# Linear-quadratic stochastic Volterra controls I: Causal feedback strategies

Yushi Hamaguchi111Corresponding Author. Graduate School of Engineering
Science, Department of Systems Innovation, Osaka University. 1-3,
Machikaneyama, Toyonaka, Osaka, Japan (Email: hmgch2950@gmail.com). This
author was supported by JSPS KAKENHI Grant Number 22K13958. and Tianxiao
Wang222School of Mathematics, Sichuan University. Chengdu, P. R. China (Email:
wtxiao2014@scu.edu.cn). This author was supported by National Natural Science
Foundation of China (No. 11971332 and 11931011) and the Science Development
Project of Sichuan University under grant 2020SCUNL201.

###### Abstract

In this paper, we formulate and investigate the notion of causal feedback
strategies arising in linear-quadratic control problems for stochastic
Volterra integral equations (SVIEs) with singular and non-convolution-type
coefficients. We show that there exists a unique solution, which we call the
causal feedback solution, to the closed-loop system of a controlled SVIE
associated with a causal feedback strategy. Furthermore, introducing two novel
equations named a Type-II extended backward stochastic Volterra integral
equation and a Lyapunov–Volterra equation, we prove a duality principle and a
representation formula for a quadratic functional of controlled SVIEs in the
framework of causal feedback strategies.

Keywords: Linear-quadratic control; stochastic Volterra integral equation;
causal feedback strategy.

2020 Mathematics Subject Classification: 60H20; 45A05; 93E20; 93B52.

##  1 Introduction

Linear-quadratic (LQ) control problems are special classes of optimal control
problems described by a linear state dynamics and a quadratic cost functional.
In the continuous time setting, the state dynamics is assumed to be governed
by a controlled differential/integral equation. In this paper, we study
controlled linear stochastic Volterra integral equations (SVIEs) of the
following form:

$\begin{split}&X(t)=x(t)+\int^{t}_{t_{0}}\\{A(t,s)X(s)+B(t,s)u(s)+b(t,s)\\}\,\mathrm{d}s+\int^{t}_{t_{0}}\\{C(t,s)X(s)+D(t,s)u(s)+\sigma(t,s)\\}\,\mathrm{d}W(s),\\\
&\hskip 227.62204ptt\in(t_{0},T),\end{split}$ (1.1)

where $u$ is a control process, $x$ is a given deterministic function called
the free term (which is also called the forcing term), $W$ is a Brownian
motion, $A,B,C$ and $D$ are matrix-valued deterministic coefficients, and $b$
and $\sigma$ are vector-valued stochastic inhomogeneous terms. The LQ control
problem for SVIE (1.1), which we call an _LQ stochastic Volterra control
problem_ , is a problem to find a control process or a strategy which minimize
a quadratic cost functional of both $u$ and $X$.

The controlled SVIE (1.1) is a Volterra-type extension of a controlled linear
stochastic differential equation (SDE):

$\begin{dcases}\,\mathrm{d}X(s)=\\{A(s)X(s)+B(s)u(s)+b(s)\\}\,\mathrm{d}s+\\{C(s)X(s)+D(s)u(s)+\sigma(s)\\}\,\mathrm{d}W(s),\
s\in[t_{0},T],\\\ X(t_{0})=x,\end{dcases}$ (1.2)

with $x$ being a constant. LQ control problems for SDEs were first studied by
Wonham [31] in 1968 and followed by many researchers; see [33, Chapter 6] and
[24] for systematic studies and the developments of LQ control theory for
SDEs. It is well-known that, in the LQ control problems for SDEs, the optimal
control $u$ has the following state-feedback representation:

$u(t)=\Xi(t)X(t)+v(t),\ t\in[t_{0},T],$ (1.3)

where $\Xi$ is a matrix-valued deterministic function, $v$ is a stochastic
inhomogeneous term, and $X$ is the optimal state process corresponding to $u$.
Therefore, given the strategy $(\Xi,v)$, for each initial condition
$(t_{0},x)$, we get the following closed-loop system:

$\begin{dcases}\,\mathrm{d}X^{t_{0},x}(s)=\\{A(s)X^{t_{0},x}(s)+B(s)u^{t_{0},x}(s)+b(s)\\}\,\mathrm{d}s\\\
\hskip
85.35826pt+\\{C(s)X^{t_{0},x}(s)+D(s)u^{t_{0},x}(s)+\sigma(s)\\}\,\mathrm{d}W(s),\
s\in[t_{0},T],\\\ X^{t_{0},x}(t_{0})=x,\\\
u^{t_{0},x}(s)=\Xi(s)X^{t_{0},x}(s)+v(s),\ s\in[t_{0},T],\end{dcases}$ (1.4)

or equivalently the SDE

$\begin{dcases}\,\mathrm{d}X^{t_{0},x}(s)=\\{(A(s)+B(s)\Xi(s))X^{t_{0},x}(s)+B(s)v(s)+b(s)\\}\,\mathrm{d}s\\\
\hskip
85.35826pt+\\{(C(s)+D(s)\Xi(s))X^{t_{0},x}(s)+D(s)v(s)+\sigma(s)\\}\,\mathrm{d}W(s),\
s\in[t_{0},T],\\\ X^{t_{0},x}(t_{0})=x.\end{dcases}$

We note that the above system is an equation for the optimal state process
$X=X^{t_{0},x}$, and the optimal control process $u=u^{t_{0},x}$ is obtained
as the outcome of the strategy $(\Xi,v)$ by inserting the solution
$X^{t_{0},x}$ into the expression $u^{t_{0},x}=\Xi X^{t_{0},x}+v$. The closed-
loop system (1.4) of the controlled SDE satisfies the flow property: for each
$0\leq t_{0}<t_{1}<t\leq T$,

$X^{t_{0},x}(t)=X^{t_{1},X^{t_{0},x}(t_{1})}(t),\
u^{t_{0},x}(t)=u^{t_{1},X^{t_{0},x}(t_{1})}(t).$

This is also called the time-consistency, which ensures that the strategy
$(\Xi,v)$ is meaningful in dynamic sense. It is well-known that the dynamic
optimality of the strategy $(\Xi,v)$ is closely related to the well-posedness
of a Riccati differential equation and a backward stochastic differential
equation (BSDE).

Compared to SDEs, the solutions of SVIEs are not Markovian or even not
semimartingales in general. These features of SVIEs make some analysis
dramatically difficult since we cannot use some fundamental tools of
stochastic calculus such as Itô’s formula directly to compute some functionals
of SVIEs. Furthermore, SVIEs do not satisfy the flow property. To illustrate
this, consider the following (uncontrolled) stochastic Volterra process:

$X^{t_{0},x}(t):=x+\int^{t}_{t_{0}}K(t,s)\,\mathrm{d}W(s),\ t\in(t_{0},T),$

where $K$ is a deterministic function, and $x$ is an initial constant at time
$t_{0}$. This is in particular the case of a fractional Brownian motion. Due
to the two time variables structure of the kernel $K$, $X$ does not satisfy
the flow property in the standard sense: for $t_{0}<t_{1}<t<T$,

$X^{t_{0},x}(t)\neq X^{t_{1},X^{t_{0},x}(t_{1})}(t)\ \ \text{where}\ \
X^{t_{1},X^{t_{0},x}(t_{1})}(t):=X^{t_{0},x}(t_{1})+\int^{t}_{t_{1}}K(t,s)\,\mathrm{d}W(s).$

The lack of the flow property gives rise to the following question: _What is
the “state space” of SVIEs?_ Actually, in the above example, the space of
constants $x$ (the space of values of $X^{t_{0},x}(t)$) is not suitable for
the state space since it is not preserved in dynamic sense. To overcome this
difficulty, [1, 17, 25, 26] considered an auxiliary process $\Theta$ defined
by

$\Theta^{t_{0},x}(s,t):=x+\int^{t}_{t_{0}}K(s,r)\,\mathrm{d}W(r),\
(s,t)\in\triangle_{2}(t_{0},T),$

where $\triangle_{2}(t_{0},T):=\\{(s,t)\,|\,t_{0}<t<s<T\\}$. Then, for any
$t_{0}<t_{1}<t<T$, we have the following generalized flow property:

$X^{t_{0},x}(t)=X^{t_{1},\Theta^{t_{0},x}(\cdot,t_{1})}(t)\ \ \text{where}\ \
X^{t_{1},\Theta^{t_{0},x}(\cdot,t_{1})}(t):=\Theta^{t_{0},x}(t,t_{1})+\int^{t}_{t_{1}}K(t,s)\,\mathrm{d}W(s).$

In the above expression,
$\Theta^{t_{0},x}(\cdot,t_{1})=(\Theta^{t_{0},x}(t,t_{1}))_{t\in(t_{1},T)}$
can be regarded as the free term of $X^{t_{1},\Theta^{t_{0},x}(\cdot,t_{1})}$
evaluated at time $t_{1}$. We note that
$\Theta^{t_{0},x}(s,\cdot)=(\Theta^{t_{0},x}(s,t))_{t\in[t_{0},s]}$ is a
semimartingale for each fixed $s\in(t_{0},T)$, and it holds that
$\Theta^{t_{0},x}(t,t)=X^{t_{0},x}(t)$. In the dynamic point of view, the
space of the auxiliary process $\Theta(\cdot,\cdot)$ can be seen as a state
space for the above example. It is worth to mention that, in the theory of
mathematical finance, the auxiliary process $\Theta$ corresponds to the so-
called forward variance which is a financial index available in the market
(see [12, 25] and references sited therein).

Let us turn to the controlled SVIE (1.1). As mentioned in the above example,
the sole space of $X(t)$ is not sufficient for describing the state space. In
view of the generalized flow property mentioned above, it is natural to
consider the following auxiliary process:

$\Theta(s,t)=x(s)+\int^{t}_{t_{0}}\\{A(s,r)X(r)+B(s,r)u(r)+b(s,r)\\}\,\mathrm{d}r+\int^{t}_{t_{0}}\\{C(s,r)X(r)+D(s,r)u(r)+\sigma(s,r)\\}\,\mathrm{d}W(r)$

for $(s,t)\in\triangle_{2}(t_{0},T)$. A particular case of
$\Theta(\cdot,\cdot)$ was introduced in [27] to treat LQ stochastic Volterra
control problems. In this paper, we call the above process a _forward state
process_ and regard the pair $(X,\Theta)$ as the (generalized) state of the
controlled SVIE (1.1). Introducing the forward state process $\Theta$ to LQ
problem also fits into the so-called causal projection approach studied by
[18, 21] in deterministic LQ Volterra control problems. From this sense, the
forward state $\Theta(s,t)$ can be seen as the causal projection of the
original controlled SVIE (1.1) which is determined by information of $X$ and
$u$ up to the current time $t$. Unlike the SDEs case, the above observation
inspires us to consider a feedback control process represented by

$u(t)=\Xi(t)X(t)+\int^{T}_{t}\Gamma(s,t)\Theta(s,t)\,\mathrm{d}s+v(t)$

for some deterministic functions $\Xi$ and $\Gamma$ and a stochastic
inhomogeneous term $v$. We call the triplet $(\Xi,\Gamma,v)$ a _causal
feedback strategy_. By inserting the above expression of the outcome $u$ into
the controlled SVIE (1.1), for each $(t_{0},x)$, we get the following closed-
loop system:

$\begin{dcases}X^{t_{0},x}(t)=x(t)+\int^{t}_{t_{0}}\\{A(t,s)X^{t_{0},x}(s)+B(t,s)u^{t_{0},x}(s)+b(t,s)\\}\,\mathrm{d}s\\\
\hskip
85.35826pt+\int^{t}_{t_{0}}\\{C(t,s)X^{t_{0},x}(s)+D(t,s)u^{t_{0},x}(s)+\sigma(t,s)\\}\,\mathrm{d}W(s),\
t\in(t_{0},T),\\\
\Theta^{t_{0},x}(s,t)=x(s)+\int^{t}_{t_{0}}\\{A(s,r)X^{t_{0},x}(r)+B(s,r)u^{t_{0},x}(r)+b(s,r)\\}\,\mathrm{d}r\\\
\hskip
85.35826pt+\int^{t}_{t_{0}}\\{C(s,r)X^{t_{0},x}(r)+D(s,r)u^{t_{0},x}(r)+\sigma(s,r)\\}\,\mathrm{d}W(r),\
(s,t)\in\triangle_{2}(t_{0},T),\\\
u^{t_{0},x}(t)=\Xi(t)X^{t_{0},x}(t)+\int^{T}_{t}\Gamma(s,t)\Theta^{t_{0},x}(s,t)\,\mathrm{d}s+v(t),\
t\in(t_{0},T).\end{dcases}$ (1.5)

We note that the above system is a coupled equation for the state process
$X^{t_{0},x}$ and the forward state process $\Theta^{t_{0},x}$, and it is
beyond the typical framework of SVIEs studied in existing works. We call the
pair $(X^{t_{0},x},\Theta^{t_{0},x})$ satisfying the above system a _causal
feedback solution_ to the controlled SVIE (1.1) at $(t_{0},x)$ corresponding
to the causal feedback strategy $(\Xi,\Gamma,v)$.

The purpose of this paper is to formulate and investigate causal feedback
solutions of controlled SVIE (1.1) systematically. On one hand, the present
paper can be seen as a preliminary of our continued paper [16], where we go
into the main topic of LQ stochastic Volterra control problems and
characterize the optimal causal feedback strategy. On the other hand, besides
the preliminary, the present paper provides a novel framework for SVIEs which
is interesting by its own right in theoretical point of view. The main
contributions of this paper are the following three points:

  * (i)

We show that there exists a unique causal feedback solution to the controlled
SVIE (1.1) for any given causal feedback strategy $(\Xi,\Gamma,v)$ and any
input condition $(t_{0},x)$ (see 2.4).

  * (ii)

We prove a _duality principle_ for the causal feedback solution (see 3.3).

  * (iii)

We derive a _representation formula for a quadratic functional_ of the causal
feedback solution in terms of the free term $x$ (see 4.11).

Let us make some comments on the contribution (i) compared to existing works.
SVIEs with regular coefficients were first studied by Berger and Mizel [6, 7]
and followed by [5, 19, 20]. Studies of SVIEs with singular coefficients
(which include fractional SDEs and fractional Brownian motion with Hurst
parameter less than $1/2$) can be found in [11, 30, 34, 35], etc. Recently, in
[15], the first author of the present paper investigated the algebraic
structure of linear SVIEs with singular coefficients and proved a variation of
constants formula. In the above works, there are no controls in SVIEs. Optimal
control problems of SVIEs were studied by [4, 9, 13, 14, 22, 23, 27, 28, 29,
32] in the open-loop framework, where feedback strategies were not considered.
In the special case of SVIEs with completely monotone and convolution-type
kernels, several kinds of feedback representations of the optimal controls
were investigated by [2, 8, 10]. Specifically, Abi Jaber, Miller and Pham [2]
studied LQ stochastic Volterra control problems with completely monotone and
convolution-type kernels. Based on the so-called Markovian lift approach,
which is an infinite-dimensional approach associated with the special
structure of the completely monotone kernel, they obtained a kind of a linear
feedback representation of the optimal control; see also [3] for the study on
the associated integral operator Riccati equation. Compared to these works, we
study linear controlled SVIEs with general (singular and non-convolution-type)
coefficients. Our definition of causal feedback strategies, which is motivated
from the generalized flow property as mentioned above, is different from the
infinite dimensional ones considered in [2, 8, 10]. It is worth to mention
that Viens and Zhang [25] proved the functional Itô formula for non-linear
SVIEs with non-convolution-type coefficients and applied it to optimal control
theory. The corresponding optimal strategy is similar to our definition of
causal feedback strategies, but they did not study the well-posedness of the
associated closed-loop system (1.5). We emphasize that the closed-loop system
(1.5) is a coupled system of the original state process $X$ and the forward
state process $\Theta$, which is beyond the framework of the existing
literature of SVIEs. Our contribution (i) ensures the well-posedness of the
closed-loop system (1.5).

Let us illustrate the motivation of the contributions (ii) and (iii). In
typical LQ control problems for SDEs, the controlled dynamics of the state
process $X$ is described by a linear SDE of the forms of (1.2) in the open-
loop framework or (1.4) in the closed-loop framework. In such problems, the
key points are to calculate (in terms of given parameters such as controls,
initial states and some inhomogeneous terms) a linear functional of the form

$\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\langle\psi(t),X(t)\rangle\,\mathrm{d}t\Bigr{]}$
(1.6)

with an adapted process $\psi$ and a quadratic functional of the form

$\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\langle
Q(t)X(t),X(t)\rangle\,\mathrm{d}t\Bigr{]}$ (1.7)

with a symmetric matrix-valued function $Q$. It is well-known that the linear
functional (1.6) for the solution $X$ to a linear SDE can be calculated by
using the duality principle in terms of a BSDE, and the quadratic functional
(1.7) can be calculated in terms of a Lyapunov differential equation (see
[24]). In contrast, in our continued paper [16], we consider LQ stochastic
Volterra control problems in the framework of causal feedback strategies,
where we are faced with the calculations of a linear functional of the form

$\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\Bigl{\\{}\langle\psi(t),X^{t_{0},x}(t)\rangle+\int^{T}_{t}\langle\chi(s,t),\Theta^{t_{0},x}(s,t)\rangle\,\mathrm{d}s\Bigr{\\}}\,\mathrm{d}t\Bigr{]}$
(1.8)

and a quadratic functional of the form

$\begin{split}&\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\Bigl{\\{}\langle
Q^{(1)}(t)X^{t_{0},x}(t),X^{t_{0},x}(t)\rangle+2\int^{T}_{t}\langle
Q^{(2)}(s,t)X^{t_{0},x}(t),\Theta^{t_{0},x}(s,t)\rangle\,\mathrm{d}s\\\
&\hskip 56.9055pt+\int^{T}_{t}\\!\\!\int^{T}_{t}\langle
Q^{(3)}(s_{1},s_{2},t)\Theta^{t_{0},x}(s_{2},t),\Theta^{t_{0},x}(s_{1},t)\rangle\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2}\Bigr{\\}}\,\mathrm{d}t\Bigr{]}.\end{split}$
(1.9)

Here, $(X^{t_{0},x},\Theta^{t_{0},x})$ is the causal feedback solution to the
controlled SVIE (1.1) corresponding to a given causal feedback strategy
$(\Xi,\Gamma,v)$, $\psi$ and $\chi$ are stochastic processes, and
$Q^{(1)},Q^{(2)}$ and $Q^{(3)}$ are matrix-valued deterministic functions. The
analyses of (1.8) and (1.9) serve as basis of the study of LQ stochastic
Volterra control problems in [16], and these are the main topics of the
present paper.

In order to calculate the linear functional (1.8), as a counterpart of a BSDE
in the Volterra setting, we introduce a novel class of backward stochastic
equations which we call a _Type-II extended backward stochastic Volterra
integral equation_ (Type-II EBSVIE), see (3.1). This is a generalization of a
class of Type-II backward stochastic Volterra integral equations (Type-II
BSVIEs) introduced by Yong [32] to the framework of causal feedback
strategies. We prove the well-posedness of Type-II EBSVIEs and derive a
duality principle between the causal feedback solution to a controlled SVIE
and the adapted solution to a Type-II EBSVIE.

In order to calculate the quadratic functional (1.9), we introduce a system of
Lyapunov-type Volterra integro-differential equations which we call a
_Lyapunov–Volterra equation_ , see (4.5). We prove the well-posedness of
Lyapunov–Volterra equations. Furthermore, we provide a representation formula
for the quadratic functional (1.9) of a causal feedback solution to a
(homogeneous) controlled SVIE in terms of the free term $x$ and the solution
to a Lyapunov–Volterra equation.

As mentioned above, the studies on causal feedback strategies including the
duality principle and the representation formula for a quadratic functional
play fundamental roles in LQ stochastic Volterra control problems. In our
continued paper [16], we characterize the optimal causal feedback strategy by
means of a _Riccati–Volterra equation_ , which is a Riccati-type equation
closely related to the Lyapunov–Volterra equation, and a Type-II EBSVIE.

The rest of this paper is organized as follows: In Section 2, we give a
standing assumption of this paper and investigate the novel notion of causal
feedback solutions to controlled SVIEs. In Section 3, we introduce Type-II
EBSVIEs and prove the duality principle. In Section 4, we derive
Lyapunov–Volterra equations and investigate quadratic functionals of
controlled SVIEs. The main theorems 2.4, 3.3 and 4.11 are proved in Appendix A
systematically.

### Notation

$(\Omega,\mathcal{F},\mathbb{P})$ is a complete probability space, and $W$ is
a one-dimensional Brownian motion. $\mathbb{F}=(\mathcal{F}_{t})_{t\geq 0}$
denotes the $\mathbb{P}$-augmented filtration generated by $W$.
$\mathbb{E}[\cdot]$ denotes the expectation, and
$\mathbb{E}_{t}[\cdot]:=\mathbb{E}[\cdot|\mathcal{F}_{t}]$ denotes the
conditional expectation given by $\mathcal{F}_{t}$ for each $t\geq 0$.
Throughout this paper, $\mathbb{E}[\cdot]^{1/2}$ denotes the square root of
the expectation $\mathbb{E}[\cdot]$, not the expectation of the square root.
For each $0\leq t_{0}<T<\infty$, we define

$\displaystyle\triangle_{2}(t_{0},T):=\\{(t,s)\in(t_{0},T)^{2}\,|\,T>t>s>t_{0}\\},\
\text{(a triangle region)}$
$\displaystyle\square_{3}(t_{0},T):=\\{(s_{1},s_{2},t)\in(t_{0},T)^{3}\,|\,t<s_{1}\wedge
s_{2}\\}.\ \text{(a square pyramid region)}$

For each matrix $M\in\mathbb{R}^{d_{1}\times d_{2}}$ with
$d_{1},d_{2}\in\mathbb{N}$, $|M|$ denotes the Frobenius norm, and
$M^{\top}\in\mathbb{R}^{d_{2}\times d_{1}}$ denotes the transpose. For each
$d\in\mathbb{N}$, $\mathbb{S}^{d}$ denotes the set of $(d\times d)$-symmetric
matrices. We define $\mathbb{R}^{d}:=\mathbb{R}^{d\times 1}$, that is, each
element of $\mathbb{R}^{d}$ is understood as a column vector. We denote by
$\langle\cdot,\cdot\rangle$ the usual inner product on a Euclidean space. For
each set $\Lambda$, $\mbox{\rm{1}}\mbox{\rm{l}}_{\Lambda}$ denotes the
characteristic function.

