$\Psi(t):=\big{[}1,p,k_{1}^{1},k_{1}^{2},\int_{\mathcal{E}_{1}}k_{2}^{1}(t,e)\nu_{1}(de),\int_{\mathcal{E}_{2}}k_{2}^{2}(t,e)\nu_{2}(de)\big{]}$
and $\Psi^{0}(t):=\big{[}1,p,k_{1}^{1},k_{1}^{2},\\\
\int_{\mathcal{E}_{1}}k_{2}^{1}(t,e)\nu_{1}(de),\int_{\mathcal{E}_{2}}k_{2}^{2}(t,e)\nu_{2}(de),0\big{]}$.

Observing the above equation, we apply Itô’s formula to $(x^{1}_{t})^{2}$ to
get

$\displaystyle d(x^{1}_{t})^{2}$
$\displaystyle=\bigg{[}2\tilde{b}_{1x}(t)(x^{1}_{t})^{2}+\sum_{i=1}^{2}\big{[}\sigma^{2}_{ix}(x^{1}_{t})^{2}+(\delta\sigma_{i}(t))^{2}\mathbbm{1}_{[\bar{t},\bar{t}+\epsilon]}+2\sigma_{ix}x^{1}_{t}\delta\sigma_{i}\mathbbm{1}_{[\bar{t},\bar{t}+\epsilon]}\big{]}$
$\displaystyle\qquad+\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\mathbb{E}_{i}\big{[}f_{ix}^{2}(t,e)\big{|}\mathcal{P}\otimes\mathcal{B}(\mathcal{E}_{i})\big{]}\nu_{i}(de)(x^{1}_{t})^{2}\bigg{]}dt+\sum_{i=1}^{2}\Big{\\{}\big{[}2\sigma_{ix}(t)(x^{1}_{t})^{2}$
$\displaystyle\qquad+2\delta\sigma_{i}x^{1}_{t}\mathbbm{1}_{[\bar{t},\bar{t}+\epsilon]}\big{]}dW^{i}_{t}+\int_{\mathcal{E}_{i}}\big{[}2f_{ix}(t,e)+f_{ix}^{2}(t,e)\big{]}(x^{1}_{t-})^{2}\tilde{N}_{i}(de,dt)\Big{\\}},$

then we introduce the adjoint equation for $(x^{1}_{t})^{2}$ as follows

$\left\\{\begin{aligned}
&-dP_{t}=\bigg{\\{}\Psi^{0}(t)D^{2}\tilde{l}(t)\Psi^{0}(t)^{\top}+s^{2}_{t}\tilde{b}_{2xx}(t)+2\sigma_{2x}(t)\tilde{b}_{2x}(t)\alpha_{t}+h_{t}\Psi(t)D^{2}g(t)\Psi(t)^{\top}\\\
&\quad+m_{t}\tilde{b}_{1xx}(t)+2\tilde{b}_{2x}(t)\beta^{2}_{t}+2\tilde{b}_{1x}(t)P_{t}+\sum^{2}_{i=1}\big{[}n^{i}_{t}\sigma_{ixx}(t)+2\sigma_{ix}(t)Q^{i}_{t}+\sigma^{2}_{ix}(t)P_{t}\big{]}\\\
&\quad+\sum^{2}_{i=1}\int_{\mathcal{E}_{i}}\Big{[}\mathbb{E}_{i}\big{[}f_{ixx}(t,e)\big{|}\mathcal{P}\otimes\mathcal{B}(\mathcal{E}_{i})\big{]}\tilde{n}^{i}_{(t,e)}+\mathbb{E}_{i}\big{[}2f_{ix}(t,e)+f^{2}_{ix}(t,e)\big{|}\mathcal{P}\otimes\mathcal{B}(\mathcal{E}_{i})\big{]}\tilde{Q}^{i}_{(t,e)}\\\
&\quad+\mathbb{E}_{i}\big{[}f^{2}_{ix}(t,e)\big{|}\mathcal{P}\otimes\mathcal{B}(\mathcal{E}_{i})\big{]}P_{t}\Big{]}\nu_{i}(de)+\int_{\mathcal{E}_{2}}\Big{[}\mathbb{E}_{2}\big{[}f_{4xx}(t,e)\big{|}\mathcal{P}\otimes\mathcal{B}(\mathcal{E}_{2})\big{]}\tilde{s}^{2}_{(t,e)}\\\
&\quad+\mathbb{E}_{2}\big{[}2f_{4x}(t,e)+2f_{2x}(t,e)f_{4x}(t,e)\big{|}\mathcal{P}\otimes\mathcal{B}(\mathcal{E}_{2})\big{]}\tilde{\beta}^{2}_{(t,e)}\\\
&\quad+\mathbb{E}_{2}\big{[}2f_{2x}(t,e)f_{4x}(t,e)\big{|}\mathcal{P}\otimes\mathcal{B}(\mathcal{E}_{2})\big{]}\alpha_{t}\Big{]}\nu_{2}(de)\bigg{\\}}dt\\\
&\quad-\sum^{2}_{i=1}Q^{i}_{t}dW^{i}_{t}-\sum^{2}_{i=1}\int_{\mathcal{E}_{i}}\tilde{Q}^{i}_{(t,e)}\tilde{N}_{i}(de,dt),\quad
t\in[0,T],\\\
&P_{T}=h_{T}\phi_{xx}(\bar{x}_{T})+\tilde{\Phi}_{xx}(\bar{x}_{T},\bar{\tilde{\Gamma}}_{T}),\end{aligned}\right.$
(3.34)

which admits a unique solution
$(P,Q^{1},Q^{2},\tilde{Q}^{1},\tilde{Q}^{2})\in\mathcal{N}^{\beta}[0,T]$.
Applying Itô’s formula to $P_{t}(x^{1}_{t})^{2}$ and combining (3.33), we have

$\displaystyle
J(u^{\epsilon})-J(\bar{u})=E\bigg{\\{}\int_{0}^{T}\bigg{[}\Big{(}-\sum_{i=1}^{2}\big{[}\tilde{l}_{z^{i}}p_{t}\delta\sigma_{i}(t)+h_{t}g_{z^{i}}p_{t}\delta\sigma_{i}(t)-n^{i}_{t}\delta\sigma_{i}(t)\big{]}$
(3.35) $\displaystyle\quad+h_{t}\delta
g(t,\Delta^{1},\Delta^{2})+\delta\tilde{l}(t,\Delta^{1},\Delta^{2})+m_{t}\delta\tilde{b}_{1}+s^{2}_{t}\delta\tilde{b}_{2}+\delta\sigma_{2}(t)\delta\tilde{b}_{2}\alpha_{t}$
$\displaystyle\quad+\frac{1}{2}(\delta\sigma_{1}(t))^{2}P_{t}+\frac{1}{2}(\delta\sigma_{2}(t))^{2}P_{t}\Big{)}\mathbbm{1}_{[\bar{t},\bar{t}+\epsilon]}+\Big{(}n^{1}_{t}\delta\sigma_{1x}x^{1}_{t}+n^{2}_{t}\delta\sigma_{2x}x^{1}_{t}+s^{2}_{t}\delta\tilde{b}_{2x}x^{1}_{t}$
$\displaystyle\quad+s^{2}_{t}\delta\tilde{b}_{2\tilde{\Gamma}}\tilde{\Gamma}^{1}_{t}+\sigma_{2x}(t)\alpha_{t}x^{1}_{t}\delta\tilde{b}_{2}+\tilde{b}_{2x}\alpha_{t}x^{1}_{t}\delta\sigma_{2}+\tilde{b}_{2\tilde{\Gamma}}\alpha_{t}\tilde{\Gamma}^{1}_{t}\delta\sigma_{2}+\delta\sigma_{1}(t)\beta^{1}_{t}\tilde{\Gamma}^{1}_{t}+\delta\tilde{b}_{2}\beta^{2}_{t}x^{1}_{t}$
$\displaystyle\quad+\delta\sigma_{2}(t)\beta^{2}_{t}\tilde{\Gamma}^{1}_{t}+\sigma_{1x}(t)x^{1}_{t}P_{t}\delta\sigma_{1}(t)+\sigma_{2x}(t)x^{1}_{t}P_{t}\delta\sigma_{2}(t)+\delta\sigma_{2}(t)Q^{2}_{t}x^{1}_{t}$
$\displaystyle\quad+\delta\sigma_{1}(t)Q^{1}_{t}x^{1}_{t}\Big{)}\mathbbm{1}_{[\bar{t},\bar{t}+\epsilon]}\bigg{]}dt\bigg{\\}}+o(\epsilon).$

We can check that

$\displaystyle E\int_{0}^{T}$
$\displaystyle\Big{(}n^{1}_{t}\delta\sigma_{1x}x^{1}_{t}+n^{2}_{t}\delta\sigma_{2x}x^{1}_{t}+s^{2}_{t}\delta\tilde{b}_{2x}x^{1}_{t}+s^{2}_{t}\delta\tilde{b}_{2\tilde{\Gamma}}\tilde{\Gamma}^{1}_{t}+\sigma_{2x}\alpha_{t}x^{1}_{t}\delta\tilde{b}_{2}+\tilde{b}_{2x}\alpha_{t}x^{1}_{t}\delta\sigma_{2}$
$\displaystyle+\tilde{b}_{2\tilde{\Gamma}}\alpha_{t}\tilde{\Gamma}^{1}_{t}\delta\sigma_{2}+\delta\sigma_{1}\beta^{1}_{t}\tilde{\Gamma}^{1}_{t}+\delta\tilde{b}_{2}\beta^{2}_{t}x^{1}_{t}+\delta\sigma_{2}\beta^{2}_{t}\tilde{\Gamma}^{1}_{t}+\sigma_{1x}x^{1}_{t}P_{t}\delta\sigma_{1}$
$\displaystyle+\sigma_{2x}x^{1}_{t}P_{t}\delta\sigma_{2}+\delta\sigma_{2}Q^{2}_{t}x^{1}_{t}+\delta\sigma_{1}Q^{1}_{t}x^{1}_{t}\Big{)}\mathbbm{1}_{[\bar{t},\bar{t}+\epsilon]}dt=o(\epsilon),$

therefore, we have

$\displaystyle
J(u^{\epsilon})-J(\bar{u})=E\bigg{\\{}\int_{0}^{T}\Big{(}-\sum^{2}_{i=1}\Big{[}\tilde{l}_{z^{i}}p_{t}\delta\sigma_{i}(t)+h_{t}g_{z^{i}}p_{t}\delta\sigma_{i}(t)-n^{i}_{t}\delta\sigma_{i}(t)-\frac{1}{2}(\delta\sigma_{i}(t))^{2}P_{t}\Big{]}$
(3.36) $\displaystyle+h_{t}\delta
g(t,\Delta^{1},\Delta^{2})+m_{t}\delta\tilde{b}_{1}+\delta\tilde{l}(t,\Delta^{1},\Delta^{2})+s^{2}_{t}\delta\tilde{b}_{2}+\delta\sigma_{2}(t)\delta\tilde{b}_{2}\alpha_{t}\Big{)}\mathbbm{1}_{[\bar{t},\bar{t}+\epsilon]}dt\bigg{\\}}+o(\epsilon)\geq
0,$

i.e.,

$\displaystyle
E\Big{[}\sum^{2}_{i=1}\big{[}-(\tilde{l}_{z^{i}}+h_{t}g_{z^{i}})p_{t}+n^{i}_{t}\big{]}\delta\sigma_{i}(t)+h_{t}\delta
g(t,\Delta^{1},\Delta^{2})+\delta\tilde{l}(t,\Delta^{1},\Delta^{2})+m_{t}\delta\tilde{b}_{1}+s^{2}_{t}\delta\tilde{b}_{2}$
(3.37)
$\displaystyle+\delta\sigma_{2}(t)\delta\tilde{b}_{2}\alpha_{t}+\sum^{2}_{i=1}\frac{1}{2}(\delta\sigma_{i}(t))^{2}P_{t}\Big{|}\mathcal{F}^{Y}_{t}\Big{]}\geq
0,\ a.e.\ t\in[0,T],\ P\mbox{-}a.s.,$

