# Stochastic Linear-quadratic Control Problems with Affine
Constraints††thanks: This work was supported by the National Natural Science
Foundation of China (11671282, 12171339).

Zhun Gou  College of Mathematics and Statistics, Chongqing Technology and
Business University, Chongqing 400067, P.R.China  Nan-jing Huang Corresponding
author<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of
Mathematics, Sichuan University, Chengdu, Sichuan 610064, P.R. China  Xian-jun
Long  College of Mathematics and Statistics, Chongqing Technology and Business
University, Chongqing 400067, P.R.China  Jian-hao Kang  School of Mathematics,
Southwest Jiaotong University, Chengdu, Sichuan 610031, P.R. China

Abstract. In this paper, we investigate the stochastic linear-quadratic
control problems with affine constraints in random coefficients case. With the
help of the Pontryagin maximum principle and stochastic Riccati equation, the
dual problem of original problem is established and the feedback solution of
the optimal control problem is obtained. Under the Slater condition, the
equivalence is proved between the solutions to the original problem and the
ones of the dual problem, and the KKT condition is also provided for the dual
problem. Finally, an invertibility assumption is given for ensuring the
uniqueness of the solutions to the dual problem.

Keywords: Stochastic control; Affine constraint; Feedback solution; KKT
condition.

2020 Mathematics Subject Classification: 49N10, 90C46, 93B52, 93E20.

##  1 Introduction

####

Linear-quadratic control problems (LQ problems for short) constitute the most
significant part of control theory and has been widely investigated in the
literature for a variety of deterministic control systems [3, 10]. There are
two main reasons. (i) With the help of Riccati equations, for such LQ
problems, we are able to find the feedback solutions (or optimal feedback
controls), which exhibit elegant properties due to their simple and nice
structures. Furthermore, feedback solutions are very useful and particularly
important in many practical control problems since they keep the corresponding
control to be robust with respect to (small) perturbation [12]. (ii) In lots
of nonlinear control problems, it is really hard or even impossible to find
the feedback solutions, while many of them can be reasonably approximated by
the LQ problems [24].

Stochastic linear-quadratic control problems (SLQ problems for short), as
important extensions of the LQ problems, can date back to the pioneered work
[22]. Since then SLQ problems have attract more and more researchers and
remain one of the most fascinating topics in control theory [5]. On the other
hand, for some practical reasons, one has to consider constraints on the state
and control when dealing with stochastic control problems, i.e., the
admissible control sets in SLQ problems with constraints are smaller than the
ones (the whole space) in SLQ problems with no constraints. Various type
constraints have been considered in SLQ problems in the existing works, such
as quadratic constraints [11], terminal state constraint [1], expectation
constraints [4, 7] and risk constraints [7, 20]. Such constraints ”shrink”
original admissible control sets and make the stochastic control problem more
complicated and more challengeable. Nevertheless, we show by Examples 2.1 and
2.2 that many ”shrunken” admissible control sets can be approximatively
described by original admissible control sets with finite many affine-type
constraints, in which the coefficients are random. Thus it would be
interesting and significant to study the stochastic control problems with
affine constraints (SLQ-AC problems for short) in random coefficients case.
However, to the best of our knowledge, there are few works studying SLQ-AC
problems (for the study of stochastic control problems with random
coefficients, we refer the reader to [15, 21, 19, 14] and the references
within).

The overarching goal of this paper is to study the SLQ-AC problems in random
coefficients case. The main contributions of this paper can be organized as
follows: (i) establishes the KKT condition including the feedback expression
of the optimal control, which has additional ”concession” terms; characterizes
the ”concession” terms explicitly with the help of backward stochastic
differential equations (BSDEs) and the Malliavin derivative; (iii) gives a
sufficient condition for ensuring the uniqueness of the optimal control of the
dual problem.

The rest of this paper is structured as follows. The next section introduces
some necessary preliminaries including the formulation of the SLQ problems
with affine constraints and two motivating examples. In Section 3, by using
the Pontryagin maximum principle, we obtain the formulation of the dual
problem to which the solutions are as well the ones to original SLQ problems
with affine constraints under the Slater condition, and make the invertibility
assumption for ensuring the Slater condition and the uniqueness of solutions
to the dual problem. Then we give the KKT condition for solving the dual
problem. Before concluding this paper in Section 5, we provide a sufficient
condition for guaranteeing the invertibility assumption, and give two examples
to show the effectiveness of our main results.

##  2 Preliminaries

In this section, we introduce some basic notations and present the formulation
of the SLQ problems with affine constraints. Then we give two motivating
examples for studying such problems.

###  2.1 Problem Formulation

Let $\mathbb{R}^{n\times m}$ be the Euclidean space of $n\times m$-matrices
$\Xi$ equipped with the norm $\|\Xi\|_{\mathbb{R}^{n\times
m}}=tr^{\frac{1}{2}}(\Xi^{\top}\Xi)$ and inner product
$\langle\Xi,\widetilde{\Xi}\rangle_{\mathbb{R}^{n\times
m}}=tr(\Xi^{\top}\widetilde{\Xi})$ ($\widetilde{\Xi}\in\mathbb{R}^{n\times
m}$), where $tr(\cdot)$ and $(\cdot)^{\top}$ represent the trace of an
$n\times n$-matrix and the transpose of an $n\times m$-matrix, respectively.
Let $\mathcal{I}_{n}$ be the $n\times n$-identity matrix, $\mathcal{S}^{n}$
the subset of symmetric matrices of $\mathbb{R}^{n\times n}$,
$\mathcal{S}_{+}^{n}$ the subset of positive definite matrices of
$\mathcal{S}^{n}$, and $\mathcal{S}_{++}^{n}$ the subset of strictly positive
definite matrices of $\mathcal{S}_{+}^{n}$.

In this paper, we consider the controlled linear stochastic differential
equation in a filtered probability space
$(\Omega,\mathfrak{F},\mathfrak{F}_{t},\mathbb{P})$ satisfying the usual
hypothesis as follows:

$\begin{cases}dX(t)=\left[A(t)X(t)+B(t)u(t)\right]dt+\left[C(t)X(t)+D(t)u(t)\right]dB_{t},\quad
t\in[s,T],\\\ X(s)=\xi\in\mathbb{R}^{n}.\end{cases}$ (1)

where $[s,T]$ denotes a finite time duration with $T>s\geq 0$; $X(\cdot)$ is
the $\mathbb{R}^{n}$-valued state variable determined by the
$\mathbb{R}^{m}$-valued control variable $u(\cdot)$; $B_{t}$ is a standard
one-dimensional Brownian motion which generates the right-continuous and
increasing $\sigma$-algebra $\mathcal{F}=(\mathcal{F}_{t})_{t\geq 0}$;
$A,C:[0,T]\times\Omega\rightarrow\mathbb{R}^{n\times n}$,
$B,D\in[0,T]\times\Omega\rightarrow\mathbb{R}^{n\times m}$ are all bounded and
$\mathcal{F}$-progressively measurable coefficients. For any given Euclidean
space $\mathcal{H}$ and $p\in[1,+\infty)$, we make use of the following
notations throughout this paper.

  * •

$\mathcal{L}^{p}([s,T],\mathcal{H})$: the space of all measurable functions
$f:[s,T]\rightarrow\mathcal{H}$ with
$\int_{s}^{T}\|f(t)\|^{p}_{\mathcal{H}}dt<\infty$.

  * •

$\mathcal{L}^{p}_{\mathcal{F}_{t}}(\Omega,\mathcal{H})$: the space of all
$\mathcal{H}$-valued $\mathcal{F}_{t}$-measurable random variables $\eta$ with
$\mathbb{E}[\|\eta\|_{\mathcal{H}}^{p}]<\infty$.

  * •

$\mathcal{M}^{p}_{\mathcal{F}}([s,T],\mathcal{H})$: the Banach space of all
$\mathcal{H}$-valued $\mathcal{F}$-progressively measurable processes
$\\{X(t),s\leq t\leq T\\}$ with
$\mathbb{E}\left[\sup\limits_{t\in[s,T]}\|X(t)\|^{p}_{\mathcal{H}}\right]<\infty$.

