{0\leq t\leq
T}|\hat{y}^{\prime}|^{2}+\sum_{i=1}^{2}\bigg{(}\int_{0}^{T}|\hat{z}^{i,\prime}|^{2}dt+\int_{0}^{T}\int_{\mathcal{E}_{i}}|\hat{\tilde{z}}^{i,\prime}_{(t,e)}|^{2}\nu_{i}(de)dt\bigg{)}\bigg{]}.$

We take Lipschitz coefficients
$L_{\sigma_{1}},L_{\sigma_{2}},\tilde{C}_{f_{1}}$ and $\tilde{C}_{f_{2}}$
small sufficient, $C$ is a constant not related to $T$ but changed every step,
therefore we can choose $0<\tilde{T}\leq T_{0}$ such that

$C(1+\tilde{T})\frac{C(\tilde{T}+\tilde{T}^{2})+\tilde{T}K^{2}+L_{\sigma_{1}}^{2}+L_{\sigma_{2}}^{2}+\tilde{C}_{f_{1}}+\tilde{C}_{f_{2}}+\tilde{T}\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\rho^{2}(e)\nu_{i}(de)}{1-\big{[}C\tilde{T}^{2}+C\tilde{T}K^{2}+C\tilde{T}\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\rho^{2}(e)\nu_{i}(de)\big{]}}<1.$
(2.20)

Then, for any $0\leq T\leq\tilde{T}$, $\mathbb{I}_{1}$ is a contraction
mapping on $[0,T]$ which has a unique fixed point
$\mathbb{I}_{1}(y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})=(y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})$.
Furthermore, FBSDEP (2.10) has a unique solution
$(x,y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})$ on small interval $[0,T]$. The
proof is complete. ∎

######  Remark 2.4.

Theorem 2.1 only gives the existence and uniqueness of solutions to fully
coupled FBSDEP when $\beta=2$. In fact, we encounter some difficulties in
obtaining the counterpart in $\mathcal{M}^{\beta}[0,T]$ for $\beta>2$.
However, a special fully coupled FBSDEP, where the jump coefficients do not
contain $(z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})$, can be proved to admit a
unique solution $(x,y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})$ in
$\mathcal{M}^{\beta}[0,T]$ for $\beta\geq 2$, and the proof is similar, we
only list the main theorem in the following and omit the details.

######  Theorem 2.2.

Suppose that (A4) holds, for any $\beta\geq 2$, then there exists a constant
$\tilde{T}>0$ depending on $K,L_{\sigma_{1}},L_{\sigma_{2}},\rho$ such that,
for every $0\leq T\leq\tilde{T}$, the fully coupled FBSDEP (2.10), in which
the jump coefficients do not contain
$(z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})$, has a unique solution in
$\mathcal{M}^{\beta}[0,T]$.

######  Remark 2.5.

As we mentioned before, we fail to obtain the unique solvability of fully-
coupled FBSDEP in $\mathcal{M}^{\beta}[0,T]$ for $\beta>2$, especially for
jump coefficient $f_{i}$ containing
$(z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})$. More specificly, the critical
obstacle appears in the norm defined in $F_{i}^{2,\beta}[0,T]$, which is not
supportive enough for us to get the $L^{\beta}$-estimate for BSDEP for
$\beta>2$. Therefore, for obtaining a better result for a more general fully-
coupled FBSDEP ($f_{i}$ can contain $\tilde{z}^{1},\tilde{z}^{2}$ but is
independent of $z^{1},z^{2}$), we should change the space
$F_{i}^{2,\beta}[0,T]$ with an another norm.

Next, we introduce an another kind of necessary solution space as follows to
replace the original space $F_{i}^{2,\beta}[0,T]$:

$\tilde{F}_{i}^{2,\beta}[0,T]:=\bigg{\\{}\tilde{z}^{i}\big{|}\tilde{z}^{i}\mbox{
is $\mathcal{E}_{i}$-predictable and
}E\bigg{[}\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{z}^{i}_{(t,e)}|^{2}N_{i}(de,dt)\bigg{)}^{\frac{\beta}{2}}\bigg{]}<\infty\bigg{\\}},\
i=1,2,$

with norm
$||\tilde{z}^{i}||_{2,\beta}^{\beta}:=E\big{[}\big{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{z}^{i}_{(t,e)}|^{2}N_{i}(de,dt)\big{)}^{\frac{\beta}{2}}\big{]}$.

We set $\tilde{\mathcal{M}}^{\beta}[0,T]:=S^{\beta}[0,T]\times
S^{\beta}[0,T]\times M_{1}^{2,\beta}[0,T]\times
M_{2}^{2,\beta}[0,T]\times\tilde{F}_{1}^{2,\beta}[0,T]\times\tilde{F}_{2}^{2,\beta}[0,T]$
and $\tilde{\mathcal{N}}^{\beta}[0,T]:=S^{\beta}[0,T]\times
M_{1}^{2,\beta}[0,T]\times
M_{2}^{2,\beta}[0,T]\times\tilde{F}_{1}^{2,\beta}[0,T]\times\tilde{F}_{2}^{2,\beta}[0,T]$.
Specially, for $\beta=2$, it is easy to find that
$\tilde{F}_{i}^{2,\beta}[0,T]=F_{i}^{2,\beta}[0,T],\tilde{\mathcal{M}}^{\beta}[0,T]=\mathcal{M}^{\beta}[0,T]$
and $\tilde{\mathcal{N}}^{\beta}[0,T]=\mathcal{N}^{\beta}[0,T]$.

(A5) For any $t\in[0,T]$, $(x,y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})\in
R^{6}$, the jump coefficient $f_{i}$ is independent of $(z^{1},z^{2})$ for
$i=1,2$.

######  Theorem 2.3.

Suppose that (A4), (A5) hold, then there exists a constant $\tilde{T}>0$
depending on $K,L_{\sigma_{1}},L_{\sigma_{2}},L_{f_{1}},L_{f_{2}},\rho$ such
that, for every $0\leq T\leq\tilde{T}$ and $\beta\geq 2$, the fully coupled
FBSDEP (2.10) has a unique solution in $\tilde{\mathcal{M}}^{\beta}[0,T]$.

Before we prove the Theorem 2.3, we should give the following necessary lemmas
about the existence and uniqueness of solution and a priori estimate of BSDEP.

######  Lemma 2.2.

Suppose that (A4) hold, then for $\beta\geq 2$, the following BSDEP

$\left\\{\begin{aligned}
-dy_{t}&=g(t,y_{t},z^{1}_{t},z^{2}_{t},\tilde{z}^{1}_{(t,e)},\tilde{z}^{2}_{(t,e)})dt-\sum_{i=1}^{2}z^{i}_{t}dW^{i}_{t}-\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\tilde{z}^{i}_{(t,e)}\tilde{N}_{i}(de,dt),\quad
t\in[0,T],\\\ y_{T}&=\xi.\end{aligned}\right.$ (2.21)

has a unique solution in $\tilde{\mathcal{N}}^{\beta}[0,T]$.

###### Proof.

We first prove the case that $g$ dose not contain
$(y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})$, then $y$ is given by the right-
continuous version of
$y_{t}=E\big{[}\xi+\int_{t}^{T}g(s)ds|\mathcal{F}_{t}\big{]}$. Thus, for
$\beta\geq 2$, the martingale representation theorem for locally square
integrable martingale in [41] implies that there exist unique predictable,
$\mathcal{E}_{1}$-predictable and $\mathcal{E}_{2}$-predictable
$(z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})\in M_{1}^{2,\beta}[0,T]\times
M_{2}^{2,\beta}[0,T]\times\tilde{F}_{1}^{2,\beta}[0,T]\times\tilde{F}_{2}^{2,\beta}[0,T]$
satisfying:

$E\bigg{[}\xi+\int_{0}^{T}g(s)ds\bigg{|}\mathcal{F}_{t}\bigg{]}=y_{0}+\sum_{i=1}^{2}\bigg{(}\int_{0}^{t}z^{i}_{s}dW_{s}+\int_{0}^{t}\int_{\mathcal{E}_{i}}\tilde{z}^{i}_{(s,e)}\tilde{N}_{i}(de,ds)\bigg{)},\
P\mbox{-}a.s.$ (2.22)

Next, we have
$|y_{t}|=E\big{[}|\xi|+\int_{0}^{T}|g(s)|ds|\mathcal{F}_{t}\big{]}$, and using
martingale inequality (Ikeda and Watanabe [12]), we get

$E\bigg{[}\sup_{0\leq t\leq T}|y_{t}|^{\beta}\bigg{]}\leq
C_{\beta}E\bigg{[}\bigg{(}|\xi|+\int_{0}^{T}|g(s)|ds\bigg{)}^{\beta}\bigg{]},$
(2.23)

where $C_{\beta}$ is a positive constant which depends only on the number
$\beta$ and not related to $T$. Due to (A4), we continue to deduce that

$E\bigg{[}\sup_{0\leq t\leq T}|y_{t}|^{\beta}\bigg{]}\leq
C_{\beta}E\bigg{[}|\xi|^{\beta}+\bigg{(}\int_{0}^{T}|g(s)|ds\bigg{)}^{\beta}\bigg{]}=C_{\beta}\bigg{[}E|\xi|^{\beta}+E\bigg{(}\int_{0}^{T}|g(s)|ds\bigg{)}^{\beta}\bigg{]}<\infty,$
(2.24)

and by B-D-G’s inequality, we have

$\displaystyle
E\bigg{[}\bigg{(}\sum_{i=1}^{2}\bigg{(}\int_{0}^{T}|z^{i}_{s}|^{2}ds+\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{z}^{i}_{(s,e)}|^{2}N_{i}(de,ds)\bigg{)}\bigg{)}^{\frac{\beta}{2}}\bigg{]}$
$\displaystyle\leq
c_{\beta}E\bigg{[}\bigg{(}\sum_{i=1}^{2}\bigg{|}\int_{0}^{T}z^{i}_{s}dW^{i}_{s}+\int_{0}^{T}\int_{\mathcal{E}_{i}}\tilde{z}^{i}_{(s,e)}\tilde{N}_{i}(de,ds)\bigg{|}\bigg{)}^{\beta}\bigg{]}$
$\displaystyle\leq c_{\beta}E\bigg{[}\bigg{|}\xi+\int_{0}^{T}g(s)ds-
y_{0}\bigg{|}^{\beta}\bigg{]}\leq
c_{\beta}E\bigg{[}|\xi|^{\beta}+\bigg{(}\int_{0}^{T}|g(s)|ds\bigg{)}^{\beta}\bigg{]},$

where $c_{\beta}$ changed every step. Then we define a mapping
$\mathcal{X}:\tilde{\mathcal{N}}^{\beta}[0,T]\rightarrow\tilde{\mathcal{N}}^{\beta}[0,T]$
and maps
$(y^{\prime},z^{1,\prime},z^{2,\prime},\tilde{z}^{1,\prime},\tilde{z}^{2,\prime})$
into the solution $(y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})$ of the BSDEP
associated with the generator
$g(t,y^{\prime},z^{1,\prime},z^{2,\prime},\tilde{z}^{1,\prime},\tilde{z}^{2,\prime})$.
Given
$(y^{1,\prime},z^{1,1,\prime},z^{2,1,\prime},\tilde{z}^{1,1,\prime},\tilde{z}^{2,1,\prime})$,
$(y^{2,\prime},z^{1,2,\prime},z^{2,2,\prime},\tilde{z}^{1,2,\prime},\tilde{z}^{2,2,\prime})\in\tilde{\mathcal{N}}^{\beta}[0,T]$,
define
$(y^{1},z^{1,1},z^{2,1},\tilde{z}^{1,1},\tilde{z}^{2,1})=\mathcal{X}(y^{1,\prime},z^{1,1,\prime},z^{2,1,\prime},\tilde{z}^{1,1,\prime},\tilde{z}^{2,1,\prime})$
and $(y^{2},z^{1,2},z^{2,2},\tilde{z}^{1,2},\tilde{z}^{2,2})\\\
=\mathcal{X}(y^{2,\prime},z^{1,2,\prime},z^{2,2,\prime},\tilde{z}^{1,2,\prime},\tilde{z}^{2,2,\prime})$,
and set

$\displaystyle\tilde{y}^{\prime}=y^{1,\prime}-y^{2,\prime},\
\tilde{z}^{1,\prime}=z^{1,1,\prime}-z^{1,2,\prime},\
\tilde{z}^{2,\prime}=z^{2,1,\prime}-z^{2,2,\prime},\
\tilde{\tilde{z}}^{1,\prime}=\tilde{z}^{1,1,\prime}-\tilde{z}^{1,2,\prime},\
\tilde{\tilde{z}}^{2,\prime}=\tilde{z}^{2,1,\prime}-\tilde{z}^{2,2,\prime},$
(2.25) $\displaystyle\tilde{y}=y^{1}-y^{2},\ \tilde{z}^{1}=z^{1,1}-z^{1,2},\
\tilde{z}^{2}=z^{2,1}-z^{2,2},\
\tilde{\tilde{z}}^{1}=\tilde{z}^{1,1}-\tilde{z}^{1,2},\
\tilde{\tilde{z}}^{2}=\tilde{z}^{2,1}-\tilde{z}^{2,2},$

where
$(\tilde{y},\tilde{z}^{1},\tilde{z}^{2},\tilde{\tilde{z}}^{1},\tilde{\tilde{z}}^{2})$
satisfies