Define the Hamiltonian function

$\displaystyle\mathcal{H}(t,x,y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2},\tilde{\Gamma};p,h,m,n^{1},n^{2},s^{2},\alpha,P,u)$
(3.38)
$\displaystyle:=\sum^{2}_{i=1}\big{[}-(\tilde{l}_{z^{i}}+h_{t}g_{z^{i}})p_{t}+n^{i}_{t}\big{]}\sigma_{i}(t,x,u)+\sum^{2}_{i=1}\Big{[}\frac{1}{2}\sigma^{2}_{i}(t,x,u)P_{t}-\sigma_{i}(t,x,u)\sigma_{i}(t)P_{t}\Big{]}$
$\displaystyle\quad+h_{t}g\big{(}t,x,y,z^{1}+p(\sigma_{1}(t,x,u)-\sigma_{1}(t)),z^{2}+p(\sigma_{2}(t,x,u)-\sigma_{2}(t)),\tilde{z}^{1},\tilde{z}^{2},u\big{)}$
$\displaystyle\quad+\tilde{l}\big{(}t,x,y,z^{1}+p(\sigma_{1}(t,x,u)-\sigma_{1}(t)),z^{2}+p(\sigma_{2}(t,x,u)-\sigma_{2}(t)),\tilde{z}^{1},\tilde{z}^{2},\tilde{\Gamma},u\big{)}$
$\displaystyle\quad+m_{t}\tilde{b}_{1}(t,x,u)+s^{2}_{t}\tilde{b}_{2}(t,x,\tilde{\Gamma},u)+(\sigma_{2}(t,x,u)-\sigma_{2}(t))(\tilde{b}_{2}(t,x,\tilde{\Gamma},u)-\tilde{b}_{2}(t))\alpha_{t},$

the maximum conditions (3.37) is equivalent to

$\displaystyle
E\big{[}\mathcal{H}(t,\bar{x},\bar{y},\bar{z}^{1},\bar{z}^{2},\bar{\tilde{z}}^{1},\bar{\tilde{z}}^{2},\bar{\tilde{\Gamma}};p,h,m,n^{1},n^{2},s^{2},\alpha,P,u)-\mathcal{H}(t)\big{|}\mathcal{F}^{Y}_{t}\big{]}\geq
0,$ (3.39) $\displaystyle\hskip 199.16928pta.e.\ t\in[0,T],\ P\mbox{-}a.s.,$

where
$\mathcal{H}(t)=\mathcal{H}(t,\bar{x},\bar{y},\bar{z}^{1},\bar{z}^{2},\bar{\tilde{z}}^{1},\bar{\tilde{z}}^{2},\bar{\tilde{\Gamma}};p,h,m,n^{1},n^{2},s^{2},\alpha,P,\bar{u})$.

The main result in this paper is the following theorem.

######  Theorem 3.1.

Let (A1)-(A3) hold, and $\bar{u}$ be an optimal control,
$(\bar{x},\bar{y},\bar{z}^{1},\bar{z}^{2},\bar{\tilde{z}}^{1},\bar{\tilde{z}}^{2},\bar{\tilde{\Gamma}})$
be the corresponding solution to the FBSDEP (2.1) and (3.26). Then the maximum
condition (3.39) holds, where $(p,q^{1},q^{2},\tilde{q}^{1},\tilde{q}^{2})$
satisfies (3.13),
$(h,m,n^{1},n^{2},\tilde{n}^{1},\tilde{n}^{2},r,s^{1},s^{2},\tilde{s}^{1},\tilde{s}^{2})$
satisfies (3.29),(3.30),
$(\alpha,\beta^{1},\beta^{2},\tilde{\beta}^{1},\tilde{\beta}^{2})$ satisfies
(3.32) and $(P,Q^{1},Q^{2},\tilde{Q}^{1},\tilde{Q}^{2})$ satisfies (3.34).

######  Remark 3.2.

Theorem 3.1 is a new partially observed maximum principle which, in fact, can
not be covered by any other results in the existing literature but can cover
many existing results in some special cases. In fact, in (2.1), (2.3) and
(2.4) of our problem,

(1) considering the LQ case, if
$\sigma_{2}=f_{1}=f_{2}=b_{2}=\sigma_{3}=f_{3}\equiv 0$ with
$z^{2}=\tilde{z}^{1}=\tilde{z}^{2}\equiv 0$, Theorem 3.1 reduces to Theorem
4.1 of [9] if $b,\sigma$ are independent of backward components $(Y,Z)$.

Interestingly, we could find some new and important relationships when making
comparisons with the existing literature.

(2) If $\sigma_{2}=f_{1}=f_{2}=b_{2}=\sigma_{3}=f_{3}=l=\Phi\equiv 0$,
$\Gamma(y)=y$, $g$ is independent of $(z^{2},\tilde{z}^{1},\tilde{z}^{2})$
with $z^{2}=\tilde{z}^{1}=\tilde{z}^{2}\equiv 0$, and the Hamiltonian function
(3.38) reduces to

$\displaystyle\tilde{\mathcal{H}}(t,x,y,z^{1};\tilde{p},h,m,n^{1},\tilde{P},u)=(-h_{t}g_{z^{1}}\tilde{p}_{t}+n^{1}_{t})\sigma_{1}(t,x,u)+m_{t}b_{1}(t,x,u)$
(3.40)
$\displaystyle+h_{t}g(t,x,y,z^{1}+\tilde{p}(\sigma_{1}(t,x,u)-\sigma_{1}(t)),u)+\frac{1}{2}\sigma_{1}^{2}(t,x,u)\tilde{P}_{t}-\sigma_{1}(t,x,u)\sigma_{1}(t)\tilde{P}_{t}.$

For avoiding confusion, we reset the solutions to (3.13) and (3.34) as
$(\tilde{p},\tilde{q}^{1})$ and $(\tilde{P},\tilde{Q}^{1})$. Then Theorem 3.1
reduces to Theorem 2 of Hu [8], the global maximum principle of completely
observed case for decoupled FBSDEs with the following relations hold:

$\displaystyle p_{t}=\tilde{p}_{t},\ q_{t}=\tilde{q}^{1}_{t},\
p_{t}h_{t}=m_{t},\ h_{t}q_{t}+f_{z}p_{t}h_{t}=n^{1}_{t},$ (3.41)
$\displaystyle P_{t}h_{t}=\tilde{P}_{t},\
h_{t}Q_{t}+f_{z}P_{t}h_{t}=\tilde{Q}^{1}_{t},$

and maximum condition have the relation

$\displaystyle\frac{1}{h_{t}}\Big{[}\tilde{\mathcal{H}}(t,\bar{x},\bar{y},\bar{z}^{1};\tilde{p},h,m,n^{1},\tilde{P},u)-\tilde{\mathcal{H}}(t,\bar{x},\bar{y},\bar{z}^{1};\tilde{p},h,m,n^{1},\tilde{P},\bar{u})\Big{]}$
(3.42)
$\displaystyle\quad=\mathcal{H}(t,\bar{x},\bar{y},\bar{z},u,p,q,P)-\mathcal{H}(t,\bar{x},\bar{y},\bar{z},\bar{u},p,q,P)\geq
0$

where the generator $g=f$,
$(\tilde{p}_{t},\tilde{q}^{1}_{t}),h_{t},(m_{t},n^{1}_{t}),(\tilde{P}_{t},\tilde{Q}^{1}_{t})$
satisfy (3.13),(3.29),(3.30) and (3.34), respectively. And
$(p_{t},q_{t}),(P_{t},Q_{t})$ satisfy (15), (16) and $\mathcal{H}$ is defined
in (44) in [8].

(3) If $\sigma_{2}=f_{1}=f_{2}=b_{2}=\sigma_{3}=f_{3}\equiv 0$,
$\phi(x)=y_{T}$(deterministic), $l,g$ are independent of
$(z^{2},\tilde{z}^{1},\tilde{z}^{2})$ with
$z^{2}=\tilde{z}^{1}=\tilde{z}^{2}\equiv 0$, and the corresponding Hamiltonian
function (with convex control domain) reduces to:

$\displaystyle\mathcal{H}(t,x,y,z^{1},p,h,m,n^{1},u)=[-(l_{z^{1}}+h_{t}g_{z^{1}})p_{t}+n^{1}_{t}]\sigma_{1}(t,x,u)+m_{t}b_{1}(t,x,u)$
(3.43)
$\displaystyle\quad+h_{t}g(t,x,y,z^{1}+p(\sigma_{1}(t,x,u)-\sigma_{1}(t)),u)+l(t,x,y,z^{1}+p(\sigma_{1}(t,x,u)-\sigma_{1}(t)),u).$

In our paper, adjoint equation $(p,q)$ is deduced by the relation between
$y^{1}$ and $x^{1}$ due to its coupling at $T$ with $y_{T}=\phi(x_{T})$.
However, when we degenerate to the case $y_{T}=y_{T}$ being deterministic, it
is natural that $p$ disappears. Then Theorem 3.1 reduces to Theorem 4.4 of
Peng [25], the local maximum principle of completely observed case for
decoupled FBSDEs with the following relations:

$\displaystyle h_{t}=q_{t},\ m_{t}=p_{t},\ n^{1}_{t}=k_{t},$ (3.44)
$\displaystyle\mathcal{H}_{u}(t,\bar{x},\bar{y},\bar{z}^{1},h,m,n^{1},\bar{u})=H_{u}(\bar{x},\bar{y},\bar{z},\bar{u},p,k,q,t),$

where $h_{t},m_{t},n^{1}_{t}$ satisfy (3.29),(3.30), and $q_{t},p_{t},k_{t}$
satisfy (4.12) and $H$ is defined in the [25]. Noting the backward state
equation, we can ensure that the relation can be built equivalently when the
generators $g$ between our setting and [25] exist a minus sign.

(4) If $\sigma_{2}=f_{2}=b_{2}=\sigma_{3}=f_{3}=g=\phi=\Gamma\equiv 0$, $l$ is
independent of $(y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})$ with
$y=z^{1}=z^{2}=\tilde{z}^{1}=\tilde{z}^{2}\equiv 0$, and the corresponding
Hamiltonian function $\mathcal{H}$ reduces to:

$\displaystyle\mathcal{H}(t,x,m,n^{1},P,u)$ (3.45)
$\displaystyle\quad=n^{1}_{t}\sigma_{1}(t,x,u)+l(t,x,u)+m_{t}b_{1}(t,x,u)+\frac{1}{2}\sigma_{1}^{2}(t,x,u)P_{t}-\sigma_{1}(t,x,u)\sigma_{1}(t)P_{t}.$

Then Theorem 3.1 reduces to Theorem 5.2 of Song et al. [38], the global
maximum principle of completely observed case for SDEPs with the following
relation

$(m,n^{1},\tilde{n}^{1})=(p,q,k),\quad(\tilde{P},Q^{1},\tilde{Q}^{1})=(P,Q,K),$
(3.46)

where, for avoiding confusion, we reset $(\tilde{P},Q^{1},\tilde{Q}^{1})$ as
the solution to the reduced second adjoint equation (3.34),
$(m,n^{1},\tilde{n}^{1})$ satisfies the redecued equation (3.30), and
$(p,q,k),(P,Q,K)$ satisfy (5.1),(5.2) in [38], respectively.

(5) If the control domain is convex,
$\sigma_{2}=f_{2}=b_{2}=\sigma_{3}=f_{3}\equiv 0$, $g,l$ are independent of
$(z^{2},\tilde{z}^{2})$ with $z^{2}=\tilde{z}^{2}\equiv 0$, Theorem 3.1
reduces to Theorem 2.1 of Shi and Wu [33], the local maximum principle of
completely observed case for FBSDEPs with
$h=-p,(m,n^{1},\tilde{n}^{1})=(q,k,R)$, where $h,(m,n^{1},\tilde{n}^{1})$
satisfy (3.29),(3.30), and $p_{t},(q_{t},k_{t},R_{(t,e)})$ satisfy (7) in
[33].