  * •

$\mathcal{L}^{\infty}_{\mathcal{F}}(\Omega,\mathcal{M}^{p}_{\mathcal{F}}([s,T],\mathcal{H}))$:
the Banach space of all $\mathcal{F}$-progressively measurable processes
$\\{X(t),s\leq t\leq T\\}$ with
$\mathop{\mbox{esssup}}\limits_{\omega\in\Omega}\left(\sup\limits_{t\in[s,T]}\|X(t)\|^{p}_{\mathcal{H}}\right)^{\frac{1}{p}}<\infty$.

  * •

$\mathcal{L}^{p}_{\mathcal{F}}([s,T],\mathcal{H})$: the space of all
$\mathcal{H}$-valued $\mathcal{F}$-progressively measurable processes
$\\{X(t),s\leq t\leq T\\}$ with
$\mathbb{E}\left[\int_{s}^{T}\|X(t)\|_{\mathcal{H}}^{p}dt\right]<\infty$.

Clearly, $\mathcal{L}^{2}_{\mathcal{F}_{t}}(\Omega,\mathcal{H})$ (resp.
$\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathcal{H})$) is a Hilbert space
equipped with the norm
$\|\cdot\|_{\mathcal{L}^{2}_{\mathcal{F}_{t}}(\Omega,\mathcal{H})}=\mathbb{E}^{\frac{1}{2}}[\|\cdot\|_{\mathcal{H}}^{2}]$
(resp.
$\|\cdot\|_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathcal{H})}=\mathbb{E}^{\frac{1}{2}}\left[\int_{s}^{T}|\cdot(t)|_{\mathcal{H}}^{2}dt\right]$)
and inner product
$\langle\cdot,\cdot\rangle_{\mathcal{L}^{2}_{\mathcal{F}_{t}(\Omega,\mathcal{H})}}$
(resp.
$\langle\cdot,\cdot\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathcal{H})}$).
Without loss of generality, we denote $\langle\cdot,\cdot\rangle$ (resp.
$|\cdot|$) the inner product (resp. the norm) in any Euclidean space. For
notational convenience we will frequently suppress the $t$-dependence of a
process involved in an equation or an integral.

The admissible control set of $u(\cdot)$ is defined as follows, where both
equality and inequality constraints are involved.

######  Definition 2.1.

For given positive integer $l,l^{\prime}\in\mathbb{R}^{*}$ with $1\leq
l^{\prime}\leq l$,
$\alpha_{i}\in\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})$,
$\beta_{i}\in\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})$,
$\gamma_{i}\in{\mathcal{L}^{2}_{\mathcal{F}_{T}}(\Omega,\mathbb{R}^{n})}$ and
$a=(a_{1},a_{2},\cdots,a_{l})^{\top}\in\mathbb{R}^{l}$, the admissible control
set is defined as follows.

$\displaystyle U^{ad}=\Bigg{\\{}$ $\displaystyle
u\in\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})\big{|}u\;\mbox{satisfies
the following affine constraints:}$ $\displaystyle\langle
X,\alpha_{i}\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})}+\langle
u,\beta_{i}\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})}+\langle
X(T),\gamma_{i}\rangle_{\mathcal{L}^{2}_{\mathcal{F}_{T}}(\Omega,\mathbb{R}^{n})}\leq
a_{i},\;i=1,2,\cdots,l^{\prime},$ $\displaystyle\langle
X,\alpha_{i}\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})}+\langle
u,\beta_{i}\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})}+\langle
X(T),\gamma_{i}\rangle_{\mathcal{L}^{2}_{\mathcal{F}_{T}}(\Omega,\mathbb{R}^{n})}=a_{i},\;i=l^{\prime}+1,l^{\prime}+2,\cdots,l.\Bigg{\\}}$
(2)

######  Remark 2.1.

For any given $u\in U^{ad}$, there exists a unique solution
$X\in\mathcal{M}^{2}_{\mathcal{F}}([s,T],\mathcal{H})$ [13].

For the controlled pair $(X,u)$ governed by SDE (1), consider the following
cost functional

$J(s,\xi,u)=\mathbb{E}\left[\displaystyle{\int_{s}^{T}}\left\langle\begin{bmatrix}E(t)&F(t)&G(t)\\\
F^{\top}(t)&I(t)&K(t)\\\
G^{\top}(t)&K^{\top}(t)&0\end{bmatrix}\begin{bmatrix}X(t)\\\ u(t)\\\
1\end{bmatrix},\begin{bmatrix}X(t)\\\ u(t)\\\ 1\end{bmatrix}\right\rangle
dt+\left\langle\begin{bmatrix}M&N\\\ N&0\end{bmatrix}\begin{bmatrix}X(T)\\\
1\end{bmatrix},\begin{bmatrix}X(T)\\\ 1\end{bmatrix}\right\rangle\right],$ (3)

where $E:[0,T]\times\Omega\rightarrow\mathcal{S}_{+}^{n}$,
$F:[0,T]\times\Omega\rightarrow\mathbb{R}^{n\times m}$,
$I:[0,T]\times\Omega\rightarrow\mathcal{S}_{++}^{m}$ and
$M:[0,T]\times\Omega\rightarrow\mathcal{S}_{+}^{n}$ are all bounded and
$\mathcal{F}$-progressively measurable. Moreover, we require
$G\in\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})$,
$K\in\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})$,
$N\in\mathcal{L}^{2}_{\mathcal{F}_{T}}(\Omega,\mathbb{R}^{n})$ and

$E-FI^{-1}F^{\top}\in\mathcal{L}^{\infty}_{\mathcal{F}}(\Omega,\mathcal{M}^{1}_{\mathcal{F}}([s,T],\mathcal{S}_{+}^{m}))$

in the sense that
$\left\langle(E-FI^{-1}F^{\top})v,v\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})}\geq
0$ for all $v\in\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})$. Then,
the SLQ-AC problems can be specified as follows.

######  Problem 2.1.

Find $\widehat{u}\in U^{ad}$ such that

$J(s,\xi,\widehat{u})=\inf\limits_{u\in U^{ad}}J(s,\xi,u),$

where $J$ is given by (3).

###  2.2 Motivating Examples

To begin with, we introduce the following theorem.

######  Theorem 2.1.

For given closed and convex set $U\subsetneqq\mathcal{H}$, where $\mathcal{H}$
is a Banach space with the dual space $\mathcal{H}^{*}$. Set
$\mathfrak{B}=\\{\nu\in\mathcal{H}^{*}|\nu(v)\leq 0,\;\forall v\in U\\}$ and
$\widetilde{U}=\\{v\in\mathcal{H}|f(v)\leq 0,\;\forall f\in\mathfrak{B}\\}$.
Then $U=\widetilde{U}$.

###### Proof.

Clearly, one has $U\subset\widetilde{U}$. We need to show that
$\widetilde{U}\subset U$. Suppose there exists a point
$v_{0}\in\widetilde{U}\setminus U$. Since $U$ is closed and convex, it follows
from the separation theorem of convex sets (see, e.g., Corollary 3.10 in [6])
that there exists a functional $\widetilde{\nu}\in\mathfrak{B}$ such that
$\widetilde{\nu}(v_{0})>0$ and $\widetilde{\nu}(v)\leq 0$ ($\forall
v\in\mathcal{H}$). On the other hand, by $v_{0}\in\widetilde{U}$ and
$\widetilde{\nu}\in\mathfrak{B}$, we have $\widetilde{\nu}(v_{0})\leq 0$,
which forms a contradiction. Summarizing the above arguments, one has
$U=\widetilde{U}$. ∎

Theorem 2.1 states that any closed and convex subset in Banach space can be
characterized by some affine constraints. We would like to provide two
practical examples for Theorem 2.1, in which and the forms of $\widetilde{U}$
are explicitly given and can be approximately described by adding finite many
affine constraints on the space
$\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})$. The first example is
inspired by [11], which is concerned with the quadratic constraints.

