# Multi-Dimensional Super-Linear Backward Stochastic Volterra Integral
Equations

Shengjun Fan111School of Mathematics, China University of Mining and
Technology, Xuzhou, China. Email: shengjunfan @cumt.edu.cn. This author is
supported by National Natural Science Foundation of China (No. 12171471).
Tianxiao Wang 222School of Mathematics, Sichuan University, Chengdu, China.
Email<EMAIL_ADDRESS>This author is supported by the NSF of China
under grants 11971332 and 11931011, the Science Development Project of Sichuan
University under grant 2020SCUNL201. and Jiongmin Yong 333Department of
Mathematics, University of Central Florida, Orlando, USA. Email:
<EMAIL_ADDRESS>This author is supported in part by NSF Grant
DMS-1812921.

###### Abstract

In this paper, a systematic investigation is carried out for the general
solvability of multi-dimensional backward stochastic Volterra integral
equations (BSVIEs) with the generators being super-linear in the adjustment
variable $Z$. Two major situations are discussed: (i) When the free term is
bounded with the dependence of the generator on $Z$ being of “diagonally
strictly” quadratic growth and being sub-quadratically coupled with off-
diagonal components; (ii) When the free term is unbounded having exponential
moments of arbitrary order with the dependence of the generator on $Z$ being
diagonally no more than quadratic and being independent of off-diagonal
components. Besides, for the case that the generator is super-quadratic in
$Z$, some negative results are presented.

Keywords. Backward stochastic Volterra integral equations, multi-dimensional
systems, unbounded free terms, diagonally quadratic generator, sub-quadratic
growth.

AMS 2020 Mathematical sciences classifications. 60H20, 45D05.

##  1 Introduction

Let $(\Omega,{\cal F},\mathbb{F},\mathbb{P})$ be a filtered complete
probability space on which a standard one-dimensional Brownian motion
$W(\cdot)$ is defined with $\mathbb{F}=\\{{\cal F}_{t}\\}_{t\geqslant 0}$
being its natural filtration augmented by all the $\mathbb{P}$-null sets.
Consider the following backward stochastic Volterra integral equation (BSVIE,
for short):

(1.1) $Y(t)=\psi(t)+\int_{t}^{T}g(t,s,Y(s),Z(t,s))ds-\int_{t}^{T}Z(t,s)dW(s),\
\ t\in[0,T],$

where
$g:\Delta[0,T]\times\mathbb{R}^{m}\times\mathbb{R}^{m}\times\Omega\to\mathbb{R}^{m}$,
satisfying proper conditions, is called the generator of BSVIE (1.1), with
$\Delta^{*}[0,T]=\\{(t,s)\in[0,T]^{2}\bigm{|}0\leqslant t<s\leqslant T\\}$
being the upper triangle domain within the square $[0,T]^{2}$, and
$\psi(\cdot)$ is called a free term. The unknown is a pair
$(Y(\cdot),Z(\cdot\,,\cdot))$ of processes having certain properties, called
an adapted solution. For convenience, we refer to $Y(\cdot)$ as the backward
term and refer to $Z(\cdot\,,\cdot)$ as the adjustment term (since such a term
makes it possible to have adaptiveness of $Y(\cdot)$). When $g(t,s,y,z)\equiv
g(s,y,z)$ and $\psi(t)\equiv\xi$ are independent of $t$, (1.1) is reduced to
the following backward stochastic differential equation (BSDE, for short) (of
the integral form):

(1.2) $Y(t)=\xi+\int_{t}^{T}g(s,Y(s),Z(s))ds-\int_{t}^{T}Z(s)dW(s),\ \
t\in[0,T].$

Linear BSDE was introduced by Bismut in 1973 ([3]) and the general nonlinear
version was introduced by Pardoux–Peng in 1990 ([29]) where the well-posedness
was established under uniform Lipschitz condition of the map $(y,z)\mapsto
g(s,y,z)$. Among a huge number of relevant papers, we mention El
Karoui–Peng–Quenez [12] and the books by Ma–Yong [28], Yong–Zhou [50], and
Zhang [51] (and the references cite therein). In 2000, Kobylanski studied one-
dimensional BSDEs having a quadratic growth of the map $z\mapsto g(s,y,z)$
(QBSDE, for short) with a bounded terminal term ([26]). This work was extended
by Briand–Hu [5, 6] allowing the terminal term to be unbounded (having
exponential moments). Further, Delbaen–Hu–Bao ([8]) and Fan–Hu ([13])
respectively investigated the (one-dimensional) cases with $z\mapsto g(s,y,z)$
being super-quadratic and sub-quadratic. For multi-dimensional situations,
some extensions for QBSDEs were carried out by Hu–Tang ([24]), and Fan–Hu–Tang
([15]) for either $z\mapsto g(s,y,z)$ being diagonal, or so-called diagonally
strictly quadratic (see below for a precise definition).

Study of BSVIEs was begun with the work of Lin in 2002 ([27]). For general
theory of BSVIEs, the readers are referred to Yong [46, 48], Wang–Zhang [45],
Shi–Wang [32], Djordjević–Janković [11], Hu–Oksendal [23], Popier [30] for the
well-posedness of BSVIEs, to Wang–Yong [42] for comparison theorems, to
Shi–Wang–Yong [33] for some extensions to mean-field BSVIEs, to Shi–Wen–Xiong
[34] for doubly BSVIEs, to Hamaguchi [19] for infinite horizonal BSVIEs, to
Wang [36] for extended BSVIEs, to Wang–Yong [38], Hernández [21],
Hernández–Possamai [22] for BSVIEs with $Z(s,s)$ appears, to Agram–Djehiche
[2] for reflected BSVIEs, to Yong [49], Wang–Yong [43], Wang [36],
Wang–Yong–Zhang [39] for connections with PDEs, to Yong [46, 48], Wang–Zhang
[44], Wang [40] for applications in stochastic optimal control theory, to Yong
[47], Wang–Shi [41], Agram [1], Beissner–Rosazza Gianin [4], Wang–Sun–Yong
[37] for connections with mathematical finance, and to Wang–Yong [38],
Hamaguchi [20] for applications to time-inconsistent stochastic optimal
controls. It is well-known by now that the well-posedness can be established
under the uniform Lipschitz conditions of $(y,z)\mapsto g(t,s,y,z)$ for the
general multi-dimensional cases. For the case of $z\mapsto g(t,s,y,z)$ being
quadratic, the one-dimensional well-posedness was established by Wang–Sun–Yong
[37] with bounded free terms, and the multi-dimensional case was studied in
Hernández [21] with essentially small bounded free terms. To our best
knowledge, it is completely blank for the general multi-dimensional super-
linear BSVIEs (including sub-quadratic, strictly quadratic, and super-
quadratic growth of $z\mapsto g(t,s,y,z)$) with bounded/unbounded free terms.
From mathematical point of view, it will be good to fill these blanks for
BSVIEs, and to explore further, even for BSDEs. This is a major motivation of
the current paper. Our main results can be briefly summarized as follows. Let
us emphasis that all results are for multi-dimensional BSVIEs. Also, although
we only consider the case of one-dimensional Brwonian motion, all the results
hold for multi-dimensional Brownian motions, just with some modification of
notations.

(i) For the case $z\mapsto g(t,s,y,z)$ being diagonally strictly quadratic,
well-posedness of BSVIEs will be established under the condition that the free
term is bounded, allowing off-diagonal components of $z$ appear sub-
quadratically (see a later section for the precise meaning). Moreover, our
results allow the generator to have a mixture of quadratic and linear growth
components. Note that in the literature, the results for QBSDEs do not cover
those for linear growth BSDEs ([24, 15]). Hence, this part of the results are
new even for BSDEs.

(ii) For the case $z\mapsto g(t,s,y,z)$ being super-linear but sub-quadratic,
we will establish the well-posedness with more general free terms than the
quadratic BSVIEs. Some componentwise extended convexity/concavity of $z\mapsto
g(t,s,y,z)$ will be assumed. This part of results are also new for BSDEs
because only one-dimensional sub-quadratic BSDEs were studied in the
literature ([13]).

(iii) For the case $z\mapsto g(t,s,y,z)$ being super-quadratic, we will
establish some negative results on the well-posedness, either no existence, or
no uniqueness. In some sense, these results are extensions of similar ones for
BSDEs ([8]).

Now, let us present another important motivation of the current paper.

A given random variable $\xi\in L^{1}_{{\cal F}_{T}}(\Omega;\mathbb{R})$ can
represent the return of some asset position at some future time $T$. A
standard dynamic entropic risk measure of $\xi$ with a risk aversion parameter
$\gamma>0$ is defined by

(1.3)
$\rho^{1}_{t}(\xi)={1\over\gamma}\log\mathbb{E}_{t}\big{[}e^{-\gamma\xi}\big{]},\
\ t\in[0,T],$

where $\mathbb{E}_{t}[\,\cdot\,]=\mathbb{E}[\,\cdot\,|\,{\cal F}_{t}]$ ([12]).
It is known ([18, 17]) that the above is a convex risk measure (not coherent),
on the set

$\mathscr{E}_{\gamma}^{1}=\Big{\\{}\xi\in L^{1}_{{\cal
F}_{T}}(\Omega;\mathbb{R})\bigm{|}\sup_{t\in[0,T]}\mathbb{E}_{t}\big{[}e^{-\gamma\xi}\big{]}<\infty\Big{\\}}.$

Clearly, $L^{\infty}_{{\cal F}_{T}}(\Omega;\mathbb{R})$ is a proper subset of
$\mathscr{E}_{\gamma}^{1}$, as the latter also contains some unbounded random
variables, besides bounded ones. Let us consider the following BSDE (of
integral form)

(1.4) $Y(t)=-\xi+\int_{t}^{T}{\gamma\over
2}|Z(s)|^{2}ds-\int_{t}^{T}Z(s)dW(s),\ \ t\in[0,T].$

Suppose $(Y(\cdot),Z(\cdot))$ is the adapted solution of this BSDE. Then, a
straightforward calculation shows that

$Y(t)={1\over\gamma}\log\mathbb{E}_{t}\big{[}e^{-\gamma\xi}\big{]}=\rho_{t}^{1}(\xi).$

This means that the backward term $Y(\cdot)$ of the adapted solution
$(Y(\cdot),Z(\cdot))$ to the BSDE (1.4) can be exactly taken as a dynamic
entropic risk measure for $\xi$, as long as $\xi\in\mathscr{E}^{1}_{\gamma}$.

Next, we want to extend the above dynamic entropic risk measure to a larger
set than $\mathscr{E}_{\gamma}^{1}$. To this end, for any $\alpha>0$, we
define

$\xi^{\\{\alpha\\}}=(\xi^{+})^{\alpha}-(\xi^{-})^{\alpha}\equiv\left\\{\begin{array}[]{ll}\vskip
3.0pt plus 1.0pt minus 1.0pt\displaystyle\xi^{\alpha},\qquad\xi\geqslant 0,\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\displaystyle-|\xi|^{\alpha},\ \
\xi<0.\end{array}\right.$

This is the odd extension of $\xi^{\alpha}$, $\xi\geqslant 0$ to $\mathbb{R}$,
symmetric with respect to the origin, which is well-defined on $\mathbb{R}$
and is strictly increasing. Now, for $\alpha,\gamma>0$, we introduce

$\mathscr{E}_{\gamma}^{\alpha}=\Big{\\{}\xi\in L^{1}_{{\cal
F}_{T}}(\Omega;\mathbb{R})\bigm{|}\sup_{t\in[0,T]}\mathbb{E}_{t}\big{[}e^{-\gamma\xi^{\\{\alpha\\}}}\big{]}<\infty\Big{\\}}.$

Since

$\mathbb{E}_{t}\big{[}e^{-\gamma\xi^{\\{\alpha\\}}}\big{]}=\mathbb{E}_{t}\big{[}e^{-\gamma\xi^{\alpha}}I_{(\xi\geqslant
0)}\big{]}+\mathbb{E}_{t}\big{[}e^{\gamma|\xi|^{\alpha}}I_{(\xi<0)}\big{]},\quad
0\leqslant\mathbb{E}_{t}\big{[}e^{-\gamma\xi^{\alpha}}I_{(\xi\geqslant
0)}\big{]}\leqslant 1,$

we see that

$\mathscr{E}^{\alpha}_{\gamma}=\Big{\\{}\xi\in L^{1}_{{\cal
F}_{T}}(\Omega;\mathbb{R})\bigm{|}\sup_{t\in[0,T]}\mathbb{E}_{t}\big{[}e^{\gamma|\xi|^{\alpha}}I_{(\xi<0)}\big{]}<\infty\Big{\\}}.$

Clearly, both $\alpha\mapsto\mathscr{E}^{\alpha}_{\gamma}$ and
$\gamma\mapsto\mathscr{E}^{\alpha}_{\gamma}$ are (strictly) decreasing, in
particular,

$\mathscr{E}^{1}_{\gamma}\varsubsetneq\mathscr{E}^{\alpha}_{\gamma},\ \
\alpha\in(0,1).$

Now, for the return
$\xi\in\mathscr{E}^{\alpha}_{\gamma}\setminus\mathscr{E}^{1}_{\gamma}$,
inspired by (1.3), we may try to define an extended dynamic entropic risk
measure of such a $\xi$ by the following:

(1.5)
$\rho_{t}^{\alpha}(\xi)=\Big{(}{1\over\gamma}\log\mathbb{E}_{t}\big{[}e^{-\gamma\xi^{\\{\alpha\\}}}\big{]}\Big{)}^{\\{{1\over\alpha}\\}},\
\ t\in[0,T].$

However, the above only satisfies the monotonicity, and it does not even have
the translation invariance. Hence, this is not a good dynamic risk measure. To
have a reasonable entropic type dynamic risk measure for
$\xi\in\mathscr{E}_{\gamma}^{\alpha}$, suggested by the case of
$\mathscr{E}_{\gamma}^{1}$, we consider the following BSDE:

(1.6) $Y(t)=-\xi+\int_{t}^{T}{\gamma\over\kappa}|Z(s)|^{2\over
2-\alpha}ds+\int_{t}^{T}Z(s)dW(s),\ \ t\in[0,T],$

for some $\kappa>1$, whose generator has a sub-quadratic growth in $Z$ (since
${2\over 2-\alpha}\in(1,2)$). According to Fan–Hu [13], for any
$\xi\in\mathscr{E}^{\alpha}_{\gamma}$, the above BSDE admits a unique adapted
solution $(Y(\cdot),Z(\cdot))$ satisfying

(1.7) $e^{\frac{\gamma}{\kappa}|Y(t)|^{\alpha}}I_{(Y(t)>0)}\leqslant
K\mathbb{E}_{t}\Big{[}e^{\gamma|\xi|^{\alpha}}I_{(\xi<0)}\Big{]}$

for some $K>1$. In the current case, the map $\xi\mapsto Y(t)\equiv Y(t;\xi)$
satisfies the monotonicity, translation invariance, and convex. Hence, we may
take such a map as an extended entropic dynamic risk measure on the set
$\mathscr{E}_{\gamma}^{\alpha}$. This leads to the study of BSDEs with the
generators being sub-quadratic in $Z$.

Next, according to Yong [47], Wang–Sun–Yong [37], in the one-dimensional case,
if instead of the return $\xi$ at the terminal time $T$, one has an ${\cal
F}_{T}$-measurable position process $\psi(\cdot)$ (not necessarily
$\mathbb{F}$-adapted), to dynamically measure the risk, it will be more proper
to consider the following BSVIE:

(1.8) $Y(t)=-\psi(t)+\int_{t}^{T}g(t,s,Z(t,s))ds-\int_{t}^{T}Z(t,s)dW(s),\ \
t\in[0,T],$

for a proper generator $g$, where $z\mapsto g(t,s,z)$ could be of linear,
super-linear (which includes sub-quadratic, quadratic, or even super-
quadratic). If $(Y(\cdot),Z(\cdot\,,\cdot))$ is an adapted solution of BSVIE
(1.8), we can call $Y(\cdot)$ an equilibrium dynamical (entropic) risk measure
of $\psi(\cdot)$. In the case that $\psi(\cdot)$ is bounded, and $z\mapsto
g(t,s,z)$ is quadratic, (the one-dimensional BSVIE) (1.8) admits a unique
adapted solution $(Y(\cdot),Z(\cdot\,,\cdot))$ so that we have an equilibrium
entropic dynamical risk measure for $\psi(\cdot)$ ([37]).

It is common that there are more than one position processes that have to be
measured their dynamic risks individually. For example, for a (commercial
and/or investment) bank, there might be some loan positions which have quite
different features. They are closely related (coupled) and the dynamical risks
should be measured separately. Therefore, we naturally have the vector-valued
(equilibrium) dynamical risk measure process which satisfies a multi-
dimensional BSDEs or BSVIEs. This consideration gives us a motivation of
studying multi-dimensional BSVIEs with the generator $g$ having various types
of growth in $z$. From this, people might see the application potential
contained in the main results of the current paper.

The rest of the paper is organized as follows. Some preliminary is presented
in Section 2, including the revisit of relevant BSDE results and some crucial
lemmas to be used later. Section 3 is devoted to the case of bounded free
terms for diagonally strictly quadratic case and the mixed linear and
quadratic case. In Section 4, we discuss BSVIEs with unbounded free for the
generators having no more than quadratic growth. Some negative results are
presented in Section 5 when the generator is super-quadratic in $Z$. Finally,
some concluding remarks are collected in Section 6.

##  2 Preliminaries

###  2.1 Some spaces

Let us first introduce some basic spaces. For each $t\in[0,T]$ and any
Euclidean space $\mathbb{H}$ with the norm $|\cdot|$ which could be
$\mathbb{R}^{n},\mathbb{R}^{n\times d}$, etc., we define the following basic
space:

$L^{p}_{{\cal
F}_{t}}(\Omega;\mathbb{H})=\Big{\\{}\xi:\Omega\to\mathbb{H}\bigm{|}\xi\hbox{
is ${\cal F}_{t}$-measurable,
}\|\xi\|_{p}\equiv\big{(}\mathbb{E}|\xi|^{p}\big{)}^{1\over
p}<\infty\Big{\\}},\quad p\in[1,\infty).$

In an obvious way, we can define $L^{\infty}_{{\cal
F}_{t}}(\Omega;\mathbb{H})$. Clearly, $L^{p}_{{\cal
F}_{t}}(\Omega;\mathbb{H})$ is a Banach space under the norm
$\|\,\cdot\,\|_{p}$, for all $p\in[1,\infty]$. When the range space
$\mathbb{H}$ is clear from the context and is not necessarily to be
emphasized, we will omit $\mathbb{H}$. In particular, we will denote
$L^{p}_{{\cal F}_{T}}(\Omega)=L^{p}_{{\cal F}_{T}}(\Omega;\mathbb{H})$.

Next, we introduce spaces of stochastic processes. In order to avoid
repetition, all processes $(t,\omega)\mapsto\varphi(t,\omega)$ are assumed to
be at least ${\cal B}[0,T]\otimes{\cal F}_{T}$-measurable without further
mentioning, where ${\cal B}[0,T]$ is the Borel $\sigma$-field of $[0,T]$. For
$p,q\in[1,\infty)$,

$\begin{array}[]{ll}\displaystyle L^{p}_{{\cal
F}_{T}}(\Omega;L^{q}(0,T;\mathbb{H}))\negthinspace=\negthinspace\Big{\\{}\varphi:[0,T]\negthinspace\times\negthinspace\Omega\to\mathbb{H}\bigm{|}\mathbb{E}\Big{(}\int_{0}^{T}\negthinspace\negthinspace\negthinspace|\varphi(t)|^{q}dt\Big{)}^{p\over
q}\negthinspace\negthinspace\negthinspace<\infty\Big{\\}},\\\ \vskip 3.0pt
plus 1.0pt minus 1.0pt\cr\displaystyle L^{p}_{{\cal
F}_{T}}(\Omega;L^{\infty}(0,\negthinspace
T;\mathbb{H}))\negthinspace=\negthinspace\Big{\\{}\varphi:[0,T]\negthinspace\times\negthinspace\Omega\to\mathbb{H}\bigm{|}\mathbb{E}\Big{(}\mathop{\rm
esssup}_{t\in[0,T]}|\varphi(t)|^{p}\Big{)}\negthinspace<\negthinspace\infty\Big{\\}}\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle L^{p}_{{\cal
F}_{T}}(\Omega;C([0,\negthinspace
T];\mathbb{H}))\negthinspace=\negthinspace\Big{\\{}\varphi:[0,T]\negthinspace\times\negthinspace\Omega\to\mathbb{H}\bigm{|}t\mapsto\varphi(t,\omega)\hbox{
is continuous,
}\mathbb{E}\Big{(}\sup_{t\in[0,T]}|\varphi(t)|^{p}\Big{)}\negthinspace<\negthinspace\infty\Big{\\}},\\\
\displaystyle L^{q}_{{\cal
F}_{T}}(0,T;L^{p}(\Omega;\mathbb{H}))\negthinspace=\negthinspace\Big{\\{}\varphi:[0,T]\negthinspace\times\negthinspace\Omega\to\mathbb{H}\bigm{|}\int_{0}^{T}\Big{(}\mathbb{E}|\varphi(t)|^{p}\Big{)}^{q\over
p}dt<\infty\Big{\\}},\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle
L^{\infty}_{{\cal
F}_{T}}(0,T;L^{p}(\Omega;\mathbb{H}))\negthinspace=\negthinspace\Big{\\{}\varphi:[0,T]\negthinspace\times\negthinspace\Omega\to\mathbb{H}\bigm{|}\mathop{\rm
esssup}_{t\in[0,T]}\Big{(}\mathbb{E}|\varphi(t)|^{p}\Big{)}^{1\over
p}<\negthinspace\infty\Big{\\}},\\\ \displaystyle C_{{\cal
F}_{T}}([0,T];L^{p}(\Omega;\mathbb{H}))\negthinspace=\negthinspace\Big{\\{}\varphi:[0,T]\negthinspace\times\negthinspace\Omega\to\mathbb{H}\bigm{|}t\mapsto\varphi(t,\omega)\hbox{
is continuous,
}\sup_{t\in[0,T]}\Big{(}\mathbb{E}|\varphi(t)|^{p}\Big{)}^{1\over
p}<\negthinspace\infty\Big{\\}}.\end{array}$

The spaces $L^{\infty}_{{\cal F}_{T}}(\Omega;L^{\infty}(0,T;\mathbb{H}))$,
$L^{\infty}_{{\cal F}_{T}}(0,T;L^{\infty}(\Omega;\mathbb{H}))$,
$L^{\infty}_{{\cal F}_{T}}(\Omega;C([0,T];\mathbb{H}))$, and $C_{{\cal
F}_{T}}([0,T];$ $L^{\infty}(\Omega;\mathbb{H}))$ can be defined in obvious
ways. For all $p\in[1,\infty)$ , we denote

(2.1) $L^{p}_{{\cal F}_{T}}(0,T;\mathbb{H})\equiv L^{p}_{{\cal
F}_{T}}(0,T;L^{p}(\Omega;\mathbb{H}))=L^{p}_{{\cal
F}_{T}}(\Omega;L^{p}(0,T;\mathbb{H}))=L^{p}((0,T)\times\Omega;\mathbb{H}).$

We point out that when $p=\infty$, the last two equalities in the above might
not be true without certain type of measurability (see Example 5.0.10 in [16]
and Example 1.42 in [31]). Therefore, we do not mention the above with
$p=\infty$ to avoid some technicalities. Note that processes in the above
spaces are not necessarily $\mathbb{F}$-adapted. The subset of, say,
$L^{p}_{{\cal F}_{T}}(\Omega;L^{q}(0,T;\mathbb{H}))$, consisting of all
$\mathbb{F}$-progressively measurable processes, is denoted by
$L^{p}_{\mathbb{F}}(\Omega;$ $L^{q}(0,T;\mathbb{H}))$ (having subscript
$\mathbb{F}$ instead of ${\cal F}_{T}$). All the $\mathbb{F}$-progressively
measurable version of the other spaces listed in the above can be denoted
similarly; for example, $C_{\mathbb{F}}([0,T];L^{p}(\Omega;\mathbb{H}))$, and
so on. Finally, in the above, $[0,T]$ can be replaced by any $[t,T]$, and
again, when the range space $\mathbb{H}$ is clear from the context, we will
omit $\mathbb{H}$; for example, $L^{p}_{\mathbb{F}}(\Omega;C[t,T])$, and so
on.