######  Remark 3.3.

As mentioned in Remark 2.2, for the partially observed case, we can also
degenerate and cover the following cases and obtain its equivalence under two
different kinds of frameworks. In fact, in (2.1), (2.3) and (2.4) of our
problem,

(1) if the control domain is convex, $f_{1}=f_{2}=f_{3}\equiv 0$,
$\sigma_{3}\equiv 1$, $b_{2}$ is independent of
$(y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2},u)$, $l,g$ are independent of
$(\tilde{z}^{1},\tilde{z}^{2})$ with $\tilde{z}^{1}=\tilde{z}^{2}\equiv 0$,
Theorem 3.1 reduces to Theorem 2.1 of Wang et al. [44], the local maximum
principle of partially observed case for FBSDEs with correlated Brownian
noises with the following important relations:

$\displaystyle\tilde{\Gamma}_{t}=Z_{t},\quad
b_{2}(\cdot,\cdot)=h(\cdot,\cdot),\quad
h_{t}=-p_{t}Z_{t},\quad(r_{t},s^{1}_{t},s^{2}_{t})=(P_{t},Q_{t},\tilde{Q}_{t}),$
(3.47) $\displaystyle m_{t}=q_{t}Z_{t},\quad n^{1}_{t}=k_{t}Z_{t},\quad
n^{2}_{t}=h(t,\bar{x})q_{t}Z_{t}+\tilde{k}_{t}Z_{t},$

and the reduced maximum conditions also have the following equivalent
relations

$\displaystyle
E\big{[}\mathcal{H}_{u}(t,\bar{x},\bar{y},\bar{z}^{1},\bar{z}^{2},\bar{\tilde{\Gamma}};h,m,n^{1},n^{2},s^{2},\bar{u})\big{|}\mathcal{F}^{Y}_{t}\big{]}$
(3.48)
$\displaystyle=\bar{E}\big{[}H_{u}(t,\bar{x},\bar{y},\bar{z},\bar{\tilde{z}},\bar{u};p,q,k,\tilde{k},\tilde{Q})\big{|}\mathcal{Y}_{t}\big{]}=0,\quad\mbox{with}\quad\bar{E}\equiv
E^{\bar{u}},$

where $\tilde{\Gamma}$ is the Radon-Nikodym derivative in the second framework
(for our setting), and $Z$ is the Radon-Nikodym derivative in the first
framework ([44]). $h,(r,s^{1},s^{2}),(m,n^{1},n^{2})$ satisfy (3.29), (3.30),
and $(P,Q,\tilde{Q}),p,(q,k,\tilde{k})$ satisfy (2.3), (2.4) in [44] in which
the Hamiltonian function $H$ is defined. Therefore, in the non-Poisson case,
the equivalence between the two kinds of frameworks can be proved with the
relations above.

(2) If $f_{1}=f_{2}=f_{3}=g=\phi=\Gamma\equiv 0$, $\sigma_{3}\equiv 1$,
$l,b_{2}$ are independent of $(y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})$
with $y=z^{1}=z^{2}=\tilde{z}^{1}=\tilde{z}^{2}\equiv 0$, Theorem 3.1 reduces
to Theorem 2.1 of Tang [39], the global maximum principle of partially
observed case for SDEs with correlated Brownian noises with the following
relations:

$\displaystyle m_{t}=\rho_{t}q_{t},\ n^{1}_{t}=\rho_{t}k_{t},\
n^{2}_{t}=\rho_{t}q_{t}h(t,\bar{x},\bar{u})+\rho_{t}\tilde{k}_{t},$ (3.49)
$\displaystyle r_{t}=r_{t},\ s^{1}_{t}=R_{t},\ s^{2}_{t}=\tilde{R}_{t},$
$\displaystyle\alpha_{t}=q_{t},\ \beta^{1}_{t}=k_{t},\
\beta^{2}_{t}=\tilde{k}_{t},$ $\displaystyle
P_{t}=\rho_{t}Q_{t},Q^{1}_{t}=\rho_{t}K_{t},\
Q^{2}_{t}=\rho_{t}Q_{t}h(t,\bar{x},\bar{u})+\rho_{t}\tilde{K}_{t},$

where
$(r,s^{1},s^{2}),(m,n^{1},n^{2}),(\alpha,\beta^{1},\beta^{2}),(P,Q^{1},Q^{2})$
satisfy (3.29),(3.30), (3.32) and (3.34), respectively and
$(r,R,\tilde{R}),(q,k,\tilde{k}),(Q,K,\tilde{K})$ satisfy (2.23), (2.24) and
(2.25) in [39], respectively.

Similarly, due to the equivalent transformation between these two kinds of
frameworks, we can direct give the following reduced cases:

(3) if $\sigma_{1}$ is independent of $u$, $\sigma_{2}=f_{1}=f_{2}=f_{3}\equiv
0$, $\sigma_{3}\equiv 1$, $l,b_{2}$ are independent of
$(y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})$, $g$ is independent of
$(z^{2},\tilde{z}^{1},\tilde{z}^{2})$ with
$z^{2}=\tilde{z}^{1}=\tilde{z}^{2}\equiv 0$, Theorem 3.1 reduces to Theorem
2.1 of Wang and Wu [43], the global maximum principle of partially observed
case for FBSDEs;

(4) if the control domain is convex, $\sigma_{2}=f_{1}=f_{2}=f_{3}\equiv 0$,
$\sigma_{3}\equiv 1$, $b_{2}$ is independent of
$(y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})$, $l,g$ are independent of
$(z^{2},\tilde{z}^{1},\tilde{z}^{2})$ with
$z^{2}=\tilde{z}^{1}=\tilde{z}^{2}\equiv 0$, Theorem 3.1 reduces to Theorem
2.3 of Wu [50], the local maximum principle of partially observed case for
FBSDEs;

(5) if the control domain is convex, $\sigma_{2}=f_{2}=f_{3}\equiv 0$,
$\sigma_{3}\equiv 1$, $l,g$ are independent of $(z^{2},\tilde{z}^{2})$,
$b_{2}$ is independent of $y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2}$ with
$z^{2}=\tilde{z}^{2}\equiv 0$, Theorem 3.1 reduces to Theorem 1 of Xiao [55]
with $h$ independent of $(y,z,r)$, the local maximum principle for FBSDEPs
with Brownian-noised observation;

(6) if the control domain is convex, $f_{2}=f_{3}=g=\phi=\Gamma\equiv 0$,
$\sigma_{3}\equiv 1$, $l,b_{2}$ are independent of
$(y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})$ with
$y=z^{1}=z^{2}=\tilde{z}^{1}=\tilde{z}^{2}\equiv 0$, Theorem 3.1 reduces to
Theorem 2 of Xiao [56], the local maximum principle for SDEPs with correlated
Brownian noise;

(7) if $\sigma_{2}=f_{1}=f_{2}=f_{3}=g=\phi=\Gamma\equiv 0$, $\sigma_{3}\equiv
1$, $l,b_{2}$ are independent of $(y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})$
with $y=z^{1}=z^{2}=\tilde{z}^{1}=\tilde{z}^{2}\equiv 0$, Theorem 3.1 reduces
to Theorem 2.1 of Li and Tang [18], the global maximum principle of partially
observed case for SDEs;

(8) if $\sigma_{1}=\sigma_{2}=f_{2}=f_{3}=g=\phi=\Gamma\equiv 0$,
$\sigma_{3}\equiv 1$, $l,b_{2}$ are independent of
$(y,z^{1},z^{2},\tilde{z}^{1},\tilde{z}^{2})$ and
$y=z^{1}=z^{2}=\tilde{z}^{1}=\tilde{z}^{2}\equiv 0$, Theorem 3.1 reduces to
Theorem 2.1 of Tang and Hou [40], the global maximum principle for point
processes with Brownian-noised observation.

######  Remark 3.4.

However, Theorem 3.1 could not cover results in Shi and Wu [33] or Hu et al.
[9], since their controlled systems are fully coupled FBSDEs. Moreover,
results in Wu [49] and Yong [66] could not be covered either, since unknown
parameters have to be involved.

##  4 Partially Observed Linear Quadratic Optimal Control Problem of FBSDEPs

In this section, as applications, we study a partially observed LQ optimal
control problem of FBSDEPs, and obtain the state estimate feedback form of the
optimal control with the filtering technique with random jumps.

###  4.1 Problem Formulation and Verification Theorem

Consider the following linear state equation

$\left\\{\begin{aligned}
dx^{u}_{t}&=\big{[}b_{11}(t)x^{u}_{t}+b_{12}(t)u_{t}+b_{13}(t)\big{]}dt+\big{[}\sigma_{11}(t)x^{u}_{t}+\sigma_{12}(t)u_{t}+\sigma_{13}(t)\big{]}dW^{1}_{t}\\\
&\quad+\big{[}\sigma_{21}(t)x^{u}_{t}+\sigma_{22}(t)u_{t}+\sigma_{23}(t)\big{]}d\tilde{W}^{2}_{t}+\int_{\mathcal{E}_{1}}\big{[}f_{11}(t,e)x^{u}_{t-}+f_{12}(t,e)\big{]}\tilde{N}_{1}(de,dt)\\\
&\quad+\int_{\mathcal{E}_{2}}\big{[}f_{21}(t,e)x^{u}_{t-}+f_{22}(t,e)\big{]}\tilde{N}^{\prime}_{2}(de,dt),\\\
-dy_{t}^{u}&=\bigg{[}g_{11}(t)x^{u}_{t}+g_{12}(t)y^{u}_{t}+g_{13}(t)z^{1,u}_{t}+g_{14}(t)z^{2,u}_{t}+\int_{\mathcal{E}_{1}}g_{15}(t,e)\tilde{z}^{1,u}_{(t,e)}\nu_{1}(de)\\\
&\qquad+\int_{\mathcal{E}_{2}}g_{16}(t,e)\tilde{z}^{2,u}_{(t,e)}\nu_{2}(de)+g_{17}(t)u_{t}+g_{18}(t)\bigg{]}dt-z^{1,u}_{t}dW^{1}_{t}-z^{2,u}_{t}dW^{2}_{t}\\\
&\quad-\int_{\mathcal{E}_{1}}\tilde{z}^{1,u}_{(t,e)}\tilde{N}_{1}(de,dt)-\int_{\mathcal{E}_{2}}\tilde{z}^{2,u}_{(t,e)}\tilde{N}_{2}(de,dt),\quad
t\in[0,T],\\\ x_{0}^{u}&=x_{0},\
y_{T}^{u}=\phi_{11}x^{u}_{T}+\phi_{12},\end{aligned}\right.$ (4.1)

and observation equation

$\left\\{\begin{aligned}
dY_{t}&=b_{22}(t)dt+\sigma_{3}(t)d\tilde{W}^{2}_{t}+\int_{\mathcal{E}_{2}}f_{3}(t,e)\tilde{N}^{\prime}_{2}(de,dt),\quad
t\in[0,T],\\\ Y_{0}&=0,\end{aligned}\right.$ (4.2)

where we set $b_{2}(t,x,u)=b_{22}(t)$, which is independent of $x$ and $u$.
The cost functional is given as a quadratic form:

$J(u)=\bar{E}\bigg{[}\int_{0}^{T}l_{11}(t)(u_{t})^{2}dt+(y_{0}^{u})^{2}\bigg{]}.$
(4.3)

Here, $l_{11}(t)>0$ and we set $U=(-\infty,-1]\cup[1,+\infty)$. We assume that
all the coefficients are deterministic functions,
$|\sigma_{3}|^{-1},|f_{3}(t,e)|^{-1}$ exist and are bounded. As Section 3,
under the new probability measure $P$, the state equation (4.1) becomes

$\left\\{\begin{aligned}
dx^{u}_{t}&=\bigg{[}\Big{(}b_{11}(t)-\sigma_{3}^{-1}(t)b_{22}(t)\sigma_{21}(t)-\int_{\mathcal{E}_{2}}(\lambda_{11}(t,e)-1)f_{21}(t,e)\nu_{2}(de)\Big{)}x^{u}_{t}\\\
&\qquad+\Big{(}b_{12}(t)-\sigma_{3}^{-1}(t)b_{22}(t)\sigma_{22}(t)\Big{)}u_{t}+b_{13}(t)-\sigma_{3}^{-1}(t)b_{22}(t)\sigma_{23}(t)\\\
&\qquad-\int_{\mathcal{E}_{2}}(\lambda_{11}(t,e)-1)f_{22}(t,e)\nu_{2}(de)\bigg{]}dt\\\
&\quad+\sum_{i=1}^{2}\big{[}\sigma_{i1}(t)x^{u}_{t}+\sigma_{i2}(t)u_{t}+\sigma_{i3}(t)\big{]}dW^{i}_{t}\\\
&\quad+\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\big{[}f_{i1}(t,e)x^{u}_{t-}+f_{i2}(t,e)\big{]}\tilde{N}_{i}(de,dt)\\\
-dy_{t}^{u}&=\bigg{[}g_{11}(t)x^{u}_{t}+g_{12}(t)y^{u}_{t}+g_{13}(t)z^{1,u}_{t}+g_{14}(t)z^{2,u}_{t}+\int_{\mathcal{E}_{1}}g_{15}(t,e)\tilde{z}^{1,u}_{(t,e)}\nu_{1}(de)\\\
&\qquad+\int_{\mathcal{E}_{2}}g_{16}(t,e)\tilde{z}^{2,u}_{(t,e)}\nu_{2}(de)+g_{17}(t)u_{t}+g_{18}(t)\bigg{]}dt-\sum_{i=1}^{2}z^{i,u}_{t}dW^{i}_{t}\\\
&\quad-\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\tilde{z}^{i,u}_{(t,e)}\tilde{N}_{i}(de,dt),\quad
t\in[0,T],\\\ x_{0}^{u}&=x_{0},\
y_{T}^{u}=\phi_{11}x^{u}_{T}+\phi_{12}.\end{aligned}\right.$ (4.4)