######  Example 2.1.

For bounded and $\mathcal{F}$-progressively measurable functionals
$\widetilde{E}:[0,T]\times\Omega\rightarrow\mathcal{S}_{+}^{n}$,
$\widetilde{I}:[0,T]\times\Omega\rightarrow\mathcal{S}_{++}^{m}$,
$\widetilde{M}:[0,T]\times\Omega\rightarrow\mathcal{S}_{+}^{n}$, consider the
following quadratic functional

$\Theta:U_{1}\rightarrow\mathbb{R},\;\Theta(u)=\langle
X,\widetilde{E}X\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})}+\langle
u,\widetilde{I}u\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})}+\langle
X(T),\widetilde{M}X(T)\rangle_{\mathcal{L}^{2}_{\mathcal{F}_{T}}(\Omega,\mathbb{R}^{n})}$

and the admissible control set

$U_{1}=\left\\{u\in\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})\big{|}\Theta(u)\leq
a_{0}\right\\},$

where $a_{0}$ is a given positive constant, and the controlled pair $(X,u)$ is
governed by (1). Then one can easily show that $U_{1}$ is closed and convex in
$\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})$. On the other hand, let

$\displaystyle
U_{2}=\Bigg{\\{}u\in\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})\big{|}$
$\displaystyle\langle
X,\widetilde{E}\widetilde{\alpha}\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})}+\langle
u,\widetilde{I}\widetilde{\beta}\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})}+\langle
X(T),\widetilde{M}\widetilde{\gamma}\rangle_{\mathcal{L}^{2}_{\mathcal{F}_{T}}(\Omega,\mathbb{R}^{n})}\leq
a_{0},$ $\displaystyle\mbox{for
all}\;(\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma})\in\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})\times\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})\times\mathcal{L}^{2}_{\mathcal{F}_{T}}(\Omega,\mathbb{R}^{n})\;\mbox{such
that}$
$\displaystyle\langle\widetilde{E}\widetilde{\alpha},\widetilde{\alpha}\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})}+\langle\widetilde{I}\widetilde{\beta},\widetilde{\beta}\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})}+\langle\widetilde{M}\widetilde{\gamma},\widetilde{\gamma}\rangle_{\mathcal{L}^{2}_{\mathcal{F}_{T}}(\Omega,\mathbb{R}^{n})}=a_{0}\Bigg{\\}}.$

Then it follows from

$\displaystyle\langle
X,\widetilde{E}\widetilde{\alpha}\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})}+\langle
u,\widetilde{I}\widetilde{\beta}\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})}+\langle
X(T),\widetilde{M}\widetilde{\gamma}\rangle_{\mathcal{L}^{2}_{\mathcal{F}_{T}}(\Omega,\mathbb{R}^{n})}$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\Theta(u)+\frac{1}{2}\left[\langle\widetilde{E}\widetilde{\alpha},\widetilde{\alpha}\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})}+\langle\widetilde{I}\widetilde{\beta},\widetilde{\beta}\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})}+\langle\widetilde{M}\widetilde{\gamma},\widetilde{\gamma}\rangle_{\mathcal{L}^{2}_{\mathcal{F}_{T}}(\Omega,\mathbb{R}^{n})}\right]\leq
a_{0}$

that $U_{2}\subseteq U_{1}$. Moreover, by choosing
$\widetilde{\alpha}=\sqrt{\frac{a_{0}}{\Theta(u)}}X$,
$\widetilde{\beta}=\sqrt{\frac{a_{0}}{\Theta(u)}}u$ and
$\widetilde{\alpha}=\sqrt{\frac{a_{0}}{\Theta(u)}}X(T)$, one has
$U_{1}\subseteq U_{2}$. Hence we conclude $U_{1}=U_{2}$. Let

$\displaystyle U^{p}_{2}=\Bigg{\\{}u\in$
$\displaystyle\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})\big{|}\langle
X,\widetilde{E}\widetilde{\alpha}^{i}\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})}+\langle
u,\widetilde{I}\widetilde{\beta}^{i}\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})}+\langle
X(T),\widetilde{M}\widetilde{\gamma}^{i}\rangle_{\mathcal{L}^{2}_{\mathcal{F}_{T}}(\Omega,\mathbb{R}^{n})}\leq
a_{0},$ $\displaystyle\mbox{for
all}\;(\widetilde{\alpha}^{i},\widetilde{\beta}^{i},\widetilde{\gamma}^{i})\in\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})\times\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})\times\mathcal{L}^{2}_{\mathcal{F}_{T}}(\Omega,\mathbb{R}^{n})\;(i=1,2,\cdots,p)$
$\displaystyle\mbox{such
that}\;\mathbb{P}\left\\{(\widetilde{\alpha}^{i},\widetilde{\beta}^{i},\widetilde{\gamma}^{i})=(\widetilde{\alpha}^{j},\widetilde{\beta}^{j},\widetilde{\gamma}^{j}),\;i\neq
j\right\\}=1\;\mbox{and}$
$\displaystyle\langle\widetilde{E}\widetilde{\alpha}^{i},\widetilde{\alpha}^{i}\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})}+\langle\widetilde{I}\widetilde{\beta}^{i},\widetilde{\beta}^{i}\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})}+\langle\widetilde{M}\widetilde{\gamma}^{i},\widetilde{\gamma}^{i}\rangle_{\mathcal{L}^{2}_{\mathcal{F}_{T}}(\Omega,\mathbb{R}^{n})}=a_{0}\Bigg{\\}}.$

Then $U_{1}$ can be approximately described by $U^{p}_{2}$ when
$p\in\mathbb{N}^{*}$ is large enough.

The second example is concerned with risk constraints (see, e.g., [7]).

######  Example 2.2.

For a constant $a_{0}>0$, let

$U_{3}=\left\\{u\in\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})\big{|}\mbox{Var}(X(T))=\mathbb{E}(X^{2}(T))-\mathbb{E}^{2}(X(T))\leq
a_{0}\right\\}$

and

$\displaystyle
U_{4}=\Bigg{\\{}u\in\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})\big{|}$
$\displaystyle\left\langle\widehat{\gamma},X(T)\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}_{T}}(\Omega,\mathbb{R}^{n})}\leq
a_{0}$ $\displaystyle\mbox{for
all}\;\widehat{\gamma}\in\mathcal{L}^{2}_{\mathcal{F}_{T}}(\Omega,\mathbb{R}^{n})\;\mbox{such
that}\;\mathbb{E}(\widehat{\gamma})=0\;\mbox{and}\;\mbox{Var}(\widehat{\gamma})=a_{0}\Bigg{\\}},$

where $\mbox{Var}(\cdot)$ denotes the variance of a random variable, and the
controlled pair $(X,u)$ is governed by (1). By similar arguments in Example
2.1, we know $U_{3}$ is convex and closed in
$\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})$ and $U_{3}=U_{4}$. Let

$\displaystyle
U^{p}_{4}=\Bigg{\\{}u\in\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})\big{|}$
$\displaystyle\left\langle\widehat{\gamma}^{i},X(T)\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}_{T}}(\Omega,\mathbb{R}^{n})}\leq
a_{0},\quad\mbox{for
all}\;\widehat{\gamma}^{i}\in\mathcal{L}^{2}_{\mathcal{F}_{T}}(\Omega,\mathbb{R}^{n})\;(i=1,2,\cdots,p)$
$\displaystyle\mbox{such
that}\;\mathbb{P}\left\\{\widetilde{\gamma}^{i}=\widetilde{\gamma}^{j},\;i\neq
j\right\\}=1,\;\mathbb{E}(\widehat{\gamma}^{i})=0\;\mbox{and}\;\mbox{Var}(\widehat{\gamma}^{i})=a_{0}\Bigg{\\}}.$

Then $U_{3}$ can be approximately described by $U^{p}_{4}$ when
$p\in\mathbb{N}^{*}$ is large enough.

##  3 Dual Problem and the KKT condition

In this section, we give the formulation of the dual problem for Problem 2.1,
and then derive the KKT condition for solving the dual problem. To begin with,
we make the following assumption for Problem 2.1.

######  Assumption 3.1.

(the Slater condition) There exists at least one $\overline{u}\in U^{ad}$ such
that

$\begin{cases}\langle\overline{X},\alpha_{i}\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})}+\langle\overline{u},\beta_{i}\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})}+\langle\overline{X}(T),\gamma_{i}\rangle_{\mathcal{L}^{2}_{\mathcal{F}_{T}}(\Omega,\mathbb{R}^{n})}<a_{i},\;i=1,2,\cdots,l^{\prime},\\\
\langle\overline{X},\alpha_{i}\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})}+\langle\overline{u},\beta_{i}\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})}+\langle\overline{X}(T),\gamma_{i}\rangle_{\mathcal{L}^{2}_{\mathcal{F}_{T}}(\Omega,\mathbb{R}^{n})}=a_{i},\;i=l^{\prime}+1,l^{\prime}+2,\cdots,l,\end{cases}$

where the controlled pair $(\overline{X},\overline{u})$ is governed by (1).