For each $0\leq t_{0}<T<\infty$ and $d_{1},d_{2}\in\mathbb{N}$, we define some
spaces of stochastic (and deterministic) processes as follows:

  * •

$(L^{2}_{\mathbb{F}}(t_{0},T;\mathbb{R}^{d_{1}\times
d_{2}}),\|\cdot\|_{L^{2}_{\mathbb{F}}(t_{0},T)})$ is the Hilbert space of
$\mathbb{R}^{d_{1}\times d_{2}}$-valued, square-integrable and
$\mathbb{F}$-progressively measurable processes on $(t_{0},T)$.

  * •

$(L^{2,1}_{\mathbb{F}}(\triangle_{2}(t_{0},T);\mathbb{R}^{d_{1}\times
d_{2}}),\|\cdot\|_{L^{2,1}_{\mathbb{F}}(\triangle_{2}(t_{0},T))})$ is the
Banach space of $\mathbb{R}^{d_{1}\times d_{2}}$-valued and measurable
processes $\xi$ on $\triangle_{2}(t_{0},T)$ such that $\xi(t,\cdot)$ is
$\mathbb{F}$-progressively measurable on $(t_{0},t)$ for each $t\in(t_{0},T)$,
and that $\|\xi\|_{L^{2,1}_{\mathbb{F}}(\triangle_{2}(t_{0},T))}<\infty$,
where

$\|\xi\|_{L^{2,1}_{\mathbb{F}}(\triangle_{2}(t_{0},T))}:=\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\Bigl{(}\int^{t}_{t_{0}}|\xi(t,s)|\,\mathrm{d}s\Bigr{)}^{2}\,\mathrm{d}t\Bigr{]}^{1/2}.$

  * •

$(L^{2}_{\mathbb{F}}(\triangle_{2}(t_{0},T);\mathbb{R}^{d_{1}\times
d_{2}}),\|\cdot\|_{L^{2}_{\mathbb{F}}(\triangle_{2}(t_{0},T))})$ is the
Hilbert space of $\xi\in
L^{2,1}_{\mathbb{F}}(\triangle_{2}(t_{0},T);\mathbb{R}^{d_{1}\times d_{2}})$
such that $\|\xi\|_{L^{2}_{\mathbb{F}}(\triangle_{2}(t_{0},T))}<\infty$, where

$\|\xi\|_{L^{2}_{\mathbb{F}}(\triangle_{2}(t_{0},T))}:=\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\\!\\!\int^{t}_{t_{0}}|\xi(t,s)|^{2}\,\mathrm{d}s\\!\,\mathrm{d}t\Bigr{]}^{1/2}.$

  * •

$(L^{2}_{\mathbb{F},\mathrm{c}}(\triangle_{2}(t_{0},T);\mathbb{R}^{d_{1}\times
d_{2}}),\|\cdot\|_{L^{2}_{\mathbb{F},\mathrm{c}}(\triangle_{2}(t_{0},T))})$ is
the Banach space of $\xi\in
L^{2}_{\mathbb{F}}(\triangle_{2}(t_{0},T);\mathbb{R}^{d_{1}\times d_{2}})$
such that $s\mapsto\xi(t,s)$ is uniformly continuous on $(t_{0},t)$ with the
limits defined by $\xi(t,t):=\lim_{s\uparrow t}\xi(t,s)$ and
$\xi(t,t_{0}):=\lim_{s\downarrow t_{0}}\xi(t,s)$ for a.e. $t\in(t_{0},T)$,
a.s., and satisfies
$\|\xi\|_{L^{2}_{\mathbb{F},\mathrm{c}}(\triangle_{2}(t_{0},T))}<\infty$,
where

$\|\xi\|_{L^{2}_{\mathbb{F},\mathrm{c}}(\triangle_{2}(t_{0},T))}:=\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\sup_{s\in[t_{0},t]}|\xi(t,s)|^{2}\,\mathrm{d}t\Bigr{]}^{1/2}.$

  * •

$(L^{\infty}(t_{0},T;\mathbb{R}^{d_{1}\times
d_{2}}),\|\cdot\|_{L^{\infty}(t_{0},T)})$ is the Banach space of
$\mathbb{R}^{d_{1}\times d_{2}}$-valued essentially bounded measurable
functions on $(t_{0},T)$.

  * •

For $\mathbb{T}=(t_{0},T),(t_{0},T)^{2},\triangle_{2}(t_{0},T)$ and
$\square_{3}(t_{0},T)$, $(L^{2}(\mathbb{T};\mathbb{R}^{d_{1}\times
d_{2}}),\|\cdot\|_{L^{2}(\mathbb{T})})$ is the Hilbert space of
$\mathbb{R}^{d_{1}\times d_{2}}$-valued and square-integrable deterministic
functions on $\mathbb{T}$.

  * •

$(L^{2,1}(\triangle_{2}(t_{0},T);\mathbb{R}^{d_{1}\times
d_{2}}),\|\cdot\|_{L^{2,1}(\triangle_{2}(t_{0},T))})$ is the Banach space of
$\mathbb{R}^{d_{1}\times d_{2}}$-valued deterministic functions $f$ on
$\triangle_{2}(t_{0},T)$ such that
$\|f\|_{L^{2,1}(\triangle_{2}(t_{0},T))}<\infty$, where

$\|f\|_{L^{2,1}(\triangle_{2}(t_{0},T))}:=\Bigl{(}\int^{T}_{t_{0}}\Bigl{(}\int^{t}_{t_{0}}|f(t,s)|\,\mathrm{d}s\Bigr{)}^{2}\,\mathrm{d}t\Bigr{)}^{1/2}.$

  * •

$(L^{2,2,1}(\square_{3}(t_{0},T);\mathbb{R}^{d_{1}\times
d_{2}}),\|\cdot\|_{L^{2,2,1}(\square_{3}(t_{0},T))})$ is the Banach space of
$\mathbb{R}^{d_{1}\times d_{2}}$-valued deterministic functions $f$ on
$\square_{3}(t_{0},T)$ such that
$\|f\|_{L^{2,2,1}(\square_{3}(t_{0},T))}<\infty$, where

$\|f\|_{L^{2,2,1}(\square_{3}(t_{0},T))}:=\Bigl{(}\int^{T}_{t_{0}}\\!\\!\int^{T}_{t_{0}}\Bigl{(}\int^{s_{1}\wedge
s_{2}}_{t_{0}}|f(s_{1},s_{2},t)|\,\mathrm{d}t\Bigr{)}^{2}\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2}\Bigr{)}^{1/2}<\infty.$

  * •

$L^{2,2,1}_{\mathrm{sym}}(\square_{3}(t_{0},T);\mathbb{R}^{d\times d})$ is the
set of $f\in L^{2,2,1}(\square_{3}(t_{0},T);\mathbb{R}^{d\times d})$ such that
$f(s_{1},s_{2},t)=f(s_{2},s_{1},t)^{\top}$ for a.e.
$(s_{1},s_{2},t)\in\square_{3}(0,T)$. It is easy to see that
$L^{2,2,1}_{\mathrm{sym}}(\square_{3}(t_{0},T);\mathbb{R}^{d\times d})$ is a
closed subspace of $L^{2,2,1}(\square_{3}(t_{0},T);\mathbb{R}^{d_{1}\times
d_{2}})$.

  * •

$\mathscr{L}^{2}(\triangle_{2}(t_{0},T);\mathbb{R}^{d_{1}\times d_{2}})$ is
the set of $f\in L^{2}(\triangle_{2}(t_{0},T);\mathbb{R}^{d_{1}\times d_{2}})$
satisfying the following two conditions:

    * –

it holds that

$\|f\|_{\mathscr{L}^{2}(\triangle_{2}(t_{0},T))}:=\underset{t\in(t_{0},T)}{\mathrm{ess\,sup}}\Bigl{(}\int^{T}_{t}|f(s,t)|^{2}\,\mathrm{d}s\Bigr{)}^{1/2}<\infty;$

    * –

for any $\varepsilon>0$, there exists a finite partition
$\\{U_{i}\\}^{m}_{i=0}$ of $(t_{0},T)$ with $t_{0}=U_{0}<U_{1}<\cdots<U_{m}=T$
such that

$\underset{t\in(U_{i},U_{i+1})}{\mathrm{ess\,sup}}\Bigl{(}\int^{U_{i+1}}_{t}|f(s,t)|^{2}\,\mathrm{d}s\Bigr{)}^{1/2}<\varepsilon$

for each $i\in\\{0,1,\dots,m-1\\}$.

It is easy to see that
$(\mathscr{L}^{2}(\triangle_{2}(t_{0},T);\mathbb{R}^{d_{1}\times
d_{2}}),\|\cdot\|_{\mathscr{L}^{2}(\triangle_{2}(t_{0},T))})$ is a Banach
space.

######  Remark 1.1.

The space $\mathscr{L}^{2}(\triangle_{2}(t_{0},T);\mathbb{R}^{d_{1}\times
d_{2}})$ includes important classes of singular kernels appearing in the study
of SVIEs. On one hand, convolution-type kernels of the form
$f(t,s)=f_{1}(t-s)f_{2}(s)$, $(t,s)\in\triangle_{2}(0,T)$, with $f_{1}\in
L^{2}(0,T;\mathbb{R})$ and $f_{2}\in L^{\infty}(0,T;\mathbb{R}^{d_{1}\times
d_{2}})$ is in $\mathscr{L}^{2}(\triangle_{2}(t_{0},T);\mathbb{R}^{d_{1}\times
d_{2}})$. On the other hand, if $f$ satisfies the $L^{p}$-integrability

$\underset{t\in(t_{0},T)}{\mathrm{ess\,sup}}\Bigl{(}\int^{T}_{t}|f(s,t)|^{p}\,\mathrm{d}s\Bigr{)}^{1/p}<\infty$

for some $p>2$, then it is in
$\mathscr{L}^{2}(\triangle_{2}(t_{0},T);\mathbb{R}^{d_{1}\times d_{2}})$ too.

Throughout this paper, $d\in\mathbb{N}$ represents the dimension of state
processes, and $\ell\in\mathbb{N}$ represents the dimension of control
processes. We fix a finite terminal time $T\in(0,\infty)$.

##  2 Causal feedback solutions

We define the set of input conditions by
$\mathcal{I}:=\\{(t_{0},x)\,|\,t_{0}\in[0,T),\,x\in
L^{2}(t_{0},T;\mathbb{R}^{d})\\}$ and the set of control processes by
$\mathcal{U}(t_{0},T):=L^{2}_{\mathbb{F}}(t_{0},T;\mathbb{R}^{\ell})$. For
each input condition $(t_{0},x)\in\mathcal{I}$ and control
$u\in\mathcal{U}(t_{0},T)$, consider the controlled linear SVIE (1.1). The
following is the standing assumption of this paper.

######  Assumption 1.

  * •

The coefficients: $A\in L^{2}(\triangle_{2}(0,T);\mathbb{R}^{d\times d})$,
$B\in L^{2}(\triangle_{2}(0,T);\mathbb{R}^{d\times\ell})$,
$C\in\mathscr{L}^{2}(\triangle_{2}(0,T);\mathbb{R}^{d\times d})$,
$D\in\mathscr{L}^{2}(\triangle_{2}(0,T);\mathbb{R}^{d\times\ell})$.

  * •

The inhomogeneous terms: $b\in
L^{2,1}_{\mathbb{F}}(\triangle_{2}(0,T);\mathbb{R}^{d})$, $\sigma\in
L^{2}_{\mathbb{F}}(\triangle_{2}(0,T);\mathbb{R}^{d})$.

######  Remark 2.1.

In the standing assumption, the coefficients and the inhomogeneous terms are
singular and of non-convolution-types. For example, $A(t,s)$ is allowed to
diverge as $s\uparrow t$, and the same is true for $B,C,D,b$ and $\sigma$. Our
framework is more general than [9] (where the coefficients are of non-
convolution-types, but $B,C$ and $D$ are essentially regular, and the
inhomogeneous terms $b$ and $\sigma$ do not appear) and [2, 3] (where the
coefficients are singular, but they are of convolution-types with completely
monotone kernels, and the inhomogeneous terms are deterministic). See 1.1. It
is also worth to mention that the assumptions for the coefficients $C$ and $D$
being in $\mathscr{L}^{2}(\triangle_{2}(t_{0},T);\mathbb{R}^{d_{1}\times
d_{2}})$ fit into the framework of the so-called $\star$-Volterra kernels
introduced in [15].

Based on the discussion in the introductory section, we define the notions of
causal feedback strategies and the associated causal feedback solutions of
linear controlled SVIEs.

######  Definition 2.2.

Each triplet
$(\Xi,\Gamma,v)\in\mathcal{S}(0,T):=L^{\infty}(0,T;\mathbb{R}^{\ell\times
d})\times L^{2}(\triangle_{2}(0,T);\mathbb{R}^{\ell\times
d})\times\mathcal{U}(0,T)$ is called a _casual feedback strategy_. For each
$(\Xi,\Gamma,v)\in\mathcal{S}(0,T)$ and $(t_{0},x)\in\mathcal{I}$, we say that
a triplet $(X^{t_{0},x},\Theta^{t_{0},x},u^{t_{0},x})\in
L^{2}_{\mathbb{F}}(t_{0},T;\mathbb{R}^{d})\times
L^{2}_{\mathbb{F},\mathrm{c}}(\triangle_{2}(t_{0},T);\mathbb{R}^{d})\times\mathcal{U}(t_{0},T)$
is a _casual feedback solution_ of controlled SVIE (1.1) at $(t_{0},x)$
corresponding to $(\Xi,\Gamma,v)$ if it satisfies the closed-loop system
(1.5). We sometimes call the pair $(X^{t_{0},x},\Theta^{t_{0},x})$ the causal
feedback solution for simplicity. The control process
$u^{t_{0},x}\in\mathcal{U}(t_{0},T)$ is called the _outcome_ of the causal
feedback strategy $(\Xi,\Gamma,v)$ at $(t_{0},x)$, and we write
$(\Xi,\Gamma,v)[t_{0},x](t):=u^{t_{0},x}(t)$.

######  Remark 2.3.

Note that if $(X^{t_{0},x},\Theta^{t_{0},x})\in
L^{2}_{\mathbb{F}}(t_{0},T;\mathbb{R}^{d})\times
L^{2}_{\mathbb{F},\mathrm{c}}(\triangle_{2}(t_{0},T);\mathbb{R}^{d})$
satisfies the closed-loop system (1.5) for a given causal feedback strategy
$(\Xi,\Gamma,v)\in\mathcal{S}(0,T)$ and an input condition
$(t_{0},x)\in\mathcal{I}$, then the outcome
$u^{t_{0},x}=(\Xi,\Gamma,v)[t_{0},x]$ is automatically in
$\mathcal{U}(t_{0},T)$. We emphasize that the causal feedback strategy
$(\Xi,\Gamma,v)$ is chosen to be independent of the input condition
$(t_{0},x)$, while the causal feedback solution
$(X^{t_{0},x},\Theta^{t_{0},x})$ and the outcome
$u^{t_{0},x}=(\Xi,\Gamma,v)[t_{0},x]$ depend on $(t_{0},x)$.

We will also consider the homogeneous version of the controlled SVIE (1.1),
where the inhomogeneous terms $b$ and $\sigma$ vanish. In this case, the
controlled SVIE (1.1) becomes

$X_{0}(t)=x(t)+\int^{t}_{t_{0}}\\{A(t,s)X_{0}(s)+B(t,s)u(s)\\}\,\mathrm{d}s+\int^{t}_{t_{0}}\\{C(t,s)X_{0}(s)+D(t,s)u(s)\\}\,\mathrm{d}W(s),\
t\in(t_{0},T).$ (2.1)

For each causal feedback strategy $(\Xi,\Gamma,v)\in\mathcal{S}(0,T)$ and
input condition $(t_{0},x)\in\mathcal{I}$, the corresponding causal feedback
solution $(X^{t_{0},x}_{0},\Theta^{t_{0},x}_{0})\in
L^{2}_{\mathbb{F}}(t_{0},T;\mathbb{R}^{d})\times
L^{2}_{\mathbb{F},\mathrm{c}}(\triangle_{2}(t_{0},T);\mathbb{R}^{d})$ of the
homogeneous controlled SVIE (2.1) satisfies

$\begin{dcases}X^{t_{0},x}_{0}(t)=x(t)+\int^{t}_{t_{0}}\\{A(t,s)X^{t_{0},x}_{0}(s)+B(t,s)u^{t_{0},x}_{0}(s)\\}\,\mathrm{d}s\\\
\hskip
56.9055pt+\int^{t}_{t_{0}}\\{C(t,s)X^{t_{0},x}_{0}(s)+D(t,s)u^{t_{0},x}_{0}(s)\\}\,\mathrm{d}W(s),\
\ t\in(t_{0},T),\\\
\Theta^{t_{0},x}_{0}(s,t)=x(s)+\int^{t}_{t_{0}}\\{A(s,r)X^{t_{0},x}_{0}(r)+B(s,r)u^{t_{0},x}_{0}(r)\\}\,\mathrm{d}r\\\
\hskip
56.9055pt+\int^{t}_{t_{0}}\\{C(s,r)X^{t_{0},x}_{0}(r)+D(s,r)u^{t_{0},x}_{0}(r)\\}\,\mathrm{d}W(r),\
\ (s,t)\in\triangle_{2}(t_{0},T),\\\
u^{t_{0},x}_{0}(t)=\Xi(t)X^{t_{0},x}_{0}(t)+\int^{T}_{t}\Gamma(s,t)\Theta^{t_{0},x}_{0}(s,t)\,\mathrm{d}s+v(t),\
t\in(t_{0},T).\end{dcases}$

We denote the outcome by
$(\Xi,\Gamma,v)^{0}[t_{0},x]:=u^{t_{0},x}_{0}\in\mathcal{U}(t_{0},T)$.

The following is concerned with the existence and uniqueness of the causal
feedback solution. This is the first main result of this paper.

######  Theorem 2.4.

For each causal feedback strategy $(\Xi,\Gamma,v)\in\mathcal{S}(0,T)$ and each
input condition $(t_{0},x)\in\mathcal{I}$, the controlled SVIE (1.1) has a
unique causal feedback solution $(X^{t_{0},x},\Theta^{t_{0},x},u^{t_{0},x})\in
L^{2}_{\mathbb{F}}(t_{0},T;\mathbb{R}^{d})\times
L^{2}_{\mathbb{F},\mathrm{c}}(\triangle_{2}(t_{0},T);\mathbb{R}^{d})\times\mathcal{U}(t_{0},T)$.
Furthermore, there exists a constant $K>0$ depending only on
$A,B,C,D,\Xi,\Gamma$ such that

$\begin{split}&\|X^{t_{0},x}\|_{L^{2}_{\mathbb{F}}(t_{0},T)}+\|\Theta^{t_{0},x}\|_{L^{2}_{\mathbb{F},\mathrm{c}}(\triangle_{2}(t_{0},T))}+\|u^{t_{0},x}\|_{L^{2}_{\mathbb{F}}(t_{0},T)}\\\
&\leq
K(\|x\|_{L^{2}(t_{0},T)}+\|b\|_{L^{2,1}_{\mathbb{F}}(\triangle_{2}(t_{0},T))}+\|\sigma\|_{L^{2}_{\mathbb{F}}(\triangle_{2}(t_{0},T))}+\|v\|_{L^{2}_{\mathbb{F}}(t_{0},T)}).\end{split}$
(2.2)

###### Proof.

See Section A.1. ∎

Let us further observe the structure of the causal feedback solution. Let a
causal feedback strategy $(\Xi,\Gamma,v)\in\mathcal{S}(0,T)$ be fixed. For
each input condition $(t_{0},x)\in\mathcal{I}$, let
$(X^{t_{0},x},\Theta^{t_{0},x},u^{t_{0},x})$ be the corresponding causal
feedback solution to the controlled SVIE (1.1). Clearly,
$(X^{t_{0},x},\Theta^{t_{0},x},u^{t_{0},x})$ does not depend on $\Xi(t)$ for
$t\in(0,t_{0})$, $\Gamma(s,t)$ for $s\in(t,T)$ and $t\in(0,t_{0})$, or $v(t)$
for $t\in(0,t_{0})$. Furthermore, for any $t_{1}\in(t_{0},T)$, we have

$\begin{dcases}X^{t_{0},x}(t)=\Theta^{t_{0},x}(t,t_{1})+\int^{t}_{t_{1}}\\{A(t,s)X^{t_{0},x}(s)+B(t,s)u^{t_{0},x}(s)+b(t,s)\\}\,\mathrm{d}s\\\
\hskip
85.35826pt+\int^{t}_{t_{1}}\\{C(t,s)X^{t_{0},x}(s)+D(t,s)u^{t_{0},x}(s)+\sigma(t,s)\\}\,\mathrm{d}W(s),\
t\in(t_{1},T),\\\
\Theta^{t_{0},x}(s,t)=\Theta^{t_{0},x}(s,t_{1})+\int^{t}_{t_{1}}\\{A(s,r)X^{t_{0},x}(r)+B(s,r)u^{t_{0},x}(r)+b(s,r)\\}\,\mathrm{d}r\\\
\hskip
85.35826pt+\int^{t}_{t_{1}}\\{C(s,r)X^{t_{0},x}(r)+D(s,r)u^{t_{0},x}(r)+\sigma(s,r)\\}\,\mathrm{d}W(r),\
(s,t)\in\triangle_{2}(t_{1},T),\\\
u^{t_{0},x}(t)=\Xi(t)X^{t_{0},x}(t)+\int^{T}_{t}\Gamma(s,t)\Theta^{t_{0},x}(s,t)\,\mathrm{d}s+v(t),\
t\in(t_{1},T).\end{dcases}$

By the uniqueness of the causal feedback solution on $(t_{1},T)$, we see that

$\begin{dcases}X^{t_{0},x}(t)=X^{t_{1},\Theta^{t_{0},x}(\cdot,t_{1})}(t),\
t\in(t_{1},T),\\\
\Theta^{t_{0},x}(s,t)=\Theta^{t_{1},\Theta^{t_{0},x}(\cdot,t_{1})}(s,t),\
(s,t)\in\triangle_{2}(t_{1},T),\\\
u^{t_{0},x}(t)=u^{t_{1},\Theta^{t_{0},x}(\cdot,t_{1})}(t),\
t\in(t_{1},T).\end{dcases}$

This is the generalized flow property of the causal feedback solution.