Applying Theorem 3.1 in the previous section, the maximum condition for the
optimal control is as follows:

$\displaystyle
E\Big{[}[n^{1}_{t}\sigma_{12}(t)+n^{2}_{t}\sigma_{22}(t)+2l_{11}(t)\bar{\tilde{\Gamma}}_{t}\bar{u}_{t}+g_{17}(t)h_{t}+(b_{12}(t)-\sigma_{3}^{-1}(t)b_{22}(t)\sigma_{22}(t))m_{t}](u-\bar{u}_{t})$
(4.5)
$\displaystyle\quad+[l_{11}(t)\bar{\tilde{\Gamma}}_{t}+\frac{1}{2}(\sigma^{2}_{12}(t)+\sigma^{2}_{22}(t))P_{t}](u-\bar{u}_{t})^{2}\Big{|}\mathcal{F}^{Y}_{t}\Big{]}\geq
0,\ a.e.\ t\in[0,T],\ P\mbox{-}a.s.$

where we only consider that $\lambda(t,x,e)=\lambda_{11}(t,e)$, which is
independent of $x$. Moreover, we know that the candidate optimal control
$\bar{u}$ should satisfy

$\displaystyle
E\Big{[}l_{11}(t)\bar{\tilde{\Gamma}}_{t}u_{t}^{2}+\big{[}\sigma_{12}(t)n^{1}_{t}+\sigma_{22}(t)n^{2}_{t}+g_{17}(t)h_{t}+(b_{12}(t)-\sigma_{3}^{-1}(t)b_{22}(t)\sigma_{22}(t))m_{t}\big{]}u_{t}$
(4.6)
$\displaystyle\quad+\frac{1}{2}\sigma^{2}_{12}(t)P_{t}u_{t}^{2}-\sigma^{2}_{12}(t)P_{t}\bar{u}_{t}u_{t}+\frac{1}{2}\sigma^{2}_{22}(t)P_{t}u_{t}^{2}-\sigma^{2}_{22}(t)P_{t}\bar{u}_{t}u_{t}\Big{|}\mathcal{F}^{Y}_{t}\Big{]}$
$\displaystyle\geq
E\bigg{[}l_{11}(t)\bar{\tilde{\Gamma}}_{t}\bar{u}_{t}^{2}+\big{[}\sigma_{12}(t)n^{1}_{t}+\sigma_{22}(t)n^{2}_{t}+g_{17}(t)h_{t}+(b_{12}(t)-\sigma_{3}^{-1}(t)b_{22}(t)\sigma_{22}(t))m_{t}\big{]}\bar{u}_{t}$
$\displaystyle\qquad-\frac{1}{2}\sigma^{2}_{12}(t)P_{t}\bar{u}_{t}^{2}-\frac{1}{2}\sigma^{2}_{22}(t)P_{t}\bar{u}_{t}^{2}\Big{|}\mathcal{F}^{Y}_{t}\Big{]},\quad\forall
u\in U,\ a.e.\ t\in[0,T],\ P\mbox{-}a.s.$

Then we have

$\bar{u}_{t}=\begin{cases}\mu_{t},\ &\mu_{t}\in(-\infty,-1]\cup[1,+\infty),\\\
1,&0\leq\mu_{t}<1,\\\ -1,&-1<\mu_{t}<0,\end{cases}$ (4.7)

with

$\mu_{t}:=-\frac{1}{2}(l_{11}(t)\bar{\tilde{\Gamma}}_{t})^{-1}\big{[}\sigma_{12}(t)\hat{n}^{1}_{t}+\sigma_{22}(t)\hat{n}^{2}_{t}+g_{17}(t)\hat{h}_{t}+(b_{12}(t)-\sigma_{3}^{-1}(t)b_{22}(t)\sigma_{22}(t))\hat{m}_{t}\big{]},$
(4.8)

where $\hat{\kappa}_{t}:=E[\kappa_{t}|\mathcal{F}_{t}^{Y}]$ are optimal
estimates for $\kappa=h,m,n^{1},n^{2}$, and
$(h,m,n^{1},n^{2},\tilde{n}^{1},\tilde{n}^{2})$ satisfies the following linear
FBSDEP:

$\left\\{\begin{aligned}
dh_{t}&=g_{12}(t)h_{t}dt+g_{13}(t)h_{t}dW^{1}_{t}+g_{14}(t)h_{t}dW^{2}_{t}+\int_{\mathcal{E}_{1}}g_{15}(t,e)h_{t-}\tilde{N}_{1}(de,dt)\\\
&\quad+\int_{\mathcal{E}_{2}}g_{16}(t,e)h_{t-}\tilde{N}_{2}(de,dt)\\\
-dm_{t}&=\bigg{\\{}g_{11}(t)h_{t}+\Big{(}b_{11}(t)-\sigma_{3}^{-1}(t)b_{22}(t)\sigma_{21}(t)-\int_{\mathcal{E}_{2}}(\lambda_{11}(t,e)-1)f_{21}(t,e)\nu_{2}(de)\Big{)}m_{t}\\\
&\quad+\sum_{i=1}^{2}\Big{[}\sigma_{i1}(t)n^{i}_{t}+\int_{\mathcal{E}_{i}}f_{i1}(t,e)\tilde{n}^{i}_{(t,e)}\nu_{i}(de)\Big{]}\bigg{\\}}dt\\\
&\quad-\sum_{i=1}^{2}n^{i}_{t}dW^{i}_{t}-\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\tilde{n}^{i}_{(t,e)}\tilde{N}_{i}(de,dt),\quad
t\in[0,T],\\\ h_{0}&=2\bar{y}_{0},\quad
m_{T}=\phi_{11}h_{T}.\end{aligned}\right.$ (4.9)

Before we represent the optimal control $\bar{u}$ as a form of state estimate
feedback based on the known information filtration $\mathcal{F}_{t}^{Y}$, we
should verify that the candidate control is indeed optimal.

######  Proposition 4.1.

The $\bar{u}_{t}$ defined by (4.7) is an optimal control.

###### Proof.

It is easy to check the optimality of (4.7) by common technique, and we omit
the details. ∎

###  4.2 State Filtering Estimate

In this section, we aim to obtain the state estimate feedback of the optimal
control $\bar{u}$ in the filtering form. Moreover, we wish to get a more
explicit form by introducing some ODEs.

First, assuming that $m_{t}=\Pi_{t}^{1}h_{t}$, $t\in[0,T]$, with
$\Pi_{T}^{1}=\phi_{11}$, and applying Itô’s formula to $\Pi_{t}^{1}h_{t}$,
where $\Pi$ is deterministic function, we achieve

$\displaystyle dm_{t}$
$\displaystyle=\big{[}\Pi_{t}^{1}g_{12}(t)h_{t}+\dot{\Pi}_{t}^{1}h_{t}\big{]}dt+\Pi_{t}^{1}g_{13}(t)h_{t}dW^{1}_{t}+\Pi_{t}^{1}g_{14}(t)h_{t}dW^{2}_{t}$
(4.10)
$\displaystyle\quad+\int_{\mathcal{E}_{1}}\Pi_{t}^{1}g_{15}(t,e)h_{t-}\tilde{N}_{1}(de,dt)+\int_{\mathcal{E}_{2}}\Pi_{t}^{1}g_{16}(t,e)h_{t-}\tilde{N}_{2}(de,dt).$

Comparing this with the coefficients in the second equation of (4.9), we have

$n_{t}^{1}=\Pi_{t}^{1}g_{13}(t)h_{t},\ n_{t}^{2}=\Pi_{t}^{1}g_{14}(t)h_{t},\
\tilde{n}^{1}_{(t,e)}=\Pi_{t}^{1}g_{15}(t,e)h_{t-},\
\tilde{n}^{2}_{(t,e)}=\Pi_{t}^{1}g_{16}(t,e)h_{t-},$ (4.11)

and

$\displaystyle\Pi_{t}^{1}g_{12}(t)h_{t}+\dot{\Pi}_{t}^{1}h_{t}+g_{11}(t)h_{t}+\Big{(}b_{11}(t)-\sigma_{3}^{-1}(t)b_{22}(t)\sigma_{21}(t)$
(4.12)
$\displaystyle-\int_{\mathcal{E}_{2}}(\lambda_{11}(t,e)-1)f_{21}(t,e)\nu_{2}(de)\Big{)}m_{t}+\sigma_{11}(t)n^{1}_{t}+\sigma_{21}(t)n^{2}_{t}$
$\displaystyle+\int_{\mathcal{E}_{1}}f_{11}(t,e)\tilde{n}^{1}_{(t,e)}\nu_{1}(de)+\int_{\mathcal{E}_{2}}f_{21}(t,e)\tilde{n}^{2}_{(t,e)}\nu_{2}(de)=0,\quad
t\in[0,T].$

Substituting (4.11) into (4.12), we have

$\displaystyle\bigg{[}\Pi_{t}^{1}g_{12}(t)+\dot{\Pi}_{t}^{1}+g_{11}(t)+\Big{(}b_{11}(t)-\sigma_{3}^{-1}(t)b_{22}(t)\sigma_{21}(t)$
(4.13)
$\displaystyle-\int_{\mathcal{E}_{2}}(\lambda_{11}(t,e)-1)f_{21}(t,e)\nu_{2}(de)\Big{)}\Pi_{t}^{1}+\sigma_{11}(t)\Pi_{t}^{1}g_{13}(t)+\sigma_{21}(t)\Pi_{t}^{1}g_{14}(t)$
$\displaystyle+\int_{\mathcal{E}_{1}}f_{11}(t,e)\Pi_{t}^{1}g_{15}(t,e)\nu_{1}(de)+\int_{\mathcal{E}_{2}}f_{21}(t,e)\Pi_{t}^{1}g_{16}(t,e)\nu_{2}(de)\bigg{]}h_{t}=0,\quad
t\in[0,T].$

If we introduce the following ODE:

$\left\\{\begin{aligned}
&\dot{\Pi}_{t}^{1}+\bigg{[}g_{12}(t)+b_{11}(t)-\sigma_{3}^{-1}(t)b_{22}(t)\sigma_{21}(t)+\int_{\mathcal{E}_{2}}(g_{16}(t,e)-\lambda_{11}(t,e)+1)f_{21}(t,e)\nu_{2}(de)\\\
&+\sigma_{11}(t)g_{13}(t)+\sigma_{21}(t)g_{14}(t)+\int_{\mathcal{E}_{1}}f_{11}(t,e)g_{15}(t,e)\nu_{1}(de)\bigg{]}\Pi_{t}^{1}+g_{11}(t)=0,\quad
t\in[0,T],\\\ &\Pi_{T}^{1}=\phi_{11},\end{aligned}\right.$ (4.14)

noting (4.7), (4.8), then the optimal control in
$U=(-\infty,-1]\cup[1,+\infty)$ becomes

$\displaystyle\bar{u}_{t}$
$\displaystyle=-\frac{1}{2}L(\Pi_{t}^{1})\hat{h}_{t},\quad a.e.\ t\in[0,T],\
P\mbox{-}a.s.,$ (4.15)

where we set

$\displaystyle L(\Pi_{t}^{1})$
$\displaystyle:=(l_{11}(t)\tilde{\Gamma}_{t})^{-1}\big{[}\sigma_{12}(t)\Pi_{t}^{1}g_{13}(t)+\sigma_{22}(t)\Pi_{t}^{1}g_{14}(t)+g_{17}(t)+(b_{12}(t)-\sigma_{3}^{-1}(t)b_{22}(t)\sigma_{22}(t))\Pi_{t}^{1}\big{]}.$