Then, we introduce the following relaxed SLQ problem.

######  Problem 3.1.

Find $u^{*}=u^{*}(\lambda)$ such that

$\widehat{J}(s,\xi,\lambda,u^{*})=\inf\limits_{u\in\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})}J(s,\xi,\lambda,u),$

where

$\widehat{J}(s,\xi,\lambda,u)=J(u)+2\sum\limits_{i=1}^{l}\lambda_{i}\left[\langle
X,\alpha_{i}\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})}+\langle
u,\beta_{i}\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})}+\langle
X(T),\gamma_{i}\rangle_{\mathcal{L}^{2}_{\mathcal{F}_{T}}(\Omega,\mathbb{R}^{n})}\right]-2\langle
a,\lambda\rangle,$

and where $\lambda=(\lambda_{1},\lambda_{2},\cdots,\lambda_{l})^{\top}$
($\lambda_{i}\geq 0$, $i=1,2,\cdots,l$).

######  Remark 3.1.

$\widehat{J}(s,\xi,\lambda,u^{*})$ is nothing but the Lagrangian dual
functional of Problem 2.1.

For Problem 3.1, we can define the Hamiltonian functional as follows:

$H=-\frac{1}{2}\left\langle\begin{bmatrix}E&F&G\\\ F^{\top}&I&K\\\
G^{\top}&K^{\top}&0\end{bmatrix}\begin{bmatrix}X\\\ u\\\
1\end{bmatrix},\begin{bmatrix}X\\\ u\\\
1\end{bmatrix}\right\rangle-\sum\limits_{i=1}^{l}\lambda_{i}\left[\langle
X,\alpha_{i}\rangle+\langle u,\beta_{i}\rangle\right]+\langle
AX+Bu,Y\rangle+\langle CX+Du,Z\rangle.$

Then, one can easily check that derivatives of $H$ read

$\begin{cases}\frac{\partial H}{\partial
X}=-\left(EX+Fu+G\right)-\sum\limits_{i=1}^{l}\lambda_{i}\alpha_{i}+A^{\top}Y+C^{\top}Z,\\\
\frac{\partial H}{\partial
u}=-\left(F^{\top}X+Iu+K\right)-\sum\limits_{i=1}^{l}\lambda_{i}\beta_{i}+B^{\top}Y+D^{\top}Z\end{cases}$

and the adjoint BSDE is

$\begin{cases}dY=\left[EX+Fu+G+\sum\limits_{i=1}^{l}\lambda_{i}\alpha_{i}-A^{\top}Y-C^{\top}Z\right]dt+ZdB_{t},\quad
t\in[s,T],\\\
Y(T)=-MX(T)-N-\sum\limits_{i=1}^{l}\lambda_{i}\gamma_{i}.\end{cases}$

According to the Pontryagin maximum princile (see, e.g., Theorem 3.3 [Chapter
3] in [24]), the optimal control of Problem 3.1 satisfies

$Iu^{*}=-\left(F^{\top}X^{*}+K\right)-\sum\limits_{i=1}^{l}\lambda_{i}\beta_{i}+B^{\top}Y^{*}+D^{\top}Z^{*}.$
(4)

Next, we search for the feedback expression of $u^{*}$. Set
$Y^{*}=-PX^{*}-Q-\sum\limits_{i=1}^{l}\lambda_{i}R_{i}$. Then one can show
that $P$ is described by the following backward stochastic Riccati equation
(BSRE)

$\begin{cases}dP=-\left[E+PA+A^{\top}P+C^{\top}\Lambda+\Lambda
C+C^{\top}PC-L^{\top}S^{-1}L\right]dt+\Lambda dB_{t},\quad t\in[s,T],\\\
P(T)=M,\end{cases}$ (5)

and $Q$, $R_{i}$ satisfy the following system of backward stochastic
differential equations (BSDEs).

$\begin{cases}dQ=-\left[A^{\top}Q+C^{\top}\pi+G-L^{\top}S^{-1}\psi\right]dt+\pi
dB_{t},\quad t\in[s,T],\\\
dR_{i}=-\left[A^{\top}R_{i}+C^{\top}r_{i}+\alpha_{i}-L^{\top}S^{-1}\rho_{i}\right]dt+r_{i}dB_{t},\quad
t\in[s,T],\\\ Q(T)=N,\quad R_{i}(T)=\gamma_{i}.\end{cases}$ (6)

Here $S=I+D^{\top}PD$, $L=F^{\top}+B^{\top}P+D^{\top}\Lambda+D^{\top}PC$,
$\psi=K+B^{\top}Q+D^{\top}\pi$ and
$\rho_{i}=\beta_{i}+B^{\top}R_{i}+D^{\top}r_{i}$. Hence we have the following
feedback expression:

$u^{*}=-S^{-1}\left(LX^{*}+\psi+\sum\limits_{i=1}^{l}\lambda_{i}\rho_{i}\right).$
(7)

The existence and uniqueness of solutions to (5) and (6) can be ensured by the
following proposition.

######  Proposition 3.1.

The following two assertions hold:

  1. (I)

There exists a unique solution
$(P,\Lambda)\in\mathcal{L}^{\infty}_{\mathcal{F}}(\Omega,\mathcal{M}^{1}_{\mathcal{F}}([s,T],\mathcal{S}^{n}))\times\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathcal{S}^{n})$
to BSRE (5) such that for some $\varepsilon>0$, the inequality

$S=I+D^{\top}PD\geq\varepsilon\mathcal{I}_{m},\qquad\mbox{a.e.}\;t\in[s,T],\;\mathbb{P}-\mbox{a.s}.$
(8)

holds in the sense that
$\left\langle(I+D^{\top}PD)v,v\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})}\geq\varepsilon\|v\|^{2}_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})}$
for all $v\in\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})$.

  2. (II)

There exists a unique solution

$(Q,\pi,R_{i},r_{i})\in\mathcal{M}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})\times\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})\times\mathcal{M}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})\times\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})$

to the system of BSDEs (6).

###### Proof.

(I) follows from Theorem 6.3 in [18] (or Theorems 5.1 and 5.2 in [25])
directly. Then, noticing (8) and the linearity of (6), (II) follows from
Proposition 3.2 in [23] immediately. ∎

######  Remark 3.2.

From (2) and (6), one can easily show that

$U^{ad}=\Bigg{\\{}u\in\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})\bigg{|}\left\langle
u,\beta_{i}+B^{\top}R_{i}+D^{\top}r_{i}\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})}\leq
a_{i},\;i=1,2,\cdots,l\Bigg{\\}}.$

Obviously, $U^{ad}$ is closed and convex in
$\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})$.

Now, we are able to give the explicit formulation of the Lagranian dual
functional $\widehat{J}(s,\xi,\lambda,u^{*})$. Applying the Itô formula to
$\left\langle
X^{*},Y^{*}-Q-\sum\limits_{i=1}^{l}\lambda_{i}R_{i}\right\rangle$ derives