$\displaystyle\tilde{y}_{t}$
$\displaystyle=\int_{t}^{T}\big{[}g(t,y^{1,\prime},z^{1,1,\prime},z^{2,1,\prime},\tilde{z}^{1,1,\prime},\tilde{z}^{2,1,\prime})-g(t,y^{2,\prime},z^{1,2,\prime},z^{2,2,\prime},\tilde{z}^{1,2,\prime},\tilde{z}^{2,2,\prime})\big{]}ds$
(2.26)
$\displaystyle\quad-\sum_{i=1}^{2}\int_{t}^{T}\tilde{z}^{i}_{s}dW^{i}_{s}-\sum_{i=1}^{2}\int_{t}^{T}\int_{\mathcal{E}_{i}}\tilde{\tilde{z}}^{i}_{s,e}\tilde{N}_{i}(de,ds),$

and the terminal value is equal to zero. For fixed
$(y^{1,\prime},z^{1,1,\prime},z^{2,1,\prime},\tilde{z}^{1,1,\prime},\tilde{z}^{2,1,\prime})$
and
$(y^{2,\prime},z^{1,2,\prime},z^{2,2,\prime},\tilde{z}^{1,2,\prime},\tilde{z}^{2,2,\prime})$,
by the above estimate, we get

$\displaystyle E\bigg{[}\sup_{0\leq t\leq
T}|\tilde{y}_{t}|^{\beta}+\sum_{i=1}^{2}\bigg{(}\bigg{(}\int_{0}^{T}|\tilde{z}^{i}_{s}|^{2}ds\bigg{)}^{\frac{\beta}{2}}+\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{\tilde{z}}^{i}|^{2}N_{i}(de,ds)\bigg{)}^{\frac{\beta}{2}}\bigg{)}\bigg{]}$
(2.27) $\displaystyle\leq
C_{\beta}E\bigg{[}\bigg{(}\int_{0}^{T}\big{|}g(t,y^{1,\prime},z^{1,1,\prime},z^{2,1,\prime},\tilde{z}^{1,1,\prime},\tilde{z}^{2,1,\prime})-g(t,y^{2,\prime},z^{1,2,\prime},z^{2,2,\prime},\tilde{z}^{1,2,\prime},\tilde{z}^{2,2,\prime})\big{|}ds\bigg{)}^{\beta}\bigg{]}$
$\displaystyle\leq
C_{\beta}E\bigg{[}\bigg{(}\int_{0}^{T}C_{g}(|\tilde{y}^{\prime}|+|\tilde{z}^{1,\prime}|+|\tilde{z}^{2,\prime}|+||\tilde{\tilde{z}}^{1,\prime}||+||\tilde{\tilde{z}}^{2,\prime}||)ds\bigg{)}^{\beta}\bigg{]}$
$\displaystyle\leq
T^{\frac{\beta}{2}}C_{\beta}E\bigg{[}\bigg{(}\int_{0}^{T}C_{g}^{2}(|\tilde{y}^{\prime}|^{2}+|\tilde{z}^{1,\prime}|^{2}+|\tilde{z}^{2,\prime}|^{2}+||\tilde{\tilde{z}}^{1,\prime}||^{2}+||\tilde{\tilde{z}}^{2,\prime}||^{2})ds\bigg{)}^{\frac{\beta}{2}}\bigg{]}$
$\displaystyle\leq
T^{\frac{\beta}{2}}C_{g}^{\beta}C_{\beta}E\bigg{[}T^{\frac{\beta}{2}}\sup_{0\leq
t\leq
T}|\tilde{y}^{\prime}|^{\beta}+\sum_{i=1}^{2}\bigg{(}\bigg{(}\int_{0}^{T}|\tilde{z}^{i,\prime}|^{2}dt\bigg{)}^{\frac{\beta}{2}}+\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{\tilde{z}}^{i,\prime}|^{2}N_{i}(de,dt)\bigg{)}^{\frac{\beta}{2}}\bigg{)}\bigg{]}.$

Choosing $T$ such that $C_{\beta}C_{g}^{\beta}T^{\frac{\beta}{2}}<1$ and
$C_{\beta}C_{g}^{\beta}T^{\beta}<1$, then the mapping, from
$\tilde{\mathcal{N}}^{\beta}[0,T]$ to itself, is a contraction, and admits a
fixed point which corresponds to the solution to BSDEP in
$\tilde{\mathcal{N}}^{\beta}[0,T]$. For arbitrary $T$, we can obtain the
unique solution $(y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})$ by subdividing
interval $[0,T]$ into finite number of pieces. The proof is complete. ∎

Moreover, we need the following a priori estimate.

######  Lemma 2.3.

Let $T>0$ and $\xi^{1},\xi^{2}\in L^{2}_{\mathcal{F}_{T}}(\Omega)$, and
$g^{1},g^{2}$ satisfy (A4), let
$(y^{1},z^{1,1},z^{2,1},\tilde{z}^{1,1},\tilde{z}^{2,1})$ and
$(y^{2},z^{1,2},z^{2,2},\tilde{z}^{1,2},\tilde{z}^{2,2})$ be the solutions to
the BSDEP (2.21) associated with the terminal conditions $\xi^{1},\xi^{2}$ and
the generators $g^{1},g^{2}$, respectively. Then for $\beta\geq 2$, there
exist some constant $C_{\beta,g,T,\delta}$ depending on $\beta,T,\delta$ and
Lipschitz constants $C_{g}$ such that:

$\displaystyle E\bigg{[}\sup_{0\leq t\leq
T}|\tilde{y}_{t}|^{\beta}\bigg{]}+E\bigg{[}\sum_{i=1}^{2}\bigg{(}\bigg{(}\int_{0}^{T}|\tilde{z}^{i}|^{2}dt\bigg{)}^{\frac{\beta}{2}}+\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{\tilde{z}}^{i}|^{2}N_{i}(de,dt)\bigg{)}^{\frac{\beta}{2}}\bigg{)}\bigg{]}$
(2.28) $\displaystyle\leq
C_{\beta,g,T,\delta}E\bigg{[}|\xi^{1}-\xi^{2}|^{\beta}+\bigg{(}\int_{0}^{T}\big{|}g^{1}(s,y^{2},z^{1,2},z^{2,2},\tilde{z}^{1,2},\tilde{z}^{2,2})$
$\displaystyle\qquad\qquad\qquad-g^{2}(t,y^{2},z^{1,2},z^{2,2},\tilde{z}^{1,2},\tilde{z}^{2,2})\big{|}ds\bigg{)}^{\beta}\bigg{]}.$

###### Proof.

It is easy to derive that
$(\tilde{y},\tilde{z}^{1},\tilde{z}^{2},\tilde{\tilde{z}}^{1},\tilde{\tilde{z}}^{2})$
satisfies:

$\left\\{\begin{aligned}
-d\tilde{y}_{t}&=\big{[}g^{1}(t,y^{1},z^{1,1},z^{2,1},\tilde{z}^{1,1},\tilde{z}^{2,1})-g^{2}(t,y^{2},z^{1,2},z^{2,2},\tilde{z}^{1,2},\tilde{z}^{2,2})\big{]}dt\\\
&\quad-\sum_{i=1}^{2}\tilde{z}^{i}dW^{i}_{t}-\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\tilde{\tilde{z}}^{i}\tilde{N}_{i}(de,dt),\quad
t\in[0,T],\\\ \tilde{y}_{T}&=\xi^{1}-\xi^{2}.\end{aligned}\right.$ (2.29)

Applying Itô’s formula to $|\tilde{y}|^{2}$, we have

$\displaystyle|\tilde{y}_{0}|^{2}+\sum_{i=1}^{2}\bigg{(}\int_{0}^{T}|\tilde{z}^{i}|^{2}dt+\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{\tilde{z}}^{i}|^{2}N_{i}(de,dt)\bigg{)}$
(2.30)
$\displaystyle=|\tilde{y}_{T}|^{2}+\int_{0}^{T}2\tilde{y}\big{[}g^{1}(t,y^{1},z^{1,1},z^{2,1},\tilde{z}^{1,1},\tilde{z}^{2,1})-g^{2}(t,y^{2},z^{1,2},z^{2,2},\tilde{z}^{1,2},\tilde{z}^{2,2})\big{]}dt$
$\displaystyle\qquad-\sum_{i=1}^{2}\int_{0}^{T}2\tilde{y}\tilde{z}^{i}dW^{i}_{t}-\sum_{i=1}^{2}\int_{0}^{T}\int_{\mathcal{E}_{i}}2\tilde{y}\tilde{\tilde{z}}^{i}\tilde{N}_{i}(de,dt),$

and

$\displaystyle\sum_{i=1}^{2}\bigg{(}\int_{0}^{T}|\tilde{z}^{i}|^{2}dt+\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{\tilde{z}}^{i}|^{2}N_{i}(de,dt)\bigg{)}$
$\displaystyle\leq|\xi^{1}-\xi^{2}|^{2}+\int_{0}^{T}2\tilde{y}\big{[}g^{1}(t,y^{1},z^{1,1},z^{2,1},\tilde{z}^{1,1},\tilde{z}^{2,1})-g^{1}(t,y^{2},z^{1,2},z^{2,2},\tilde{z}^{1,2},\tilde{z}^{2,2})$
$\displaystyle\quad+g^{1}(t,y^{2},z^{1,2},z^{2,2},\tilde{z}^{1,2},\tilde{z}^{2,2})-g^{2}(t,y^{2},z^{1,2},z^{2,2},\tilde{z}^{1,2},\tilde{z}^{2,2})\big{]}dt$
$\displaystyle\quad-\sum_{i=1}^{2}\int_{0}^{T}2\tilde{y}\tilde{z}^{i}dW^{i}_{t}-\sum_{i=1}^{2}\int_{0}^{T}\int_{\mathcal{E}_{i}}2\tilde{y}\tilde{\tilde{z}}^{i}\tilde{N}_{i}(de,dt)$
$\displaystyle\leq|\xi^{1}-\xi^{2}|^{2}+\int_{0}^{T}2\tilde{y}\big{[}C_{g}(|\tilde{y}|+|\tilde{z}^{1}|+|\tilde{z}^{2}|+||\tilde{\tilde{z}}^{1}||+||\tilde{\tilde{z}}^{2}||)\big{]}dt$
$\displaystyle\quad+\int_{0}^{T}2\tilde{y}\big{[}g^{1}(t,y^{2},z^{1,2},z^{2,2},\tilde{z}^{1,2},\tilde{z}^{2,2})-g^{2}(t,y^{2},z^{1,2},z^{2,2},\tilde{z}^{1,2},\tilde{z}^{2,2})\big{]}dt$
$\displaystyle\quad+\sum_{i=1}^{2}\bigg{(}\bigg{|}\int_{0}^{T}2\tilde{y}\tilde{z}^{i}dW^{i}_{t}\bigg{|}+\bigg{|}\int_{0}^{T}\int_{\mathcal{E}_{i}}2\tilde{y}\tilde{\tilde{z}}^{i}\tilde{N}_{i}(de,dt)\bigg{|}\bigg{)}$
$\displaystyle\leq|\xi^{1}-\xi^{2}|^{2}+\int_{0}^{T}\big{[}2C_{g}|\tilde{y}|^{2}+4C_{g}^{2}|\tilde{y}|^{2}+\frac{1}{2}|\tilde{z}^{1}|^{2}+\frac{1}{2}|\tilde{z}^{2}|^{2}+\frac{1}{2}||\tilde{\tilde{z}}^{1}||^{2}+\frac{1}{2}||\tilde{\tilde{z}}^{2}||^{2}\big{]}dt$
(2.31) $\displaystyle\quad+2\sup_{0\leq t\leq
T}|\tilde{y}|\int_{0}^{T}\big{[}g^{1}(t,y^{2},z^{1,2},z^{2,2},\tilde{z}^{1,2},\tilde{z}^{2,2})-g^{2}(t,y^{2},z^{1,2},z^{2,2},\tilde{z}^{1,2},\tilde{z}^{2,2})\big{]}dt$
$\displaystyle\quad+\sum_{i=1}^{2}\bigg{(}\bigg{|}\int_{0}^{T}2\tilde{y}\tilde{z}_{i}dW^{i}_{t}\bigg{|}+\bigg{|}\int_{0}^{T}\int_{\mathcal{E}_{i}}2\tilde{y}\tilde{\tilde{z}}^{i}\tilde{N}_{i}(de,dt)\bigg{|}\bigg{)}$
$\displaystyle\leq|\xi^{1}-\xi^{2}|^{2}+(2C_{g}+4C_{g}^{2})T\sup_{0\leq t\leq
T}|\tilde{y}|^{2}+\frac{1}{2}\bigg{[}\sum_{i=1}^{2}\bigg{(}\int_{0}^{T}|\tilde{z}^{i}|^{2}dt+\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{\tilde{z}}^{i}|^{2}N_{i}(de,dt)\bigg{)}\bigg{]}$
$\displaystyle\quad+2\delta\sup_{0\leq t\leq
T}|\tilde{y}|^{2}+\frac{1}{2\delta}\bigg{(}\int_{0}^{T}[g^{1}(t,y^{2},z^{1,2},z^{2,2},\tilde{z}^{1,2},\tilde{z}^{2,2})-g^{2}(t,y^{2},z^{1,2},z^{2,2},\tilde{z}^{1,2},\tilde{z}^{2,2})]dt\bigg{)}^{2}$
$\displaystyle\quad+\sum_{i=1}^{2}\bigg{(}\bigg{|}\int_{0}^{T}2\tilde{y}\tilde{z}^{i}dW^{i}_{t}\bigg{|}+\bigg{|}\int_{0}^{T}\int_{\mathcal{E}_{i}}2\tilde{y}\tilde{\tilde{z}}^{i}\tilde{N}_{i}(de,dt)\bigg{|}\bigg{)}.$