Putting (4.15) into (4.4), we get

$\left\\{\begin{aligned}
d\bar{x}_{t}&=\bigg{\\{}\Big{[}b_{11}(t)-\sigma_{3}^{-1}(t)b_{22}(t)\sigma_{21}(t)-\int_{\mathcal{E}_{2}}(\lambda_{11}(t,e)-1)f_{21}(t,e)\nu_{2}(de)\Big{]}\bar{x}_{t}\\\
&\qquad-\frac{1}{2}\big{[}b_{12}(t)-\sigma_{3}^{-1}(t)b_{22}(t)\sigma_{22}(t)\big{]}L(\Pi_{t}^{1})\hat{h}_{t}+b_{13}(t)-\sigma_{3}^{-1}(t)b_{22}(t)\sigma_{23}(t)\\\
&\qquad-\int_{\mathcal{E}_{2}}(\lambda_{11}(t,e)-1)f_{22}(t,e)\nu_{2}(de)\bigg{\\}}dt+\sum_{i=1}^{2}\bigg{[}\sigma_{i1}(t)\bar{x}_{t}-\frac{1}{2}\sigma_{i2}(t)L(\Pi_{t}^{1})\hat{h}_{t}\\\
&\qquad+\sigma_{i3}(t)\bigg{]}dW^{i}_{t}+\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\big{[}f_{i1}(t,e)\bar{x}_{t-}+f_{i2}(t,e)\big{]}\tilde{N}_{i}(de,dt),\\\
\bar{x}_{0}&=x_{0},\end{aligned}\right.$ (4.16) $\left\\{\begin{aligned}
-d\bar{y}_{t}&=\bigg{[}g_{11}(t)\bar{x}_{t}+g_{12}(t)\bar{y}_{t}+g_{13}(t)\bar{z}^{1}_{t}+g_{14}(t)\bar{z}^{2}_{t}+\int_{\mathcal{E}_{1}}g_{15}(t,e)\bar{\tilde{z}}^{1}_{(t,e)}\nu_{1}(de)\\\
&\qquad+\int_{\mathcal{E}_{2}}g_{16}(t,e)\bar{\tilde{z}}^{2}_{(t,e)}\nu_{2}(de)-\frac{1}{2}g_{17}(t)L(\Pi_{t}^{1})\hat{h}_{t}+g_{18}(t)\bigg{]}dt\\\
&\quad-\sum_{i=1}^{2}\bar{z}^{i}_{t}dW^{i}_{t}-\sum_{i=1}^{2}\int_{\mathcal{E}_{i}}\bar{\tilde{z}}^{i}_{(t,e)}\tilde{N}_{i}(de,dt),\quad
t\in[0,T],\\\
\bar{y}_{T}&=\phi_{11}\bar{x}_{T}+\phi_{12},\end{aligned}\right.$ (4.17)

and the observation equation becomes

$\left\\{\begin{aligned}
&dY_{t}=\sigma_{3}(t)dW_{t}^{2}+\int_{\mathcal{E}_{2}}f_{3}(t,e)\tilde{N}_{2}(de,dt),\quad
t\in[0,T],\\\ &Y_{0}=0.\end{aligned}\right.$ (4.18)

We conjecture that

$\bar{y}_{t}=\Pi_{t}^{2}\bar{x}_{t}+\Pi_{t}^{3}h_{t}+\eta_{t},$ (4.19)

with $\Pi_{T}^{2}=\phi_{11},\Pi_{T}^{3}=0,\eta_{T}=\phi_{12}$. Applying Itô’s
formula, we have

$\displaystyle d\bar{y}_{t}$
$\displaystyle=\bigg{\\{}\bigg{[}b_{11}(t)-\sigma_{3}^{-1}(t)b_{22}(t)\sigma_{21}(t)-\int_{\mathcal{E}_{2}}(\lambda_{11}(t,e)-1)f_{21}(t,e)\nu_{2}(de)\bigg{]}\Pi_{t}^{2}\bar{x}_{t}$
$\displaystyle\qquad-\frac{1}{2}\big{[}b_{12}(t)-\sigma_{3}^{-1}(t)b_{22}(t)\sigma_{22}(t)\big{]}\Pi_{t}^{2}L(\Pi_{t}^{1})\hat{h}_{t}+\bigg{[}b_{13}(t)-\sigma_{3}^{-1}(t)b_{22}(t)\sigma_{23}(t)$
$\displaystyle\qquad-\int_{\mathcal{E}_{2}}(\lambda_{11}(t,e)-1)f_{22}(t,e)\nu_{2}(de)\bigg{]}\Pi_{t}^{2}+\dot{\Pi}_{t}^{2}\bar{x}_{t}+\Pi_{t}^{3}g_{12}(t)h_{t}+\dot{\Pi}_{t}^{3}h_{t}+\dot{\eta}_{t}\bigg{\\}}dt$
$\displaystyle\quad+\bigg{[}\sigma_{11}(t)\Pi_{t}^{2}\bar{x}_{t}-\frac{1}{2}\Pi_{t}^{2}\sigma_{12}(t)L(\Pi_{t}^{1})\hat{h}_{t}+\sigma_{13}(t)\Pi_{t}^{2}+\Pi_{t}^{3}g_{13}(t)h_{t}\bigg{]}dW_{t}^{1}$
$\displaystyle\quad+\bigg{[}\sigma_{21}(t)\Pi_{t}^{2}\bar{x}_{t}-\frac{1}{2}\Pi_{t}^{2}\sigma_{22}(t)L(\Pi_{t}^{1})\hat{h}_{t}+\sigma_{23}(t)\Pi_{t}^{2}+\Pi_{t}^{3}g_{14}(t)h_{t}\bigg{]}dW_{t}^{2}$
$\displaystyle\quad+\int_{\mathcal{E}_{1}}\big{[}\Pi_{t}^{2}f_{11}(t,e)\bar{x}_{t-}+f_{12}(t,e)\Pi_{t}^{2}+g_{15}(t,e)\Pi_{t}^{3}h_{t-}\big{]}\tilde{N}_{1}(de,dt)$
$\displaystyle\quad+\int_{\mathcal{E}_{2}}\big{[}\Pi_{t}^{2}f_{21}(t,e)\bar{x}_{t-}+f_{22}(t,e)\Pi_{t}^{2}+g_{16}(t,e)\Pi_{t}^{3}h_{t-}\big{]}\tilde{N}_{2}(de,dt).$
(4.20)

Comparing this with the second equation in (4.17), we get

$\displaystyle\bar{z}^{1}_{t}=\sigma_{11}(t)\Pi_{t}^{2}\bar{x}_{t}-\frac{1}{2}\Pi_{t}^{2}\sigma_{12}(t)L(\Pi_{t}^{1})\hat{h}_{t}+\sigma_{13}(t)\Pi_{t}^{2}+\Pi_{t}^{3}g_{13}(t)h_{t},$
(4.21)
$\displaystyle\bar{z}^{2}_{t}=\sigma_{21}(t)\Pi_{t}^{2}\bar{x}_{t}-\frac{1}{2}\Pi_{t}^{2}\sigma_{22}(t)L(\Pi_{t}^{1})\hat{h}_{t}+\sigma_{23}(t)\Pi_{t}^{2}+\Pi_{t}^{3}g_{14}(t)h_{t},$
$\displaystyle\bar{\tilde{z}}^{1}_{(t,e)}=\Pi_{t}^{2}f_{11}(t,e)\bar{x}_{t-}+f_{12}(t,e)\Pi_{t}^{2}+g_{15}(t,e)\Pi_{t}^{3}h_{t-},$
$\displaystyle\bar{\tilde{z}}^{2}_{(t,e)}=\Pi_{t}^{2}f_{21}(t,e)\bar{x}_{t-}+f_{22}(t,e)\Pi_{t}^{2}+g_{16}(t,e)\Pi_{t}^{3}h_{t-},$

and

$\displaystyle\bigg{[}b_{11}(t)-\sigma_{3}^{-1}(t)b_{22}(t)\sigma_{21}(t)-\int_{\mathcal{E}_{2}}(\lambda_{11}(t,e)-1)f_{21}(t,e)\nu_{2}(de)\bigg{]}\Pi_{t}^{2}\bar{x}_{t}$
(4.22)
$\displaystyle-\frac{1}{2}\big{[}b_{12}(t)-\sigma_{3}^{-1}(t)b_{22}(t)\sigma_{22}(t)\big{]}\Pi_{t}^{2}L(\Pi_{t}^{1})\hat{h}_{t}+\bigg{[}b_{13}(t)-\sigma_{3}^{-1}(t)b_{22}(t)\sigma_{23}(t)$
$\displaystyle-\int_{\mathcal{E}_{2}}(\lambda_{11}(t,e)-1)f_{22}(t,e)\nu_{2}(de)\bigg{]}\Pi_{t}^{2}+\dot{\Pi}_{t}^{2}\bar{x}_{t}+\Pi_{t}^{3}g_{12}(t)h_{t}+\dot{\Pi}_{t}^{3}h_{t}+\dot{\eta}_{t}+g_{11}(t)\bar{x}_{t}$
$\displaystyle+g_{12}(t)\big{[}\Pi_{t}^{2}\bar{x}_{t}+\Pi_{t}^{3}h_{t}+\eta_{t}\big{]}+g_{13}(t)\big{[}\sigma_{11}(t)\Pi_{t}^{2}\bar{x}_{t}-\frac{1}{2}\Pi_{t}^{2}\sigma_{12}(t)L(\Pi_{t}^{1})\hat{h}_{t}+\sigma_{13}(t)\Pi_{t}^{2}$
$\displaystyle+\Pi_{t}^{3}g_{13}(t)h_{t}\big{]}+g_{14}(t)\big{[}\sigma_{21}(t)\Pi_{t}^{2}\bar{x}_{t}-\frac{1}{2}\Pi_{t}^{2}\sigma_{22}(t)L(\Pi_{t}^{1})\hat{h}_{t}+\sigma_{23}(t)\Pi_{t}^{2}+\Pi_{t}^{3}g_{14}(t)h_{t}\big{]}$
$\displaystyle+\int_{\mathcal{E}_{1}}g_{15}(t,e)\big{[}\Pi_{t}^{2}f_{11}(t,e)\bar{x}_{t}+f_{12}(t,e)\Pi_{t}^{2}+g_{15}(t,e)\Pi_{t}^{3}h_{t}\big{]}\nu_{1}(de)+\int_{\mathcal{E}_{2}}g_{16}(t,e)\big{[}\Pi_{t}^{2}f_{21}(t,e)\bar{x}_{t}$
$\displaystyle+f_{22}(t,e)\Pi_{t}^{2}+g_{16}(t,e)\Pi_{t}^{3}h_{t}\big{]}\nu_{2}(de)-\frac{1}{2}g_{17}(t)L(\Pi_{t}^{1})\hat{h}_{t}+g_{18}(t)=0,\quad
t\in[0,T].$

Taking $E[\cdot|\mathcal{F}_{t}^{Y}]$ on both sides of (4.19), (4.21) and
(4.22), we have

$\displaystyle\hat{\bar{y}}_{t}=\Pi_{t}^{2}\hat{\bar{x}}_{t}+\Pi_{t}^{3}\hat{h}_{t}+\eta_{t},$
(4.23)
$\displaystyle\hat{\bar{z}}^{1}_{t}=\sigma_{11}(t)\Pi_{t}^{2}\hat{\bar{x}}_{t}+(\Pi_{t}^{3}g_{13}(t)-\frac{1}{2}\Pi_{t}^{2}\sigma_{12}(t)L(\Pi_{t}^{1}))\hat{h}_{t}+\sigma_{13}(t)\Pi_{t}^{2},$
$\displaystyle\hat{\bar{z}}^{2}_{t}=\sigma_{21}(t)\Pi_{t}^{2}\hat{\bar{x}}_{t}+(\Pi_{t}^{3}g_{14}(t)-\frac{1}{2}\Pi_{t}^{2}\sigma_{22}(t)L(\Pi_{t}^{1}))\hat{h}_{t}+\sigma_{23}(t)\Pi_{t}^{2},$
$\displaystyle\hat{\bar{\tilde{z}}}^{1}_{(t,e)}=\Pi_{t}^{2}f_{11}(t,e)\hat{\bar{x}}_{t-}+f_{12}(t,e)\Pi_{t}^{2}+g_{15}(t,e)\Pi_{t}^{3}\hat{h}_{t-},$
$\displaystyle\hat{\bar{\tilde{z}}}^{2}_{(t,e)}=\Pi_{t}^{2}f_{21}(t,e)\hat{\bar{x}}_{t-}+f_{22}(t,e)\Pi_{t}^{2}+g_{16}(t,e)\Pi_{t}^{3}\hat{h}_{t-},$