$\displaystyle\quad\;\left\langle
X^{*}(T),Y^{*}(T)-Q(T)-\sum\limits_{i=1}^{l}\lambda_{i}R_{i}(T)\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}_{T}}(\Omega,\mathbb{R}^{n})}-\left\langle\xi,Y^{*}(s)-Q(s)-\sum\limits_{i=1}^{l}\lambda_{i}R_{i}(s)\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}_{s}}(\Omega,\mathbb{R}^{n})}$
$\displaystyle=\left\langle
X^{*},EX^{*}+Fu^{*}+\sum\limits_{i=1}^{l}\lambda_{i}\alpha_{i}+G+A^{\top}Q+C^{\top}\pi+G-A^{\top}Y^{*}-C^{\top}Z^{*}-L^{\top}S^{-1}\psi\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})}$
$\displaystyle\quad\mbox{}+\left\langle
X^{*},\sum\limits_{i=1}^{l}\lambda_{i}\left(A^{\top}R_{i}+C^{\top}r_{i}+\alpha_{i}-L^{\top}S^{-1}\rho_{i}\right)\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})}$
$\displaystyle\quad\mbox{}+\left\langle
AX^{*}+Bu^{*},Y^{*}-Q-\sum\limits_{i=1}^{l}\lambda_{i}R_{i}\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})}+\left\langle
CX^{*}+Du^{*},Z^{*}-\pi-\sum\limits_{i=1}^{l}\lambda_{i}r_{i}\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})}$
$\displaystyle=\left\langle
X^{*},EX^{*}+2G-L^{\top}S^{-1}\left(\psi+\sum\limits_{i=1}^{l}\lambda_{i}\rho_{i}\right)+2\sum\limits_{i=1}^{l}\lambda_{i}\alpha_{i}\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})}+\left\langle
X^{*},Fu^{*}\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})}$
$\displaystyle\quad\mbox{}+\left\langle
u^{*},B^{\top}Y^{*}+D^{\top}Z^{*}-B^{\top}Q-D^{\top}\pi-\sum\limits_{i=1}^{l}\lambda_{i}\left(B^{\top}R_{i}+D^{\top}r_{i}\right)\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})}$
$\displaystyle=\left\langle
X^{*},EX^{*}+2Fu^{*}+2G\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})}+\left\langle
u^{*},Iu^{*}+2K\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})}$
$\displaystyle\quad\mbox{}-\left\langle
u^{*}+S^{-1}LX^{*},\psi+\sum\limits_{i=1}^{l}\lambda_{i}\rho_{i}\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})}+2\sum\limits_{i=1}^{l}\lambda_{i}\left[\left\langle
X^{*},\alpha_{i}\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})}+\left\langle
u^{*},\beta_{i}\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})}\right],$

where we have used (4). Combining (7), one has

$\displaystyle\widehat{J}(s,\xi,\lambda,u^{*})=$
$\displaystyle\left\langle\xi,P(s)\xi+2Q(s)+2\sum\limits_{i=1}^{l}\lambda_{i}R_{i}(s)\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}_{s}}(\Omega,\mathbb{R}^{n})}$
$\displaystyle\mbox{}-\left\langle
S^{-1}\left[\psi+\sum\limits_{i=1}^{l}\lambda_{i}\rho_{i}\right],\psi+\sum\limits_{i=1}^{l}\lambda_{i}\rho_{i}\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})}-2\langle\lambda,a\rangle$
$\displaystyle=$
$\displaystyle\left\langle\xi,P(s)\xi+2Q(s)\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}_{s}}(\Omega,\mathbb{R}^{n})}-\left\langle
S^{-1}\psi,\psi\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})}$
$\displaystyle\mbox{}+2\left\langle
R^{\top}(s)\xi-a,\lambda\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}_{s}}(\Omega,\mathbb{R}^{l})}-2\left\langle\rho^{\top}S^{-1}\psi,\lambda\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{l})}-\left\langle\rho^{\top}S^{-1}\rho\lambda,\lambda\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{l})},$

which is quadratic with respect to $\lambda$. Here
$\rho=(\rho_{1},\rho_{2},\cdots,\rho_{l})^{\top}$ and
$R=(R_{1},R_{2},\cdots,R_{l})^{\top}$. We summarize above arguments as the
following theorem.

######  Theorem 3.1.

The optimal control of Problem 3.1 is

$u^{*}=-S^{-1}\left(LX^{*}+\psi+\sum\limits_{i=1}^{l}\lambda_{i}\rho_{i}\right).$

Furthermore, the Lagrangian dual functional of Problem 2.1 reads

$\displaystyle\widehat{J}(s,\xi,\lambda,u^{*})=$
$\displaystyle\left\langle\xi,P(s)\xi+2Q(s)\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}_{s}}(\Omega,\mathbb{R}^{n})}-\left\langle
S^{-1}\psi,\psi\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})}$
$\displaystyle\mbox{}+2\left\langle
R^{\top}(s)\xi-a,\lambda\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}_{s}}(\Omega,\mathbb{R}^{l})}-2\left\langle\rho^{\top}S^{-1}\psi,\lambda\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{l})}-\left\langle\rho^{\top}S^{-1}\rho\lambda,\lambda\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{l})}.$
(9)

In the following, we would like to transform the constraints of $u$ (see
Remark 3.2) into the ones of $\lambda$. Applying the Itô formula to
$\left\langle X^{*},R_{i}\right\rangle$ derives

$\displaystyle\quad\;\left\langle
X^{*}(T),\gamma_{i}\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}_{T}}(\Omega,\mathbb{R}^{n})}-\left\langle\xi,R_{i}(s)\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}_{s}}(\Omega,\mathbb{R}^{n})}$
$\displaystyle=-\left\langle
X^{*},\alpha_{i}\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})}+\left\langle
S^{-1}LX^{*},\beta_{i}\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})}-\left\langle
S^{-1}\left[\psi+\sum\limits_{i=1}^{l}\lambda_{i}\rho_{i}\right],B^{\top}R_{i}+D^{\top}r_{i}\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})}$
$\displaystyle=\left\langle
S^{-1}\left[LX^{*}+\psi+\sum\limits_{i=1}^{l}\lambda_{i}\rho_{i}\right],\beta_{i}\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})}-\left\langle
S^{-1}\left[\psi+\rho\lambda\right],\rho_{i}\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})}-\left\langle
X^{*},\alpha_{i}\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})}$
$\displaystyle=-\left\langle
X^{*},\alpha_{i}\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})}-\left\langle
u^{*},\beta_{i}\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})}-\left\langle
S^{-1}\left[\psi+\rho\lambda\right],\rho_{i}\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})}.$

Therefore, one has

$\displaystyle\delta_{i}(\lambda)$ $\displaystyle=\langle
X,\alpha_{i}\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})}+\langle
u,\beta_{i}\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})}+\langle
X(T),\gamma_{i}\rangle_{\mathcal{L}^{2}_{\mathcal{F}_{T}}(\Omega,\mathbb{R}^{n})}$
$\displaystyle=\left\langle\xi,R_{i}(s)\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}_{s}}(\Omega,\mathbb{R}^{n})}-\left\langle
S^{-1}\psi,\rho_{i}\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})}-\left\langle\rho^{\top}S^{-1}\rho_{i},\lambda\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{l})}$
(10)

and so the dual problem can be given as follows.

######  Problem 3.2.

Find $\lambda^{*}\in U_{0}$ such that

$\widehat{J}(s,\xi,\lambda^{*},u^{**})=\sup\limits_{\lambda\in
U_{0}}\widehat{J}(s,\xi,\lambda,u^{*}),$

where $u^{**}=u^{*}(\lambda^{*})$, the Lagrangian dual functional
$\widehat{J}(s,\xi,\lambda,u^{*})$ is given by (9), and

$U_{0}=\left\\{\lambda\in\mathbb{R}^{l}\big{|}\lambda_{i}\geq
0\;(i=1,2,\cdots,l),\;\delta_{i}(\lambda)\leq
a_{i}\;(i=1,2,\cdots,l^{\prime})\mbox{ and
}\delta_{i}(\lambda)=a_{i}\;(i=l^{\prime}+1,l^{\prime}+2,\cdots,l)\right\\}.$

From Assumption 3.1, it follows that $U_{0}$ is nonempty, closed and convex in
$\mathbb{R}^{l}$. Furthermore, it is well known that the strong duality of
Problem 3.2 can be obtained under Assumption 3.1 (see [2, 8]), i.e.,
$\widehat{J}(s,\xi,\lambda^{*},u^{**})=J(s,\xi,\widehat{u})$. Therefore,
$u^{**}\in U^{ad}$ is an optimal control of Problem 2.1. Moreover, if the
invertibility assumption

$\mathbb{E}\left[\int_{s}^{T}\rho^{\top}(t)S^{-1}(t)\rho(t)dt\right]\in\mathcal{S}_{++}^{l}$
(11)

holds, then we have the following apparent results: (i) the mapping
$\delta:U_{0}\rightarrow U_{S}$ defined by
$\delta(\lambda)=\left(\delta_{1}(\lambda),\delta_{2}(\lambda),\cdots,\delta_{l}(\lambda)\right)^{\top}$
is bijective, where
$U_{S}=\left\\{\varrho=(\varrho_{1},\varrho_{2},\cdots,\varrho_{l})^{\top}\in\mathbb{R}^{l}|\varrho_{i}\leq
a_{i},\;i=1,2,\cdots,l\right\\}$, and so the Slater condition holds if and
only if $int(U_{0})=\delta^{-1}(U_{S})\neq\emptyset$; (ii) the function
$\widehat{J}(s,\xi,\lambda,u^{*})$ is uniformly concave with respect to
$\lambda$, which indicates the uniqueness of
$\lambda^{*}=(\lambda^{*}_{1},\lambda^{*}_{2},\cdots,\lambda_{l}^{*})^{\top}$.

