# A Risk-Sensitive Global Maximum Principle for Controlled Fully Coupled
FBSDEs with Applications ††thanks: This work is supported by National Key
Research $\&$ Development Program of China (2022YFA1006104), National Natural
Science Foundations of China (11971266, 11831010), and Shandong Provincial
Natural Science Foundations (ZR2022JQ01, ZR2020ZD24, ZR2019ZD42).

Jingtao Lin , 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: This paper is concerned with a kind of risk-sensitive optimal
control problem for fully coupled forward-backward stochastic systems. The
control variable enters the diffusion term of the state equation and the
control domain is not necessarily convex. A new global maximum principle is
obtained without assuming that the value function is smooth. The maximum
condition, the first- and second-order adjoint equations heavily depend on the
risk-sensitive parameter. An optimal control problem with a fully coupled
linear forward-backward stochastic system and an exponential-quadratic cost
functional is discussed. The optimal feedback control and optimal cost are
obtained by using Girsanov’s theorem and completion-of-squares approach via
risk-sensitive Riccati equations. A local solvability result of coupled risk-
sensitive Riccati equations is given by Picard-Lindel$\ddot{o}$f’s Theorem.

Keywords: forward-backward stochastic differential equation, risk-sensitive
control, maximum principle, linear-exponential-quadratic control.

Mathematics Subject Classification: 93E20, 60H10, 35K15

##  1 Introduction

Originating from the concept of utility, risk sensitivity is widely related to
mathematical finance, such as optimal investment (Fleming and Sheu [14, 15]),
asset management (Bielecki and Pliska [4], Davis and Lleo [10]) and portfolio
optimization (Nagai and Peng [30], Nagai [29]), etc. When taking into account
the controller’s risk preference, it is natural to consider the risk-sensitive
optimal control problems, where usually some risk-sensitive parameters/indices
are introduced. Risk-sensitive optimal control problems have close connections
with linear-exponential-quadratic Gaussian (LEQG, for short) problems
(Jacobson [20], Whittle [37], Bensoussan and van Schuppen [3], Duncan [12]),
robust $H_{\infty}$ control problems (Başar [2], Lim and Zhou [23]) and
differential games (James [21]). The topic of risk-sensitive optimal control
problems remains one of most active research fields in optimal control theory
for a long time.

Among others, Pontryagin’s maximum principle is a useful tool to solve risk-
sensitive optimal control problems. Pioneer work for stochastic systems can be
seen in Whittle [38, 39], Charalambous and Hiley [8, 9]. Assuming the
smoothness of the value function and using the relation between the global
stochastic maximum principle of Peng [31] and the dynamic programming
principle of Yong and Zhou [43], Lim and Zhou [24] established a global
maximum principle for the risk-sensitive optimal control problem, with the
following exponential-of-integral cost functional

$J^{\theta}(u(\cdot))=\mathbb{E}\Big{[}e^{\theta\big{[}\int_{0}^{T}f(t,x(t),u(t))\,dt+\phi(x(T))\big{]}}\Big{]},$
(1.1)

where $\theta\in\mathbb{R}$ is the risk-sensitive index. The state process
$x(\cdot)$ follows a controlled stochastic differential equation (SDE, for
short):

$\left\\{\begin{aligned}
dx(t)&=b(t,x(t),u(t))\,dt+\sigma(t,x(t),u(t))\,dW(t),\quad t\in[0,T],\\\
x(0)&=x_{0},\end{aligned}\right.$ (1.2)

and the control domain $U$ is not necessarily convex. The result in [24] is
applied to an optimal portfolio choice problem by Shi and Wu [35], and
extended to the jump-diffusion case by Shi and Wu [34]. By introducing a non-
linear transformation from El Karoui and Hamadène [13], Djehiche et al. [11]
eliminated the smoothness assumption of the value function, and obtained a
risk-sensitive global maximum principle for optimal control of SDEs of mean-
field type, where the expectations of the state and control variables are
included in the drift and diffusion terms. The result in [24] is also extended
to the optimal control of backward SDEs (BSDEs, for short) by Chala [5], where
a risk-sensitive local maximum principle is proved. (The word “local” here
means that the control domain is assumed to be convex.) In [5], the controlled
BSDE is of the following Pardoux-Peng’s form

$\left\\{\begin{aligned} -dy(t)&=-g(t,y(t),z(t),u(t))\,dt-z(t)\,dW(t),\quad
t\in[0,T],\\\ y(T)&=\xi,\end{aligned}\right.$ (1.3)

and the exponential-of-integral cost functional is the following

$J^{\theta}(u(\cdot))=\mathbb{E}\Big{[}e^{\theta\big{[}\int_{0}^{T}f(t,y(t),z(t),u(t))\,dt+\psi(y(0))\big{]}}\Big{]}.$
(1.4)

Some sufficient conditions are supplemented in Chala [6].

Recently, motivated by some cash flow modeling and controlling problems,
Khallout and Chala [22] considered a risk-sensitive optimal control problem,
where the state equation is the following controlled fully coupled forward-
backward SDE (FBSDE, for short):

$\left\\{\begin{aligned}
dx(t)&=b(t,x(t),y(t),z(t),u(t))\,dt+\sigma(t,x(t),y(t),z(t),u(t))\,dW(t),\\\
-dy(t)&=g(t,x(t),y(t),z(t),u(t))\,dt-z(t)\,dW(t),\quad t\in[0,T],\\\
x(0)&=x_{0},\quad y(T)=\xi,\end{aligned}\right.$ (1.5)

with the exponential-of-integral cost functional

$J^{\theta}(u(\cdot))=\mathbb{E}\Big{[}e^{\theta\big{[}\int_{0}^{T}f(t,x(t),y(t),z(t),u(t))\,dt+\phi(x(T))+\psi(y(0))\big{]}}\Big{]}.$
(1.6)

By assuming that the control domain is convex, necessary and sufficient
maximum principles are obtained. Results for mean-field type control is soon
given in Chala and Hafayed [7]. Both these results generalized the pioneer
work of Wu [40] from risk-neutral case to risk-sensitive case. Risk-sensitive
zero-sum stochastic differential games are studied by Moon et al. [28].

When the control domain is not necessarily convex, Moon [27] extended previous
works of Yong [42] and Wu [41] from risk-neutral case to the risk-sensitive
case. However, unknown parameters are unavoidable when applying Ekeland’s
variational principle to treat some embedded problems with state constraints,
which means that the results in [27] are not ideal.

Recently, Hu et al. [17] obtained a novel global maximum principle for optimal
control of fully coupled forward-backward stochastic systems, where the state
equation is the same as (1.5), with the recursive cost functional, that is,

$J(u(\cdot))=y(0).$ (1.7)

The main techniques in [17] is a new first- and second-order spike variational
method which was first introduced by Hu [16] and a new decoupling relation
between first- and second-order variations of $x(\cdot)$, $y(\cdot)$ and
$z(\cdot)$. For ease of use, very recently, Lin and Shi [25] extended the
above result from the recursive cost functional (1.7) to the case with the
cost functional of the following general form:

$J(u(\cdot))=\mathbb{E}\bigg{[}\int_{0}^{T}l(t,x(t),y(t),z(t),u(t))\,dt+f(x(T),y(0))\bigg{]}.$
(1.8)

Motivated by the above literatures, in this paper, we consider the global
maximum principle for the risk-sensitive optimal control problem, with the
controlled fully coupled FBSDE:

$\left\\{\begin{aligned}
dX(t)&=b(t,X(t),Y(t),Z(t),u(t))\,dt+\sigma(t,X(t),Y(t),Z(t),u(t))\,dW(t),\\\
-dY(t)&=g(t,X(t),Y(t),Z(t),u(t))\,dt-Z(t)\,dW(t),\quad t\in[0,T],\\\
X(0)&=x_{0},\quad Y(T)=\phi(X(T)),\end{aligned}\right.$ (1.9)

and the general exponential-of-integral cost functional:

$J^{\theta}(u(\cdot))=\mathbb{E}\Big{[}e^{\theta\big{[}\int_{0}^{T}l(t,X(t),Y(t),Z(t),u(t))\,dt+f(X(T),Y(0))\big{]}}\Big{]}.$
(1.10)

To the best of the authors’ knowledge, the risk-sensitive global maximum
principle for fully coupled FBSDEs obtained in this paper has not been studied
in the existing literature. In Section 3 of this paper, new first- and second-
order adjoint equations are obtained. Especially, the new first-order adjoint
equation (3.16) is a non-coupled FBSDE.

For linear-quadratic (LQ, for short) optimal control problems in risk-neutral
case, there exists a long list of literatures. However, for LQ optimal control
problems of fully coupled forward-backward stochastic systems, few papers
exist, as far as we know, except Wang et al. [36], Shi and Wu [32, 33], Huang
and Shi [19], and Hu et al. [17] for some special examples. Very recently,
motivated by optimization under rational expectations, Hu et al. [18] gave a
systematic research for the LQ optimal control problem of fully coupled
forward-backward stochastic systems, and a new decoupling technique is
proposed to obtain the feedback optimal control.

Linear-exponential-quadratic (LEQ, for short) problem, introduced by
Jacobson[20], is the risk-sensitive counterpart of LQ optimal control
problems. The reader is referred to Djehiche et al. [11], Duncan [12], Lim and
Zhou [24], Moon et al. [28] , Shi and Wu [34], Ma and Wang [26] and the
references therein for the theory of LEQ problem and its wide ranges of
applications. In Section 5 of this paper, distinguished from the above
literature reviewed, we consider an LEQ problem of fully coupled forward-
backward stochastic systems. Especially, all square terms and cross terms of
$X$, $Y$, $Z$ and $u$ can appear in the cost functional. Inspired by Djehiche
et al. [11] and Duncan [12], the optimal feedback control and optimal cost are
obtained by using Girsanov’s Theorem and completion-of-squares approach via
risk-sensitive Riccati equations.

Compared with the existing results, the innovation of this paper can be
summarized along the following lines:

  * •

The control system is a fully coupled FBSDE, the cost functional is a general
exponential-of-integral form, the control domain is not necessarily convex,
and the assumption of smoothness of value function is not needed.

  * •

By introducing a non-linear transformation, new first- and second-order
adjoint equations, which depend on the risk-sensitive parameter, are obtained.
Especially, the first-order adjoint equations are decoupled forward-backward
SDEs, which are more useful in practice.

  * •

The global maximum principle does not contain unknown parameters, which
suggests that our result is more convenient to apply.

  * •

An LEQ problem of fully coupled linear forward-backward stochastic systems is
discussed. All square terms and cross terms of $X$, $Y$, $Z$ and $u$ can
appear in the cost functional. The optimal feedback control and optimal cost
are obtained by using Girsanov’s Theorem and completion-of-squares approach
via risk-sensitive Riccati equations. A local solvability result of coupled
risk-sensitive Riccati equations (5.10) and (5.15) is given by Picard-
Lindel$\ddot{o}$f’s Theorem.

The rest of this paper is organized as follows. In Section 2, we state the
problem and assumptions. The main result is established in Section 3 and its
detailed proof is given in Section 4. In Section 5, an LEQ problem of fully
coupled forward-backward stochastic systems is discussed. Finally, some
concluding remarks are given in Section 6.

Notations. In this paper, the $n$-dimensional Euclidian space is denoted by
$\mathbb{R}^{n}$, and $x^{\top}$ is the transpose of $x\in\mathbb{R}^{n}$. The
inner product of $x,y\in\mathbb{R}^{n}$ is denoted by $\langle x,y\rangle$,
and the norm of $x\in\mathbb{R}^{n}$ is denoted by $|x|=\langle
x,x\rangle^{\frac{1}{2}}$. Let $\mathcal{S}^{n}$, $\mathcal{S}^{n}_{+}$,
$\bar{\mathcal{S}}^{n}_{+}$ be the spaces of $n\times n$ dimensional
symmetric, symmetric positive definite, symmetric positive semi-definite
matrices, respectively. $D\psi$ is the gradient of the function $\psi(x,y,z)$
with respect to $x,y,z$, and $D^{2}\psi$ is the Hessian matrix of it with
respect to $x,y,z$.

##  2 Problem formulation

Let $T\in(0,+\infty)$ be fixed and
$(\Omega,\mathcal{F},\mathbb{P},\left\\{\mathcal{F}_{t}\right\\}_{t\geq 0})$
be a complete filtered probability space on which a one-dimensional standard
Brownian motion $W(\cdot)$ is defined such that
$\mathbb{F}:=\left\\{\mathcal{F}_{t}\right\\}_{t\geq 0}$ is the natural
filtration generated by $W(\cdot)$, augmented by all $\mathbb{P}$-null sets in
$\mathcal{F}$. We introduced some notations of spaces used throughout this
paper.

$L^{p}_{\mathcal{F}_{T}}(\Omega;\mathbb{R}^{n})$: the space of
$\mathcal{F}_{T}$-measurable $\mathbb{R}^{n}$-valued random variables $\eta$
such that

$||\eta||_{p}:=\big{(}\mathbb{E}[|\eta|^{p}]\big{)}^{\frac{1}{p}}<\infty.$

$L^{p}_{\mathcal{F}}([0,T];\mathbb{R}^{n})$: the space of $\mathbb{F}$-adapted
and $p$th integrable stochastic processes $f(\cdot)$ on $[0,T]$ such that

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

$L^{\infty}_{\mathcal{F}}([0,T];\mathbb{R}^{n})$: the space of
$\mathbb{F}$-adapted and uniformly bounded stochastic processes on $[0,T]$
such that

$||f||_{\infty}:=\mathop{\mathrm{esssup}}\limits_{(t,\omega)\,\in[0,T]\times\Omega}|f(t)|<\infty.$

$L^{p}_{\mathcal{F}}(\Omega;C([0,T];\mathbb{R}^{n}))$: the space of
$\mathbb{F}$-adapted continuous stochastic processes $f(\cdot)$ on $[0,T]$
such that

$\mathbb{E}\bigg{[}\sup\limits_{0\leq t\leq T}|f(t)|^{p}\bigg{]}<\infty.$

In this paper, we study a risk-sensitive optimal control problem for a fully
coupled forward-backward stochastic system. The state equation is:

$\left\\{\begin{aligned}
dX(t)&=b(t,X(t),Y(t),Z(t),u(t))\,dt+\sigma(t,X(t),Y(t),Z(t),u(t))\,dW(t),\\\
-dY(t)&=g(t,X(t),Y(t),Z(t),u(t))\,dt-Z(t)\,dW(t),\quad t\in[0,T],\\\
X(0)&=x_{0},\quad Y(T)=\phi(X(T)),\end{aligned}\right.$ (2.1)

where $(X(\cdot),Y(\cdot),Z(\cdot))$ is the state process triple valued in
$\mathbb{R}^{n}\times\mathbb{R}\times\mathbb{R}$. The coefficients
$b,\sigma:[0,T]\times\mathbb{R}^{n}\times\mathbb{R}\times\mathbb{R}\times
U\to\mathbb{R}^{n}$,
$g:[0,T]\times\mathbb{R}^{n}\times\mathbb{R}\times\mathbb{R}\times
U\to\mathbb{R}$ and $\phi:\mathbb{R}^{n}\to\mathbb{R}$ are given functions.
For $b=b,\sigma$, set
$b(\cdot)=(b^{(1)}(\cdot),b^{(2)}(\cdot),\cdots,b^{(n)}(\cdot))^{\top}\in\mathbb{R}^{n}$.

An admissible control $u(\cdot)$ is an $\mathbb{F}$-adapted process valued in
the nonempty control domain $U\subseteq\mathbb{R}^{k}$ such that

$\sup\limits_{0\leq t\leq T}\mathbb{E}\big{[}|u(t)|^{8}\big{]}<\infty.$

Denote the admissible control set by $\mathcal{U}[0,T]$.

In risk-sensitive optimal control problems, it is taken into account the risk
attitude (risk-seeking, risk-averse or risk-neutral) of the controller in
choosing an appropriate criterion to minimize. In this paper, we consider the
cost functional of the exponential-of integral type:

$J^{RS}(u(\cdot)):=\mathbb{E}\Big{[}e^{\theta\big{[}\int_{0}^{T}l(t,X(t),Y(t),Z(t),u(t))\,dt+f(X(T),Y(0))\big{]}}\Big{]},$
(2.2)

where $l:[0,T]\times\mathbb{R}^{n}\times\mathbb{R}\times\mathbb{R}\times
U\to\mathbb{R}^{n}$, $f:\mathbb{R}^{n}\times\mathbb{R}\to\mathbb{R}$ are given
functions, and $\theta$ is the risk-sensitive index.