and

$\displaystyle\bigg{\\{}\bigg{[}b_{11}(t)-\sigma_{3}^{-1}(t)b_{22}(t)\sigma_{21}(t)-\int_{\mathcal{E}_{2}}(\lambda_{11}(t,e)-1)f_{21}(t,e)\nu_{2}(de)\bigg{]}\Pi_{t}^{2}+\dot{\Pi}_{t}^{2}+g_{11}(t)$
$\displaystyle\quad+g_{13}(t)\sigma_{11}(t)\Pi_{t}^{2}+g_{14}(t)\sigma_{21}(t)\Pi_{t}^{2}+\int_{\mathcal{E}_{1}}g_{15}(t,e)f_{11}(t,e)\nu_{1}(de)\Pi_{t}^{2}$
$\displaystyle\quad+\int_{\mathcal{E}_{2}}g_{16}(t,e)f_{21}(t,e)\nu_{2}(de)\Pi_{t}^{2}+g_{12}(t)\Pi_{t}^{2}\bigg{]}\hat{\bar{x}}_{t}$
$\displaystyle+\bigg{[}\dot{\Pi}_{t}^{3}+\Pi_{t}^{3}g_{12}(t)-\frac{1}{2}(b_{12}(t)-\sigma_{3}^{-1}(t)b_{22}(t)\sigma_{22}(t))\Pi_{t}^{2}L(\Pi_{t}^{1})-\frac{1}{2}g_{13}(t)\Pi_{t}^{2}\sigma_{12}(t)L(\Pi_{t}^{1})$
(4.24)
$\displaystyle\quad-\frac{1}{2}g_{14}(t)\Pi_{t}^{2}\sigma_{22}(t)L(\Pi_{t}^{1})+g^{2}_{13}(t)\Pi_{t}^{3}+g^{2}_{14}(t)\Pi_{t}^{3}+\int_{\mathcal{E}_{1}}g^{2}_{15}(t,e)\nu_{1}(de)\Pi_{t}^{3}$
$\displaystyle\quad+\int_{\mathcal{E}_{2}}g^{2}_{16}(t,e)\nu_{2}(de)\Pi_{t}^{3}-\frac{1}{2}g_{17}(t)L(\Pi_{t}^{1})+g_{12}(t)\Pi_{t}^{3}\bigg{]}\hat{h}_{t}$
$\displaystyle+\bigg{[}b_{13}(t)-\sigma_{3}^{-1}(t)b_{22}(t)\sigma_{23}(t)-\int_{\mathcal{E}_{2}}(\lambda_{11}(t,e)-1)f_{22}(t,e)\nu_{2}(de)\bigg{]}\Pi_{t}^{2}$
$\displaystyle+\dot{\eta}_{t}+g_{12}(t)\eta_{t}+g_{13}(t)\sigma_{13}(t)\Pi_{t}^{2}+g_{14}(t)\sigma_{23}(t)\Pi_{t}^{2}+\int_{\mathcal{E}_{1}}g_{15}(t,e)f_{12}(t,e)\nu_{1}(de)\Pi_{t}^{2}$
$\displaystyle+\int_{\mathcal{E}_{2}}g_{16}(t,e)f_{22}(t,e)\nu_{2}(de)\Pi_{t}^{2}+g_{18}(t)=0,\quad
t\in[0,T].$

Applying Theorem A.2 in the Appendix to (4.16),(4.17) and (4.9), we get the
forward-backward stochastic differential filtering equations (FBSDFEs for
short) of
$(\hat{\bar{x}},\hat{\bar{y}},\hat{\bar{z}}^{2},\hat{\bar{\tilde{z}}}^{2},\hat{h})$:

$\left\\{\begin{aligned}
d\hat{\bar{x}}_{t}&=\bigg{\\{}\bigg{[}b_{11}(t)-\sigma_{3}^{-1}(t)b_{22}(t)\sigma_{21}(t)-\int_{\mathcal{E}_{2}}(\lambda_{11}(t,e)-1)f_{21}(t,e)\nu_{2}(de)\bigg{]}\hat{\bar{x}}_{t}\\\
&\qquad-\frac{1}{2}\big{[}b_{12}(t)-\sigma_{3}^{-1}(t)b_{22}(t)\sigma_{22}(t)\big{]}L(\Pi_{t}^{1})\hat{h}_{t}+b_{13}(t)-\sigma_{3}^{-1}(t)b_{22}(t)\sigma_{23}(t)\\\
&\qquad-\int_{\mathcal{E}_{2}}(\lambda_{11}(t,e)-1)f_{22}(t,e)\nu_{2}(de)\bigg{\\}}dt\\\
&\quad+\Big{[}\sigma_{21}(t)\hat{\bar{x}}_{t}-\frac{1}{2}\sigma_{22}(t)L(\Pi_{t}^{1})\hat{h}_{t}+\sigma_{23}(t)\Big{]}dW^{2}_{t}\\\
&\quad+\int_{\mathcal{E}_{2}}\big{[}f_{21}(t,e)\hat{\bar{x}}_{t-}+f_{22}(t,e)\big{]}\tilde{N}_{2}(de,dt),\\\
d\hat{h}_{t}&=g_{12}(t)\hat{h}_{t}dt+g_{14}(t)\hat{h}_{t}dW^{2}_{t}+\int_{\mathcal{E}_{2}}g_{16}(t,e)\hat{h}_{t-}\tilde{N}_{2}(de,dt),\\\
-d\hat{\bar{y}}_{t}&=\bigg{\\{}\Big{[}g_{11}(t)+g_{13}(t)\sigma_{11}(t)\Pi_{t}^{2}+\int_{\mathcal{E}_{1}}g_{15}(t,e)f_{11}(t,e)\nu_{1}(de)\Pi_{t}^{2}\Big{]}\hat{\bar{x}}_{t}+g_{12}(t)\hat{\bar{y}}_{t}\\\
&\qquad+\Big{[}g_{13}(t)\sigma_{13}(t)+\int_{\mathcal{E}_{1}}g_{15}(t,e)f_{12}(t,e)\nu_{1}(de)\Big{]}\Pi_{t}^{2}+g_{14}(t)\hat{\bar{z}}^{2}_{t}\\\
&\qquad+\int_{\mathcal{E}_{2}}g_{16}(t,e)\hat{\bar{\tilde{z}}}^{2}_{(t,e)}\nu_{2}(de)+\Big{[}\Big{(}g^{2}_{13}(t)+\int_{\mathcal{E}_{1}}g^{2}_{15}(t,e)\nu_{1}(de)\Big{)}\Pi_{t}^{3}\\\
&\qquad-\frac{1}{2}g_{13}(t)\sigma_{12}(t)L(\Pi_{t}^{1})\Pi_{t}^{2}-\frac{1}{2}g_{17}(t)L(\Pi_{t}^{1})\Big{]}\hat{h}_{t}+g_{18}(t)\bigg{\\}}dt\\\
&\quad-\hat{\bar{z}}^{2}_{t}dW^{2}_{t}-\int_{\mathcal{E}_{2}}\hat{\bar{\tilde{z}}}^{2}_{(t,e)}\tilde{N}_{2}(de,dt),\quad
t\in[0,T],\\\ \hat{\bar{x}}_{0}&=x_{0},\ \hat{h}_{0}=2\hat{\bar{y}}_{0},\
\hat{\bar{y}}_{T}=\phi_{11}\hat{\bar{x}}_{T}+\phi_{12}.\end{aligned}\right.$
(4.25)

Note that the innovation process $\bar{W}^{2}\equiv W^{2}$. Moreover,
$\Pi^{2},\Pi^{3}$ and $\eta$ are the solutions to the following ODEs,
respectively:

$\left\\{\begin{aligned}
&\dot{\Pi}_{t}^{2}+\bigg{[}b_{11}(t)-\sigma_{3}^{-1}(t)b_{22}(t)\sigma_{21}(t)-\int_{\mathcal{E}_{2}}(\lambda_{11}(t,e)-1)f_{21}(t,e)\nu_{2}(de)+g_{13}(t)\sigma_{11}(t)\\\
&+g_{14}(t)\sigma_{21}(t)+\int_{\mathcal{E}_{1}}g_{15}(t,e)f_{11}(t,e)\nu_{1}(de)+\int_{\mathcal{E}_{2}}g_{16}(t,e)f_{21}(t,e)\nu_{2}(de)+g_{12}(t)\bigg{]}\Pi_{t}^{2}\\\
&+g_{11}(t)=0,\quad t\in[0,T],\\\ &\Pi_{T}^{2}=\phi_{11},\end{aligned}\right.$
(4.26) $\left\\{\begin{aligned}
&\dot{\Pi}_{t}^{3}+\bigg{[}2g_{12}(t)+g^{2}_{13}(t)+g^{2}_{14}(t)+\int_{\mathcal{E}_{1}}g^{2}_{15}(t,e)\nu_{1}(de)+\int_{\mathcal{E}_{2}}g^{2}_{16}(t,e)\nu_{2}(de)\bigg{]}\Pi_{t}^{3}\\\
&-\frac{1}{2}\bigg{[}b_{12}(t)-\sigma_{3}^{-1}(t)b_{22}(t)\sigma_{22}(t)+g_{13}(t)\sigma_{12}(t)+g_{14}(t)\sigma_{22}(t)\bigg{]}L(\Pi_{t}^{1})\Pi_{t}^{2}\\\
&-\frac{1}{2}g_{17}(t)L(\Pi_{t}^{1})=0,\quad t\in[0,T],\\\
&\Pi_{T}^{3}=0,\end{aligned}\right.$ (4.27) $\left\\{\begin{aligned}
&\dot{\eta}_{t}+g_{12}(t)\eta_{t}+\bigg{[}b_{13}(t)-\sigma_{3}^{-1}(t)b_{22}(t)\sigma_{23}(t)-\int_{\mathcal{E}_{2}}(\lambda_{11}(t,e)-1)f_{22}(t,e)\nu_{2}(de)\bigg{]}\Pi_{t}^{2}\\\
&+g_{13}(t)\sigma_{13}(t)\Pi_{t}^{2}+g_{14}(t)\sigma_{23}(t)\Pi_{t}^{2}+\int_{\mathcal{E}_{1}}g_{15}(t,e)f_{12}(t,e)\nu_{1}(de)\Pi_{t}^{2}\\\
&+\int_{\mathcal{E}_{2}}g_{16}(t,e)f_{22}(t,e)\nu_{2}(de)\Pi_{t}^{2}+g_{18}(t)=0,\quad
t\in[0,T],\\\ &\eta_{T}=\phi_{12}.\end{aligned}\right.$ (4.28)

From (4.14) and (4.26), we find that $\Pi_{t}^{1}=\Pi_{t}^{2}$, for
$t\in[0,T]$. Therefore, we have

$\hat{\bar{y}}_{0}=\Pi_{0}^{1}x_{0}+\Pi_{0}^{3}\hat{h}_{0}+\eta_{0}=\Pi_{0}^{1}x_{0}+2\Pi_{0}^{3}\hat{\bar{y}}_{0}+\eta_{0},$
(4.29)

i.e.,

$\hat{\bar{y}}_{0}=(1-2\Pi_{0}^{3})^{-1}(\Pi_{0}^{1}x_{0}+\eta_{0}),\quad\hat{h}_{0}=2(1-2\Pi_{0}^{3})^{-1}(\Pi_{0}^{1}x_{0}+\eta_{0}).$
(4.30)

And it is not hard to get the explicit form of $\hat{h}_{t}$:

$\displaystyle\hat{h}_{t}$
$\displaystyle=\hat{h}_{0}\exp\bigg{\\{}\int_{0}^{t}\bigg{[}g_{12}(s)-\frac{1}{2}g^{2}_{14}(s)+\int_{\mathcal{E}_{2}}(\ln(1+g_{16}(s,e))-g_{16}(s,e))\nu_{2}(de)\bigg{]}ds$
(4.31)
$\displaystyle\qquad\qquad+\int_{0}^{t}g_{14}(s)dW^{2}_{s}+\int_{0}^{t}\int_{\mathcal{E}_{2}}\ln(1+g_{16}(s,e))\tilde{N}_{2}(de,ds)\bigg{\\}}.$

Then, the optimal control (4.15) in $U=(-\infty,-1]\cup[1,+\infty)$ has an
explicit form:

$\displaystyle\bar{u}_{t}$
$\displaystyle=-L(\Pi_{t}^{1})(1-2\Pi_{0}^{3})^{-1}(\Pi_{0}^{1}x_{0}+\eta_{0})\exp\bigg{\\{}\int_{0}^{t}\bigg{[}g_{12}(s)-\frac{1}{2}g^{2}_{14}(s)+\int_{\mathcal{E}_{2}}(\ln(1+g_{16}(s,e))$
(4.32) $\displaystyle\qquad-
g_{16}(s,e))\nu_{2}(de)\bigg{]}ds+\int_{0}^{t}g_{14}(s)dW^{2}_{s}+\int_{0}^{t}\int_{\mathcal{E}_{2}}\ln(1+g_{16}(s,e))\tilde{N}_{2}(de,ds)\bigg{\\}},$
$\displaystyle\hskip 199.16928pt\quad a.e.\ t\in[0,T],\ P\mbox{-}a.s.,$

where $\Pi^{1},\Pi^{3},\eta$ are the solutions to (4.14), (4.27), (4.28),
respectively, and $\Pi_{0}^{1},\Pi_{0}^{3},\eta_{0}$ are the corresponding
values at initial time $t=0$.