Summarizing the above arguments, we have the following theorem.

######  Theorem 3.2.

The solution $u^{**}=u^{*}(\lambda^{*})$ to Problem 3.2 is as well the
solution to Problem 2.1. Moreover, if the invertibility assumption (11) holds,
then

  1. (I)

the Slater condition holds if and only if $int(U_{0})\neq\emptyset$;

  2. (II)

Problem 3.2 is uniquely solvable.

######  Remark 3.3.

From Proposition 3.1, one has $S\in\mathcal{S}_{++}^{l}$. Hence invertibility
assumption (11) can be satisfied under some rank conditions for $\rho$ and
$D$. We provide a sufficient condition for ensuring the invertibility in the
next section.

Combining (9) and (10), one has
$\delta(\lambda)-a=\frac{1}{2}\frac{d}{d\lambda}\widehat{J}(s,\xi,\lambda,u^{*})$.
Therefore, by standard arguments, we are now able to give the KKT condition
for the optimal control of Problem 2.1 as follows.

######  Theorem 3.3.

$\widehat{u}$ is an optimal control of Problem 2.1 if and only if the
following KKT condition holds:

$\begin{cases}\widehat{u}=-S^{-1}\left(L\widehat{X}+\psi+\rho^{\top}\lambda^{*}\right),\quad\lambda^{*}\in
U_{0},\\\
\lambda^{*}_{i}\left(\delta_{i}(\lambda^{*})-a_{i}\right)=0\;(i=1,2,\cdots,l^{\prime}),\quad\delta_{i}(\lambda^{*})=a_{i}\;(i=l^{\prime}+1,l^{\prime}+2,\cdots,l).\end{cases}$
(12)

######  Remark 3.4.

If $\rho=0^{m\times l}$, then (12) reduces to
$\widehat{u}=-S^{-1}\left(L\widehat{X}+\psi\right)$, i.e., $\widehat{u}$ is
indeed the optimal control of the SLQ problem with no constraints. In other
words, such an optimal control naturally satisfies all the affine constraints.

##  4 Sufficient Condition for Invertibility

In this section, we provide a sufficient condition for ensuring the
invertibility assumption (11). Then, with the help of Malliavin calculus, we
give the explicit expression of
$\rho_{i}=\beta_{i}+B^{\top}R_{i}+D^{\top}r_{i}$.

Recall the BSDE in (6):

$\begin{cases}dR_{i}=-\left[A^{\top}R_{i}+C^{\top}r_{i}+\alpha_{i}-L^{\top}S^{-1}\rho_{i}\right]dt+r_{i}dB_{t},\quad
t\in[s,T],\\\ R_{i}(T)=\gamma_{i},\end{cases}$ (13)

We analyse the solutions to (13) in cases $D\neq 0^{n\times m}$ and
$D=0^{n\times m}$, where $0^{n\times m}$ is an $n\times m$-null matrix.

###  4.1 Case I: $D\neq 0^{n\times m}$

######  Assumption 4.1.

For $D=(D_{1},D_{2},\cdots,D_{m})$, we assume that:

  * •

$\mbox{rank}(D)=m$, i.e., the vectors $\left\\{D_{i}\right\\}_{i=1}^{m}$ are
linearly independent in the sense that if there exists a vector
$k=(k_{1},k_{2},\cdots,k_{m})^{\top}\in\mathbb{R}^{m}$ such that
$\sum\limits_{i=1}^{m}k_{i}D_{i}=0^{n\times 1}$, $\forall
t\in[s,T],\;\mathbb{P}$-a.s., then $k=0^{m\times 1}$.

  * •

There exists a positive constant $\varepsilon$ such that
$tr(D^{\top}D)\geq\varepsilon>0$, $\forall t\in[s,T],\;\mathbb{P}$-a.s..

Consider the following SDE:

$\begin{cases}dV=(VA^{\top}+\mathcal{P}B^{\top})dt+(VC^{\top}+\mathcal{P}D^{\top})dB_{t},\quad
t\in[s,T],\\\ V(s)=\mathcal{I}_{n},\end{cases}$ (14)

where
$\mathcal{P}=V\left[D(D^{\top}D)^{-1}B^{\top}-C^{\top}\right]D(D^{\top}D)^{-1}$
and $\mathcal{Q}=VD(D^{\top}D)^{-1}$. Clearly, (14) admits a unique solution
$V\in\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n\times n})$ which is
invertible. Then, applying the Itô formula to $VR_{i}$ derives

$\begin{cases}d(VR_{i})=\left[(\mathcal{P}+VL^{\top}S^{-1})(B^{\top}R_{i}+D^{\top}r_{i})-V\alpha_{i}+VL^{\top}S^{-1}\beta_{i}\right]dt+\mathcal{Q}(B^{\top}R_{i}+D^{\top}r_{i})dB_{t},\\\
V(T)R_{i}(T)=V(T)\gamma_{i}.\end{cases}$ (15)

By setting $\overline{\pi}=\sum\limits_{i=1}^{l}k_{i}\pi_{i}$
($\pi\in\\{\alpha_{i},\beta_{i},\rho_{i},\gamma_{i},R_{i},r_{i}\\}$), it
follows from (15) that

$\begin{cases}d(V\overline{R})=\left[(\mathcal{P}+VL^{\top}S^{-1})(B^{\top}\overline{R}+D^{\top}\overline{r})-V\overline{\alpha}+VL^{\top}S^{-1}\overline{\beta}\right]dt+\mathcal{Q}(B^{\top}\overline{R}+D^{\top}\overline{r})dB_{t},\\\
V(T)\overline{R}_{i}(T)=V(T)\overline{\gamma}_{i}.\end{cases}$ (16)

Concerning on the BSDE (16), we have the following proposition.

######  Proposition 4.1.

Under Assumption 4.1, the following three statements are equivalent.

  1. (I)

$B^{\top}\overline{R}+D^{\top}\overline{r}+\overline{\beta}=0^{n\times
1}\qquad\mbox{a.e.}\;t\in[s,T],\;\mathbb{P}-\mbox{a.s}.$,

  2. (II)

$V(t)\overline{R}(t)=V(T)\overline{\gamma}+\int_{t}^{T}\left[\mathcal{P}(\tau)\overline{\beta}(\tau)+V(\tau)\overline{\alpha}(\tau)\right]d\tau+\int_{t}^{T}\mathcal{Q}(\tau)\overline{\beta}(\tau)dB_{\tau},\quad\forall
t\in[s,T]$, $\mathbb{P}$-a.s.,

  3. (III)

$\mathbb{E}\left[V(T)\overline{\gamma}+\int_{t}^{T}\left[\mathcal{P}(\tau)\overline{\beta}(\tau)+V(\tau)\overline{\alpha}(\tau)\right]d\tau\Big{|}\mathcal{F}_{t}\right]=V(T)\overline{\gamma}+\int_{t}^{T}\left[\mathcal{P}(\tau)\overline{\beta}(\tau)+V(\tau)\overline{\alpha}(\tau)\right]d\tau+\int_{t}^{T}\mathcal{Q}(\tau)\overline{\beta}(\tau)dB_{\tau}$,
$\forall t\in[s,T]$, $\mathbb{P}$-a.s..

###### Proof.

(I)$\Rightarrow$(II): Substituting (I) into (16), we have (II) immediately.

(II)$\Rightarrow$(III): Taking conditional expectation on both sides in (II)
derives

$\mathbb{E}\left[V(t)\overline{R}(t)\Big{|}\mathcal{F}_{t}\right]=V(t)\overline{R}(t)=\mathbb{E}\left[V(T)\overline{\gamma}+\int_{t}^{T}\mathcal{P}(\tau)\overline{\beta}(\tau)+V(\tau)\overline{\alpha}(\tau)d\tau\Big{|}\mathcal{F}_{t}\right],$

which implies (III).