##  3 Type-II EBSVIEs and duality principle

In this section, for a given causal feedback strategy
$(\Xi,\Gamma,v)\in\mathcal{S}(0,T)$ and $(\psi,\chi)\in
L^{2}_{\mathbb{F}}(0,T;\mathbb{R}^{d})\times
L^{2,1}_{\mathbb{F}}(\triangle_{2}(0,T);\mathbb{R}^{d})$, we investigate the
linear functional (1.8) which we rewrite here for readers’ convenience:

$\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\Bigl{\\{}\langle\psi(t),X^{t_{0},x}(t)\rangle+\int^{T}_{t}\langle\chi(s,t),\Theta^{t_{0},x}(s,t)\rangle\,\mathrm{d}s\Bigr{\\}}\,\mathrm{d}t\Bigr{]}.$

Here, $(X^{t_{0},x},\Theta^{t_{0},x})\in
L^{2}_{\mathbb{F}}(t_{0},T;\mathbb{R}^{d})\times
L^{2}_{\mathbb{F},\mathrm{c}}(\triangle_{2}(t_{0},T);\mathbb{R}^{d})$ is the
causal feedback solution to the controlled SVIE (1.1) at
$(t_{0},x)\in\mathcal{I}$ corresponding to $(\Xi,\Gamma,v)$. By the linearity
of $(X^{t_{0},x},\Theta^{t_{0},x})$ with respect to the tuple
$(x,b,\sigma,v)\in L^{2}(t_{0},T;\mathbb{R}^{d})\times
L^{2,1}_{\mathbb{F}}(\triangle_{2}(t_{0},T);\mathbb{R}^{d})\times
L^{2}_{\mathbb{F}}(\triangle_{2}(t_{0},T);\mathbb{R}^{d})\times
L^{2}_{\mathbb{F}}(t_{0},T;\mathbb{R}^{d})$, together with the estimate (2.2),
the above term can be seen as a bounded linear functional $\Psi$ on the
product space:

$\Psi(x,b,\sigma,v):=\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\Bigl{\\{}\langle\psi(t),X^{t_{0},x}(t)\rangle+\int^{T}_{t}\langle\chi(s,t),\Theta^{t_{0},x}(s,t)\rangle\,\mathrm{d}s\Bigr{\\}}\,\mathrm{d}t\Bigr{]}.$

Note that the bounded linear functional $\Psi$ is determined by
$A,B,C,D,\Xi,\Gamma$ and $(\psi,\chi)$. Our purpose is to derive the dual
representation of $\Psi$.

To do so, we introduce the following (linear) _Type-II extended backward
stochastic Volterra integral equation_ (Type-II EBSVIE):

$\begin{dcases}\,\mathrm{d}\eta(t,s)=-\Bigl{\\{}\chi(t,s)+\Gamma(t,s)^{\top}\int^{T}_{s}B(r,s)^{\top}\eta(r,s)\,\mathrm{d}r+\Gamma(t,s)^{\top}\int^{T}_{s}D(r,s)^{\top}\zeta(r,s)\,\mathrm{d}r\Bigr{\\}}\,\mathrm{d}s\\\
\hskip 85.35826pt+\zeta(t,s)\,\mathrm{d}W(s),\ (t,s)\in\triangle_{2}(0,T),\\\
\eta(t,t)=\psi(t)+\int^{T}_{t}(A+B\triangleright\Xi)(r,t)^{\top}\eta(r,t)\,\mathrm{d}r+\int^{T}_{t}(C+D\triangleright\Xi)(r,t)^{\top}\zeta(r,t)\,\mathrm{d}r,\
t\in(0,T),\end{dcases}$ (3.1)

with $A+B\triangleright\Xi\in L^{2}(\triangle_{2}(0,T);\mathbb{R}^{d\times
d})$ and
$C+D\triangleright\Xi\in\mathscr{L}^{2}(\triangle_{2}(0,T);\mathbb{R}^{d\times
d})$ defined by

$(A+B\triangleright\Xi)(t,s):=A(t,s)+B(t,s)\Xi(s),\
(C+D\triangleright\Xi)(t,s):=C(t,s)+D(t,s)\Xi(s)$

for $(t,s)\in\triangle_{2}(0,T)$.

######  Definition 3.1.

We say that a pair $(\eta,\zeta)\in
L^{2}_{\mathbb{F},\mathrm{c}}(\triangle_{2}(0,T);\mathbb{R}^{d})\times
L^{2}_{\mathbb{F}}(\triangle_{2}(0,T);\mathbb{R}^{d})$ is an _adapted
solution_ to the Type-II EBSVIE (3.1) if it satisfies

$\begin{dcases}\eta(t,\theta)=\eta(t,t)+\int^{t}_{\theta}\Bigl{\\{}\chi(t,s)+\Gamma(t,s)^{\top}\int^{T}_{s}B(r,s)^{\top}\eta(r,s)\,\mathrm{d}r+\Gamma(t,s)^{\top}\int^{T}_{s}D(r,s)^{\top}\zeta(r,s)\,\mathrm{d}r\Bigr{\\}}\,\mathrm{d}s\\\
\hskip 113.81102pt-\int^{t}_{\theta}\zeta(t,s)\,\mathrm{d}W(s),\\\
\eta(t,t)=\psi(t)+\int^{T}_{t}(A+B\triangleright\Xi)(r,t)^{\top}\eta(r,t)\,\mathrm{d}r+\int^{T}_{t}(C+D\triangleright\Xi)(r,t)^{\top}\zeta(r,t)\,\mathrm{d}r,\end{dcases}$

for a.e. $t\in(0,T)$ and any $\theta\in[0,t]$, a.s.

######  Remark 3.2.

The Type-II EBSVIE (3.1) is a generalization of a class of linear Type-II
BSVIEs introduced by Yong [32]. Indeed, if $\chi=0$ and $(\Xi,\Gamma)=(0,0)$,
then we see from the first equation in (3.1) that

$\eta(t,s)=\mathbb{E}_{s}[\eta(t)],\
\eta(t)=\mathbb{E}[\eta(t)]+\int^{t}_{0}\zeta(t,s)\,\mathrm{d}W(s),$

for $(t,s)\in\triangle_{2}(0,T)$, where $\eta(t):=\eta(t,t)$. Thus, the second
equation in (3.1) becomes

$\eta(t)=\psi(t)+\int^{T}_{t}\\{A(r,t)^{\top}\mathbb{E}_{t}[\eta(r)]+C(r,t)^{\top}\zeta(r,t)\\}\,\mathrm{d}r,\
t\in(0,T),$

or equivalently, by means of the martingale representation theorem,

$\eta(t)=\psi(t)+\int^{T}_{t}\\{A(r,t)^{\top}\eta(r)+C(r,t)^{\top}\zeta(r,t)\\}\,\mathrm{d}r-\int^{T}_{t}\zeta(t,r)\,\mathrm{d}W(r),\
t\in(0,T),$

for a suitable choice of $\zeta(t,r)$ for $(r,t)\in\triangle_{2}(0,T)$. This
is a linear Type-II BSVIE. In this case, by [15, 32], the following duality
principle holds:

$\displaystyle\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\langle\psi(t),X(t)\rangle\,\mathrm{d}t\Bigr{]}=\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\langle\eta(t),\varphi^{x,b,\sigma,v}(t)\rangle\,\mathrm{d}t\Bigr{]}$
$\displaystyle=\int^{T}_{t_{0}}\langle\mathbb{E}[\eta(t)],x(t)\rangle\,\mathrm{d}t+\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\Bigl{\\{}\int^{T}_{t}\langle\mathbb{E}_{t}[\eta(s)],b(s,t)\rangle\,\mathrm{d}s+\int^{T}_{t}\langle\zeta(s,t),\sigma(s,t)\rangle\,\mathrm{d}s\Bigr{\\}}\,\mathrm{d}t\Bigr{]}$
$\displaystyle\hskip
28.45274pt+\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\Bigl{\langle}\int^{T}_{t}\\{B(s,t)^{\top}\mathbb{E}_{t}[\eta(s)]+D(s,t)^{\top}\zeta(s,t)\\}\,\mathrm{d}s,v(t)\Bigr{\rangle}\,\mathrm{d}t\Bigr{]},$

where $X\in L^{2}_{\mathbb{F}}(t_{0},T;\mathbb{R}^{d})$ is the solution to the
SVIE

$X(t)=\varphi^{x,b,\sigma,v}(t)+\int^{t}_{t_{0}}A(t,s)X(s)\,\mathrm{d}s+\int^{t}_{t_{0}}C(t,s)X(s)\,\mathrm{d}W(s),\
t\in(t_{0},T),$

with the free term $\varphi^{x,b,\sigma,v}(t)\in
L^{2}_{\mathbb{F}}(t_{0},T;\mathbb{R}^{d})$ defined by

$\varphi^{x,b,\sigma,v}(t):=x(t)+\int^{t}_{t_{0}}\\{B(t,s)v(s)+b(t,s)\\}\,\mathrm{d}s+\int^{t}_{t_{0}}\\{D(t,s)v(s)+\sigma(t,s)\\}\,\mathrm{d}W(s),\
t\in(t_{0},T).$

The following theorem generalizes the above duality principle to the framework
of causal feedback solutions of controlled SVIEs. This is the second main
result of the present paper.

######  Theorem 3.3.

Let $(\Xi,\Gamma)\in L^{\infty}(0,T;\mathbb{R}^{\ell\times d})\times
L^{2}(\triangle_{2}(0,T);\mathbb{R}^{\ell\times d})$ be fixed. For any
$(\psi,\chi)\in L^{2}_{\mathbb{F}}(0,T;\mathbb{R}^{d})\times
L^{2,1}_{\mathbb{F}}(\triangle_{2}(0,T);\mathbb{R}^{d})$, there exists a
unique adapted solution $(\eta,\zeta)\in
L^{2}_{\mathbb{F},\mathrm{c}}(\triangle_{2}(0,T);\mathbb{R}^{d})\times
L^{2}_{\mathbb{F}}(\triangle_{2}(0,T);\mathbb{R}^{d})$ to the Type-II EBSVIE
(3.1). Furthermore, for any $v\in\mathcal{U}(0,T)$ and
$(t_{0},x)\in\mathcal{I}$, the following _duality principle_ holds:

$\begin{split}&\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\Bigl{\\{}\langle\psi(t),X^{t_{0},x}(t)\rangle+\int^{T}_{t}\langle\chi(s,t),\Theta^{t_{0},x}(s,t)\rangle\,\mathrm{d}s\Bigr{\\}}\,\mathrm{d}t\Bigr{]}\\\
&=\int^{T}_{t_{0}}\langle\mathbb{E}[\eta(t,t_{0})],x(t)\rangle\,\mathrm{d}t+\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\Bigl{\\{}\int^{T}_{t}\langle\eta(s,t),b(s,t)\rangle\,\mathrm{d}s+\int^{T}_{t}\langle\zeta(s,t),\sigma(s,t)\rangle\,\mathrm{d}s\Bigr{\\}}\,\mathrm{d}t\Bigr{]}\\\
&\hskip
28.45274pt+\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\Bigl{\langle}\int^{T}_{t}\\{B(s,t)^{\top}\eta(s,t)+D(s,t)^{\top}\zeta(s,t)\\}\,\mathrm{d}s,v(t)\Bigr{\rangle}\,\mathrm{d}t\Bigr{]},\end{split}$
(3.2)

where $(X^{t_{0},x},\Theta^{t_{0},x})\in
L^{2}_{\mathbb{F}}(t_{0},T;\mathbb{R}^{d})\times
L^{2}_{\mathbb{F},\mathrm{c}}(\triangle_{2}(t_{0},T);\mathbb{R}^{d})$ is the
causal feedback solution to the controlled SVIE (1.1) at $(t_{0},x)$
corresponding to the causal feedback strategy
$(\Xi,\Gamma,v)\in\mathcal{S}(0,T)$.

###### Proof.

See Section A.2. ∎

##  4 Lyapunov–Volterra equations and quadratic functionals

In this section, we investigate the quadratic functional (1.9) with respect to
the homogeneous controlled SVIE (2.1). Specifically, for each fixed
$(\Xi,\Gamma)\in L^{\infty}(0,T;\mathbb{R}^{\ell\times d})\times
L^{2}(\triangle_{2}(0,T);\mathbb{R}^{\ell\times d})$ and
$(Q^{(1)},Q^{(2)},Q^{(3)})\in L^{\infty}(0,T;\mathbb{S}^{d})\times
L^{2}(\triangle_{2}(0,T);\mathbb{R}^{d\times d})\times
L^{2,2,1}_{\mathrm{sym}}(\square_{3}(0,T);\mathbb{R}^{d\times d})$, consider
the following term:

$\begin{split}&\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\Bigl{\\{}\langle
Q^{(1)}(t)X^{t_{0},x}_{0}(t),X^{t_{0},x}_{0}(t)\rangle+2\int^{T}_{t}\langle
Q^{(2)}(s,t)X^{t_{0},x}_{0}(t),\Theta^{t_{0},x}_{0}(s,t)\rangle\,\mathrm{d}s\\\
&\hskip 42.67912pt+\int^{T}_{t}\\!\\!\int^{T}_{t}\langle
Q^{(3)}(s_{1},s_{2},t)\Theta^{t_{0},x}_{0}(s_{2},t),\Theta^{t_{0},x}_{0}(s_{1},t)\rangle\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2}\Bigr{\\}}\,\mathrm{d}t\Bigr{]},\end{split}$
(4.1)

where $(X^{t_{0},x}_{0},\Theta^{t_{0},x}_{0})\in
L^{2}_{\mathbb{F}}(t_{0},T;\mathbb{R}^{d})\times
L^{2}_{\mathbb{F},\mathrm{c}}(\triangle_{2}(t_{0},T);\mathbb{R}^{d})$ is the
causal feedback solution to the homogeneous controlled SVIE (2.1) at
$(t_{0},x)\in\mathcal{I}$ corresponding to the causal feedback strategy
$(\Xi,\Gamma,0)\in\mathcal{S}(0,T)$. (4.1) can be written as
$\Phi^{t_{0}}(x,x)$ with

$\displaystyle\Phi^{t_{0}}(x,y)$
$\displaystyle:=\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\Bigl{\\{}\langle
Q^{(1)}(t)X^{t_{0},x}_{0}(t),X^{t_{0},y}_{0}(t)\rangle$ $\displaystyle\hskip
56.9055pt+\int^{T}_{t}\langle
Q^{(2)}(s,t)X^{t_{0},x}_{0}(t),\Theta^{t_{0},y}_{0}(s,t)\rangle\,\mathrm{d}s+\int^{T}_{t}\langle
Q^{(2)}(s,t)X^{t_{0},y}_{0}(t),\Theta^{t_{0},x}_{0}(s,t)\rangle\,\mathrm{d}s$
$\displaystyle\hskip 56.9055pt+\int^{T}_{t}\\!\\!\int^{T}_{t}\langle
Q^{(3)}(s_{1},s_{2},t)\Theta^{t_{0},x}_{0}(s_{2},t),\Theta^{t_{0},y}_{0}(s_{1},t)\rangle\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2}\Bigr{\\}}\,\mathrm{d}t\Bigr{]}$

for $x,y\in L^{2}(t_{0},T;\mathbb{R}^{d})$. For each fixed $t_{0}\in[0,T)$,
noting the linearity of $(X^{t_{0},x}_{0},\Theta^{t_{0},x}_{0})$ with respect
to $x\in L^{2}(t_{0},T;\mathbb{R}^{d})$, the estimate (2.2) and the
symmetricity of $(Q^{(1)},Q^{(2)},Q^{(3)})$, we see that $\Phi^{t_{0}}$ is a
bounded symmetric bilinear form on the Hilbert space
$L^{2}(t_{0},T;\mathbb{R}^{d})$. Therefore, there exists a unique self-adjoint
bounded linear operator $\mathcal{P}^{t_{0}}$ on
$L^{2}(t_{0},T;\mathbb{R}^{d})$ such that

$\displaystyle\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\Bigl{\\{}\langle
Q^{(1)}(t)X^{t_{0},x}_{0}(t),X^{t_{0},x}_{0}(t)\rangle+2\int^{T}_{t}\langle
Q^{(2)}(s,t)X^{t_{0},x}_{0}(t),\Theta^{t_{0},x}_{0}(s,t)\rangle\,\mathrm{d}s$
$\displaystyle\hskip 42.67912pt+\int^{T}_{t}\\!\\!\int^{T}_{t}\langle
Q^{(3)}(s_{1},s_{2},t)\Theta^{t_{0},x}_{0}(s_{2},t),\Theta^{t_{0},x}_{0}(s_{1},t)\rangle\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2}\Bigr{\\}}\,\mathrm{d}t\Bigr{]}$
$\displaystyle=\Phi^{t_{0}}(x,x)=\langle\mathcal{P}^{t_{0}}x,x\rangle_{L^{2}(t_{0},T)}$

for any $x\in L^{2}(t_{0},T;\mathbb{R}^{d})$, where
$\langle\cdot,\cdot\rangle_{L^{2}(t_{0},T)}$ denotes the inner product on
$L^{2}(t_{0},T;\mathbb{R}^{d})$. We consider the following natural ansatz of
the bounded self-adjoint operator $\mathcal{P}^{t_{0}}$:

$(\mathcal{P}^{t_{0}}x)(t):=P^{(1)}(t)x(t)+\int^{T}_{t_{0}}P^{(2)}(t,r,t_{0})x(r)\,\mathrm{d}r,\
t\in(t_{0},T),$

with suitable choice of matrix-valued deterministic functions $P^{(1)}$ and
$P^{(2)}$. The above ansatz is a combination of a multiplicative operator and
a Hilbert–Schmidt integral operator. The purpose of this section is to derive
a Lyapunov-type dynamics of the pair $(P^{(1)},P^{(2)})$.

###  4.1 Preliminaries

First, we introduce a suitable space of the pairs $(P^{(1)},P^{(2)})$ and
investigate some fundamental properties.

######  Definition 4.1.

We denote by $\Pi(0,T)$ the set of pairs $(P^{(1)},P^{(2)})$ with
$P^{(1)}:(0,T)\to\mathbb{R}^{d\times d}$ and
$P^{(2)}:\square_{3}(t_{0},T)\to\mathbb{R}^{d\times d}$ such that

  * •

$P^{(1)}\in L^{\infty}(0,T;\mathbb{S}^{d})$;

  * •

for a.e. $(s_{1},s_{2})\in(0,T)^{2}$, $t\mapsto P^{(2)}(s_{1},s_{2},t)$ is
absolutely continuous on $(0,s_{1}\wedge s_{2})$;

  * •

the function $P^{(2)}(s_{1},s_{2},s_{1}\wedge s_{2}):=\lim_{t\uparrow
s_{1}\wedge s_{2}}P(s_{1},s_{2},t)$, $(s_{1},s_{2})\in(0,T)^{2}$, is in
$L^{2}((0,T)^{2};\mathbb{R}^{d\times d})$;

  * •

the function $\dot{P}^{(2)}(s_{1},s_{2},t):=\frac{\partial P^{(2)}}{\partial
t}(s_{1},s_{2},t)$, $(s_{1},s_{2},t)\in\square_{3}(0,T)$, is in
$L^{2,2,1}(\square_{3}(0,T);\mathbb{R}^{d\times d})$;

  * •

for a.e. $(s_{1},s_{2},t)\in\square_{3}(0,T)$, it holds that
$P^{(2)}(s_{1},s_{2},t)=P^{(2)}(s_{2},s_{1},t)^{\top}$.