For each $t\in[0,T]$, $\mu,\theta>0$, $q\geqslant 1$, we also introduce the
following spaces of processes:

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus 1.0pt\displaystyle{\cal
E}_{{\cal F}_{t}}^{\mu,\theta}(\Omega;\mathbb{R}^{n})=\Big{\\{}\xi\in
L^{1}_{{\cal
F}_{t}}(\Omega;\mathbb{R}^{n})\bigm{|}\mathbb{E}\big{[}\exp\big{(}\mu|\xi|^{\theta}\big{)}\big{]}<\infty\Big{\\}},\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\displaystyle{\cal E}^{\mu,\theta}_{{\cal
F}_{T}}(\Omega;L^{q}(t,T;\mathbb{R}^{n}))=\Big{\\{}\varphi(\cdot)\in
L^{q}_{{\cal F}_{T}}(t,T;\mathbb{R}^{n})\bigm{|}\varphi(\cdot)\hbox{ is ${\cal
B}[t,T]\otimes{\cal F}_{T}$-measurable},\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\Big{(}\int_{t}^{T}|\varphi(s)|^{q}ds\Big{)}^{1\over
q}\in{\cal E}^{\mu,\theta}_{{\cal F}_{T}}(\Omega;\mathbb{R})\Big{\\}},\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\displaystyle{\cal
E}_{\mathbb{F}}^{\mu,\theta}(\Omega;L^{q}(t,T;\mathbb{R}^{n}))=\Big{\\{}\varphi(\cdot)\in{\cal
E}^{\mu,\theta}_{{\cal
F}_{T}}(\Omega;L^{q}(t,T;\mathbb{R}^{n}))\bigm{|}\varphi(\cdot)\hbox{ is
$\mathbb{F}$-progressively measurable}\Big{\\}}.\end{array}$

The spaces ${\cal E}_{{\cal
F}_{T}}^{\mu,\theta}(\Omega;L^{\infty}(t,T;\mathbb{R}^{n}))$ and ${\cal
E}_{\mathbb{F}}^{\mu,\theta}(\Omega;L^{\infty}(t,T;\mathbb{R}^{n}))$ can be
defined in an obvious way. Further,

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus 1.0pt\displaystyle{\cal
E}^{\mu,\theta}_{{\cal
F}_{T}}(\Omega;C([t,T];\mathbb{R}^{n}))=\Big{\\{}\varphi(\cdot)\in{\cal
E}^{\mu,\theta}_{{\cal
F}_{T}}(\Omega;L^{\infty}(t,T;\mathbb{R}^{n}))\bigm{|}\varphi(\cdot)\hbox{ is
continuous on $[t,T]$}\Big{\\}},\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle{\cal
E}_{\mathbb{F}}^{\mu,\theta}(\Omega;C([t,T];\mathbb{R}^{n}))=\Big{\\{}\varphi(\cdot)\in{\cal
E}^{\mu,\theta}_{{\cal
F}_{T}}(\Omega;C([t,T];\mathbb{R}^{n}))\bigm{|}\varphi(\cdot)\hbox{ is
$\mathbb{F}$-progressively measurable}\Big{\\}}.\end{array}$

Note that for each $t\in[0,T]$,

(2.2) $L^{\infty}_{{\cal F}_{t}}(\Omega;\mathbb{R}^{n})\varsubsetneq{\cal
E}^{\mu,\theta}_{{\cal
F}_{t}}(\Omega;\mathbb{R}^{n})\varsubsetneq\bigcap_{p\geqslant 1}L^{p}_{{\cal
F}_{t}}(\Omega;\mathbb{R}^{n}),\quad\forall\mu,\theta>0,$

and ${\cal E}^{\mu,\theta}_{{\cal F}_{t}}(\Omega;\mathbb{R}^{n})$ is
decreasing in $\mu$ and $\theta$, respectively. Similar inclusions and
monotonicity also hold for the other spaces defined above with parameters
$\mu$ and $\theta$.

Next, for each $t\in[0,T)$, denote

$\Delta^{*}[t,T]=\big{\\{}(r,s)\in[t,T]^{2}\bigm{|}0\leqslant t\leqslant
r<s\leqslant T\big{\\}}.$

For $p,q\in[1,+\infty)$, we introduce the following space for
$Z(\cdot\,,\cdot)$:

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus 1.0pt\displaystyle
L^{p}_{\mathbb{F}}(\Omega;L^{q}(\Delta^{*}[t,T];\mathbb{R}^{n}))=\Big{\\{}\zeta:\Delta^{*}[t,T]\times\Omega\to\mathbb{R}^{n}\bigm{|}\forall
r\in[t,T],\,\zeta(r,\cdot)\in
L^{p}_{\mathbb{F}}(\Omega;L^{q}(r,T;\mathbb{R}^{n})),\\\ \vskip 3.0pt plus
1.0pt minus
1.0pt\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\mathbb{E}\int_{t}^{T}\Big{(}\int_{r}^{T}|\zeta(r,s)|^{q}ds\Big{)}^{p\over
q}dr<\infty\Big{\\}}.\end{array}$

Also, we let

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus 1.0pt\displaystyle
L^{\infty}(0,T;L^{p}_{\mathbb{F}}(\Omega;L^{q}(\cdot\,,T;\mathbb{R}^{n})))=\Big{\\{}\varphi:\Delta^{*}[0,T]\times\Omega\to\mathbb{R}^{n}\bigm{|}\forall
t\in[0,T],\,\varphi(t,\cdot)\in
L^{p}_{\mathbb{F}}(\Omega;L^{q}(t,T;\mathbb{R}^{n})),\\\ \vskip 3.0pt plus
1.0pt minus
1.0pt\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\mathop{\rm
esssup}_{t\in[0,T]}\mathbb{E}\Big{(}\int_{t}^{T}|\varphi(t,s)|^{q}ds\Big{)}^{p\over
q}<\infty\Big{\\}}.\end{array}$

The case that $p$ and/or $q$ is equal to $\infty$ can be defined in an obvious
way. Further, for each $t\in[0,T]$, denote by $\mathscr{T}[t,T]$ the set of
all $\mathbb{F}$-stopping times $\tau$ valued in $[t,T]$. A uniformly
integrable $\mathbb{F}$-martingale $M=\\{M(t):0\leqslant t\leqslant T\\}$ with
$M_{0}=0$ is called a BMO martingale on $[0,T]$ if

$\|M(\cdot)\|^{2}_{{\rm
BMO}[0,T]}\triangleq\sup_{\tau\in\mathscr{T}[0,T]}\Big{\|}\mathbb{E}_{\tau}\big{[}|M(T)-M(\tau)|^{2}\big{]}\Big{\|}_{\infty}<\infty.$

It is known (see [25]) that for any given $K>0$, there are constants
$c_{1},c_{2}>0$ depending only on $K$ such that for any BMO martingale
$N(\cdot)$ with

$\big{\|}N(\cdot)\big{\|}_{{\rm BMO}_{\mathbb{P}}[0,T]}\leqslant K,$

by setting

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle
d\widetilde{\mathbb{P}}:=\exp\Big{\\{}\int_{0}^{T}N(t)dW(t)-\frac{1}{2}\int_{0}^{T}|N(t)|^{2}dt\Big{\\}}d\mathbb{P},\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\widetilde{M}(\cdot):=M(\cdot)-\langle
M,N\rangle(\cdot),\end{array}$

one has that $\widetilde{M}(\cdot)$ is a BMO martingale under
$\widetilde{\mathbb{P}}$, and

(2.3) $c_{1}\big{\|}M(\cdot)\big{\|}_{{\rm
BMO}_{\mathbb{P}}[0,T]}\leqslant\big{\|}\widetilde{M}(\cdot)\big{\|}_{{\rm
BMO}_{\widetilde{\mathbb{P}}}[0,T]}\leqslant
c_{2}\big{\|}M(\cdot)\big{\|}_{{\rm BMO}_{\mathbb{P}}[0,T]}.$

This means that the BMO norm of the BMO martingale $M(\cdot)$ under
$\mathbb{P}$ is equivalent to that of BMO martingale $\widetilde{M}(\cdot)$
under $\widetilde{\mathbb{P}}$. We now introduce

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\overline{{\rm
BMO}}([t,T];\mathbb{R}^{n})\triangleq\Big{\\{}\varphi:[t,T]\times\Omega\to\mathbb{R}^{n}\bigm{|}\varphi(\cdot)\hbox{
is $\mathbb{F}$--progressively measurable and}\\\ \vskip 3.0pt plus 1.0pt
minus
1.0pt\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\big{\|}\varphi(\cdot)\big{\|}^{2}_{\overline{{\rm
BMO}}[t,T]}=\sup_{\tau\in\mathscr{T}[t,T]}\Big{\|}\mathbb{E}_{\tau}\Big{[}\int_{\tau}^{T}|\varphi(s)|^{2}ds\Big{]}\Big{\|}_{\infty}<\infty\Big{\\}},\end{array}$

and

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\overline{{\rm
BMO}}(\Delta^{*}[0,T];\mathbb{R}^{n})\triangleq\Big{\\{}\varphi:\Delta^{*}[0,T]\times\Omega\to\mathbb{R}^{n}\bigm{|}\varphi(t,\cdot)\hbox{
is $\mathbb{F}$--progressively measurable}\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\hbox{on
$[t,T]$, $\hbox{\rm a.s.}~{}t\in[0,T]$, and}\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\big{\|}\varphi(\cdot\,,\cdot)\big{\|}^{2}_{\overline{{\rm
BMO}}(\Delta^{*}[0,T])}\negthinspace\negthinspace=\mathop{\rm
esssup}_{t\in[0,T]}\big{\|}\varphi(t,\cdot)\big{\|}^{2}_{\overline{{\rm
BMO}}[t,T]}\negthinspace<\negthinspace\infty\Big{\\}}.\end{array}$

Clearly,

(2.4) $\overline{\rm BMO}([t,T];\mathbb{R}^{n})\supseteq
L^{\infty}_{\mathbb{F}}(\Omega;L^{2}(t,T;\mathbb{R}^{n})).$

On the other hand, for good enough process $\varphi(\cdot\,,\cdot)$, we have

(2.5) $\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\|\varphi\|_{L^{\infty}(0,T;L^{2}_{\mathbb{F}}(\Omega;L^{2}(\cdot\,,T;\mathbb{R}^{n})))}^{2}=\mathop{\rm
esssup}_{t\in[0,T]}\mathbb{E}\int_{t}^{T}|\varphi(t,s)|^{2}ds=\mathop{\rm
esssup}_{t\in[0,T]}\mathbb{E}\Big{[}\mathbb{E}_{t}\Big{(}\int_{t}^{T}|\varphi(t,s)|^{2}ds\Big{)}\Big{]}\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle\leqslant\mathop{\rm
esssup}_{t\in[0,T]}\Big{\|}\mathbb{E}_{t}\Big{[}\int_{t}^{T}|\varphi(t,s)|^{2}ds\Big{]}\Big{\|}_{\infty}\leqslant\mathop{\rm
esssup}_{t\in[0,T]}\sup_{\tau\in\mathscr{T}[t,T]}\Big{\|}\mathbb{E}_{\tau}\Big{[}\int_{\tau}^{T}|\varphi(t,s)|^{2}ds\Big{]}\Big{\|}_{\infty}\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle=\big{\|}\varphi(\cdot\,,\cdot)\big{\|}^{2}_{\overline{{\rm
BMO}}(\Delta^{*}[0,T])}.\end{array}$

Hence,

(2.6) $\overline{\rm BMO}(\Delta^{*}[0,T];\mathbb{R}^{n})\subseteq
L^{\infty}(0,T;L^{2}_{\mathbb{F}}(\Omega;L^{2}(\cdot\,,T;\mathbb{R}^{n}))).$

At the moment, it is not clear if the equalities in (2.4) and (2.6) hold. They
will again be related to the question we faced for (2.1) with $p=\infty$.
Fortunately it is irrelevant whether there are such equalities.

In what follows, for simplicity of the presentation, we will misuse the same
notation for different range spaces which can be easily identified from the
contexts.

######  Definition 2.1.

A pair $(Y(\cdot),Z(\cdot\,,\cdot))\in
L^{1}_{\mathbb{F}}(0,T;\mathbb{R}^{n})\times
L^{2}_{\mathbb{F}}(\Omega;L^{2}(\Delta^{*}[0,T];\mathbb{R}^{n}))$ is called an
adapted solution of BSVIE (1.1) if the equation is satisfied in the usual
Itô’s sense.

Note that although in the above definition, we only require
$(Y(\cdot),Z(\cdot\,,\cdot))$ to be in a very large space, in our main
results, under proper conditions, our adapted solutions will have much better
regularity/integrability.

###  2.2 Results for BSDEs revisited

In this subsection, we recall some relevant results for BSDEs. Consider the
following BSDE of integral form (which could be one-dimensional or multi-
dimensional):

(2.7) $Y(t)=\xi+\int_{t}^{T}f(s,Y(s),Z(s))ds-\int_{t}^{T}Z(s)dW(s),\ \
t\in[0,T].$

We look at the following classes of problems.

(i) Quadratic BSDEs with bounded terminal terms.

The main feature of such a BSDE is that $z\mapsto f(t,y,z)$ has a quadratic
growth. Because of this, such an equation is called a quadratic BSDE (QBSDE,
for short). We introduce the following hypothesis.

(H2.1) The map
$f:[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\Omega\to\mathbb{R}^{n}$
is measurable such that $t\mapsto f(t,y,z)$ is $\mathbb{F}$-progressively
measurable, for all $(y,z)\in\mathbb{R}^{n}\times\mathbb{R}^{n}$. There exists
a non-negative valued process $\alpha(\cdot)\in
L^{\infty}_{\mathbb{F}}(\Omega;L^{1}(0,T;\mathbb{R}))$, a continuous, monotone
increasing function $\phi:[0,\infty)\to[0,\infty)$ with $\phi(0)=0$, and
constants $\beta,\lambda\geqslant 0$, $\gamma\geqslant\bar{\gamma}>0$,
$\delta\in(0,1)$ such that

(2.8) $\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\hbox{\rm
sgn$\,$}(y^{i})f^{i}(t,y,z)\leqslant\alpha(t)\negthinspace+\negthinspace\beta|y|\negthinspace+\negthinspace\lambda\sum_{j\neq
i}|z^{j}|^{1+\delta}\negthinspace+\negthinspace{\gamma\over 2}|z^{i}|^{2},\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall(t,y,z)\in[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n},\quad
1\leqslant i\leqslant n,\end{array}$

and either

(2.9) $\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle-\alpha(t)\negthinspace-\negthinspace\beta|y|\negthinspace-\negthinspace\lambda\sum_{j\neq
i}|z^{j}|^{1+\delta}\negthinspace+\negthinspace{\bar{\gamma}\over
2}|z^{i}|^{2}\leqslant f^{i}(t,y,z),\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall(t,y,z)\in[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n},\quad
1\leqslant i\leqslant n,\end{array}$

or

(2.10) $\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\alpha(t)\negthinspace+\negthinspace\beta|y|\negthinspace+\negthinspace\lambda\sum_{j\neq
i}|z^{j}|^{1+\delta}\negthinspace-\negthinspace{\bar{\gamma}\over
2}|z^{i}|^{2}\geqslant f^{i}(t,y,z),\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall(t,y,z)\in[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n},\quad
1\leqslant i\leqslant n.\end{array}$

Further, it holds that

(2.11) $\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle|f^{i}(t,y,z)-f^{i}(t,\bar{y},\bar{z})|\\\ \vskip 3.0pt plus
1.0pt minus
1.0pt\displaystyle\leqslant\phi(|y|\vee|\bar{y}|)\Big{[}\big{(}1+|z|+|\bar{z}|\big{)}\big{(}|y-\bar{y}|+|z^{i}-\bar{z}^{i}|\big{)}+\big{(}1+|z|^{\delta}+|\bar{z}|^{\delta}\big{)}\sum_{j\neq
i}|z^{j}-\bar{z}^{j}|\Big{]},\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall
t\in[0,T],~{}y,\bar{y},z,\bar{z}\in\mathbb{R}^{n},~{}i=1,2,\cdots,n.\end{array}$

Note that in the whole paper, we use the notation $\hbox{\rm
sgn$\,$}(x):=I_{(x>0)}-I_{(x\leqslant 0)}$. Condition (2.9)–(2.10) are
referred to as the diagonally strictly quadratic condition for the generator
$f$ in $z$. This condition excludes the case that $z^{i}\mapsto f^{i}(t,y,z)$
is of linear growth. The following is Theorem 2.5 of Fan–Hu–Tang [15] which
gives the existence and uniqueness of adapted solutions to QBSDEs with bounded
terminal terms.

######  Proposition 2.2.

Let (H2.1) hold. Then for any $\xi\in L^{\infty}_{{\cal
F}_{T}}(\Omega;\mathbb{R}^{n})$, BSDE (2.7) admits a unique adapted solution
$(Y(\cdot),Z(\cdot))\in
L^{\infty}_{\mathbb{F}}(\Omega;C([0,T];\mathbb{R}^{n}))\times\overline{\rm
BMO}([0,T];\mathbb{R}^{n})$.

It is very natural to ask what happens if the terminal state is unbounded for
QBSDEs. This leads to the second situation as follows.

(ii) QBSDEs with unbounded terminal terms.

For such a situation, to get a positive result, the following hypothesis has
been assumed in the literature.

(H2.2) Let the measurability condition in (H2.1) hold for
$f:[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\Omega\to\mathbb{R}^{n}$.
For each $i=1,2,\cdots,n$, $f^{i}$ only depends on $(t,y,z^{i},\omega)$
(independent of $z^{j}$, $j\neq i$) and $z^{i}\mapsto f^{i}(t,y,z^{i})$ is
convex for all $(t,y)\in[0,T]\times\mathbb{R}^{n}$. There exists some
$\alpha(\cdot)\in\cap_{\mu\geqslant 1}{\cal
E}^{\mu,1}_{\mathbb{F}}(\Omega;L^{1}(0,T;\mathbb{R}))$ and constants
$\beta,\gamma>0$, such that

(2.12) $|f^{i}(t,y,z^{i})|\leqslant\alpha(t)+\beta|y|+{\gamma\over
2}|z^{i}|^{2},\quad\forall(t,y,z^{i})\in[0,T]\times\mathbb{R}^{n}\times\mathbb{R}$

and

(2.13) $|f(t,y,z)-f(t,\bar{y},z)|\leqslant\beta|y-\bar{y}|,\quad\forall
t\in[0,T],~{}y,\bar{y},z\in\mathbb{R}^{n}.\vspace{0.2cm}$

The following is Theorem 2.9 of Fan–Hu–Tang [15].

######  Proposition 2.3.

Let (H2.2) hold. Then, for any $\xi\in\bigcap_{\mu\geqslant 1}{\cal E}_{{\cal
F}_{T}}^{\mu,1}(\Omega;\mathbb{R}^{n})$, BSDE (2.7) admits a unique adapted
solution

$(Y(\cdot),Z(\cdot))\in\Big{[}\bigcap_{\mu\geqslant 1}{\cal
E}^{\mu,1}_{\mathbb{F}}(\Omega;C(0,T];\mathbb{R}^{n}))\Big{]}\times\Big{[}\bigcap_{p\geqslant
1}L^{p}_{\mathbb{F}}(\Omega;L^{2}(0,T;\mathbb{R}^{n}))\Big{]}.$

By (2.2), we see that the terminal term in the above could be unbounded.
However, the conditions assumed for the generator $f$ in (H2.2) is much
stronger than that assumed in (H2.1), especially $f^{i}$ has to be independent
of $z^{j}$, $j\neq i$, although the condition on $\alpha(\cdot)$ in (H2.2) is
a little weaker. Also, due to the assumption that $z\mapsto f(t,y,z)$ is
diagonally dependent, the diagonally strictly quadratic growth condition
(which is used to take care the sub-quadratic growth of off-diagonal
component) is not necessary.

Next, if $z\mapsto f(t,y,z)$ is growing superlinearly but subquadratically,
and diagonally dependent, then Proposition 2.3 should still be true. It is
expected that we might be able to relax the condition on the terminal term.
This will lead to the third situation.

(iii) Sub-quadratic BSDEs with unbounded terminal term.

For such a situation, there were only some results for one-dimensional BSDEs,
under the following hypothesis:

(H2.3) Let the measurability condition in (H2.1) hold for
$f:[0,T]\times\mathbb{R}\times\mathbb{R}\times\Omega\to\mathbb{R}$. There
exist some constants $\delta\in(0,1)$, $\beta\geqslant 0$, $\gamma>0$, and
$\alpha(\cdot)\in\bigcap_{\mu\geqslant 1}{\cal E}^{\mu,{2\delta\over
1+\delta}}_{\mathbb{F}}(\Omega;L^{1}(0,T;\mathbb{R}))$ such that

(2.14)
$|f(t,y,z)|\leqslant\alpha(t)+\beta|y|+\gamma|z|^{1+\delta},\quad\forall(t,y,z)\in[0,T]\times\mathbb{R}\times\mathbb{R}.$

Moreover, $(y,z)\mapsto f(t,y,z)$ is convex for all $t\in[0,T]$.

The following is from Theorem 3.9 in Fan-Hu [13].

######  Proposition 2.4.

Let (H2.3) hold. Then for any $\xi\in\bigcap_{\mu\geqslant 1}{\cal E}_{{\cal
F}_{T}}^{\mu,{2\delta\over 1+\delta}}(\Omega;\mathbb{R})$, BSDE (2.7) admits a
unique adapted solution

$(Y(\cdot),Z(\cdot))\in\Big{[}\bigcap_{\mu\geqslant 1}{\cal
E}^{\mu,{2\delta\over
1+\delta}}_{\mathbb{F}}(\Omega;C([0,T];\mathbb{R}))\Big{]}\times\Big{[}\bigcap_{p\geqslant
1}L^{p}_{\mathbb{F}}(\Omega;L^{2}(0,T;\mathbb{R}))\Big{]}.$

Since for $\delta\in(0,1)$, one has ${2\delta\over 1+\delta}<1$. Consequently,

${\cal E}^{\mu,1}_{\mathbb{F}}(\Omega;L^{1}(0,T;\mathbb{R}))\subseteq{\cal
E}_{\mathbb{F}}^{\mu,{2\delta\over 1+\delta}}(\Omega;L^{1}(0,T;\mathbb{R})).$

This implies that the condition on $\alpha(\cdot)$ assumed in (H2.3) is weaker
than that in (H2.2). Also,

${\cal E}^{\mu,1}_{{\cal F}_{T}}(\Omega;\mathbb{R})\subseteq{\cal E}_{{\cal
F}_{T}}^{\mu,{2\delta\over 1+\delta}}(\Omega;\mathbb{R}).$

Hence, the condition assumed for $\xi$ in Proposition 2.4 is weaker than that
in Proposition 2.3. The trade-off is that Proposition 2.4 is only for one-
dimensional BSDEs.