Next we explain the risk-sensitive index $\theta$ in detail. Define

$L_{T}(u(\cdot)):=\int_{0}^{T}l(t,X(t),Y(t),Z(t),u(t))\,dt+f(X(T),Y(0)),$

then the certainty-equivalent expectation is given by

$\displaystyle\Theta_{\theta}(u(\cdot))$ $\displaystyle:=\frac{1}{\theta}\ln
J^{RS}(u(\cdot))=\frac{1}{\theta}\ln\mathbb{E}\big{[}e^{\theta
L_{T}(u(\cdot))}\big{]}.$

For a sufficiently small $\theta$, $\Theta_{\theta}(u(\cdot))$ has the
following Taylor’s expansion:

$\Theta_{\theta}(u(\cdot))=\mathbb{E}\big{[}L_{T}(u(\cdot))\big{]}+\frac{\theta}{2}\mbox{var}\big{[}L_{T}(u(\cdot))\big{]}+o(\theta).$

If $\theta<0$, the term $\mbox{var}\big{[}\Theta_{T}(u(\cdot))\big{]}$, as a
measure of risk, improves the performance $\Theta_{\theta}(u(\cdot))$. Hence,
in this case, the controller is called risk-seeking. Similarly, $\theta>0$
corresponds to the risk-averse controller.

In this paper, the problem is to find an optimal control $\bar{u}(\cdot)$ over
$\mathcal{U}[0,T]$ to optimize the cost functional (2.2), i.e.,

$J^{RS}(\bar{u}(\cdot))=\inf\limits_{u(\cdot)\in\,\mathcal{U}[0,T]}J^{RS}(u(\cdot)).$
(2.3)

The corresponding $(\bar{X}(\cdot),\bar{Y}(\cdot),\bar{Z}(\cdot))$ is called
the optimal state process triple, and $(\bar{X}(\cdot),\bar{Y}(\cdot)$,
$\bar{Z}(\cdot),\bar{u}(\cdot))$ is the optimal quadruple.

We need the following hypotheses.

######  Assumption 1.

  * The control domain $U$ is a nonempty separable subset of $\mathbb{R}^{k}$.

######  Assumption 2.

  * (i)

For $b=b$, $\sigma$, $g$, $\phi$, all $b$, $b_{x}$, $b_{y}$, $b_{z}$ are
continuous in $(x,y,z,u)$; $b_{x}$, $b_{y}$, $b_{z}$ are bounded; there exists
some constant $L>0$ such that for any $t\in[0,T]$, $x\in\mathbb{R}^{n},$
$y\in\mathbb{R}$, $z\in\mathbb{R}$ and $u,u^{\prime}\in U$, it holds that

$\begin{array}[]{cr}|b(t,x,y,z,u)|\leq L(1+|x|+|y|+|z|+|u|),\\\
|\sigma(t,0,0,z,u)-\sigma(t,0,0,z,u^{\prime})|\leq
L(1+|u|+|u^{\prime}|).\end{array}$

  * (ii)

$f$ is continuous in $(x,y)$, and there exists some constant $L>0$ such that
for any $x\in\mathbb{R}^{n},$ $y\in\mathbb{R}$, $|f(t,x,y)|\leq
L(1+|x|^{2}+|y|^{2})$; $l$ is continuous in $(x,y,z,u)$, and there exists some
constant $L>0$ such that for any $x\in\mathbb{R}^{n},$ $y\in\mathbb{R}$,
$z\in\mathbb{R}$ and $u\in U$, $|l(t,x,y,z,u)|\leq
L(1+|x|^{2}+|y|^{2}+|z|^{2}+|u|^{2})$.

  * (iii)

For any $2\leq\beta\leq 8$,
$\Lambda_{\beta}:=C_{\beta}2^{\beta+1}(1+T^{\beta})c_{1}^{\beta}<1$, where
$c_{1}=max\\{L_{2},L_{3}\\}$,
$L_{2}=max\\{||b_{y}||_{\infty},||b_{z}||_{\infty},||\sigma_{y}||_{\infty}\\}$,
$L_{3}=||\sigma_{z}||_{\infty}$, $C_{\beta}$ is defined in Lemma A.1 in the
appendix of [17] for
$L_{1}=max\\{||b_{x}||_{\infty},||\sigma_{x}||_{\infty},||g_{x}||_{\infty},||g_{y}||_{\infty},||g_{z}||_{\infty},||\phi_{x}||_{\infty}\\}$.

  * (iv)

For $b=b$, $\sigma$, $g$, $\phi$, all $b_{xx}$, $b_{xy}$, $b_{yy}$, $b_{xz}$,
$b_{yz}$, $b_{zz}$ are continuous in $(x,y,z,u)$; $b_{xx}$, $b_{xy}$,
$b_{yy}$, $b_{xz}$, $b_{yz}$, $b_{zz}$ are bounded.

For any given control $u(\cdot)\in\mathcal{U}[0,T]$, Assumption 2 guarantees
that the state equation (2.1) and all the first- and second-order adjoint
equations in Section 3 are uniquely solvable. In fact, some of Assumption 2
can be relaxed, but we do not give details since the topic of the well-
posedness of FBSDEs is not the main target of this paper.

##  3 Main result

In this section, the main result, a Pontryagin’s type necessary condition of
the risk-sensitive optimal control problem (2.3), is established. The main
idea is to transfer (2.3) into an equivalent risk-neutral one, by introducing
an auxiliary process. First, we recall a risk-neutral global MP of fully
coupled FBSDEs in our accompanying paper [25].

###  3.1 A risk-neutral global maximum principle

In this section, we consider the state equation (2.1) with the cost functional
of the following Mayer’s type:

$J^{RN}(u(\cdot))=\mathbb{E}\big{\\{}\varphi(X(T),Y(0))\big{\\}},$ (3.1)

where $\varphi:\mathbb{R}^{n}\times\mathbb{R}\to\mathbb{R}$ is a given
function satisfies the same assumptions as $f$ in Assumption 2.

Let $(\bar{X}(\cdot),\bar{Y}(\cdot),\bar{Z}(\cdot),\bar{u}(\cdot))$ be an
optimal quadruple. Let $E_{\epsilon}\subset[0,T]$ is a measurable set with
$|E_{\epsilon}|=\epsilon\in(0,T)$. For any $u(\cdot)\in\mathcal{U}[0,T]$, the
spike variation of $\bar{u}(\cdot)$ is defined as

$u^{\epsilon}(t):=\bar{u}(t)I_{E^{c}_{\epsilon}}(t)+u(t)I_{E_{\epsilon}}(t).$
(3.2)

Let $(X^{\epsilon}(\cdot),Y^{\epsilon}(\cdot),Z^{\epsilon}(\cdot))$ be the
corresponding state process triple. For $b=b$, $\sigma$, $g$, $\phi$,
$\varphi$ and $x=x$, $y$, $z$, we introduce the following notations:

$\displaystyle b(t)$
$\displaystyle:=\psi(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),\bar{u}(t)),\quad
b_{x}(t):=b_{x}(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),\bar{u}(t)),$
$\displaystyle Db(t)$
$\displaystyle:=Db(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),\bar{u}(t)),\quad
D^{2}b(t):=D^{2}b(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),\bar{u}(t)),$
$\displaystyle\delta b(t)$
$\displaystyle:=b(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),u(t))-b(t),\quad\delta
b_{x}(t):=b_{x}(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),u(t))-b_{x}(t),$
$\displaystyle\delta b(t,\Delta)$
$\displaystyle:=\psi(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t)+\Delta(t),u(t))-b(t),$
$\displaystyle\delta b_{x}(t,\Delta)$
$\displaystyle:=b_{x}(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t)+\Delta(t),u(t))-b_{x}(t).$

In the above, $\Delta(\cdot)$ is an $\mathbb{F}$-adapted process which is the
solution of the following algebra equation:

$\Delta(t)=p^{\top}(t)\delta\sigma(t,\Delta(t)),$ (3.3)

where $p(\cdot)$ is the solution of the following (3.4). The detail can be
found in [17].

We first introduce the first- and second-order decoupling fields, to describe
the relation among $\bar{X}(\cdot)$, $\bar{Y}(\cdot)$ and $\bar{Z}(\cdot)$.
Interested readers may refer to [17] and [25] for more details. In this paper,
for $\Phi(\cdot)\in\mathbb{R}^{d}$, $\Psi(\cdot)\in\mathbb{R}$,
$x\in\mathbb{R}^{n}$ and $y\in\mathbb{R}$, we have the following notions:

$\Phi_{x}=\begin{bmatrix}\Phi^{1}_{x_{1}}&\cdots&\Phi^{1}_{x_{n}}\\\
\vdots&\ddots&\vdots\\\ \Phi^{d}_{x_{1}}&\cdots&\Phi^{d}_{x_{n}}\\\
\end{bmatrix},\ \Phi_{y}=\begin{bmatrix}\Phi^{1}_{y}\\\ \vdots\\\
\Phi^{d}_{y}\\\ \end{bmatrix},\ \Psi_{x}=\begin{bmatrix}\Psi_{x_{1}}\\\
\vdots\\\ \Psi_{x_{n}}\\\ \end{bmatrix},\
\Psi_{xx}=\begin{bmatrix}\Psi_{x_{1}x_{1}}&\cdots&\Psi_{x_{1}x_{n}}\\\
\vdots&\ddots&\vdots\\\ \Psi_{x_{n}x_{1}}&\cdots&\Psi_{x_{n}x_{n}}\\\
\end{bmatrix}.$

We introduce the following first-order decoupling field:

$\left\\{\begin{aligned}
-dp(t)&=\Big{\\{}g_{x}(t)+p(t)g_{y}(t)+K_{1}(t)g_{z}(t)+b^{\top}_{x}(t)p(t)+p(t)b^{\top}_{y}(t)p(t)\\\
&\qquad+K_{1}(t)b^{\top}_{z}(t)p(t)+\sigma^{\top}_{x}(t)q(t)+p(t)\sigma^{\top}_{y}(t)q(t)+K_{1}(t)\sigma^{\top}_{z}(t)q(t)\Big{\\}}\,dt\\\
&\quad-q(t)\,dW(t),\quad t\in[0,T],\\\
p(T)&=\phi_{x}(\bar{X}(T)),\end{aligned}\right.$ (3.4)

which admits a unique solution $(p(\cdot),q(\cdot))\in
L^{2}_{\mathcal{F}}(\Omega;C([0,T];\mathbb{R}^{n}))\times
L^{2}_{\mathcal{F}}([0,T];\mathbb{R}^{n})$, where

$\mathbb{R}^{n}\ni
K_{1}(t):=(1-p^{\top}(t)\sigma_{z}(t))^{-1}\big{[}\sigma^{\top}_{x}(t)p(t)+p(t)\sigma^{\top}_{y}(t)p(t)+q(t)\big{]},$
(3.5)

and the following second-order decoupling field:

$\left\\{\begin{aligned}
-dP(t)&=\Big{\\{}G_{y}(t)P(t)+G_{z}(t)K_{2}(t)+\big{[}b^{\top}_{x}(t)+p(t)b^{\top}_{y}(t)+K_{1}(t)b^{\top}_{z}(t)\big{]}P(t)\\\
&\quad+P(t)\big{[}b_{x}(t)+b_{y}(t)p^{\top}(t)+b_{z}(t)K^{\top}_{1}(t)\big{]}\\\
&\quad+\big{[}\sigma^{\top}_{x}(t)+p(t)\sigma^{\top}_{y}(t)+K_{1}(t)\sigma^{\top}_{z}(t)\big{]}Q(t)\\\
&\quad+Q(t)\big{[}\sigma_{x}(t)+\sigma_{y}(t)p^{\top}(t)+\sigma_{z}(t)K^{\top}_{1}(t)\big{]}\\\
&\quad+\big{[}\sigma^{\top}_{x}(t)+p(t)\sigma^{\top}_{y}(t)+K_{1}(t)\sigma^{\top}_{z}(t)\big{]}P(t)\big{[}\sigma_{x}(t)+\sigma_{y}(t)p^{\top}(t)+\sigma_{z}(t)K^{\top}_{1}(t)\big{]}\\\
&\quad+[I_{n},p(t),K_{1}(t)]D^{2}G(t)[I_{n},p(t),K_{1}(t)]^{\top}\Big{\\}}\,dt-Q(t)\,dW(t),\quad
t\in[0,T],\\\ P(T)&=\phi_{xx}(\bar{X}(T)),\end{aligned}\right.$ (3.6)

which admits a unique solution $(P(\cdot),Q(\cdot))\in
L^{2}_{\mathcal{F}}(\Omega;C([0,T];\mathcal{S}^{n}))\times
L^{2}_{\mathcal{F}}([0,T];\mathcal{S}^{n})$, and

$\displaystyle\mathbb{R}^{n\times n}\ni K_{2}(t)$
$\displaystyle:=(1-p^{\top}(t)\sigma_{z}(t))^{-1}\Big{\\{}p^{\top}(t)\sigma_{y}(t)P(t)+Q(t)+\big{[}\sigma^{\top}_{x}(t)+p(t)\sigma^{\top}_{y}(t)$
(3.7)
$\displaystyle\qquad+K_{1}(t)\sigma^{\top}_{z}(t)\big{]}P(t)+P(t)\big{[}\sigma_{x}(t)+\sigma_{y}(t)p^{\top}(t)+\sigma_{z}(t)K^{\top}_{1}(t)\big{]}$
$\displaystyle\qquad+p^{\top}(t)[I_{n},p,K_{1}]D^{2}\sigma(t)[I_{n},p,K_{1}]^{\top}\Big{\\}},$

where

$G(t,x,y,z,u,p,q):=g(t,x,y,z,u)+p^{\top}b(t,x,y,z,u)+q^{\top}\sigma(t,x,y,z,u).$
(3.8)

Moreover, in this paper, we have the following notions:

$p^{\top}(t)[I_{n},p,K_{1}]D^{2}b(t)[I_{n},p,K_{1}]^{\top}:=\sum_{i=1}^{n}p^{(i)}(t)[I_{n},p,K_{1}]D^{2}b^{(i)}(t)[I_{n},p,K_{1}]^{\top},$

and $[I_{n},p,K_{1}]D^{2}b^{(i)}[I_{n},p,K_{1}]^{\top}$, $i=1,2,\cdots,n,$
denotes the $n\times n$ matrix:

$\displaystyle D^{2}b^{(i)}[I_{n},p,K_{1}]^{\top}$
$\displaystyle:=I_{n}b^{(i)}_{xx}I_{n}+I_{n}b^{(i)}_{xy}p^{\top}+I_{n}b^{(i)}_{xz}K_{1}^{\top}+pb^{(i)}_{yx}I_{n}+pb^{(i)}_{yy}p^{\top}$
(3.9) $\displaystyle\quad\
+pb^{(i)}_{yz}K_{1}^{\top}+K_{1}b^{(i)}_{zx}I_{n}+K_{1}b^{(i)}_{zy}p^{\top}+K_{1}b^{(i)}_{zz}K_{1}^{\top},$

similar as $[I_{n},p,K_{1}]D^{2}\sigma^{(i)}(t)[I_{n},p,K_{1}]^{\top}$ and
$[I_{n},p,K_{1}]D^{2}g(t)[I_{n},p,K_{1}]^{\top}$.

To establish estimations of variations, the following technical assumption is
needed (see [17]).

######  Assumption 3.

The solution $q(\cdot)$ of (3.4) is a bounded process.