(III)$\Rightarrow$(I): Combining (16) and (III), we have the following linear
BSDE:

$\begin{cases}d\sigma_{1}=(\mathcal{P}+VL^{\top}S^{-1})(\mathcal{Q}^{\top}\mathcal{Q})^{-1}\mathcal{Q}^{\top}\sigma_{2}dt+\sigma_{2}dB_{t},\quad
t\in[s,T],\\\ \sigma_{1}(T)=0,\end{cases}$ (17)

where
$\sigma_{1}(t)=V(t)R(t)-\mathbb{E}\left[V(T)\overline{\gamma}+\int_{t}^{T}\mathcal{P}(\tau)\overline{\beta}(\tau)+V(\tau)\overline{\alpha}(\tau)d\tau\Big{|}\mathcal{F}_{t}\right]$
and
$\sigma_{2}=\mathcal{Q}(B^{\top}\overline{R}+D^{\top}\overline{r}+\overline{\beta})$.
Since
$(\mathcal{P}+VL^{\top}S^{-1})(\mathcal{Q}^{\top}\mathcal{Q})^{-1}\mathcal{Q}^{\top}\in\mathcal{L}^{\infty}_{\mathcal{F}}(\Omega,\mathcal{M}^{1}_{\mathcal{F}}([s,T],\mathbb{R}^{n\times
m}))$ (or by using Theorem 4.2 in [13]), BSDE (17) admits a unique solution
$(\sigma_{1},\sigma_{2})=(0^{n\times 1},0^{n\times 1})$ in the space
$\mathcal{M}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})\times\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})$
and so

$\sigma_{2}=\mathcal{Q}(B^{\top}\overline{R}+D^{\top}\overline{r}+\overline{\beta})=0^{m\times
1}\qquad\mbox{a.e.}\;t\in[s,T],\;\mathbb{P}-\mbox{a.s}..$

Finally, it follows from $\mbox{rank}(\mathcal{Q})=m$ that

$B^{\top}\overline{R}+D^{\top}\overline{r}+\overline{\beta}=(\mathcal{Q}^{\top}\mathcal{Q})^{-1}\mathcal{Q}^{\top}0^{m\times
1}=0^{n\times 1}\qquad\mbox{a.e.}\;t\in[s,T],\;\mathbb{P}-\mbox{a.s}.,$

which ends the proof. ∎

According to Proposition 4.1, we immediately have the following theorem.

######  Theorem 4.1.

If Assumption 4.1 holds, then $\mbox{rank}(\rho)=l\leq m$ if and only if the
vector group

$\left\\{\mathbb{E}\left[\mathfrak{L}_{t}(\alpha_{i},\beta_{i},\gamma_{i})\Big{|}\mathcal{F}_{t}\right]-\mathfrak{L}_{t}(\alpha_{i},\beta_{i},\gamma_{i})-\int_{t}^{T}\mathcal{Q}(\tau)\beta_{i}(\tau)dB_{\tau}\right\\}_{i=1}^{l}$

is linearly independent, where
$\mathfrak{L}_{t}(\alpha_{i},\beta_{i},\gamma_{i})=V(T)\gamma_{i}+\int_{t}^{T}V(\tau)\left[L^{\top}(\tau)S^{-1}(\tau)\beta_{i}(\tau)-\alpha_{i}(\tau)\right]d\tau$.

Now we would like to find the expression of $B^{\top}R_{i}+D^{\top}r_{i}$. Let
$\widetilde{\sigma}=(\mathcal{P}+VL^{\top}S^{-1})(\mathcal{Q}^{\top}\mathcal{Q})^{-1}\mathcal{Q}^{\top}$
and $\sigma_{i}=\mathcal{Q}(B^{\top}R_{i}+D^{\top}r_{i})$. Consider the
following SDE:

$\begin{cases}dW=-W\widetilde{\sigma}dB_{t},\quad t\in[s,T],\\\
W(s)=\mathcal{I}_{n}.\end{cases}$ (18)

It follows from the Itô formula that

$\begin{cases}d(WVR_{i})=WV(L^{\top}S^{-1}\beta_{i}-\alpha_{i})dt+W\left(\sigma_{i}-\widetilde{\sigma}VR_{i}\right)dB_{t},\quad
t\in[s,T],\\\ W(T)V(T)R_{i}(T)=W(T)V(T)\gamma_{i}\end{cases}$ (19)

and so
$W(t)V(t)R_{i}(t)=\mathbb{E}\left[W(T)V(T)\gamma_{i}+\int_{t}^{T}W(\tau)V(\tau)\left[L^{\top}(\tau)S^{-1}(\tau)\beta_{i}(\tau)-\alpha_{i}(\tau)\right]d\tau\Big{|}\mathcal{F}_{t}\right]$.
By means of Malliavin Calculus, we have the following result.

######  Theorem 4.2.

If Assumption 4.1 holds and

$\widetilde{\mathfrak{L}}_{t}(\alpha_{i},\beta_{i},\gamma_{i})=W(T)V(T)\gamma_{i}+\int_{t}^{T}\left[W(\tau)V(\tau)L^{\top}(\tau)S^{-1}(\tau)\beta_{i}(\tau)-W(\tau)V(\tau)\alpha_{i}(\tau)\right]d\tau$

admits the Malliavin derivative
$\mathfrak{D}_{t}\widetilde{\mathfrak{L}}_{t}(\alpha_{i},\beta_{i},\gamma_{i})$
with
$\mathbb{E}\left[\int_{s}^{T}|\mathfrak{D}_{t}\widetilde{\mathfrak{L}}_{t}(\alpha_{i},\beta_{i},\gamma_{i})|^{2}dt\right]<\infty$,
where $V$ and $W$ are given by (14) and (18), respectively. Then

$R_{i}=V^{-1}W^{-1}\mathbb{E}\left[\widetilde{\mathfrak{L}}_{\cdot}(\alpha_{i},\beta_{i},\gamma_{i})\Big{|}\mathcal{F}_{\cdot}\right]$

and

$\displaystyle\rho_{i}=$
$\displaystyle\beta_{i}+(\mathcal{Q}^{\top}\mathcal{Q})^{-1}\mathcal{Q}^{\top}W^{-1}\mathbb{E}\left[\mathfrak{D}_{\cdot}\widetilde{\mathfrak{L}}_{\cdot}(\alpha_{i},\beta_{i},\gamma_{i})\Big{|}\mathcal{F}_{\cdot}\right]$
$\displaystyle\mbox{}+(\mathcal{Q}^{\top}\mathcal{Q})^{-1}\mathcal{Q}^{\top}(\mathcal{P}+VL^{\top}S^{-1})(\mathcal{Q}^{\top}\mathcal{Q})^{-1}\mathcal{Q}^{\top}W^{-1}\mathbb{E}\left[\widetilde{\mathfrak{L}}_{\cdot}(\alpha_{i},\beta_{i},\gamma_{i})\Big{|}\mathcal{F}_{\cdot}\right].$
(20)

###### Proof.

It is easy to derive the first equality. According to Remark 4.3 in [17],
$\mathbb{E}\left[\mathfrak{D}_{\cdot}\widetilde{\mathfrak{L}}_{\cdot}(\alpha_{i},\beta_{i},\gamma_{i})\Big{|}\mathcal{F}_{\cdot}\right]$
admits a predictable modification. Therefore one can choose
$\mathbb{E}\left[\mathfrak{D}_{\cdot}\widetilde{\mathfrak{L}}_{\cdot}(\alpha_{i},\beta_{i},\gamma_{i})\Big{|}\mathcal{F}_{\cdot}\right]\in\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{n})$
. From (19) and Clark-Ocone’s formula (see, e.g., Theorem 4.1 in [17]), we see
that

$\displaystyle\widetilde{\mathfrak{L}}_{t}(\alpha_{i},\beta_{i},\gamma_{i})-\mathbb{E}\left[\widetilde{\mathfrak{L}}_{t}(\alpha_{i},\beta_{i},\gamma_{i})\Big{|}\mathcal{F}_{t}\right]$
$\displaystyle=\int_{t}^{T}\left[W(\tau)\sigma_{i}(\tau)-W(\tau)\widetilde{\sigma}(\tau)V(\tau)R_{i}(\tau)\right]dB_{\tau}$
$\displaystyle=\int_{t}^{T}\mathbb{E}\left[\mathfrak{D}_{\tau}\widetilde{\mathfrak{L}}_{\tau}(\alpha_{i},\beta_{i},\gamma_{i})\Big{|}\mathcal{F}_{\tau}\right]dB_{\tau}.$

And so for $\tau\in[s,T]$, one has

$\displaystyle\mathbb{E}\left[\mathfrak{D}_{\tau}\widetilde{\mathfrak{L}}_{\tau}(\alpha_{i},\beta_{i},\gamma_{i})\Big{|}\mathcal{F}_{\tau}\right]$
$\displaystyle=W(\tau)\sigma_{i}(\tau)-W(\tau)\widetilde{\sigma}(\tau)V(\tau)R_{i}(\tau)$
$\displaystyle=W(\tau)\mathcal{Q}(B^{\top}R_{i}+D^{\top}r_{i})-W(\tau)\widetilde{\sigma}(\tau)W^{-1}(\tau)\mathbb{E}\left[\widetilde{\mathfrak{L}}_{\tau}(\alpha_{i},\beta_{i},\gamma_{i})\Big{|}\mathcal{F}_{\tau}\right],$

which indicates the second equality (20). ∎

An illustrative example for Theorem 4.2 is given as follows.