######  Remark 4.2.

Observe that $\Pi(0,T)$ is a Banach space with the norm

$\displaystyle\|P\|_{\Pi(0,T)}:=\underset{t\in(0,T)}{\mathrm{ess\,sup}}|P^{(1)}(t)|+\Bigl{(}\int^{T}_{0}\\!\\!\int^{T}_{0}|P^{(2)}(s_{1},s_{2},s_{1}\wedge
s_{2})|^{2}\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2}\Bigr{)}^{1/2}$
$\displaystyle\hskip
113.81102pt+\Bigl{(}\int^{T}_{0}\\!\\!\int^{T}_{0}\Bigl{(}\int^{s_{1}\wedge
s_{2}}_{0}|\dot{P}^{(2)}(s_{1},s_{2},t)|\,\mathrm{d}t\Bigr{)}^{2}\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2}\Bigr{)}^{1/2}$

for $P=(P^{(1)},P^{(2)})\in\Pi(0,T)$. Furthermore, the following holds:

$\underset{t\in(0,T)}{\mathrm{ess\,sup}}|P^{(1)}(t)|+\Bigl{(}\int^{T}_{0}\\!\\!\int^{T}_{0}\sup_{t\in[0,s_{1}\wedge
s_{2}]}|P^{(2)}(s_{1},s_{2},t)|^{2}\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2}\Bigr{)}^{1/2}\leq\|P\|_{\Pi(0,T)}<\infty.$

######  Lemma 4.3.

Let $P=(P^{(1)},P^{(2)})\in\Pi(0,T)$. Then for each $t_{0}\in[0,T)$, the map
$\mathcal{P}^{t_{0}}:L^{2}(t_{0},T;\mathbb{R}^{d})\to
L^{2}(t_{0},T;\mathbb{R}^{d})$ defined by

$(\mathcal{P}^{t_{0}}x)(t):=P^{(1)}(t)x(t)+\int^{T}_{t_{0}}P^{(2)}(t,r,t_{0})x(r)\,\mathrm{d}r,\
t\in(t_{0},T),$

for $x\in L^{2}(t_{0},T;\mathbb{R}^{d})$, is a self-adjoint bounded linear
operator on the Hilbert space $L^{2}(t_{0},T;\mathbb{R}^{d})$.

###### Proof.

It is clear that $\mathcal{P}^{t_{0}}$ is a bounded linear operator on
$L^{2}(t_{0},T;\mathbb{R}^{d})$. Noting that $P^{(1)}(t)\in\mathbb{S}^{d}$ and
$P^{(2)}(s_{1},s_{2},t)=P^{(2)}(s_{2},s_{1},t)^{\top}$, for each $x,y\in
L^{2}(t_{0},T;\mathbb{R}^{d})$,

$\displaystyle\langle\mathcal{P}^{t_{0}}x,y\rangle_{L^{2}(t_{0},T)}$
$\displaystyle=\int^{T}_{t_{0}}\langle(\mathcal{P}^{t_{0}}x)(t),y(t)\rangle\,\mathrm{d}t$
$\displaystyle=\int^{T}_{t_{0}}\langle
P^{(1)}(t)x(t),y(t)\rangle\,\mathrm{d}t+\int^{T}_{t_{0}}\\!\\!\int^{T}_{t_{0}}\langle
P^{(2)}(s_{1},s_{2},t_{0})x(s_{2}),y(s_{1})\rangle\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2}$
$\displaystyle=\int^{T}_{t_{0}}\langle
x(t),P^{(1)}(t)y(t)\rangle\,\mathrm{d}t+\int^{T}_{t_{0}}\\!\\!\int^{T}_{t_{0}}\langle
x(s_{2}),P^{(2)}(s_{2},s_{1},t_{0})y(s_{1})\rangle\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2}$
$\displaystyle=\int^{T}_{t_{0}}\langle
x(t),(\mathcal{P}^{t_{0}}y)(t)\rangle\,\mathrm{d}t=\langle
x,\mathcal{P}^{t_{0}}y\rangle_{L^{2}(t_{0},T)}.$

Thus, $\mathcal{P}^{t_{0}}$ is self-adjoint. ∎

######  Lemma 4.4.

Let $P=(P^{(1)},P^{(2)})\in\Pi(0,T)$. Assume that

$\int^{T}_{t_{0}}\langle
P^{(1)}(t)x(t),x(t)\rangle\,\mathrm{d}t+\int^{T}_{t_{0}}\\!\\!\int^{T}_{t_{0}}\langle
P^{(2)}(s_{1},s_{2},t_{0})x(s_{2}),x(s_{1})\rangle\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2}=0$

for any $(t_{0},x)\in\mathcal{I}$. Then $P=0$ in the sense that $P^{(1)}(t)=0$
for a.e. $t\in(0,T)$ and $P^{(2)}(s_{1},s_{2},t)=0$ for any
$t\in[0,s_{1}\wedge s_{2}]$ for a.e. $(s_{1},s_{2})\in(0,T)^{2}$.

###### Proof.

It suffices to show that $P^{(1)}=0$. Then $P^{(2)}=0$ follows easily.
Considering $t_{0}=0$ and
$x(t):=\sqrt{N}\mbox{\rm{1}}\mbox{\rm{l}}_{[\tau,\tau+1/N]}(t)y$, $t\in(0,T)$,
with $\tau\in(0,T)$, $N\in\mathbb{N}$ with $\tau+1/N<T$ and
$y\in\mathbb{R}^{d}$ being arbitrary, we obtain

$N\int^{\tau+1/N}_{\tau}\langle
P^{(1)}(t)y,y\rangle\,\mathrm{d}t+N\int^{\tau+1/N}_{\tau}\\!\\!\int^{\tau+1/N}_{\tau}\langle
P^{(2)}(s_{1},s_{2},0)y,y\rangle\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2}=0.$

Thus, we have

$N\int^{\tau+1/N}_{\tau}P^{(1)}(t)\,\mathrm{d}t+N\int^{\tau+1/N}_{\tau}\\!\\!\int^{\tau+1/N}_{\tau}P^{(2)}(s_{1},s_{2},0)\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2}=0.$

By the Lebesgue differentiation theorem, for a.e. $\tau\in(0,T)$, the first
term tends to $P^{(1)}(\tau)$ as $N\to\infty$. On the other hand, for any
$\tau\in(0,T)$, the second term can be estimated as

$\Bigl{|}N\int^{\tau+1/N}_{\tau}\\!\\!\int^{\tau+1/N}_{\tau}P^{(2)}(s_{1},s_{2},0)\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2}\Bigr{|}\leq\Bigl{(}\int^{\tau+1/N}_{\tau}\\!\\!\int^{\tau+1/N}_{\tau}|P^{(2)}(s_{1},s_{2},0)|^{2}\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2}\Bigr{)}^{1/2},$

and the right-hand side tends to zero as $N\to\infty$. Thus, we have
$P^{(1)}(\tau)=0$ for a.e. $\tau\in(0,T)$. This completes the proof. ∎

The following notations are useful for capturing the structure of some
expressions arising in calculations of quadratic functionals of SVIEs.

######  Definition 4.5.

Let $P=(P^{(1)},P^{(2)})\in\Pi(0,T)$. For each
$M_{1}:\triangle_{2}(0,T)\to\mathbb{R}^{d_{1}\times d}$ and
$M_{2}:\triangle_{2}(0,T)\to\mathbb{R}^{d\times d_{2}}$ with
$d_{1},d_{2}\in\mathbb{N}$, we define

$\displaystyle(M_{1}{\ltimes}P)(s,t):=M_{1}(s,t)P^{(1)}(s)+\int^{T}_{t}M_{1}(r,t)P^{(2)}(r,s,t)\,\mathrm{d}r,\
(s,t)\in\triangle_{2}(0,T),$
$\displaystyle(P{\rtimes}M_{2})(s,t):=P^{(1)}(s)M_{2}(s,t)+\int^{T}_{t}P^{(2)}(s,r,t)M_{2}(r,t)\,\mathrm{d}r,\
(s,t)\in\triangle_{2}(0,T),$

and

$(M_{1}{\ltimes}P{\rtimes}M_{2})(t):=\\!\int^{T}_{t}M_{1}(s,t)P^{(1)}(s)M_{2}(s,t)\,\mathrm{d}s+\\!\int^{T}_{t}\\!\\!\int^{T}_{t}\\!M_{1}(s_{1},t)P^{(2)}(s_{1},s_{2},t)M_{2}(s_{2},t)\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2},\
t\in(0,T).$

######  Lemma 4.6.

Let $P=(P^{(1)},P^{(2)})\in\Pi(0,T)$. Fix $d_{1},d_{2}\in\mathbb{N}$.

  * (i)

For each $M\in\mathcal{L}(\triangle_{2}(0,T);\mathbb{R}^{d\times d_{2}})$ with
$\mathcal{L}$ being one of $L^{2,1}_{\mathbb{F}}$, $L^{2}_{\mathbb{F}}$,
$L^{2,1}$, $L^{2}$ and $\mathscr{L}^{2}$, it holds that
$P{\rtimes}M\in\mathcal{L}(\triangle_{2}(0,T);\mathbb{R}^{d\times d_{2}})$ and

$\|P{\rtimes}M\|_{\mathcal{L}(\triangle_{2}(0,T))}\leq\|P\|_{\Pi(0,T)}\|M\|_{\mathcal{L}(\triangle_{2}(0,T))}.$

Furthermore, $(P{\rtimes}M)^{\top}=M^{\top}{\ltimes}P$.

  * (ii)

For each $M_{1}\in\mathscr{L}^{2}(\triangle_{2}(0,T);\mathbb{R}^{d_{1}\times
d})$ and $M_{2}\in\mathscr{L}^{2}(\triangle_{2}(0,T);\mathbb{R}^{d\times
d_{2}})$, it holds that $M_{1}{\ltimes}P{\rtimes}M_{2}\in
L^{\infty}(0,T;\mathbb{R}^{d_{1}\times d_{2}})$ and

$\|M_{1}{\ltimes}P{\rtimes}M_{2}\|_{L^{\infty}(0,T)}\leq\|M_{1}\|_{\mathscr{L}^{2}(\triangle_{2}(0,T))}\|P\|_{\Pi(0,T)}\|M_{2}\|_{\mathscr{L}^{2}(\triangle_{2}(0,T))}.$

Furthermore,
$(M_{1}{\ltimes}P{\rtimes}M_{2})^{\top}=M^{\top}_{2}{\ltimes}P{\rtimes}M^{\top}_{1}$.
In particular, $M^{\top}_{2}{\ltimes}P{\rtimes}M_{2}\in
L^{\infty}(0,T;\mathbb{S}^{d_{2}})$.

  * (iii)

For each $M\in\mathscr{L}^{2}(\triangle_{2}(0,T);\mathbb{R}^{d_{1}\times d})$
and $\xi\in L^{2}_{\mathbb{F}}(\triangle_{2}(0,T);\mathbb{R}^{d\times
d_{2}})$, it holds that

$M{\ltimes}P{\rtimes}\xi\in L^{2}_{\mathbb{F}}(0,T;\mathbb{R}^{d_{1}\times
d_{2}}),\ \xi^{\top}{\ltimes}P{\rtimes}\xi\in
L^{1}_{\mathbb{F}}(0,T;\mathbb{S}^{d_{2}}).$

###### Proof.

This lemma can be proved by the Cauchy–Schwarz inequality and the
symmetricity, and we omit the details. ∎

###  4.2 Derivations of Lyapunov–Volterra equations

By using the notations defined in the last subsection, we derive a Lyapunov-
type dynamics of $P=(P^{(1)},P^{(2)})\in\Pi(0,T)$ and investigate the
quadratic functional (4.1). Before stating the main theorem, we prove the
following key lemma, which is helpful not only for understanding how to derive
the Lyapunov-type equation, but also for the study of LQ stochastic Volterra
control problems.

######  Lemma 4.7.

Let $P=(P^{(1)},P^{(2)})\in\Pi(0,T)$, $(t_{0},x)\in\mathcal{I}$ and
$u\in\mathcal{U}(t_{0},T)$. Then

$\displaystyle\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\Bigl{\\{}\langle
P^{(1)}(t)X(t),X(t)\rangle+2\int^{T}_{t}\langle
P^{(2)}(s,t,t)X(t),\Theta(s,t)\rangle\,\mathrm{d}s\Bigr{\\}}\,\mathrm{d}t\Bigr{]}$
$\displaystyle=\int^{T}_{t_{0}}\langle
P^{(1)}(t)x(t),x(t)\rangle\,\mathrm{d}t+\int^{T}_{t_{0}}\\!\\!\int^{T}_{t_{0}}\langle
P^{(2)}(s_{1},s_{2},t_{0})x(s_{2}),x(s_{1})\rangle\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2}+\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}(\sigma^{\top}{\ltimes}P{\rtimes}\sigma)(t)\,\mathrm{d}t\Bigr{]}$
$\displaystyle\hskip
14.22636pt+\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\Bigl{\\{}\langle(D^{\top}{\ltimes}P{\rtimes}D)(t)u(t),u(t)\rangle$
$\displaystyle\hskip
56.9055pt+2\Bigl{\langle}(D^{\top}{\ltimes}P{\rtimes}C)(t)X(t)+\int^{T}_{t}(B^{\top}{\ltimes}P)(s,t)\Theta(s,t)\,\mathrm{d}s+(D^{\top}{\ltimes}P{\rtimes}\sigma)(t),u(t)\Bigr{\rangle}$
$\displaystyle\hskip
56.9055pt+\langle(C^{\top}{\ltimes}P{\rtimes}C)(t)X(t),X(t)\rangle+2\int^{T}_{t}\langle(P{\rtimes}A)(s,t)X(t),\Theta(s,t)\rangle\,\mathrm{d}s$
$\displaystyle\hskip
56.9055pt+\int^{T}_{t}\\!\\!\int^{T}_{t}\langle\dot{P}^{(2)}(s_{1},s_{2},t)\Theta(s_{2},t),\Theta(s_{1},t)\rangle\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2}$
$\displaystyle\hskip
56.9055pt+2\langle(C^{\top}{\ltimes}P{\rtimes}\sigma)(t),X(t)\rangle+2\int^{T}_{t}\langle(P{\rtimes}b)(s,t),\Theta(s,t)\rangle\,\mathrm{d}s\Bigr{\\}}\,\mathrm{d}t\Bigr{]},$

where $X$ and $\Theta$ are the state process and the forward state process,
respectively, corresponding to the input condition $(t_{0},x)$ and the control
$u$.

###### Proof.

First, we note that $P^{(1)}(t)\in\mathbb{S}^{d}$ for a.e. $t\in(0,T)$.
Applying Itô’s formula to $\langle
P^{(1)}(t)\Theta(t,\cdot),\Theta(t,\cdot)\rangle$ on $(t_{0},t)$ for a.e.
$t\in(t_{0},T)$, we have

$\displaystyle\mathbb{E}[\langle
P^{(1)}(t)X(t),X(t)\rangle]=\mathbb{E}[\langle
P^{(1)}(t)\Theta(t,t),\Theta(t,t)\rangle]$ $\displaystyle=\langle
P^{(1)}(t)x(t),x(t)\rangle$ $\displaystyle\hskip
14.22636pt+\mathbb{E}\Bigl{[}\int^{t}_{t_{0}}\Bigl{\\{}2\langle
P^{(1)}(t)\Theta(t,s),A(t,s)X(s)+B(t,s)u(s)+b(t,s)\rangle$
$\displaystyle\hskip 56.9055pt+\langle
P^{(1)}(t)(C(t,s)X(s)+D(t,s)u(s)+\sigma(t,s)),C(t,s)X(s)+D(t,s)u(s)+\sigma(t,s)\rangle\Bigr{\\}}\,\mathrm{d}s\Bigr{]}$
$\displaystyle=\langle
P^{(1)}(t)x(t),x(t)\rangle+\mathbb{E}\Bigl{[}\int^{t}_{t_{0}}\sigma(t,s)^{\top}P^{(1)}(t)\sigma(t,s)\,\mathrm{d}s\Bigr{]}$
$\displaystyle\hskip
14.22636pt+\mathbb{E}\Bigl{[}\int^{t}_{t_{0}}\Bigl{\\{}\langle
D(t,s)^{\top}P^{(1)}(t)D(t,s)u(s),u(s)\rangle$ $\displaystyle\hskip
56.9055pt+2\langle
D(t,s)^{\top}P^{(1)}(t)C(t,s)X(s)+B(t,s)^{\top}P^{(1)}(t)\Theta(t,s)+D(t,s)^{\top}P^{(1)}(t)\sigma(t,s),u(s)\rangle$
$\displaystyle\hskip 56.9055pt+\langle
C(t,s)^{\top}P^{(1)}(t)C(t,s)X(s),X(s)\rangle+2\langle
P^{(1)}(t)A(t,s)X(s),\Theta(t,s)\rangle$ $\displaystyle\hskip
56.9055pt+2\langle C(t,s)^{\top}P^{(1)}(t)\sigma(t,s),X(s)\rangle+2\langle
P^{(1)}(t)b(t,s),\Theta(t,s)\rangle\Bigr{\\}}\,\mathrm{d}s\Bigr{]}.$

Integrating both sides with respect to $t\in(t_{0},T)$ and applying Fubini’s
theorem, we get

$\begin{split}&\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\langle
P^{(1)}(t)X(t),X(t)\rangle\,\mathrm{d}t\Bigr{]}\\\ &=\int^{T}_{t_{0}}\langle
P^{(1)}(t)x(t),x(t)\rangle\,\mathrm{d}t+\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\\!\\!\int^{T}_{t}\sigma(s,t)^{\top}P^{(1)}(s)\sigma(s,t)\,\mathrm{d}s\\!\,\mathrm{d}t\Bigr{]}\\\
&\hskip
14.22636pt+\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\Bigl{\\{}\Bigl{\langle}\int^{T}_{t}D(s,t)^{\top}P^{(1)}(s)D(s,t)\,\mathrm{d}s\,u(t),u(t)\Bigr{\rangle}\\\
&\hskip
56.9055pt+2\Bigl{\langle}\int^{T}_{t}D(s,t)^{\top}P^{(1)}(s)C(s,t)\,\mathrm{d}s\,X(t)+\int^{T}_{t}B(s,t)^{\top}P^{(1)}(s)\Theta(s,t)\,\mathrm{d}s\\\
&\hskip
113.81102pt+\int^{T}_{t}D(s,t)^{\top}P^{(1)}(s)\sigma(s,t)\,\mathrm{d}s,u(t)\Bigr{\rangle}\\\
&\hskip
56.9055pt+\Bigl{\langle}\int^{T}_{t}C(s,t)^{\top}P^{(1)}(s)C(s,t)\,\mathrm{d}s\,X(t),X(t)\Bigr{\rangle}+2\int^{T}_{t}\langle
P^{(1)}(s)A(s,t)X(t),\Theta(s,t)\rangle\,\mathrm{d}s\\\ &\hskip
56.9055pt+2\Bigl{\langle}\int^{T}_{t}C(s,t)^{\top}P^{(1)}(s)\sigma(s,t)\,\mathrm{d}s,X(t)\Bigr{\rangle}+2\int^{T}_{t}\langle
P^{(1)}(s)b(s,t),\Theta(s,t)\rangle\,\mathrm{d}s\Bigr{\\}}\,\mathrm{d}t\Bigr{]}.\end{split}$
(4.2)

Next, we note that $P^{(2)}(s_{1},s_{2},t)=P^{(2)}(s_{2},s_{1},t)^{\top}$ for
any $t\in[0,s_{1}\wedge s_{2}]$ for a.e. $(s_{1},s_{2})\in(0,T)^{2}$. Applying
Itô’s formula to $\langle
P^{(2)}(s,t,\cdot)\Theta(t,\cdot),\Theta(s,\cdot)\rangle$ on $(t_{0},t)$ for
a.e. $(s,t)\in\triangle_{2}(t_{0},T)$, we have