Furthermore, we note that for one-dimensional BSDEs with the dependence of the
generator on $Z$ being no more than quadratic, in Fan-Hu-Tang [14] and Fan-Hu
[13], the authors also proved a general existence and uniqueness result under
the following hypothesis:

(H2.4) Let the measurability condition in (H2.1) hold for
$f:[0,T]\times\mathbb{R}\times\mathbb{R}\times\Omega\to\mathbb{R}$. There
exist some constants $\beta\geqslant 0$, $\gamma>0$, $\delta\in(0,1]$, and
$\alpha(\cdot)\in\cap_{\mu\geqslant 1}{\cal E}^{\mu,{2\delta\over
1+\delta}}_{\mathbb{F}}(\Omega;L^{1}(0,T;\mathbb{R}))$ such that

(2.15)
$|f(t,y,z)|\leqslant\alpha(t)+\beta|y|+\gamma|z|^{1+\delta},\quad\forall(t,y,z)\in[0,T]\times\mathbb{R}\times\mathbb{R},$
(2.16)
$|f(t,y,z)-f(t,\bar{y},z)|\leqslant\beta|y-\bar{y}|,\quad\forall(t,y,\bar{y},z)\in[0,T]\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}$

and

(2.17) $\begin{array}[]{ll}\displaystyle
f(t,y,(1-\theta)z+\theta\bar{z})-\theta
f(t,y,\bar{z})\leqslant(1-\theta)\big{(}\alpha(t)+\beta|y|+\gamma|z|^{1+\delta}\big{)},\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\qquad\qquad\qquad\qquad\qquad\forall(t,y,z,\bar{z})\in[0,T]\times\mathbb{R}\times\mathbb{R}\times\mathbb{R},~{}\theta\in(0,1).\end{array}$

Moreover, $z\mapsto f(t,y,z)$ is continuous for all
$(t,y)\in[0,T]\times\mathbb{R}$.

Observe that if (2.16) and (2.17) hold, then for each
$(t,y_{1},y_{2},z_{1},z_{2})\in[0,T]\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}$
and each $\theta\in(0,1)$, we have

$\begin{array}[]{ll}\displaystyle g(t,y_{1},z_{1})-\theta
g(t,y_{2},z_{2})\leqslant|g(t,y_{1},z_{1})-g(t,y_{2},z_{1})|+g(t,y_{2},(1-\theta)\hat{z}_{\theta}+\theta
z_{2})-\theta g(t,y_{2},z_{2})\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\hskip
106.69783pt\leqslant\beta|y_{1}-y_{2}|+(1-\theta)(\alpha(t)+\beta|y_{2}|+\gamma|\hat{z}_{\theta}|^{1+\delta})\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle\hskip
106.69783pt\leqslant(1-\theta)(\beta|\hat{y}_{\theta}|+\beta|y_{2}|+\alpha(t)+\beta|y_{2}|+\gamma|\hat{z}_{\theta}|^{1+\delta})\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle\hskip
106.69783pt=(1-\theta)(\alpha(t)+2\beta|y_{2}|+\beta|\hat{y}_{\theta}|+\gamma|\hat{z}_{\theta}|^{1+\delta}),\end{array}$

where

$\hat{y}_{\theta}:=\frac{y_{1}-\theta y_{2}}{1-\theta}\qquad{\rm
and}\qquad\hat{z}_{\theta}:=\frac{z_{1}-\theta
z_{2}}{1-\theta}.\vspace{0.1cm}$

Then, by virtue of (i) of Theorem 5 in Fan-Hu-Tang [14] for the case
$\delta=1$ and Theorem 3.9 in Fan-Hu [13] for the case $\delta\in(0,1)$, the
following proposition follows immediately.

######  Proposition 2.5.

Let (H2.4) hold. Then for any $\xi\in\bigcap_{\mu\geqslant 1}{\cal E}_{{\cal
F}_{T}}^{\mu,{2\delta\over 1+\delta}}(\Omega;\mathbb{R})$, BSDE (2.7) admits a
unique adapted solution

$(Y(\cdot),Z(\cdot))\in\Big{[}\bigcap_{\mu\geqslant 1}{\cal
E}^{\mu,{2\delta\over
1+\delta}}_{\mathbb{F}}(\Omega;C([0,T];\mathbb{R}))\Big{]}\times\Big{[}\bigcap_{p\geqslant
1}L^{p}_{\mathbb{F}}(\Omega;L^{2}(0,T;\mathbb{R}))\Big{]}.\vspace{0.1cm}$

######  Remark 2.6.

If the generator $g$ satisfies assumption (H2.4) with (2.17) replaced with

(2.18) $\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle
f(t,y,(1-\theta)z+\theta\bar{z})-\theta
f(t,y,\bar{z})\geqslant-(1-\theta)\big{(}\alpha(t)+\beta|y|+\gamma|z|^{1+\delta}\big{)},\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\qquad\qquad\qquad\qquad\qquad\forall(t,y,z,\bar{z})\in[0,T]\times\mathbb{R}\times\mathbb{R}\times\mathbb{R},~{}\theta\in(0,1),\end{array}$

then it is easy to verify that $\bar{g}$ must satisfy assumption (H2.4), where

$\bar{g}(t,y,z):=-g(t,-y,-z),\quad\forall(t,y,z)\in[0,T]\times\mathbb{R}\times\mathbb{R}.$

Consequently, when (2.17) in (H2.4) is replaced with (2.18), the conclusion of
Proposition 2.5 still holds.

Furthermore, it is well known that a convex function with linear growth must
be a Lipschitz function. And, it is not hard to check that if $z\mapsto
f(t,y,z)$ is convex for all $(t,y)\in[0,T]\times\mathbb{R}$ and (2.15) holds,
then (2.17) is also true. Then, (H2.4) is weaker than (H2.3) and (H2.2) in the
one-dimensional setting. Hence, Proposition 2.5 generalizes Proposition 2.4
and Proposition 2.3 for the one-dimensional case.

In addition, in this paper we will show that (2.17) (also (2.18)) can imply
the continuity of $g$ with respect to $z$ in (H2.4), see Section 4 for
details.

In what follows, we will substantially extend the above results to BSVIEs, by
which we mean that some of the extensions are new even for BSDEs.

###  2.3  A crucial lemma

We now present a lemma which has played a crucial rule in proving the above
results, and it will play some important role below when we discuss the
situation of unbounded free terms.

######  Lemma 2.7.

Suppose that $\xi$ is an ${\cal F}_{T}$-measurable random variable and random
field
$f:[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\Omega\to\mathbb{R}^{n}$
is measurable such that for each $(y,z)\in\mathbb{R}^{n}\times\mathbb{R}^{n}$,
$t\mapsto f(t,y,z)$ is $\mathbb{F}$-progressively measurable. Let
$(Y(\cdot),Z(\cdot))$ be an adapted solution of BSDE (2.7) such that
$Y(\cdot)\in\bigcap_{\mu\geqslant 1}{\cal
E}^{\mu,\frac{2\delta}{1+\delta}}_{\mathbb{F}}(\Omega;C([0,T];\mathbb{R}^{n}))$
and

(2.19) $\hbox{\rm
sgn$\,$}\big{(}Y^{i}(s)\big{)}f^{i}(s,Y(s),Z(s))\leqslant\alpha_{0}(s)+\beta|Z^{i}(s)|^{1+\delta},\quad
s\in[0,T],~{}1\leqslant i\leqslant n,$

for some constants $\beta>0$, $\delta\in(0,1]$, and some map
$\alpha_{0}(\cdot)\in\bigcap_{\mu\geqslant 1}{\cal
E}^{\mu,\frac{2\delta}{1+\delta}}_{\mathbb{F}}(\Omega;L^{1}(0,T;\mathbb{R}_{+}))$.
Then there exists a constant $K>0$ depending only on $(\delta,\beta,T)$ such
that for each $i=1,\cdots,n$,

(2.20) $e^{|Y^{i}(t)|^{2\delta\over
1+\delta}}+\mathbb{E}_{t}\Big{[}\int_{t}^{T}|Z^{i}(s)|^{2}ds\Big{]}\leqslant
K\mathbb{E}_{t}\Big{[}e^{K\big{(}|\xi^{i}|+\int_{t}^{T}\alpha_{0}(\tau)d\tau\big{)}^{2\delta\over
1+\delta}}\Big{]},\ \ t\in[0,T],$

and for each $p>0$, there exists a constant $K_{p}>0$ depending only on
$(\delta,\beta,T)$ and $p$ such that for each $i=1,\cdots,n$,

(2.21)
$\displaystyle\mathbb{E}\Big{[}\sup_{s\in[t,T]}e^{p|Y^{i}(s)|^{2\delta\over
1+\delta}}\Big{]}+\mathbb{E}\Big{[}\Big{(}\int_{t}^{T}|Z^{i}(s)|^{2}ds\Big{)}^{p\over
2}\Big{]}\leqslant
K_{p}\mathbb{E}\Big{[}e^{K_{p}\big{(}|\xi^{i}|+\int_{t}^{T}\alpha_{0}(\tau)d\tau\big{)}^{2\delta\over
1+\delta}}\Big{]},\ \ t\in[0,T].$

Proof. Let $t\in[0,T)$ be fixed, $(Y(\cdot),Z(\cdot))$ be the adapted solution
of BSDE (2.7). Let

$\varphi(r,\rho)=e^{\mu(r)\rho^{\theta}},\qquad(r,\rho)\in[t,T]\times[\rho_{0},\infty),$

with $\mu(\cdot)\geqslant 1$, $\mu^{\prime}(r)>0$, $\rho_{0}>1$, and
$\theta\in(0,1]$, all being undetermined. Note that the smaller the $\theta$,
the smaller the $\varphi(r,\rho)$. We calculate

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\varphi_{r}(r,\rho)=\varphi(r,\rho)\mu^{\prime}(r)\rho^{\theta},\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\varphi_{\rho}(r,\rho)=\varphi(r,\rho)\theta\mu(r)\rho^{\theta-1},\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\varphi_{\rho\rho}(r,\rho)=\varphi(r,\rho)\Big{(}\theta^{2}\mu(r)^{2}\rho^{2\theta-2}+\theta(\theta-1)\mu(r)\rho^{\theta-2}\Big{)}\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\qquad\qquad=\varphi(r,\rho)\theta\mu(r)\rho^{\theta-2}\Big{(}\theta\mu(r)\rho^{\theta}-(1-\theta)\Big{)}\geqslant\
\varphi(r,\rho)\theta\mu(r)\rho^{\theta-2}(\theta\rho_{0}^{\theta}-1)>0,\end{array}$

provided $\rho_{0}>\theta^{-{1\over\theta}}$. For such a case,
$\rho\mapsto\varphi(r,\rho)$ is strictly convex on $[\rho_{0},\infty)$. Note
that $\displaystyle\lim_{\theta\downarrow 0}\theta^{-{1\over\theta}}=+\infty$.
Thus, in this approach, when $\theta$ is getting small, $\rho_{0}$ is getting
large. For $i=1,\cdots,n$, define

${\cal Y}^{i}(r)=|Y^{i}(r)|+k+\int_{t}^{r}\alpha_{0}(s)ds,\ \ r\in[t,T],$

with $k\geqslant\rho_{0}$, and let $L^{i}(\cdot)$ denote the local time of
$Y^{i}(\cdot)$ at $0$. In view of (2.19), applying Itô-Tanaka’s formula to
$s\mapsto\varphi(s,{\cal Y}^{i}(s))$, one has

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus 1.0pt\displaystyle
d\varphi\big{(}s,{\cal Y}^{i}(s)\big{)}=\Big{(}\varphi_{s}\big{(}s,{\cal
Y}^{i}(s)\big{)}+\varphi_{\rho}\big{(}s,{\cal
Y}^{i}(s)\big{)}\big{(}-\hbox{\rm
sgn$\,$}\big{(}Y^{i}(s)\big{)}f^{i}(s,Y(s),Z(s))+\alpha_{0}(s)\big{)}\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\displaystyle\qquad\qquad\qquad\ \ +{1\over
2}\varphi_{\rho\rho}\big{(}s,{\cal
Y}^{i}(s)\big{)}|Z^{i}(s)|^{2}\Big{)}ds+\varphi_{\rho}\big{(}s,{\cal
Y}^{i}(s)\big{)}dL^{i}(s)+\varphi_{\rho}\big{(}s,{\cal
Y}^{i}(s)\big{)}\hbox{\rm sgn$\,$}\big{(}Y^{i}(s)\big{)}Z^{i}(s)dW(s)\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\displaystyle\hskip
55.48277pt\geqslant\varphi\big{(}s,{\cal
Y}^{i}(s)\big{)}\Big{[}\mu^{\prime}(s){\cal
Y}^{i}(s)^{\theta}-\theta\mu(s){\cal
Y}^{i}(s)^{\theta-1}\beta|Z^{i}(s)|^{1+\delta}\\\ \vskip 3.0pt plus 1.0pt
minus 1.0pt\displaystyle\hskip 122.34692pt+{\theta\mu(s)\over 2}{\cal
Y}^{i}(s)^{\theta-2}\Big{(}\theta\mu(s){\cal
Y}^{i}(s)^{\theta}-(1-\theta)\Big{)}|Z^{i}(s)|^{2}\Big{]}ds\\\ \vskip 3.0pt
plus 1.0pt minus 1.0pt\displaystyle\hskip 65.44142pt+\varphi\big{(}s,{\cal
Y}^{i}(s)\big{)}\theta\mu(s){\cal Y}^{i}(s)^{\theta-1}\hbox{\rm
sgn$\,$}\big{(}Y^{i}(s)\big{)}Z^{i}(s)dW(s),\ \ s\in[t,T].\end{array}$

For each $r\in[t,T]$ and $m\geqslant 1$, define the following stopping time:

$\sigma_{m}^{r}:=\inf\Big{\\{}s\in[r,T]\big{|}|Y(s)|+\int_{r}^{s}|Z(\tau)|^{2}d\tau\geqslant
m\Big{\\}}\wedge T$

with the convention $\inf\emptyset=+\infty$. Then, by integrating on
$[r,\sigma_{m}^{r}]$ we get that for each $m\geqslant 1$ and $r\in[t,T]$,

(2.22) $\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\varphi\big{(}r,{\cal
Y}^{i}(r)\big{)}+X_{r}^{m}\leqslant\varphi\big{(}\sigma_{m}^{r},{\cal
Y}^{i}(\sigma_{m}^{r})\big{)}+\int_{r}^{\sigma_{m}^{r}}\varphi\big{(}s,{\cal
Y}^{i}(s)\big{)}\Big{[}-\mu^{\prime}(s){\cal Y}^{i}(s)^{\theta}\\\ \vskip
3.0pt plus 1.0pt minus 1.0pt\displaystyle\qquad\qquad\qquad+{\theta\mu(s)\over
2}{\cal Y}^{i}(s)^{\theta-2}\Big{(}2\beta{\cal
Y}^{i}(s)|Z^{i}(s)|^{1+\delta}-\theta\mu(s){\cal
Y}^{i}(s)^{\theta}|Z^{i}(s)|^{2}+(1-\theta)|Z^{i}(s)|^{2}\Big{)}\Big{]}ds,\end{array}$

where

$X_{r}^{m}:=\int_{r}^{\sigma_{m}^{r}}\varphi\big{(}s,{\cal
Y}^{i}(s)\big{)}\theta\mu(s){\cal Y}^{i}(s)^{\theta-1}\hbox{\rm
sgn$\,$}\big{(}Y^{i}(s)\big{)}Z^{i}(s)dW(s).$

We split the rest into two cases.

Case 1. For $\delta\in(0,1)$, one has

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus 1.0pt\displaystyle
2\beta{\cal Y}^{i}(s)|Z^{i}(s)|^{1+\delta}=2\beta\Big{(}{\cal
Y}^{i}(s)^{\theta}|Z^{i}(s)|^{2}\Big{)}^{1+\delta\over 2}{\cal
Y}^{i}(s)^{1-{\theta{1+\delta\over 2}}}\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\leqslant{\theta\over 2}{\cal
Y}^{i}(s)^{\theta}|Z^{i}(s)|^{2}+K_{0}{\cal Y}^{i}(s)^{(1-\theta{1+\delta\over
2}){2\over 1-\delta}}={\theta\over 2}{\cal
Y}^{i}(s)^{\theta}|Z^{i}(s)|^{2}+K_{0}{\cal Y}^{i}(s)^{2-\theta(1+\delta)\over
1-\delta},\ \ s\in[t,T].\end{array}$

In the above, $K_{0}>0$ is an absolute constant, only depending on
$\beta,\delta,\theta$. We now require that

${2-\theta(1+\delta)\over 1-\delta}\leqslant 2,$

which is equivalent to $\theta\geqslant{2\delta\over 1+\delta}$. Hence, the
best possibility is to choose $\theta={2\delta\over 1+\delta}$. Then with such
a $\theta$, we have (noting $\mu(s)\geqslant 1$ and ${\cal Y}^{i}(s)\geqslant
k$) that for each $m\geqslant 1$ and $r\in[t,T]$,

(2.23) $\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\varphi(r,{\cal
Y}^{i}(r))+X_{r}^{m}\leqslant\varphi\big{(}\sigma_{m}^{r},{\cal
Y}^{i}(\sigma_{m}^{r})\big{)}+\int_{r}^{\sigma_{m}^{r}}\varphi\big{(}s,{\cal
Y}^{i}(s)\big{)}{\cal Y}^{i}(s)^{\theta}\Big{[}-\Big{(}\mu^{\prime}(s)-{\theta
K_{0}\over 2}\mu(s)\Big{)}\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\hskip 170.71652pt-{\theta\mu(s)\over 2}\Big{(}{\theta\over
2}{\cal Y}^{i}(s)^{\theta}-(1-\theta)\Big{)}|Z^{i}(s)|^{2}\Big{)}\Big{]}ds\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\displaystyle\hskip
76.82234pt\leqslant\varphi\big{(}\sigma_{m}^{r},{\cal
Y}^{i}(\sigma_{m}^{r})\big{)}+\int_{r}^{\sigma_{m}^{r}}\varphi\big{(}s,{\cal
Y}^{i}(s)\big{)}{\cal Y}^{i}(s)^{\theta}\Big{[}-\Big{(}\mu^{\prime}(s)-{\theta
K_{0}\over 2}\mu(s)\Big{)}\Big{)}\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\hskip 170.71652pt-{\theta\mu(s)\over 2}\Big{(}{\theta\over
2}k^{\theta}-1\Big{)}|Z^{i}(s)|^{2}\Big{)}\Big{]}ds.\end{array}$

Now, we choose $k\geqslant 1+\theta^{-{1\over\theta}}$ and
$\mu(\cdot)\geqslant 1$ so that

(2.24) ${\theta\over 2}\Big{(}{\theta\over 2}k^{\theta}-1\Big{)}\geqslant
1,\qquad\mu^{\prime}(s)\geqslant{\theta K_{0}\over 2}\mu(s),\quad s\in[t,T].$

Then, for each $m\geqslant 1$ and $r\in[t,T]$, in view of
$\mathbb{E}_{r}[X_{r}^{m}]=0$, by taking conditional expectation with respect
to ${\cal F}_{r}$ in (2.23) we can obtain that

(2.25) $\varphi(r,{\cal
Y}^{i}(r))+\mathbb{E}_{r}\Big{[}\int_{r}^{\sigma_{m}^{r}}|Z^{i}(s)|^{2}ds\Big{]}\leqslant\mathbb{E}_{r}\big{[}\varphi\big{(}\sigma_{m}^{r},{\cal
Y}^{i}(\sigma_{m}^{r})\big{)}\big{]},$

with $k$ depending on $\delta$. By Fatou’s lemma and Lebesgue’s dominated
convergence theorem, as well as the integrability of $Y(\cdot)$ and
$\alpha_{0}(\cdot)$, sending $m$ to infinity in (2.24) yields the following

$e^{\mu(r)\big{(}|Y^{i}(r)|+k+\int_{t}^{r}\alpha_{0}(\tau)d\tau\big{)}^{2\delta\over
1+\delta}}+\mathbb{E}_{r}\Big{[}\int_{r}^{T}|Z^{i}(s)|^{2}ds\Big{]}\leqslant\mathbb{E}_{r}\Big{[}e^{\mu(T)\big{(}|\xi^{i}|+k+\int_{t}^{T}\alpha_{0}(\tau)d\tau\big{)}^{2\delta\over
1+\delta}}\Big{]},\ \ r\in[t,T],$

which is just the following result

(2.26) $e^{\mu(t)\big{(}|Y^{i}(t)|+k\big{)}^{2\delta\over
1+\delta}}+\mathbb{E}_{t}\Big{[}\int_{t}^{T}|Z^{i}(s)|^{2}ds\Big{]}\leqslant\mathbb{E}_{t}\Big{[}e^{\mu(T)\big{(}|\xi^{i}|+k+\int_{t}^{T}\alpha_{0}(\tau)d\tau\big{)}^{2\delta\over
1+\delta}}\Big{]}$

for the case $\delta\in(0,1)$ when $r=t$. Furthermore, it follows from Doob’s
martingale inequality that for each $p>1$,

(2.27)
$\begin{array}[]{ll}\displaystyle\mathbb{E}\Big{[}\sup_{r\in[t,T]}e^{p\mu(r)\big{(}|Y^{i}(r)|+k+\int_{t}^{r}\alpha_{0}(\tau)d\tau\big{)}^{2\delta\over
1+\delta}}\Big{]}\leqslant\mathbb{E}\Big{[}\sup_{r\in[t,T]}\Big{\\{}\mathbb{E}_{r}\Big{[}e^{\mu(T)\big{(}|\xi^{i}|+k+\int_{t}^{T}\alpha_{0}(\tau)d\tau\big{)}^{2\delta\over
1+\delta}}\Big{]}\Big{\\}}^{p}\Big{]}\\\ \hskip
176.407pt\displaystyle\leqslant\big{(}{p\over
p-1}\big{)}^{p}\mathbb{E}\Big{[}e^{p\mu(T)\big{(}|\xi^{i}|+k+\int_{t}^{T}\alpha_{0}(\tau)d\tau\big{)}^{2\delta\over
1+\delta}}\Big{]}.\end{array}$

On the other hand, by (2.23) and (2.24) we can deduce that for each
$m\geqslant 1$ and $p>1$,

$\mathbb{E}\Big{[}\big{(}\int_{r}^{\sigma_{m}^{r}}|Z^{i}(s)|^{2}ds\big{)}^{p\over
2}\Big{]}\leqslant
2^{p-1}\Big{\\{}\mathbb{E}\Big{[}\big{(}\varphi\big{(}\sigma_{m}^{r},{\cal
Y}^{i}(\sigma_{m}^{r})\big{)}\big{)}^{p\over
2}\Big{]}+\mathbb{E}\big{[}|X_{r}^{m}|^{p\over 2}\big{]}\Big{\\}},\ \
r\in[t,T].$