##  5 Concluding remarks

In this paper, we have derived a global maximum principle for progressive
optimal control of partially observed forward-backward stochastic systems with
random jumps. Compared to [8], we extend it to the case with partial
information and random jumps. Compared to [38], we extend it to the partially
observed forward-backward stochastic system. Compared to [18], we extend it to
the forward-backward system with random jumps. Compared to [40], we extend it
to the forward-backward system with both Brownian motion and Poisson
martingale measure. We should notice that, different from [18] and [40], we
extend the continuous observation equation into discontinuous case, in which
Brownian motion and Poisson random measure are both considered. Meanwhile, by
using the technique introduced by [21], and extending it into the case with
jumps, then combining with the random field method, the $L^{\beta}$-solution
to fully coupled FBSDEPs and its $L^{\beta}(\beta\geq 2)$-estimate under some
assumptions are given to prepare for the maximum principle, and the non-linear
filtering equation for partially observed stochastic system with random jumps,
Theorem A.2, is derived for obtaining the optimal control’s state estimate
feedback for the LQ case. Also, different from the work in [57], we extend it
to the general non-linear forward-backward system and observation equation
with Poisson random measure, where the control domain is non-convex, instead
of the LQ stochastic system with Poisson process. Moreover, we have given an
explicit form of the optimal control by introducing some ODEs.

Global maximum principles for optimal control of fully coupled FBSDEPs and of
mean-field’s type FBSDEs are very challenging and interesting to research. We
will work on these topics in the near future.

## Appendix: Non-linear Filtering Equation of SDEPs

Consider the following signal-observation system:

$\left\\{\begin{aligned} h_{t}&=h_{0}+\int_{0}^{t}H_{s}ds+y_{t},\\\
Y_{t}&=Y_{0}+\int_{0}^{t}A_{s}(\omega)ds+\int_{0}^{t}B_{s}(Y)dW^{2}_{s}+\int_{0}^{t}\int_{\mathcal{E}_{2}}C_{s-}(Y,e)\tilde{N}_{2}(de,ds),\end{aligned}\right.$
(A.1)

where

$y_{t}\equiv\sum_{i=1}^{2}\int_{0}^{t}b_{i,s}(\omega)dW^{i}_{s}+\sum_{i=1}^{2}\int_{0}^{t}\int_{\mathcal{E}_{i}}c_{i,s}(\omega,e)\tilde{N}_{i}(de,ds)$

is a one-dimensional RCLL square integrable martingale with $y_{0}=0$ and
$b_{1,s}(\omega)(b_{2,s}(\omega)):[0,T]\times\Omega\rightarrow\mathbb{R}$ is
$\mathcal{G}/\mathcal{B}(\mathbb{R})$ measurable,
$c_{1,s}(\omega,e)(c_{2,s}(\omega,e)):[0,T]\times\Omega\times\mathcal{E}_{1}(\mathcal{E}_{2})\rightarrow\mathbb{R}$
is
$\mathcal{G}\otimes\mathcal{B}(\mathcal{E}_{1})(\mathcal{B}(\mathcal{E}_{2}))/\mathcal{B}(\mathbb{R})$
measurable.

On a given filtered probability space
$(\Omega,\mathcal{F},(\mathcal{F}_{t}),P)$, $(W^{1}_{t},W^{2}_{t})$ is a two-
dimensional standard $\mathcal{F}_{t}$-Brownian motion and
$(\tilde{N}_{1}(de,dt),\tilde{N}_{2}(de,dt))$ is a standard Poisson martingale
measure with $\tilde{N}_{i}(de,dt)=N_{i}(de,dt)-\nu_{i}(de)dt$,
$\nu_{i}(A_{i})<\infty,\forall A_{i}\in\mathcal{E}_{i}$, where $N_{i}(de,dt)$
is Poisson random measure with $EN_{i}(de,dt)=\nu_{i}(de)dt$, i=1,2.

We need the following assumption.

(H1) $A_{t}(\omega):[0,T]\times\Omega\rightarrow\mathbb{R}$ is
$\mathcal{G}/\mathcal{B}(\mathcal{\mathbb{R}})$ measurable,
$B_{t}(Y):=B(t,Y_{r},r\leq t):[0,T]\times
D([0,T];\mathbb{R})\rightarrow\mathbb{R}$ is
$\mathcal{G}\otimes\mathcal{B}(D([0,T];\mathbb{R}))/\mathcal{B}(R)$-measurable
and for each $t\geq 0$, $B_{t}(Y)$ is $\mathcal{B}(D([0,T];\\\
\mathbb{R}))/\mathcal{B}(\mathbb{R})$-measurable.
$C_{t}(Y,e):=C(t,Y_{r},e,r\leq t):[0,T]\times
D([0,T];\mathbb{R})\times\mathcal{E}_{2}\rightarrow\mathbb{R}$ is
$\mathcal{G}\otimes\mathcal{B}(D([0,T];\mathbb{R}))\otimes\mathcal{B}(\mathcal{E}_{2})/\mathcal{B}(\mathbb{R})$-measurable.
We set $C_{t-}(Y,e):=C(t,Y_{r},e,r<t)$, where $Y$ is an
$\mathcal{F}_{t}$-adapted RCLL process, then $C_{t-}(Y,e)$ is
$\mathcal{F}_{t}$-$\mathcal{E}_{2}$-predictable, where we define
$D([0,T];\mathbb{R})$ as the totality of RCLL maps from $[0,T]$ to
$\mathbb{R}$.

The following result belongs to Situ [37].

######  Theorem A.1.

Assume that $(h,Y)$ is an $\mathcal{F}_{t}$-adapted solution to (A.1) and for
all $T>0$,

$E\int_{0}^{T}|A_{s}(\omega)|ds<\infty,\
E\int_{0}^{T}|B_{s}(Y)|^{2}ds<\infty,\
E\int_{0}^{T}\int_{\mathcal{E}_{2}}|C_{s}(Y,e)|^{2}\nu_{2}(de)ds<\infty,$
(A.2)

and $B^{-1}$ exists and is bounded. Then

(1) $\bar{M}_{t}:=\int_{0}^{t}B^{-1}_{s-}(Y)\big{[}dY_{s}-\pi_{s}(A)ds\big{]}$
is a locally square integrable martingale on the probability space
$(\Omega,\mathcal{F},(\mathcal{F}_{t}^{Y}),P)$ and the observation equation of
(A.1) can be rewritten as

$Y_{t}=Y_{0}+\int_{0}^{t}\pi_{s}(A)ds+\int_{0}^{t}B_{s-}(Y)d\bar{M}_{s}.$
(A.3)

(2) Furthermore,
$\bar{M}_{t}=\bar{W}^{2}_{t}+\int_{0}^{t}\int_{\mathcal{E}_{2}}B_{s-}^{-1}(Y)C_{s-}(Y,e)\tilde{N}_{2}(de,ds)$,
where $\pi_{s}(A):=E\big{[}A_{s}|\mathcal{F}_{s}^{Y}\big{]}$,
$\bar{W}^{2}_{t}=\int_{0}^{t}B_{s}^{-1}(Y)(A_{s}-\pi_{s}(A))ds+W^{2}_{t}$ is a
Brownian motion on $(\Omega,\mathcal{F},(\mathcal{F}_{t}^{Y}),P)$, and
$\tilde{N}_{2}(de,dt)$ is an $\mathcal{F}_{t}^{Y}$-adapted standard Poisson
martingale measure.

Usually, we call $\bar{W}^{2}_{t}$ and $\tilde{N}_{2}(de,ds)$ the innovation
processes for $W^{2}_{t}$ and $\tilde{N}_{2}(de,ds)$. Therefore, the
observation equation becomes

$Y_{t}=Y_{0}+\int_{0}^{t}\pi_{s}(A)ds+\int_{0}^{t}B_{s}(Y)d\bar{W}^{2}_{s}+\int_{0}^{t}\int_{\mathcal{E}_{2}}C_{s-}(Y,e)\tilde{N}_{2}(de,ds).$
(A.4)

Before giving our main theorem, we need the following lemmas, similar to [37].

######  Lemma A.1.

$\big{\\{}E\big{[}y_{t}\big{|}\mathcal{F}_{t}^{Y}]\big{\\}}_{t\geq 0}$ is an
$\mathcal{F}_{t}^{Y}$-adapted square integrable martingale.

######  Lemma A.2.

$\big{\\{}E\big{[}\int_{0}^{t}H_{s}ds\big{|}\mathcal{F}_{t}^{Y}\big{]}-\int_{0}^{t}\pi_{s}(H)ds\big{\\}}_{t\geq
0}$ is an $\mathcal{F}_{t}^{Y}$-adapted square integrable martingale.

######  Lemma A.3.

There exists an $f(s,\omega)\in M^{2}_{\mathcal{F}_{t}^{Y}}[0,t]$ and a
$g(s,\omega,e)\in F^{2}_{\mathcal{F}_{t}^{Y}}[0,t]$ such that

$\displaystyle
E\big{[}h_{0}\big{|}\mathcal{F}_{t}^{Y}\big{]}+E\bigg{[}\int_{0}^{t}H_{s}ds\bigg{|}\mathcal{F}_{t}^{Y}\bigg{]}+E\big{[}y_{t}\big{|}\mathcal{F}_{t}^{Y}\big{]}-\int_{0}^{t}\pi_{s}(H)ds$
(A.5)
$\displaystyle=\pi_{0}(h)+\int_{0}^{t}f(s,\omega)d\bar{W}^{2}_{s}+\int_{0}^{t}\int_{\mathcal{E}_{2}}g(s,\omega,e)\tilde{N}_{2}(de,ds).$

Hence

$\displaystyle\pi_{t}(h)$
$\displaystyle=E\big{[}h_{0}\big{|}\mathcal{F}_{t}^{Y}\big{]}+E\bigg{[}\int_{0}^{t}H_{s}ds\bigg{|}\mathcal{F}_{t}^{Y}\bigg{]}+E\big{[}y_{t}\big{|}\mathcal{F}_{t}^{Y}\big{]}$
(A.6)
$\displaystyle=\pi_{0}(h)+\int_{0}^{t}\pi_{s}(H)ds+\int_{0}^{t}f(s,\omega)d\bar{W}^{2}_{s}+\int_{0}^{t}\int_{\mathcal{E}_{2}}g(s,\omega,e)\tilde{N}_{2}(de,ds),$

where $M^{2}_{\mathcal{F}_{t}^{Y}}[0,t]$ is the space of
$\mathcal{F}_{t}^{Y}$-predictable stochastic processes satisfying
$E\int_{0}^{t}|f(s,\omega)|^{2}ds\\\ <\infty$, and
$F^{2}_{\mathcal{F}_{t}^{Y}}[0,t]$ is that of
$\mathcal{F}_{t}^{Y}$-$\mathcal{E}_{2}$-predictable stochastic processes
satisfying $E\int_{0}^{t}\int_{\mathcal{E}_{2}}|g(s,\omega,e)|^{2}\\\
\nu_{2}(de)ds<\infty$.

Next, we give the non-linear filtering equation of partially observed SDEP
system as follows.

######  Theorem A.2.

Assume that (1) $|B_{t}(x)|^{2}+||C_{t}(x,e)||^{2}\leq
c_{0}(1+|x|^{2}),\forall x\in D([0,T];\mathbb{R})$ with some constant $c_{0}$,
and $C^{-1}$ exists. For any $T<\infty$,
$E\int_{0}^{T}|A_{s}(\omega)|^{2}ds<\infty$.
(2) $B^{-1}$ exists and is bounded.
$|B_{t}(x_{1})-B_{t}(x_{2})|^{2}+||C_{t}(x_{1},e)-C_{t}(x_{2},e)||^{2}\leq
k|x_{1}-x_{2}|^{2}$ for any $x_{1},x_{2}\in D([0,T];\mathbb{R})$ and $t\geq
0$.
(3) $E|h_{0}|^{2}<\infty$, $E|Y_{0}|^{2}<\infty$,
$E\int_{0}^{T}|H_{s}|^{2}ds<\infty$, $E\big{[}\sup_{0\leq t\leq
T}|h_{t}|^{2}\big{]}<\infty$, for all $T<\infty$. Then the optimal filter
$\pi_{t}(h):=E\big{[}h_{t}\big{|}\mathcal{F}_{t}^{Y}\big{]}$ will satisfy the
following non-linear filtering equation:

$\displaystyle\pi_{t}(h)$
$\displaystyle=\pi_{0}(h)+\int_{0}^{t}\pi_{s}(H)ds+\int_{0}^{t}\big{[}\pi_{s}(b_{2})+B_{s}^{-1}(Y)(\pi_{s}(Ah)-\pi_{s}(A)\pi_{s}(h))\big{]}d\bar{W}^{2}_{s}$
(A.7)
$\displaystyle\quad+\int_{0}^{t}\int_{\mathcal{E}_{2}}\pi_{s-}(c_{2})\tilde{N}_{2}(de,ds),\quad
P\mbox{-}a.s.$

###### Proof.

Let
$\tilde{y}_{t}:=\int_{0}^{t}f(s,\omega)d\bar{W}^{2}_{s}+\int_{0}^{t}\int_{\mathcal{E}_{2}}g(s,\omega,e)\tilde{N}_{2}(de,ds)$,
and let

$z_{t}:=\int_{0}^{t}\lambda^{1}_{s}(Y)d\bar{W}^{2}_{s}\\\
+\int_{0}^{t}\int_{\mathcal{E}_{2}}\lambda^{2}_{s}(Y,e)\tilde{N}_{2}(de,ds),$

where $\lambda^{1}_{t}(Y)$ is a scalar-valued bounded
$\mathcal{F}_{t}^{Y}$-adapted stochastic process and $\lambda^{2}_{s}(Y,e)$ is
any given one-dimensional $\mathcal{F}_{t}^{Y}$-$\mathcal{E}_{2}$-predictable
stochastic process. Then $z$ is an $\mathcal{F}_{t}^{Y}$-predictable square
integrable martingale.