$\displaystyle 2\mathbb{E}[\langle
P^{(2)}(s,t,t)X(t),\Theta(s,t)\rangle]=2\mathbb{E}[\langle
P^{(2)}(s,t,t)\Theta(t,t),\Theta(s,t)\rangle]$ $\displaystyle=2\langle
P^{(2)}(s,t,t_{0})x(t),x(s)\rangle$ $\displaystyle\hskip
14.22636pt+2\mathbb{E}\Bigl{[}\int^{t}_{t_{0}}\Bigl{\\{}\langle\dot{P}^{(2)}(s,t,r)\Theta(t,r),\Theta(s,r)\rangle$
$\displaystyle\hskip 56.9055pt+\langle
P^{(2)}(s,t,r)\Theta(t,r),A(s,r)X(r)+B(s,r)u(r)+b(s,r)\rangle$
$\displaystyle\hskip 56.9055pt+\langle
P^{(2)}(s,t,r)(A(t,r)X(r)+B(t,r)u(r)+b(t,r)),\Theta(s,r)\rangle$
$\displaystyle\hskip 56.9055pt+\langle
P^{(2)}(s,t,r)(C(t,r)X(r)+D(t,r)u(r)+\sigma(t,r)),C(s,r)X(r)+D(s,r)u(r)+\sigma(s,r)\rangle\Bigr{\\}}\,\mathrm{d}r\Bigr{]}$
$\displaystyle=\langle P^{(2)}(s,t,t_{0})x(t),x(s)\rangle+\langle
P^{(2)}(t,s,t_{0})x(s),x(t)\rangle$ $\displaystyle\hskip
14.22636pt+\mathbb{E}\Bigl{[}\int^{t}_{t_{0}}\\{\sigma(s,r)^{\top}P^{(2)}(s,t,r)\sigma(t,r)+\sigma(t,r)^{\top}P^{(2)}(t,s,r)\sigma(s,r)\\}\,\mathrm{d}r\Bigr{]}$
$\displaystyle\hskip
14.22636pt+\mathbb{E}\Bigl{[}\int^{t}_{t_{0}}\Bigl{\\{}\langle(D(s,r)^{\top}P^{(2)}(s,t,r)D(t,r)+D(t,r)^{\top}P^{(2)}(t,s,r)D(s,r))u(r),u(r)\rangle$
$\displaystyle\hskip
56.9055pt+2\langle(D(s,r)^{\top}P^{(2)}(s,t,r)C(t,r)+D(t,r)^{\top}P^{(2)}(t,s,r)C(s,r))X(r)$
$\displaystyle\hskip
113.81102pt+B(s,r)^{\top}P^{(2)}(s,t,r)\Theta(t,r)+B(t,r)^{\top}P^{(2)}(t,s,r)\Theta(s,r)$
$\displaystyle\hskip
113.81102pt+D(s,r)^{\top}P^{(2)}(s,t,r)\sigma(t,r)+D(t,r)^{\top}P^{(2)}(t,s,r)\sigma(s,r),u(r)\rangle$
$\displaystyle\hskip
56.9055pt+\langle(C(s,r)^{\top}P^{(2)}(s,t,r)C(t,r)+C(t,r)^{\top}P^{(2)}(t,s,r)C(s,r))X(r),X(r)\rangle$
$\displaystyle\hskip 56.9055pt+2\langle
P^{(2)}(s,t,r)A(t,r)X(r),\Theta(s,r)\rangle+2\langle
P^{(2)}(t,s,r)A(s,r)X(r),\Theta(t,r)\rangle$ $\displaystyle\hskip
56.9055pt+\langle\dot{P}^{(2)}(s,t,r)\Theta(t,r),\Theta(s,r)\rangle+\langle\dot{P}^{(2)}(t,s,r)\Theta(s,r),\Theta(t,r)\rangle$
$\displaystyle\hskip 56.9055pt+2\langle
C(s,r)^{\top}P^{(2)}(s,t,r)\sigma(t,r)+C(t,r)^{\top}P^{(2)}(t,s,r)\sigma(s,r),X(r)\rangle$
$\displaystyle\hskip 56.9055pt+2\langle
P^{(2)}(s,t,r)b(t,r),\Theta(s,r)\rangle+2\langle
P^{(2)}(t,s,r)b(s,r),\Theta(t,r)\rangle\Bigr{\\}}\,\mathrm{d}r\Bigr{]}.$

We integrate both sides with respect to $(s,t)\in\triangle_{2}(t_{0},T)$ and
apply Fubini’s theorem. Noting that

$\int^{T}_{t_{0}}\\!\\!\int^{T}_{t}\\{f(s,t)+f(t,s)\\}\,\mathrm{d}s\\!\,\mathrm{d}t=\int_{\triangle_{2}(t_{0},T)}\\{f(s,t)+f(t,s)\\}\,\mathrm{d}(s,t)=\int^{T}_{t_{0}}\\!\\!\int^{T}_{t_{0}}f(s_{1},s_{2})\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2}$

for each function $f$ on $(t_{0},T)^{2}$, we obtain

$\begin{split}&2\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\\!\\!\int^{T}_{t}\langle
P^{(2)}(s,t,t)X(t),\Theta(s,t)\rangle\,\mathrm{d}s\\!\,\mathrm{d}t\Bigr{]}\\\
&=\int^{T}_{t_{0}}\\!\\!\int^{T}_{t_{0}}\langle
P^{(2)}(s_{1},s_{2},t_{0})x(s_{2}),x(s_{1})\rangle\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2}+\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\int^{T}_{t}\\!\\!\int^{T}_{t}\sigma(s_{1},t)^{\top}P^{(2)}(s_{1},s_{2},t)\sigma(s_{2},t)\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2}\,\mathrm{d}t\Bigr{]}\\\
&\hskip
14.22636pt+\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\Bigl{\\{}\Bigl{\langle}\int^{T}_{t}\\!\\!\int^{T}_{t}D(s_{1},t)^{\top}P^{(2)}(s_{1},s_{2},t)D(s_{2},t)\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2}\,u(t),u(t)\Bigr{\rangle}\\\
&\hskip
56.9055pt+2\Bigl{\langle}\int^{T}_{t}\\!\\!\int^{T}_{t}D(s_{1},t)^{\top}P^{(2)}(s_{1},s_{2},t)C(s_{2},t)\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2}\,X(t)\\\
&\hskip
113.81102pt+\int^{T}_{t}\\!\\!\int^{T}_{t}B(r,t)^{\top}P^{(2)}(r,s,t)\,\mathrm{d}r\,\Theta(s,t)\,\mathrm{d}s\\\
&\hskip
113.81102pt+\int^{T}_{t}\\!\\!\int^{T}_{t}D(s_{1},t)^{\top}P^{(2)}(s_{1},s_{2},t)\sigma(s_{2},t)\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2},u(t)\Bigr{\rangle}\\\
&\hskip
56.9055pt+\Bigl{\langle}\int^{T}_{t}\\!\\!\int^{T}_{t}C(s_{1},t)^{\top}P^{(2)}(s_{1},s_{2},t)C(s_{2},t)\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2}\,X(t),X(t)\Bigr{\rangle}\\\
&\hskip
56.9055pt+2\int^{T}_{t}\Bigl{\langle}\int^{T}_{t}P^{(2)}(s,r,t)A(r,t)\,\mathrm{d}r\,X(t),\Theta(s,t)\Bigr{\rangle}\,\mathrm{d}s\\\
&\hskip
56.9055pt+\int^{T}_{t}\\!\\!\int^{T}_{t}\langle\dot{P}^{(2)}(s_{1},s_{2},t)\Theta(s_{2},t),\Theta(s_{1},t)\rangle\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2}\\\
&\hskip
56.9055pt+2\Bigl{\langle}\int^{T}_{t}\\!\\!\int^{T}_{t}C(s_{1},t)^{\top}P^{(2)}(s_{1},s_{2},t)\sigma(s_{2},t)\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2},X(t)\Bigr{\rangle}\\\
&\hskip
56.9055pt+2\int^{T}_{t}\Bigl{\langle}\int^{T}_{t}P^{(2)}(s,r,t)b(r,t)\,\mathrm{d}r,\Theta(s,t)\Bigr{\rangle}\,\mathrm{d}s\Bigr{\\}}\,\mathrm{d}t\Bigr{]}.\end{split}$
(4.3)

By (4.2) and (4.3), noting 4.5, we get the assertion. ∎

######  Remark 4.8.

A key point in the above proof is the use of Itô’s formula to some quadratic
form of $\Theta(\cdot,\cdot)$, from which one obtains the corresponding
quadratic form of $X(\cdot)$. This kind of techniques also appeared in [27]
where LQ stochastic Volterra control problems were treated in the open-loop
framework.

######  Corollary 4.9.

Let $P=(P^{(1)},P^{(2)})\in\Pi(0,T)$ and $(\Xi,\Gamma)\in
L^{\infty}(0,T;\mathbb{R}^{\ell\times d})\times
L^{2}(\triangle_{2}(0,T);\mathbb{R}^{\ell\times d})$ be given. For each
$(t_{0},x)\in\mathcal{I}$, let $(X^{t_{0},x}_{0},\Theta^{t_{0},x}_{0})\in
L^{2}_{\mathbb{F}}(t_{0},T;\mathbb{R}^{d})\times
L^{2}_{\mathbb{F},\mathrm{c}}(\triangle_{2}(t_{0},T);\mathbb{R}^{d})$ be the
causal feedback solution to the homogeneous controlled SVIE (2.1) at
$(t_{0},x)\in\mathcal{I}$ corresponding to the causal feedback strategy
$(\Xi,\Gamma,0)\in\mathcal{S}(0,T)$. Then it holds that

$\displaystyle\int^{T}_{t_{0}}\langle
P^{(1)}(t)x(t),x(t)\rangle\,\mathrm{d}t+\int^{T}_{t_{0}}\\!\\!\int^{T}_{t_{0}}\langle
P^{(2)}(s_{1},s_{2},t_{0})x(s_{2}),x(s_{1})\rangle\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2}$
$\displaystyle=\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\Bigl{\\{}\langle(P^{(1)}(t)-F^{(1)}[\Xi;P](t))X^{t_{0},x}_{0}(t),X^{t_{0},x}_{0}(t)\rangle$
$\displaystyle\hskip
56.9055pt+2\int^{T}_{t}\langle(P^{(2)}(s,t,t)-F^{(2)}[\Xi,\Gamma;P](s,t))X^{t_{0},x}_{0}(t),\Theta^{t_{0},x}_{0}(s,t)\rangle\,\mathrm{d}s$
$\displaystyle\hskip
56.9055pt-\int^{T}_{t}\\!\\!\int^{T}_{t}\langle(\dot{P}^{(2)}(s_{1},s_{2},t)+F^{(3)}[\Gamma;P](s_{1},s_{2},t))\Theta^{t_{0},x}_{0}(s_{2},t),\Theta^{t_{0},x}_{0}(s_{1},t)\rangle\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2}\Bigr{\\}}\,\mathrm{d}t\Bigr{]},$

where

$\begin{split}&F^{(1)}[\Xi;P](t):=(C^{\top}{\ltimes}P{\rtimes}C)(t)+\Xi(t)^{\top}(D^{\top}{\ltimes}P{\rtimes}C)(t)+(C^{\top}{\ltimes}P{\rtimes}D)(t)\Xi(t)+\Xi(t)^{\top}(D^{\top}{\ltimes}P{\rtimes}D)(t)\Xi(t),\\\
&\hskip 170.71652ptt\in(0,T),\\\
&F^{(2)}[\Xi,\Gamma;P](s,t):=(P{\rtimes}A)(s,t)+(P{\rtimes}B)(s,t)\Xi(t)+\Gamma(s,t)^{\top}(D^{\top}{\ltimes}P{\rtimes}C)(t)+\Gamma(s,t)^{\top}(D^{\top}{\ltimes}P{\rtimes}D)(t)\Xi(t),\\\
&\hskip 170.71652pt(s,t)\in\triangle_{2}(0,T),\\\
&F^{(3)}[\Gamma;P](s_{1},s_{2},t):=\Gamma(s_{1},t)^{\top}(B^{\top}{\ltimes}P)(s_{2},t)+(P{\rtimes}B)(s_{1},t)\Gamma(s_{2},t)+\Gamma(s_{1},t)^{\top}(D^{\top}{\ltimes}P{\rtimes}D)(t)\Gamma(s_{2},t),\\\
&\hskip 170.71652pt(s_{1},s_{2},t)\in\square_{3}(0,T).\end{split}$ (4.4)

###### Proof.

Denote the outcome by $u^{t_{0},x}_{0}:=(\Xi,\Gamma,0)^{0}[t_{0},x]$. By using
4.7 with $(b,\sigma)=(0,0)$, we have

$\displaystyle\int^{T}_{t_{0}}\langle
P^{(1)}(t)x(t),x(t)\rangle\,\mathrm{d}t+\int^{T}_{t_{0}}\\!\\!\int^{T}_{t_{0}}\langle
P^{(2)}(s_{1},s_{2},t_{0})x(s_{2}),x(s_{1})\rangle\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2}$
$\displaystyle=\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\Bigl{\\{}\langle
P^{(1)}(t)X^{t_{0},x}_{0}(t),X^{t_{0},x}_{0}(t)\rangle+2\int^{T}_{t}\langle
P^{(2)}(s,t,t)X^{t_{0},x}_{0}(t),\Theta^{t_{0},x}_{0}(s,t)\rangle\,\mathrm{d}s\Bigr{\\}}\,\mathrm{d}t\Bigr{]}$
$\displaystyle\hskip
14.22636pt-\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\Bigl{\\{}\langle(D^{\top}{\ltimes}P{\rtimes}D)(t)u^{t_{0},x}_{0}(t),u^{t_{0},x}_{0}(t)\rangle$
$\displaystyle\hskip
56.9055pt+2\Bigl{\langle}(D^{\top}{\ltimes}P{\rtimes}C)(t)X^{t_{0},x}_{0}(t)+\int^{T}_{t}(B^{\top}{\ltimes}P)(s,t)\Theta^{t_{0},x}_{0}(s,t)\,\mathrm{d}s,u^{t_{0},x}_{0}(t)\Bigr{\rangle}$
$\displaystyle\hskip
56.9055pt+\langle(C^{\top}{\ltimes}P{\rtimes}C)(t)X^{t_{0},x}_{0}(t),X^{t_{0},x}_{0}(t)\rangle+2\int^{T}_{t}\langle(P{\rtimes}A)(s,t)X^{t_{0},x}_{0}(t),\Theta^{t_{0},x}_{0}(s,t)\rangle\,\mathrm{d}s$
$\displaystyle\hskip
56.9055pt+\int^{T}_{t}\\!\\!\int^{T}_{t}\langle\dot{P}^{(2)}(s_{1},s_{2},t)\Theta^{t_{0},x}_{0}(s_{2},t),\Theta^{t_{0},x}_{0}(s_{1},t)\rangle\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2}\Bigr{\\}}\,\mathrm{d}t\Bigr{]}.$

By using the relation
$u^{t_{0},x}_{0}(t)=\Xi(t)X^{t_{0},x}_{0}(t)+\int^{T}_{t}\Gamma(s,t)\Theta^{t_{0},x}_{0}(s,t)\,\mathrm{d}s$
and noting the definition (4.4) of $(F^{(1)},F^{(2)},F^{(3)})$, we get the
assertion. ∎

######  Remark 4.10.

By the definition (4.4), together with 4.6, we see that the map

$P\mapsto(F^{(1)}[\Xi;P],F^{(2)}[\Xi,\Gamma;P],F^{(3)}[\Gamma;P]).$

is a bounded linear operator from $\Pi(0,T)$ to
$L^{\infty}(0,T;\mathbb{S}^{d})\times
L^{2}(\triangle_{2}(0,T);\mathbb{R}^{d\times d})\times
L^{2,2,1}_{\mathrm{sym}}(\square_{3}(0,T);\mathbb{R}^{d\times d})$.

Now we derive a dynamics for $P=(P^{(1)},P^{(2)})\in\Pi(0,T)$ which represents
the quadratic functional (4.1). Let $(Q^{(1)},Q^{(2)},Q^{(3)})\in
L^{\infty}(0,T;\mathbb{S}^{d})\times
L^{2}(\triangle_{2}(0,T);\mathbb{R}^{d\times d})\times
L^{2,2,1}_{\mathrm{sym}}(\square_{3}(0,T);\mathbb{R}^{d\times d})$ be given.
From 4.9, it is natural to introduce the following equation:

$\begin{dcases}P^{(1)}(t)=F^{(1)}[\Xi;P](t)+Q^{(1)}(t),\ t\in(0,T),\\\
P^{(2)}(s,t,t)=P^{(2)}(t,s,t)^{\top}=F^{(2)}[\Xi,\Gamma;P](s,t)+Q^{(2)}(s,t),\
(s,t)\in\triangle_{2}(0,T),\\\
\dot{P}^{(2)}(s_{1},s_{2},t)+F^{(3)}[\Gamma;P](s_{1},s_{2},t)+Q^{(3)}(s_{1},s_{2},t)=0,\
(s_{1},s_{2},t)\in\square_{3}(0,T),\end{dcases}$ (4.5)

with $(F^{(1)}[\Xi;P],F^{(2)}[\Xi,\Gamma;P],F^{(3)}[\Gamma;P])$ defined by
(4.4). In the integral notation, (4.5) is rewritten as

$\begin{dcases}P^{(1)}(t)=\int^{T}_{t}(C(s,t)+D(s,t)\Xi(t))^{\top}P^{(1)}(s)(C(s,t)+D(s,t)\Xi(t))\,\mathrm{d}s\\\
\hskip
56.9055pt+\int^{T}_{t}\\!\\!\int^{T}_{t}(C(s_{1},t)+D(s_{1},t)\Xi(t))^{\top}P^{(2)}(s_{1},s_{2},t)(C(s_{2},t)+D(s_{2},t)\Xi(t))\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2}\\\
\hskip 56.9055pt+Q^{(1)}(t),\ t\in(0,T),\\\
P^{(2)}(s,t,t)=P^{(2)}(t,s,t)^{\top}\\\
=P^{(1)}(s)(A(s,t)+B(s,t)\Xi(t))+\Gamma(s,t)^{\top}\int^{T}_{t}D(r,t)^{\top}P^{(1)}(r)(C(r,t)+D(r,t)\Xi(t))\,\mathrm{d}r\\\
\hskip
14.22636pt+\int^{T}_{t}P^{(2)}(s,r,t)(A(r,t)+B(r,t)\Xi(t))\,\mathrm{d}r\\\
\hskip
14.22636pt+\Gamma(s,t)^{\top}\int^{T}_{t}\\!\\!\int^{T}_{t}D(r_{1},t)^{\top}P^{(2)}(r_{1},r_{2},t)(C(r_{2},t)+D(r_{2},t)\Xi(t))\,\mathrm{d}r_{1}\\!\,\mathrm{d}r_{2}\\\
\hskip 14.22636pt+Q^{(2)}(s,t),\ (s,t)\in\triangle_{2}(0,T),\\\
\dot{P}^{(2)}(s_{1},s_{2},t)+\Gamma(s_{1},t)^{\top}B(s_{2},t)^{\top}P^{(1)}(s_{2})+P^{(1)}(s_{1})B(s_{1},t)\Gamma(s_{2},t)\\\
\hskip
5.69046pt+\Gamma(s_{1},t)^{\top}\int^{T}_{t}D(r,t)^{\top}P^{(1)}(r)D(r,t)\,\mathrm{d}r\,\Gamma(s_{2},t)\\\
\hskip
5.69046pt+\Gamma(s_{1},t)^{\top}\int^{T}_{t}B(r,t)^{\top}P^{(2)}(r,s_{2},t)\,\mathrm{d}r+\int^{T}_{t}P^{(2)}(s_{1},r,t)B(r,t)\,\mathrm{d}r\,\Gamma(s_{2},t)\\\
\hskip
5.69046pt+\Gamma(s_{1},t)^{\top}\int^{T}_{t}\\!\\!\int^{T}_{t}D(r_{1},t)^{\top}P^{(2)}(r_{1},r_{2},t)D(r_{2},t)\,\mathrm{d}r_{1}\\!\,\mathrm{d}r_{2}\,\Gamma(s_{2},t)\\\
\hskip 5.69046pt+Q^{(3)}(s_{1},s_{2},t)=0,\
(s_{1},s_{2},t)\in\square_{3}(0,T).\end{dcases}$

The above is a coupled system of Lyapunov-type (backward) Volterra integro-
differential equations for the pair $P=(P^{(1)},P^{(2)})\in\Pi(0,T)$ of
matrix-valued deterministic functions. In this paper, we call the above
equation a _Lyapunov–Volterra equation_. We say that
$P=(P^{(1)},P^{(2)})\in\Pi(0,T)$ satisfying the above equalities a solution to
the Lyapunov–Volterra equation (4.5).

Now we are ready to state the third main result of this paper.

######  Theorem 4.11.

Let $(\Xi,\Gamma)\in L^{\infty}(0,T;\mathbb{R}^{\ell\times d})\times
L^{2}(\triangle_{2}(0,T);\mathbb{R}^{\ell\times d})$. For each
$(Q^{(1)},Q^{(2)},Q^{(3)})\in L^{\infty}(0,T;\mathbb{S}^{d})\times
L^{2}(\triangle_{2}(0,T);\mathbb{R}^{d\times d})\times
L^{2,2,1}_{\mathrm{sym}}(\square_{3}(0,T);\mathbb{R}^{d\times d})$, there
exists a unique solution $P=(P^{(1)},P^{(2)})\in\Pi(0,T)$ to the
Lyapunov–Volterra equation (4.5). Furthermore, for any
$(t_{0},x)\in\mathcal{I}$, the following holds:

$\begin{split}&\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\Bigl{\\{}\langle
Q^{(1)}(t)X^{t_{0},x}_{0}(t),X^{t_{0},x}_{0}(t)\rangle+2\int^{T}_{t}\langle
Q^{(2)}(s,t)X^{t_{0},x}_{0}(t),\Theta^{t_{0},x}_{0}(s,t)\rangle\,\mathrm{d}s\\\
&\hskip 56.9055pt+\int^{T}_{t}\\!\\!\int^{T}_{t}\langle
Q^{(3)}(s_{1},s_{2},t)\Theta^{t_{0},x}_{0}(s_{2},t),\Theta^{t_{0},x}_{0}(s_{1},t)\rangle\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2}\Bigr{\\}}\,\mathrm{d}t\Bigr{]}\\\
&=\int^{T}_{t_{0}}\langle
P^{(1)}(t)x(t),x(t)\rangle\,\mathrm{d}t+\int^{T}_{t_{0}}\\!\\!\int^{T}_{t_{0}}\langle
P^{(2)}(s_{1},s_{2},t_{0})x(s_{2}),x(s_{1})\rangle\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2},\end{split}$
(4.6)

where $(X^{t_{0},x}_{0},\Theta^{t_{0},x}_{0})$ is the causal feedback solution
to the homogeneous controlled SVIE (2.1) at $(t_{0},x)$ corresponding to
$(\Xi,\Gamma,0)\in\mathcal{S}(0,T)$.