By virtue of BDG’s inequality, the definition of $X_{r}^{m}$ and Hölder’s
inequality and in view of ${\cal Y}^{i}(s)\geqslant k\geqslant 1$ and
$\theta\in(0,1)$, observe that

$\begin{array}[]{ll}\displaystyle 2^{p-1}\mathbb{E}\big{[}|X_{r}^{m}|^{p\over
2}\big{]}\leqslant 2^{p-1}(\mu(T))^{p\over
2}\mathbb{E}\Big{[}\Big{(}\int_{r}^{\sigma_{m}^{r}}\big{[}\varphi\big{(}s,{\cal
Y}^{i}(s)\big{)}\big{]}^{2}|Z^{i}(s)|^{2}ds\Big{)}^{p\over 4}\Big{]}\\\ \hskip
71.13188pt\displaystyle\leqslant 2^{p-1}(\mu(T))^{p\over
2}\mathbb{E}\Big{[}\sup_{s\in[r,\sigma_{m}^{r}]}\big{[}\big{(}\varphi\big{(}s,{\cal
Y}^{i}(s)\big{)}\big{)}^{p\over
2}\big{]}\Big{(}\int_{r}^{\sigma_{m}^{r}}|Z^{i}(s)|^{2}ds\Big{)}^{p\over
4}\Big{]}\\\ \hskip 71.13188pt\displaystyle\leqslant
2^{2p-3}(\mu(T))^{p}\mathbb{E}\Big{[}\sup_{s\in[r,\sigma_{m}^{r}]}\big{(}\varphi\big{(}s,{\cal
Y}^{i}(s)\big{)}\big{)}^{p}\Big{]}+{1\over
2}\mathbb{E}\Big{[}\Big{(}\int_{r}^{\sigma_{m}^{r}}|Z^{i}(s)|^{2}ds\Big{)}^{p\over
2}\Big{]},\ \ r\in[t,T].\end{array}$

We have

$\begin{array}[]{ll}\displaystyle\mathbb{E}\Big{[}\Big{(}\int_{r}^{\sigma_{m}^{r}}|Z^{i}(s)|^{2}ds\Big{)}^{p\over
2}\Big{]}\leqslant
2^{p}\mathbb{E}\Big{[}\big{(}\varphi\big{(}\sigma_{m}^{r},{\cal
Y}^{i}(\sigma_{m}^{r})\big{)}\big{)}^{p\over
2}\Big{]}+2^{2p-2}(\mu(T))^{p}\mathbb{E}\Big{[}\sup_{s\in[r,\sigma_{m}^{r}]}\big{(}\varphi\big{(}s,{\cal
Y}^{i}(s)\big{)}\big{)}^{p}\Big{]}\\\ \hskip
105.2751pt\displaystyle\leqslant(8\mu(T))^{p}\mathbb{E}\Big{[}\sup_{s\in[r,\sigma_{m}^{r}]}\big{(}\varphi\big{(}s,{\cal
Y}^{i}(s)\big{)}\big{)}^{p}\Big{]},\qquad r\in[t,T],\end{array}$

and then (letting $m\rightarrow\infty$ and $r=t$)

(2.28)
$\begin{array}[]{ll}\displaystyle\mathbb{E}\Big{[}\Big{(}\int_{t}^{T}|Z^{i}(s)|^{2}ds\Big{)}^{p\over
2}\Big{]}\leqslant(8\mu(T))^{p}\mathbb{E}\Big{[}\sup_{s\in[t,T]}\big{(}\varphi\big{(}s,{\cal
Y}^{i}(s)\big{)}\big{)}^{p}\Big{]}\\\ \hskip
99.58464pt\displaystyle=(8\mu(T))^{p}\mathbb{E}\Big{[}\sup_{r\in[t,T]}e^{p\mu(r)\big{(}|Y^{i}(r)|+k+\int_{t}^{r}\alpha_{0}(\tau)d\tau\big{)}^{2\delta\over
1+\delta}}\Big{]}.\end{array}$

Combining (2.27) and (2.28) yields

(2.29)
$\begin{array}[]{ll}\displaystyle\mathbb{E}\Big{[}\sup_{s\in[t,T]}e^{p\mu(s)\big{(}|Y^{i}(s)|+k+\int_{t}^{s}\alpha_{0}(\tau)d\tau\big{)}^{2\delta\over
1+\delta}}\Big{]}+\mathbb{E}\Big{[}\Big{(}\int_{t}^{T}|Z^{i}(s)|^{2}ds\Big{)}^{p\over
2}\Big{]}\\\ \displaystyle\qquad\leqslant\big{(}{p\over
p-1}\big{)}^{p}\big{(}(8\mu(T))^{p}+1\big{)}\mathbb{E}\Big{[}e^{p\mu(T)\big{(}|\xi^{i}|+k+\int_{t}^{T}\alpha_{0}(\tau)d\tau\big{)}^{2\delta\over
1+\delta}}\Big{]},\ \ \forall p>1.\end{array}$

for the case $\delta\in(0,1)$. Finally, (2.20) and (2.21) follows from (2.26)
and (2.29) as well for the case of $\delta\in(0,1)$.

Case 2. For $\delta=1$, we have that $\theta={2\delta\over 1+\delta}=1$, and
it follows from (2.22) that for each $m\geqslant 1$,

(2.30) $\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\varphi(r,{\cal
Y}^{i}(r))+X_{r}^{m}\leqslant\varphi\big{(}\sigma_{m}^{r},{\cal
Y}^{i}(\sigma_{m}^{r})\big{)}+\int_{r}^{\sigma_{m}^{r}}\varphi\big{(}s,{\cal
Y}^{i}(s)\big{)}\Big{[}-{\cal Y}^{i}(s)\mu^{\prime}(s)\\\ \vskip 3.0pt plus
1.0pt minus 1.0pt\displaystyle\hskip 156.49014pt+{\mu(s)\over 2}{\cal
Y}^{i}(s)^{-1}\Big{(}2\beta{\cal Y}^{i}(s)|Z^{i}(s)|^{2}-\mu(s){\cal
Y}^{i}(s)|Z^{i}(s)|^{2}\Big{)}\Big{]}ds\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\ \ =\varphi\big{(}\sigma_{m}^{r},{\cal
Y}^{i}(\sigma_{m}^{r})\big{)}+\int_{r}^{\sigma_{m}^{r}}\varphi\big{(}s,{\cal
Y}^{i}(s)\big{)}\Big{[}-{\cal Y}^{i}(s)\mu^{\prime}(s)+{\mu(s)\over
2}\big{(}2\beta-\mu(s)\big{)}|Z^{i}(s)|^{2}\Big{]}ds,\ \
r\in[t,T].\end{array}$

Hence, by choosing

$\mu^{\prime}(s)>0,\qquad{\mu(s)\over 2}\big{(}\mu(s)-2\beta\big{)}\geqslant
1,\quad s\in[0,T],$

and taking conditional expectation with respect to ${\cal F}_{r}$ in (2.30) we
get (2.25) for the case $\delta=1$, and sending $m$ to infinity and letting
$r=t$ yields (2.26) for the case $\delta=1$. Then, a similar argument to that
from (2.27) to (2.28) yields (2.29) for the case $\delta=1$. Finally, (2.20)
and (2.21) follows from (2.26) and (2.29) as well for the case of $\delta=1$.

######  Remark 2.8.

The following assertion can be obtained by a similar proof to above. Let the
assumptions in Lemma 2.7 hold with (2.19) replaced by

$I_{(Y^{i}(s)>0)}f^{i}(s,Y(s),Z(s))\leqslant\alpha_{0}(s)+\beta|Z^{i}(s)|^{1+\delta},\quad\hbox{\rm
a.s.}.$

Then there exist a $k\geqslant 1$ and a differentiable function
$\mu:[0,T]\to[1,\infty)$ such that for each $i=1,\cdots,n$,

$e^{\mu(t)\big{(}(Y^{i}(t))^{+}+k\big{)}^{2\delta\over
1+\delta}}+\mathbb{E}_{t}\Big{[}\int_{t}^{T}I_{(Y^{i}(s)>0)}|Z^{i}(s)|^{2}ds\Big{]}\leqslant\mathbb{E}_{t}\Big{[}e^{\mu(T)\big{(}(\xi^{i})^{+}+k+\int_{t}^{T}\alpha_{0}(\tau)d\tau\big{)}^{2\delta\over
1+\delta}}\Big{]},\ \ t\in[0,T].$

Consequently, there exists a constant $K>0$ depending only on
$(\delta,\beta,T)$ such that for each $i=1,\cdots,n$,

$e^{[(Y^{i}(t))^{+}]^{2\delta\over
1+\delta}}+\mathbb{E}_{t}\Big{[}\int_{t}^{T}I_{(Y^{i}(s)>0)}|Z^{i}(s)|^{2}ds\Big{]}\leqslant
K\mathbb{E}_{t}\Big{[}e^{K\big{(}(\xi^{i})^{+}+\int_{t}^{T}\alpha_{0}(\tau)d\tau\big{)}^{2\delta\over
1+\delta}}\Big{]},\ \ t\in[0,T].$

##  3 BSVIEs with Bounded Free Terms

For convenience, we rewrite BSVIE (1.1) here

(3.1)
$Y(t)=\psi(t)+\int_{t}^{T}g(t,s,Y(s),Z(t,s))ds-\int_{t}^{T}Z(t,s)dW(s),\quad
t\in[0,T].$

In this section, we consider the case of coupled multi-dimensional BSVIEs with
bounded free terms. In what follows, we use $K>0$ to represent a generic
constant which could be different from line to line.

###  3.1 Diagonally strictly quadratic case

Let us begin with the following hypothesis on the generator $g$ of BSVIE (3.1)
which is comparable with (H2.1) for BSDEs.

(H3.1) Map
$g:\Delta^{*}[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\Omega\to\mathbb{R}^{n}$
is measurable so that $s\mapsto g(t,s,y,z)$ is $\mathbb{F}$-progressively
measurable on $[t,T]$ for each
$(t,y,z)\in[0,T)\times\mathbb{R}^{n}\times\mathbb{R}^{n}$. There exist a non-
negative valued process $\alpha(\cdot\,,\cdot)\in
L^{\infty}(0,T;L^{\infty}_{\mathbb{F}}(\Omega;L^{1}(\cdot\,,T;\mathbb{R})))$,
a continuous, monotone increasing function $\phi:[0,\infty)\to[0,\infty)$ with
$\phi(0)=0$, and constants $\beta\geqslant\gamma>0$, $\delta\in[0,1)$ such
that

(3.2)
$\begin{array}[]{ll}\displaystyle-\alpha(t,s)-\beta\Big{(}|y|+\sum_{j\neq
i}|z^{j}|^{1+\delta}\Big{)}+\gamma|z^{i}|^{2}\leqslant
g^{i}(t,s,y,z)\leqslant\alpha(t,s)+\beta\Big{(}|y|+\sum_{j\neq
i}|z^{j}|^{1+\delta}+|z^{i}|^{2}\Big{)},\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall(t,s,y,z)\in\Delta^{*}[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n},~{}i=1,2,\cdots,n.\end{array}$

Further, it holds that

(3.3) $\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle|g^{i}(t,s,y,z)-g^{i}(t,s,\bar{y},\bar{z})|\leqslant\phi(|y|\vee|\bar{y}|)\Big{[}\big{(}1+|z|+|\bar{z}|\big{)}\big{(}|y-\bar{y}|+|z^{i}-\bar{z}^{i}|\big{)}\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\big{(}1+|z|^{\delta}+|\bar{z}|^{\delta}\big{)}\sum_{j\neq
i}|z^{j}-\bar{z}^{j}|\Big{]},\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall(t,s)\in\Delta^{*}[0,T],~{}y,\bar{y},z,\bar{z}\in\mathbb{R}^{n},~{}i=1,2,\cdots,n.\end{array}$

Let us introduce the following seemingly a little more general hypothesis than
the above (H3.1).

(H3.1)′ Let (H3.1) hold with (3.2) replaced by the following: For each
$i=1,2,\cdots,n$, either

(3.4) $\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle-\alpha(t,s)-\beta\Big{(}|y|+\sum_{j\neq
i}|z^{j}|^{1+\delta}\Big{)}+\gamma|z^{i}|^{2}\leqslant
g^{i}(t,s,y,z)\leqslant\alpha(t,s)+\beta\Big{(}|y|+\sum_{j\neq
i}|z^{j}|^{1+\delta}+|z^{i}|^{2}\Big{)},\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall(t,s,y,z)\in\Delta^{*}[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n},\end{array}$

or

(3.5) $\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\alpha(t,s)+\beta\Big{(}|y|+\sum_{j\neq
i}|z^{j}|^{1+\delta}\Big{)}-\gamma|z^{i}|^{2}\geqslant
g^{i}(t,s,y,z)\geqslant-\alpha(t,s)-\beta\Big{(}|y|+\sum_{j\neq
i}|z^{j}|^{1+\delta}+|z^{i}|^{2}\Big{)},\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall(t,s,y,z)\in\Delta^{*}[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}.\end{array}$

We claim that (H3.1) and (H3.1)′ are essentially equivalent. In fact, if
(H3.1)′ holds, by relabeling, we may assume that for some $k=0,1,2,\cdots,n$,
one has (3.4) holds for $1\leqslant i\leqslant k$, and (3.5) holds for
$k+1\leqslant i\leqslant n$. If $k=n$, it means that for all $i=1,2,\cdots,n$,
(3.4) holds, and if $k=0$, it means that for all $i=1,2,\cdots,n$, (3.5)
holds. Now, we define

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\bar{z}\equiv(\bar{z}^{1},\cdots,\bar{z}^{k},\bar{z}^{k+1},\cdots,\bar{z}^{n})=(z^{1},\cdots,z^{k},-z^{k+1},\cdots,-z^{n}),\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\bar{y}\equiv(\bar{y}^{1},\cdots,\bar{y}^{k},\bar{y}^{k+1},\cdots,\bar{y}^{n})=(y^{1},\cdots,y^{k},-y^{k+1},\cdots,-y^{n}),\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\bar{g}^{i}(t,s,y,z)=g^{i}(t,s,y,z),\qquad~{}~{}1\leqslant
i\leqslant k,\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\bar{g}^{i}(t,s,y,z)=-g^{i}(t,s,\bar{y},\bar{z}),\qquad
k+1\leqslant i\leqslant n.\end{array}$

Then (H3.1) holds for $\bar{g}(t,s,y,z)$. Thus, (H3.1) and (H3.1)′ are
equivalent in such a sense. In what follows, we will use (H3.1) instead of
(H3.1)′.

Next, condition (3.2) means that the $i$-th component $g^{i}$ of the generator
$g$ has exactly quadratic growth in the $i$-th component $z^{i}$ of $z$, and
has sub-quadratical growth in the other components of $z$. Such a condition is
referred to as the diagonally strictly quadratic condition of $z\mapsto
g(t,s,y,z)$, which was introduced for BSDEs the first time in [14]. Roughly
speaking, (3.2) amounts to saying that

$0<\gamma\leqslant\mathop{\underline{\rm
lim}}_{|z^{i}|\to\infty}{g^{i}(t,s,y,z)\over|z^{i}|^{2}}\leqslant\mathop{\overline{\rm
lim}}_{|z^{i}|\to\infty}{g^{i}(t,s,y,z)\over|z^{i}|^{2}}\leqslant\beta.$

Hypothesis (H3.1) is satisfied by many examples. Here is a typical one:

$g^{i}(t,s,y,z)=\sqrt{(|y|+1)(|z|+1)}+|z|\cos|z^{i}|-|z|^{1+\delta}+|z^{i}|^{2},\
\ i=1,\cdots,n.$

To establish the existence and uniqueness of adapted solution
$(Y(\cdot),Z(\cdot\,,\cdot))$ to our BSVIE (1.1), we first present the
following result.

######  Proposition 3.1.

Let (H3.1) hold. Then, for each $\psi(\cdot)\in L_{{\cal
F}_{T}}^{\infty}(0,T;\mathbb{R}^{n})$, $y(\cdot)\in
L^{\infty}_{\mathbb{F}}(0,T;\mathbb{R}^{n})$, the following BSVIE

(3.6) $Y(t)=\psi(t)+\int_{t}^{T}g(t,s,y(s),Z(t,s))ds-\int_{t}^{T}Z(t,s)dW(s),\
\ t\in[0,T]$

admits a unique adapted solution $(Y(\cdot),Z(\cdot\,,\cdot))\in
L^{\infty}_{\mathbb{F}}(0,T;\mathbb{R}^{n})\times\overline{{\rm
BMO}}(\Delta^{*}[0,T];\mathbb{R}^{n})$. Furthermore, there exists a constant
$K_{0}>0$ depending on
$(\psi(\cdot),\alpha(\cdot\,,\cdot),\beta,\gamma,\delta,n,T)$ such that if

(3.7) $\|y(t)\|_{\infty}\leqslant K_{0}e^{K_{0}(T-t)},\qquad\hbox{\rm
a.e.}\;t\in[0,T],$

then

(3.8) $\|Y(t)\|_{\infty}\leqslant K_{0}e^{K_{0}(T-t)},\qquad\hbox{\rm
a.e.}\;t\in[0,T],$

and

(3.9) $\|Z(\cdot\,,\cdot)\|_{\overline{\rm
BMO}(\Delta^{*}[0,T])}\leqslant\sqrt{nK_{0}}\exp\Big{(}K_{0}e^{K_{0}T}\Big{)}\equiv\bar{K}_{0}.$

Proof. For given $\psi(\cdot)\in L_{{\cal
F}_{T}}^{\infty}(0,T;\mathbb{R}^{n})$ and $y(\cdot)\in
L_{\mathbb{F}}^{\infty}(0,T;\mathbb{R}^{n})$, it follows from Proposition 2.2
(or [15, Theorem 2.5]) that for $\hbox{\rm a.e.}\,t\in[0,T]$, the following
BSDE

(3.10)
$\eta(t,r)=\psi(t)+\int_{r}^{T}g(t,s,y(s),\zeta(t,s))ds-\int_{r}^{T}\zeta(t,s)dW(s),\
\ r\in[t,T]$

admits a unique adapted solution $(\eta(t,\cdot),\zeta(t,\cdot))\in
L^{\infty}_{\mathbb{F}}(\Omega;C([t,T];\mathbb{R}^{n}))\times\overline{{\rm
BMO}}([t,T];\mathbb{R}^{n})$. Moreover, there exists a constant $K>0$
depending on $(\alpha(\cdot\,,\cdot),\beta,\gamma,\delta,n,T)$ and
$\|y(\cdot)\|_{L^{\infty}_{\mathbb{F}}(0,T;\mathbb{R}^{n})}$ such that

(3.11)
$\|\eta(t,\cdot)\|_{L^{\infty}_{\mathbb{F}}(\Omega;C([t,T];\mathbb{R}^{n}))}+\|\zeta(t,\cdot)\|_{\overline{{\rm
BMO}}[t,T]}\leqslant K,\quad\hbox{\rm a.e.}~{}t\in[0,T].$

Let

$Y(t)=\eta(t,t),\qquad Z(t,s)=\zeta(t,s),\qquad(t,s)\in\Delta[0,T].$

Then $(Y(\cdot),Z(\cdot\,,\cdot))\in
L^{\infty}_{\mathbb{F}}(0,T;\mathbb{R}^{n})\times\overline{{\rm
BMO}}(\Delta^{*}[0,T];\mathbb{R}^{n})$ is an adapted solution of BSVIE (3.6).

Suppose now that $(\widetilde{Y}(\cdot),\widetilde{Z}(\cdot\,,\cdot))\in
L^{\infty}_{\mathbb{F}}(0,T;\mathbb{R}^{n})\times\overline{{\rm
BMO}}(\Delta^{*}[0,T];\mathbb{R}^{n})$ is another adapted solution of BSVIE
(3.6). That is,

(3.12)
$\widetilde{Y}(t)=\psi(t)+\int_{t}^{T}g(t,s,y(s),\widetilde{Z}(t,s))ds-\int_{t}^{T}\widetilde{Z}(t,s)dW(s),\
\ t\in[0,T].$

With the given $(\widetilde{Y}(\cdot),\widetilde{Z}(\cdot\,,\cdot))$, for
$\hbox{\rm a.e.}\;t\in[0,T]$, the following BSDE:

(3.13)
$\widetilde{\eta}(t,r)=\psi(t)+\int_{r}^{T}g(t,s,y(s),\widetilde{Z}(t,s))ds-\int_{r}^{T}\widetilde{\zeta}(t,s)dW(s),\
\ r\in[t,T]$

admits a unique solution
$(\widetilde{\eta}(t,\cdot),\widetilde{\zeta}(t,\cdot))\in
L^{\infty}_{\mathbb{F}}(\Omega;C([t,T];\mathbb{R}^{n}))\times\overline{{\rm
BMO}}([t,T];\mathbb{R}^{n})$ (see Proposition 2.2 or [15, Theorem 2.5]). By
(3.12) and (3.13), it is immediate that for ${\rm a.e.}\ t\in[0,T]$,

(3.14)
$\widetilde{Y}(t)=\mathbb{E}_{t}\Big{[}\psi(t)+\int_{t}^{T}g(t,s,y(s),\widetilde{Z}(t,s))ds\Big{]}=\widetilde{\eta}(t,t)$

and

$\int_{t}^{T}\widetilde{Z}(t,s)dW(s)=\int_{t}^{T}\widetilde{\zeta}(t,s)dW(s),$

which implies

(3.15) $\widetilde{Z}(t,s)=\widetilde{\zeta}(t,s),\ \ {\rm a.e.}\ s\in(t,T].$

It then follows from (3.13) that for $\hbox{\rm a.e.}\;t\in[0,T]$,

$\widetilde{\eta}(t,r)=\psi(t)+\int_{r}^{T}g(t,s,y(s),\widetilde{\zeta}(t,s))ds-\int_{r}^{T}\widetilde{\zeta}(t,s)dW(s),\
\ r\in[t,T],$

which means that $(\widetilde{\eta}(t,\cdot),\widetilde{\zeta}(t,\cdot))$ is
an adapted solution of BSDE (3.10) in
$L^{\infty}_{\mathbb{F}}(\Omega;C([t,T];\mathbb{R}^{n}))\times\overline{{\rm
BMO}}([t,T];$ $\mathbb{R}^{n})$, and then

(3.16)
$\widetilde{\eta}(t,\cdot)=\eta(t,\cdot),\qquad\widetilde{\zeta}(t,\cdot)=\zeta(t,\cdot).$

Combining (3.14), (3.15) and (3.16) yields that

$\widetilde{Y}(t)=\eta(t,t),\qquad\widetilde{Z}(t,s)=\zeta(t,s),\qquad\hbox{\rm
a.e.}\;(t,s)\in\Delta^{*}[0,T].$

Consequently, BSVIE (3.6) admits a unique adapted solution
$(Y(\cdot),Z(\cdot\,,\cdot))\in
L^{\infty}_{\mathbb{F}}(\Omega;C([t,T];\mathbb{R}^{n}))\times\overline{{\rm
BMO}}$ $(\Delta^{*}[0,T];\mathbb{R}^{n})$.