Next, we recall $(\lambda(\cdot),\xi(\cdot),\eta(\cdot))\in
L^{2}_{\mathcal{F}}(\Omega;C([0,T];\mathbb{R}^{n}))\times
L^{2}_{\mathcal{F}}(\Omega;C([0,T];\mathbb{R}^{n}))\times
L^{2}_{\mathcal{F}}([0,T];\mathbb{R}^{n})$ and
$(\Gamma(\cdot),\Lambda(\cdot))\in
L^{2}_{\mathcal{F}}(\Omega;C([0,T];\mathcal{S}^{n}))\times
L^{2}_{\mathcal{F}}([0,T];\mathcal{S}^{n})$, which are the solutions of the
following first- and second-order adjoint equations, respectively:

$\left\\{\begin{aligned}
d\lambda(t)&=\Big{\\{}\big{[}\xi^{\top}(t)b_{z}(t)+\eta^{\top}(t)\sigma_{z}(t)\big{]}(1-p^{\top}(t)\sigma_{z}(t))^{-1}p^{\top}(t)\sigma_{y}(t)+\xi^{\top}(t)b_{y}(t)\\\
&\qquad+\eta^{\top}(t)\sigma_{y}(t)+\lambda(t)\big{[}G_{y}(t)+(1-p^{\top}(t)\sigma_{z}(t))^{-1}g_{z}(t)p^{\top}(t)\sigma_{y}(t)\big{]}\Big{\\}}\,dt\\\
&\quad+\Big{\\{}\big{[}\xi^{\top}(t)b_{z}(t)+\eta^{\top}(t)\sigma_{z}(t)\big{]}(1-p^{\top}(t)\sigma_{z}(t))^{-1}p^{\top}(t)\sigma_{z}(t)+\xi^{\top}(t)b_{z}(t)\\\
&\qquad+\eta^{\top}(t)\sigma_{z}(t)+\lambda(t)\big{[}G_{z}(t)+(1-p^{\top}(t)\sigma_{z}(t))^{-1}g_{z}(t)p^{\top}(t)\sigma_{z}(t)\big{]}\Big{\\}}\,dW(t),\\\
-d\xi(t)&=\Big{\\{}\big{[}b^{\top}_{x}(t)+p(t)b^{\top}_{y}(t)+K_{1}(t)b^{\top}_{z}(t)\big{]}\xi(t)+\big{[}\sigma^{\top}_{x}(t)+p(t)\sigma^{\top}_{y}(t)\\\
&\qquad+K_{1}(t)\sigma^{\top}_{z}(t)\big{]}\eta(t)\Big{\\}}\,dt-\eta(t)\,dW(t),\quad
t\in[0,T],\\\
\lambda(0)&=\varphi_{y}(\bar{X}(T),\bar{Y}(0)),\quad\xi(T)=\varphi_{x}(\bar{X}(T),\bar{Y}(0)),\end{aligned}\right.$
(3.10) $\left\\{\begin{aligned}
-d\Gamma(t)&=\Big{\\{}\Gamma(t)\big{[}b_{x}(t)+b_{y}(t)p^{\top}(t)+b_{z}(t)K^{\top}_{1}(t)\big{]}+\big{[}b^{\top}_{x}(t)+p(t)b^{\top}_{y}(t)\\\
&\qquad+K_{1}(t)b^{\top}_{z}(t)\big{]}\Gamma(t)+\big{[}\sigma^{\top}_{x}(t)+p(t)\sigma^{\top}_{y}(t)+K_{1}(t)\sigma^{\top}_{z}(t)\big{]}\Gamma(t)\big{[}\sigma_{x}(t)\\\
&\qquad+\sigma_{y}(t)p^{\top}(t)+\sigma_{z}(t)K^{\top}_{1}(t)\big{]}+\big{[}\sigma^{\top}_{x}(t)+p(t)\sigma^{\top}_{y}(t)+K_{1}(t)\sigma^{\top}_{z}(t)\big{]}\Lambda(t)\\\
&\qquad+\Lambda(t)\big{[}\sigma_{x}(t)+\sigma_{y}(t)p^{\top}(t)+\sigma_{z}(t)K_{1}^{\top}(t)\big{]}+\xi^{\top}(t)\big{[}b_{y}(t)P(t)\\\
&\qquad+b_{z}(t)K_{2}(t)+[I_{n},p(t),K_{1}(t)]D^{2}b(t)[I_{n},p(t),K_{1}(t)]^{\top}\big{]}+\eta^{\top}(t)\big{[}\sigma_{y}(t)P(t)\\\
&\qquad+\sigma_{z}(t)K_{2}(t)+[I_{n},p(t),K_{1}(t)]D^{2}\sigma(t)[I_{n},p(t),K_{1}(t)]^{\top}\big{]}\Big{\\}}\,dt\\\
&\quad-\Lambda(t)\,dW(t),\quad t\in[0,T],\\\
\Gamma(T)&=\varphi_{xx}(\bar{X}(T),\bar{Y}(0)).\end{aligned}\right.$ (3.11)

######  Remark 3.1.

[17] first established (3.4) and (3.6) in one-dimensional form to decouple the
first- and second-order variational equations. The above high-dimension form
is given in [25]. Under some mild conditions, the unique solvability of (3.3)
and (3.4) had been proved in Theorem 3.6 and Lemma 3.9 of [17]. (3.10) is a
non-coupled FBSDE and Assumption 2 guarantees its unique solvability. (3.6) is
a linear BSDE with uniformly Lipschitz coefficients, which has a unique
solution.

We define the risk-neutral Hamiltonian function
$\mathcal{H}^{RN}:[0,T]\times\mathbb{R}^{n}\times\mathbb{R}\times\mathbb{R}\times
U\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathcal{S}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathcal{S}^{n}\to\mathbb{R}$:

$\displaystyle\mathcal{H}$
$\displaystyle{}^{RN}(t,x,y,z,u,p,q,P,\lambda,\xi,\eta,\Gamma)$ (3.12)
$\displaystyle:=\big{(}\xi^{\top}+\lambda
p^{\top}\big{)}b(t,x,y,z+\Delta(t),u)+\big{(}\eta^{\top}+\lambda
q^{\top}\big{)}\sigma(t,x,y,z+\Delta(t),u)$ $\displaystyle\quad+\lambda
g(t,x,y,z+\Delta(t),u)+\frac{1}{2}\big{[}\sigma(t,x,y,z+\Delta(t),u)-\sigma(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),\bar{u}(t))\big{]}^{\top}$
$\displaystyle\quad\times(\Gamma+\lambda
P)\big{[}\sigma(t,x,y,z+\Delta(t),u)-\sigma(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),\bar{u}(t))\big{]},$

where $\Delta(\cdot)$ is the solution of (3.3) corresponding to $u(t)=u$.

The following lemma is from [25].

######  Lemma 3.1.

Suppose Assumption 1, Assumption 2 and Assumption 3 hold. Let (3.4) and (3.3)
have unique solutions $(p(\cdot),q(\cdot))\in
L^{2}_{\mathcal{F}}(\Omega;C([0,T],\mathbb{R}))\times
L^{2}_{\mathcal{F}}([0,T];\mathbb{R})$ and $\Delta(\cdot)\in
L^{2}_{\mathcal{F}}([0,T];\mathbb{R}^{n})$ respectively. Let
$\bar{u}(\cdot)\in\mathcal{U}[0,T]$ be optimal and
$(\bar{X}(\cdot),\bar{Y}(\cdot),\bar{Z}(\cdot))$ be the corresponding state
processes of (2.1). Then

$\displaystyle\mathcal{H}^{RN}\big{(}t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),u,p(t),q(t),P(t),\lambda(t),\xi(t),\eta(t),\Gamma(t)\big{)}$
(3.13)
$\displaystyle\quad\geq\mathcal{H}^{RN}\big{(}t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),\bar{u}(t),p(t),q(t),P(t),\lambda(t),\xi(t),\eta(t),\Gamma(t)\big{)},$
$\displaystyle\hskip 142.26378pt\forall u\in U,\quad
a.e.,\quad\mathbb{P}\mbox{-}a.s.,$

where $\Delta(\cdot)$ satisfies (3.3), and $(P(\cdot),Q(\cdot))$,
$(\lambda(\cdot),\xi(\cdot),\eta(\cdot))$, $(\Gamma(\cdot),\Lambda(\cdot))$
are solutions of (3.6), (3.10) and (3.11) respectively.

###  3.2 Risk-sensitive global maximum principle

In this section, we give the risk-sensitive global maximum principle, which is
the main result of this paper. The proof will be given in the next section.

Consider the state equation (2.1) combined with the risk-sensitive cost
functional (2.2). The first-order decoupling field is given as:

$\left\\{\begin{aligned}
-dp(t)&=\Big{\\{}g_{x}(t)+p(t)g_{y}(t)+K_{1}(t)g_{z}(t)+b^{\top}_{x}(t)p(t)+p(t)b^{\top}_{y}(t)p(t)\\\
&\qquad+K_{1}(t)b^{\top}_{z}(t)p(t)+\sigma^{\top}_{x}(t)q(t)+p(t)\sigma^{\top}_{y}(t)q(t)+K_{1}(t)\sigma^{\top}_{z}(t)q(t)\Big{\\}}\,dt\\\
&\quad-q(t)\,dW(t),\quad t\in[0,T],\\\
p(T)&=\phi_{x}(\bar{X}(T)),\end{aligned}\right.$ (3.14)

which admits a unique solution $(p(\cdot),q(\cdot))\in
L^{2}_{\mathcal{F}}(\Omega;C([0,T];\mathbb{R}^{n}))\times
L^{2}_{\mathcal{F}}([0,T];\mathbb{R}^{n})$, and $K_{1}(\cdot)$ is the process
defined in (3.5). The second-order risk-sensitive decoupling field is:

$\left\\{\begin{aligned}
-dP(t)&=\Big{\\{}G_{y}(t)P(t)+G_{z}(t)K_{2}(t)+\big{[}b^{\top}_{x}(t)+p(t)b^{\top}_{y}(t)+K_{1}(t)b^{\top}_{z}(t)\big{]}P(t)\\\
&\quad\
+P(t)\big{[}b_{x}(t)+b_{y}(t)p^{\top}(t)+b_{z}(t)K^{\top}_{1}(t)\big{]}+\big{[}\sigma^{\top}_{x}(t)+p(t)\sigma^{\top}_{y}(t)\\\
&\quad\
+K_{1}(t)\sigma^{\top}_{z}(t)\big{]}Q(t)+Q(t)\big{[}\sigma_{x}(t)+\sigma_{y}(t)p^{\top}(t)+\sigma_{z}(t)K^{\top}_{1}(t)\big{]}+\big{[}\sigma^{\top}_{x}(t)\\\
&\quad\
+p(t)\sigma^{\top}_{y}(t)+K_{1}(t)\sigma^{\top}_{z}(t)\big{]}P(t)\big{[}\sigma_{x}(t)+\sigma_{y}(t)p^{\top}(t)+\sigma_{z}(t)K^{\top}_{1}(t)\big{]}\\\
&\quad\
+[I_{n},p(t),K_{1}(t)]D^{2}G(t)[I_{n},p(t),K_{1}(t)]^{\top}\Big{\\}}\,dt-Q(t)\,dW(t),\quad
t\in[0,T],\\\ P(T)&=\phi_{xx}(\bar{X}(T)),\end{aligned}\right.$ (3.15)

which admits unique solution $(P(\cdot),Q(\cdot))\in
L^{2}_{\mathcal{F}}([0,T];\mathcal{S}^{n})\times
L^{2}_{\mathcal{F}}([0,T];\mathcal{S}^{n})$. Further, $K_{2}(\cdot)$ and
$G(\cdot,x,y,z,u,p,q)$ are given in (3.7) and (3.8) respectively.

Similar as the risk-neutral case, we introduce the first-order risk-sensitive
adjoint processes $(k(\cdot),m(\cdot),n(\cdot))\in
L^{2}_{\mathcal{F}}(\Omega;C([0,T];\mathbb{R}^{n}))\times
L^{2}_{\mathcal{F}}(\Omega;C([0,T];\mathbb{R}^{n}))\times
L^{2}_{\mathcal{F}}([0,T];\mathbb{R}^{n})$, which is the unique solution of
the following non-coupled FBSDE:

$\left\\{\begin{aligned}
dk(t)&=\Big{\\{}\big{[}m^{\top}(t)b_{z}(t)+l_{z}(t)+(n^{\top}(t)+\theta\kappa(t)m^{\top}(t))\sigma_{z}(t)\big{]}(1-p^{\top}(t)\sigma_{z}(t))^{-1}\\\
&\qquad\times(p^{\top}(t)\sigma_{y}(t)-\theta\kappa(t))+m^{\top}(t)b_{y}(t)+l_{y}(t)+(n^{\top}(t)+\theta\kappa(t)m^{\top}(t))\sigma_{y}(t)\\\
&\qquad+k(t)\big{[}g_{y}(t)+p^{\top}(t)b_{y}(t)+l_{y}(t)+q^{\top}(t)\sigma_{y}(t)+(1-p^{\top}(t)\sigma_{z}(t))^{-1}\\\
&\qquad\times
g_{z}(t)p^{\top}(t)\sigma_{y}(t)\big{]}+\theta^{2}\kappa^{2}(t)k(t)-\theta\kappa(t)k(t)\big{[}g_{z}(t)+p^{\top}(t)b_{z}(t)+l_{z}(t)\\\
&\qquad+q^{\top}(t)\sigma_{z}(t)+(1-p^{\top}(t)\sigma_{z}(t))^{-1}g_{z}(t)p^{\top}(t)\sigma_{z}(t)\big{]}\Big{\\}}\,dt\\\
&\quad+\Big{\\{}\big{[}m^{\top}(t)b_{z}(t)+l_{z}(t)+(n^{\top}(t)+\theta\kappa(t)m^{\top}(t))\sigma_{z}(t)\big{]}(1-p^{\top}(t)\sigma_{z}(t))^{-1}\\\
&\qquad\
+k(t)\big{[}g_{z}(t)+p^{\top}(t)b_{z}(t)+l_{z}(t)+q^{\top}(t)\sigma_{z}(t)+(1-p^{\top}(t)\sigma_{z}(t))^{-1}\\\
&\qquad\ \times
g_{z}(t)p^{\top}(t)\sigma_{z}(t)\big{]}-\theta\kappa(t)k(t)\Big{\\}}\,dW(t),\\\
\end{aligned}\right.$ $\left\\{\begin{aligned}
-dm(t)&=\Big{\\{}\big{[}b^{\top}_{x}(t)+p(t)b^{\top}_{y}(t)+K_{1}(t)b^{\top}_{z}(t)\big{]}m(t)+l^{\top}_{x}(t)+p(t)l^{\top}_{y}(t)\\\
&\quad\
+K_{1}(t)l^{\top}_{z}(t)+\big{[}\sigma^{\top}_{x}(t)+p(t)\sigma^{\top}_{y}(t)+K_{1}(t)\sigma^{\top}_{z}(t)\big{]}(\theta\kappa(t)m(t)+n(t))\\\
&\quad\ +\theta\kappa(t)n(t)\Big{\\}}\,dt-n(t)\,dW(t),\quad t\in[0,T],\\\
k(0)&=f_{y}(\bar{X}(T),\bar{Y}(0)),\quad
m(T)=f_{x}(\bar{X}(T),\bar{Y}(0)),\end{aligned}\right.$ (3.16)

where $(\gamma(\cdot),\kappa(\cdot))\in\mathbb{R}\times\mathbb{R}$ is the
unique solution of (4.10), a BSDE with quadratic generator. The unique
solvability of (3.16) is discussed in Remark 4.1.

Next, we introduce the second-order risk-sensitive adjoint processes
$(M(\cdot),N(\cdot))\in L^{2}_{\mathcal{F}}([0,T];\\\ \mathcal{S}^{n})\times
L^{2}_{\mathcal{F}}([0,T];\mathcal{S}^{n})$, which satisfies the following
matrix-valued linear BSDE:

$\left\\{\begin{aligned}
-dM(t)&=\Big{\\{}M(t)\big{[}b_{x}(t)+b_{y}(t)p^{\top}(t)+b_{z}(t)K^{\top}_{1}(t)\big{]}\\\
&\qquad+\big{[}b^{\top}_{x}(t)+p(t)b^{\top}_{y}(t)+K_{1}(t)b^{\top}_{z}(t)\big{]}M(t)\\\
&\qquad+\big{[}\sigma^{\top}_{x}(t)+p(t)\sigma^{\top}_{y}(t)+K_{1}(t)\sigma^{\top}_{z}(t)\big{]}\big{[}M(t)+\theta
m(t)m^{\top}(t)\big{]}\\\
&\qquad\times\big{[}\sigma_{x}(t)+\sigma_{y}(t)p^{\top}(t)+\sigma_{z}(t)K^{\top}_{1}(t)\big{]}\\\
&\qquad+\big{[}\sigma^{\top}_{x}(t)+p(t)\sigma^{\top}_{y}(t)+K_{1}(t)\sigma^{\top}_{z}(t)\big{]}\big{[}N(t)+\theta\kappa(t)M(t)+\theta
m(t)n^{\top}(t)\big{]}\\\ &\qquad+\big{[}N(t)+\theta\kappa(t)M(t)+\theta
n(t)m^{\top}(t)\big{]}\big{[}\sigma_{x}(t)+\sigma_{y}(t)p^{\top}(t)+\sigma_{z}(t)K_{1}^{\top}(t)\big{]}\\\
&\qquad+\big{[}m^{\top}(t)b_{y}(t)+l_{y}(t)\big{]}P(t)+\big{[}m^{\top}(t)b_{z}(t)+l_{z}(t)\big{]}K_{2}(t)+\theta\kappa(t)N(t)\\\
&\qquad-\theta
n(t)n^{\top}(t)+m^{\top}(t)[I_{n},p(t),K_{1}(t)]D^{2}b(t)[I_{n},p(t),K_{1}(t)]^{\top}\\\
&\qquad+[I_{n},p(t),K_{1}(t)]D^{2}l(t)[I_{n},p(t),K_{1}(t)]^{\top}\\\
&\qquad+\big{[}n^{\top}(t)+\theta\kappa(t)m^{\top}(t)\big{]}[I_{n},p(t),K_{1}(t)]D^{2}\sigma(t)[I_{n},p(t),K_{1}(t)]^{\top}\Big{\\}}\,dt\\\
&\quad-N(t)\,dW(t),\quad t\in[0,T],\\\
M(T)&=f_{xx}(\bar{X}(T),\bar{Y}(0)).\end{aligned}\right.$ (3.17)

Define the risk-sensitive Hamiltonian
$\mathcal{H}^{RS}:[0,T]\times\mathbb{R}^{n}\times\mathbb{R}\times\mathbb{R}\times
U\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathcal{S}^{n}\times\mathcal{S}^{n}\times\mathbb{R}\to\mathbb{R}$:

$\displaystyle\mathcal{H}_{\theta}^{RS}(t,x,y,z,u,p,q,k,m,n,P,M,\kappa)$
(3.18)
$\displaystyle:=\big{(}m^{\top}+kp^{\top}\big{)}b(t)+l(t)+\big{(}n^{\top}+\theta\kappa
m^{\top}+kq^{\top}\big{)}\sigma(t)+kg(t)$
$\displaystyle\qquad+\frac{1}{2}\big{(}\sigma(t)-\sigma(t,\bar{X},\bar{Y},\bar{Z},\bar{u})\big{)}^{\top}\big{(}M+\theta
mm^{\top}+kP\big{)}\big{(}\sigma(t)-\sigma(t,\bar{X},\bar{Y},\bar{Z},\bar{u})\big{)},$

where $b(t)\equiv b(t,x,y,z+\Delta(t),u)$ for simplification, and $\sigma(t)$,
$g(t)$ and $l(t)$ are the same. $\Delta(\cdot)$ is the solution of (3.3)
corresponding to $u(t)=u$.