###### Proof.

See Section A.3. ∎

##  Appendix A Appendix: Proofs of the main results

In this appendix, we prove our main Theorems 2.4, 3.3 and 4.11. The following
lemma is an abstract formulation of the so-called _method of continuation_ for
(non-linear) equations on a Banach space. This is crucial for the proof of the
well-posedness of several kinds of equations arising in this paper.

######  Lemma A.1.

Let $(\mathcal{X},\|\cdot\|_{\mathcal{X}})$ be a Banach space,
$(\mathcal{Y},\|\cdot\|_{\mathcal{Y}})$ be a normed space, and
$\mathcal{T}:\mathcal{Y}\to\mathcal{X}$ be a continuous map. Suppose that a
map $F:\mathcal{X}\to\mathcal{Y}$ satisfies the following two properties:

  * •

Lipschitz continuity: There exists a constant $L>0$ such that

$\|F(X_{1})-F(X_{2})\|_{\mathcal{Y}}\leq L\|X_{1}-X_{2}\|_{\mathcal{X}}$

for any $X_{1},X_{2}\in\mathcal{X}$.

  * •

A priori estimate: There exists a constant $K>0$ such that, for any
$\varphi_{1},\varphi_{2}\in\mathcal{Y}$ and $\lambda\in[0,1]$, two solutions
$X_{1},X_{2}\in\mathcal{X}$ of the equations

$X_{i}=\mathcal{T}(\lambda F(X_{i})+\varphi_{i}),\ i=1,2,$

satisfy

$\|X_{1}-X_{2}\|_{\mathcal{X}}\leq
K\|\varphi_{1}-\varphi_{2}\|_{\mathcal{Y}}.$

Then, for any $\varphi\in\mathcal{Y}$, there exists a unique solution
$X\in\mathcal{X}$ to the equation

$X=\mathcal{T}(F(X)+\varphi).$

###### Proof.

The uniqueness of the solution follows from the a priori estimate. We prove
the existence by _the method of continuation_. Let $N\in\mathbb{N}$ be a
number such that $N\geq 2(1+KL)$. For each $n\in\\{0,1,\dots,N\\}$, we say
that (Pn) holds if for any $\varphi\in\mathcal{Y}$, there exists a unique
solution $X\in\mathcal{X}$ to the equation
$X=\mathcal{T}(\frac{n}{N}F(X)+\varphi)$. We shall show that (PN) holds.
Clearly, (P0) holds. Assume that (Pn) holds for some
$n\in\\{0,1,\dots,N-1\\}$. Fix an arbitrary $\varphi\in\mathcal{Y}$. By virtue
of the assumption (Pn), we can define a sequence $X_{i}\in\mathcal{X}$ with
$i\in\mathbb{N}$ inductively as the solution to the equation

$X_{i}=\mathcal{T}\Bigl{(}\frac{n}{N}F(X_{i})+\frac{1}{N}F(X_{i-1})+\varphi\Bigr{)},$

with the convention that $X_{0}:=0$. By the a priori estimate and the
Lipschitz continuity of $F$, for each $i\in\mathbb{N}$, we have

$\displaystyle\|X_{i+1}-X_{i}\|_{\mathcal{X}}\leq\frac{K}{N}\|F(X_{i})-F(X_{i-1})\|_{\mathcal{Y}}\leq\frac{KL}{N}\|X_{i}-X_{i-1}\|_{\mathcal{X}}\leq\frac{1}{2}\|X_{i}-X_{i-1}\|_{\mathcal{X}},$

where the last inequality follows from $N\geq 2(1+KL)$. The above estimate
implies that $\\{X_{i}\\}_{i\in\mathbb{N}}$ is a Cauchy sequence on the Banach
space $\mathcal{X}$, and thus the limit $X:=\lim_{i\to\infty}X_{i}$ exists.
Noting the continuity of $F$ and $\mathcal{T}$, we obtain

$X=\mathcal{T}\Bigl{(}\frac{n}{N}F(X)+\frac{1}{N}F(X)+\varphi\Bigr{)}=\mathcal{T}\Bigl{(}\frac{n+1}{N}F(X)+\varphi\Bigr{)}.$

Since $\varphi\in\mathcal{Y}$ is arbitrary, we see that (Pn+1) holds. By the
induction, we see that (PN) holds. This completes the proof. ∎

######  Remark A.2.

We will use the above abstract lemma to show the existence and uniqueness of
the solutions to SVIEs, Type-II EBSVIEs and Lyapunov–Volterra equations. In
the framework of this paper, we typically consider the case where
$\mathcal{T}$ and $F$ are bounded linear operators. In this case, the
Lipschitz continuity of $F$ is trivial, and the a priori estimate of the above
form is a key step to show the well-posedness of the equations.

###  A.1 Proof of 2.4: Causal feedback solutions of controlled SVIEs

First, we prove the well-posedness of a linear (uncontrolled) SVIE. Although
the following lemma follows from the general theory of [15], we give a simple
proof here by means of A.1 for the sake of self-containedness.

######  Lemma A.3.

Let $A\in L^{2}(\triangle_{2}(0,T);\mathbb{R}^{d\times d})$ and
$C\in\mathscr{L}^{2}(\triangle_{2}(0,T);\mathbb{R}^{d\times d})$ be given. For
any $\varphi\in L^{2}_{\mathbb{F}}(t_{0},T;\mathbb{R}^{d})$ with
$t_{0}\in[0,T)$, there exists a unique solution $X\in
L^{2}_{\mathbb{F}}(t_{0},T;\mathbb{R}^{d})$ to the SVIE

$X(t)=\varphi(t)+\int^{t}_{t_{0}}A(t,s)X(s)\,\mathrm{d}s+\int^{t}_{t_{0}}C(t,s)X(s)\,\mathrm{d}W(s),\
t\in(t_{0},T).$

Furthermore, the following estimate holds:

$\|X\|_{L^{2}_{\mathbb{F}}(t_{0},T)}\leq
2m^{2}(1+2L)^{m-1}\|\varphi\|_{L^{2}_{\mathbb{F}}(t_{0},T)},$

where
$L:=\|A\|_{L^{2}(\triangle_{2}(0,T))}+\|C\|_{\mathscr{L}^{2}(\triangle_{2}(0,T))}$,
and $m\in\mathbb{N}$ is the number of division intervals
$0=U_{0}<U_{1}<\cdots<U_{m}=T$ of $(0,T)$ satisfying

$\Bigl{(}\int^{U_{i+1}}_{U_{i}}\\!\\!\int^{t}_{U_{i}}|A(t,s)|^{2}\,\mathrm{d}s\\!\,\mathrm{d}t\Bigr{)}^{1/2}+\underset{t\in(U_{i},U_{i+1})}{\mathrm{ess\,sup}}\Bigl{(}\int^{U_{i+1}}_{t}|C(s,t)|^{2}\,\mathrm{d}s\Bigr{)}^{1/2}\leq\frac{1}{2}$

for $i=0,1,\dots,m-1$.

###### Proof.

It suffices to show the theorem for $t_{0}=0$. Define
$F:L^{2}_{\mathbb{F}}(0,T;\mathbb{R}^{d})\to
L^{2}_{\mathbb{F}}(0,T;\mathbb{R}^{d})$ by

$F(X)(t):=\int^{t}_{0}A(t,s)X(s)\,\mathrm{d}s+\int^{t}_{0}C(t,s)X(s)\,\mathrm{d}W(s),\
t\in(0,T),$

for $X\in L^{2}_{\mathbb{F}}(0,T;\mathbb{R}^{d})$. We prove that, for any
$\varphi\in L^{2}_{\mathbb{F}}(0,T;\mathbb{R}^{d})$, the equation
$X=F(X)+\varphi$ has a unique solution $X\in
L^{2}_{\mathbb{F}}(0,T;\mathbb{R}^{d})$. It is easy to see that $F$ is a
bounded linear operator on $L^{2}_{\mathbb{F}}(0,T;\mathbb{R}^{d})$. We prove
the a priori estimate. Let
$L:=\|A\|_{L^{2}(\triangle_{2}(0,T))}+\|C\|_{\mathscr{L}^{2}(\triangle_{2}(0,T))}$
and $0=U_{0}<U_{1}<\cdots<U_{m}=T$ be a partition of $(0,T)$ satisfying the
requirement in the assertion. We emphasize that such a partition exists since
$A\in L^{2}(\triangle_{2}(0,T);\mathbb{R}^{d\times d})$ and
$C\in\mathscr{L}^{2}(\triangle_{2}(0,T);\mathbb{R}^{d\times d})$. Let
$\varphi_{1},\varphi_{2}\in L^{2}_{\mathbb{F}}(0,T;\mathbb{R}^{d})$ and
$\lambda\in[0,1]$ be fixed, and assume that $X_{1},X_{2}\in
L^{2}_{\mathbb{F}}(0,T;\mathbb{R}^{d})$ satisfy $X_{i}=\lambda
F(X_{i})+\varphi_{i}$ for $i=1,2$. Define $\bar{X}:=X_{1}-X_{2}$ and
$\bar{\varphi}:=\varphi_{1}-\varphi_{2}$. Then we have $\bar{X}=\lambda
F(\bar{X})+\bar{\varphi}$. For $i\in\\{0,1,\dots,m-1\\}$, we have

$\bar{X}(t)=\bar{\varphi}(t)+\lambda\\{F^{1}_{i}(\bar{X})(t)+F^{2}_{i}(\bar{X})(t)\\},\
t\in(U_{i},U_{i+1}),$

where

$\displaystyle
F^{1}_{i}(\bar{X})(t):=\int^{U_{i}}_{0}A(t,s)\bar{X}(s)\,\mathrm{d}s+\int^{U_{i}}_{0}C(t,s)\bar{X}(s)\,\mathrm{d}W(s),\
t\in(U_{i},U_{i+1}),$ $\displaystyle
F^{2}_{i}(\bar{X})(t):=\int^{t}_{U_{i}}A(t,s)\bar{X}(s)\,\mathrm{d}s+\int^{t}_{U_{i}}C(t,s)\bar{X}(s)\,\mathrm{d}W(s),\
t\in(U_{i},U_{i+1}).$

It is easy to see that

$\|F^{1}_{i}(\bar{X})\|_{L^{2}_{\mathbb{F}}(U_{i},U_{i+1})}\leq
L\|\bar{X}\|_{L^{2}_{\mathbb{F}}(0,U_{i})},\
\|F^{2}_{i}(\bar{X})\|_{L^{2}_{\mathbb{F}}(U_{i},U_{i+1})}\leq\frac{1}{2}\|\bar{X}\|_{L^{2}_{\mathbb{F}}(U_{i},U_{i+1})}.$

Thus, we have

$\|\bar{X}\|_{L^{2}_{\mathbb{F}}(U_{i},U_{i+1})}\leq
2\|\bar{\varphi}\|_{L^{2}_{\mathbb{F}}(U_{i},U_{i+1})}+2L\|\bar{X}\|_{L^{2}_{\mathbb{F}}(0,U_{i})}\leq
2\|\bar{\varphi}\|_{L^{2}_{\mathbb{F}}(0,T)}+2L\sum^{i-1}_{j=0}\|\bar{X}\|_{L^{2}_{\mathbb{F}}(U_{j},U_{j+1})}.$

By the discrete Gronwall inequality, we have

$\max_{i\in\\{0,1,\dots,m-1\\}}\|\bar{X}\|_{L^{2}_{\mathbb{F}}(U_{i},U_{i+1})}\leq
2\|\bar{\varphi}\|_{L^{2}_{\mathbb{F}}(0,T)}\sum^{m}_{j=1}(1+2L)^{j-1}\leq
2m(1+2L)^{m-1}\|\bar{\varphi}\|_{L^{2}_{\mathbb{F}}(0,T)},$

and thus

$\|\bar{X}\|_{L^{2}_{\mathbb{F}}(0,T)}\leq\sum^{m-1}_{i=0}\|\bar{X}\|_{L^{2}_{\mathbb{F}}(U_{i},U_{i+1})}\leq
2m^{2}(1+2L)^{m-1}\|\bar{\varphi}\|_{L^{2}_{\mathbb{F}}(0,T)}.$

Therefore, the a priori estimate holds. By A.1, for any $\varphi\in
L^{2}_{\mathbb{F}}(0,T;\mathbb{R}^{d})$, the SVIE $X=F(X)+\varphi$ has a
unique solution $X\in L^{2}_{\mathbb{F}}(0,T;\mathbb{R}^{d})$. ∎

The following technical lemma reveals the structure of the causal feedback
solution of a linear controlled SVIE.

######  Lemma A.4.

Let $(\Xi,\Gamma,v)\in\mathcal{S}(0,T)$ be fixed. For each
$(t_{0},x)\in\mathcal{I}$, a triplet
$(\Xi^{t_{0},x},\Theta^{t_{0},x},u^{t_{0},x})\in
L^{2}_{\mathbb{F}}(t_{0},T;\mathbb{R}^{d})\times
L^{2}_{\mathbb{F},\mathrm{c}}(\triangle_{2}(t_{0},T);\mathbb{R}^{d})\times\mathcal{U}(t_{0},T)$
is a causal feedback solution to the controlled SVIE (1.1) if and only if the
following holds:

$\begin{dcases}\begin{pmatrix}X^{t_{0},x}(t)\\\
u^{t_{0},x}(t)\end{pmatrix}=\mathbb{X}^{t_{0},x}(t),\ t\in(t_{0},T),\\\
\Theta^{t_{0},x}(s,t)=x(s)+\int^{t}_{t_{0}}\\{(A(s,r),B(s,r))\mathbb{X}^{t_{0},x}(r)+b(s,r)\\}\,\mathrm{d}r\\\
\hskip
85.35826pt+\int^{t}_{t_{0}}\\{(C(s,r),D(s,r))\mathbb{X}^{t_{0},x}(r)+\sigma(s,r)\\}\,\mathrm{d}W(r),\
(s,t)\in\triangle_{2}(t_{0},T),\end{dcases}$ (A.1)

where $\mathbb{X}^{t_{0},x}\in
L^{2}_{\mathbb{F}}(t_{0},T;\mathbb{R}^{d+\ell})$ is the solution to the SVIE

$\mathbb{X}^{t_{0},x}(t)=\Phi(t)+\int^{t}_{t_{0}}\mathbb{A}(t,s)\mathbb{X}^{t_{0},x}(s)\,\mathrm{d}s+\int^{t}_{t_{0}}\mathbb{C}(t,s)\mathbb{X}^{t_{0},x}(s)\,\mathrm{d}W(s),\
t\in(t_{0},T).$ (A.2)

Here, $\Phi\in L^{2}_{\mathbb{F}}(t_{0},T;\mathbb{R}^{d+\ell})$,
$\mathbb{A}\in L^{2}(\triangle_{2}(0,T);\mathbb{R}^{(d+\ell)\times(d+\ell)})$
and
$\mathbb{C}\in\mathscr{L}^{2}(\triangle_{2}(0,T);\mathbb{R}^{(d+\ell)\times(d+\ell)})$
are defined by

$\displaystyle\Phi(t):=\begin{pmatrix}x(t)+\int^{t}_{t_{0}}b(t,s)\,\mathrm{d}s+\int^{t}_{t_{0}}\sigma(t,s)\,\mathrm{d}W(s)\\\
x^{\Xi,\Gamma}(t)+\int^{t}_{t_{0}}b^{\Xi,\Gamma}(t,s)\,\mathrm{d}s+\int^{t}_{t_{0}}\sigma^{\Xi,\Gamma}(t,s)\,\mathrm{d}W(s)+v(t)\end{pmatrix},\
t\in(t_{0},T),$ $\displaystyle\mathbb{A}(t,s):=\begin{pmatrix}A(t,s)&B(t,s)\\\
A^{\Xi,\Gamma}(t,s)&B^{\Xi,\Gamma}(t,s)\end{pmatrix},\
\mathbb{C}(t,s):=\begin{pmatrix}C(t,s)&D(t,s)\\\
C^{\Xi,\Gamma}(t,s)&D^{\Xi,\Gamma}(t,s)\end{pmatrix},\
(t,s)\in\triangle_{2}(0,T),$

with

$x^{\Xi,\Gamma}(t):=\Xi(t)x(t)+\int^{T}_{t}\Gamma(s,t)x(s)\,\mathrm{d}s,\
t\in(t_{0},T),$

and

$f^{\Xi,\Gamma}(t,s):=\Xi(t)f(t,s)+\int^{T}_{t}\Gamma(r,t)f(r,s)\,\mathrm{d}r,\
(t,s)\in\triangle_{2}(0,T),$

for $f=b,\sigma,A,B,C,D$.

###### Proof.

By using the Cauchy–Schwarz inequality, we can easily show that

$\Phi\in L^{2}_{\mathbb{F}}(t_{0},T;\mathbb{R}^{d+\ell}),\ \mathbb{A}\in
L^{2}(\triangle_{2}(0,T);\mathbb{R}^{(d+\ell)\times(d+\ell)})\ \text{and}\
\mathbb{C}\in\mathscr{L}^{2}(\triangle_{2}(0,T);\mathbb{R}^{(d+\ell)\times(d+\ell)}).$

Let $(X^{t_{0},x},\Theta^{t_{0},x},u^{t_{0},x})\in
L^{2}_{\mathbb{F}}(t_{0},T;\mathbb{R}^{d})\times
L^{2}_{\mathbb{F},\mathrm{c}}(\triangle_{2}(t_{0},T);\mathbb{R}^{d})\times\mathcal{U}(t_{0},T)$
be a causal feedback solution to the controlled SVIE (1.1) at
$(t_{0},x)\in\mathcal{I}$ corresponding to the causal feedback strategy
$(\Xi,\Gamma,v)\in\mathcal{S}(0,T)$. By using the (stochastic) Fubini theorem,
we have

$\displaystyle u^{t_{0},x}(t)$
$\displaystyle=\Xi(t)X^{t_{0},x}(t)+\int^{T}_{t}\Gamma(s,t)\Theta^{t_{0},x}(s,t)\,\mathrm{d}s+v(t)$
$\displaystyle=\Xi(t)x(t)+\Xi(t)\int^{t}_{t_{0}}\\{A(t,s)X^{t_{0},x}(s)+B(t,s)u^{t_{0},x}(s)+b(t,s)\\}\,\mathrm{d}s$
$\displaystyle\hskip
14.22636pt+\Xi(t)\int^{t}_{t_{0}}\\{C(t,s)X^{t_{0},x}(s)+D(t,s)u^{t_{0},x}(s)+\sigma(t,s)\\}\,\mathrm{d}W(s)$
$\displaystyle\hskip
14.22636pt+\int^{T}_{t}\Gamma(s,t)x(s)\,\mathrm{d}s+\int^{T}_{t}\Gamma(s,t)\int^{t}_{t_{0}}\\{A(s,r)X^{t_{0},x}(r)+B(s,r)u^{t_{0},x}(r)+b(s,r)\\}\,\mathrm{d}r\,\mathrm{d}s$
$\displaystyle\hskip
14.22636pt+\int^{T}_{t}\Gamma(s,t)\int^{t}_{t_{0}}\\{C(s,r)X^{t_{0},x}(r)+D(s,r)u^{t_{0},x}(r)+\sigma(s,r)\\}\,\mathrm{d}W(r)\,\mathrm{d}s+v(t)$
$\displaystyle=x^{\Xi,\Gamma}(t)+\int^{t}_{t_{0}}b^{\Xi,\Gamma}(t,s)\,\mathrm{d}s+\int^{t}_{t_{0}}\sigma^{\Xi,\Gamma}(t,s)\,\mathrm{d}W(s)+v(t)$
$\displaystyle\hskip
14.22636pt+\int^{t}_{t_{0}}(A^{\Xi,\Gamma}(t,s),B^{\Xi,\Gamma}(t,s))\begin{pmatrix}X^{t_{0},x}(s)\\\
u^{t_{0},x}(s)\end{pmatrix}\,\mathrm{d}s+\int^{t}_{t_{0}}(C^{\Xi,\Gamma}(t,s),D^{\Xi,\Gamma}(t,s))\begin{pmatrix}X^{t_{0},x}(s)\\\
u^{t_{0},x}(s)\end{pmatrix}\,\mathrm{d}W(s)$

for a.e. $t\in(t_{0},T)$, a.s. Therefore, the process $\mathbb{X}^{t_{0},x}\in
L^{2}_{\mathbb{F}}(t_{0},T;\mathbb{R}^{d+\ell})$ defined by the first equality
in (A.1) satisfies the SVIE (A.2), and the second equality in (A.1) holds.

Conversely, if $\mathbb{X}^{t_{0},x}\in
L^{2}_{\mathbb{F}}(t_{0},T;\mathbb{R}^{d+\ell})$ solves the SVIE (A.2), then
the triplet $(X^{t_{0},x},\Theta^{t_{0},x},u^{t_{0},x})\in
L^{2}_{\mathbb{F}}(t_{0},T;\mathbb{R}^{d})\times
L^{2}_{\mathbb{F},\mathrm{c}}(\triangle_{2}(t_{0},T);\mathbb{R}^{d})\times\mathcal{U}(t_{0},T)$
defined by (A.1) is the causal feedback solution of the controlled SVIE (1.1).
This completes the proof. ∎

Now we are ready to prove 2.4.