We now turn to the proof of the second part of Proposition 3.1. First of all,
for any given $k\geqslant 1$, by Young’s inequality that

(3.17)
$kn^{1+\delta}|\zeta(t,s)|^{1+\delta}\leqslant{\frac{1}{\widetilde{M}}}|\zeta(t,s)|^{2}+K,\
\ s\in[t,T].$

where $K$ is a constant depending on $\widetilde{M}$, $\delta$ and $k$, and

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\widetilde{M}\buildrel\triangle\over{=}2\sup_{1\leqslant
i\leqslant n}\mathop{\rm esssup}_{t\in[0,T]}\mathop{\rm
esssup}_{\tau}\Big{\|}\mathbb{E}_{\tau}\int_{\tau}^{T}|\zeta^{i}(t,s)|^{2}ds\Big{\|}_{\infty}<\infty,\end{array}$

Define process $\widetilde{\zeta}^{i}(t,\cdot)$ as
$\frac{1}{\sqrt{\widetilde{M}}}\zeta^{i}(t,\cdot)$ on $[t,T]$, we see that
$\big{\|}\widetilde{\zeta}^{i}(t,\cdot)\big{\|}_{\overline{\rm
BMO}[t,T]}\leqslant\frac{1}{2}.$ By John-Nirenberg inequality, we see that

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\mathbb{E}_{t}\Big{[}\exp\Big{(}\frac{1}{\widetilde{M}}\int_{t}^{T}|\zeta^{i}(t,s)|^{2}ds\Big{)}\Big{]}\leqslant
2.\end{array}$

Therefore, for any $k\geqslant 1$,

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\mathbb{E}_{t}\Big{[}\exp\Big{(}k\int_{t}^{T}|\zeta(t,s)|^{1+\delta}ds\Big{)}\Big{]}\leqslant\mathbb{E}_{t}\Big{[}\exp\Big{(}kn^{\delta}\sum_{i=1}^{n}\int_{t}^{T}|\zeta^{i}(t,s)|^{1+\delta}ds\Big{)}\Big{]}\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle\ \
=\mathbb{E}_{t}\Big{[}\prod_{i=1}^{n}\exp\Big{(}kn^{\delta}\int_{t}^{T}|\zeta^{i}(t,s)|^{1+\delta}ds\Big{)}\Big{]}\leqslant\prod_{i=1}^{n}\Big{\\{}\mathbb{E}_{t}\Big{[}\exp\Big{(}kn^{1+\delta}\int_{t}^{T}|\zeta^{i}(t,s)|^{1+\delta}ds\Big{)}\Big{]}\Big{\\}}^{\frac{1}{n}}\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle\ \
\leqslant\prod_{i=1}^{n}\Big{\\{}\mathbb{E}_{t}\Big{[}\exp\Big{(}\int_{t}^{T}\frac{|\zeta^{i}(t,s)|^{2}}{\widetilde{M}}ds\Big{)}\Big{]}\Big{\\}}^{\frac{1}{n}}e^{K}\leqslant
2e^{K}<\infty.\end{array}$

This together with the integrability of $\alpha(t,\cdot)$, $y(\cdot)$
indicates that $\alpha_{0}(t,\cdot)\in\bigcap_{\mu\geqslant 1}{\cal
E}^{\mu,1}_{\mathbb{F}}(\Omega;L^{1}(t,T;\mathbb{R}_{+}))$, where

$\alpha_{0}(t,s)\buildrel\triangle\over{=}\alpha(t,s)+\beta\big{(}|y(s)|+|\zeta(t,s)|^{1+\delta}\big{)},\vspace{0.1cm}$

Taking into account (H3.1), together with BSDE (3.10), by virtue of Lemma 2.7
(or [14, Proposition 2.1]), we have for each $i=1,2,\cdots,n$, $\hbox{\rm
a.e.}\;t\in[0,T]$ and each $r\in[t,T]$, noting that $\psi(\cdot)$ is bounded,

(3.18) $\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\exp\Big{(}\gamma|\eta^{i}(t,r)|\Big{)}+\mathbb{E}_{r}\Big{[}\int_{r}^{T}|\zeta^{i}(t,s)|^{2}ds\Big{]}\\\
\ \ \displaystyle\leqslant
K\mathbb{E}_{r}\Big{[}\exp\Big{(}K|\psi^{i}(t)|+K\int_{r}^{T}\big{[}\alpha(t,s)+\beta\big{(}|y(s)|+|\zeta(t,s)|^{1+\delta}\big{)}\big{]}ds\Big{)}\Big{]}\\\
\ \ \displaystyle\leqslant
K\exp\Big{(}K\int_{r}^{T}\|y(s)\|_{\infty}ds\Big{)}\mathbb{E}_{r}\Big{[}\exp\Big{(}K\int_{r}^{T}|\zeta(t,s)|^{1+\delta}ds\Big{)}\Big{]}.\end{array}$

Note that on the right-hand side, the term
$\int_{r}^{T}|\zeta(t,s)|^{1+\delta}ds$ appears in the exponent, which cannot
be directly controlled by the term $\int_{r}^{T}|\zeta(t,s)|^{2}ds$. We now
make use of the diagonally strictly quadratic condition (3.2). Let
$i=1,\cdots,n$, and $m\geqslant 1$, For almost every $t\in[0,T]$ and each
$r\in[t,T]$, we define the following $\mathbb{F}$-stopping time

$\sigma_{m}^{t,r,i}=\inf\Big{\\{}\tau\in[r,T]\bigm{|}\int_{r}^{\tau}|\zeta^{i}(t,s)|^{2}ds\geqslant
m\Big{\\}}\land T,$

with the convention $\inf\varnothing=+\infty$, and define

$X_{m}^{t,r,i}=\eta^{i}(t,r)-\eta^{i}(t,\sigma_{m}^{t,r,i})+\int_{r}^{\sigma_{m}^{t,r,i}}\big{[}\alpha(t,s)+\beta\big{(}|y(s)|+|\zeta(t,s)|^{1+\delta}\big{)}\big{]}ds.$

Then it follows from the first inequality in (3.2) and BSDE (3.10) that

(3.19)
$\begin{array}[]{ll}\displaystyle\gamma\int_{r}^{\sigma^{t,r,i}_{m}}\negthinspace\negthinspace\negthinspace\negthinspace\negthinspace|\zeta^{i}(t,s)|^{2}ds\leqslant\int_{r}^{\sigma_{m}^{t,r,i}}\negthinspace\negthinspace\negthinspace\big{[}\alpha(t,s)+\beta\big{(}|y(s)|+|\zeta(t,s)|^{1+\delta}\big{)}+g^{i}(t,s,y(s),\zeta(t,s))\big{]}ds\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\displaystyle\
\qquad\qquad\qquad\qquad=X_{m}^{t,r,i}+\int_{r}^{\sigma_{m}^{t,r,i}}\zeta^{i}(t,s)dW(s),\
\ r\in[t,T].\end{array}$

This yields that for each $\varepsilon>0$,

(3.20)
$\begin{array}[]{ll}\displaystyle\exp\Big{(}\gamma\varepsilon\int_{r}^{\sigma_{m}^{t,r,i}}|\zeta^{i}(t,s)|^{2}ds\Big{)}\leqslant\exp\Big{(}\varepsilon
X_{m}^{t,r,i}\Big{)}\cdot\exp\Big{(}{3\over
2}\varepsilon^{2}\int_{r}^{\sigma_{m}^{t,r,i}}|\zeta^{i}(t,s)|^{2}ds\Big{)}\\\
\hskip
56.9055pt\displaystyle\qquad\cdot\exp\Big{(}\varepsilon\int_{r}^{\sigma_{m}^{t,r,i}}\zeta^{i}(t,s)dW(s)-{3\over
2}\varepsilon^{2}\int_{r}^{\sigma_{m}^{t,r,i}}|\zeta^{i}(t,s)|^{2}ds\Big{)},\
\ r\in[t,T].\end{array}$

Observe that the process

$H_{m}^{t,r,i}(\tau):=\exp\Big{(}3\varepsilon\int_{r}^{\tau\land\sigma_{m}^{t,r,i}}\zeta^{i}(t,s)dW(s)-{9\over
2}\varepsilon^{2}\int_{r}^{\tau\land\sigma_{m}^{t,r,i}}|\zeta^{i}(t,s)|^{2}ds\Big{)}$

is a positive martingale with $H_{m}^{t,r,i}(r)=1$. Taking conditional
expectation on both sides of (3.20) and applying Hölder’s inequality, we
obtain that for each $r\in[t,T]$,

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\mathbb{E}_{r}\Big{[}\exp\Big{(}\gamma\varepsilon\int_{r}^{\sigma_{m}^{t,r,i}}\negthinspace\negthinspace\negthinspace\negthinspace\negthinspace\negthinspace|\zeta^{i}(t,s)|^{2}ds\Big{)}\Big{]}\negthinspace\leqslant\negthinspace\Big{(}\mathbb{E}_{r}\Big{[}\exp\Big{(}3\varepsilon
X_{m}^{t,r,i}\Big{)}\Big{]}\Big{)}^{1\over
3}\Big{(}\mathbb{E}_{r}\Big{[}\exp\Big{(}{9\over
2}\varepsilon^{2}\int_{r}^{\sigma_{m}^{t,r,i}}\negthinspace\negthinspace\negthinspace\negthinspace\negthinspace\negthinspace|\zeta^{i}(t,s)|^{2}ds\Big{)}\Big{]}\Big{)}^{1\over
3}.\end{array}$

Consequently, for $\varepsilon\leqslant{2\gamma\over 9}$, we have

$\Big{\\{}\mathbb{E}_{r}\Big{[}\exp\Big{(}\gamma\varepsilon\int_{r}^{\sigma_{m}^{t,r,i}}|\zeta^{i}(t,s)|^{2}ds\Big{)}\Big{]}\Big{\\}}^{2\over
3}\leqslant\Big{\\{}\mathbb{E}_{r}\Big{[}\exp\Big{(}3\varepsilon
X_{m}^{t,r,i}\Big{)}\Big{]}\Big{\\}}^{1\over 3},\ \ r\in[t,T].$

In view of the definitions of $\sigma_{m}^{t,r,i}$ and $X_{m}^{t,r,i}$,
sending $m\to\infty$ in the above inequality yields that for each
$\varepsilon\in(0,{2\gamma\over 9}]$ and almost every $t\in[0,T]$,

(3.21)
$\begin{array}[]{ll}\displaystyle\Big{\\{}\mathbb{E}_{r}\Big{[}\exp\Big{(}\gamma\varepsilon\int_{r}^{T}|\zeta^{i}(t,s)|^{2}ds\Big{)}\Big{]}\Big{\\}}^{2}\leqslant\lim\limits_{m\rightarrow\infty}\mathbb{E}_{r}\Big{[}\exp\Big{(}3\varepsilon
X_{m}^{t,r,i}\Big{)}\Big{]}\\\ \ \
\displaystyle=\mathbb{E}_{r}\Big{\\{}\exp\Big{[}3\varepsilon\Big{(}\eta^{i}(t,r)-\psi^{i}(t)+\int_{r}^{T}\big{[}\alpha(t,s)+\beta\big{(}|y(s)|+|\zeta(t,s)|^{1+\delta}\big{)}\big{]}ds\Big{)}\Big{]}\Big{\\}}\\\
\ \ \displaystyle\leqslant
K\exp\Big{\\{}3\varepsilon\|\eta(t,r)\|_{\infty}+K\int_{r}^{T}\|y(s)\|_{\infty}ds\Big{\\}}\mathbb{E}_{r}\Big{[}\exp\Big{(}3\varepsilon\beta\int_{r}^{T}|\zeta(t,s)|^{1+\delta}ds\Big{)}\Big{]},\
\ r\in[t,T].\end{array}$

Thus, it follows from Hölder’s inequality that for almost every $t\in[0,T]$,
with $\varepsilon_{0}:={2\gamma\over 9}$,

(3.22)
$\begin{array}[]{ll}\displaystyle\Big{\\{}\mathbb{E}_{r}\Big{[}\exp\Big{(}{\gamma\varepsilon_{0}\over
n}\int_{r}^{T}|\zeta(t,s)|^{2}ds\Big{)}\Big{]}\Big{\\}}^{2}=\Big{\\{}\mathbb{E}_{r}\Big{[}\exp\Big{(}{\gamma\varepsilon_{0}\over
n}\sum_{i=1}^{n}\int_{r}^{T}|\zeta^{i}(t,s)|^{2}ds\Big{)}\Big{]}\Big{\\}}^{2}\\\
\ \
\displaystyle=\Big{\\{}\mathbb{E}_{r}\Big{[}\prod_{i=1}^{n}\exp\Big{(}{\gamma\varepsilon_{0}\over
n}\int_{r}^{T}|\zeta^{i}(t,s)|^{2}ds\Big{)}\Big{]}\Big{\\}}^{2}\\\ \ \
\displaystyle\leqslant\Big{\\{}\prod_{i=1}^{n}\Big{\\{}\mathbb{E}_{r}\Big{[}\exp\Big{(}{\gamma\varepsilon_{0}\over
n}\int_{r}^{T}|\zeta^{i}(t,s)|^{2}ds\Big{)}\Big{]}^{n}\Big{\\}}^{1\over
n}\Big{\\}}^{2}\\\ \ \
\displaystyle=\Big{\\{}\prod_{i=1}^{n}\Big{\\{}\mathbb{E}_{r}\Big{[}\exp\Big{(}\gamma\varepsilon_{0}\int_{r}^{T}|\zeta^{i}(t,s)|^{2}ds\Big{)}\Big{]}\Big{\\}}^{1\over
n}\Big{\\}}^{2}\\\ \ \ \displaystyle\leqslant\negthinspace
K\negthinspace\exp\negthinspace\Big{[}3\varepsilon_{0}\|\eta(t,r)\|_{\infty}\negthinspace\negthinspace+\negthinspace\negthinspace
K\int_{r}^{T}\negthinspace\negthinspace\negthinspace\|y(s)\|_{\infty}ds\Big{]}\Big{\\{}\mathbb{E}_{r}\Big{[}\exp\Big{(}3\varepsilon_{0}\beta\negthinspace\int_{r}^{T}\negthinspace\negthinspace\negthinspace|\zeta(t,s)|^{1+\delta}ds\Big{)}\Big{]}\Big{\\}},\
\ r\in[t,T].\end{array}$

By Young’s inequality that

(3.23)
$3\varepsilon_{0}\beta|\zeta(t,s)|^{1+\delta}\leqslant{\gamma\varepsilon_{0}\over
n}|\zeta(t,s)|^{2}+K,\ \ s\in[t,T].$

Coming back to (3.22), by (3.23) we deduce that for $\hbox{\rm
a.e.}\,t\in[0,T]$,

(3.24)
$\begin{array}[]{ll}\displaystyle\Big{\\{}\mathbb{E}_{r}\Big{[}\exp\Big{(}{\gamma\varepsilon_{0}\over
n}\int_{r}^{T}|\zeta(t,s)|^{2}ds\Big{)}\Big{]}\Big{\\}}^{2}\\\ \ \
\displaystyle\leqslant
K\exp\Big{(}3\varepsilon_{0}\|\eta(t,r)\|_{\infty}\negthinspace\negthinspace+\negthinspace
K\int_{r}^{T}\negthinspace\negthinspace\negthinspace\|y(s)\|_{\infty}ds\Big{)}\mathbb{E}_{r}\Big{[}\exp\Big{(}{\gamma\varepsilon_{0}\over
n}\int_{r}^{T}|\zeta(t,s)|^{2}ds\Big{)}\Big{]},\ \ r\in[t,T],\end{array}$

which leads to

(3.25) $\mathbb{E}_{r}\Big{[}\exp\Big{(}{\gamma\varepsilon_{0}\over
n}\int_{r}^{T}|\zeta(t,s)|^{2}ds\Big{)}\Big{]}\leqslant
K\exp\Big{(}3\varepsilon_{0}\|\eta(t,r)\|_{\infty}+K\int_{r}^{T}\|y(s)\|_{\infty}ds\Big{)},\
\ r\in[t,T].$

Therefore, for any $\nu>0$, noting (by Young’s inequality again)

$\nu|\zeta(t,s)|^{1+\delta}\leqslant{\gamma\varepsilon_{0}\over
n^{2}}|\zeta(t,s)|^{2}+K,$

by Hölder’s inequality and (3.25) we have

(3.26)
$\begin{array}[]{ll}\displaystyle\mathbb{E}_{r}\Big{[}\exp\Big{(}\nu\int_{r}^{T}|\zeta(t,s)|^{1+\delta}ds\Big{)}\Big{]}\leqslant
K\mathbb{E}_{r}\Big{[}\exp\Big{(}{\gamma\varepsilon_{0}\over
n^{2}}\int_{r}^{T}|\zeta(t,s)|^{2}ds\Big{)}\Big{]}\\\ \ \
\displaystyle\leqslant
K\Big{\\{}\mathbb{E}_{r}\Big{[}\exp\Big{(}{\gamma\varepsilon_{0}\over
n}\int_{r}^{T}|\zeta(t,s)|^{2}ds\Big{)}\Big{]}\Big{\\}}^{1\over n}\\\ \ \
\displaystyle\leqslant K\exp\Big{[}{3\varepsilon_{0}\over
n}\|\eta(t,r)\|_{\infty}+K\int_{r}^{T}\|y(s)\|_{\infty}ds\Big{]},\ \
r\in[t,T].\end{array}$

Hence, it follows from (3.18) and (3.26) that for $i=1,\cdots,n$ and
$\hbox{\rm a.e.}\,t\in[0,T]$,

(3.27)
$\begin{array}[]{ll}\displaystyle\exp\Big{(}\gamma|\eta^{i}(t,r)|\Big{)}+\mathbb{E}_{r}\Big{[}\int_{r}^{T}|\zeta^{i}(t,s)|^{2}ds\Big{]}\\\
\ \ \displaystyle\leqslant
K\exp\Big{(}K\int_{r}^{T}\|y(s)\|_{\infty}ds\Big{)}\mathbb{E}_{r}\Big{[}\exp\Big{(}K\int_{r}^{T}|\zeta(t,s)|^{1+\delta}ds\Big{)}\Big{]}\\\
\ \ \displaystyle\leqslant
K\exp\Big{(}K\int_{r}^{T}\|y(s)\|_{\infty}ds\Big{)}\exp\Big{[}{3\varepsilon_{0}\over
n}\|\eta(t,r)\|_{\infty}+K\int_{r}^{T}\|y(s)\|_{\infty}ds\Big{]}\\\ \ \
\displaystyle\leqslant K\exp\Big{(}{3\varepsilon_{0}\over
n}\|\eta(t,r)\|_{\infty}+K\int_{r}^{T}\|y(s)\|_{\infty}ds\Big{)},\ \
r\in[t,T].\end{array}$

Note that $3\varepsilon_{0}={2\gamma\over 3}<\gamma$. The above leads to

(3.28) $\|\eta(t,r)\|_{\infty}\leqslant
K_{0}\Big{(}1+\int_{r}^{T}\|y(s)\|_{\infty}ds\Big{)},\ \ r\in[t,T],$

for some absolute constant $K_{0}>0$, independent of $y(\cdot)$, in
particular. Also, by enlarging $K_{0}>0$ if necessary, we have that (see
(3.27))

(3.29)
$\mathbb{E}_{r}\Big{(}\int_{r}^{T}|\zeta^{i}(t,s)|^{2}ds\Big{)}\leqslant
K_{0}\exp\Big{(}2K_{0}\int_{r}^{T}\|y(s)\|_{\infty}ds\Big{)},\ \ r\in[t,T].$

Now, if

$\|y(s)\|_{\infty}\leqslant K_{0}e^{K_{0}(T-s)},\quad\hbox{\rm
a.e.}\;s\in[0,T],$

then it follows from (3.28) that for a.e. $t\in[0,T]$,

(3.30) $\|\eta(t,r)\|_{\infty}\leqslant
K_{0}+K_{0}\int_{r}^{T}K_{0}e^{K_{0}(T-s)}ds=K_{0}e^{K_{0}(T-r)},\ \
r\in[t,T].$

Then, (3.8) follows by letting $r=t$ in (3.30). Furthermore, in view of
(3.29), one has that for $\hbox{\rm a.e.}\,t\in[0,T]$,

$\big{\|}\zeta(t,\cdot)\big{\|}^{2}_{\overline{{\rm BMO}}[t,T]}\leqslant
nK_{0}\exp\Big{(}2K_{0}e^{K_{0}(T-t)}\Big{)}.$

Hence, (3.9) holds since

$\big{\|}Z(\cdot\,,\cdot)\big{\|}_{\overline{{\rm
BMO}}(\Delta^{*}[0,T])}=\mathop{\rm
esssup}_{t\in[0,T]}\big{\|}\zeta(t,\cdot)\big{\|}_{\overline{{\rm
BMO}}[t,T]}\leqslant\sqrt{nK_{0}}\exp\Big{(}K_{0}e^{K_{0}T}\Big{)}.$

The proof of Proposition 3.1 is then complete.

We are now ready to state and prove the main result of this section.

######  Theorem 3.2.

Let (H3.1) hold. Then for any $\psi(\cdot)\in L_{{\cal
F}_{T}}^{\infty}(0,T;\mathbb{R}^{n})$, BSVIE (1.1) admits a unique adapted
solution $(Y(\cdot),Z(\cdot\,,\cdot))\in
L^{\infty}_{\mathbb{F}}(0,T;\mathbb{R}^{n})\times\overline{{\rm
BMO}}(\Delta^{*}[0,T];\mathbb{R}^{n})$.