Then we get

$\displaystyle
E\bigg{[}\sum_{i=1}^{2}\bigg{(}\bigg{(}\int_{0}^{T}|\tilde{z}^{i}|^{2}dt\bigg{)}^{\frac{\beta}{2}}+\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{\tilde{z}}^{i}|^{2}N_{i}(de,dt)\bigg{)}^{\frac{\beta}{2}}\bigg{)}\bigg{]}$
(2.32) $\displaystyle\leq
C_{\beta}\bigg{\\{}E|\xi^{1}-\xi^{2}|^{\beta}+(2C_{g}+4C_{g}^{2})^{\frac{\beta}{2}}T^{\frac{\beta}{2}}E\bigg{[}\sup_{0\leq
t\leq
T}|\tilde{y}|^{\beta}\bigg{]}+\delta^{\frac{\beta}{2}}E\bigg{[}\sup_{0\leq
t\leq T}|\tilde{y}|^{\beta}\bigg{]}$
$\displaystyle\quad+\frac{1}{\delta^{\frac{\beta}{2}}}E\bigg{(}\int_{0}^{T}[g^{1}(t,y^{2},z^{1,2},z^{2,2},\tilde{z}^{1,2},\tilde{z}^{2,2})-g^{2}(t,y^{2},z^{1,2},z^{2,2},\tilde{z}^{1,2},\tilde{z}^{2,2})]dt\bigg{)}^{\beta}$
$\displaystyle\quad+E\sum_{i=1}^{2}\bigg{|}\int_{0}^{T}\tilde{y}\tilde{z}^{i}dW^{i}_{t}\bigg{|}^{\frac{\beta}{2}}+E\sum_{i=1}^{2}\bigg{|}\int_{0}^{T}\int_{\mathcal{E}_{i}}\tilde{y}\tilde{\tilde{z}}^{i}\tilde{N}_{i}(de,dt)\bigg{|}^{\frac{\beta}{2}}\bigg{\\}}.$

Note that

$\displaystyle
E\bigg{|}\int_{0}^{T}\tilde{y}\tilde{z}^{i}dW^{i}_{t}\bigg{|}^{\frac{\beta}{2}}$
$\displaystyle\leq
CE\bigg{(}\int_{0}^{T}|\tilde{y}|^{2}|\tilde{z}^{i}|^{2}dt\bigg{)}^{\frac{\beta}{4}}\leq
CE\bigg{[}\sup_{0\leq t\leq
T}|\tilde{y}_{t}|^{\frac{\beta}{2}}\bigg{(}\int_{0}^{T}|\tilde{z}^{i}|^{2}dt\bigg{)}^{\frac{\beta}{4}}\bigg{]}$
(2.33) $\displaystyle\leq\frac{C^{2}}{2}E\bigg{[}\sup_{0\leq t\leq
T}|\tilde{y}_{t}|^{\beta}\bigg{]}+\frac{1}{2}E\bigg{[}\bigg{(}\int_{0}^{T}|\tilde{z}^{i}|^{2}dt\bigg{)}^{\frac{\beta}{2}}\bigg{]},$

and

$\displaystyle
E\bigg{|}\int_{0}^{T}\int_{\mathcal{E}_{i}}\tilde{y}\tilde{\tilde{z}}^{i}\tilde{N}_{i}(de,dt)\bigg{|}^{\frac{\beta}{2}}\leq
CE\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{y}|^{2}|\tilde{\tilde{z}}^{i}|^{2}N_{i}(de,dt)\bigg{)}^{\frac{\beta}{4}}$
(2.34) $\displaystyle\leq CE\bigg{[}\sup_{0\leq t\leq
T}|\tilde{y}_{t}|^{\frac{\beta}{2}}\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{\tilde{z}}^{i}|^{2}N_{i}(de,dt)\bigg{)}^{\frac{\beta}{4}}\bigg{]}$
$\displaystyle\leq\frac{C^{2}}{2}E\bigg{[}\sup_{0\leq t\leq
T}|\tilde{y}_{t}|^{\beta}\bigg{]}+\frac{1}{2}E\bigg{[}\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{\tilde{z}}^{i}|^{2}N_{i}(de,dt)\bigg{)}^{\frac{\beta}{2}}\bigg{]}.$

Then the inequality (2.32) becomes

$\displaystyle
E\bigg{[}\sum_{i=1}^{2}\bigg{(}\bigg{(}\int_{0}^{T}|\tilde{z}^{i}|^{2}dt\bigg{)}^{\frac{\beta}{2}}+\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{\tilde{z}}^{i}|^{2}N_{i}(de,dt)\bigg{)}^{\frac{\beta}{2}}\bigg{)}\bigg{]}$
(2.35) $\displaystyle\leq
C_{\beta}\bigg{\\{}E|\xi^{1}-\xi^{2}|^{\beta}+\big{[}(2C_{g}+4C_{g}^{2})^{\frac{\beta}{2}}T^{\frac{\beta}{2}}+\delta^{\frac{\beta}{2}}+C^{2}\big{]}E\bigg{[}\sup_{0\leq
t\leq T}|\tilde{y}_{t}|^{\beta}\bigg{]}$
$\displaystyle\quad+\frac{1}{\delta^{\frac{\beta}{2}}}E\bigg{(}\int_{0}^{T}[g^{1}(t,y^{2},z^{1,2},z^{2,2},\tilde{z}^{1,2},\tilde{z}^{2,2})-g^{2}(t,y^{2},z^{1,2},z^{2,2},\tilde{z}^{1,2},\tilde{z}^{2,2})]dt\bigg{)}^{\beta}\bigg{\\}}.$

Meanwhile, we can obtain from the integral form of (2.29) that

$\displaystyle|\tilde{y}|^{\beta}$ $\displaystyle\leq
C_{\beta}\bigg{\\{}|\xi^{1}-\xi^{2}|^{\beta}+\bigg{|}\int_{t}^{T}[g^{1}(s,y^{1},z^{1,1},z^{2,1},\tilde{z}^{1,1},\tilde{z}^{2,1})-g^{2}(t,y^{2},z^{1,2},z^{2,2},\tilde{z}^{1,2},\tilde{z}^{2,2})]ds\bigg{|}^{\beta}$
(2.36)
$\displaystyle\qquad+\sum_{i=1}^{2}\bigg{(}\bigg{|}\int_{t}^{T}\tilde{z}^{i}dW^{i}_{s}\bigg{|}^{\beta}+\bigg{|}\int_{t}^{T}\int_{\mathcal{E}_{i}}\tilde{\tilde{z}}^{i}\tilde{N}_{i}(de,ds)\bigg{|}^{\beta}\bigg{)}\bigg{\\}}.$

By B-D-G’s inequality, we have

$\displaystyle E\bigg{[}\sup_{0\leq t\leq
T}|\tilde{y}_{t}|^{\beta}\bigg{]}\leq
C_{\beta}\bigg{\\{}E|\xi^{1}-\xi^{2}|^{\beta}+E\bigg{|}\int_{0}^{T}[g^{1}(s,y^{1},z^{1,1},z^{2,1},\tilde{z}^{1,1},\tilde{z}^{2,1})$
(2.37)
$\displaystyle\qquad-g^{2}(t,y^{2},z^{1,2},z^{2,2},\tilde{z}^{1,2},\tilde{z}^{2,2})]ds\bigg{|}^{\beta}+E\sum_{i=1}^{2}\bigg{(}\int_{0}^{T}|\tilde{z}^{i}|^{2}ds\bigg{)}^{\frac{\beta}{2}}$
$\displaystyle\qquad+E\sum_{i=1}^{2}\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{\tilde{z}}^{i}|^{2}N_{i}(de,ds)\bigg{)}^{\frac{\beta}{2}}\bigg{\\}}$
$\displaystyle\leq
C_{\beta}\bigg{\\{}E|\xi^{1}-\xi^{2}|^{\beta}+E\bigg{(}\int_{0}^{T}C_{g}(|\tilde{y}|+|\tilde{z}^{1}|+|\tilde{z}^{2}|+||\tilde{\tilde{z}}^{1}||+||\tilde{\tilde{z}}^{2}||)ds\bigg{)}^{\beta}$
$\displaystyle\qquad+E\bigg{(}\int_{0}^{T}|g^{1}(s,y^{2},z^{1,2},z^{2,2},\tilde{z}^{1,2},\tilde{z}^{2,2})-g^{2}(t,y^{2},z^{1,2},z^{2,2},\tilde{z}^{1,2},\tilde{z}^{2,2})|ds\bigg{)}^{\beta}$
$\displaystyle\qquad+E\sum_{i=1}^{2}\bigg{(}\int_{0}^{T}|\tilde{z}^{i}|^{2}ds\bigg{)}^{\frac{\beta}{2}}+E\sum_{i=1}^{2}\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{\tilde{z}}^{i}|^{2}N_{i}(de,ds)\bigg{)}^{\frac{\beta}{2}}\bigg{\\}}.$

Substituting (2.35), we can rearrange the above inequality as follows

$\displaystyle E\bigg{[}\sup_{0\leq t\leq T}|\tilde{y}_{t}|^{\beta}\bigg{]}$
$\displaystyle\leq
C_{\beta,g,T}E|\xi^{1}-\xi^{2}|^{\beta}+C_{\beta,g,T,\delta}E\bigg{(}\int_{0}^{T}|g^{1}(s,y^{2},z^{1,2},z^{2,2},\tilde{z}^{1,2},\tilde{z}^{2,2})$
(2.38)
$\displaystyle\quad-g^{2}(t,y^{2},z^{1,2},z^{2,2},\tilde{z}^{1,2},\tilde{z}^{2,2})|ds\bigg{)}^{\beta}+C_{\beta,g,T,\delta}E\bigg{[}\sup_{0\leq
t\leq T}|\tilde{y}_{t}|^{\beta}\bigg{]}.$

Let $1-C_{\beta,g,T,\delta}>0$ and we get

$\displaystyle E\bigg{[}\sup_{0\leq t\leq T}|\tilde{y}_{t}|^{\beta}\bigg{]}$
$\displaystyle\leq
C_{\beta,g,T,\delta}\bigg{[}E|\xi^{1}-\xi^{2}|^{\beta}+E\bigg{(}\int_{0}^{T}|g^{1}(s,y^{2},z^{1,2},z^{2,2},\tilde{z}^{1,2},\tilde{z}^{2,2})$
(2.39)
$\displaystyle\qquad\qquad\quad-g^{2}(t,y^{2},z^{1,2},z^{2,2},\tilde{z}^{1,2},\tilde{z}^{2,2})|ds\bigg{)}^{\beta}\bigg{]}.$

Finally, the a priori estimate is given in the following

$\displaystyle E\bigg{[}\sup_{0\leq t\leq
T}|\tilde{y}_{t}|^{\beta}\bigg{]}+E\bigg{[}\sum_{i=1}^{2}\bigg{(}\bigg{(}\int_{0}^{T}|\tilde{z}^{i}|^{2}dt\bigg{)}^{\frac{\beta}{2}}+\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{\tilde{z}}^{i}|^{2}N_{i}(de,dt)\bigg{)}^{\frac{\beta}{2}}\bigg{)}\bigg{]}$
(2.40) $\displaystyle\leq
C_{\beta,g,T,\delta}E\bigg{[}|\xi^{1}-\xi^{2}|^{\beta}+\bigg{(}\int_{0}^{T}|g^{1}(s,y^{2},z^{1,2},z^{2,2},\tilde{z}^{1,2},\tilde{z}^{2,2})$
$\displaystyle\qquad\qquad\qquad-g^{2}(t,y^{2},z^{1,2},z^{2,2},\tilde{z}^{1,2},\tilde{z}^{2,2})|ds\bigg{)}^{\beta}\bigg{]}.$

The proof is complete. ∎

We now back to the proof of Theorem 2.3.