Now we give the risk-sensitive global maximum principle, whose proof will be
given in the next section.

######  Theorem 3.1.

Suppose Assumption 1, Assumption 2 and Assumption 3 hold. Let (3.14) and (3.3)
have unique solutions $(p(\cdot),q(\cdot))\in
L^{2}_{\mathcal{F}}(\Omega;C([0,T],\mathbb{R}))\times
L^{2}_{\mathcal{F}}([0,T];\mathbb{R})$ and $\Delta(\cdot)\in
L^{2}_{\mathcal{F}}([0,T];\mathbb{R}^{n})$ respectively. Let
$\bar{u}(\cdot)\in\mathcal{U}[0,T]$ be optimal and
$(\bar{X}(\cdot),\bar{Y}(\cdot),\bar{Z}(\cdot))$ be the corresponding state
processes of (2.1). Then

$\displaystyle\theta\,\mathcal{H}_{\theta}^{RS}\big{(}t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),u,p(t),q(t),k(t),m(t),n(t),P(t),M(t),\kappa(t)\big{)}$
(3.19)
$\displaystyle\geq\theta\,\mathcal{H}_{\theta}^{RS}\big{(}t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),\bar{u}(t),p(t),q(t),k(t),m(t),n(t),P(t),M(t),\kappa(t)\big{)},$
$\displaystyle\hskip 170.71652pt\forall u\in U,\quad
a.e.,\quad\mathbb{P}\mbox{-}a.s.,$

where $(P(\cdot),Q(\cdot))$, $(k(\cdot),m(\cdot),n(\cdot))$,
$(M(\cdot),N(\cdot))$ and $\kappa(\cdot)$ are solutions of (3.14), (3.15),
(3.16), (3.17) and (4.10) respectively.

##  4 Proof of the main result

The exponential-of-integral in the cost functional (2.2) makes the risk-
sensitive problem different from the risk-neutral one. Hence, the risk-neutral
global maximum principle can not be used directly, to give the necessary
conditions of the optimal control. Inspired by Lim and Zhou [24], we
reformulate the risk-sensitive stochastic optimal control problem (2.3) to the
risk-neutral form with the help of a one-dimensional auxiliary process
$\tilde{X}(\cdot)$, which is the solution of the following equation:

$\left\\{\begin{aligned} d\tilde{X}(t)&=l(t,X(t),Y(t),Z(t),u(t))\,dt,\quad
t\in[0,T],\\\ \tilde{X}(0)&=0,\end{aligned}\right.$

where $l$ is defined in (2.2). Set

$\displaystyle\mathbbm{X}(t):=\begin{bmatrix}X(t)\\\
\tilde{X}(t)\end{bmatrix},\quad\mathbbm{B}(t):=\begin{bmatrix}b(t)\\\
l(t)\end{bmatrix},\quad\Sigma(t):=\begin{bmatrix}\sigma(t)\\\ 0\end{bmatrix},$

in which we omit $\mathbbm{X}(t)$, $Y(t)$, $Z(t)$ and $u(t)$ in the above maps
for simplify, for $t\in[0,T]$.

Thus we can reformulate the problem (2.3) as:

$\left\\{\begin{aligned}
d\mathbbm{X}(t)&=\mathbbm{B}(t,\mathbbm{X}(t),Y(t),Z(t),u(t))\,dt+\Sigma(t,\mathbbm{X}(t),Y(t),Z(t),u(t))\,dW(t),\\\
-dY(t)&=g(t,X(t),Y(t),Z(t),u(t))\,dt-Z(t)\,dW(t),\quad t\in[0,T],\\\
\mathbbm{X}(0)&=\begin{bmatrix}x^{\top}_{0}&0\end{bmatrix}^{\top}\in\mathbb{R}^{n+1},\quad
Y(T)=\phi(X(T)),\end{aligned}\right.$ (4.1)

with the cost functional:

$J^{RS}(u(\cdot))=\mathbb{E}\big{[}e^{\theta[\,\tilde{X}(T)+f(X(T),Y(0))\,]}\big{]}.$
(4.2)

###  4.1 Applying risk-neutral global maximum principle

Since (4.2) only has the terminal term, the results in Section 3.1 can be
applied now. For the new state equation (4.1) and cost functional (4.2), we
recall (3.4) as:

$\left\\{\begin{aligned}
-d\mathbbm{p}(t)&=\Big{\\{}g_{\mathbbm{x}}(t)+\mathbbm{p}(t)g_{y}(t)+\mathbbm{K}_{1}(t)g_{z}(t)+\mathbbm{B}^{\top}_{\mathbbm{x}}(t)\mathbbm{p}(t)+\mathbbm{p}(t)\mathbbm{B}^{\top}_{y}(t)\mathbbm{p}(t)\\\
&\qquad+\mathbbm{K}_{1}(t)\mathbbm{B}^{\top}_{z}(t)\mathbbm{p}(t)+\Sigma^{\top}_{x}(t)\mathbbm{q}(t)+\mathbbm{p}(t)\Sigma^{\top}_{y}(t)\mathbbm{q}(t)+\mathbbm{K}_{1}(t)\Sigma^{\top}_{z}(t)\mathbbm{q}(t)\Big{\\}}\,dt\\\
&\quad-\mathbbm{q}(t)\,dW(t),\quad t\in[0,T],\\\
\mathbbm{p}(T)&=\begin{bmatrix}\phi_{x}(\bar{X}(T))\\\
0\end{bmatrix},\end{aligned}\right.$ (4.3)

where the $\mathbb{R}^{n+1}$-valued process

$\mathbbm{K}_{1}(t):=(1-\mathbbm{p}^{\top}(t)\Sigma_{z}(t))^{-1}\big{[}\Sigma^{\top}_{\mathbbm{x}}(t)\mathbbm{p}(t)+\mathbbm{p}(t)\Sigma^{\top}_{y}(t)\mathbbm{p}(t)+\mathbbm{q}(t)\big{]}.$

Set

$\mathbbm{p}:=\begin{bmatrix}p\\\
\tilde{p}\end{bmatrix},\quad\mathbbm{q}:=\begin{bmatrix}q\\\
\tilde{q}\end{bmatrix},\quad\mathbbm{K}_{1}:=\begin{bmatrix}K_{1}\\\
\tilde{K}_{1}\end{bmatrix}.$

We can deduce that (4.3) has a unique solution $(p(\cdot),q(\cdot))\in
L^{2}_{\mathcal{F}}(\Omega;C([0,T];\mathbb{R}^{n}))\times
L^{2}_{\mathcal{F}}([0,T];\mathbb{R}^{n})$ and
$(\tilde{p}(\cdot),\tilde{q}(\cdot))\in
L^{2}_{\mathcal{F}}(\Omega;C([0,T];\mathbb{R}))\times
L^{2}_{\mathcal{F}}([0,T];\mathbb{R})$, and $K_{1}(\cdot)$ and
$\tilde{K}_{1}(\cdot)$ are processes value in $\mathbb{R}^{n}$ and
$\mathbb{R}$ respectively.

Recall (3.6) as the following:

$\left\\{\begin{aligned}
-d\mathbbm{P}(t)&=\Big{\\{}G_{y}(t)\mathbbm{P}(t)+G_{z}(t)\mathbbm{K}_{2}(t)+\big{[}\mathbbm{B}^{\top}_{\mathbbm{x}}(t)+\mathbbm{p}(t)\mathbbm{B}^{\top}_{y}(t)+\mathbbm{K}_{1}(t)\mathbbm{B}^{\top}_{z}(t)\big{]}\mathbbm{P}(t)\\\
&\qquad+\mathbbm{P}(t)\big{[}\mathbbm{B}_{\mathbbm{x}}(t)+\mathbbm{B}_{y}(t)\mathbbm{p}^{\top}(t)+\mathbbm{B}_{z}(t)\mathbbm{K}^{\top}_{1}(t)\big{]}\\\
&\qquad+\big{[}\Sigma^{\top}_{\mathbbm{x}}(t)+\mathbbm{p}(t)\Sigma^{\top}_{y}(t)+\mathbbm{K}_{1}(t)\Sigma^{\top}_{z}(t)\big{]}\mathbbm{Q}(t)\\\
&\qquad+\mathbbm{Q}(t)\big{[}\Sigma_{\mathbbm{x}}(t)+\Sigma_{y}(t)\mathbbm{p}^{\top}(t)+\Sigma_{z}(t)\mathbbm{K}^{\top}_{1}(t)\big{]}\\\
&\qquad+\big{[}\Sigma^{\top}_{\mathbbm{x}}(t)+\mathbbm{p}(t)\Sigma^{\top}_{y}(t)+\mathbbm{K}_{1}(t)\Sigma^{\top}_{z}(t)\big{]}\mathbbm{P}(t)\big{[}\Sigma_{\mathbbm{x}}(t)+\Sigma_{y}(t)\mathbbm{p}^{\top}(t)\\\
&\qquad+\Sigma_{z}(t)\mathbbm{K}^{\top}_{1}(t)\big{]}+[I_{n+1},\mathbbm{p}(t),\mathbbm{K}_{1}(t)]D^{2}\mathbbm{G}(t)[I_{n+1},\mathbbm{p}(t),\mathbbm{K}_{1}(t)]^{\top}\Big{\\}}\,dt\\\
&\quad-\mathbbm{Q}(t)\,dW(t),\quad t\in[0,T],\\\
\mathbbm{P}(T)&=\begin{bmatrix}\phi_{xx}(\bar{X}(T))&0_{n}\\\
0_{n}^{\top}&0\end{bmatrix},\end{aligned}\right.$ (4.4)

where

$\mathbbm{G}(t,\mathbbm{x},y,z,u,\mathbbm{p},\mathbbm{q}):=g(t,\mathbbm{x},y,z,u)+\mathbbm{p}^{\top}\mathbbm{B}(t,\mathbbm{x},y,z,u)+\mathbbm{q}^{\top}\Sigma(t,\mathbbm{x},y,z,u),$

and the $\mathcal{S}^{n+1}$-valued process

$\displaystyle\mathbbm{K}_{2}(t)$
$\displaystyle:=(1-\mathbbm{p}^{\top}(t)\Sigma_{z}(t))^{-1}\Big{\\{}\mathbbm{p}^{\top}(t)\Sigma_{y}(t)\mathbbm{P}(t)+\mathbbm{Q}(t)+\big{[}\Sigma^{\top}_{\mathbbm{x}}+\mathbbm{p}\Sigma^{\top}_{y}+\mathbbm{K}_{1}\Sigma^{\top}_{z}\big{]}\mathbbm{P}(t)$
$\displaystyle\qquad+\mathbbm{P}(t)\big{[}\Sigma_{\mathbbm{x}}+\Sigma_{y}\mathbbm{p}^{\top}+\Sigma_{z}\mathbbm{K}^{\top}_{1}\big{]}+\mathbbm{p}^{\top}(t)[I_{n+1},\mathbbm{p},\mathbbm{K}_{1}]D^{2}\Sigma(t)[I_{n+1},\mathbbm{p},\mathbbm{K}_{1}]^{\top}\Big{\\}}.$

We know that (4.4) admits a unique solution
$(\mathbbm{P}(\cdot),\mathbbm{Q}(\cdot))\in
L^{2}_{\mathcal{F}}(\Omega;C([0,T];\mathcal{S}^{n+1}))\times
L^{2}_{\mathcal{F}}([0,T];\\\ \mathcal{S}^{n+1})$. Set

$\mathbbm{P}:=\begin{bmatrix}P&P_{2}\\\
P_{2}^{\top}&P_{4}\end{bmatrix},\quad\mathbbm{Q}:=\begin{bmatrix}Q&Q_{2}\\\
Q_{2}^{\top}&Q_{4}\end{bmatrix},\quad\mathbbm{K}_{2}:=\begin{bmatrix}K_{2}&\tilde{K}_{2}\\\
\tilde{K}_{2}^{\top}&\tilde{\tilde{K}}_{2}\end{bmatrix},$

where $P(\cdot)$, $Q(\cdot)$ and $K_{2}(\cdot)$ are $n$-dimensional matrix-
valued processes.

After some calculation, we can conclude that:

$\left\\{\begin{aligned}
-d\tilde{p}(t)&=\Big{\\{}\tilde{p}(t)g_{y}(t)+\tilde{K}_{1}(t)g_{z}(t)+\tilde{p}(t)\big{[}b^{\top}_{y}(t)p(t)+l_{y}(t)\tilde{p}(t)\big{]}\\\
&\qquad+\tilde{p}(t)\sigma^{\top}_{y}(t)q(t)+\tilde{K}_{1}(t)\sigma_{z}^{\top}(t)q(t)\Big{\\}}\,dt-\tilde{q}(t)\,dW(t),\quad
t\in[0,T],\\\ \tilde{p}(T)&=0,\end{aligned}\right.$ (4.5)

with

$\tilde{K}_{1}(t)=(1-\mathbbm{p}^{\top}(t)\Sigma_{z}(t))^{-1}\big{[}\tilde{p}(t)\sigma^{\top}_{y}(t)p(t)+\tilde{q}(t)\big{]}.$

Combining with the unique solvability of BSDE (4.5), we can arrive at
$(\tilde{p}(\cdot),\tilde{q}(\cdot))=(0,0)$, $\tilde{K}_{1}(\cdot)=0$. Using
the similar method, we also drive that $(P_{2}(\cdot),Q_{2}(\cdot))=(0,0)$,
$(P_{4}(\cdot),Q_{4}(\cdot))=(0,0)$, $\tilde{K}_{2}(\cdot)=0$ and
$\tilde{\tilde{K}}_{2}(\cdot)=0$.

Thus, from (4.3) and (4.4), we have:

$\left\\{\begin{aligned}
-dp(t)&=\Big{\\{}g_{x}(t)+p(t)g_{y}(t)+K_{1}(t)g_{z}(t)+b^{\top}_{x}(t)p(t)+p(t)b^{\top}_{y}(t)p(t)\\\
&\qquad+K_{1}(t)b^{\top}_{z}(t)p(t)+\sigma^{\top}_{x}(t)q(t)+p(t)\sigma^{\top}_{y}(t)q(t)+K_{1}(t)\sigma^{\top}_{z}(t)q(t)\Big{\\}}\,dt\\\
&\quad-q(t)\,dW(t),\quad t\in[0,T],\\\
p(T)&=\phi_{x}(\bar{X}(T)),\end{aligned}\right.$

and

$\left\\{\begin{aligned}
-dP(t)&=\Big{\\{}G_{y}(t)P(t)+G_{z}(t)K_{2}(t)+\big{[}b^{\top}_{x}(t)+p(t)b^{\top}_{y}(t)+K_{1}(t)b^{\top}_{z}(t)\big{]}P(t)\\\
&\qquad+P(t)\big{[}b_{x}(t)+b_{y}(t)p^{\top}(t)+b_{z}(t)K^{\top}_{1}(t)\big{]}\\\
&\qquad+\big{[}\sigma^{\top}_{x}(t)+p(t)\sigma^{\top}_{y}(t)+K_{1}(t)\sigma^{\top}_{z}(t)\big{]}Q(t)\\\
&\qquad+Q(t)\big{[}\sigma_{x}(t)+\sigma_{y}(t)p^{\top}(t)+\sigma_{z}(t)K^{\top}_{1}(t)\big{]}\\\
&\qquad+\big{[}\sigma^{\top}_{x}(t)+p(t)\sigma^{\top}_{y}(t)+K_{1}(t)\sigma^{\top}_{z}(t)\big{]}P(t)\big{[}\sigma_{x}(t)+\sigma_{y}(t)p^{\top}(t)+\sigma_{z}(t)K^{\top}_{1}(t)\big{]}\\\
&\qquad+[I_{n},p(t),K_{1}(t)]D^{2}G(t)[I_{n},p(t),K_{1}(t)]^{\top}\Big{\\}}\,dt-Q(t)\,dW(t),\quad
t\in[0,T],\\\ P(T)&=\phi_{xx}(\bar{X}(T)),\end{aligned}\right.$

which coincide with (3.14) and (3.15) respectively.