Proof. Let $K_{0},\bar{K}_{0}>0$ be the constants in the above proposition.
Let

$\begin{array}[]{ll}\mathscr{B}\triangleq\Big{\\{}(U(\cdot),V(\cdot\,,\cdot))\in
L_{\mathbb{F}}^{\infty}(0,T;\mathbb{R}^{n})\times\overline{{\rm
BMO}}(\Delta^{*}[0,T];\mathbb{R}^{n})\bigm{|}\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\qquad\qquad\qquad\|U(t)\|_{\infty}\leqslant
K_{0}e^{K_{0}(T-t)}\ \ {\rm a.e.}\ t\in[0,T]\ \ {\rm and}\ \
\|V(\cdot\,,\cdot)\|_{\overline{{\rm
BMO}}(\Delta^{*}[0,T])}\leqslant\bar{K}_{0}\Big{\\}}\end{array}$

which is a convex closed set in the Banach space
$L_{\mathbb{F}}^{\infty}(0,T;\mathbb{R}^{n})\times\overline{{\rm
BMO}}(\Delta^{*}[0,T];\mathbb{R}^{n})$. By Proposition 3.1 we know that for
each $(y(\cdot),z(\cdot\,,\cdot))\in\mathscr{B}$, the following BSVIE

(3.31) $\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus 1.0pt\displaystyle
Y(t)=\psi(t)+\int_{t}^{T}g(t,s,y(s),Z(t,s))ds-\int_{t}^{T}Z(t,s)dW(s),\ \
t\in[0,T]\end{array}$

admits a unique adapted solution $(Y(\cdot),Z(\cdot\,,\cdot))\in\mathscr{B}$.
That is to say, the map

$\Gamma\big{(}y(\cdot),z(\cdot\,,\cdot)\big{)}\triangleq\big{(}Y(\cdot),Z(\cdot\,,\cdot)\big{)},\quad\big{(}y(\cdot),z(\cdot\,,\cdot)\big{)}\in\mathscr{B}$

is well-defined and maps from $\mathscr{B}$ to itself. In order to prove the
desired result, it suffices to prove that for some $k>0$, the map $\Gamma$ is
contract in $\mathscr{B}$ with the following equivalent norm:

(3.32)
$\Big{\|}\big{(}U(\cdot),V(\cdot\,,\cdot)\big{)}\Big{\|}_{\mathscr{B}_{k}}\triangleq\sqrt{\mathop{\rm
esssup}_{t\in[0,T]}\Big{(}\Big{\|}e^{{kt\over
2}}U(t)\Big{\|}^{2}_{\infty}+\Big{\|}e^{{k\cdot\over
2}}V(t,\cdot)\Big{\|}^{2}_{\overline{{\rm BMO}}[t,T]}\Big{)}}.$

Now, for any
$(y(\cdot),z(\cdot\,,\cdot)),(\widetilde{y}(\cdot),\widetilde{z}(\cdot\,,\cdot))\in\mathscr{B}$,
set

$(Y(\cdot),Z(\cdot\,,\cdot))=\Gamma(y(\cdot),z(\cdot\,,\cdot)),\qquad(\widetilde{Y}(\cdot),\widetilde{Z}(\cdot\,,\cdot))=\Gamma(\widetilde{y}(\cdot),\widetilde{z}(\cdot\,,\cdot)).$

It follows from Proposition 3.1 that

(3.33) $Y(t)=\eta(t,t),\ Z(t,s)=\zeta(t,s),\
\widetilde{Y}(t)=\widetilde{\eta}(t,t),\
\widetilde{Z}(t,s)=\widetilde{\zeta}(t,s),\ \ (t,s)\in\Delta^{*}[0,T],$

where for almost every $t\in[0,T]$, $(\eta(t,\cdot),\zeta(t,\cdot))$ and
$(\widetilde{\eta}(t,\cdot),\widetilde{\zeta}(t,\cdot))$ are respectively the
unique solutions of the following BSDEs in
$L_{\mathbb{F}}^{\infty}(t,T;\mathbb{R}^{n})\times\overline{{\rm
BMO}}([t,T];\mathbb{R}^{n})$,

$\left\\{\negthinspace\negthinspace\begin{array}[]{ll}\displaystyle\eta(t,r)=\psi(t)+\int_{r}^{T}g(t,s,y(s),\zeta(t,s))ds-\int_{r}^{T}\zeta(t,s)dW(s),\
\ r\in[t,T],\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\widetilde{\eta}(t,r)=\psi(t)+\int_{r}^{T}g(t,s,\widetilde{y}(s),\widetilde{\zeta}(t,s))ds-\int_{r}^{T}\widetilde{\zeta}(t,s)dW(s),\
\ r\in[t,T].\end{array}\right.$

By the definition of $\mathscr{B}$ and (3.33), for almost every $t\in[0,T]$,
we have

(3.34)
$\phi(|y(t)|)\leqslant\phi(|y(t)|\vee|\widetilde{y}(t)|)\leqslant\phi\big{(}K_{0}e^{K_{0}(T-t)}\big{)}\leqslant\phi\big{(}K_{0}e^{K_{0}T}\big{)}$

and

(3.35) $\big{\|}\zeta(t,\cdot)\big{\|}_{\overline{\rm
BMO}[t,T]},\big{\|}\widetilde{\zeta}(t,\cdot)\big{\|}_{\overline{\rm
BMO}[t,T]}\leqslant\bar{K}_{0}.$

The rest of the proof is divided into three steps.

Step 1. Girsanov’s transform.

For $(t,s)\in\Delta^{*}[0,T]$, denote

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus 1.0pt\displaystyle\Delta
y(s)=y(s)-\widetilde{y}(s),\quad\Delta\eta(t,s)=\eta(t,s)-\widetilde{\eta}(t,s),\quad\Delta\zeta(t,s)=\zeta(t,s)-\widetilde{\zeta}(t,s),\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\Delta\eta^{i}(t,s)\negthinspace=\negthinspace\eta^{i}(t,s)\negthinspace-\negthinspace\widetilde{\eta}^{i}(t,s),\quad\Delta\zeta^{i}(t,s)\negthinspace=\negthinspace\zeta^{i}(t,s)\negthinspace-\negthinspace\widetilde{\zeta}^{i}(t,s),\quad
1\leqslant i\leqslant n.\end{array}$

Let

$\widehat{\zeta}_{i}(t,s)=(\zeta^{1}(t,s),\cdots,\zeta^{i-1}(t,s),\widetilde{\zeta}^{i}(t,s),\zeta^{i+1}(t,s),\cdots,\zeta^{n}(t,s))^{\top},\quad
1\leqslant i\leqslant n.$

Then

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\zeta(t,s)-\widehat{\zeta}_{i}(t,s)=(0,\cdots,0,\Delta\zeta^{i}(t,s),0,\cdots,0)^{\top},\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\widehat{\zeta}_{i}(t,s)-\widetilde{\zeta}(t,s)=(\Delta\zeta^{1}(t,s),\cdots,\Delta\zeta^{i-1}(t,s),0,\Delta\zeta^{i+1}(t,s),\cdots,\Delta\zeta^{n}(t,s))^{\top}.\end{array}$

Observe that

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus 1.0pt\displaystyle
g^{i}(t,s,y(s),\zeta(t,s))-g^{i}(t,s,\widetilde{y}(s),\widetilde{\zeta}(t,s))\\\
\ \
\displaystyle=g^{i}(t,s,y(s),\zeta(t,s))-g^{i}(t,s,y(s),\widehat{\zeta}_{i}(t,s))+g^{i}(t,s,y(s),\widehat{\zeta}_{i}(t,s))-g^{i}(t,s,\widetilde{y}(s),\widetilde{\zeta}(t,s))\\\
\ \ \displaystyle=g^{i}_{z}(t,s)\Delta\zeta^{i}(t,s)+{\cal
G}^{i}(t.s),\end{array}$

where

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus 1.0pt\displaystyle
g^{i}_{z}(t,s)\buildrel\triangle\over{=}{g^{i}(t,s,y(s),\zeta(t,s))-g^{i}(t,s,y(s),\widehat{\zeta}_{i}(t,s))\over\Delta\zeta^{i}(t,s)}{\bf
1}_{\\{\Delta\zeta^{i}(t,s)\neq 0\\}},\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle{\cal
G}^{i}(t,s)\buildrel\triangle\over{=}g^{i}(t,s,y(s),\widehat{\zeta}_{i}(t,s))-g^{i}(t,s,\widetilde{y}(s),\widetilde{\zeta}(t,s)).\end{array}$

By (3.3) and (3.34), we have

(3.36) $\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle|g^{i}_{z}(t,s)|\leqslant\phi\big{(}K_{0}e^{K_{0}T}\big{)}\Big{(}1+|\zeta(t,s)|+|\widehat{\zeta}_{i}(t,s)|\Big{)},\qquad
j\neq i,\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\displaystyle|{\cal
G}^{i}(t,s)|\leqslant\phi\big{(}K_{0}e^{K_{0}T}\big{)}\Big{[}\Big{(}1+|\widehat{\zeta}_{i}(t,s)|+|\widetilde{\zeta}(t,s)|\Big{)}|\Delta
y(s)|\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\qquad\qquad\qquad\qquad\qquad+\sum_{j\neq
i}\Big{(}1+|\widehat{\zeta}_{i}(t,s)|^{\delta}+|\widetilde{\zeta}(t,s)|^{\delta}\Big{)}|\Delta\zeta^{j}(t,s)|\Big{]}.\end{array}$

According to the above notation, we have

(3.37) $\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\Delta\eta^{i}(t,r)=\negthinspace\negthinspace\int_{r}^{T}\negthinspace\negthinspace\negthinspace\Big{(}g^{i}_{z}(t,s)\Delta\zeta^{i}(t,s)+{\cal
G}^{i}(t,s)\Big{)}ds-\negthinspace\negthinspace\int_{r}^{T}\negthinspace\negthinspace\negthinspace\Delta\zeta^{i}(t,s)dW(s),\
\ r\in[t,T].\end{array}$

Extend $g^{i}_{z}(t,s)$ to be zero for $s\in[0,t)$ and let

$N_{i}(t,r):=\int_{0}^{r}g^{i}_{z}(t,s)dW(s),\ \ r\in[0,T].$

In view of (3.35) and (3.36), we know that $N_{i}(t,\cdot)$ is a BMO
martingale under $\mathbb{P}$, and that

(3.38) $\|N_{i}(t,\cdot)\|_{{\rm BMO}_{\mathbb{P}}[0,T]}\leqslant
K\Big{(}1+2\big{\|}\zeta(t,\cdot)\big{\|}_{\overline{{\rm
BMO}}[t,T]}+\big{\|}\widetilde{\zeta}(t,\cdot)\big{\|}_{\overline{{\rm
BMO}}[t,T]}\Big{)}\leqslant K.$

Then, by [25, Theorem 2.3] we know that the stochastic exponential
$\mathscr{E}[N_{i}(t,\cdot)](\cdot)$ of $N_{i}(t,\cdot)$, given by

$\mathscr{E}[N_{i}(t,\cdot)](r)=\exp\Big{\\{}\int_{0}^{r}g^{i}_{z}(t,s)dW(s)-{1\over
2}\int_{0}^{r}|g^{i}_{z}(t,s)|^{2}ds\Big{\\}},\ \ r\in[0,T],$

is a uniformly integrable martingale under $\mathbb{P}$.

Now, we define the probability measure $\mathbb{P}^{t,i}$ on $(\Omega,{\cal
F}_{T})$ by the following:

$d\mathbb{P}^{t,i}=\mathscr{E}[N_{i}(t,\cdot)](T)d\mathbb{P}.$

Note that

$M_{i}(t,r)=\int_{0}^{r}\Delta\zeta^{i}(t,s)dW(s),\ \ r\in[0,T]$

is also a BMO martingale under $\mathbb{P}$. It follows from [25, Theorem 3.6]
that the process

(3.39) $\widetilde{W}_{i}(t,r)=W(r)-\int_{0}^{r}g^{i}_{z}(t,s)ds,\ \
r\in[0,T]$

is a standard Brownian motion under the probability measure
$\mathbb{P}^{t,i}$, and that the process

$\widetilde{M}_{i}(t,r)=\int_{0}^{r}\Delta\zeta^{i}(t,s)d\widetilde{W}_{i}(t,s),\
\ r\in[0,T]$

the Girsanov’s transform of $M_{i}(t,\cdot)$, is a BMO martingale under the
probability measure $\mathbb{P}^{t,i}$. Moreover, by (3.37) and (3.39) we know
that for $i=1,\cdots,n$,

(3.40)
$\Delta\eta^{i}(t,r)+\int_{r}^{T}\Delta\zeta^{i}(t,s)d\widetilde{W}_{i}(t,s)=\int_{r}^{T}{\cal
G}^{i}(t,s)ds,\ \ r\in[t,T].\vspace{0.2cm}$

Step 2. Basic estimate.

Fix $i=1,\cdots,n$, $0\leqslant t\leqslant r\leqslant T$ and $k>0$. Let
$\mathbb{E}^{t,i}_{r}[\xi]$ denote the conditional expectation of the random
variable $\xi$ with respect to ${\cal F}_{r}$ under the probability measure
$\mathbb{P}^{t,i}$. Taking square and $\mathbb{E}^{t,i}_{r}[\cdot]$ in both
sides of (3.40) yields that

(3.41)
$|\Delta\eta^{i}(t,r)|^{2}+\mathbb{E}^{t,i}_{r}\Big{(}\int_{r}^{T}|\Delta\zeta^{i}(t,s)|^{2}ds\Big{)}=\mathbb{E}^{t,i}_{r}\Big{(}\Big{|}\int_{r}^{T}{\cal
G}^{i}(t,s)ds\Big{|}^{2}\Big{)}.$

which implies that

(3.42)
$e^{kr}|\Delta\eta^{i}(t,r)|^{2}+\mathbb{E}^{t,i}_{r}\Big{(}\int_{r}^{T}e^{kr}|\Delta\zeta^{i}(t,s)|^{2}ds\Big{)}=\mathbb{E}^{t,i}_{r}\Big{[}e^{kr}\Big{(}\int_{r}^{T}|{\cal
G}^{i}(t,s)|ds\Big{)}^{2}\Big{]}.$

Note that

$\int_{r}^{T}ke^{ks}\Big{(}\int_{s}^{T}|\Delta\zeta^{i}(t,u)|^{2}du\Big{)}ds=\Big{(}e^{ks}\int_{s}^{T}|\Delta\zeta^{i}(t,u)|^{2}du\Big{)}\Big{|}^{T}_{r}+\int_{r}^{T}e^{ks}|\Delta\zeta^{i}(t,s)|^{2}ds.$

As a result,

(3.43)
$\int_{r}^{T}e^{ks}|\Delta\zeta^{i}(t,s)|^{2}ds=\int_{r}^{T}e^{kr}|\Delta\zeta^{i}(t,s)|^{2}ds+k\int_{r}^{T}\Big{(}\int_{s}^{T}e^{ks}|\Delta\zeta^{i}(t,u)|^{2}du\Big{)}ds.$

Combining (3.42) and (3.43), we deduce that

(3.44) $\begin{array}[]{ll}\displaystyle
e^{kr}|\Delta\eta^{i}(t,r)|^{2}+\mathbb{E}^{t,i}_{r}\Big{(}\int_{r}^{T}e^{ks}|\Delta\zeta^{i}(t,s)|^{2}ds\Big{)}\\\
\ \ \displaystyle\leqslant
2\mathbb{E}^{t,i}_{r}\Big{[}e^{kr}\Big{(}\int_{r}^{T}\big{|}{\cal
G}^{i}(t,s)\big{|}ds\Big{)}^{2}\Big{]}+k\mathbb{E}^{t,i}_{r}\Big{\\{}\int_{r}^{T}\mathbb{E}^{t,i}_{s}\Big{[}e^{ks}\Big{(}\int_{s}^{T}\big{|}{\cal
G}^{i}(t,u)\big{|}du\Big{)}^{2}\Big{]}ds\Big{\\}}\\\ \ \
\displaystyle\leqslant
2\mathbb{E}^{t,i}_{r}\Big{[}e^{kr}\Big{(}\int_{r}^{T}\big{|}{\cal
G}^{i}(t,s)\big{|}ds\Big{)}^{2}\Big{]}+\mathbb{E}^{t,i}_{r}\Big{[}\int_{r}^{T}ke^{ks}\Big{(}\int_{s}^{T}\big{|}{\cal
G}^{i}(t,u)\big{|}du\Big{)}^{2}ds\Big{]}.\end{array}$

Furthermore, observe by (3.36) and Hölder’s inequality that

$\begin{array}[]{ll}\displaystyle\Big{(}\int_{r}^{T}|{\cal
G}^{i}(t,s)|ds\Big{)}^{2}\leqslant
K\Big{\\{}\int_{r}^{T}\Big{[}\Big{(}1+|\zeta(t,s)|+|\widetilde{\zeta}(t,s)|\Big{)}|\Delta
y(s)|\\\
\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\sum_{j\neq
i}\Big{(}1+|\zeta(t,s)|^{\delta}+|\widetilde{\zeta}(t,s)|^{\delta}\Big{)}|\Delta\zeta^{j}(t,s)|\Big{]}ds\Big{\\}}^{2}\\\
\displaystyle\leqslant K\Big{\\{}\Big{[}\int_{r}^{T}e^{-{k\over 2}s}e^{{k\over
2}s}\Big{(}1+|\zeta(t,s)|+|\widetilde{\zeta}(t,s)|\Big{)}|\Delta
y(s)|ds\Big{]}^{2}\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\qquad\qquad\qquad+\sum_{j\neq
i}\Big{[}\int_{r}^{T}e^{-{k\over 2}s}e^{{k\over
2}s}\Big{(}1+|\zeta(t,s)|^{\delta}+|\widetilde{\zeta}(t,s)|^{\delta}\Big{)}|\Delta\zeta^{j}(t,s)|ds\Big{]}^{2}\Big{\\}}\\\
\displaystyle\leqslant
K\Big{[}\Big{(}\int_{r}^{T}e^{-ks}ds\Big{)}\Big{(}\int_{r}^{T}e^{ks}\big{(}1+|\zeta(t,s)|+|\widetilde{\zeta}(t,s)|\big{)}^{2}|\Delta
y(s)|^{2}ds\Big{)}\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\qquad\qquad+\sum_{j\neq
i}\Big{(}\int_{r}^{T}e^{-ks}\Big{(}1+|\zeta(t,s)|^{\delta}+|\widetilde{\zeta}(t,s)|^{\delta}\Big{)}^{2}ds\Big{)}\Big{(}\int_{r}^{T}e^{ks}|\Delta\zeta^{j}(t,s)|^{2}ds\Big{)}\Big{]}\\\
\displaystyle\leqslant{Ke^{-kr}\over k}\int_{r}^{T}e^{ks}\vartheta(t,s)|\Delta
y(s)|^{2}ds\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\qquad\qquad+K\Big{(}\int_{r}^{T}e^{-{k\over
1-\delta}s}ds\Big{)}^{1-\delta}\Big{(}\int_{r}^{T}|\vartheta(t,s)ds\Big{)}^{\delta}\Big{(}\int_{r}^{T}e^{ks}|\Delta\zeta(t,s)|^{2}ds\Big{)}\\\
\displaystyle\leqslant{Ke^{-kr}\over k}\int_{r}^{T}e^{ks}\vartheta(t,s)|\Delta
y(s)|^{2}ds+{Ke^{-kr}\over
k^{1-\delta}}\Big{(}\int_{r}^{T}\vartheta(t,s)ds\Big{)}^{\delta}\Big{(}\int_{r}^{T}e^{ks}|\Delta\zeta(t,s)|^{2}ds\Big{)},\end{array}$

where $\vartheta(t,s):=1+|\zeta(t,s)|^{2}+|\widetilde{\zeta}(t,s)|^{2},\
s\in[t,T]$. Then,

(3.45) $\displaystyle e^{kr}\Big{(}\int_{r}^{T}\big{|}{\cal
G}^{i}(t,s)\big{|}ds\Big{)}^{2}\displaystyle\leqslant{K\over
k}\int_{r}^{T}e^{ks}\vartheta(t,s)|\Delta y(s)|^{2}ds+{K\over
k^{1-\delta}}\Big{(}1+\int_{r}^{T}\vartheta(t,s)ds\Big{)}\int_{r}^{T}e^{ks}|\Delta\zeta(t,s)|^{2}ds.$

Thus, we also have that for each $s\in[t,T]$,

$\displaystyle e^{ks\over 2}\Big{(}\int_{s}^{T}\big{|}{\cal
G}^{i}(t,u)\big{|}du\Big{)}^{2}\displaystyle\leqslant{K\over
k}\int_{s}^{T}e^{ku\over 2}\vartheta(t,u)|\Delta y(u)|^{2}du+{K\over
k^{1-\delta}}\Big{(}1+\int_{s}^{T}\vartheta(t,u)du\Big{)}\int_{s}^{T}e^{ku\over
2}|\Delta\zeta(t,u)|^{2}du.$

It then follows from the above inequality that

(3.46)
$\begin{array}[]{ll}\displaystyle\int_{r}^{T}ke^{ks}\Big{(}\int_{s}^{T}\big{|}{\cal
G}^{i}(t,u)\big{|}du\Big{)}^{2}ds\leqslant K\int_{r}^{T}\Big{(}e^{ks\over
2}\int_{s}^{T}e^{ku\over 2}\vartheta(t,u)|\Delta y(u)|^{2}du\Big{)}ds\\\
\displaystyle\qquad\qquad\qquad+{K\over
k^{1-\delta}}\Big{(}1+\int_{r}^{T}\vartheta(t,s)ds\Big{)}\int_{r}^{T}\Big{(}ke^{ks\over
2}\int_{s}^{T}e^{ku\over 2}|\Delta\zeta(t,u)|^{2}du\Big{)}ds.\\\ \ \
\displaystyle={2K\over k}\Big{(}e^{ks\over 2}\int_{s}^{T}e^{ku\over
2}\vartheta(t,u)|\Delta
y(u)|^{2}du\Big{|}^{T}_{r}+\int_{r}^{T}e^{ks}\vartheta(t,s)|\Delta
y(s)|^{2}ds\Big{)}\\\ \displaystyle\qquad+{K\over
k^{1-\delta}}\Big{(}1+\int_{r}^{T}\vartheta(t,s)ds\Big{)}\Big{(}2e^{ks\over
2}\int_{s}^{T}e^{ku\over
2}|\Delta\zeta(t,u)|^{2}du\Big{|}^{T}_{r}+2\int_{r}^{T}e^{ks}|\Delta\zeta(t,s)|^{2}ds\Big{)}\\\
\ \ \displaystyle\leqslant{K\over k}\int_{r}^{T}e^{ks}\vartheta(t,s)|\Delta
y(s)|^{2}ds+{K\over
k^{1-\delta}}\Big{(}1+\int_{r}^{T}\vartheta(t,s)ds\Big{)}\int_{r}^{T}e^{ks}|\Delta\zeta(t,s)|^{2}ds.\end{array}$

Now, we substitute (3.45) and (3.46) into (3.44) to obtain that

(3.47) $\begin{array}[]{ll}\displaystyle
e^{kr}|\Delta\eta^{i}(t,r)|^{2}+\mathbb{E}^{t,i}_{r}\Big{[}\int_{r}^{T}e^{ks}|\Delta\zeta^{i}(t,s)|^{2}ds\Big{]}\\\
\ \ \displaystyle\leqslant{K\over
k}\mathbb{E}^{t,i}_{r}\Big{[}\int_{r}^{T}e^{ks}\vartheta(t,s)|\Delta
y(s)|^{2}ds\Big{]}+{K\over
k^{1-\delta}}\mathbb{E}^{t,i}_{r}\Big{[}\Big{(}1+\int_{r}^{T}\vartheta(t,s)ds\Big{)}\int_{r}^{T}e^{ks}|\Delta\zeta(t,s)|^{2}ds\Big{]}.\vspace{0.2cm}\end{array}$

Step 3. Contraction mapping.