###### Proof.

For any
$(y^{\prime},z^{1,\prime},z^{2,\prime},\tilde{z}^{1,\prime},\tilde{z}^{2,\prime})\in\tilde{\mathcal{N}}^{\beta}[0,T]$,
it is easy to get a unique solution
$(y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})\\\
\in\tilde{\mathcal{N}}^{\beta}[0,T]$ to the decoupled FBSDEP as follows:

$\left\\{\begin{aligned}
dx_{t}&=b(t,x_{t},y^{\prime}_{t},z^{1,\prime}_{t},z^{2,\prime}_{t},\tilde{z}^{1,\prime}_{(t,e)},\tilde{z}^{2,\prime}_{(t,e)})dt+\sum_{i=1}^{2}\sigma_{i}(t,x_{t},y^{\prime}_{t},z^{1,\prime}_{t},z^{2,\prime}_{t},\tilde{z}^{1,\prime}_{(t,e)},\tilde{z}^{2,\prime}_{(t,e)})dW^{i}_{t}\\\
&\quad+\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}f_{i}(t,x_{t},y^{\prime}_{t},\tilde{z}^{1,\prime}_{(t,e)},\tilde{z}^{2,\prime}_{(t,e)},e)\tilde{N}_{i}(de,dt),\\\
-dy_{t}&=g(t,\Theta(t))dt-\sum_{i=1}^{2}z^{i}_{t}dW^{i}_{t}-\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\tilde{z}^{i}_{(t,e)}\tilde{N}_{i}(de,dt),\quad
t\in[0,T],\\\ x_{0}&=x_{0},\ \ y_{T}=\phi(x_{T}),\end{aligned}\right.$ (2.41)

by solving forward SDEP first in $S^{\beta}[0,T]$ and then solving BSDEP in
$\tilde{\mathcal{N}}^{\beta}[0,T]$, sequentially. Define a map
$\mathcal{X}_{1}$ from $\tilde{\mathcal{N}}^{\beta}[0,T]$ to itself. Let
$(y^{\prime}_{i},z^{1,\prime}_{i},z^{2,\prime}_{i},\tilde{z}^{1,\prime}_{i},\tilde{z}^{2,\prime}_{i})\in\tilde{\mathcal{N}}^{\beta}[0,T]$
and
$(y_{i},z^{1}_{i},z^{2}_{i},\tilde{z}^{1}_{i},\tilde{z}^{2}_{i})=\mathcal{X}_{1}(y^{\prime}_{i},z^{1,\prime}_{i},z^{2,\prime}_{i},\tilde{z}^{1,\prime}_{i},\tilde{z}^{2,\prime}_{i}),i=1,2$
and define

$\displaystyle\hat{y}^{\prime}=y^{\prime}_{1}-y^{\prime}_{2},\
\hat{z}^{1,\prime}=z^{1,\prime}_{1}-z^{1,\prime}_{2},\
\hat{z}^{2,\prime}=z^{2,\prime}_{1}-z^{2,\prime}_{2},\
\hat{\tilde{z}}^{1,\prime}=\tilde{z}^{1,\prime}_{1}-\tilde{z}^{1,\prime}_{2},\
\hat{\tilde{z}}^{2,\prime}=\tilde{z}^{2,\prime}_{1}-\tilde{z}^{2,\prime}_{2},$
(2.42) $\displaystyle\hat{y}=y_{1}-y_{2},\ \hat{z}^{1}=z^{1}_{1}-z^{1}_{2},\
\hat{z}^{2}=z^{2}_{1}-z^{2}_{2},\
\hat{\tilde{z}}^{1}=\tilde{z}^{1}_{1}-\tilde{z}^{1}_{2},\
\hat{\tilde{z}}^{2}=\tilde{z}^{2}_{1}-\tilde{z}^{2}_{2},\
\hat{x}=x_{1}-x_{2}.$

By (A4), we get

$\displaystyle|\hat{x}_{t}|^{\beta}$ $\displaystyle\leq
C_{\beta}\bigg{\\{}\bigg{|}\int_{0}^{t}C\big{(}|\hat{x}|+|\hat{y}^{\prime}|+|\hat{z}^{1,\prime}|+|\hat{z}^{2,\prime}|+||\hat{\tilde{z}}^{1,\prime}||+||\hat{\tilde{z}}^{2,\prime}||\big{)}ds\bigg{|}^{\beta}$
(2.43)
$\displaystyle\qquad\quad+\sum_{i=1}^{2}\bigg{|}\int_{0}^{t}\big{[}K(|\hat{x}|+|\hat{y}^{\prime}|)+L_{\sigma_{i}}(|\hat{z}^{1,\prime}|+|\hat{z}^{2,\prime}|+||\hat{\tilde{z}}^{1,\prime}||+||\hat{\tilde{z}}^{2,\prime}||)\big{]}dW^{i}_{s}\bigg{|}^{\beta}$
$\displaystyle\qquad\quad+\sum_{i=1}^{2}\bigg{|}\int_{0}^{t}\int_{\mathcal{E}_{i}}\big{[}\rho(e)(|\hat{x}|+|\hat{y}^{\prime}|)+L_{f_{i}}(e)(|\hat{\tilde{z}}^{1,\prime}|+|\hat{\tilde{z}}^{2,\prime}|)\big{]}\tilde{N}_{i}(de,ds)\bigg{|}^{\beta}\bigg{\\}},$

and

$\displaystyle E\bigg{[}\sup_{0\leq t\leq T}|\hat{x}_{t}|^{\beta}\bigg{]}\leq
C_{\beta}\bigg{\\{}(CT^{\frac{\beta}{2}}+L_{\sigma_{1}}^{\beta}+L_{\sigma_{2}}^{\beta})\sum_{i=1}^{2}E\bigg{(}\int_{0}^{T}|\hat{z}^{i,\prime}|^{2}dt\bigg{)}^{\frac{\beta}{2}}$
(2.44)
$\displaystyle\qquad+\big{(}CT^{\frac{\beta}{2}}+L_{\sigma_{1}}+L_{\sigma_{2}}+\tilde{C}_{f_{1}}+\tilde{C}_{f_{2}}\big{)}\sum_{i=1}^{2}E\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\hat{\tilde{z}}^{i,\prime}|^{2}N_{i}(de,dt)\bigg{)}^{\frac{\beta}{2}}$
$\displaystyle\qquad+\bigg{(}CT^{\beta}+K^{\beta}T^{\frac{\beta}{2}}+T\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\rho^{\beta}(e)\nu_{i}(de)\bigg{)}E\bigg{[}\sup_{0\leq
t\leq T}|\hat{x}_{t}|^{\beta}+\sup_{0\leq t\leq
T}|\hat{y}^{\prime}_{t}|^{\beta}\bigg{]}\bigg{\\}}.$

And there exists a $\hat{T}$ such that
$1-C_{\beta}\bigg{(}C\hat{T}^{\beta}+K^{\beta}\hat{T}^{\frac{\beta}{2}}+\hat{T}\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\rho^{\beta}(e)\nu_{i}(de)\bigg{)}>0$,
then we have, for $T\leq\hat{T}$,

$\displaystyle E\bigg{[}\sup_{0\leq t\leq
T}|\hat{x}_{t}|^{\beta}\bigg{]}\leq\frac{C_{\beta}(CT^{\frac{\beta}{2}}+L_{\sigma_{1}}^{\beta}+L_{\sigma_{2}}^{\beta})}{1-C_{\beta}\bigg{(}CT^{\beta}+K^{\beta}T^{\frac{\beta}{2}}+T\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\rho^{\beta}(e)\nu_{i}(de)\bigg{)}}\sum_{i=1}^{2}E\bigg{(}\int_{0}^{T}|\hat{z}^{i,\prime}|^{2}dt\bigg{)}^{\frac{\beta}{2}}$
(2.45)
$\displaystyle\quad+\frac{C_{\beta}(CT^{\frac{\beta}{2}}+L_{\sigma_{1}}^{\beta}+L_{\sigma_{2}}^{\beta}+\tilde{C}_{f_{1}}+\tilde{C}_{f_{2}})}{1-C_{\beta}\bigg{(}CT^{\beta}+K^{\beta}T^{\frac{\beta}{2}}+T\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\rho^{\beta}(e)\nu_{i}(de)\bigg{)}}\sum_{i=1}^{2}E\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\hat{\tilde{z}}^{i,\prime}|^{2}N_{i}(de,dt)\bigg{)}^{\frac{\beta}{2}}$
$\displaystyle\quad+\frac{C_{\beta}\bigg{(}CT^{\beta}+K^{\beta}T^{\frac{\beta}{2}}+T\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\rho^{\beta}(e)\nu_{i}(de)\bigg{)}}{1-C_{\beta}\bigg{(}CT^{\beta}+K^{\beta}T^{\frac{\beta}{2}}+T\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\rho^{\beta}(e)\nu_{i}(de)\bigg{)}}E\bigg{[}\sup_{0\leq
t\leq T}|\hat{y}^{\prime}_{t}|^{\beta}\bigg{]}.$

Next, we can get the BSDEP of
$(\hat{y},\hat{z}^{1},\hat{z}^{2},\hat{\tilde{z}}^{1},\hat{\tilde{z}}^{2})$ in
the following:

$\left\\{\begin{aligned}
-d\hat{y}_{t}&=\big{[}g(t,\Theta_{1})-g(t,\Theta_{2})\big{]}dt-\sum_{i=1}^{2}\hat{z}^{i}_{t}dW^{i}_{t}-\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\hat{\tilde{z}}^{i}_{(t,e)}\tilde{N}_{i}(de,dt),\quad
t\in[0,T],\\\ \hat{y}_{T}&=\phi(x_{1T})-\phi(x_{2T}).\end{aligned}\right.$
(2.46)

By the a priori estimate (2.28) in Lemma 2.3, we have

$\displaystyle E\bigg{[}\sup_{0\leq t\leq
T}|\hat{y}_{t}|^{\beta}\bigg{]}+E\bigg{[}\sum_{i=1}^{2}\bigg{(}\bigg{(}\int_{0}^{T}|\hat{z}^{i}|^{2}dt\bigg{)}^{\frac{\beta}{2}}+\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\hat{\tilde{z}}^{i}|^{2}N_{i}(de,dt)\bigg{)}^{\frac{\beta}{2}}\bigg{)}\bigg{]}$
(2.47) $\displaystyle\leq
C\bigg{[}E|\phi(x_{1T})-\phi(x_{2T})|^{\beta}+E\bigg{(}\int_{0}^{T}|g(t,x_{1},y_{1},z^{1}_{1},z^{2}_{1},\tilde{z}^{1}_{1},\tilde{z}^{2}_{1})-g(t,x_{2},y_{1},z^{1}_{1},z^{2}_{1},\tilde{z}^{1}_{1},\tilde{z}^{2}_{1})|dt\bigg{)}^{\beta}\bigg{]}$
$\displaystyle\leq
C\bigg{[}E\big{[}C\hat{x}_{T}\big{]}^{\beta}+E\bigg{(}\int_{0}^{T}\hat{x}_{t}dt\bigg{)}^{\beta}\bigg{]}\leq
C\big{(}1+T^{\beta}\big{)}E\bigg{[}\sup_{0\leq t\leq
T}|\hat{x}|^{\beta}\bigg{]}.$

Then substituting (2.45) into (2.47), we can obtain

$\displaystyle E\bigg{[}\sup_{0\leq t\leq
T}|\hat{y}_{t}|^{\beta}\bigg{]}+E\bigg{[}\sum_{i=1}^{2}\bigg{(}\bigg{(}\int_{0}^{T}|\hat{z}^{i}|^{2}dt\bigg{)}^{\frac{\beta}{2}}+\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\hat{\tilde{z}}^{i}|^{2}N_{i}(de,dt)\bigg{)}^{\frac{\beta}{2}}\bigg{)}\bigg{]}$
(2.48)
$\displaystyle\leq\frac{C_{\beta}\big{(}1+T^{\beta}\big{)}\Big{[}C(T^{\frac{\beta}{2}}+T^{\beta})+L_{\sigma_{1}}^{\beta}+L_{\sigma_{2}}^{\beta}+\tilde{C}_{f_{1}}+\tilde{C}_{f_{2}}+K^{\beta}T^{\frac{\beta}{2}}+T\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\rho^{\beta}(e)\nu_{i}(de)\Big{]}}{1-C_{\beta}\Big{[}CT^{\beta}+K^{\beta}T^{\frac{\beta}{2}}+T\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\rho^{\beta}(e)\nu_{i}(de)\Big{]}}$
$\displaystyle\quad\times
E\bigg{[}\sum_{i=1}^{2}\bigg{(}\bigg{(}\int_{0}^{T}|\hat{z}^{i,\prime}|^{2}dt\bigg{)}^{\frac{\beta}{2}}+\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\hat{\tilde{z}}^{i,\prime}|^{2}N_{i}(de,dt)\bigg{)}^{\frac{\beta}{2}}\bigg{)}+\sup_{0\leq
t\leq T}|\hat{y}^{\prime}_{t}|^{\beta}\bigg{]}.$

We take Lipschitz coefficients
$L_{\sigma_{1}},L_{\sigma_{2}},\tilde{C}_{f_{1}}$ and $\tilde{C}_{f_{2}}$
small enough, $C$ is a constant not related to $T$ but changed every step.
Thus, there exists a $\tilde{T}\leq\hat{T}$ small enough such that

$\displaystyle\frac{C_{\beta}\big{(}1+\tilde{T}^{\beta}\big{)}\Big{[}C(\tilde{T}^{\frac{\beta}{2}}+\tilde{T}^{\beta})+L_{\sigma_{1}}^{\beta}+L_{\sigma_{2}}^{\beta}+\tilde{C}_{f_{1}}+\tilde{C}_{f_{2}}+K^{\beta}\tilde{T}^{\frac{\beta}{2}}+\tilde{T}\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\rho^{\beta}(e)\nu_{i}(de)\Big{]}}{1-C_{\beta}\Big{[}C\tilde{T}^{\beta}+K^{\beta}\tilde{T}^{\frac{\beta}{2}}+\tilde{T}\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\rho^{\beta}(e)\nu_{i}(de)\Big{]}}<1.$
(2.49)

Then for any $0\leq T\leq\tilde{T}$, $\mathcal{X}_{1}$ is a contraction on
$[0,T]$, which has a unique fixed point
$\mathcal{X}_{1}(y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})=(y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})$.
Moreover, FBSDEP (2.10) admits a unique solution
$(x,y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})$ on small interval $[0,T]$. The
proof is complete. ∎

The following assumption is needed.

(A6) For any $t\in[0,T]$, $(x,y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})\in
R^{6}$, the jump coefficient $f_{i}$ is independent of
$(z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})$ for $i=1,2$.