###### Proof of 2.4.

By A.3, the SVIE (A.2) admits a unique solution $\mathbb{X}^{t_{0},x}\in
L^{2}_{\mathbb{F}}(t_{0},T;\mathbb{R}^{d+\ell})$. By A.4, the causal feedback
solution $(X^{t_{0},x},\Theta^{t_{0},x},u^{t_{0},x})\in
L^{2}_{\mathbb{F}}(t_{0},T;\mathbb{R}^{d})\times
L^{2}_{\mathbb{F},\mathrm{c}}(\triangle_{2}(t_{0},T);\mathbb{R}^{d})\times\mathcal{U}(t_{0},T)$
to the controlled SVIE (1.1) uniquely exists. Furthermore, the following
holds:

$\|\mathbb{X}^{t_{0},x}\|_{L^{2}_{\mathbb{F}}(t_{0},T)}\leq
2m^{2}(1+2L)\|\Phi\|_{L^{2}_{\mathbb{F}}(t_{0},T)}$

with
$L:=\|\mathbb{A}\|_{L^{2}(\triangle_{2}(0,T))}+\|\mathbb{C}\|_{\mathscr{L}^{2}(\triangle_{2}(0,T))}$
and $m$ being a number of division intervals $0=U_{0}<U_{1}<\cdots<U_{m}=T$ of
$(0,T)$ satisfying

$\Bigl{(}\int^{U_{i+1}}_{U_{i}}\\!\\!\int^{t}_{U_{i}}|\mathbb{A}(t,s)|^{2}\,\mathrm{d}s\\!\,\mathrm{d}t\Bigr{)}^{1/2}+\underset{t\in(U_{i},U_{i+1})}{\mathrm{ess\,sup}}\Bigl{(}\int^{U_{i+1}}_{t}|\mathbb{C}(s,t)|^{2}\,\mathrm{d}s\Bigr{)}^{1/2}\leq\frac{1}{2}$

for $i=0,1,\dots,m-1$. Observe that

$\|\Phi\|_{L^{2}(t_{0},T)}\leq(1+\|\Xi\|_{L^{\infty}(0,T)}+\|\Gamma\|_{L^{2}(\triangle_{2}(0,T))})(\|x\|_{L^{2}(t_{0},T)}+\|b\|_{L^{2,1}_{\mathbb{F}}(\triangle_{2}(t_{0},T))}+\|\sigma\|_{L^{2}_{\mathbb{F}}(\triangle_{2}(t_{0},T))})+\|v\|_{L^{2}_{\mathbb{F}}(t_{0},T)}.$

Furthermore, noting the second equality in (A.1) and using Doob’s inequality,
we have

$\displaystyle\|\Theta^{t_{0},x}\|_{L^{2}_{\mathbb{F},\mathrm{c}}(\triangle_{2}(t_{0},T))}=\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\sup_{t\in[t_{0},s]}|\Theta^{t_{0},x}(s,t)|^{2}\,\mathrm{d}s\Bigr{]}^{1/2}$
$\displaystyle\leq\|x\|_{L^{2}(t_{0},T)}+(\|A\|_{L^{2}(\triangle_{2}(0,T))}+\|B\|_{L^{2}(\triangle_{2}(0,T))})\|\mathbb{X}^{t_{0},x}\|_{L^{2}_{\mathbb{F}}(t_{0},T)}+\|b\|_{L^{2,1}_{\mathbb{F}}(\triangle_{2}(t_{0},T))}$
$\displaystyle\hskip
14.22636pt+2(\|C\|_{\mathscr{L}^{2}(\triangle_{2}(0,T))}+\|D\|_{\mathscr{L}^{2}(\triangle_{2}(0,T))})\|\mathbb{X}^{t_{0},x}\|_{L^{2}_{\mathbb{F}}(t_{0},T)}+2\|\sigma\|_{L^{2}_{\mathbb{F}}(\triangle_{2}(t_{0},T))}.$

Therefore, the estimate (2.2) holds. This completes the proof. ∎

######  Remark A.5.

From the above proof, the constant $K>0$ in the estimate (2.2) can be chosen
to be uniform with respect to small scaling of the coefficients. More
precisely, for each $(\Xi,\Gamma,v)\in\mathcal{S}(0,T)$ and $(t_{0},x)$, the
corresponding causal feedback solution
$(X^{t_{0},x},\Theta^{t_{0},x},u^{t_{0},x})$ to the controlled SVIE (1.1) with
the coefficients $A,B,C,D$ replaced by
$\lambda_{A}A,\lambda_{B}B,\lambda_{C}C,\lambda_{D}D$ for some constants
$\lambda_{A},\lambda_{B},\lambda_{C},\lambda_{D}\in[0,1]$ satisfies the
estimate (2.2) with the same constant $K>0$.

###  A.2 Proof of 3.3: Type-II EBSVIEs

First, we prove the well-posedness of a trivial Type-II EBSVIE where $\Xi$ and
$\Gamma$ vanish.

######  Lemma A.6.

For any
$(\psi,\chi)\in\mathcal{Y}:=L^{2}_{\mathbb{F}}(0,T;\mathbb{R}^{d})\times
L^{2,1}_{\mathbb{F}}(\triangle_{2}(0,T);\mathbb{R}^{d})$, there exists a
unique adapted solution
$(\eta,\zeta)\in\mathcal{X}:=L^{2}_{\mathbb{F},\mathrm{c}}(\triangle_{2}(0,T);\mathbb{R}^{d})\times
L^{2}_{\mathbb{F}}(\triangle_{2}(0,T);\mathbb{R}^{d})$ to the trivial Type-II
EBSVIE

$\begin{dcases}\mathrm{d}\eta(t,s)=-\chi(t,s)\,\mathrm{d}s+\zeta(t,s)\,\mathrm{d}W(s),\
(t,s)\in\triangle_{2}(0,T),\\\ \eta(t,t)=\psi(t),\ t\in(0,T).\end{dcases}$

Furthermore, the solution map $\mathcal{T}:(\psi,\chi)\mapsto(\eta,\zeta)$ is
a bounded linear operator from $\mathcal{Y}$ to $\mathcal{X}$.

###### Proof.

We construct the adapted solution. Define $\eta\in
L^{2}_{\mathbb{F},\mathrm{c}}(\triangle_{2}(0,T);\mathbb{R}^{d})$ by

$\eta(t,s):=\mathbb{E}_{s}\Bigl{[}\psi(t)+\int^{t}_{s}\chi(t,r)\,\mathrm{d}r\Bigr{]},\
(t,s)\in\triangle_{2}(0,T).$

Also, for a.e. $t\in(0,T)$, define $\zeta(t,\cdot)\in
L^{2}_{\mathbb{F}}(0,t;\mathbb{R}^{d})$ via the martingale representation
theorem:

$\psi(t)+\int^{t}_{0}\chi(t,r)\,\mathrm{d}r=\mathbb{E}\Bigl{[}\psi(t)+\int^{t}_{0}\chi(t,r)\,\mathrm{d}r\Bigr{]}+\int^{t}_{0}\zeta(t,s)\,\mathrm{d}W(s).$

Then we have $\eta(t,t)=\psi(t)$, $t\in(0,T)$, and

$\eta(t,s)=\eta(t,t)+\int^{t}_{s}\chi(t,r)\,\mathrm{d}r-\int^{t}_{s}\zeta(t,r)\,\mathrm{d}W(r),\
(t,s)\in\triangle_{2}(0,T).$

Thus, the pair
$(\eta,\zeta)\in\mathcal{X}=L^{2}_{\mathbb{F},\mathrm{c}}(\triangle_{2}(0,T);\mathbb{R}^{d})\times
L^{2}_{\mathbb{F}}(\triangle_{2}(0,T);\mathbb{R}^{d})$ is an adapted solution
to the trivial Type-II EBSVIE. From the construction, we get

$\displaystyle\|\zeta\|_{L^{2}_{\mathbb{F}}(\triangle_{2}(0,T))}$
$\displaystyle=\mathbb{E}\Bigl{[}\int^{T}_{0}\\!\\!\int^{t}_{0}|\zeta(t,s)|^{2}\,\mathrm{d}s\\!\,\mathrm{d}t\Bigr{]}^{1/2}=\mathbb{E}\Bigl{[}\int^{T}_{0}\Bigl{|}\int^{t}_{0}\zeta(t,s)\,\mathrm{d}W(s)\Bigr{|}^{2}\,\mathrm{d}t\Bigr{]}^{1/2}$
$\displaystyle\leq\mathbb{E}\Bigl{[}\int^{T}_{0}\Bigl{|}\psi(t)+\int^{t}_{0}\chi(t,s)\,\mathrm{d}s\Bigr{|}^{2}\,\mathrm{d}t\Bigr{]}^{1/2}$
$\displaystyle\leq\|\psi\|_{L^{2}_{\mathbb{F}}(0,T;\mathbb{R}^{d})}+\|\chi\|_{L^{2,1}_{\mathbb{F}}(\triangle_{2}(0,T))}.$

Also, by Doob’s inequality,

$\displaystyle\|\eta\|_{L^{2}_{\mathbb{F},\mathrm{c}}(\triangle_{2}(0,T))}$
$\displaystyle=\mathbb{E}\Bigl{[}\int^{T}_{0}\sup_{s\in[0,t]}|\eta(t,s)|^{2}\,\mathrm{d}t\Bigr{]}^{1/2}=\Bigl{(}\int^{T}_{0}\mathbb{E}\Bigl{[}\sup_{s\in[0,t]}|\eta(t,s)|^{2}\Bigr{]}\,\mathrm{d}t\Bigr{)}^{1/2}$
$\displaystyle=\Bigl{(}\int^{T}_{0}\mathbb{E}\Bigl{[}\sup_{s\in[0,t]}\Bigl{|}\mathbb{E}_{s}\Bigl{[}\psi(t)+\int^{t}_{s}\chi(t,r)\,\mathrm{d}r\Bigr{]}\Bigr{|}^{2}\Bigr{]}\,\mathrm{d}t\Bigr{)}^{1/2}$
$\displaystyle\leq
2\|\psi\|_{L^{2}_{\mathbb{F}}(0,T)}+2\|\chi\|_{L^{2,1}_{\mathbb{F}}(\triangle_{2}(0,T))}.$

The uniqueness of the adapted-solution and the linearity of the solution map
$\mathcal{T}:\mathcal{Y}\to\mathcal{X}$ are clear. The boundedness of
$\mathcal{T}$ follows from the above estimates. This completes the proof. ∎

###### Proof of 3.3.

We first prove the duality principle (3.2). After that, we provide a priori
estimate of the adapted solution of Type-II EBSVIE (3.1) by using the duality
principle and the estimate (2.2) of the causal feedback solution to the
controlled SVIE. Then, the existence and uniqueness of the adapted solution
will follow from A.1.

The duality principle (3.2): Suppose that $(\eta,\zeta)\in
L^{2}_{\mathbb{F},\mathrm{c}}(\triangle_{2}(0,T);\mathbb{R}^{d})\times
L^{2}_{\mathbb{F}}(\triangle_{2}(0,T);\mathbb{R}^{d})$ is an adapted solution
to the Type-II EBSVIE (3.1). For each $(t_{0},x)\in\mathcal{I}$ and
$v\in\mathcal{U}(0,T)$, denote by
$(X,\Theta,u):=(X^{t_{0},x},\Theta^{t_{0},x},u^{t_{0},x})\in
L^{2}_{\mathbb{F}}(t_{0},T;\mathbb{R}^{d})\times
L^{2}_{\mathbb{F},\mathrm{c}}(\triangle_{2}(t_{0},T);\mathbb{R}^{d})\times\mathcal{U}(t_{0},T)$
the causal feedback solution to the controlled SVIE (1.1) at
$(t_{0},x)\in\mathcal{I}$ corresponding to the causal feedback strategy
$(\Xi,\Gamma,v)\in\mathcal{S}(0,T)$. By the second equality in (3.1), we have

$\begin{split}&\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\langle\psi(t),X(t)\rangle\,\mathrm{d}t\Bigr{]}\\\
&=\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\Bigl{\\{}\langle\eta(t,t),X(t)\rangle-\Bigl{\langle}\int^{T}_{t}(A+B\triangleright\Xi)(s,t)^{\top}\eta(s,t)\,\mathrm{d}s+\int^{T}_{t}(C+D\triangleright\Xi)(s,t)^{\top}\zeta(s,t)\,\mathrm{d}s,X(t)\Bigr{\rangle}\Bigr{\\}}\,\mathrm{d}t\Bigr{]}.\end{split}$
(A.3)

For a.e. $t\in(t_{0},T)$, noting that $\Theta(t,t)=X(t)$ and applying Iô’s
formula to $\langle\eta(t,\cdot),\Theta(t,\cdot)\rangle$ on $(t_{0},t)$, we
have

$\displaystyle\mathbb{E}[\langle\eta(t,t),X(t)\rangle]=\mathbb{E}[\langle\eta(t,t),\Theta(t,t)\rangle]$
$\displaystyle=\langle\mathbb{E}[\eta(t,t_{0})],x(t)\rangle$
$\displaystyle\hskip
14.22636pt+\mathbb{E}\Bigl{[}\int^{t}_{t_{0}}\Bigl{\\{}-\Bigl{\langle}\chi(t,s)+\Gamma(t,s)^{\top}\int^{T}_{s}B(r,s)^{\top}\eta(r,s)\,\mathrm{d}r+\Gamma(t,s)^{\top}\int^{T}_{s}D(r,s)^{\top}\zeta(r,s)\,\mathrm{d}r,\Theta(t,s)\Bigr{\rangle}$
$\displaystyle\hskip
85.35826pt+\langle\eta(t,s),A(t,s)X(s)+B(t,s)u(s)+b(t,s)\rangle$
$\displaystyle\hskip
85.35826pt+\langle\zeta(t,s),C(t,s)X(s)+D(t,s)u(s)+\sigma(t,s)\rangle\Bigr{\\}}\,\mathrm{d}s\Bigr{]}$
$\displaystyle=\langle\mathbb{E}[\eta(t,t_{0})],x(t)\rangle+\mathbb{E}\Bigl{[}\int^{t}_{t_{0}}\\{-\langle\chi(t,s),\Theta(t,s)\rangle+\langle\eta(t,s),b(t,s)\rangle+\langle\zeta(t,s),\sigma(t,s)\rangle\\}\,\mathrm{d}s\Bigr{]}$
$\displaystyle\hskip
14.22636pt+\mathbb{E}\Bigl{[}\int^{t}_{t_{0}}\langle(A+B\triangleright\Xi)(t,s)^{\top}\eta(t,s)+(C+D\triangleright\Xi)(t,s)^{\top}\zeta(t,s),X(s)\rangle\,\mathrm{d}s\Bigr{]}$
$\displaystyle\hskip
14.22636pt+\mathbb{E}\Bigl{[}\int^{t}_{t_{0}}\Bigl{\\{}\langle
B(t,s)^{\top}\eta(t,s)+D(t,s)^{\top}\zeta(t,s),u(s)-\Xi(s)X(s)\rangle$
$\displaystyle\hskip
85.35826pt-\Bigl{\langle}\int^{T}_{s}\\{B(r,s)^{\top}\eta(r,s)+D(r,s)^{\top}\zeta(r,s)\\}\,\mathrm{d}r,\Gamma(t,s)\Theta(t,s)\Bigr{\rangle}\Bigr{\\}}\,\mathrm{d}s\Bigr{]}.$

Integrating both sides with respect to $t\in(t_{0},T)$ and applying Fubini’s
theorem, we get

$\begin{split}&\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\langle\eta(t,t),X(t)\rangle\,\mathrm{d}t\Bigr{]}\\\
&=\int^{T}_{t_{0}}\langle\mathbb{E}[\eta(t,t_{0})],x(t)\rangle\,\mathrm{d}t\\\
&\hskip
14.22636pt+\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\Bigl{\\{}-\int^{T}_{t}\langle\chi(s,t),\Theta(s,t)\rangle\,\mathrm{d}s+\int^{T}_{t}\langle\eta(s,t),b(s,t)\rangle\,\mathrm{d}s+\int^{T}_{t}\langle\zeta(s,t),\sigma(s,t)\rangle\,\mathrm{d}s\Bigr{\\}}\,\mathrm{d}t\Bigr{]}\\\
&\hskip
14.22636pt+\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\Bigl{\langle}\int^{T}_{t}(A+B\triangleright\Xi)(s,t)^{\top}\eta(s,t)\,\mathrm{d}s+\int^{T}_{t}(C+D\triangleright\Xi)(s,t)^{\top}\zeta(s,t)\,\mathrm{d}s,X(t)\Bigr{\rangle}\,\mathrm{d}t\Bigr{]}\\\
&\hskip
14.22636pt+\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\Bigl{\langle}\int^{T}_{t}\\{B(s,t)^{\top}\eta(s,t)+D(s,t)^{\top}\zeta(s,t)\\}\,\mathrm{d}s,u(t)-\Xi(t)X(t)-\int^{T}_{t}\Gamma(r,t)\Theta(r,t)\,\mathrm{d}r\Bigr{\rangle}\,\mathrm{d}t\Bigr{]}\end{split}$
(A.4)

By (A.3), (A.4) and the relation
$u(t)=\Xi(t)X(t)+\int^{T}_{t}\Gamma(r,t)\Theta(r,t)\,\mathrm{d}r+v(t)$, we
obtain the duality principle (3.2).

Existence and uniqueness of the adapted solution: We define a map $F$ from
$\mathcal{X}:=L^{2}_{\mathbb{F},\mathrm{c}}(\triangle_{2}(0,T);\mathbb{R}^{d})\times
L^{2}_{\mathbb{F}}(\triangle_{2}(0,T);\mathbb{R}^{d})$ to
$\mathcal{Y}:=L^{2}_{\mathbb{F}}(0,T;\mathbb{R}^{d})\times
L^{2,1}_{\mathbb{F}}(\triangle_{2}(0,T);\mathbb{R}^{d})$ by

$\displaystyle F(\eta,\zeta):=(F_{1}(\eta,\zeta),F_{2}(\eta,\zeta)),$
$\displaystyle
F_{1}(\eta,\zeta)(t):=\int^{T}_{t}(A+B\triangleright\Xi)(r,t)^{\top}\eta(r,t)\,\mathrm{d}r+\int^{T}_{t}(C+D\triangleright\Xi)(r,t)^{\top}\zeta(r,t)\,\mathrm{d}r,\
t\in(0,T),$ $\displaystyle
F_{2}(\eta,\zeta)(t,s):=\Gamma(t,s)^{\top}\int^{T}_{s}B(r,s)^{\top}\eta(r,s)\,\mathrm{d}r+\Gamma(t,s)^{\top}\int^{T}_{s}D(r,s)^{\top}\zeta(r,s)\,\mathrm{d}r,\
(t,s)\in\triangle_{2}(0,T),$

for $(\eta,\zeta)\in\mathcal{X}$. It is easy to see that
$F:\mathcal{X}\to\mathcal{Y}$ is a bounded linear operator. Also, consider the
bounded linear operator $\mathcal{T}:\mathcal{Y}\to\mathcal{X}$ defined in
A.6. Observe that, for each $(\psi,\chi)\in\mathcal{Y}$,
$(\eta,\zeta)\in\mathcal{X}$ is an adapted solution to the Type-II EBSVIE
(3.1) if and only if the following holds:

$(\eta,\zeta)=\mathcal{T}(F(\eta,\zeta)+(\psi,\chi))$

In order to apply A.1, we have to show the a priori estimate.