Applying Itô’s formula to $\tilde{y}_{t}z_{t}$, we have

$E[\tilde{y}_{t}z_{t}]=E\int_{0}^{t}\lambda_{s}^{1}(Y)f(s,\omega)ds+E\int_{0}^{t}\int_{\mathcal{E}_{2}}\lambda_{s}^{2}(Y,e)g(s,\omega,e)\nu_{2}(de)ds.$
(A.8)

On the other hand, since
$\tilde{y}_{t}=\pi_{t}(h)-\pi_{0}(h)-\int_{0}^{t}\pi_{s}(H)ds\in\mathcal{F}_{t}^{Y}$,
then we have

$\displaystyle E[\tilde{y}_{t}z_{t}]$
$\displaystyle=E[\pi_{t}(h)z_{t}]-E[\pi_{0}(h)z_{t}]-E\bigg{[}\int_{0}^{t}\pi_{s}(H)dsz_{t}\bigg{]}$
(A.9) $\displaystyle=E\bigg{[}h_{t}z_{t}-\int_{0}^{t}H_{s}z_{s}ds\bigg{]}.$

Since
$\bar{W}^{2}_{t}=\int_{0}^{t}B_{s-}^{-1}(Y)(A_{s}-\pi_{s}(A))ds+W^{2}_{t}$, we
get

$\displaystyle
z_{t}=\tilde{z}_{t}+\int_{0}^{t}\lambda^{1}_{s}(Y)B_{s-}^{-1}(Y)(A_{s}-\pi_{s}(A))ds,$
(A.10)

where
$\tilde{z}_{t}=\int_{0}^{t}\lambda^{1}_{s}(Y)dW^{2}_{s}+\int_{0}^{t}\int_{\mathcal{E}_{2}}\lambda^{2}_{s}(Y,e)\tilde{N}_{2}(de,ds)$.
Then (A.9) becomes

$\displaystyle E[\tilde{y}_{t}z_{t}]$
$\displaystyle=E\bigg{[}h_{t}\tilde{z}_{t}-\int_{0}^{t}H_{s}\tilde{z}_{s}ds\bigg{]}+E\bigg{[}h_{t}\int_{0}^{t}\lambda^{1}_{s}(Y)B^{-1}_{s-}(Y)(A_{s}-\pi_{s}(A))ds$
(A.11)
$\displaystyle\qquad-\int_{0}^{t}H_{s}\int_{0}^{s}\lambda^{1}_{u}(Y)B^{-1}_{u-}(Y)(A_{u}-\pi_{u}(A))duds\bigg{]}.$

Here, we should note that $\tilde{z}$ is an $\mathcal{F}_{t}$-adapted square
integrable martingale w.r.t. $P$. Note that

$E[\tilde{z}_{t}h_{0}]=E\big{[}E[\tilde{z}_{t}h_{0}|\mathcal{F}_{0}]\big{]}=E[\tilde{z}_{0}h_{0}]=0,$
(A.12)

and

$\displaystyle E\bigg{[}\int_{0}^{t}\tilde{z}_{s}H_{s}ds\bigg{]}$
$\displaystyle=E\bigg{[}\int_{0}^{t}E[\tilde{z}_{t}|\mathcal{F}_{s}]H_{s}ds\bigg{]}=E\bigg{[}\int_{0}^{t}E[\tilde{z}_{t}H_{s}|\mathcal{F}_{s}]ds\bigg{]}$
(A.13)
$\displaystyle=E\bigg{[}\int_{0}^{t}\tilde{z}_{t}H_{s}ds\bigg{]}=E\bigg{[}\tilde{z}_{t}\int_{0}^{t}H_{s}ds\bigg{]}.$

Therefore, we have

$\displaystyle
E\bigg{[}h_{t}\tilde{z}_{t}-\int_{0}^{t}H_{s}\tilde{z}_{s}ds\bigg{]}=E\bigg{[}\tilde{z}_{t}(h_{t}-h_{0})-\tilde{z}_{t}\int_{0}^{t}H_{s}ds\bigg{]}$
(A.14)
$\displaystyle=E\bigg{[}\tilde{z}_{t}(h_{t}-h_{0}-\int_{0}^{t}H_{s}ds)\bigg{]}=E[\tilde{z}_{t}y_{t}]$
$\displaystyle=E\bigg{[}\tilde{z}_{t}\sum_{i=1}^{2}\bigg{(}\int_{0}^{t}b_{i,s}(\omega)dW^{i}_{s}+\int_{0}^{t}\int_{\mathcal{E}_{i}}c_{i,s}(\omega,e)\tilde{N}_{i}(de,ds)\bigg{)}\bigg{]}$
$\displaystyle=E\bigg{\langle}\int_{0}^{t}\lambda^{1}_{s}(Y)dW^{2}_{s},\int_{0}^{t}b_{2,s}(\omega)dW^{2}_{s}\bigg{\rangle}$
$\displaystyle\qquad+E\bigg{\langle}\int_{0}^{t}\int_{\mathcal{E}_{2}}\lambda^{2}_{s}(Y,e)\tilde{N}_{2}(de,ds),\int_{0}^{t}\int_{\mathcal{E}_{2}}c_{2,s}(\omega,e)\tilde{N}_{2}(de,ds)\bigg{\rangle}$
$\displaystyle=E\int_{0}^{t}\lambda^{1}_{s}(Y)b_{2,s}(\omega)ds+E\bigg{[}\int_{0}^{t}\int_{\mathcal{E}_{2}}\lambda^{2}_{s}(Y,e)\mathbb{E}_{2}\big{[}c_{2,s}(\omega,e)\big{|}\mathcal{P}\otimes\mathcal{B}(\mathcal{E}_{2})\big{]}\nu_{2}(de)ds\bigg{]},$

and

$\displaystyle
E\bigg{[}h_{t}\int_{0}^{t}\lambda^{1}_{s}(Y)B^{-1}_{s}(Y)(A_{s}-\pi_{s}(A))ds\bigg{]}$
$\displaystyle=E\bigg{[}\int_{0}^{t}\lambda^{1}_{s}(Y)B^{-1}_{s}(Y)(A_{s}-\pi_{s}(A))(h_{t}-h_{s})ds\bigg{]}$
$\displaystyle\quad+E\bigg{[}\int_{0}^{t}\lambda^{1}_{s}(Y)B^{-1}_{s}(Y)(A_{s}h_{s}-\pi_{s}(A)h_{s})ds\bigg{]}$
$\displaystyle=E\bigg{[}\int_{0}^{t}\lambda^{1}_{s}(Y)B^{-1}_{s}(Y)(A_{s}-\pi_{s}(A))(h_{t}-h_{s})ds\bigg{]}$
(A.15)
$\displaystyle\quad+E\bigg{[}\int_{0}^{t}\lambda^{1}_{s}(Y)B^{-1}_{s}(Y)(\pi_{s}(Ah)-\pi_{s}(A)\pi_{s}(h))ds\bigg{]}.$

Notice that $h_{t}-h_{s}=\int_{s}^{t}H_{u}du+y_{t}-y_{s}$, then we have

$\displaystyle
E\bigg{[}\int_{0}^{t}\lambda^{1}_{s}(Y)B^{-1}_{s}(Y)(A_{s}-\pi_{s}(A))(h_{t}-h_{s})ds\bigg{]}$
$\displaystyle=E\bigg{[}\int_{0}^{t}\lambda^{1}_{s}(Y)B^{-1}_{s}(Y)(A_{s}-\pi_{s}(A))(y_{t}-y_{s})ds\bigg{]}$
$\displaystyle\quad+E\bigg{[}\int_{0}^{t}\lambda^{1}_{s}(Y)B^{-1}_{s}(Y)(A_{s}-\pi_{s}(A))\int_{s}^{t}H_{u}duds\bigg{]}$
(A.16)
$\displaystyle=E\bigg{[}\int_{0}^{t}\lambda^{1}_{s}(Y)B^{-1}_{s}(Y)(A_{s}-\pi_{s}(A))E\big{[}(y_{t}-y_{s})|\mathcal{F}_{s}\big{]}ds\bigg{]}$
$\displaystyle\quad+E\bigg{[}\int_{0}^{t}\int_{0}^{u}\lambda^{1}_{s}(Y)B^{-1}_{s}(Y)(A_{s}-\pi_{s}(A))dsH_{u}du\bigg{]}$
$\displaystyle=E\bigg{[}\int_{0}^{t}\int_{0}^{u}\lambda^{1}_{s}(Y)B^{-1}_{s}(Y)(A_{s}-\pi_{s}(A))dsH_{u}du\bigg{]}.$

Combining (A.15) with (A.16), we get

$\displaystyle
E\bigg{[}h_{t}\int_{0}^{t}\lambda^{1}_{s}(Y)B^{-1}_{s}(Y)(A_{s}-\pi_{s}(A))ds\bigg{]}-E\bigg{[}\int_{0}^{t}\int_{0}^{u}\lambda^{1}_{s}(Y)B^{-1}_{s}(Y)(A_{s}-\pi_{s}(A))dsH_{u}du\bigg{]}$
(A.17)
$\displaystyle=E\bigg{[}\int_{0}^{t}\lambda^{1}_{s}(Y)B^{-1}_{s}(Y)(\pi_{s}(Ah)-\pi_{s}(A)\pi_{s}(h))ds\bigg{]}.$

Substituting (A.14) and (A.17) into (A.11), we have

$\displaystyle E[\tilde{y}_{t}z_{t}]$
$\displaystyle=E\bigg{[}\int_{0}^{t}\lambda^{1}_{s}(Y)\bigg{(}b_{2,s}(\omega)+B^{-1}_{s}(Y)(\pi_{s}(Ah)-\pi_{s}(A)\pi_{s}(h))\bigg{)}ds\bigg{]}$
(A.18)
$\displaystyle\quad+E\bigg{[}\int_{0}^{t}\int_{\mathcal{E}_{2}}\lambda^{2}_{s}(Y,e)\mathbb{E}_{2}[c_{2,s}(\omega,e)|\mathcal{P}\otimes\mathcal{B}(\mathcal{E}_{2})]\nu_{2}(de)ds\bigg{]}$
$\displaystyle=E\bigg{[}\int_{0}^{t}\lambda^{1}_{s}(Y)\bigg{(}\pi_{s}(b_{2})+B^{-1}_{s}(Y)(\pi_{s}(Ah)-\pi_{s}(A)\pi_{s}(h))\bigg{)}ds\bigg{]}$
$\displaystyle\quad+E\bigg{[}\int_{0}^{t}\int_{\mathcal{E}_{2}}\lambda^{2}_{s}(Y,e)\mathbb{E}_{2}[\pi_{s}(c_{2})|\mathcal{P}\otimes\mathcal{B}(\mathcal{E}_{2})]\nu_{2}(de)ds\bigg{]}.$

Compared this to (A.8), we get

$\displaystyle f(s,Y)$
$\displaystyle=\pi_{s}(b_{2})+B^{-1}_{s}(Y)\big{(}\pi_{s}(Ah)-\pi_{s}(A)\pi_{s}(h)\big{)},$
(A.19) $\displaystyle g(s,Y,e)$
$\displaystyle=\mathbb{E}_{2}\big{[}\pi_{s-}(c_{2})\big{|}\mathcal{P}\otimes\mathcal{B}(\mathcal{E}_{2})\big{]}=\pi_{s-}(c_{2}).$

By Lemma A.3, we can derive (A.7). The proof is complete. ∎

## Acknowledgement

Many thanks for discussion and suggestions with Prof. Jie Xiong at Southern
University of Science and Technology, Postdoc. Yuanzhuo Song at Chinese
Academy of Sciences and Dr. Weijun Meng at Shandong University.

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