######  Example 4.1.

Suppose that $[s,T]=[0,1]$, $l^{\prime}=1$, $\xi=(1,1)^{\top}$,
$\varsigma_{1}=(1,0)^{\top}$, $\varsigma_{2}=(0,1)^{\top}$, $i\in\\{1,2\\}$,
$a=(2-e,e)^{\top}$,
$2A(t)=C(t)=D(t)=I(t)=\frac{1}{2}E(t)=F(t)=\mathcal{I}_{2}$,
$B(t)=M=0^{2\times 2}$, $G(t)=K(t)=\alpha_{i}(t)=\beta_{i}(t)=N=0^{2\times
1}$, $\gamma_{i}=e^{B_{1}-1}\varsigma_{i}$. Then we have from (5) and (6) that
$P(t)=(e^{1-t}-1)\mathcal{I}_{2}$, $L(t)=S(t)=e^{1-t}\mathcal{I}_{2}$,
$\Lambda(t)=0^{2\times 2}$, $Q(t)=\pi(t)=0^{2\times 1}$. According to Theorem
4.2, one has $R_{i}(t)=r_{i}(t)=\rho_{i}(t)=e^{B_{t}-t}\varsigma_{i}$ and
$R_{i}(0)=\varsigma_{i}$. Hence

$\delta_{i}=1-\lambda_{i}\langle
e^{1-t}\rho_{i},\rho_{i}\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})}=1-(e-1)\lambda_{i}.$

Next, by the KKT condition (12), we have

$\begin{cases}\widehat{u}=-\widehat{X}-e^{B_{t}-t}\lambda^{*},\quad\lambda^{*}_{1}\geq
1\\\ \lambda^{*}_{1}(1-\lambda^{*}_{1})=0,\quad
1-(e-1)\lambda^{*}_{2}=e\end{cases}$

and so $\lambda^{*}=(1,-1)^{\top}$. Finally, by
$\mathbb{E}\left[\int_{0}^{1}\rho^{\top}(t)S^{-1}(t)\rho(t)dt\right]=(e-1)\mathcal{I}_{2}\in\mathcal{S}_{++}^{2}$
and Theorems 3.2 and 3.3, the unique feedback optimal control reads

$\widehat{u}=-\widehat{X}-e^{B_{t}-t}\begin{bmatrix}1\\\ -1\end{bmatrix},\quad
t\in[0,1].$

###  4.2 Case II: $D=0^{n\times m}$

In the case $D=0^{n\times m}$, by similar discussion as in Subsection 4.1, we
have the following results.

######  Theorem 4.3.

If $D=0^{n\times m}$, then
$R_{i}=\widetilde{V}^{-1}\mathbb{E}\left[\mathfrak{L}_{\cdot}(\alpha_{i},\beta_{i},\gamma_{i})\Big{|}\mathcal{F}_{\cdot}\right]$,
where

$\mathfrak{L}_{t}(\alpha_{i},\beta_{i},\gamma_{i})=\widetilde{V}(T)\gamma_{i}+\int_{t}^{T}\widetilde{V}(\tau)\left[L^{\top}(\tau)S^{-1}(\tau)\beta_{i}(\tau)-\alpha_{i}(\tau)\right]d\tau$

and $\widetilde{V}$ is a solution to the following equation:

$\begin{cases}d\widetilde{V}=\widetilde{V}(A^{\top}-L^{\top}S^{-1}B^{\top})dt+\widetilde{V}C^{\top}dB_{t},\quad
t\in[s,T],\\\ \widetilde{V}(s)=\mathcal{I}_{n}.\end{cases}$

Moreover, $\mbox{rank}(\rho)=l\leq m$ if and only if the vector group
$\left\\{B^{\top}\widetilde{V}^{-1}\mathbb{E}\left[\mathfrak{L}_{\cdot}(\alpha_{i},\beta_{i},\gamma_{i})\Big{|}\mathcal{F}_{\cdot}\right]+\beta_{i}\right\\}_{i=1}^{l}$
is linearly independent.

An illustrative example for Theorem 4.3 is given as follows.

######  Example 4.2.

Assume that $[s,T]=[0,1]$, $l^{\prime}=2$, $\xi=(1,1)^{\top}$,
$\varsigma_{1}=(1,0)^{\top}$, $\varsigma_{2}=(0,1)^{\top}$, $i\in\\{1,2\\}$,
$a=(-3,1)^{\top}$, $-2A(t)=B(t)=C(t)=I(t)=M=\mathcal{I}_{2}$,
$D(t)=E(t)=F(t)=0^{2\times 2}$,
$G(t)=K(t)=\alpha_{i}(t)=\beta_{i}(t)=0^{2\times 1}$,
$N=\frac{2}{2-t}e^{B_{t}-t}\xi$, $\gamma_{i}=2e^{B_{1}-1}\varsigma_{i}$. Then
we have from (5) and (6) that $P(t)=L(t)=\frac{1}{2-t}\mathcal{I}_{2}$,
$S(t)=\mathcal{I}_{2}$, $\Lambda(t)=0^{2\times 2}$. According to Theorem 4.3,
one has $R_{i}(t)=r_{i}(t)=\rho_{i}(t)=\frac{2}{2-t}e^{B_{t}-t}\varsigma_{i}$
and $R_{i}(0)=\varsigma_{i}$. Similarly, we have
$Q(t)=\pi(t)=\frac{2}{2-t}e^{B_{t}-t}\xi$. Hence

$\delta_{i}=1-\left\langle
Q,\rho_{i}\right\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})}-\lambda_{i}\langle\rho_{i},\rho_{i}\rangle_{\mathcal{L}^{2}_{\mathcal{F}}([s,T],\mathbb{R}^{m})}=-1-2\lambda_{i}.$

Next, by the KKT condition (12), we have

$\begin{cases}\widehat{u}=-\frac{1}{2-t}(\widehat{X}+2e^{B_{t}-t}\lambda^{*}),\quad\lambda^{*}_{1}\geq
1,\quad\lambda^{*}_{2}\geq 0,\\\
\lambda^{*}_{1}(2-2\lambda^{*}_{1})=0,\quad\lambda^{*}_{2}(-2-2\lambda^{*}_{2})=0\end{cases}$

and so $\lambda^{*}=(1,0)^{\top}$. Finally, by
$\mathbb{E}\left[\int_{0}^{1}\rho^{\top}(t)S^{-1}(t)\rho(t)dt\right]=2\mathcal{I}_{2}\in\mathcal{S}_{++}^{2}$
and Theorems 3.2 and 3.3, the unique feedback optimal control reads

$\widehat{u}=-\frac{1}{2-t}\left(\widehat{X}+e^{B_{t}-t}\begin{bmatrix}2\\\
0\end{bmatrix}\right),\quad t\in[0,1].$

##  5 Conclusions

This paper is devoted to studying the SLQ problems with affine constraints in
random coefficients case. The contribution of this paper can be organized as
follows. (I) The KKT condition was established for the dual problem of
original problem. (II) Feedback expression of the optimal control is obtained.
(III) A sufficient condition is provided for ensuring the uniqueness of
optimal control.

It is worth mentioning the SLQ probelms with quadratic constraints (see
Example 2.1) or risk constraints (see Example 2.2) call for further research.
Moreover, extension of our problem formulation, including the SLQ problems
with delay/jumps/partial information [9, 16, 26], promises to be interesting
and important topics. We plan to address these problems as we continue our
research.

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