Next, we set

$\tilde{\mathbbm{m}}:=\begin{bmatrix}\tilde{m}_{n}\\\
\tilde{m}_{2}\end{bmatrix}_{n+1},\quad\tilde{\mathbbm{n}}:=\begin{bmatrix}\tilde{n}_{n}\\\
\tilde{n}_{2}\end{bmatrix}_{n+1},\quad\tilde{\mathbbm{M}}:=\begin{bmatrix}\tilde{M}_{n\times
n}&\tilde{M}_{2}\\\
\tilde{M}^{\top}_{2}&\tilde{M}_{4}\end{bmatrix}_{(n+1)^{2}},\quad\tilde{\mathbbm{N}}:=\begin{bmatrix}\tilde{N}_{n\times
n}&\tilde{N}_{2}\\\
\tilde{N}^{\top}_{2}&\tilde{N}_{4}\end{bmatrix}_{(n+1)^{2}}.$

In the above, we use the subscript $n$, $n\times n$ and so on, to denote the
dimensions of related matrix-valued processes.

Recall (3.10) as:

$\left\\{\begin{aligned}
d\tilde{k}(t)&=\Big{\\{}\big{[}\tilde{\mathbbm{m}}^{\top}(t)\mathbbm{B}_{z}(t)+\tilde{\mathbbm{n}}^{\top}(t)\Sigma_{z}(t)\big{]}(1-\mathbbm{p}^{\top}(t)\Sigma_{z}(t))^{-1}\mathbbm{p}^{\top}(t)\Sigma_{y}(t)+\tilde{\mathbbm{m}}^{\top}(t)\mathbbm{B}_{y}(t)\\\
&\qquad+\tilde{\mathbbm{n}}^{\top}(t)\Sigma_{y}(t)+\tilde{k}(t)\big{[}\mathbbm{G}_{y}(t)+(1-\mathbbm{p}^{\top}(t)\Sigma_{z}(t))^{-1}g_{z}(t)\mathbbm{p}^{\top}(t)\Sigma_{y}(t)\big{]}\Big{\\}}\,dt\\\
&\quad+\Big{\\{}\big{[}\tilde{\mathbbm{m}}^{\top}(t)\mathbbm{B}_{z}(t)+\tilde{\mathbbm{n}}^{\top}(t)\Sigma_{z}(t)\big{]}(1-\mathbbm{p}^{\top}(t)\Sigma_{z}(t))^{-1}\mathbbm{p}^{\top}(t)\Sigma_{z}(t)\\\
&\qquad\
+\tilde{\mathbbm{m}}^{\top}(t)\mathbbm{B}_{z}(t)+\tilde{\mathbbm{n}}^{\top}(t)\Sigma_{z}(t)\\\
&\qquad\
+\tilde{k}(t)\big{[}\mathbbm{G}_{z}(t)+(1-\mathbbm{p}^{\top}(t)\Sigma_{z}(t))^{-1}g_{z}(t)\mathbbm{p}^{\top}(t)\Sigma_{z}(t)\big{]}\Big{\\}}\,dW(t),\\\
-d\tilde{\mathbbm{m}}(t)&=\Big{\\{}\big{[}\mathbbm{B}^{\top}_{\mathbbm{x}}(t)+\mathbbm{p}(t)\mathbbm{B}^{\top}_{y}(t)+\mathbbm{K}_{1}(t)\mathbbm{B}^{\top}_{z}(t)\big{]}\tilde{\mathbbm{m}}(t)\\\
&\qquad+\big{[}\Sigma^{\top}_{\mathbbm{x}}(t)+\mathbbm{p}(t)\Sigma^{\top}_{y}(t)+\mathbbm{K}_{1}(t)\Sigma^{\top}_{z}(t)\big{]}\tilde{\mathbbm{n}}(t)\Big{\\}}\,dt-\tilde{\mathbbm{n}}(t)\,dW(t),\quad
t\in[0,T],\\\ \tilde{k}(0)&=\theta
e^{\theta[\,\bar{\tilde{X}}(T)+f(\bar{X}(T),\bar{Y}(0))\,]}f_{y}(\bar{X}(T),\bar{Y}(0)),\\\
\tilde{\mathbbm{m}}(T)&=\theta
e^{\theta[\,\bar{\tilde{X}}(T)+f(\bar{X}(T),\bar{Y}(0))\,]}\begin{bmatrix}f_{x}(\bar{X}(T),\bar{Y}(0))\\\
1\end{bmatrix},\\\ \end{aligned}\right.$ (4.6)

which admits a unique solution
$(\tilde{k}(\cdot),\tilde{\mathbbm{m}}(\cdot),\tilde{\mathbbm{n}}(\cdot))\in
L^{2}_{\mathcal{F}}(\Omega;C([0,T];\mathbb{R}^{n+1}))\times
L^{2}_{\mathcal{F}}(\Omega;C([0,T];\\\ \mathbb{R}^{n+1}))\times
L^{2}_{\mathcal{F}}([0,T];\mathbb{R}^{n+1})$, and recall (3.11) as:

$\left\\{\begin{aligned}
-d\tilde{\mathbbm{M}}(t)&=\Big{\\{}\tilde{\mathbbm{M}}(t)\big{[}\mathbbm{B}_{\mathbbm{x}}(t)+\mathbbm{B}_{y}(t)\mathbbm{p}^{\top}(t)+\mathbbm{B}_{z}(t)\mathbbm{K}^{\top}_{1}(t)\big{]}\\\
&\qquad+\big{[}\mathbbm{B}^{\top}_{\mathbbm{x}}(t)+\mathbbm{p}(t)\mathbbm{B}^{\top}_{y}(t)+\mathbbm{K}_{1}(t)\mathbbm{B}^{\top}_{z}(t)\big{]}\tilde{\mathbbm{M}}(t)\\\
&\qquad+\big{[}\Sigma^{\top}_{\mathbbm{x}}(t)+\mathbbm{p}(t)\Sigma^{\top}_{y}(t)+\mathbbm{K}_{1}(t)\Sigma^{\top}_{z}(t)\big{]}\tilde{\mathbbm{M}}(t)\big{[}\Sigma_{\mathbbm{x}}(t)+\Sigma_{y}(t)\mathbbm{p}^{\top}(t)\\\
&\qquad+\Sigma_{z}(t)\mathbbm{K}^{\top}_{1}(t)\big{]}+\tilde{\mathbbm{m}}^{\top}(t)\big{[}\mathbbm{B}_{y}(t)\mathbbm{P}(t)+\mathbbm{B}_{z}(t)\mathbbm{K}_{2}(t)\\\
&\qquad+[I_{n+1},\mathbbm{p}(t),\mathbbm{K}_{1}(t)]D^{2}\mathbbm{B}(t)[I_{n+1},\mathbbm{p}(t),\mathbbm{K}_{1}(t)]^{\top}\big{]}+\tilde{\mathbbm{n}}^{\top}(t)\big{[}\Sigma_{y}(t)\mathbbm{P}(t)\\\
&\qquad+\Sigma_{z}(t)\mathbbm{K}_{2}(t)+[I_{n+1},\mathbbm{p}(t),\mathbbm{K}_{1}(t)]D^{2}\Sigma(t)[I_{n+1},\mathbbm{p}(t),\mathbbm{K}_{1}(t)]^{\top}\big{]}\Big{\\}}\,dt\\\
&\quad-\tilde{\mathbbm{N}}(t)\,dW(t),\quad t\in[0,T],\\\
\tilde{\mathbbm{M}}(T)&=\theta
e^{\theta[\,\bar{\tilde{X}}(T)+f(\bar{X}(T),\bar{Y}(0))\,]}\begin{bmatrix}\bar{f}_{xx}+\theta\bar{f}_{x}\bar{f}_{x}^{\top}&\theta\bar{f}_{x}\\\
\theta\bar{f}_{x}^{\top}&\theta\end{bmatrix},\end{aligned}\right.$ (4.7)

which admits a unique solution
$(\tilde{\mathbbm{M}}(\cdot),\tilde{\mathbbm{N}}(\cdot))\in
L^{2}_{\mathcal{F}}(\Omega;C([0,T];\mathcal{S}^{n+1}))\times
L^{2}_{\mathcal{F}}([0,T];\mathcal{S}^{n+1})$.

###  4.2 Non-linear transformations

We can get the risk-sensitive global maximum principle from Lemma 3.1
directly, since we have established the first- and second-order adjoint
equations we need. However, it is not an ideal one since the adjoint variables
involve auxiliary and unnecessary components. It is to say that
$\tilde{\mathbbm{m}}(\cdot)$, $\tilde{\mathbbm{n}}(\cdot)$,
$\tilde{\mathbbm{M}}(\cdot)$ and $\tilde{\mathbbm{N}}(\cdot)$ are of dimension
$n+1$ whereas state $X(\cdot)$ is of dimension $n$.

In this section, we introduce some non-linear transformations to eliminate the
additional parts of $(\tilde{\mathbbm{m}}(\cdot),\tilde{\mathbbm{n}}(\cdot))$
and $(\tilde{\mathbbm{M}}(\cdot),\tilde{\mathbbm{N}}(\cdot))$, which finally
lead to Theorem 3.1.

####  4.2.1 Transformation of the first-order processes

Form (4.6), we can easily conclude that:

$\left\\{\begin{aligned} d\tilde{m}_{2}(t)&=\tilde{n}_{2}(t)\,dW(t),\quad
t\in[0,T],\\\ \tilde{m}_{2}(T)&=\theta
e^{\theta\big{[}\,\bar{\tilde{X}}(T)+f(\bar{X}(T),\bar{Y}(0))\,\big{]}}.\end{aligned}\right.$
(4.8)

As in Moon [27], its $\mathcal{F}_{t}$-adapted solution can be written as:

$\tilde{m}_{2}(t)=\mathbb{E}\Big{[}\theta
e^{\theta\big{[}\,\bar{\tilde{X}}(T)+f(\bar{X}(T),\bar{Y}(0))\,\big{]}}\,\Big{|}\,\mathcal{F}_{t}\Big{]}:=\theta\zeta(t),$

where $\zeta(\cdot)$ is a martingale values in $\mathbb{R}$ such that
$\zeta(t)>0$, $\mathbb{P}$-a.s., for $t\in[0,T]$.

Introducing the following non-linear transformation

$\displaystyle\mathbbm{m}(t)=\frac{\tilde{\mathbbm{m}}(t)}{\theta\zeta(t)}:=\begin{bmatrix}m(t)\\\
m_{2}(t)\end{bmatrix},$ (4.9)

which makes the terminal condition of (4.6) become

$\mathbbm{m}(T)=\begin{bmatrix}f_{x}(\bar{X}(T),\bar{Y}(0))\\\
1\end{bmatrix}.$

Before we can apply Itô’s formula to $\mathbbm{m}(\cdot)$, we first have to
compute $d\frac{1}{\zeta(\cdot)}$, in other words, $d\zeta(\cdot)$. In fact,
we have

$\displaystyle\zeta(t)$
$\displaystyle=\mathbb{E}\Big{[}e^{\theta\big{[}\,\bar{\tilde{X}}(T)+f(\bar{X}(T),\bar{Y}(0))\,\big{]}}\,\Big{|}\,\mathcal{F}_{t}\Big{]}$
$\displaystyle=e^{\theta\int_{0}^{t}l(s,\bar{X},\bar{Y},\bar{Z},\bar{u})\,ds}\times\mathbb{E}\Big{[}e^{\theta\big{[}\,\int_{t}^{T}l(s,\bar{X},\bar{Y},\bar{Z},\bar{u})\,ds+f(\bar{X}(T),\bar{Y}(0))\,\big{]}}\,\Big{|}\,\mathcal{F}_{t}\Big{]}:=e^{\theta\int_{0}^{t}l(s,\bar{X},\bar{Y},\bar{Z},\bar{u})\,ds}\bar{\zeta}(t).$

Assuming that $\bar{\zeta}(t)=e^{\theta\gamma(t)}$ and

$\left\\{\begin{aligned} d\gamma(t)&=\mathcal{A}(t)\,dt+\kappa(t)\,dW(t),\quad
t\in[0,T],\\\ \gamma(T)&=f(\bar{X}(T),\bar{Y}(0)),\end{aligned}\right.$

where $\mathcal{A}(\cdot)$ is to be determined. Define
$\bar{\gamma}(t):=e^{\theta\gamma(t)}$, we now choose suitable
$\mathcal{A}(t)$ such that $\bar{\gamma}(t)=\bar{\zeta}(t)$. By Itô’s formula,

$d\bar{\gamma}(t)=\theta\bar{\gamma}(t)\Big{\\{}\Big{[}\mathcal{A}(t)+\frac{\theta}{2}\kappa^{2}(t)\Big{]}\,dt+\kappa(t)\,dW(t)\Big{\\}}.$

Set $\rho(t):=e^{\theta\int_{0}^{t}l(s,\bar{X},\bar{Y},\bar{Z},\bar{u})\,ds}$.
Then we can easily deduce that

$d\rho(t)\bar{\gamma}(t)=\theta\rho(t)\bar{\gamma}(t)\Big{\\{}\Big{[}\mathcal{A}(t)+l(t,\bar{X},\bar{Y},\bar{Z},\bar{u})+\frac{\theta}{2}\kappa^{2}(t)\Big{]}\,dt+\kappa(t)\,dW(t)\Big{\\}}.$

Integral from $t$ to $T$, we have

$\displaystyle\rho(T)\bar{\gamma}(T)-\rho(t)\bar{\gamma}(t)$
$\displaystyle=\int_{t}^{T}\theta\rho(s)\bar{\gamma}(s)\Big{[}\mathcal{A}(s)+l(s,\bar{X},\bar{Y},\bar{Z},\bar{u})+\frac{\theta}{2}\kappa^{2}(s)\Big{]}\,ds$
$\displaystyle\quad+\int_{t}^{T}\theta\rho(s)\bar{\gamma}(s)\kappa(s)\,dW(s).$

Notice that
$\rho(T)\bar{\gamma}(T)=e^{\theta\int_{0}^{T}l(t,\bar{X},\bar{Y},\bar{Z},\bar{u})\,dt+\theta
f(\bar{X}(T),\bar{Y}(0))}$, we arrive at

$\displaystyle\bar{\gamma}(t)$
$\displaystyle=e^{\theta\int_{t}^{T}l(s,\bar{X},\bar{Y},\bar{Z},\bar{u})\,ds+\theta
f(\bar{X}(T),\bar{Y}(0))}-\rho^{-1}(t)\int_{t}^{T}\theta\rho(s)\bar{\gamma}(s)\kappa(s)\,dW(s)$
$\displaystyle\quad-\rho^{-1}(t)\int_{t}^{T}\theta\rho(s)\bar{\gamma}(s)\Big{[}\mathcal{A}(s)+l(s,\bar{X},\bar{Y},\bar{Z},\bar{u})+\frac{\theta}{2}\kappa^{2}(s)\Big{]}\,ds.$

Since $\bar{\gamma}(t)=e^{\theta\gamma(t)}$ is $\mathcal{F}_{t}$-adapted, then
we have

$\displaystyle\bar{\gamma}(t)$
$\displaystyle=\mathbb{E}\Big{[}e^{\theta\int_{t}^{T}l(s,\bar{X},\bar{Y},\bar{Z},\bar{u})\,ds+\theta
f(\bar{X}(T),\bar{Y}(0))}\,\Big{|}\,\mathcal{F}_{t}\Big{]}$
$\displaystyle\quad-\mathbb{E}\bigg{[}\rho^{-1}(t)\int_{t}^{T}\theta\rho(s)\bar{\gamma}(s)\big{[}\mathcal{A}(s)+l(s,\bar{X},\bar{Y},\bar{Z},\bar{u})+\frac{\theta}{2}\kappa^{2}(s)\big{]}\,ds\,\Big{|}\,\mathcal{F}_{t}\bigg{]}$
$\displaystyle=\bar{\zeta}(t)-\mathbb{E}\bigg{[}\rho^{-1}(t)\int_{t}^{T}\theta\rho(s)\bar{\gamma}(s)\big{[}\mathcal{A}(s)+l(s,\bar{X},\bar{Y},\bar{Z},\bar{u})+\frac{\theta}{2}\kappa^{2}(s)\big{]}\,ds\,\Big{|}\,\mathcal{F}_{t}\bigg{]}.$