By the definitions of $\vartheta(\cdot\,,\cdot)$,
$\mathbb{E}^{t,i}[\,\cdot\,]$ and $\mathbb{P}^{t,i}$ together with (2.3),
(3.35), (3.38) and the energy inequality, it follows that for each
$i=1,\cdots,n$, each $t\in[0,T]$ and each $\tau\in\mathscr{T}[t,T]$,

(3.48) $\Big{\|}e^{k\cdot\over
2}\Delta\zeta^{i}(t,\cdot)\Big{\|}^{2}_{\overline{{\rm BMO}}[t,T]}\leqslant
K\sup_{\tau\in\mathscr{T}[t,T]}\left\|\mathbb{E}^{t,i}_{\tau}\Big{[}\int_{\tau}^{T}e^{ks}|\Delta\zeta^{i}(t,s)|^{2}ds\Big{]}\right\|_{\infty},$

and

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\mathbb{E}^{t,i}_{\tau}\Big{[}\int_{\tau}^{T}\vartheta(t,s)ds\Big{]}\leqslant
K\Big{(}1+\big{\|}\zeta(t,\cdot)\big{\|}^{2}_{\overline{{\rm
BMO}}[t,T]}+\big{\|}\widetilde{\zeta}(t,\cdot)\big{\|}^{2}_{\overline{{\rm
BMO}}[t,T]}\Big{)}\leqslant K,\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\mathbb{E}^{t,i}_{\tau}\Big{[}\Big{(}1+\int_{\tau}^{T}\vartheta(t,s)ds\Big{)}^{2}\Big{]}\leqslant
K\Big{(}1+\big{\|}\zeta(t,\cdot)\big{\|}^{4}_{\overline{{\rm
BMO}}[t,T]}+\big{\|}\widetilde{\zeta}(t,\cdot\big{\|}^{4}_{\overline{{\rm
BMO}}[t,T]}\Big{)}\leqslant K,\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\mathbb{E}^{t,i}_{\tau}\Big{[}\Big{(}\int_{\tau}^{T}e^{ks}|\Delta\zeta(t,s)|^{2}ds\Big{)}^{2}\Big{]}\leqslant
K\Big{\|}e^{k\cdot\over 2}\Delta\zeta(t,\cdot)\Big{\|}^{4}_{\overline{{\rm
BMO}}[t,T]},\end{array}$

and then, by virtue of Hölder’s inequality, we have

(3.49)
$\mathbb{E}^{t,i}_{\tau}\Big{[}\int_{\tau}^{T}e^{ks}\vartheta(t,s)|\Delta
y(s)|^{2}ds\Big{]}\leqslant K\Big{\|}e^{k\cdot\over 2}\Delta
y(\cdot)\Big{\|}^{2}_{L^{\infty}_{\mathbb{F}}(t,T;\mathbb{R}^{n})},$

and

(3.50)
$\mathbb{E}^{t,i}_{\tau}\Big{[}\Big{(}1+\int_{\tau}^{T}\vartheta(t,s)ds\Big{)}\int_{\tau}^{T}e^{ks}|\Delta\zeta(t,s)|^{2}ds\Big{]}\leqslant
K\Big{\|}e^{k\cdot\over 2}\Delta\zeta(t,\cdot)\Big{\|}^{2}_{\overline{{\rm
BMO}}[t,T]}.$

Note that (3.47) holds still when $r$ is replaced with any
$\tau\in\mathscr{T}[t,T]$. It follows from (3.48)–(3.50) that for almost every
$t\in[0,T]$ and $i=1,\cdots,n$,

$\displaystyle e^{kt}|\Delta\eta^{i}(t,t)|^{2}+\Big{\|}e^{k\cdot\over
2}\Delta\zeta^{i}(t,\cdot)\Big{\|}^{2}_{\overline{{\rm
BMO}}[t,T]}\displaystyle\leqslant{K\over k}\Big{\|}e^{k\cdot\over 2}\Delta
y(\cdot)\Big{\|}^{2}_{L_{\mathbb{F}}^{\infty}(t,T;\mathbb{R}^{n})}+{K\over
k^{1-\delta}}\Big{\|}e^{k\cdot\over
2}\Delta\zeta(t,\cdot)\Big{\|}^{2}_{\overline{{\rm BMO}}[t,T]}.$

Picking $k$ large enough, we obtain

$\big{\|}(Y(\cdot)-\widetilde{Y}(\cdot),Z(\cdot\,,\cdot)-\widetilde{Z}(\cdot\,,\cdot))\big{\|}_{\mathscr{B}_{k}}\leqslant{1\over
2}\big{\|}(y(\cdot)-\widetilde{y}(\cdot),z(\cdot\,,\cdot)-\widetilde{z}(\cdot\,,\cdot))\big{\|}_{\mathscr{B}_{k}}.$

This means that $\Gamma$ is a contraction mapping on the convex closed
$\mathscr{B}$ under the norm $\|\cdot\|_{\mathscr{B}_{k}}$, which is just the
desired result. The proof is then complete.

######  Remark 3.3.

Let us make some comparisons with the relevant literature on quadratic growth
BSVIEs.

(i) In [21], multi-dimensional BSVIEs were studied when the generator is
allowed to depend on the diagonal value $Z(s,s)$. Certain differentiability on
the generator was assumed. In addition, the approach used relies on the
interesting ideas developed in [35] where the relevant data are assumed to be
bounded and sufficiently small.

(ii) In [37], one-dimensional BSVIEs (1.1) with quadratic growth were
investigated. In contrast, our assumption (3.3) in Theorem 3.1 is weaker than
that in [37, Theorem 3.5] even in the one-dimensional framework.

######  Remark 3.4.

One crucial step for the proof of Proposition 3.1 is the estimate (3.47). To
establish that, we have borrowed some interesting ideas in [32, Lemma 3.1]. In
the above proof, we have imposed the element in $\mathscr{B}$ with a new
equivalent norm in (3.32). This helps us bypass the using stochastic Fredholm
integral equations, and simplify the arguments as in [37] and [48]. Such
equivalent norm techniques also appeared in e.g. [45]. However, we have to
introduce more delicate arguments due to our quadratic growth framework.

###  3.2 A mixed case of linear and quadratic growth

Note that due to the diagonally strictly quadratic condition in (H3.1),
Theorem 3.2 could not cover the classical case of $z\mapsto g(t,s,y,z)$
growing linearly. To fill this gap, in this subsection, we are going to
consider a class of multi-dimensional coupled BSVIEs for which some components
of the generator have linear growth $z$, and others have diagonally strictly
quadratic growth.

To begin with, let us first recall the multi-dimensional BSDEs:

(3.51) $\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus 1.0pt\displaystyle
Y(t)=\xi+\int_{t}^{T}f(s,Y(s),Z(s))ds-\int_{t}^{T}Z(s)dW(s),\ \
t\in[0,T].\end{array}$

where the free term $\xi$ is an ${\cal F}_{T}$-measurable
$\mathbb{R}^{n}$-valued random variable, and the generator
$f:[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\Omega\to\mathbb{R}^{n}$
has the property that $s\mapsto f(s,y,z)$ is $\mathbb{F}$-progressively
measurable for all $(y,z)\in\mathbb{R}^{n}\times\mathbb{R}^{n}$. We emphasize
that in the literature, with bounded and/or some unbounded terminal states,
only the following three mutually exclusive cases were studied: For the map
$z\mapsto f(s,y,z)$,

$\bullet$ It has a linear growth (see the classical literature [29, 12]);

$\bullet$ It is diagonally strictly quadratic, and sub-quadratically for off-
diagonal components (see [15]);

$\bullet$ It is diagonally no more than quadratic growth and with bounded for
off-diagonal components (see [24, 15]).

We now would like to look at the mixed case of BSVIEs by which we mean that
some components of the generator $f$ have linear growth in $z$ and the other
components have at most quadratic growth (not necessarily strict) in $z$ in a
special way. More precisely, after a possible relabeling, we have the
following assumptions on the generator $f$ of BSDE (3.51).

(H3.2) There exist a non-negative valued process $\alpha(\cdot)\in
L^{\infty}_{\mathbb{F}}(\Omega;L^{1}(0,T;\mathbb{R}))$, an increasing
deterministic function $\phi:\mathbb{R}\to\mathbb{R}_{+}$ and a constant
$\beta>0$ such that for all
$(t,y,z)\in[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}$,

(3.52)
$|f^{i}(t,y,z)|\leqslant\alpha(t)+\phi(|y|)+\beta\big{(}|z|+|z^{i}|^{2}\big{)},\quad
1\leqslant i\leqslant n,$

and for some $k=0,1,2,\cdots,n$,

(3.53) $\left\\{\negthinspace\negthinspace\begin{array}[]{ll}\vskip 3.0pt plus
1.0pt minus 1.0pt\displaystyle\hbox{\rm
sgn$\,$}(y^{i})f^{i}(t,y,z)\leqslant\alpha(t)+\beta\big{(}|y|+|z^{i}|^{2}\big{)},\qquad
1\leqslant i\leqslant k,\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\hbox{\rm
sgn$\,$}(y^{i})f^{i}(t,y,z)\leqslant\alpha(t)+\beta\Big{(}|y|+\sum_{j=k+1}^{n}|z^{j}|\Big{)},\qquad
k+1\leqslant i\leqslant n.\end{array}\right.$

Here, $k=0$ means that the first condition in (3.53) is absent, and $k=n$
means that the second condition in (3.53) is absent. In addition, there exists
a $\delta\in[0,1)$ such that for each $t\in[0,T]$ and
$y,\bar{y},z,\bar{z}\in\mathbb{R}^{n}$,

(3.54) $\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle|f^{i}(t,y,z)-f^{i}(t,\bar{y},\bar{z})|\leqslant\phi(|y|\vee|\bar{y}|)\Big{[}\big{(}1+|z|+|\bar{z}|\big{)}\big{(}|y-\bar{y}|+|z^{i}-\bar{z}^{i}|\big{)}\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\big{(}1+|z|^{\delta}+|\bar{z}|^{\delta}\big{)}\sum_{j\neq
i}|z^{j}-\bar{z}^{j}|\Big{]},\qquad 1\leqslant i\leqslant n.\end{array}$

Inspired by Lemma 2.7, the following establishes an a priori estimate on the
solution of BSDE (3.51) under hypothesis (H3.2), and it is comparable with
[15, Lemma 4.1].

######  Proposition 3.5.

Let (H3.2) hold and $\xi\in L^{\infty}_{{\cal F}_{T}}(\Omega;\mathbb{R}^{n})$.
Assume that for some $T_{0}\in[0,T)$, BSDE (3.51) has an adapted solution
$(Y(\cdot),Z(\cdot))\in
L^{\infty}_{\mathbb{F}}(\Omega;C([T_{0},T];\mathbb{R}^{n}))\times\overline{{\rm
BMO}}([T_{0},T];\mathbb{R}^{n})$ on time interval $[T_{0},T]$. Then, there
exists an absolute constant $K$ depending only on
$(n,\beta,T,\xi,\alpha(\cdot))$ and being independent of $T_{0}$ such that

$\|Y(\cdot)\|^{2}_{L^{\infty}_{\mathbb{F}}(\Omega;C[T_{0},T])}+\|Z(\cdot)\|^{2}_{\overline{\rm
BMO}[T_{0},T]}\leqslant K.$

Proof. First of all, taking into account the first inequality in (3.53),
together with BSDE (3.51), by virtue of Lemma 2.7 (or [14, Proposition 2.1]),
taking $\delta=1$ and

$\alpha_{0}(s)=\alpha(s)+\beta|Y(s)|,$

we have that for each $i=1,2,\cdots,k$ and $t\in[T_{0},T]$,

(3.55) $\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\exp\Big{(}|Y^{i}(t)|\Big{)}+\mathbb{E}_{t}\Big{(}\int_{t}^{T}|Z^{i}(s)|^{2}ds\Big{)}\leqslant\displaystyle
K\mathbb{E}_{t}\Big{[}\exp\Big{(}K|\xi^{i}|+K\int_{t}^{T}\Big{(}\alpha(s)+\beta|Y(s)|\Big{)}ds\Big{)}\Big{]}\\\
\hskip 156.49014pt\leqslant\displaystyle
K\exp\Big{(}K\|\xi^{i}\|_{\infty}+K\int_{t}^{T}\|Y(\cdot)\|_{L^{\infty}_{\mathbb{F}}(\Omega;C[s,T])}ds\Big{)},\end{array}$

and then

(3.56) $|Y^{i}(t)|\leqslant
K+K\int_{t}^{T}\|Y(\cdot)\|_{L^{\infty}_{\mathbb{F}}(\Omega;C[s,T])}ds.$
(3.57) $\sum_{j=1}^{k}\|Z^{j}(\cdot)\|^{2}_{\overline{\rm BMO}[t,T]}\leqslant
K\exp\Big{(}K\int_{t}^{T}\|Y(\cdot)\|_{L^{\infty}_{\mathbb{F}}(\Omega;C[s,T])}ds\Big{)}.$

Furthermore, in view of the second inequality of in (3.53), using Itô’s
formula, Young’s inequality and Hölder’s inequality, we can deduce that for
each $i=k+1,\cdots,n$ and $t\in[T_{0},T]$,

(3.58)
$\begin{array}[]{ll}\displaystyle|Y^{i}(t)|^{2}+\mathbb{E}_{t}\Big{[}\int_{t}^{T}|Z^{i}(s)|^{2}ds\Big{]}=\mathbb{E}_{t}|\xi^{i}|^{2}+\mathbb{E}_{t}\Big{[}\int_{t}^{T}2Y^{i}(s)f^{i}(s,Y(s),Z(s))ds\Big{]}\\\
\ \ \displaystyle\leqslant\|\xi\|_{\infty}^{2}+{1\over
6}\|Y^{i}(\cdot)\|^{2}_{L^{\infty}_{\mathbb{F}}(\Omega;C[t,T])}+6\mathbb{E}_{t}\Big{\\{}\Big{[}\int_{t}^{T}\negthinspace\negthinspace\negthinspace\Big{(}\alpha(s)+\beta|Y(s)|\negthinspace+\negthinspace\beta\negthinspace\negthinspace\negthinspace\sum_{j=k+1}^{n}|Z^{j}(s)|\Big{)}ds\Big{]}^{2}\Big{\\}}\\\
\ \ \displaystyle\leqslant K+{1\over
6}\|Y^{i}(\cdot)\|^{2}_{L^{\infty}_{\mathbb{F}}(\Omega;C[t,T])}+K\int_{t}^{T}\sum_{j=1}^{n}|Y^{j}(s)|^{2}ds+K\mathbb{E}_{t}\Big{[}\Big{(}\int_{t}^{T}\sum_{j=k+1}^{n}|Z^{j}(s)|ds\Big{)}^{2}\Big{]}.\vspace{0.2cm}\end{array}$

With the last inequality (3.58) in hand, using a similar computation to step 2
in the proof of Theorem 3.2 and using Hölder’s inequality we deduce that for
each $i=k+1,\cdots,n$ and $t\in[T_{0},T]$,

$\begin{array}[]{ll}\displaystyle|\bar{Y}^{i}(t)|^{2}+\mathbb{E}_{t}\Big{[}\int_{t}^{T}|\bar{Z}^{i}(s)|^{2}ds\Big{]}\\\
\ \ \displaystyle\leqslant 3Ke^{\rho
T}+\frac{1}{2}\|\bar{Y}^{i}(\cdot)\|^{2}_{L^{\infty}_{\mathbb{F}}(\Omega;C[t,T])}\negthinspace+\negthinspace
K\negthinspace\negthinspace\int_{t}^{T}\negthinspace\negthinspace\sum_{j=1}^{n}|\bar{Y}^{j}(s)|^{2}ds+{1\over
2n^{2}}\mathbb{E}_{t}\Big{[}\int_{t}^{T}\negthinspace\negthinspace\negthinspace\Big{(}\sum_{j=k+1}^{n}|\bar{Z}^{j}(s)|\Big{)}^{2}ds\Big{]}\\\
\ \ \displaystyle\leqslant
K+\frac{1}{2}\|\bar{Y}^{i}(\cdot)\|^{2}_{L^{\infty}_{\mathbb{F}}(\Omega;C[t,T])}+K\int_{t}^{T}\sum_{j=1}^{n}\|\bar{Y}^{j}(\cdot)\|^{2}_{L^{\infty}_{\mathbb{F}}(\Omega;C[s,T])}ds+{1\over
2n}\sum_{j=k+1}^{n}\|\bar{Z}^{j}(\cdot)\|^{2}_{\overline{{\rm
BMO}}[t,T]},\end{array}$

where $\rho>0$ only depends on $n$, $\beta$,

$\ \bar{Y}^{i}(s):=e^{\rho s\over 2}Y^{i}(s)\ \ {\rm and}\ \
\bar{Z}^{i}(s):=e^{\rho s\over 2}Z^{i}(s),\ \ s\in[t,T].$

Thus, for each $i=k+1,\cdots,n$, and each $t\in[T_{0},T]$, we have

$\begin{array}[]{ll}\displaystyle{1\over
2}\|\bar{Y}^{i}(\cdot)\|^{2}_{L^{\infty}_{\mathbb{F}}(\Omega;C[t,T])}+\|\bar{Z}^{i}(\cdot)\|^{2}_{\overline{{\rm
BMO}}[t,T]}\\\ \ \ \displaystyle\leqslant
K+K\int_{t}^{T}\sum_{j=1}^{n}\|\bar{Y}^{j}(\cdot)\|^{2}_{L^{\infty}_{\mathbb{F}}(\Omega;C[s,T])}ds\displaystyle+{1\over
2n}\sum_{j=k+1}^{n}\|\bar{Z}^{j}(\cdot)\|^{2}_{\overline{{\rm
BMO}}[t,T]},\end{array}$

and then

(3.59)
$\displaystyle\sum_{j=k+1}^{n}\|\bar{Y}^{j}(\cdot)\|^{2}_{L^{\infty}_{\mathbb{F}}(\Omega;C[t,T])}+\sum_{j=k+1}^{n}\|\bar{Z}^{j}(\cdot)\|^{2}_{\overline{{\rm
BMO}}([t,T])}\leqslant
K+K\int_{t}^{T}\sum_{j=1}^{n}\|\bar{Y}^{j}(\cdot)\|^{2}_{L^{\infty}_{\mathbb{F}}(\Omega;C[s,T])}ds.\vspace{0.2cm}$

On the other hand, by (3.56) and Hölder’s inequality we deduce that for each
$i=1,\cdots,k$ and $t\in[T_{0},T]$,

$|\bar{Y}^{i}(t)|^{2}\leqslant
K+K\int_{t}^{T}\sum_{j=1}^{n}\|\bar{Y}^{j}(\cdot)\|^{2}_{L^{\infty}_{\mathbb{F}}(\Omega;C[s,T])}ds,$

and then

(3.60)
$\sum_{j=1}^{k}\|\bar{Y}^{j}(\cdot)\|^{2}_{L^{\infty}_{\mathbb{F}}(\Omega;C[t,T])}\leqslant
K+K\int_{t}^{T}\sum_{j=1}^{n}\|\bar{Y}^{j}(\cdot)\|^{2}_{L^{\infty}_{\mathbb{F}}(\Omega;C[s,T])}ds.\vspace{0.2cm}$

Combining (3.59) and (3.60) yields that

(3.61) $\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\sum_{j=1}^{n}\|\bar{Y}^{j}(\cdot)\|^{2}_{L^{\infty}_{\mathbb{F}}(\Omega;C[t,T])}\leqslant
K+K\int_{t}^{T}\sum_{j=1}^{n}\|\bar{Y}^{j}(\cdot)\|^{2}_{L^{\infty}_{\mathbb{F}}(\Omega;C[s,T])}ds,\
\ t\in[T_{0},T].\end{array}$

It then follows from Gronwall’s inequality that

(3.62)
$\sum_{j=1}^{n}\|\bar{Y}^{j}(\cdot)\|^{2}_{L^{\infty}_{\mathbb{F}}(\Omega;C[t,T])}\leqslant
K,\ \ t\in[T_{0},T].$

Finally, combining (3.57), (3.59) and (3.62) yields that

(3.63) $\sum_{j=1}^{n}\|\bar{Z}^{j}(\cdot)\|^{2}_{\overline{\rm
BMO}[t,T]}\leqslant K,\ \ t\in[T_{0},T].$

Thus, the desired conclusion follows from (3.62)–(3.63) immediately.

The following existence and uniqueness result on the bounded solution of
multi-dimensional BSDEs strengthens [15, Theorem 2.4], which amounts to saying
$k=n$ in (H3.2)

######  Theorem 3.6.

Let (H3.2) hold. Then, for any $\xi\in L^{\infty}_{{\color[rgb]{1,0,0}{\cal
F}_{T}}}(\Omega;\mathbb{R}^{n})$, BSDE (3.51) admits a unique adapted solution
$(Y(\cdot),Z(\cdot))\in
L^{\infty}_{\mathbb{F}}(\Omega;C([0,T];\mathbb{R}^{n}))\times\overline{{\rm
BMO}}([0,T];\mathbb{R}^{n})$.

Proof. In view of [15, Theorem 2.1], Proposition 3.5 and the proof of [15,
Theorem 2.4], the desired conclusion follows immediately.

Now, let us come back the study on BSVIEs, and introduce the following
assumption on the generator $g$ of BSVIE (1.1), which is comparable with
(H3.2).

(H3.3) Let (H3.1) hold with (3.2) replaced by the following: For some
$k=0,1,2,\cdots,n$ and each
$(t,s,y,z)\in\Delta^{*}[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}$, it
holds that

(3.64)
$\left\\{\negthinspace\negthinspace\begin{array}[]{ll}\displaystyle|g^{i}(t,s,y,z)|\leqslant\alpha(t,s)+\beta\big{(}|y|+|z^{i}|^{2}\big{)},\qquad
1\leqslant i\leqslant k,\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle|g^{i}(t,s,y,z)|\leqslant\alpha(t,s)+\beta\Big{(}|y|+\sum_{j=k+1}^{n}|z^{j}|\Big{)},\qquad
k+1\leqslant i\leqslant n.\end{array}\right.$

Like (H3.2), when $k=0$, the first condition in (3.64) is absent and when
$k=n$, the second condition of (3.64) is absent. The following Proposition 3.7
is comparable with the previous Proposition 3.1.

######  Proposition 3.7.

Let (H3.3) hold. Then, for each $\psi(\cdot)\in
L^{\infty}(0,T;\mathbb{R}^{n})$ and $y(\cdot)\in
L^{\infty}_{\mathbb{F}}(0,T;\mathbb{R}^{n})$, BSVIE (3.6) admits a unique
adapted solution $(Y(\cdot),Z(\cdot\,,\cdot))\in
L^{\infty}_{\mathbb{F}}(0,T;\mathbb{R}^{n})\times\overline{{\rm
BMO}}(\Delta^{*}[0,T];\mathbb{R}^{n})$. Furthermore, there exists a constant
$\rho>0$ depending on $(n,\beta)$, and a constant $K_{0}>0$ depending only on
$(\psi(\cdot),\alpha(\cdot,\cdot))$ and $(\beta,n,T)$ such that with

$\bar{y}(t):=e^{\rho s\over 2}y(t),\quad\bar{Y}(t):=e^{\rho s\over
2}Y(t),\quad\bar{Z}(t,s):=e^{\rho s\over
2}Z(t,s),\quad(t,s)\in\Delta^{*}[0,T],$

as long as

$\|\bar{y}(\cdot)\|^{2}_{L^{\infty}_{\mathbb{F}}(t,T;\mathbb{R}^{n})}\leqslant
K_{0}e^{K_{0}(T-t)},\quad\hbox{\rm a.e.}\ t\in[0,T],$

it holds that

$\|\bar{Y}(\cdot)\|^{2}_{L^{\infty}_{\mathbb{F}}(t,T;\mathbb{R}^{n})}\leqslant
K_{0}e^{K_{0}(T-t)},\quad\hbox{\rm a.e.}\ t\in[0,T],$

and

$\|\bar{Z}(\cdot\,,\cdot)\|^{2}_{\overline{{\rm
BMO}}(\Delta^{*}[0,T])}\leqslant
K_{0}\exp\big{(}K_{0}e^{K_{0}T}\big{)}+K_{0}e^{K_{0}T}=:\bar{K}_{0}.$

Proof. In view of (H3.3), it follows from Theorem 3.6 that for almost every
$t\in[0,T]$, the following BSDE

(3.65) $\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\eta(t,r)=\psi(t)+\int_{r}^{T}g(t,s,y(s),\zeta(t,s))ds-\int_{r}^{T}\zeta(t,s)dW(s),\
\ r\in[t,T]\end{array}$

admits a unique solution $(\eta(t,\cdot),\zeta(t,\cdot))\in
L^{\infty}_{\mathbb{F}}(\Omega;C([t,T];\mathbb{R}^{n}))\times\overline{{\rm
BMO}}([t,T];\mathbb{R}^{n})$. Moreover, by Proposition 3.5 we know that there
exists a constant $K>0$ depending only on
$\|y(\cdot)\|_{L^{\infty}_{\mathbb{F}}(0,T;\mathbb{R}^{n})}$ and
$(\alpha(\cdot\,,\cdot),\psi(\cdot),\beta,n,T)$ such that

$\|\eta(t,\cdot)\|_{L^{\infty}_{\mathbb{F}}(\Omega;C([t,T])}+\|\zeta(t,\cdot)\|_{\overline{{\rm
BMO}}[t,T]}\leqslant K,\quad\hbox{\rm a.e.}\;t\in[0,T].$

Let

(3.66) $Y(t)=\eta(t,t),\quad Z(t,s)=\zeta(t,s),\quad(t,s)\in\Delta^{*}[0,T].$

Then $(Y(\cdot),Z(\cdot\,,\cdot))\in
L^{\infty}_{\mathbb{F}}(0,T;\mathbb{R}^{n})\times\overline{{\rm
BMO}}(\Delta^{*}[0,T];\mathbb{R}^{n})$ is an adapted solution of BSVIE (3.6).
Moreover, in view of Theorem 3.6 again, a same argument as in Proposition 3.1
yields the uniqueness of the solution of BSVIE (3.6).