A priori estimate of the adapted solution: Let $\lambda\in[0,1]$ be fixed. For
each $(\psi_{i},\chi_{i})\in\mathcal{Y}$ with $i=1,2$, assume that
$(\eta_{i},\zeta_{i})\in\mathcal{X}$ satisfy the equations

$(\eta_{i},\zeta_{i})=\mathcal{T}(\lambda
F(\eta_{i},\zeta_{i})+(\psi_{i},\chi_{i})),\ i=1,2.$

Define
$(\bar{\eta},\bar{\zeta}):=(\eta_{1}-\eta_{2},\zeta_{1}-\zeta_{2})\in\mathcal{X}$
and
$(\bar{\psi},\bar{\chi}):=(\psi_{1}-\psi_{2},\chi_{1}-\chi_{2})\in\mathcal{Y}$.
We will show the estimate

$\|(\bar{\eta},\bar{\zeta})\|_{\mathcal{X}}\leq
K\|(\bar{\psi},\bar{\chi})\|_{\mathcal{Y}}.$

Here and the rest of this proof, $K>0$ denotes a universal constant which does
not depend on $\lambda$ or $(\psi_{i},\chi_{i})$. Observe that
$(\bar{\eta},\bar{\zeta})$ is an adapted solution to the Type-II EBSVIE (3.1)
with the coefficients $A,B,C,D$ replaced by $\lambda A,\lambda B,\lambda
C,\lambda D$ and $(\psi,\chi)$ replaced by $(\bar{\psi},\bar{\chi})$. Let
$(X_{1},\Theta_{1})\in L^{2}_{\mathbb{F}}(0,T;\mathbb{R}^{d})\times
L^{2}_{\mathbb{F},\mathrm{c}}(\triangle_{2}(0,T);\mathbb{R}^{d})$ be the
causal feedback solution to the homogeneous controlled SVIE (2.1) with the
coefficients $(\lambda A,\lambda B,\lambda C,\lambda D)$, the zero input
condition $(t_{0},x)=(0,0)\in\mathcal{I}$, and a causal feedback strategy
$(\Xi,\Gamma,v)\in\mathcal{S}(0,T)$ with $v\in\mathcal{U}(0,T)$ being
arbitrary. By the duality principle (3.2), we have

$\displaystyle\mathbb{E}\Bigl{[}\int^{T}_{0}\Bigl{\langle}\int^{T}_{t}\\{\lambda
B(s,t)^{\top}\bar{\eta}(s,t)+\lambda
D(s,t)^{\top}\bar{\zeta}(s,t)\\}\,\mathrm{d}s,v(t)\Bigr{\rangle}\,\mathrm{d}t\Bigr{]}$
$\displaystyle=\mathbb{E}\Bigl{[}\int^{T}_{0}\Bigl{\\{}\langle\bar{\psi}(t),X_{1}(t)\rangle+\int^{T}_{t}\langle\bar{\chi}(s,t),\Theta_{1}(s,t)\rangle\,\mathrm{d}s\Bigr{\\}}\,\mathrm{d}t\Bigr{]}.$

By 2.4, noting A.5, the right hand side of the above equality is estimated as

$\displaystyle\Bigl{|}\mathbb{E}\Bigl{[}\int^{T}_{0}\Bigl{\\{}\langle\bar{\psi}(t),X_{1}(t)\rangle+\int^{T}_{t}\langle\bar{\chi}(s,t),\Theta_{1}(s,t)\rangle\,\mathrm{d}s\Bigr{\\}}\,\mathrm{d}t\Bigr{]}\Bigr{|}$
$\displaystyle\leq\|\bar{\psi}\|_{L^{2}_{\mathbb{F}}(0,T)}\|X_{1}\|_{L^{2}_{\mathbb{F}}(0,T)}+\|\bar{\chi}\|_{L^{2,1}_{\mathbb{F}}(\triangle_{2}(0,T))}\|\Theta_{1}\|_{L^{2}_{\mathbb{F},\mathrm{c}}(\triangle_{2}(0,T))}$
$\displaystyle\leq
K\|(\bar{\psi},\bar{\chi})\|_{\mathcal{Y}}\|v\|_{L^{2}_{\mathbb{F}}(0,T)}.$

Therefore, we have the following estimate:

$\displaystyle\mathbb{E}\Bigl{[}\int^{T}_{0}\Bigl{|}\int^{T}_{t}\\{\lambda
B(s,t)^{\top}\bar{\eta}(s,t)+\lambda
D(s,t)^{\top}\bar{\zeta}(s,t)\\}\,\mathrm{d}s\Bigr{|}^{2}\,\mathrm{d}t\Bigr{]}^{1/2}$
$\displaystyle=\sup_{\begin{subarray}{c}v\in
L^{2}_{\mathbb{F}}(0,T;\mathbb{R}^{\ell})\\\
\|v\|_{L^{2}_{\mathbb{F}}(0,T)}\leq
1\end{subarray}}\Bigl{|}\mathbb{E}\Bigl{[}\int^{T}_{0}\Bigl{\langle}\int^{T}_{t}\\{\lambda
B(s,t)^{\top}\bar{\eta}(s,t)+\lambda
D(s,t)^{\top}\bar{\zeta}(s,t)\\}\,\mathrm{d}s,v(t)\Bigr{\rangle}\,\mathrm{d}t\Bigr{]}\Bigr{|}$
$\displaystyle\leq K\|(\bar{\psi},\bar{\chi})\|_{\mathcal{Y}}.$

By using the Cauchy–Schwarz inequality, we get

$\displaystyle\|\lambda
F_{2}(\bar{\eta},\bar{\zeta})\|_{L^{2,1}_{\mathbb{F}}(\triangle_{2}(0,T))}$
$\displaystyle\leq\|\Gamma\|_{L^{2}(\triangle_{2}(0,T))}\mathbb{E}\Bigl{[}\int^{T}_{0}\Bigl{|}\int^{T}_{t}\\{\lambda
B(s,t)^{\top}\bar{\eta}(s,t)+\lambda
D(s,t)^{\top}\bar{\zeta}(s,t)\\}\,\mathrm{d}s\Bigr{|}^{2}\,\mathrm{d}t\Bigr{]}^{1/2}$
$\displaystyle\leq K\|(\bar{\psi},\bar{\chi})\|_{\mathcal{Y}}.$

Next, for a given $\varphi\in L^{2}_{\mathbb{F}}(0,T;\mathbb{R}^{d})$, let
$(X_{2},\Theta_{2})\in L^{2}_{\mathbb{F}}(0,T;\mathbb{R}^{d})\times
L^{2}_{\mathbb{F},\mathrm{c}}(\triangle_{2}(0,T);\mathbb{R}^{d})$ be the
causal feedback solution to the controlled SVIE (1.1) with the coefficients
$(\lambda A,\lambda B,\lambda C,\lambda D)$, the inhomogeneous terms

$b(t,s)=\lambda(A+B\triangleright\Xi)(t,s)\varphi(s),\
\sigma(t,s)=\lambda(C+D\triangleright\Xi)(t,s)\varphi(s),\
(t,s)\in\triangle_{2}(0,T),$

the input condition $(t_{0},x)=(0,0)$, and the causal feedback strategy
$(\Xi,\Gamma,0)\in\mathcal{S}(0,T)$. Then the duality principle (3.2) yields
that

$\displaystyle\mathbb{E}\Bigl{[}\int^{T}_{0}\Bigl{\langle}\int^{T}_{t}\\{\lambda(A+B\triangleright\Xi)(s,t)^{\top}\bar{\eta}(s,t)+\lambda(C+D\triangleright\Xi)(s,t)^{\top}\bar{\zeta}(s,t)\\}\,\mathrm{d}s,\varphi(t)\Bigr{\rangle}\,\mathrm{d}t\Bigr{]}$
$\displaystyle=\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\Bigl{\\{}\int^{T}_{t}\langle\bar{\eta}(s,t),b(s,t)\rangle\,\mathrm{d}s+\int^{T}_{t}\langle\bar{\zeta}(s,t),\sigma(s,t)\rangle\,\mathrm{d}s\Bigr{\\}}\,\mathrm{d}t\Bigr{]}$
$\displaystyle=\mathbb{E}\Bigl{[}\int^{T}_{0}\Bigl{\\{}\langle\bar{\psi}(t),X_{2}(t)\rangle+\int^{T}_{t}\langle\bar{\chi}(s,t),\Theta_{2}(s,t)\rangle\,\mathrm{d}s\Bigr{\\}}\,\mathrm{d}t\Bigr{]}.$

Noting that
$\|b\|_{L^{2,1}_{\mathbb{F}}(\triangle_{2}(0,T))}+\|\sigma\|_{L^{2}_{\mathbb{F}}(\triangle_{2}(0,T))}\leq
K\|\varphi\|_{L^{2}_{\mathbb{F}}(0,T)}$, by the same arguments as above, we
obtain

$\displaystyle\|\lambda
F_{1}(\bar{\eta},\bar{\zeta})\|_{L^{2}_{\mathbb{F}}(0,T)}$
$\displaystyle=\mathbb{E}\Bigl{[}\int^{T}_{0}\Bigl{|}\int^{T}_{t}\\{\lambda(A+B\triangleright\Xi)(s,t)^{\top}\bar{\eta}(s,t)+\lambda(C+D\triangleright\Xi)(s,t)^{\top}\bar{\zeta}(s,t)\\}\,\mathrm{d}s\Bigr{|}^{2}\,\mathrm{d}t\Bigr{]}^{1/2}$
$\displaystyle\leq K\|(\bar{\psi},\bar{\chi})\|_{\mathcal{Y}}.$

From the above estimates, we have

$\|\lambda F(\bar{\eta},\bar{\zeta})\|_{\mathcal{Y}}\leq
K\|(\bar{\psi},\bar{\chi})\|_{\mathcal{Y}},$

and hence the desired a priori estimate holds:

$\|(\bar{\eta},\bar{\zeta})\|_{\mathcal{X}}=\|\mathcal{T}(\lambda
F(\bar{\eta},\bar{\zeta})+(\bar{\psi},\bar{\chi}))\|_{\mathcal{X}}\leq
K\|(\bar{\psi},\bar{\chi})\|_{\mathcal{Y}}.$

By A.1, for any
$(\psi,\chi)\in\mathcal{Y}=L^{2}_{\mathbb{F}}(0,T;\mathbb{R}^{d})\times
L^{2,1}_{\mathbb{F}}(\triangle_{2}(0,T);\mathbb{R}^{d})$, there exists a
unique adapted solution
$(\eta,\zeta)\in\mathcal{X}=L^{2}_{\mathbb{F},\mathrm{c}}(\triangle_{2}(0,T);\mathbb{R}^{d})\times
L^{2}_{\mathbb{F}}(\triangle_{2}(0,T);\mathbb{R}^{d})$ to the Type-II EBSVIE
(3.1). This completes the proof. ∎

###  A.3 Proof of 4.11: Lyapunov–Volterra equations

We prove 4.11. The representation formula (4.6) follows from 4.9. The idea of
the proof of the well-posedness of the Lyapunov–Volterra equation (4.5) is
similar to the one of 3.3.

###### Proof of 4.11.

The representation formula (4.6): By 4.9, it is trivial that if
$P=(P^{(1)},P^{(2)})\in\Pi(0,T)$ satisfies the Lyapunov–Volterra equation
(4.5), then the representation formula (4.6) holds.

Existence and uniqueness of the solution: Let $\mathcal{X}:=\Pi(0,T)$ and

$\mathcal{Y}:=L^{\infty}(0,T;\mathbb{S}^{d})\times
L^{2}(\triangle_{2}(0,T);\mathbb{R}^{d\times d})\times
L^{2,2,1}_{\mathrm{sym}}(\square_{3}(0,T);\mathbb{R}^{d\times d}).$

Define $\mathcal{T}:\mathcal{Y}\to\mathcal{X}$ by
$\mathcal{T}(Q):=(\mathcal{T}_{1}(Q),\mathcal{T}_{2}(Q))$ with

$\displaystyle\mathcal{T}_{1}(Q)(t):=Q^{(1)}(t),\ t\in(0,T),$
$\displaystyle\mathcal{T}_{2}(Q)(s_{1},s_{2},t):=Q^{(2)}(s_{1},s_{2})\mbox{\rm{1}}\mbox{\rm{l}}_{\triangle_{2}(0,T)}(s_{1},s_{2})+Q^{(2)}(s_{2},s_{1})^{\top}\mbox{\rm{1}}\mbox{\rm{l}}_{\triangle_{2}(0,T)}(s_{2},s_{1})-\int^{s_{1}\wedge
s_{2}}_{t}Q^{(3)}(s_{1},s_{2},\tau)\,\mathrm{d}\tau,$ $\displaystyle\hskip
170.71652pt(s_{1},s_{2},t)\in\square_{3}(0,T),$

for $Q=(Q^{(1)},Q^{(2)},Q^{(3)})\in\mathcal{Y}$. Clearly,
$\mathcal{T}:\mathcal{Y}\to\mathcal{X}$ is a bounded linear operator. Also,
define $F:\mathcal{X}\to\mathcal{Y}$ by

$\displaystyle F(P):=(F^{(1)}[\Xi;P],F^{(2)}[\Xi,\Gamma;P],F^{(3)}[\Gamma;P])$

for $P=(P^{(1)},P^{(2)})\in\mathcal{X}=\Pi(0,T)$. By 4.6,
$F:\mathcal{X}\to\mathcal{Y}$ is a bounded linear operator. For each
$Q=(Q^{(1)},Q^{(2)},Q^{(3)})\in\mathcal{Y}$,
$P=(P^{(1)},P^{(2)})\in\mathcal{X}$ is a solution to the Lyapunov–Volterra
equation (4.5) if and only if the following equation holds:

$P=\mathcal{T}(F(P)+Q).$

In order to apply A.1, we have to show the a priori estimate.

A priori estimate of the solution: Let $\lambda\in[0,1]$ be fixed. For each
$Q_{i}=(Q^{(1)}_{i},Q^{(2)}_{i},Q^{(3)}_{i})\in\mathcal{Y}$ with $i=1,2$,
assume that $P_{i}=(P^{(1)}_{i},P^{(2)}_{i})\in\mathcal{X}$ satisfy the
equations

$P_{i}=\mathcal{T}(\lambda F(P_{i})+Q_{i}),\ i=1,2.$

Define
$\bar{P}:=(\bar{P}^{(1)},\bar{P}^{(2)}):=(P^{(1)}_{1}-P^{(1)}_{2},P^{(2)}_{1}-P^{(2)}_{2})\in\mathcal{X}$
and
$\bar{Q}:=(\bar{Q}^{(1)},\bar{Q}^{(2)},\bar{Q}^{(3)}):=(Q^{(1)}_{1}-Q^{(1)}_{2},Q^{(2)}_{1}-Q^{(2)}_{2},Q^{(3)}_{1}-Q^{(3)}_{2})\in\mathcal{Y}$.
We will show the estimate

$\|\bar{P}\|_{\mathcal{X}}\leq K\|\bar{Q}\|_{\mathcal{Y}}.$

Here and the rest of this proof, $K>0$ denotes a universal constant which does
not depend on $\lambda$ or $Q_{i}$. Observe that
$\bar{P}=(\bar{P}^{(1)},\bar{P}^{(2)})$ is a solution to the Lyapunov–Volterra
equation (4.5) with the coefficients $A,B,C,D$ replaced by $\lambda A,\lambda
B,\sqrt{\lambda}C,\sqrt{\lambda}D$ and $Q=(Q^{(1)},Q^{(2)},Q^{(3)})$ replaced
by $\bar{Q}=(\bar{Q}^{(1)},\bar{Q}^{(2)},\bar{Q}^{(3)})$. For each
$t_{0}\in[0,T)$, define
$\bar{\mathcal{P}}^{t_{0}}:L^{2}(t_{0},T;\mathbb{R}^{d})\to
L^{2}(t_{0},T;\mathbb{R}^{d})$ by

$\bar{\mathcal{P}}^{t_{0}}:=\bar{P}^{(1)}(t)x(t)+\int^{T}_{t_{0}}\bar{P}^{(2)}(t,r,t_{0})x(r)\,\mathrm{d}r,\
t\in(t_{0},T),$

for $x\in L^{2}(t_{0},T;\mathbb{R}^{d})$. By 4.3, $\bar{\mathcal{P}}^{t_{0}}$
is a self-adjoint bounded linear operator on the Hilbert space
$L^{2}(t_{0},T;\mathbb{R}^{d})$. Now let $(X,\Theta)\in
L^{2}_{\mathbb{F}}(t_{0},T;\mathbb{R}^{d})\times
L^{2}_{\mathbb{F},\mathrm{c}}(\triangle_{2}(t_{0},T);\mathbb{R}^{d})$ be the
causal feedback solution to the homogeneous controlled SVIE (2.1) with the
coefficients $(\lambda A,\lambda B,\sqrt{\lambda}C,\sqrt{\lambda}D)$, a given
input condition $(t_{0},x)\in\mathcal{I}$, and the causal feedback strategy
$(\Xi,\Gamma,0)\in\mathcal{S}(0,T)$. By the representation formula (4.6), we
have

$\displaystyle\langle\bar{\mathcal{P}}^{t_{0}}x,x\rangle_{L^{2}(t_{0},T)}$
$\displaystyle=\int^{T}_{t_{0}}\langle\bar{P}^{(1)}(t)x(t),x(t)\rangle\,\mathrm{d}t+\int^{T}_{t_{0}}\\!\\!\int^{T}_{t_{0}}\langle\bar{P}^{(2)}(s_{1},s_{2},t_{0})x(s_{2}),x(s_{1})\rangle\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2}$
$\displaystyle=\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\Bigl{\\{}\langle\bar{Q}^{(1)}(t)X(t),X(t)\rangle+2\int^{T}_{t}\langle\bar{Q}^{(2)}(s,t)X(t),\Theta(s,t)\rangle\,\mathrm{d}s$
$\displaystyle\hskip
113.81102pt+\int^{T}_{t}\\!\\!\int^{T}_{t}\langle\bar{Q}^{(3)}(s_{1},s_{2},t)\Theta(s_{2},t),\Theta(s_{1},t)\rangle\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2}\Bigr{\\}}\,\mathrm{d}t\Bigr{]}.$

By 2.4, noting A.5, the right hand side of the above equality is estimated as

$\displaystyle\Bigl{|}\mathbb{E}\Bigl{[}\int^{T}_{t_{0}}\Bigl{\\{}\langle\bar{Q}^{(1)}(t)X(t),X(t)\rangle+2\int^{T}_{t}\langle\bar{Q}^{(2)}(s,t)X(t),\Theta(s,t)\rangle\,\mathrm{d}s$
$\displaystyle\hskip
113.81102pt+\int^{T}_{t}\\!\\!\int^{T}_{t}\langle\bar{Q}^{(3)}(s_{1},s_{2},t)\Theta(s_{2},t),\Theta(s_{1},t)\rangle\,\mathrm{d}s_{1}\\!\,\mathrm{d}s_{2}\Bigr{\\}}\,\mathrm{d}t\Bigr{]}\Bigr{|}$
$\displaystyle\leq\|\bar{Q}^{(1)}\|_{L^{\infty}(0,T)}\|X\|^{2}_{L^{2}_{\mathbb{F}}(t_{0},T)}+2\|\bar{Q}^{(2)}\|_{L^{2}(\triangle_{2}(0,T))}\|X\|_{L^{2}_{\mathbb{F}}(t_{0},T)}\|\Theta\|_{L^{2}_{\mathbb{F},\mathrm{c}}(\triangle_{2}(t_{0},T))}$
$\displaystyle\hskip
113.81102pt+\|\bar{Q}^{(3)}\|_{L^{2,2,1}_{\mathrm{sym}}(\square_{3}(0,T))}\|\Theta\|^{2}_{L^{2}_{\mathbb{F},\mathrm{c}}(\triangle_{2}(t_{0},T))}$
$\displaystyle\leq K\|\bar{Q}\|_{\mathcal{Y}}\|x\|^{2}_{L^{2}(t_{0},T)}.$

Therefore, the operator norm $\|\bar{\mathcal{P}}^{t_{0}}\|_{\mathrm{op}}$ of
$\bar{\mathcal{P}}^{t_{0}}$ is estimated as

$\|\bar{\mathcal{P}}^{t_{0}}\|_{\mathrm{op}}=\sup_{\begin{subarray}{c}x\in
L^{2}(t_{0},T;\mathbb{R}^{d})\\\ \|x\|_{L^{2}(t_{0},T)}\leq
1\end{subarray}}|\langle\bar{\mathcal{P}}^{t_{0}}x,x\rangle_{L^{2}(t_{0},T)}|\leq
K\|\bar{Q}\|_{\mathcal{Y}}.$

We note that, for any $M=(M_{1},\dots,M_{d_{1}})\in
L^{2}(\triangle_{2}(0,T);\mathbb{R}^{d\times d})$, $\bar{P}{\rtimes}M\in
L^{2}(\triangle_{2}(0,T);\mathbb{R}^{d\times d})$ is written by

$(\bar{P}{\rtimes}M)(s,t)=((\bar{\mathcal{P}}^{t}M_{1}(\cdot,t))(s),\dots,(\bar{\mathcal{P}}^{t}M_{d}(\cdot,t))(s)),\
(s,t)\in\triangle_{2}(0,T),$

and thus

$\|\bar{P}{\rtimes}M\|_{L^{2}(\triangle_{2}(0,T))}\leq
K\|\bar{Q}\|_{\mathcal{Y}}\|M\|_{L^{2}(\triangle_{2}(0,T))}.$

Similarly, for any
$M=(M_{1},\dots,M_{d})\in\mathscr{L}^{2}(\triangle_{2}(0,T);\mathbb{R}^{d\times
d})$ and
$N=(N_{1},\dots,N_{d})\in\mathscr{L}^{2}(\triangle_{2}(0,T);\mathbb{R}^{d\times
d})$, $M^{\top}{\ltimes}\bar{P}{\rtimes}N\in
L^{\infty}(0,T;\mathbb{R}^{d\times d})$ is written by

$(M^{\top}{\ltimes}\bar{P}{\rtimes}N)(t)=(\langle
M_{k}(\cdot,t),\bar{\mathcal{P}}^{t}N_{\ell}(\cdot,t)\rangle_{L^{2}(t,T)})_{k,\ell},\
t\in(0,T),$

and thus

$\|M^{\top}{\ltimes}\bar{P}{\rtimes}N\|_{L^{\infty}(0,T)}\leq
K\|\bar{Q}\|_{\mathcal{Y}}\|M\|_{\mathscr{L}^{2}(\triangle_{2}(0,T))}\|N\|_{\mathscr{L}^{2}(\triangle_{2}(0,T))}.$

Noting the definition (4.4) of the coefficients $(F^{(1)},F^{(2)},F^{(3)})$,
we see that

$\|F(\bar{P})\|_{\mathcal{Y}}=\|(F^{(1)}[\Xi;\bar{P}],F^{(2)}[\Xi,\Gamma;\bar{P}],F^{(3)}[\Gamma;\bar{P}])\|_{\mathcal{Y}}\leq
K\|\bar{Q}\|_{\mathcal{Y}},$

and thus the desired a priori estimate holds:

$\|\bar{P}\|_{\mathcal{X}}=\|\mathcal{T}(\lambda
F(\bar{P})+\bar{Q})\|_{\mathcal{X}}\leq K\|\bar{Q}\|_{\mathcal{Y}}.$

By A.1, for any
$Q=(Q^{(1)},Q^{(2)},Q^{(3)})\in\mathcal{Y}=L^{\infty}(0,T;\mathbb{S}^{d})\times
L^{2}(\triangle_{2}(0,T);\mathbb{R}^{d\times d})\times
L^{2,2,1}_{\mathrm{sym}}(\square_{3}(0,T);\mathbb{R}^{d\times d})$, there
exists a unique solution $P=(P^{(1)},P^{(2)})\in\mathcal{X}=\Pi(0,T)$ to the
Lyapunov–Volterra equation (4.5). This completes the proof. ∎

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