Set
$\mathcal{A}(t):=-\big{[}l(t,\bar{X},\bar{Y},\bar{Z},\bar{u})+\frac{\theta}{2}\kappa^{2}(t)\big{]}$,
then we finally conclude that
$\bar{\gamma}(t)=\bar{\zeta}(t)=e^{\theta\gamma(t)}$, where $\gamma(\cdot)$ is
the solution of the following quadratic growth BSDE, whose unique solvability
can be found in El Karoui and Hamadene [13] or Zhang [44]:

$\left\\{\begin{aligned}
d\gamma(t)&=-\Big{[}l(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),\bar{u}(t))+\frac{\theta}{2}\kappa^{2}(t)\Big{]}\,dt+\kappa(t)\,dW(t),\quad
t\in[0,T],\\\ \gamma(T)&=f(\bar{X}(T),\bar{Y}(0)).\end{aligned}\right.$ (4.10)

From Itô’s formula, we arrive at

$\left\\{\begin{aligned} d\zeta(t)&=\theta\zeta(t)\kappa(t)\,dW(t),\quad
t\in[0,T],\\\
\zeta(T)&=e^{\theta\big{[}\,\bar{\tilde{X}}(T)+f(\bar{X}(T),\bar{Y}(0))\,\big{]}},\end{aligned}\right.$
(4.11)

and

$d\frac{1}{\zeta(t)}=\frac{\theta^{2}\kappa^{2}(t)}{\zeta(t)}\,dt-\frac{\theta\kappa(t)}{\zeta(t)}\,dW(t).$

Hence, by using Itô’s formula again, we get

$\left\\{\begin{aligned}
-d\mathbbm{m}(t)&=\Big{\\{}\big{[}\mathbbm{B}^{\top}_{\mathbbm{x}}(t)+\mathbbm{p}(t)\mathbbm{B}^{\top}_{y}(t)+\mathbbm{K}_{1}(t)\mathbbm{B}^{\top}_{z}(t)\big{]}\mathbbm{m}(t)+\big{[}\Sigma^{\top}_{\mathbbm{x}}(t)+\mathbbm{p}(t)\Sigma^{\top}_{y}(t)+\mathbbm{K}_{1}(t)\\\
&\qquad\times\Sigma^{\top}_{z}(t)\big{]}(\theta\kappa(t)\mathbbm{m}(t)+\mathbbm{n}(t))+\theta\kappa(t)\mathbbm{n}(t)\Big{\\}}\,dt-\mathbbm{n}(t)\,dW(t),\
t\in[0,T],\\\ \mathbbm{m}(T)&=\begin{bmatrix}f_{x}(\bar{X}(T),\bar{Y}(0))\\\
1\end{bmatrix},\end{aligned}\right.$ (4.12)

where

$\mathbbm{m}(t)=\frac{\tilde{\mathbbm{m}}(t)}{\theta\zeta(t)}:=\begin{bmatrix}m(t)\\\
m_{2}(t)\end{bmatrix},\quad\mathbbm{n}(t)=\frac{\tilde{\mathbbm{n}}(t)}{\theta\zeta(t)}-\theta\kappa(t)\mathbbm{m}(t):=\begin{bmatrix}n(t)\\\
n_{2}(t)\end{bmatrix}.$ (4.13)

As before, consider the scalar components $(m_{2}(\cdot),n_{2}(\cdot))$, we
have

$\left\\{\begin{aligned} -dm_{2}(t)&=\theta\kappa(t)n_{2}(t)\,dt-
n_{2}(t)\,dW(t),\quad t\in[0,T],\\\ m_{2}(T)&=1.\end{aligned}\right.$

Then the unique solvability of BSDEs guarantees that
$(m_{2}(t),n_{2}(t))=(1,0)$, $\mathbb{P}$-a.s., $t\in[0,T]$.

Therefore, from (4.12), $(m(\cdot),n(\cdot))$ satisfies

$\left\\{\begin{aligned}
-dm(t)&=\Big{\\{}\big{[}b^{\top}_{x}(t)+p(t)b^{\top}_{y}(t)+K_{1}(t)b^{\top}_{z}(t)\big{]}m(t)+\big{[}l^{\top}_{x}(t)+p(t)l^{\top}_{y}(t)+K_{1}(t)l^{\top}_{z}(t)\big{]}\\\
&\qquad+\big{[}\sigma^{\top}_{x}(t)+p(t)\sigma^{\top}_{y}(t)+K_{1}(t)\sigma^{\top}_{z}(t)\big{]}(\theta\kappa(t)m(t)+n(t))\\\
&\qquad+\theta\kappa(t)n(t)\Big{\\}}\,dt-n(t)\,dW(t),\quad t\in[0,T],\\\
m(T)&=f_{x}(\bar{X}(T),\bar{Y}(0)),\end{aligned}\right.$

which coincides with the backward equation of (3.16).

Moreover, set $k(t)=\frac{\tilde{k}(t)}{\theta\zeta(t)}$. Then by Itô’s
formula, we deduce that

$\left\\{\begin{aligned}
dk(t)&=\Big{\\{}\big{[}m^{\top}(t)b_{z}(t)+l_{z}(t)+(n^{\top}(t)+\theta\kappa(t)m^{\top}(t))\sigma_{z}(t)\big{]}(1-p^{\top}(t)\sigma_{z}(t))^{-1}p^{\top}(t)\sigma_{y}(t)\\\
&\qquad+m^{\top}(t)b_{y}(t)+l_{y}(t)+(n^{\top}(t)+\theta\kappa(t)m^{\top}(t))\sigma_{y}(t)+k(t)\big{[}g_{y}(t)\\\
&\qquad+p^{\top}(t)b_{y}(t)+l_{y}(t)+q^{\top}(t)\sigma_{y}(t)+(1-p^{\top}(t)\sigma_{z}(t))^{-1}g_{z}(t)p^{\top}(t)\sigma_{y}(t)\big{]}\\\
&\qquad+\theta^{2}\kappa^{2}(t)k(t)-\theta\kappa(t)\big{[}m^{\top}(t)b_{z}(t)+l_{z}(t)+(n^{\top}(t)+\theta\kappa(t)m^{\top}(t))\sigma_{z}(t)\big{]}\\\
&\qquad-\theta\kappa(t)\big{[}m^{\top}(t)b_{z}(t)+l_{z}(t)+(n^{\top}(t)+\theta\kappa(t)m^{\top}(t))\sigma_{z}(t)\big{]}\\\
&\qquad\times(1-p^{\top}(t)\sigma_{z}(t))^{-1}p^{\top}(t)\sigma_{z}(t)-\theta\kappa(t)k(t)\big{[}g_{z}(t)+p^{\top}(t)b_{z}(t)+l_{z}(t)\\\
&\qquad+q^{\top}(t)\sigma_{z}(t)+(1-p^{\top}(t)\sigma_{z}(t))^{-1}g_{z}(t)p^{\top}(t)\sigma_{z}(t)\big{]}\Big{\\}}\,dt\\\
&\quad+\Big{\\{}\big{[}m^{\top}(t)b_{z}(t)+l_{z}(t)+(n^{\top}(t)+\theta\kappa(t)m^{\top}(t))\sigma_{z}(t)\big{]}\\\
&\qquad\
\times(1-p^{\top}(t)\sigma_{z}(t))^{-1}p^{\top}(t)\sigma_{z}(t)+k(t)\big{[}g_{z}(t)+p^{\top}(t)b_{z}(t)+l_{z}(t)\\\
&\qquad\
+q^{\top}(t)\sigma_{z}(t)+(1-p^{\top}(t)\sigma_{z}(t))^{-1}g_{z}(t)p^{\top}(t)\sigma_{z}(t)\big{]}+m^{\top}(t)b_{z}(t)\\\
&\qquad\
+l_{z}(t)+(n^{\top}(t)+\theta\kappa(t)m^{\top}(t))\sigma_{z}(t)-\theta\kappa(t)k(t)\Big{\\}}\,dW(t),\quad
t\in[0,T],\\\ k(0)&=f_{y}(\bar{X}(T),\bar{Y}(0)),\end{aligned}\right.$

which coincides with the forward equation of (3.16).

######  Remark 4.1.

(4.6) is a non-coupled FBSDE. With the help of classical BSDE theory, (4.6)
has a unique solution
$(\tilde{\mathbbm{m}}(\cdot),\tilde{\mathbbm{n}}(\cdot))$ since it is not
coupled with the forward term $\tilde{k}(\cdot)$. Hence, the unique
solvability of (3.16) is guaranteed by the unique solvability of (4.6) and
(4.11).

####  4.2.2 Transformation of the second-order processes and maximum condition

In this section, we consider the transformation of the second-order processes
in (4.7).

Similar as the previous section, we introduce the transformation

$\mathbbm{M}(t)=\frac{\tilde{\mathbbm{M}}(t)}{\theta\zeta(t)}-\theta\mathbbm{m}(t)\mathbbm{m}^{\top}(t):=\begin{bmatrix}M(t)&M_{2}(t)\\\
M_{2}^{\top}(t)&M_{4}(t)\end{bmatrix},$ (4.14)

and

$\displaystyle\mathbbm{N}(t)$
$\displaystyle=\frac{\tilde{\mathbbm{N}}(t)}{\theta\zeta(t)}-\theta\kappa(t)\big{(}\mathbbm{M}(t)+\theta\mathbbm{m}(t)\mathbbm{m}^{\top}(t)\big{)}-\theta\mathbbm{m}(t)\mathbbm{n}^{\top}(t)-\theta\mathbbm{n}(t)\mathbbm{m}^{\top}(t)$
(4.15) $\displaystyle:=\begin{bmatrix}N(t)&N_{2}(t)\\\
N_{2}^{\top}(t)&N_{4}(t)\end{bmatrix}.$

By using Itô’s formula, we have

$\left\\{\begin{aligned}
-d\mathbbm{M}(t)&=\Big{\\{}\mathbbm{M}(t)\big{[}\mathbbm{B}_{\mathbbm{x}}(t)+\mathbbm{B}_{y}(t)\mathbbm{p}^{\top}(t)+\mathbbm{B}_{z}(t)\mathbbm{K}^{\top}_{1}(t)\big{]}+\big{[}\mathbbm{B}^{\top}_{\mathbbm{x}}(t)+\mathbbm{p}(t)\mathbbm{B}^{\top}_{y}(t)\\\
&\qquad+\mathbbm{K}_{1}(t)\mathbbm{B}^{\top}_{z}(t)\big{]}\mathbbm{M}(t)+\big{[}\Sigma^{\top}_{\mathbbm{x}}(t)+\mathbbm{p}(t)\Sigma^{\top}_{y}(t)+\mathbbm{K}_{1}(t)\Sigma^{\top}_{z}(t)\big{]}\big{[}\mathbbm{M}(t)\\\
&\qquad+\theta\mathbbm{m}(t)\mathbbm{m}^{\top}(t)\big{]}\big{[}\Sigma_{\mathbbm{x}}(t)+\Sigma_{y}(t)\mathbbm{p}^{\top}(t)+\Sigma_{z}(t)\mathbbm{K}^{\top}_{1}(t)\big{]}\\\
&\qquad+\big{[}\Sigma^{\top}_{\mathbbm{x}}(t)+\mathbbm{p}(t)\Sigma^{\top}_{y}(t)+\mathbbm{K}_{1}(t)\Sigma^{\top}_{z}(t)\big{]}\big{[}\mathbbm{N}(t)+\theta\kappa(t)\mathbbm{M}(t)+\theta\mathbbm{m}(t)\mathbbm{n}^{\top}(t)\big{]}\\\
&\qquad+\big{[}\mathbbm{N}(t)+\theta\kappa(t)\mathbbm{M}(t)+\theta\mathbbm{n}(t)\mathbbm{m}^{\top}(t)\big{]}\big{[}\Sigma_{\mathbbm{x}}(t)+\Sigma_{y}(t)\mathbbm{p}^{\top}(t)+\Sigma_{z}(t)\mathbbm{K}_{1}^{\top}(t)\big{]}\\\
&\qquad+\mathbbm{m}^{\top}(t)\big{[}\mathbbm{B}_{y}(t)\mathbbm{P}(t)+\mathbbm{B}_{z}(t)\mathbbm{K}_{2}(t)\\\
&\qquad\quad+[I_{n+1},\mathbbm{p}(t),\mathbbm{K}_{1}(t)]D^{2}\mathbbm{B}(t)[I_{n+1},\mathbbm{p}(t),\mathbbm{K}_{1}(t)]^{\top}\big{]}\\\
&\qquad+(\mathbbm{n}(t)+\theta\kappa(t)\mathbbm{m}(t))^{\top}\big{[}\Sigma_{y}(t)\mathbbm{P}(t)+\Sigma_{z}(t)\mathbbm{K}_{2}(t)\\\
&\qquad\quad+[I_{n+1},\mathbbm{p}(t),\mathbbm{K}_{1}(t)]D^{2}\Sigma(t)[I_{n+1},\mathbbm{p}(t),\mathbbm{K}_{1}(t)]^{\top}\big{]}\\\
&\qquad+\theta\kappa(t)\mathbbm{N}(t)-\theta\mathbbm{n}(t)\mathbbm{n}^{\top}(t)\Big{\\}}\,dt-\mathbbm{N}(t)\,dW(t),\quad
t\in[0,T],\\\
\mathbbm{M}(T)&=\begin{bmatrix}f_{xx}(\bar{X}(T),\bar{Y}(0))&0_{n}\\\
0_{n}^{\top}&0\end{bmatrix},\end{aligned}\right.$ (4.16)

which is uniquely solvable as discussed in Remark 4.1.

Consider the matrix form here, we can show that

$\left\\{\begin{aligned} -dM_{4}(t)&=\theta\kappa(t)N_{4}(t)\,dt-
N_{4}(t)\,dW(t),\quad t\in[0,T],\\\ M_{4}(T)&=0,\end{aligned}\right.$

which has the unique solution $(M_{4}(\cdot),N_{4}(\cdot))=(0,0)$.

Further, the equation of $(M_{2}(\cdot),N_{2}(\cdot))$ is the following:

$\left\\{\begin{aligned}
-dM_{2}(t)&=\Big{\\{}\big{[}b_{x}^{\top}(t)+p(t)b_{y}^{\top}(t)+K_{1}(t)b_{z}^{\top}(t)\big{]}M_{2}(t)+\big{[}\sigma_{x}^{\top}(t)+p(t)\sigma_{y}^{\top}(t)\\\
&\quad+K_{1}(t)\sigma_{z}^{\top}(t)\big{]}\big{[}N_{2}(t)+\theta\kappa(t)M_{2}(t)\big{]}+\theta\kappa(t)N_{2}(t)\Big{\\}}\,dt-
N_{2}(t)\,dW(t),\quad t\in[0,T],\\\ M_{2}(T)&=0,\end{aligned}\right.$

which also implies that $(M_{2}(\cdot),N_{2}(\cdot))=(0,0)$ is the unique
solution.

Hence, $(M(\cdot),N(\cdot))$ is the unique solution given by

$\left\\{\begin{aligned}
-dM(t)&=\Big{\\{}M(t)\big{[}b_{x}(t)+b_{y}(t)p^{\top}(t)+b_{z}(t)K^{\top}_{1}(t)\big{]}\\\
&\qquad+\big{[}b^{\top}_{x}(t)+p(t)b^{\top}_{y}(t)+K_{1}(t)b^{\top}_{z}(t)\big{]}M(t)\\\
&\qquad+\big{[}\sigma^{\top}_{x}(t)+p(t)\sigma^{\top}_{y}(t)+K_{1}(t)\sigma^{\top}_{z}(t)\big{]}\big{[}M(t)+\theta
m(t)m^{\top}(t)\big{]}\\\
&\qquad\quad\times\big{[}\sigma_{x}(t)+\sigma_{y}(t)p^{\top}(t)+\sigma_{z}(t)K^{\top}_{1}(t)\big{]}\\\
&\qquad+\big{[}\sigma^{\top}_{x}(t)+p(t)\sigma^{\top}_{y}(t)+K_{1}(t)\sigma^{\top}_{z}(t)\big{]}\big{[}N(t)+\theta\kappa(t)M(t)+\theta
m(t)n^{\top}(t)\big{]}\\\ &\qquad+\big{[}N(t)+\theta\kappa(t)M(t)+\theta
n(t)m^{\top}(t)\big{]}\big{[}\sigma_{x}(t)+\sigma_{y}(t)p^{\top}(t)+\sigma_{z}(t)K_{1}^{\top}(t)\big{]}\\\
&\qquad+\big{[}m^{\top}(t)b_{y}(t)+l_{y}(t)\big{]}P(t)+\big{[}m^{\top}(t)b_{z}(t)+l_{z}(t)\big{]}K_{2}(t)\\\
&\qquad+\theta\kappa(t)N(t)-\theta
n(t)n^{\top}(t)+m^{\top}(t)[I_{n},p(t),K_{1}(t)]D^{2}b(t)[I_{n},p(t),K_{1}(t)]^{\top}\\\
&\qquad+[I_{n},p(t),K_{1}(t)]D^{2}l(t)[I_{n},p(t),K_{1}(t)]^{\top}\\\
&\qquad+(n(t)+\theta\kappa(t)m(t))^{\top}[I_{n},p(t),K_{1}(t)]D^{2}\sigma(t)[I_{n},p(t),K_{1}(t)]^{\top}\Big{\\}}\,dt\\\
&\quad-N(t)\,dW(t),\quad t\in[0,T],\\\
M(T)&=f_{xx}(\bar{X}(T),\bar{Y}(0)),\end{aligned}\right.$

which is equivalent to (3.17).