If we consider (2.10) in a parameterized form by initial condition
$(t,\xi)\in[0,T]\times L^{2}(\Omega,\mathcal{F}_{t},P;\\\ \mathbb{R})$:

$\left\\{\begin{aligned}
dx^{t,\xi}_{s}&=b(s,\Theta_{s}^{t,\xi})ds+\sum_{i=1}^{2}\sigma_{i}(s,\Theta_{s}^{t,\xi})dW^{i}_{s}+\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}f_{i}(s,\Theta_{s-}^{t,\xi},e)\tilde{N}_{i}(de,ds),\\\
-dy^{t,\xi}_{s}&=g(s,\Theta_{s}^{t,\xi})ds-\sum_{i=1}^{2}z^{i,t,\xi}_{s}dW^{i}_{s}-\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\tilde{z}^{i,t,\xi}_{(s,e)}\tilde{N}_{i}(de,ds),\quad
s\in[t,t+\delta],\\\ x^{t,\xi}_{t}&=\xi,\ \
y^{t,\xi}_{t+\delta}=\phi(x^{t,\xi}_{t+\delta}),\end{aligned}\right.$ (2.50)

where
$\Theta_{s}^{t,\xi}=(x^{t,\xi}_{s},y^{t,\xi}_{s},z^{1,t,\xi}_{s},z^{2,t,\xi}_{s},\tilde{z}^{1,t,\xi}_{(s,e)},\tilde{z}^{2,t,\xi}_{(s,e)})$.
Then, similar to Theorem 3.4 in Li and Wei [16], we can also obtain the
following estimate with respect to the initial value in a conditional
expectation form.

######  Lemma 2.4.

Let assumption (A4),(A6) hold. Then, for every $\beta\geq 2$, there exists a
sufficiently small constant $\tilde{\delta}$ depending on
$(K,L_{\sigma_{1}},L_{\sigma_{2}},\rho,L_{f_{1}},L_{f_{2}})$ and constant
$C_{\beta,K,\rho,\sigma_{1},\sigma_{2},f_{1},f_{2},L}$ depending on
$(\beta,K,\rho,L_{\sigma_{1}},L_{\sigma_{2}},L_{f_{1}},L_{f_{2}},L)$ such that
for every $0\leq\delta\leq\tilde{\delta}$ and $\xi\in
L^{\beta}(\Omega,\mathcal{F}_{t},P;\mathbb{R})$,

$\displaystyle E\bigg{[}\sup_{t\leq s\leq
t+\delta}|x^{t,\xi}_{s}|^{\beta}+\sup_{t\leq s\leq
t+\delta}|y^{t,\xi}_{s}|^{\beta}+\sum_{i=1}^{2}\bigg{(}\int_{t}^{t+\delta}|z^{i,t,\xi}_{s}|^{2}ds\bigg{)}^{\frac{\beta}{2}}$
(2.51)
$\displaystyle\quad+\sum_{i=1}^{2}\bigg{(}\int_{t}^{t+\delta}\int_{\mathcal{E}_{i}}|\tilde{z}^{i,t,\xi}_{(s,e)}|^{2}\nu_{i}(de)ds\bigg{)}^{\frac{\beta}{2}}\bigg{|}\mathcal{F}_{t}\bigg{]}\leq
C_{\beta,K,\rho,\sigma_{1},\sigma_{2},f_{1},f_{2},L}(1+|\xi|^{\beta}),\
P\mbox{-}a.s..$

Motivated by the Theorem 2.1 in the recent result in [21], by the critical and
thought-provoking technique in their paper, we give an extension result to the
case with Poisson jumps. Then we introduce the first main theorem in our
paper.

######  Theorem 2.4.

Let (A4),(A6) hold, and assume that, for every $\xi,\tilde{\xi}\in
L^{\beta}(\Omega,\mathcal{F}_{t},P;\mathbb{R})$, the $L^{2}$-estimations of
FBSDEP (2.50) are given as follows:

$\mathrm{(i)}$

$\displaystyle E\bigg{[}\sup_{t\leq s\leq T}|x^{t,\xi}_{s}|^{2}+\sup_{t\leq
s\leq
T}|y^{t,\xi}_{s}|^{2}+\sum_{i=1}^{2}\int_{t}^{T}|z^{i,t,\xi}_{s}|^{2}ds+\sum_{i=1}^{2}\int_{t}^{T}\int_{\mathcal{E}_{i}}|\tilde{z}^{i,t,\xi}_{(s,e)}|^{2}\nu_{i}(de)ds\bigg{|}\mathcal{F}_{t}\bigg{]}$
(2.52) $\displaystyle\qquad\leq K_{1}(1+|\xi|^{2}),$

$\mathrm{(ii)}$

$\displaystyle E\bigg{[}\sup_{t\leq s\leq
T}|x^{t,\xi}_{s}-x^{t,\tilde{\xi}}_{s}|^{2}+\sup_{t\leq s\leq
T}|y^{t,\xi}_{s}-y^{t,\tilde{\xi}}_{s}|^{2}+\sum_{i=1}^{2}\int_{t}^{T}|z^{i,t,\xi}_{s}-z^{i,t,\tilde{\xi}}_{s}|^{2}ds$
(2.53)
$\displaystyle\qquad+\sum_{i=1}^{2}\int_{t}^{T}\int_{\mathcal{E}_{i}}|\tilde{z}^{i,t,\xi}_{(s,e)}-\tilde{z}^{i,t,\tilde{\xi}}_{(s,e)}|^{2}\nu_{i}(de)ds\bigg{|}\mathcal{F}_{t}\bigg{]}\leq
K_{1}|\xi-\tilde{\xi}|^{2},$

where constant $K_{1}$ is positive and independent of $t\in[0,T]$.

Then FBSDEP (2.50) admits a unique $L^{\beta}(\beta>2)$-solution with $t=0$
and any given terminal time $T$, i.e.,

$\displaystyle E\bigg{[}\sup_{0\leq s\leq
T}|x^{0,\xi}_{s}|^{\beta}+\sup_{0\leq s\leq
T}|y^{0,\xi}_{s}|^{\beta}+\sum_{i=1}^{2}\bigg{(}\int_{0}^{T}|z^{i,0,\xi}_{s}|^{2}ds\bigg{)}^{\frac{\beta}{2}}+\sum_{i=1}^{2}\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{z}^{i,0,\xi}_{(s,e)}|^{2}\nu_{i}(de)ds\bigg{)}^{\frac{\beta}{2}}\bigg{]}$
(2.54) $\displaystyle\qquad\leq K_{2}(1+|\xi|^{\beta}),$ $\displaystyle
E\bigg{[}\sup_{0\leq s\leq
T}|x^{0,\xi}_{s}-x^{0,\tilde{\xi}}_{s}|^{\beta}+\sup_{0\leq s\leq
T}|y^{0,\xi}_{s}-y^{0,\tilde{\xi}}_{s}|^{\beta}+\sum_{i=1}^{2}\bigg{(}\int_{0}^{T}|z^{i,0,\xi}_{s}-z^{i,0,\tilde{\xi}}_{s}|^{2}ds\bigg{)}^{\frac{\beta}{2}}$
(2.55)
$\displaystyle\qquad+\sum_{i=1}^{2}\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{z}^{i,0,\xi}_{(s,e)}-\tilde{z}^{i,0,\tilde{\xi}}_{(s,e)}|^{2}\nu_{i}(de)ds\bigg{)}^{\frac{\beta}{2}}\bigg{]}\leq
K_{2}|\xi-\tilde{\xi}|^{\beta},$

###### Proof.

First, the given $L^{2}$-estimations $\mathrm{(i),(ii)}$ come to show the
existence and uniqueness of $L^{2}$-solution to FBSDE (2.50) on $[t,T]$ for
any given initial time and its corresponding initial value $(t,\xi)\in
L^{2}(\Omega,\mathcal{F}_{t},P;\mathbb{R})$.

Next, we investigate the FBSDEP (2.50) at the initial time $t=0$,

$\left\\{\begin{aligned}
dx^{0,\xi}_{s}&=b(s,\Theta_{s}^{0,\xi})ds+\sum_{i=1}^{2}\sigma_{i}(s,\Theta_{s}^{0,\xi})dW^{i}_{s}+\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}f_{i}(s,\Theta_{s-}^{0,\xi},e)\tilde{N}_{i}(de,ds),\\\
-dy^{0,\xi}_{s}&=g(s,\Theta_{s}^{0,\xi})ds-\sum_{i=1}^{2}z^{i,0,\xi}_{s}dW^{i}_{s}-\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\tilde{z}^{i,0,\xi}_{(s,e)}\tilde{N}_{i}(de,ds),\quad
s\in[0,T],\\\ x^{0,\xi}_{0}&=\xi,\ \
y^{0,\xi}_{T}=\phi(x^{0,\xi}_{T}).\end{aligned}\right.$ (2.56)

Observing the estimation (2.53), a key derivation shows

$\displaystyle E\bigg{[}\sup_{t\leq s\leq
T}|x^{0,\xi}_{s}-x^{0,\tilde{\xi}}_{s}|^{2}+\sup_{t\leq s\leq
T}|y^{0,\xi}_{s}-y^{0,\tilde{\xi}}_{s}|^{2}+\sum_{i=1}^{2}\int_{t}^{T}|z^{i,0,\xi}_{s}-z^{i,0,\tilde{\xi}}_{s}|^{2}ds$
(2.57)
$\displaystyle\qquad+\sum_{i=1}^{2}\int_{t}^{T}\int_{\mathcal{E}_{i}}|\tilde{z}^{i,0,\xi}_{(s,e)}-\tilde{z}^{i,0,\tilde{\xi}}_{(s,e)}|^{2}\nu_{i}(de)ds\bigg{|}\mathcal{F}_{t}\bigg{]}\leq
K_{1}|x^{0,\xi}_{t}-x^{0,\tilde{\xi}}_{t}|^{2},$

then we get

$\displaystyle|y^{0,\xi}_{t}-y^{0,\tilde{\xi}}_{t}|^{2}\leq
E\bigg{[}\sup_{t\leq s\leq
T}|y^{0,\xi}_{s}-y^{0,\tilde{\xi}}_{s}|^{2}\bigg{|}\mathcal{F}_{t}\bigg{]}\leq
K_{1}|x^{0,\xi}_{t}-x^{0,\tilde{\xi}}_{t}|^{2},$ (2.58)

which implies the uniformly Lipschitz condition of $y^{0,\xi}_{t}$ with
respect to $x^{0,\xi}_{t}$ for any $t\in[0,T]$, i.e.,

$\displaystyle|y^{0,\xi}_{t}-y^{0,\tilde{\xi}}_{t}|\leq\sqrt{K_{1}}|x^{0,\xi}_{t}-x^{0,\tilde{\xi}}_{t}|,\
t\in[0,T].$ (2.59)

Then, for any given $T$, we divide the whole interval $[0,T]$ into several
small intervals with same interval length $\delta$, i.e.,
$[0,\delta],[\delta,2\delta],\dots,[(k-1)\delta,k\delta]$, and without loss of
generality, we assume $T=k\delta$ and $k$ is positive integer. For the first
small interval $[0,\delta]$, under assumption (A4),(A6) and uniformly
Lipschitz condition (2.59) with terminal time $\delta$, then, by Lemma 2.4,
there exists a constant $\delta>0$ such that

$\displaystyle E\bigg{[}\sup_{0\leq
s\leq\delta}|x^{0,\xi}_{s}|^{\beta}+\sup_{0\leq
s\leq\delta}|y^{0,\xi}_{s}|^{\beta}+\sum_{i=1}^{2}\bigg{(}\bigg{(}\int_{0}^{\delta}|z^{i,0,\xi}_{s}|^{2}ds\bigg{)}^{\frac{\beta}{2}}+\bigg{(}\int_{0}^{\delta}\int_{\mathcal{E}_{i}}|\tilde{z}^{i,0,\xi}_{(s,e)}|^{2}\nu_{i}(de)ds\bigg{)}^{\frac{\beta}{2}}\bigg{)}\bigg{|}\mathcal{F}_{0}\bigg{]}$
(2.60) $\displaystyle\leq
C_{K,\rho,\sigma_{1},\sigma_{2},f_{1},f_{2},L,K_{1}}(1+|x^{0,\xi}_{0}|^{\beta}),$

where the constant $C_{K,\rho,\sigma_{1},\sigma_{2},f_{1},f_{2},L,K_{1}}$
depends on $K_{1},L$ and Lipschitz constants
$(K,\rho,L_{\sigma_{1}},L_{\sigma_{2}},\\\ L_{f_{1}},L_{f_{2}})$. Note that
the coefficients of FBSDEP (2.50) satisfy the same assumption in the interval
$[0,T]$. Therefore, by the induction for the following interval
$[(j-1)\delta,j\delta],1\leq j\leq k$ and combining the uniformly Lipschitz
condition (2.59) for terminal time $2\delta,3\delta,\cdots,k\delta$,
respectively, we have, for $1\leq j\leq k$,

$\displaystyle
E\bigg{[}\sup_{s\in[(j-1)\delta,j\delta]}|x^{0,\xi}_{s}|^{\beta}+\sup_{s\in[(j-1)\delta,j\delta]}|y^{0,\xi}_{s}|^{\beta}+\sum_{i=1}^{2}\bigg{(}\int_{(j-1)\delta}^{j\delta}|z^{i,0,\xi}_{s}|^{2}ds\bigg{)}^{\frac{\beta}{2}}$
(2.61)
$\displaystyle\qquad+\sum_{i=1}^{2}\bigg{(}\int_{(j-1)\delta}^{j\delta}\int_{\mathcal{E}_{i}}|\tilde{z}^{i,0,\xi}_{(s,e)}|^{2}\nu_{i}(de)ds\bigg{)}^{\frac{\beta}{2}}\bigg{|}\mathcal{F}_{(j-1)\delta}\bigg{]}\leq
C_{K,\rho,\sigma_{1},\sigma_{2},f_{1},f_{2},L,K_{1}}(1+|x^{0,\xi}_{(j-1)\delta}|^{\beta}).$

Now, by (2.59), we have obtained these $L^{\beta}$-estimates by Lemma 2.4 for
small intervals $[(j-1)\delta,j\delta],1\leq j\leq k$, then we show that how
to connect them together.

When considering the case of $j=1,2$, respectively, we have

$\displaystyle E\bigg{[}\sup_{0\leq
s\leq\delta}|x^{0,\xi}_{s}|^{\beta}+\sup_{0\leq
s\leq\delta}|y^{0,\xi}_{s}|^{\beta}+\sum_{i=1}^{2}\bigg{(}\bigg{(}\int_{0}^{\delta}|z^{i,0,\xi}_{s}|^{2}ds\bigg{)}^{\frac{\beta}{2}}$
(2.62)
$\displaystyle\quad+\bigg{(}\int_{0}^{\delta}\int_{\mathcal{E}_{i}}|\tilde{z}^{i,0,\xi}_{(s,e)}|^{2}\nu_{i}(de)ds\bigg{)}^{\frac{\beta}{2}}\bigg{)}\bigg{|}\mathcal{F}_{0}\bigg{]}\leq
C_{K,\rho,\sigma_{1},\sigma_{2},f_{1},f_{2},L,K_{1}}(1+|\xi|^{\beta}),$

and

$\displaystyle E\bigg{[}\sup_{\delta\leq s\leq
2\delta}|x^{0,\xi}_{s}|^{\beta}+\sup_{\delta\leq s\leq
2\delta}|y^{0,\xi}_{s}|^{\beta}+\sum_{i=1}^{2}\bigg{(}\bigg{(}\int_{\delta}^{2\delta}|z^{i,0,\xi}_{s}|^{2}ds\bigg{)}^{\frac{\beta}{2}}$
(2.63)
$\displaystyle\quad+\bigg{(}\int_{\delta}^{2\delta}\int_{\mathcal{E}_{i}}|\tilde{z}^{i,0,\xi}_{(s,e)}|^{2}\nu_{i}(de)ds\bigg{)}^{\frac{\beta}{2}}\bigg{)}\bigg{|}\mathcal{F}_{\delta}\bigg{]}\leq
C_{K,\rho,\sigma_{1},\sigma_{2},f_{1},f_{2},L,K_{1}}(1+|x^{0,\xi}_{\delta}|^{\beta}),$

we consider

$\displaystyle
E\big{[}C_{K,\rho,\sigma_{1},\sigma_{2},f_{1},f_{2},L,K_{1}}(1+|x^{0,\xi}_{\delta}|^{\beta})\big{|}\mathcal{F}_{0}\big{]}$
(2.64) $\displaystyle\leq
C_{K,\rho,\sigma_{1},\sigma_{2},f_{1},f_{2},L,K_{1}}\big{[}1+C_{K,\rho,\sigma_{1},\sigma_{2},f_{1},f_{2},L,K_{1}}(1+|\xi|^{\beta})\big{]}$
$\displaystyle\leq(C_{K,\rho,\sigma_{1},\sigma_{2},f_{1},f_{2},L,K_{1}}+C_{K,\rho,\sigma_{1},\sigma_{2},f_{1},f_{2},L,K_{1}}^{2})(1+|\xi|^{\beta}),$

and

$\displaystyle
C_{K,\rho,\sigma_{1},\sigma_{2},f_{1},f_{2},L,K_{1}}(1+|\xi|^{\beta})+E\big{[}C_{K,\rho,\sigma_{1},\sigma_{2},f_{1},f_{2},L,K_{1}}(1+|x^{0,\xi}_{\delta}\big{|}^{\beta})|\mathcal{F}_{0}\big{]}$
(2.65) $\displaystyle\leq
C^{(2)}_{K,\rho,\sigma_{1},\sigma_{2},f_{1},f_{2},L,K_{1}}(1+|\xi|^{\beta})$

where
$C^{(2)}_{K,\rho,\sigma_{1},\sigma_{2},f_{1},f_{2},L,K_{1}}=2C_{K,\rho,\sigma_{1},\sigma_{2},f_{1},f_{2},L,K_{1}}+C_{K,\rho,\sigma_{1},\sigma_{2},f_{1},f_{2},L,K_{1}}^{2}$.

Taking conditional expectation $E[\cdot|\mathcal{F}_{0}]$ on both sides of
(2.63), then adding it to (2.62), we get

$\displaystyle E\bigg{[}\sup_{0\leq s\leq
2\delta}|x^{0,\xi}_{s}|^{\beta}+\sup_{0\leq s\leq
2\delta}|y^{0,\xi}_{s}|^{\beta}+\sum_{i=1}^{2}\bigg{(}\bigg{(}\int_{0}^{\delta}|z^{i,0,\xi}_{s}|^{2}ds\bigg{)}^{\frac{\beta}{2}}+\bigg{(}\int_{\delta}^{2\delta}|z^{i,0,\xi}_{s}|^{2}ds\bigg{)}^{\frac{\beta}{2}}\bigg{)}$
(2.66)
$\displaystyle\qquad+\sum_{i=1}^{2}\bigg{(}\bigg{(}\int_{0}^{\delta}\int_{\mathcal{E}_{i}}|\tilde{z}^{i,0,\xi}_{(s,e)}|^{2}\nu_{i}(de)ds\bigg{)}^{\frac{\beta}{2}}+\bigg{(}\int_{\delta}^{2\delta}\int_{\mathcal{E}_{i}}|\tilde{z}^{i,0,\xi}_{(s,e)}|^{2}\nu_{i}(de)ds\bigg{)}^{\frac{\beta}{2}}\bigg{)}\bigg{|}\mathcal{F}_{0}\bigg{]}$
$\displaystyle\leq
C^{(2)}_{K,\rho,\sigma_{1},\sigma_{2},f_{1},f_{2},L,K_{1}}(1+|\xi|^{\beta}),$

by utilizing the inequality

$(a+b)^{k}\leq 2^{k}(a^{k}+b^{k}),\ a,b\geq 0,\ k\geq 1.$ (2.67)

Then we have, for $i=1,2$,

$\displaystyle
2^{-\frac{\beta}{2}}\bigg{(}\int_{0}^{2\delta}|z^{i,0,\xi}_{s}|^{2}ds\bigg{)}^{\frac{\beta}{2}}\leq\bigg{(}\int_{0}^{\delta}|z^{i,0,\xi}_{s}|^{2}ds\bigg{)}^{\frac{\beta}{2}}+\bigg{(}\int_{\delta}^{2\delta}|z^{i,0,\xi}_{s}|^{2}ds\bigg{)}^{\frac{\beta}{2}},$
(2.68) $\displaystyle
2^{-\frac{\beta}{2}}\bigg{(}\int_{0}^{2\delta}\int_{\mathcal{E}_{i}}|\tilde{z}^{i,0,\xi}_{(s,e)}|^{2}\nu_{i}(de)ds\bigg{)}^{\frac{\beta}{2}}\leq\bigg{(}\int_{0}^{\delta}\int_{\mathcal{E}_{i}}|\tilde{z}^{i,0,\xi}_{(s,e)}|^{2}\nu_{i}(de)ds\bigg{)}^{\frac{\beta}{2}}+\bigg{(}\int_{\delta}^{2\delta}\int_{\mathcal{E}_{i}}|\tilde{z}^{i,0,\xi}_{(s,e)}|^{2}\nu_{i}(de)ds\bigg{)}^{\frac{\beta}{2}}.$

Let
$\tilde{C}^{(2)}_{K,\rho,\sigma_{1},\sigma_{2},f_{1},f_{2},L,K_{1}}=2^{\frac{\beta}{2}}C^{(2)}_{K,\rho,\sigma_{1},\sigma_{2},f_{1},f_{2},L,K_{1}}$,
it follows that

$\displaystyle E\bigg{[}\sup_{0\leq s\leq
2\delta}|x^{0,\xi}_{s}|^{\beta}+\sup_{0\leq s\leq
2\delta}|y^{0,\xi}_{s}|^{\beta}+\sum_{i=1}^{2}\bigg{(}\bigg{(}\int_{0}^{2\delta}|z^{i,0,\xi}_{s}|^{2}ds\bigg{)}^{\frac{\beta}{2}}$
(2.69)
$\displaystyle\qquad+\bigg{(}\int_{0}^{2\delta}\int_{\mathcal{E}_{i}}|\tilde{z}^{i,0,\xi}_{(s,e)}|^{2}\nu_{i}(de)ds\bigg{)}^{\frac{\beta}{2}}\bigg{)}\bigg{|}\mathcal{F}_{0}\bigg{]}\leq\tilde{C}^{(2)}_{K,\rho,\sigma_{1},\sigma_{2},f_{1},f_{2},L,K_{1}}(1+|\xi|^{\beta}).$

Next, we continue to consider the case of $j=3$,

$\displaystyle E\bigg{[}\sup_{2\delta\leq s\leq
3\delta}|x^{0,\xi}_{s}|^{\beta}+\sup_{2\delta\leq s\leq
3\delta}|y^{0,\xi}_{s}|^{\beta}+\sum_{i=1}^{2}\bigg{(}\bigg{(}\int_{2\delta}^{3\delta}|z^{i,0,\xi}_{s}|^{2}ds\bigg{)}^{\frac{\beta}{2}}$
(2.70)
$\displaystyle\qquad+\bigg{(}\int_{2\delta}^{3\delta}\int_{\mathcal{E}_{i}}|\tilde{z}^{i,0,\xi}_{(s,e)}|^{2}\nu_{i}(de)ds\bigg{)}^{\frac{\beta}{2}}\bigg{)}\bigg{|}\mathcal{F}_{2\delta}\bigg{]}\leq
C_{K,\rho,\sigma_{1},\sigma_{2},f_{1},f_{2},L,K_{1}}(1+|x^{0,\xi}_{2\delta}|^{\beta}).$

Similarly, we have

$\displaystyle E\bigg{[}\sup_{0\leq s\leq
3\delta}|x^{0,\xi}_{s}|^{\beta}+\sup_{0\leq s\leq
3\delta}|y^{0,\xi}_{s}|^{\beta}+\sum_{i=1}^{2}\bigg{(}\bigg{(}\int_{0}^{3\delta}|z^{i,0,\xi}_{s}|^{2}ds\bigg{)}^{\frac{\beta}{2}}$
(2.71)
$\displaystyle\qquad+\bigg{(}\int_{0}^{3\delta}\int_{\mathcal{E}_{i}}|\tilde{z}^{i,0,\xi}_{(s,e)}|^{2}\nu_{i}(de)ds\bigg{)}^{\frac{\beta}{2}}\bigg{)}\bigg{|}\mathcal{F}_{0}\bigg{]}\leq\tilde{C}^{(3)}_{K,\rho,\sigma_{1},\sigma_{2},f_{1},f_{2},L,K_{1}}(1+|\xi|^{\beta}).$

By induction for $j\geq 4$, we obtain

$\displaystyle E\bigg{[}\sup_{0\leq s\leq
T}|x^{0,\xi}_{s}|^{\beta}+\sup_{0\leq s\leq
T}|y^{0,\xi}_{s}|^{\beta}+\sum_{i=1}^{2}\bigg{(}\bigg{(}\int_{0}^{T}|z^{i,0,\xi}_{s}|^{2}ds\bigg{)}^{\frac{\beta}{2}}$
(2.72)
$\displaystyle\qquad+\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{z}^{i,0,\xi}_{(s,e)}|^{2}\nu_{i}(de)ds\bigg{)}^{\frac{\beta}{2}}\bigg{)}\bigg{|}\mathcal{F}_{0}\bigg{]}\leq\tilde{C}^{(k)}_{K,\rho,\sigma_{1},\sigma_{2},f_{1},f_{2},L,K_{1}}(1+|\xi|^{\beta}).$

Finally, by taking
$K_{2}=\tilde{C}^{(k)}_{K,\rho,\sigma_{1},\sigma_{2},f_{1},f_{2},L,K_{1}}$,
then we prove the $L^{\beta}$-estimation (2.54) and can also obtain (2.55)
similarly, which guarantee the existence and uniqueness of
$L^{\beta}$-solution to FBSDEP (2.50) with $t=0$, respectively. ∎

Now, we can give the $L^{\beta}(\beta>2)$-solution and its estimate for FBSDEP
(2.50) in $\tilde{\mathcal{M}}^{\beta}[0,T]$. The following lemma can be
obtained similar to Lemma 2.4 by referring to the result in [16].

######  Lemma 2.5.

Let assumption (A4),(A5) hold. Then, for every $\beta\geq 2$, there exists a
sufficiently small constant $\tilde{\delta}$ depending on
$(K,L_{\sigma_{1}},L_{\sigma_{2}},\rho,L_{f_{1}},L_{f_{2}})$ and constant
$C_{\beta,K,\rho,\sigma_{1},\sigma_{2},f_{1},f_{2},L}$ depending on
$(\beta,K,\rho,L_{\sigma_{1}},L_{\sigma_{2}},L_{f_{1}},L_{f_{2}},L)$ such that
for every $0\leq\delta\leq\tilde{\delta}$ and $\xi\in
L^{\beta}(\Omega,\mathcal{F}_{t},P;\mathbb{R})$,

$\displaystyle E\bigg{[}\sup_{t\leq s\leq
t+\delta}|x^{t,\xi}_{s}|^{\beta}+\sup_{t\leq s\leq
t+\delta}|y^{t,\xi}_{s}|^{\beta}+\sum_{i=1}^{2}\bigg{(}\bigg{(}\int_{t}^{t+\delta}|z^{i,t,\xi}_{s}|^{2}ds\bigg{)}^{\frac{\beta}{2}}$
(2.73)
$\displaystyle\qquad+\bigg{(}\int_{t}^{t+\delta}\int_{\mathcal{E}_{i}}|\tilde{z}^{i,t,\xi}_{(s,e)}|^{2}N_{i}(de,ds)\bigg{)}^{\frac{\beta}{2}}\bigg{)}\bigg{|}\mathcal{F}_{t}\bigg{]}\leq
C_{\beta,K,\rho,\sigma_{1},\sigma_{2},f_{1},f_{2},L}(1+|\xi|^{\beta}),\
P\mbox{-}a.s..$

Similarly, we can also obtain the following important theorem based on Theorem
2.1 in [21], which shows that, for any given terminal time $T$, under some
$L^{2}$-estimate, the $L^{2}$-solution are the $L^{\beta}$-solution in
$\tilde{\mathcal{M}}^{\beta}[0,T](\beta>2)$. So we omit the detailed proof.