Now, we consider the maximum condition. Recall the risk-neutral Hamiltonian
function (3.12) and combine with transformations given above, we have

$\displaystyle\tilde{\mathcal{H}}^{RN}(t,x,y,z,u,\mathbbm{p},\mathbbm{q},\mathbbm{P},\tilde{k},\tilde{\mathbbm{m}},\tilde{\mathbbm{n}},\tilde{\mathbbm{M}})$
(4.17)
$\displaystyle:=(\tilde{\mathbbm{m}}^{\top}+\tilde{k}\mathbbm{p}^{\top})\mathbbm{B}(t,x,y,z+\Delta,u)+(\tilde{\mathbbm{n}}^{\top}+\tilde{k}\mathbbm{q}^{\top})\Sigma(t,x,y,z+\Delta,u)$
$\displaystyle\quad+\tilde{k}g(t,x,y,z+\Delta,u)+\frac{1}{2}\big{[}\Sigma(t,x,y,z+\Delta,u)-\Sigma(t,\bar{X},\bar{Y},\bar{Z},\bar{u})\big{]}^{\top}$
$\displaystyle\quad\times(\tilde{\mathbbm{M}}+\tilde{k}\mathbbm{P})\big{[}\Sigma(t,x,y,z+\Delta,u)-\Sigma(t,\bar{X},\bar{Y},\bar{Z},\bar{u})\big{]}$
$\displaystyle=\theta\zeta(t)\Big{\\{}(m^{\top}+kp^{\top})b(t)+l(t)+(n^{\top}(t)+\theta\kappa
m^{\top}+kq^{\top})\sigma(t)+kg(t)$
$\displaystyle\qquad+\frac{1}{2}\big{[}\sigma(t,x,y,z+\Delta,u)-\sigma(t,\bar{X},\bar{Y},\bar{Z},\bar{u})\big{]}^{\top}(M+\theta
mm^{\top}+kP)$
$\displaystyle\qquad\times\big{[}\sigma(t,x,y,z+\Delta,u)-\sigma(t,\bar{X},\bar{Y},\bar{Z},\bar{u})\big{]}\Big{\\}}$
$\displaystyle:=\theta\zeta(t)\mathcal{H}_{\theta}^{RS}(t,x,y,z,u,p,q,k,m,n,P,M,\kappa).$

By Lemma 3.1, we arrive at

$\displaystyle\tilde{\mathcal{H}}^{RN}(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),u,\mathbbm{p}(t),\mathbbm{q}(t),\mathbbm{P}(t),\tilde{k}(t),\tilde{\mathbbm{m}}(t),\tilde{\mathbbm{n}}(t),\tilde{\mathbbm{M}}(t))$
$\displaystyle-\tilde{\mathcal{H}}^{RN}(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),\bar{u}(t),\mathbbm{p}(t),\mathbbm{q}(t),\mathbbm{P}(t),\tilde{k}(t),\tilde{\mathbbm{m}}(t),\tilde{\mathbbm{n}}(t),\tilde{\mathbbm{M}}(t))\geq
0,$ $\displaystyle\hskip 170.71652pt\forall u\in U,\quad
a.e.,\quad\mathbb{P}\mbox{-}a.s..$

Notice that $\zeta(t)>0$, $\mathbb{P}$-a.s., $t\in[0,T]$, then we get the
maximum condition (3.19):

$\displaystyle\theta\,\mathcal{H}_{\theta}^{RS}(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),u,p(t),q(t),m(t),n(t),k(t),P(t),M(t),\kappa(t))$
$\displaystyle\geq\theta\,\mathcal{H}_{\theta}^{RS}(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),\bar{u}(t),p(t),q(t),m(t),n(t),k(t),P(t),M(t),\kappa(t)),$
$\displaystyle\hskip 170.71652pt\forall u\in U,\quad
a.e.,\quad\mathbb{P}\mbox{-}a.s..$

This completes the proof of Theorem 3.1.

##  5 A linear-exponential-quadratic example

In this section, we discuss an LEQ problem. By using Girsanov’s Theorem and
completion-of-squares technique, optimal feedback control and optimal cost are
obtained. The Riccati equations and related ordinary differential equations
(ODEs, for short) are established in risk-sensitive forms, which means the
risk-sensitive parameter appears in their equations.

For convenience, we only consider the one-dimension case and the time variable
may be omitted when no confusion appears. The state equation is the following
fully coupled linear FBSDE:

$\left\\{\begin{aligned}
dX(t)&=[A_{1}(t)X(t)+B_{1}(t)Y(t)+C_{1}(t)Z(t)+D_{1}(t)u(t)+b(t)]\,dt+\sigma(t)\,dW(t),\\\
-dY(t)&=[A_{2}(t)X(t)+B_{2}(t)Y(t)+C_{2}(t)Z(t)+D_{2}(t)u(t)+g(t)]\,dt\\\
&\quad-Z(t)\,dW(t),\quad t\in[0,T],\\\ X(0)&=x_{0},\quad
Y(T)=GX(T).\end{aligned}\right.$ (5.1)

Introduce the following cost functional, where all square terms and cross
terms of $X$, $Y$, $Z$ and $u$ can appear in:

$\displaystyle
J(u(\cdot))=\mathbb{E}\Bigg{[}\exp\bigg{\\{}\frac{\theta}{2}\bigg{(}S_{1}X^{2}(T)+S_{2}Y^{2}(0)$
(5.2)
$\displaystyle\quad+\int_{0}^{T}\begin{pmatrix}X(t),Y(t),Z(t),u(t)\end{pmatrix}\begin{pmatrix}R_{11}(t)&R_{12}(t)&R_{13}(t)&R_{14}(t)\\\
R_{12}(t)&R_{22}(t)&R_{23}(t)&R_{24}(t)\\\
R_{13}(t)&R_{23}(t)&R_{33}(t)&R_{34}(t)\\\
R_{14}(t)&R_{24}(t)&R_{34}(t)&R_{44}(t)\end{pmatrix}\begin{pmatrix}X(t)\\\
Y(t)\\\ Z(t)\\\ u(t)\end{pmatrix}\,dt\bigg{)}\bigg{\\}}\Bigg{]}.$

Accordingly, the following assumptions are imposed.

######  Assumption 4.

  * (i)

$\varphi_{i}(\cdot)$, $\varphi=A,B,C,D,i=1,2$, $\sigma(\cdot)$,
$R_{ij}(\cdot)$, $i,j=1,2,3,4$ are deterministic and bounded, and $x_{0}$,
$G$, $S_{1}$ and $S_{2}$ are constants;

  * (ii)

$R_{44}(\cdot)$ is uniformly positive definite; $\theta\textgreater 0$ is
fixed.

Now we use the completion-of-squares technique to determine the optimal
feedback control as well as the optimal cost. First, for an admissible
quadruple $(\bar{X}(\cdot),\bar{Y}(\cdot),\bar{Z}(\cdot),\bar{u}(\cdot))$, we
decouple the state equation (5.1) with

$\bar{Y}(\cdot)=\beta_{1}(\cdot)\bar{X}(\cdot)+\beta_{2}(\cdot),$ (5.3)

where the equations that $\beta_{1}(\cdot)$ and $\beta_{2}(\cdot)$ satisfy
will be determined later such that $\beta_{1}(T)=G$, $\beta_{2}(T)=0$. By
Itô’s formula, we can deduce that
$\bar{Z}(\cdot)=\beta_{1}(\cdot)\sigma(\cdot)$, $\mathbb{P}\mbox{-}a.s..$
Hence, the SDE satisfied by $\bar{X}(\cdot)$ takes the following form:

$\left\\{\begin{aligned}
d\bar{X}(t)&=\big{[}(A_{1}(t)+B_{1}(t)\beta_{1}(t))\bar{X}(t)+B_{1}(t)\beta_{2}(t)+C_{1}(t)\beta_{1}(t)\sigma(t)+D_{1}(t)\bar{u}(t)+b(t)\big{]}\,dt\\\
&\quad+\sigma(t)\,dW(t),\quad t\in[0,T],\\\ X(0)&=x_{0}.\end{aligned}\right.$
(5.4)

As mentioned before, (4.10) plays an important role in solving the risk-
sensitive optimal control problem. In this LEQ problem of the present section,
(4.10) writes

$\left\\{\begin{aligned}
d\gamma(t)&=-\Bigg{\\{}\frac{1}{2}\begin{pmatrix}\bar{X}\\\
\beta_{1}\bar{X}+\beta_{2}\\\ \beta_{1}\sigma\\\
\bar{u}\end{pmatrix}^{\top}\begin{pmatrix}R_{11}&R_{12}&R_{13}&R_{14}\\\
R_{12}&R_{22}&R_{23}&R_{24}\\\ R_{13}&R_{23}&R_{33}&R_{34}\\\
R_{14}&R_{24}&R_{34}&R_{44}\end{pmatrix}\begin{pmatrix}\bar{X}\\\
\beta_{1}\bar{X}+\beta_{2}\\\ \beta_{1}\sigma\\\
\bar{u}\end{pmatrix}+\frac{\theta}{2}\kappa^{2}(t)\Bigg{\\}}\,dt\\\
&\quad+\kappa(t)\,dW(t),\quad t\in[0,T],\\\
\gamma(T)&=\frac{1}{2}S_{1}\bar{X}^{2}(T)+\frac{1}{2}S_{2}\bar{Y}^{2}(0).\end{aligned}\right.$
(5.5)

Suppose we can express $\gamma(\cdot)$ as an explicit function of
$\bar{X}(\cdot)$ as follows:

$\gamma(\cdot)=\frac{\alpha_{1}(\cdot)}{2}\bar{X}^{2}(\cdot)+\alpha_{2}(\cdot)\bar{X}(\cdot)+\alpha_{3}(\cdot),$
(5.6)

where $\alpha_{1}(\cdot)$, $\alpha_{2}(\cdot)$ and $\alpha_{3}(\cdot)$ are
deterministic functions such that $\alpha_{1}(T)=S_{1}$, $\alpha_{2}(T)=0$ and
$\alpha_{3}(T)=\frac{1}{2}S_{2}Y(0)^{2}=\frac{1}{2}S_{2}(\beta_{1}(0)x_{0}+\beta_{2}(0))^{2}$.

In view of (5.4), applying Itô’s formula to (5.6) and identifying the
coefficients of diffusion terms, we conclude that

$\kappa(\cdot)=\sigma(\cdot)\big{[}\alpha_{1}(\cdot)X(\cdot)+\alpha_{2}(\cdot)\big{]},\quad\mathbb{P}\mbox{-}a.s..$
(5.7)

Substituting (5.7) into the drift term, and using the completion-of-squares
technique, the following equality is obtained:

$\displaystyle 0$
$\displaystyle=\Big{[}\frac{1}{2}\dot{\alpha}_{1}+\alpha_{1}(A_{1}+B_{1}\beta_{1})+\frac{1}{2}(R_{11}+2R_{12}\beta_{1}+R_{22}\beta_{1}^{2})-\frac{1}{2}R_{44}^{-1}(\alpha_{1}D_{1}+R_{14}+R_{24}\beta_{1})^{2}$
(5.8) $\displaystyle\quad\
+\frac{\theta}{2}\sigma^{2}\alpha_{1}^{2}\Big{]}X^{2}+\Big{[}\dot{\alpha}_{2}+\alpha_{2}(A_{1}+B_{1}\beta_{1})+\alpha_{1}(B_{1}\beta_{2}+C_{1}\beta_{1}\sigma+b)$
$\displaystyle\quad\
+R_{12}\beta_{2}+R_{13}\beta_{1}\sigma+R_{22}\beta_{1}\beta_{2}+R_{23}\beta_{1}^{2}\sigma+\theta\sigma^{2}\alpha_{1}\alpha_{2}$
$\displaystyle\quad\
-R_{44}^{-1}(\alpha_{1}D_{1}+R_{14}+R_{24}\beta_{1})(\alpha_{2}D_{1}+R_{24}\beta_{2}+R_{34}\beta_{1}\sigma)\Big{]}X$
$\displaystyle\ \
+\Big{[}\dot{\alpha}_{3}+\frac{1}{2}\alpha_{1}\sigma^{2}+\alpha_{2}(B_{1}\beta_{2}+C_{1}\beta_{1}\sigma+b)+\frac{1}{2}R_{22}\beta_{2}^{2}+R_{23}\sigma\beta_{1}\beta_{2}+\frac{1}{2}R_{33}\sigma^{2}\beta_{1}^{2}$
$\displaystyle\quad\
+\frac{\theta}{2}\sigma^{2}\alpha_{2}^{2}-\frac{1}{2}R_{44}^{-1}(\alpha_{2}D_{1}+R_{24}\beta_{2}+R_{34}\beta_{1}\sigma)\Big{]}$
$\displaystyle\ \
+\frac{1}{2}R_{44}\Big{\\{}u+R_{44}^{-1}\big{[}(\alpha_{1}D_{1}+R_{14}+R_{24}\beta_{1})X+(\alpha_{2}D_{1}+R_{24}\beta_{2}+R_{34}\beta_{1}\sigma)\big{]}\Big{\\}}^{2}.$

From above, by simply calculating, we can deduce the feedback representation
of $\bar{u}(\cdot)$:

$\bar{u}=-R_{44}^{-1}\big{[}(\alpha_{1}D_{1}+R_{14}+R_{24}\beta_{1})\bar{X}+(\alpha_{2}D_{1}+R_{24}\beta_{2}+R_{34}\beta_{1}\sigma)\big{]},\quad
a.e.,\quad\mathbb{P}\mbox{-}a.s.,$ (5.9)

as well as a risk-sensitive Riccati equation along with two risk-sensitive
ODEs:

$\left\\{\begin{aligned}
&\dot{\alpha}_{1}(t)+2\mathcal{C}_{1}(t)\alpha_{1}(t)+2\mathcal{C}_{2}(t)\beta_{1}(t)+2\mathcal{C}_{3}(t)\alpha_{1}(t)\beta_{1}(t)+\mathcal{C}_{4}(t)\alpha_{1}^{2}(t)\\\
&\quad+\mathcal{C}_{5}(t)\beta_{1}^{2}(t)+\mathcal{C}_{6}(t)=0,\quad
t\in[0,T],\\\ &\alpha_{1}(T)=S_{1},\end{aligned}\right.$ (5.10)
$\left\\{\begin{aligned}
&\dot{\alpha}_{2}(t)+\mathcal{C}_{7}(t)\alpha_{2}(t)+\mathcal{C}_{8}(t)\beta_{2}(t)+\mathcal{C}_{9}(t)=0,\quad
t\in[0,T],\\\ &\alpha_{2}(T)=0,\end{aligned}\right.$ (5.11)
$\left\\{\begin{aligned} &\dot{\alpha}_{3}(t)+\mathcal{C}_{10}(t)=0,\quad
t\in[0,T],\\\
&\alpha_{3}(T)=\frac{1}{2}S_{2}(\beta_{1}(0)x_{0}+\beta_{2}(0))^{2},\end{aligned}\right.$
(5.12)

where

$\displaystyle\mathcal{C}_{1}$
$\displaystyle:=A_{1}-R^{-1}_{44}D_{1}R_{14},\quad\mathcal{C}_{2}:=R_{12}-R^{-1}_{44}R_{14}R_{24},\quad\mathcal{C}_{3}:=B_{1}-R^{-1}_{44}D_{1}R_{24},$
$\displaystyle\mathcal{C}_{4}$
$\displaystyle:=\theta\sigma^{2}-R^{-1}_{44}D_{1}^{2},\quad\mathcal{C}_{5}:=R_{22}-R^{-1}_{44}R^{2}_{24},\quad\mathcal{C}_{6}:=R_{11}-R^{-1}_{44}R^{2}_{14},$
$\displaystyle\mathcal{C}_{7}$
$\displaystyle:=A_{1}+B_{1}\beta_{1}+\theta\sigma^{2}\alpha_{1}-R^{-1}_{44}D_{1}(D_{1}\alpha_{1}+R_{14}+R_{24}\beta_{1}),$
$\displaystyle\mathcal{C}_{8}$
$\displaystyle:=B_{1}\alpha_{1}+R_{12}+R_{22}\beta_{1}-R^{-1}_{44}R_{24}(D_{1}\alpha_{1}+R_{14}+R_{24}\beta_{1}),$
$\displaystyle\mathcal{C}_{9}$
$\displaystyle:=C_{1}\sigma\alpha_{1}\beta_{1}+\alpha_{1}b+R_{13}\sigma\beta_{1}+R_{23}\sigma\beta_{1}^{2}-R^{-1}_{44}R_{34}\sigma(D_{1}\alpha_{1}+R_{14}+R_{24}\beta_{1})\beta_{1},$
$\displaystyle\mathcal{C}_{10}$
$\displaystyle:=\frac{1}{2}\alpha_{1}\sigma^{2}+\alpha_{2}(B_{1}\beta_{2}+C_{1}\beta_{1}\sigma+b)+\frac{1}{2}R_{22}\beta_{2}^{2}+R_{23}\sigma\beta_{1}\beta_{2}+\frac{1}{2}R_{33}\sigma^{2}\beta_{1}^{2}$
$\displaystyle\quad+\frac{\theta}{2}\sigma^{2}\alpha_{2}^{2}-\frac{1}{2}R_{44}^{-1}(\alpha_{2}D_{1}+R_{24}\beta_{2}+R_{34}\beta_{1}\sigma)^{2}.$