######  Theorem 2.5.

Let (A4),(A5) hold, and assume that, for every $\xi,\tilde{\xi}\in
L^{\beta}(\Omega,\mathcal{F}_{t},P;\mathbb{R})$, the $L^{2}$-estimations of
FBSDEP (2.50) are given as follows:

$\mathrm{(i)}$

$\displaystyle E\bigg{[}\sup_{t\leq s\leq T}|x^{t,\xi}_{s}|^{2}+\sup_{t\leq
s\leq
T}|y^{t,\xi}_{s}|^{2}+\sum_{i=1}^{2}\bigg{(}\int_{t}^{T}|z^{i,t,\xi}_{s}|^{2}ds+\int_{t}^{T}\int_{\mathcal{E}_{i}}|\tilde{z}^{i,t,\xi}_{(s,e)}|^{2}N_{i}(de,ds)\bigg{)}\bigg{|}\mathcal{F}_{t}\bigg{]}$
(2.74) $\displaystyle\qquad\leq K_{1}(1+|\xi|^{2}),$

$\mathrm{(ii)}$

$\displaystyle E\bigg{[}\sup_{t\leq s\leq
T}|x^{t,\xi}_{s}-x^{t,\tilde{\xi}}_{s}|^{2}+\sup_{t\leq s\leq
T}|y^{t,\xi}_{s}-y^{t,\tilde{\xi}}_{s}|^{2}+\sum_{i=1}^{2}\bigg{(}\int_{t}^{T}|z^{i,t,\xi}_{s}-z^{i,t,\tilde{\xi}}_{s}|^{2}ds$
(2.75)
$\displaystyle\qquad+\int_{t}^{T}\int_{\mathcal{E}_{i}}|\tilde{z}^{i,t,\xi}_{(s,e)}-\tilde{z}^{i,t,\tilde{\xi}}_{(s,e)}|^{2}N_{i}(de,ds)\bigg{)}\bigg{|}\mathcal{F}_{t}\bigg{]}\leq
K_{1}|\xi-\tilde{\xi}|^{2},$

where constant $K_{1}$ is positive and independent of $t\in[0,T]$.

Then FBSDEP (2.50) admits a unique $L^{\beta}(\beta>2)$-solution with $t=0$
and any given terminal time $T$, i.e.,

$\displaystyle E\bigg{[}\sup_{0\leq s\leq
T}|x^{0,\xi}_{s}|^{\beta}+\sup_{0\leq s\leq
T}|y^{0,\xi}_{s}|^{\beta}+\sum_{i=1}^{2}\bigg{(}\bigg{(}\int_{0}^{T}|z^{i,0,\xi}_{s}|^{2}ds\bigg{)}^{\frac{\beta}{2}}$
(2.76)
$\displaystyle\qquad+\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{z}^{i,0,\xi}_{(s,e)}|^{2}N_{i}(de,ds)\bigg{)}^{\frac{\beta}{2}}\bigg{)}\bigg{]}\leq
K_{2}(1+|\xi|^{\beta}),$ $\displaystyle E\bigg{[}\sup_{0\leq s\leq
T}|x^{0,\xi}_{s}-x^{0,\tilde{\xi}}_{s}|^{\beta}+\sup_{0\leq s\leq
T}|y^{0,\xi}_{s}-y^{0,\tilde{\xi}}_{s}|^{\beta}+\sum_{i=1}^{2}\bigg{(}\bigg{(}\int_{0}^{T}|z^{i,0,\xi}_{s}-z^{i,0,\tilde{\xi}}_{s}|^{2}ds\bigg{)}^{\frac{\beta}{2}}$
(2.77)
$\displaystyle\qquad+\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{z}^{i,0,\xi}_{(s,e)}-\tilde{z}^{i,0,\tilde{\xi}}_{(s,e)}|^{2}N_{i}(de,ds)\bigg{)}^{\frac{\beta}{2}}\bigg{)}\bigg{]}\leq
K_{2}|\xi-\tilde{\xi}|^{\beta}.$

Then we give the $L^{\beta}(\beta\geq 2)$-estimate for fully-coupled FBSDEP.

######  Theorem 2.6.

Suppose that (A4), (A6) hold. Then, for any $\beta\geq 2$, suppose that
$(x,y,z^{1},z^{2},\tilde{z}^{1},\\\ \tilde{z}^{2})$ is the solution to (2.10),
there exist a sufficiently small constant $\tilde{\tilde{T}}>0$ depending on
$K,L_{\sigma_{1}},L_{\sigma_{2}},\rho$ and some constant
$C_{\beta,K,\rho,\sigma_{1},\sigma_{2}}$ depending on $\beta$ and Lipschitz
constants $K,\rho,L_{\sigma_{1}}$, $L_{\sigma_{2}}$, such that, for every
$0\leq T\leq\tilde{\tilde{T}}$,

$\displaystyle E\bigg{[}\sup_{0\leq t\leq T}|x_{t}|^{\beta}+\sup_{0\leq t\leq
T}|y_{t}|^{\beta}+\sum_{i=1}^{2}\bigg{(}\bigg{(}\int_{0}^{T}|z^{i}_{t}|^{2}dt\bigg{)}^{\frac{\beta}{2}}+\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{z}^{i}|^{2}\nu_{i}(de)dt\bigg{)}^{\frac{\beta}{2}}\bigg{)}\bigg{]}$
(2.78) $\displaystyle\leq
C_{\beta,K,\rho,\sigma_{1},\sigma_{2}}E\bigg{[}|x_{0}|^{\beta}+|\phi(0)|^{\beta}+\bigg{(}\int_{0}^{T}|g(t,0,0,0,0,0,0)|dt\bigg{)}^{\beta}+\bigg{(}\int_{0}^{T}|b(t,0,0,0,0,0,0)|dt\bigg{)}^{\beta}$
$\displaystyle\quad+\sum_{i=1}^{2}\bigg{(}\bigg{(}\int_{0}^{T}|\sigma_{i}(t,0,0,0,0,0,0)|^{2}dt\bigg{)}^{\frac{\beta}{2}}+\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|f_{i}(t,0,0,e)|^{2}N_{i}(de,dt)\bigg{)}^{\frac{\beta}{2}}\bigg{)}\bigg{]}.$

Moreover, Let
$(x^{1},y^{1},z^{1,1},z^{2,1},\tilde{z}^{1,1},\tilde{z}^{2,1}),(x^{2},y^{2},z^{1,2},z^{2,2},\tilde{z}^{1,2},\tilde{z}^{2,2})$
be two solutions with different initial values $x^{1}_{0},x^{2}_{0}$,
respectively. Then we also give an estimate with respect to the initial value
in the following

$\displaystyle E\bigg{[}\sup_{0\leq t\leq
T}|x^{1}_{t}-x^{2}_{t}|^{\beta}+\sup_{0\leq t\leq
T}|y^{1}_{t}-y^{2}_{t}|^{\beta}+\sum_{i=1}^{2}\bigg{(}\bigg{(}\int_{0}^{T}|z^{i,1}_{t}-z^{i,2}_{t}|^{2}dt\bigg{)}^{\frac{\beta}{2}}$
(2.79)
$\displaystyle\quad+\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{z}^{i,1}-\tilde{z}^{i,2}|^{2}\nu_{i}(de)dt\bigg{)}^{\frac{\beta}{2}}\bigg{)}\bigg{]}\leq
C_{\beta,K,\rho,\sigma_{1},\sigma_{2}}|x^{1}_{0}-x^{2}_{0}|^{\beta}.$

###### Proof.

The second estimate (2.79) are the direct result of (2.78), so we mainly prove
the first estimate. We should note that, as we analyzed in Remark 2.3, under
(A6), we have obtained the existence and uniqueness of the solution to the
fully coupled FBSDEP (2.10), which can guarantee the $\beta$-integrability in
the following. From the backward equation in (2.10), $y$ is given by the
right-continuous version of
$y_{t}=E\big{[}\phi(x_{T})+\int_{t}^{T}g(t,x,y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})ds|\mathcal{F}_{t}\big{]}$.
Since $\beta\geq 2$, by the martingale representation theorem in [41] for
local square integrable martingale,
$(z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})$ corresponds to the unique pair of
$\mathcal{E}_{1},\mathcal{E}_{2}$-predictable processes satisfying

$\displaystyle
E\Big{[}\phi(x_{T})+\int_{0}^{T}g(t,x,y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})ds\big{|}\mathcal{F}_{t}\Big{]}$
$\displaystyle\qquad=y_{0}+\sum_{i=1}^{2}\bigg{(}\int_{0}^{t}z^{i}_{s}dW^{i}_{s}+\int_{0}^{t}\int_{\mathcal{E}_{i}}\tilde{z}^{i}_{(s,e)}\tilde{N}_{i}(de,ds)\bigg{)},\
P\mbox{-}a.s.,$

we have $|y_{t}|\leq
E\big{[}|\phi(x_{T})|+\int_{0}^{T}|g(t,x,y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})|ds|\mathcal{F}_{t}\big{]}$.
Using martingale inequality, we get

$\displaystyle E\Big{[}\sup_{0\leq t\leq T}|y_{t}|^{\beta}\Big{]}\leq
E\bigg{[}\sup_{0\leq t\leq
T}\bigg{|}E\Big{[}|\phi(x_{T})|+\int_{0}^{T}|g(t,\Theta)|ds\Big{|}\mathcal{F}_{t}\Big{]}\bigg{|}^{\beta}\bigg{]}$
(2.80) $\displaystyle\leq
C_{\beta}E\bigg{[}\Big{|}E\Big{[}|\phi(x_{T})|+\int_{0}^{T}|g(t,\Theta)|ds\Big{|}\mathcal{F}_{T}\Big{]}\Big{|}^{\beta}\bigg{]}$
$\displaystyle\leq
C_{\beta}E\Big{[}\Big{|}|\phi(x_{T})|+\int_{0}^{T}|g(t,\Theta)|ds\Big{|}^{\beta}\Big{]}$
$\displaystyle\leq
C_{\beta}E\bigg{[}|\phi(x_{T})-\phi(0)+\phi(0)|^{\beta}+\Big{|}\int_{0}^{T}|g(t,\Theta)-g(t,0,0,0,0,0,0)+g(t,0,0,0,0,0,0)|ds\Big{|}^{\beta}\bigg{]}$
$\displaystyle\leq
C_{\beta}E\bigg{[}\big{|}C|x_{T}|+\phi(0)\big{|}^{\beta}+\Big{|}\int_{0}^{T}|C(|x|+|y|+|z^{1}+|z^{2}|+||\tilde{z}^{1}||+||\tilde{z}^{2}||)$
$\displaystyle\qquad\qquad+g(t,0,0,0,0,0,0)|ds\Big{|}^{\beta}\bigg{]}$
$\displaystyle\leq
C_{\beta}E\bigg{[}C^{\beta}|x_{T}|^{\beta}+|\phi(0)|^{\beta}+\Big{|}\int_{0}^{T}C(|x|+|y|+|z^{1}|+|z^{2}|+||\tilde{z}^{1}||+||\tilde{z}^{2}||)ds\Big{|}^{\beta}$
$\displaystyle\qquad+\Big{|}\int_{0}^{T}|g(t,0,0,0,0,0,0)|ds\Big{|}^{\beta}\bigg{]}$
$\displaystyle\leq C_{\beta}E\bigg{[}C^{\beta}\sup_{0\leq t\leq
T}|x_{t}|^{\beta}+|\phi(0)|^{\beta}+\Big{|}\int_{0}^{T}C|x|ds\bigg{|}^{\beta}+\bigg{|}\int_{0}^{T}C|y|ds\Big{|}^{\beta}$
$\displaystyle\qquad\quad+\sum_{i=1}^{2}\bigg{(}\Big{|}\int_{0}^{T}C|z^{i}|ds\Big{|}^{\beta}+\Big{|}\int_{0}^{T}C||\tilde{z}^{i}||ds\Big{|}^{\beta}\bigg{)}+\Big{|}\int_{0}^{T}|g(t,0,0,0,0,0,0)|ds\Big{|}^{\beta}\bigg{]}$
$\displaystyle\leq C_{\beta}E\bigg{[}(C^{\beta}+T^{\beta}C^{\beta})\sup_{0\leq
t\leq T}|x_{t}|^{\beta}+|\phi(0)|^{\beta}+T^{\beta}C^{\beta}\sup_{0\leq t\leq
T}|y_{t}|^{\beta}+C^{\beta}T^{\frac{\beta}{2}}\sum_{i=1}^{2}\Big{|}\int_{0}^{T}|z^{i}|^{2}ds\Big{|}^{\frac{\beta}{2}}$
$\displaystyle\qquad\quad+C^{\beta}T^{\frac{\beta}{2}}\sum_{i=1}^{2}\Big{|}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{z}^{i}|^{2}\nu_{i}(de)ds\Big{|}^{\frac{\beta}{2}}+\Big{|}\int_{0}^{T}|g(t,0,0,0,0,0,0)|ds\Big{|}^{\beta}\bigg{]}.$

We need the following lemma ([16], Lemma 3.1).

######  Lemma 2.6.

For any $\beta\geq 2$, we have

$E\bigg{[}\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{z}^{i}|^{2}\nu_{i}(de)ds\bigg{)}^{\frac{\beta}{2}}\bigg{]}\leq
C_{\beta}E\bigg{[}\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{z}^{i}|^{2}N_{i}(de,ds)\bigg{)}^{\frac{\beta}{2}}\bigg{]},\
i=1,2.$ (2.81)

Then, by Lemma 2.2 and B-D-G’s inequality, we can get the estimate

$\displaystyle
E\bigg{[}\sum_{i=1}^{2}\bigg{(}\bigg{(}\int_{0}^{T}|z^{i}|^{2}ds\bigg{)}^{\frac{\beta}{2}}+\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{z}^{i}|^{2}\nu_{i}(de)ds\bigg{)}^{\frac{\beta}{2}}\bigg{)}\bigg{]}$
(2.82) $\displaystyle\leq
E\bigg{[}\sum_{i=1}^{2}\bigg{(}\int_{0}^{T}|z^{i}|^{2}ds\bigg{)}^{\frac{\beta}{2}}\bigg{]}+C_{\beta}E\bigg{[}\sum_{i=1}^{2}\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{z}^{i}|^{2}N_{i}(de,ds)\bigg{)}^{\frac{\beta}{2}}\bigg{]}$
$\displaystyle\leq C_{\beta}E\bigg{[}\sup_{0\leq t\leq
T}\bigg{|}\sum_{i=1}^{2}\int_{0}^{t}z^{i}_{s}dW^{i}_{s}+\sum_{i=1}^{2}\int_{0}^{t}\int_{\mathcal{E}_{i}}\tilde{z}^{i}_{(s,e)}\tilde{N}_{i}(de,ds)\bigg{|}^{\beta}\bigg{]}$
$\displaystyle\leq C_{\beta}E\bigg{[}\sup_{0\leq t\leq
T}\bigg{(}|y_{t}|^{\beta}+|y_{0}|^{\beta}+\bigg{|}\int_{0}^{t}g(s,\Theta(s))ds\bigg{|}^{\beta}\bigg{)}\bigg{]}$
$\displaystyle\leq C_{\beta}E\bigg{[}\sup_{0\leq t\leq
T}|y_{t}|^{\beta}+\bigg{(}\int_{0}^{T}|g(s,\Theta(s))|ds\bigg{)}^{\beta}\bigg{]}+E\bigg{[}|\phi(x_{T})|^{\beta}+\bigg{|}\int_{0}^{T}g(s,\Theta(s))ds\bigg{|}^{\beta}\bigg{]}$
$\displaystyle\leq C_{\beta}E\bigg{[}\sup_{0\leq t\leq
T}|y_{t}|^{\beta}+\bigg{(}\int_{0}^{T}|g(s,\Theta)-g(s,0,0,0,0,0,0)+g(s,0,0,0,0,0,0)|ds\bigg{)}^{\beta}\bigg{]}$
$\displaystyle\quad+E\bigg{[}C^{\beta}|x_{T}|^{\beta}+|\phi(0)|^{\beta}+\bigg{|}\int_{0}^{T}g(s,\Theta)-g(s,0,0,0,0,0,0)+g(s,0,0,0,0,0,0)ds\bigg{|}^{\beta}\bigg{]}$
$\displaystyle\leq C_{\beta}E\bigg{[}(1+C^{\beta}T^{\beta})\sup_{0\leq t\leq
T}|y_{t}|^{\beta}+(C^{\beta}+C^{\beta}T^{\beta})\sup_{0\leq t\leq
T}|x_{t}|^{\beta}+|\phi(0)|^{\beta}$
$\displaystyle\quad+C^{\beta}T^{\frac{\beta}{2}}\sum_{i=1}^{2}\bigg{(}\bigg{(}\int_{0}^{T}|z^{i}|^{2}ds\bigg{)}^{\frac{\beta}{2}}+\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{z}^{i}|^{2}\nu_{i}(de)ds\bigg{)}^{\frac{\beta}{2}}\bigg{)}$
$\displaystyle\quad+\bigg{(}\int_{0}^{T}|g(s,0,0,0,0,0,0)|ds\bigg{)}^{\beta}\bigg{]}.$

From (2.80), we have

$\displaystyle(1-T^{\beta}C_{\beta})E\bigg{[}\sup_{0\leq t\leq
T}|y_{t}|^{\beta}\bigg{]}\leq
C_{\beta}E\bigg{[}(C^{\beta}+T^{\beta}C^{\beta})\sup_{0\leq t\leq
T}|x_{t}|^{\beta}+|\phi(0)|^{\beta}$
$\displaystyle\qquad+C^{\beta}T^{\frac{\beta}{2}}\sum_{i=1}^{2}\bigg{(}\bigg{|}\int_{0}^{T}|z^{i}|^{2}ds\bigg{|}^{\frac{\beta}{2}}+\bigg{|}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{z}^{i}|^{2}\nu_{i}(de)ds\bigg{|}^{\frac{\beta}{2}}\bigg{)}+\bigg{|}\int_{0}^{T}|g(t,0,0,0,0,0,0)|ds\bigg{|}^{\beta}\bigg{]}.$

Choosing $T_{1}$ small enough such that $T_{1}^{\beta}C_{\beta}<1$, and for
every $0\leq T\leq T_{1}$, we get

$\displaystyle E\bigg{[}\sup_{0\leq t\leq T}|y_{t}|^{\beta}\bigg{]}\leq
C_{\beta,T}E\bigg{[}(C^{\beta}+T^{\beta}C^{\beta})\sup_{0\leq t\leq
T}|x_{t}|^{\beta}+|\phi(0)|^{\beta}+C^{\beta}T^{\frac{\beta}{2}}\sum_{i=1}^{2}\bigg{(}\bigg{|}\int_{0}^{T}|z^{i}|^{2}ds\bigg{|}^{\frac{\beta}{2}}$
(2.83)
$\displaystyle\qquad+\bigg{|}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{z}^{i}|^{2}\nu_{i}(de)ds\bigg{|}^{\frac{\beta}{2}}\bigg{)}+\bigg{|}\int_{0}^{T}|g(t,0,0,0,0,0,0)|ds\bigg{|}^{\beta}\bigg{]}.$

From (2.82), we also have

$\displaystyle(1-C_{\beta}T^{\frac{\beta}{2}})E\bigg{[}\sum_{i=1}^{2}\bigg{(}\bigg{(}\int_{0}^{T}|z^{i}|^{2}ds\bigg{)}^{\frac{\beta}{2}}+\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{z}^{i}|^{2}\nu_{i}(de)ds\bigg{)}^{\frac{\beta}{2}}\bigg{)}\bigg{]}$
$\displaystyle\leq C_{\beta}E\bigg{[}(1+C^{\beta}T^{\beta})\sup_{0\leq t\leq
T}|y_{t}|^{\beta}+(C^{\beta}+C^{\beta}T^{\beta})\sup_{0\leq t\leq
T}|x_{t}|^{\beta}$
$\displaystyle\qquad\quad+|\phi(0)|^{\beta}+\bigg{(}\int_{0}^{T}|g(s,0,0,0,0,0,0)|ds\bigg{)}^{\beta}\bigg{]}.$

Then combining with the estimate (2.83), we get

$\displaystyle(1-C_{\beta,T}T^{\frac{\beta}{2}})E\bigg{[}\sum_{i=1}^{2}\bigg{(}\bigg{(}\int_{0}^{T}|z^{i}|^{2}ds\bigg{)}^{\frac{\beta}{2}}+\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{z}^{i}|^{2}\nu_{i}(de)ds\bigg{)}^{\frac{\beta}{2}}\bigg{)}\bigg{]}$
$\displaystyle\leq
C_{\beta,T}E\bigg{[}(C^{\beta}+T^{\beta}C^{\beta})\sup_{0\leq t\leq
T}|x_{t}|^{\beta}+|\phi(0)|^{\beta}+\bigg{|}\int_{0}^{T}|g(t,0,0,0,0,0,0)|ds\bigg{|}^{\beta}\bigg{]}.$

Choosing $0\leq T_{2}\leq T_{1}$ small enough such that
$C_{\beta,T_{2}}T_{2}^{\frac{\beta}{2}}<1$, and for every $0\leq T\leq T_{2}$,
we get

$\displaystyle
E\bigg{[}\sum_{i=1}^{2}\bigg{(}\bigg{(}\int_{0}^{T}|z^{i}|^{2}ds\bigg{)}^{\frac{\beta}{2}}+\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|\tilde{z}^{i}|^{2}\nu_{i}(de)ds\bigg{)}^{\frac{\beta}{2}}\bigg{)}\bigg{]}$
(2.84) $\displaystyle\leq
C_{\beta,T}E\bigg{[}(C^{\beta}+T^{\beta}C^{\beta})\sup_{0\leq t\leq
T}|x_{t}|^{\beta}+|\phi(0)|^{\beta}+\bigg{|}\int_{0}^{T}|g(t,0,0,0,0,0,0)|ds\bigg{|}^{\beta}\bigg{]},$

where the constant $C_{\beta,T}$ changed every step. Next, we begin to deal
with the forward equation in (2.10). First, since

$\displaystyle|x_{t}|^{\beta}$
$\displaystyle=C_{\beta}\bigg{[}|x_{0}|^{\beta}+\bigg{|}\int_{0}^{t}b(s,\Theta)ds\bigg{|}^{\beta}+\sum_{i=1}^{2}\bigg{|}\int_{0}^{t}\sigma_{i}(s,\Theta)dW^{i}_{s}\bigg{|}^{\beta}$
$\displaystyle\qquad+\sum_{i=1}^{2}\bigg{|}\int_{0}^{t}\int_{\mathcal{E}_{i}}f_{i}(s,x,y,e)\tilde{N}_{i}(de,ds)\bigg{|}^{\beta}\bigg{]},$

by B-D-G’s inequality and Hölder’s inequality, we have

$\displaystyle E\bigg{[}\sup_{0\leq t\leq T}|x_{t}|^{\beta}\bigg{]}\leq
C_{\beta}E\big{[}|x_{0}|^{\beta}\big{]}+C_{\beta}E\bigg{[}\int_{0}^{T}|b(s,\Theta)|ds\bigg{]}^{\beta}$
$\displaystyle\quad+C_{\beta}E\bigg{[}\sum_{i=1}^{2}\bigg{(}\int_{0}^{T}|\sigma_{i}(s,\Theta)|^{2}ds\bigg{)}^{\frac{\beta}{2}}\bigg{]}+C_{\beta}E\bigg{[}\sum_{i=1}^{2}\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|f_{i}(s,x,y,e)|^{2}N_{i}(de,ds)\bigg{)}^{\frac{\beta}{2}}\bigg{]}$
$\displaystyle\leq
C_{\beta}E\big{[}|x_{0}|^{\beta}\big{]}+C_{\beta}E\bigg{[}\int_{0}^{T}|b(s,\Theta)-b(s,0,0,0,0,0,0)+b(s,0,0,0,0,0,0)|ds\bigg{]}^{\beta}$
$\displaystyle\quad+C_{\beta}E\bigg{[}\sum_{i=1}^{2}\bigg{(}\int_{0}^{T}|\sigma_{i}(s,\Theta)-\sigma_{i}(s,0,0,0,0,0,0)+\sigma_{i}(s,0,0,0,0,0,0)|^{2}ds\bigg{)}^{\frac{\beta}{2}}\bigg{]}$
$\displaystyle\quad+C_{\beta}E\bigg{[}\sum_{i=1}^{2}\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}|f_{i}(s,x,y,e)-f_{i}(s,0,0,e)+f_{i}(s,0,0,e)|^{2}N_{i}(de,ds)\bigg{)}^{\frac{\beta}{2}}\bigg{]}$
$\displaystyle\leq
C_{\beta}E[|x_{0}|^{\beta}]+C_{\beta}E\bigg{[}\bigg{(}\int_{0}^{T}C(|x|+|y|+|z^{1}|+|z^{2}|+||\tilde{z}^{1}||+||\tilde{z}^{2}||)ds\bigg{)}^{\beta}\bigg{]}$
(2.85)
$\displaystyle\quad+C_{\beta}E\bigg{[}\bigg{(}\int_{0}^{T}|b(s,0,0,0,0,0,0)|ds\bigg{)}^{\beta}\bigg{]}+C_{\beta}E\bigg{[}\sum_{i=1}^{2}\bigg{(}\int_{0}^{T}K^{2}(|x|^{2}+|y|^{2})$
$\displaystyle\quad+L^{2}_{\sigma_{i}}(|z^{1}|^{2}+|z^{2}|^{2}+||\tilde{z}^{1}||^{2}+||\tilde{z}^{2}||^{2})+|\sigma_{i}(s,0,0,0,0,0,0)|^{2}ds\bigg{)}^{\frac{\beta}{2}}\bigg{]}$
$\displaystyle\quad+C_{\beta}E\bigg{[}\sum_{i=1}^{2}\bigg{(}\int_{0}^{T}\int_{\mathcal{E}_{i}}\rho^{2}(e)(|x|^{2}+|y|^{2})+|f_{i}(s,0,0,e)|^{2}N_{i}(de,ds)\bigg{)}^{\frac{\beta}{2}}\bigg{]}$
$\displaystyle\leq
C_{\beta}E\big{[}|x_{0}|^{\beta}\big{]}+C_{\beta}E\bigg{[}C^{\beta}\bigg{(}T^{\beta}\sup_{0\leq