Taking the state feedback representation (5.9) and the decoupling relation
(5.3) into (5.1), the optimal dynamics solves the linear FBSDE:

$\left\\{\begin{aligned}
d\bar{X}(t)&=[\mathcal{A}_{1}(t)\bar{X}(t)+\mathcal{B}_{1}(t)]\,dt+\sigma(t)\,dW(t),\\\
-d\bar{Y}(t)&=[\mathcal{A}_{2}(t)\bar{X}(t)+\mathcal{B}_{2}(t)]\,dt-\bar{Z}(t)\,dW(t),\quad
t\in[0,T],\\\ X(0)&=x_{0},\quad Y(T)=G\bar{X}(T),\end{aligned}\right.$ (5.13)

where

$\displaystyle\mathcal{A}_{1}$
$\displaystyle:=A_{1}+B_{1}\beta_{1}-R_{44}^{-1}D_{1}(\alpha_{1}D_{1}+R_{14}+R_{24}\beta_{1}),$
$\displaystyle\mathcal{B}_{1}$
$\displaystyle:=B_{1}\beta_{2}+C_{1}\beta_{1}\sigma+b-R_{44}^{-1}D_{1}(\alpha_{2}D_{1}+R_{24}\beta_{2}+R_{34}\beta_{1}\sigma),$
$\displaystyle\mathcal{A}_{2}$
$\displaystyle:=A_{2}+B_{2}\beta_{1}-R_{44}^{-1}D_{2}(\alpha_{1}D_{1}+R_{14}+R_{24}\beta_{1}),$
$\displaystyle\mathcal{B}_{2}$
$\displaystyle:=B_{2}\beta_{2}+C_{2}\beta_{1}\sigma+g-R_{44}^{-1}D_{2}(\alpha_{2}D_{1}+R_{24}\beta_{2}+R_{34}\beta_{1}\sigma).$

Applying Itô’s formula to (5.3) and identifying the coefficients of drift
terms, we have

$\big{[}\dot{\beta}_{1}(t)+\beta_{1}(t)\mathcal{A}_{1}(t)+\mathcal{A}_{2}(t)\big{]}\bar{X}(t)+\dot{\beta}_{2}(t)+\beta_{1}(t)\mathcal{B}_{1}(t)+\mathcal{B}_{2}(t)=0,\quad
a.e..$ (5.14)

Let the coefficients for $\bar{X}(\cdot)$ and the constant term be 0, then the
decoupling Riccati equation and ODE that $\beta_{1}(\cdot)$ and
$\beta_{2}(\cdot)$ satisfy take the following form:

$\left\\{\begin{aligned}
&\dot{\beta}_{1}(t)+\mathcal{C}_{11}(t)\beta_{1}(t)-\mathcal{C}_{12}(t)\alpha_{1}(t)-\mathcal{C}_{13}(t)\alpha_{1}(t)\beta_{1}(t)\\\
&\quad+\mathcal{C}_{14}(t)\beta_{1}^{2}(t)+\mathcal{C}_{15}(t)=0,\quad
t\in[0,T],\\\ &\beta_{1}(T)=G,\end{aligned}\right.$ (5.15)

and

$\left\\{\begin{aligned}
&\dot{\beta}_{2}(t)+\mathcal{C}_{16}(t)\beta_{2}(t)-\mathcal{C}_{17}(t)\alpha_{2}(t)+\mathcal{C}_{18}(t)=0,\quad
t\in[0,T],\\\ &\beta_{2}(T)=0,\end{aligned}\right.$ (5.16)

respectively, where

$\displaystyle\mathcal{C}_{11}$
$\displaystyle=A_{1}+B_{2}-R^{-1}_{44}(D_{1}R_{14}+D_{2}R_{24}),\quad\mathcal{C}_{12}=R^{-1}_{44}D_{1}D_{2},$
$\displaystyle\mathcal{C}_{13}$
$\displaystyle=R^{-1}_{44}D_{1}^{2},\quad\mathcal{C}_{14}=B_{1}-R^{-1}_{44}D_{1}R_{24},\quad\mathcal{C}_{15}=A_{2}-R^{-1}_{44}D_{2}R_{14},$
$\displaystyle\mathcal{C}_{16}$
$\displaystyle=B_{1}\beta_{1}+B_{2}-R^{-1}_{44}R_{24}(D_{1}\beta_{1}+D_{2}),\quad\mathcal{C}_{17}=R^{-1}_{44}D_{1}(D_{1}\beta_{1}+D_{2}),$
$\displaystyle\mathcal{C}_{18}$
$\displaystyle=(C_{1}\beta_{1}+C_{2})\sigma\beta_{1}+b\beta_{1}+g-R^{-1}_{44}R_{34}\sigma\beta_{1}(D_{1}\beta_{1}+D_{2}).$

######  Remark 5.1.

Let us give some remarks on the coupled feature of the above equations. In
fact, if the coupled Riccati equations (5.10), (5.15) admit unique solutions
$\alpha_{1}(\cdot)$, $\beta_{1}(\cdot)$ respectively, then
$\mathcal{C}_{7}(\cdot)$, $\mathcal{C}_{8}(\cdot)$, $\mathcal{C}_{9}(\cdot)$,
$\mathcal{C}_{16}(\cdot)$, $\mathcal{C}_{17}(\cdot)$ and
$\mathcal{C}_{18}(\cdot)$ become determined functions. As a result, (5.11) and
(5.16) become coupled first-order linear ODEs, whose solvability are easy to
get. Finally, (5.12) is also solvable since $\mathcal{C}_{10}(\cdot)$ is
known. A local solvability result of (5.10) and (5.15) will be given later by
Picard-Lindel$\ddot{o}$f’s Theorem.

Now we present the main result of this section.

######  Theorem 5.1.

Suppose Assumption 4 holds. Let (5.10)-(5.12), (5.15) and (5.16) have unique
solutions $\alpha_{1}(\cdot)$, $\alpha_{2}(\cdot)$, $\alpha_{3}(\cdot)$,
$\beta_{1}(\cdot)$ and $\beta_{2}(\cdot)$ respectively. Then the LEQ problem
given by (5.1) and (5.2) has a unique optimal feedback control:

$\displaystyle\bar{u}(t)=$ $\displaystyle-
R_{44}^{-1}(t)\Big{\\{}\big{[}\alpha_{1}(t)D_{1}(t)+R_{14}(t)+R_{24}(t)\beta_{1}(t)\big{]}\bar{X}(t)$
(5.17)
$\displaystyle\qquad+\alpha_{2}(t)D_{1}(t)+R_{24}(t)\beta_{2}(t)+R_{34}(t)\beta_{1}(t)\sigma(t)\Big{\\}},\quad
a.e.,\quad\mathbb{P}\mbox{-}a.s..$

Furthermore, the optimal cost is

$J(\bar{u}(\cdot))=\exp\bigg{\\{}\theta\bigg{(}\frac{1}{2}\alpha_{1}(0)x_{0}^{2}+\alpha_{2}(0)x_{0}+\alpha_{3}(0)\bigg{)}\bigg{\\}}.$
(5.18)

###### Proof.

For any given admissible control $u(\cdot)$, denoting
$(X^{u}(\cdot),Y^{u}(\cdot),Z^{u}(\cdot))$ as the corresponding solution of
(5.1). Applying Itô’s formula to
$\frac{\alpha_{1}(\cdot)}{2}(X^{u}(\cdot))^{2}+\alpha_{2}(\cdot)X^{u}(\cdot)+\alpha_{3}(\cdot)$
and integrating it from $0$ to $T$, it follows that

$\frac{S_{1}}{2}(X^{u}(T))^{2}+\frac{S_{2}}{2}(Y^{u}(0))^{2}=\frac{1}{\theta}\ln\Upsilon(0)+\int_{0}^{T}\,d\bigg{(}\frac{\alpha_{1}(t)}{2}(X^{u}(t))^{2}+\alpha_{2}(t)X^{u}(t)+\alpha_{3}(t)\bigg{)},$
(5.19)

where
$\Upsilon(0):=\exp\big{\\{}\theta\big{(}\frac{1}{2}\alpha_{1}(0)x_{0}^{2}+\alpha_{2}(0)x_{0}+\alpha_{3}(0)\big{)}\big{\\}}$
is a constant.

In view of (5.4) and (5.10)-(5.12), substituting (5.19) into (5.2) and using
the completion-of-squares technique, the corresponding cost becomes:

$\displaystyle J(u(\cdot))$
$\displaystyle=\Upsilon(0)\mathbb{E}\bigg{[}\exp\bigg{\\{}\theta\int_{0}^{T}\bigg{[}-\frac{\theta}{2}\sigma^{2}\alpha_{1}^{2}(X^{u})^{2}-\theta\sigma^{2}\alpha_{1}\alpha_{2}X^{u}-\frac{\theta}{2}\sigma^{2}\alpha_{2}^{2}$
$\displaystyle\qquad+\frac{1}{2}R_{44}\big{\\{}u+R_{44}^{-1}\big{[}(\alpha_{1}D_{1}+R_{14}+R_{24}\beta_{1})X^{u}+\alpha_{2}D_{1}+R_{24}\beta_{2}+R_{34}\beta_{1}\sigma\big{]}\big{\\}}^{2}\bigg{]}\,dt$
$\displaystyle\qquad+\int_{0}^{T}\theta\sigma(\alpha_{1}X^{u}+\alpha_{2})\,dW(t)\bigg{\\}}\bigg{]}$
$\displaystyle=\Upsilon(0)\mathbb{E}\bigg{[}\exp\bigg{\\{}\theta\int_{0}^{T}\bigg{[}\frac{1}{2}R_{44}\big{\\{}u+R_{44}^{-1}\big{[}(\alpha_{1}D_{1}+R_{14}+R_{24}\beta_{1})X^{u}$
$\displaystyle\qquad+\alpha_{2}D_{1}+R_{24}\beta_{2}+R_{34}\beta_{1}\sigma\big{]}\big{\\}}^{2}\bigg{]}\,dt$
$\displaystyle\qquad+\int_{0}^{T}\theta\sigma(\alpha_{1}X^{u}+\alpha_{2})\,dW(t)-\frac{1}{2}\int_{0}^{T}\theta^{2}\sigma^{2}(\alpha_{1}X^{u}+\alpha_{2})^{2}\,dt\bigg{\\}}\bigg{]}$
$\displaystyle=\Upsilon(0)\mathbb{E}^{u}\bigg{[}\exp\bigg{\\{}\theta\int_{0}^{T}\bigg{[}\frac{1}{2}R_{44}\big{\\{}u+R_{44}^{-1}\big{[}(\alpha_{1}D_{1}+R_{14}+R_{24}\beta_{1})X^{u}$
$\displaystyle\qquad+\alpha_{2}D_{1}+R_{24}\beta_{2}+R_{34}\beta_{1}\sigma\big{]}\big{\\}}^{2}\bigg{]}\,dt\bigg{\\}}\bigg{]},$

where $\mathbb{E}^{u}$ is the expectation with respect to $\mathbb{P}^{u}$
given by

$\frac{d\mathbb{P}^{u}}{d\mathbb{P}}:=\exp\bigg{\\{}\int_{0}^{T}\theta\sigma(t)\big{(}\alpha_{1}(t)X^{u}(t)+\alpha_{2}(t)\big{)}\,dW(t)-\frac{1}{2}\int_{0}^{T}\theta^{2}\sigma^{2}(t)\big{(}\alpha_{1}(t)X^{u}(t)+\alpha_{2}(t)\big{)}^{2}\,dt\bigg{\\}}.$

From the above calculation, it is obvious that for $\bar{u}(\cdot)$ given by
(5.17), the related optimal cost is $J(\bar{u}(\cdot))=\Upsilon(0)$. Hence,
for any given admissible $u(\cdot)$, we can conclude that

$J(u(\cdot))\geq J(\bar{u}(\cdot)).$

The proof is complete. ∎

As evident from the previous discussions, the optimal control and optimal cost
depend heavily on solutions of (5.10)-(5.12) and (5.15)-(5.16). From Remark
5.1, we only need to discuss the unique solvability of coupled Riccati
equations (5.10) and (5.15). Noting that (5.10) and (5.15) are not in
classical form, we do not have results about their solvability in general now.
However, next we investigate the local solvability of them. Set

$K:=\begin{bmatrix}\alpha_{1}&0\\\ 0&\beta_{1}\end{bmatrix},\
J:=\begin{bmatrix}0&1\\\ 1&0\end{bmatrix},\ K_{T}:=\begin{bmatrix}S_{1}&0\\\
0&G\end{bmatrix},\ \mathcal{D}_{1}:=\begin{bmatrix}\mathcal{C}_{1}&0\\\
0&\frac{1}{2}\mathcal{C}_{11}\end{bmatrix},\
\mathcal{D}_{2}:=\begin{bmatrix}2\mathcal{C}_{2}&0\\\
0&-\mathcal{C}_{12}\end{bmatrix},$
$\mathcal{D}_{3}:=\begin{bmatrix}2\mathcal{C}_{3}&0\\\
0&-\mathcal{C}_{13}\end{bmatrix},\
\mathcal{D}_{4}:=\begin{bmatrix}\mathcal{C}_{4}&0\\\
0&\frac{1}{2}\mathcal{C}_{14}\end{bmatrix},\
\mathcal{D}_{5}:=\begin{bmatrix}\mathcal{C}_{5}&0\\\ 0&0\end{bmatrix},\
\mathcal{D}_{6}:=\begin{bmatrix}\mathcal{C}_{6}&0\\\
0&\mathcal{C}_{15}\end{bmatrix}.$

Then equations (5.10) and (5.15) are equal to the following

$\left\\{\begin{aligned}
\dot{K}&(t)+\mathcal{D}_{1}(t)K(t)+K(t)\mathcal{D}_{1}(t)+\mathcal{D}_{2}(t)JK(t)J+K(t)J\mathcal{D}_{3}(t)K(t)J\\\
&+K\mathcal{D}_{4}(t)K(t)+JK(t)J\mathcal{D}_{5}(t)JK(t)J+\mathcal{D}_{6}(t)=0,\quad
t\in[0,T],\\\ K&(T)=K_{T}.\end{aligned}\right.$ (5.20)

Since
$\mathcal{G}(t,K):=\mathcal{D}_{1}K+K\mathcal{D}_{1}+\mathcal{D}_{2}JKJ+KJ\mathcal{D}_{3}KJ+K\mathcal{D}_{4}K+JKJ\mathcal{D}_{5}JKJ+\mathcal{D}_{6}$
is continuous in $t$ and uniformly locally Lipschitz continuous in $K$.
According to Picard-Lindel$\ddot{o}$f’s Theorem (See, for example, Theorem
15.2 in Agarwal and O’Regan [1]), we know that there exists an interval
$[T-\eta,T]\subseteq[0,T]$ such that the terminal value problem (5.20) as well
as (5.10) and (5.15) has a unique solution on $[T-\eta,T]$.

##  6 Concluding remarks

In this paper, we have obtained a global maximum principle for a kind of risk-
sensitive optimal control problem for fully coupled forward-backward
stochastic systems. The control variable can enter the diffusion term of the
state equation and the control domain is not necessarily convex. A new global
maximum principle is obtained without assuming that the value function is
smooth. The maximum condition, the first- and second-order adjoint equations
heavily depend on the risk-sensitive index. An optimal control problem with a
fully coupled linear forward-backward stochastic system and an exponential-
quadratic cost functional is discussed. The optimal feedback control and
optimal cost are obtained by using Girsanov’s Theorem and completion-of-
squares approach via risk-sensitive Riccati equations. A local solvability
result of coupled risk-sensitive Riccati equations (5.10) and (5.15) is given
by Picard-Lindel$\ddot{o}$f’s Theorem.

Possible future research includes the partially observed problems, mean field
type control and differential games. We will consider these topics in the near
future.

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