Finally, in view of (3.64), using a similar argument to obtain (3.57), (3.59)
and (3.61), we can deduce that there exists a constant $K_{0}>0$ depending
only on $(\alpha(\cdot\,,\cdot),\psi(\cdot),\beta,n,T)$ such that for almost
every $t\in[0,T]$,

(3.67) $\sum_{j=1}^{k}\|\zeta^{j}(t,\cdot)\|^{2}_{\overline{{\rm
BMO}}[t,T]}\leqslant
K_{0}\exp\Big{(}K_{0}\int_{t}^{T}\sum_{j=1}^{n}\|y^{j}(\cdot)\|_{L^{\infty}_{\mathbb{F}}(s,T;\mathbb{R})}ds\Big{)},$
(3.68)
$\begin{array}[]{ll}\displaystyle\sum_{j=k+1}^{n}\|\bar{\zeta}^{j}(t,\cdot)\|^{2}_{\overline{{\rm
BMO}}[t,T]}\leqslant
K_{0}+K_{0}\int_{t}^{T}\sum_{j=1}^{n}\|\bar{y}^{j}(\cdot)\|^{2}_{L^{\infty}_{\mathbb{F}}(s,T;\mathbb{R})}ds\vspace{0.2cm}\end{array}$

and

(3.69)
$\sum_{j=1}^{n}\|\bar{\eta}^{j}(t,\cdot)\|^{2}_{L^{\infty}_{\mathbb{F}}(\Omega;C[t,T])}\leqslant
K_{0}+K_{0}\int_{t}^{T}\sum_{j=1}^{n}\|\bar{y}^{j}(\cdot)\|^{2}_{L^{\infty}_{\mathbb{F}}(s,T;\mathbb{R})}ds,$

where for each $j=1,\cdots,n$,

$\bar{y}^{j}(s):=e^{\rho s\over 2}y^{j}(s),\ \ \bar{\eta}^{j}(t,s):=e^{\rho
s\over 2}\eta^{j}(t,s)\ \ {\rm and}\ \ \bar{\zeta}^{j}(t,s):=e^{\rho s\over
2}\zeta^{j}(t,s),\ \ s\in(t,T].$

Thus, in view of (3.30), the desired assertion follows from (3.66)–(3.69)
immediately.

The following Theorem 3.8 is comparable with the previous Theorem 3.2, which
is the main result of this subsection.

######  Theorem 3.8.

Let (H3.3) hold. Then, for each $\psi(\cdot)\in
L^{\infty}(0,T;\mathbb{R}^{n})$, BSVIE (1.1) admits a unique adapted solution
$(Y(\cdot),Z(\cdot\,,\cdot))\in
L^{\infty}_{\mathbb{F}}(0,T;\mathbb{R}^{n})\times\overline{{\rm
BMO}}(\Delta^{*}[0,T];\mathbb{R}^{n})$.

Proof. Let $K_{0}$, $\bar{K}_{0}$ and $\rho$ be the constants in Proposition
3.7. It is clear that

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\mathscr{B}:=\Big{\\{}U(\cdot),V(\cdot\,,\cdot))\in
L^{\infty}_{\mathbb{F}}(0,T;\mathbb{R}^{n})\times\overline{{\rm
BMO}}(\Delta^{*}[0,T];\mathbb{R}^{n})\bigm{|}\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\qquad\qquad\|\bar{U}(\cdot)\|^{2}_{L^{\infty}_{\mathbb{F}}(t,T;\mathbb{R}^{n})}\leqslant
K_{0}e^{K_{0}(T-t)},\ \ \hbox{\rm a.e.}\ t\in[0,T]\ \ {\rm and}\ \
\|\bar{V}(\cdot\,,\cdot)\|^{2}_{\overline{{\rm
BMO}}(\Delta^{*}[0,T])}\leqslant\bar{K}_{0},\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\qquad\qquad\hbox{where for each}\ t\in[0,T],\ \
\bar{U}(s):=e^{ks\over 2}U(s)\ \ {\rm and}\ \ \bar{V}(t,s):=e^{ks\over
2}V(t,s),\ s\in(t,T]\Big{\\}}\end{array}$

is a convex closed set in Banach space
$L^{\infty}_{\mathbb{F}}(0,T;\mathbb{R}^{n})\times\overline{{\rm
BMO}}(\Delta^{*}[0,T];\mathbb{R}^{n})$. By Proposition 3.7 we know that for
each $(y(\cdot),z(\cdot\,,\cdot))\in\mathscr{B}$, the following BSVIE

$\displaystyle
Y(t)=\psi(t)+\int_{t}^{T}g(t,s,y(s),Z(t,s))ds-\int_{t}^{T}Z(t,s)dW(s),\ \
t\in[0,T]$

admits a unique adapted solution $(Y(\cdot),Z(\cdot,\cdot))\in\mathscr{B}$.
That is to say, the map

$\Gamma\big{(}y(\cdot),z(\cdot,\cdot)\big{)}:=\big{(}Y(\cdot),Z(\cdot,\cdot)\big{)},\
\ \forall\big{(}y(\cdot),z(\cdot,\cdot)\big{)}\in\mathscr{B}$

is well defined and stable in $\mathscr{B}$. The rest proof runs as in Theorem
3.2.

##  4 Multi-dimensional BSVIEs with Unbounded Free Terms

In this section, we are going to investigate multi-dimensional BSVIEs with
unbounded free terms. We will treat the quadratic and sub-quadratic cases in a
unified framework. Therefore, in this section, we let $\delta\in(0,1]$,
allowing $\delta=1$ (which corresponds to the quadratic case).

We first introduce the following assumption on the generator $g$ of BSVIEs
(1.1) which is comparable with assumptions (H2.2)–(H2.4) for BSDEs.

(H4.1) The map
$g:\Delta^{*}[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$
is measurable so that $s\mapsto g(t,s,y,z)$ is $\mathbb{F}$-progressively
measurable on $[t,T]$ for each
$(t,y,z)\in[0,T)\times\mathbb{R}^{n}\times\mathbb{R}^{n}$. There exist
constants $\beta>0$, $\delta\in(0,1]$ and a non-negative process
$\alpha:\Delta^{*}[0,T]\times\Omega\to\mathbb{R}_{+}$ with
$s\mapsto\alpha(t,s)$ being $\mathbb{F}$-progressively measurable on $[t,T]$,
and with

$\mathbb{E}\Big{\\{}\mathop{\rm
esssup}_{t\in[0,T]}\Big{[}\exp\Big{(}p\,\Big{|}\int_{t}^{T}\negthinspace\negthinspace\alpha(t,s)ds\Big{|}^{2\delta\over
1+\delta}\Big{)}\Big{]}\Big{\\}}<\infty,\quad\forall p\geqslant 1,$

such that $g(t,s,y,z):=(g^{1}(t,s,y,z^{1}),\cdots,g^{n}(t,s,y,z^{n}))^{\top}$
with

(4.1) $\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle|g^{i}(t,s,y,z^{i})|\negthinspace\leqslant\negthinspace\alpha(t,s)\negthinspace+\negthinspace\beta\big{(}|y|\negthinspace+\negthinspace|z^{i}|^{1+\delta}\big{)},\qquad\forall(t,s,y,z)\negthinspace\in\negthinspace\Delta^{*}[0,T]\negthinspace\times\negthinspace\mathbb{R}^{n}\negthinspace\times\negthinspace\mathbb{R}^{n},~{}1\leqslant
i\leqslant n,\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle|g(t,s,y,z)-g(t,s,\bar{y},z)|\leqslant\beta|y-\bar{y}|,\qquad\quad\forall(t,s)\in\Delta^{*}[0,T],~{}y,\bar{y},z\in\mathbb{R}^{n},\end{array}$

and

(4.2) $\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus 1.0pt\displaystyle
g^{i}(t,s,y,(1-\theta)z^{i}+\theta\bar{z}^{i})-\theta
g^{i}(t,s,y,\bar{z}^{i})\leqslant(1-\theta)\big{[}\alpha(t,s)+\beta\big{(}|y|+|z^{i}|^{1+\delta}\big{)}\big{]},\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\qquad\qquad\qquad\qquad\qquad\forall(t,s)\in\Delta^{*}[0,T],~{}y,z,\bar{z}\in\mathbb{R}^{n},~{}\theta\in(0,1),~{}1\leqslant
i\leqslant n.\end{array}$

Relevant to the above, let us introduce the following hypothesis.

(H4.1)′ Let (H4.1) hold with (4.2) replaced by the following: For each
$i=1,2,\cdots,n$, either

(4.3) $\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus 1.0pt\displaystyle
g^{i}(t,s,y,(1-\theta)z^{i}+\theta\bar{z}^{i})-\theta
g^{i}(t,s,y,\bar{z}^{i})\leqslant(1-\theta)\big{[}\alpha(t,s)+\beta\big{(}|y|+|z^{i}|^{1+\delta}\big{)}\big{]},\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\forall(t,s)\in\Delta^{*}[0,T],~{}y,z,\bar{z}\in\mathbb{R}^{n},~{}\theta\in(0,1),\end{array}$

or

(4.4) $\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus 1.0pt\displaystyle
g^{i}(t,s,y,(1-\theta)z^{i}+\theta\bar{z}^{i})-\theta
g^{i}(t,s,y,\bar{z}^{i})\geqslant-(1-\theta)\big{[}\alpha(t,s)+\beta\big{(}|y|+|z^{i}|^{1+\delta}\big{)}\big{]},\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\forall(t,s)\in\Delta^{*}[0,T],~{}y,z,\bar{z}\in\mathbb{R}^{n},~{}\theta\in(0,1).\end{array}$

Although (H4.1)′ seems to be a little general than (H4.1), we now show that
actually they are equivalent after some minor modifications. In fact, by
relabeling, we may assume that for some $k=0,1,2,\cdots,n$, one has (4.3)
holds for $1\leqslant i\leqslant k$, and (4.4) holds for $k+1\leqslant
i\leqslant n$. If $k=0$, it means that for all $i=1,2,\cdots,n$, (4.4) holds,
and if $k=n$, it means that for all $i=1,2,\cdots,n$, (4.3) holds. Now, we
define

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\bar{z}=(\bar{z}^{1},\cdots,\bar{z}^{k},\bar{z}^{k+1},\cdots,\bar{z}^{n})=(z^{1},\cdots,z^{k},-z^{k+1},\cdots,-z^{n}),\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\bar{y}=(\bar{y}^{1},\cdots,\bar{y}^{k},\bar{y}^{k+1},\cdots,\bar{y}^{n})=(y^{1},\cdots,y^{k},-y^{k+1},\cdots,-y^{n}),\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\bar{g}^{i}(t,s,y,{\color[rgb]{0,0,1}z^{i}})=g^{i}(t,s,\bar{y},{\color[rgb]{0,0,1}\bar{z}^{i}}),\quad
1\leqslant i\leqslant k,\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\bar{g}^{i}(t,s,y,{\color[rgb]{0,0,1}z^{i}})=-g^{i}(t,s,\bar{y},{\color[rgb]{0,0,1}\bar{z}^{i}}),\quad
k+1\leqslant i\leqslant n.\end{array}$

Then (H4.1)′ holds for $g(t,s,y,z)$ if and only if (H4.1) holds for
$\bar{g}(t,s,\bar{y},\bar{z})$.

Next, we introduce the following hypothesis.

(H4.2) Let (H4.1) hold with (4.2) replaced by the following: for each
$i=1,2,\cdots,n$, and each $(t,s,y)\in\Delta^{*}[0,T]\times\mathbb{R}^{n}$,
$z^{i}\mapsto g^{i}(t,s,y,z^{i})$ be convex, or concave.

It is clear that when (H4.2) holds, then (H4.1)′ must hold for $g$. Hence,
(H4.1) (or (H4.1)′) is not very restrictive.

Further, if the generator $g$ satisfies (4.3) or (4.4), then
$g^{i}(t,s,y,\cdot\,)$ is locally Lipschitz continuous for each
$(t,s,y)\in\Delta^{*}[0,T]\times\mathbb{R}^{n}$ and each $i=1,2,\cdots,n$. We
only verify the case of (4.3). The other case can be proved in the same way.
Suppose that (4.3) holds. For each fixed $z_{0}\in\mathbb{R}$, let

$\widetilde{g}^{\,i}(t,s,y,z):=g^{i}(t,s,y,z_{0}+z)-g^{i}(t,s,y,z_{0}),\quad\forall(t,s,y,z)\in\Delta^{*}[0,T]\times\mathbb{R}^{n}\times\mathbb{R},\
\ 1\leqslant i\leqslant n.$

By (4.3), for each $(z_{1},z_{2})\in\mathbb{R}\times\mathbb{R}$ and
$\theta\in(0,1)$, we have $\widetilde{g}^{i}(t,s,y,0)=0$ and

(4.5) $\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\widetilde{g}^{\,i}(t,s,y,(1-\theta)z_{1}+\theta
z_{2})-\theta\widetilde{g}^{\,i}(t,s,y,z_{2})\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle=g^{i}(t,s,y,(1-\theta)(z_{0}+z_{1})+\theta(z_{0}+z_{2}))-\theta
g^{i}(t,s,y,z_{0}+z_{2})-(1-\theta)g^{i}(t,s,y,z_{0})\\\ \vskip 3.0pt plus
1.0pt minus
1.0pt\cr\displaystyle\leqslant(1-\theta)\big{(}\alpha(t,s)+\beta|y|+\beta|z_{0}+z_{1}|^{1+\delta}\big{)}+(1-\theta)|g^{i}(t,s,y,z_{0})|\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\leqslant(1-\theta)\big{(}K(t,s,y,z_{0})+2^{\delta}\beta|z_{1}|^{1+\delta}\big{)},\end{array}$

with

$K(t,s,y,z_{0})=\alpha(t,s)+\beta|y|+2^{\delta}\beta|z_{0}|^{1+\delta}+|g^{i}(t,s,y,z_{0})|.\vspace{0.2cm}$

Letting $z_{1}=1,z_{2}=0$ and $z_{1}=1,z_{2}=1-1/\theta$ in (4.5) respectively
yields that for each $\theta\in(0,1)$,

$\widetilde{g}^{\,i}(t,s,y,1-\theta)\leqslant\big{(}K(t,s,y,z_{0})+2^{\delta}\beta\big{)}(1-\theta)$

and

$\widetilde{g}^{\,i}\big{(}t,s,y,1-{1\over\theta}\big{)}\geqslant-\big{(}K(t,s,y,z_{0})+2^{\delta}\beta)\Big{|}1-{1\over\theta}\Big{|},$

which means that

$|g^{i}(t,s,y,z_{0}+z)-g^{i}(t,s,y,z_{0})|=|\widetilde{g}^{\,i}(t,s,y,z)|\leqslant\big{(}K(t,s,y,z_{0})+2^{\delta}\beta\big{)}|z|,\quad\forall
z\in(-1,1).$

This gives the local Lipschitz of $g^{i}(t,s,y,\cdot)$.

In what follows, we use $K>0$ to represent a generic positive constant
depending only on $(\beta,\delta,T,n)$ which could be different from line to
line, and if it also depends on $p$, we will denote it by $K_{p}$. In
addition, for notational convenience, we will frequently use the following
function

(4.6) $\Phi_{\lambda}(x;\mu):=\exp(\mu x^{\lambda}),\quad\forall
x,\mu\geqslant 0,\ \ \lambda>0.$

Note that for each $x_{1},x_{2},\mu\geqslant 0$ and $\lambda\in(0,1]$,

(4.7)
$\Phi_{\lambda}(x_{1}+x_{2};\mu)\leqslant\Phi_{\lambda}(x_{1};\mu)\Phi_{\lambda}(x_{2};\mu).$

######  Proposition 4.1.

Let (H4.1) hold and $\lambda\buildrel\triangle\over{=}{2\delta\over
1+\delta}$. Then, for any $\psi(\cdot)\in\cap_{p\geqslant 1}{\cal
E}^{p,\lambda}_{{\cal F}_{T}}(\Omega;L^{\infty}(0,T;\mathbb{R}^{n}))$ and
$y(\cdot)\in\cap_{p\geqslant 1}{\cal
E}_{\mathbb{F}}^{p,\lambda}(\Omega;L^{\infty}(0,T;\mathbb{R}^{n}))$, the
following BSVIE

(4.8) $Y(t)=\psi(t)+\int_{t}^{T}g(t,s,y(s),Z(t,s))ds-\int_{t}^{T}Z(t,s)dW(s),\
\ t\in[0,T]$

admits a unique adapted solution

$(Y(\cdot),Z(\cdot\,,\cdot))\in\Big{[}\bigcap_{p\geqslant 1}{\cal
E}_{\mathbb{F}}^{p,\lambda}(\Omega;L^{\infty}(0,T;\mathbb{R}^{n}))\Big{]}\times\Big{[}\bigcap_{p\geqslant
1}L^{\infty}(0,T;L^{p}_{\mathbb{F}}(\Omega;L^{2}(\cdot\,,T;\mathbb{R}^{n})))\Big{]},$

such that $\eta(t,\cdot)\in\cap_{p\geqslant 1}{\cal
E}_{\mathbb{F}}^{p,\lambda}(\Omega;C([t,T];\mathbb{R}^{n}))$ for almost every
$t\in[0,T]$, where

(4.9)
$\eta(t,r):=Y(t)-\int_{t}^{r}g(t,s,y(s),Z(t,s))ds+\int_{t}^{r}Z(t,s)dW(s),\ \
r\in[t,T].$

Proof. Fix $y(\cdot)\in\cap_{p\geqslant 1}{\cal
E}_{\mathbb{F}}^{p,\lambda}(\Omega;L^{\infty}(0,T;\mathbb{R}^{n}))$. For each
$i=1,\cdots,n$, and almost every $t\in[0,T]$, it follows from (4.1) and (4.2)
that

(4.10)
$|g^{i}(t,s,y(s),z)|\leqslant\alpha(t,s)+\beta|y(s)|+\beta|z|^{1+\delta},\ \
\forall(s,z)\in[t,T]\times\mathbb{R}.$

and

(4.11) $\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus 1.0pt\displaystyle
g^{i}(t,s,y(s),(1-\theta)z^{i}+\theta\bar{z}^{i})-\theta
g^{i}(t,s,y(s),\bar{z}^{i})\leqslant(1-\theta)\big{[}\alpha(t,s)+\beta|y(s)|+\beta|z^{i}|^{1+\delta}\big{]},\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\qquad\qquad\qquad\qquad\qquad\forall(t,s)\in\Delta^{*}[0,T],~{}z,\bar{z}\in\mathbb{R}^{n},~{}\theta\in(0,1),~{}1\leqslant
i\leqslant n.\end{array}$

Furthermore, due to integrability of $\alpha(\cdot\,,\cdot)$, $\psi(\cdot)$
and $y(\cdot)$, by Hölder’s inequality we have

(4.12)
$\mathbb{E}\Big{[}\Phi_{\lambda}\Big{(}|\psi^{i}(t)|+\int_{t}^{T}\big{(}\alpha(t,s)+\beta|y(s)|\big{)}ds;p\Big{)}\Big{]}<+\infty,\
\ \forall p>1.$

Then, by virtue of Proposition 2.5 we know that the following one-dimensional
parameterized BSDE

(4.13)
$\eta^{i}(t,r)=\psi^{i}(t)+\int_{r}^{T}g^{i}(t,s,y(s),\zeta^{i}(t,s))ds-\int_{r}^{T}\zeta^{i}(t,s)dW(s),\
\ r\in[t,T]$

admits a unique adapted solution $(\eta^{i}(t,\cdot),\zeta^{i}(t,\cdot))$,
taking value in $\mathbb{R}^{2}$ such that

$\mathbb{E}\Big{[}\Phi_{\lambda}\Big{(}\sup_{s\in[t,T]}|\eta^{i}(t,s)|;p\Big{)}+\Big{(}\int_{t}^{T}|\zeta^{i}(t,s)|^{2}dt\Big{)}^{p\over
2}\Big{]}<+\infty,\ \ \forall p>1.$

Let

(4.14) $\eta(t,\cdot):=(\eta^{1}(t,\cdot),\cdots,\eta^{n}(t,\cdot))^{\top}\ \
{\rm and}\ \
\zeta(t,\cdot):=(\zeta^{1}(t,\cdot),\cdots,\zeta^{n}(t,\cdot)^{\top}.\vspace{0.1cm}$

Then, by virtue of Hölder’s inequality we know that for almost every
$t\in[0,T]$, $(\eta(t,\cdot),\zeta(t,\cdot))$ is the unique solution in the
space $[\bigcap_{p\geqslant 1}{\cal
E}_{\mathbb{F}}^{p,\lambda}(\Omega;C([t,T];\mathbb{R}^{n}))]\times[\bigcap_{p\geqslant
1}L^{p}_{\mathbb{F}}(\Omega;L^{2}(t,T;\mathbb{R}^{n}))]$ of the following BSDE

(4.15)
$\eta(t,r)=\psi(t)+\int_{r}^{T}g(t,s,y(s),\zeta(t,s))ds-\int_{r}^{T}\zeta(t,s)dW(s),\
\ r\in[t,T].$

Let

(4.16) $\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus 1.0pt\displaystyle
Y(t)=\eta(t,t),\ \ Z(t,s)=\zeta(t,s),\ \ \
(t,s)\in\Delta^{*}[0,T].\end{array}$

Then, it follows from (4.15) and (4.16) that $(Y(\cdot),Z(\cdot,\cdot))$
solves BSVIE (4.8), and that the $\eta(t,\cdot)$ defined in (4.9) equals to
that in (4.15) and then it belongs to $\bigcap_{p\geqslant 1}{\cal
E}_{\mathbb{F}}^{p,\lambda}(\Omega;C([t,T];\mathbb{R}^{n}))$.

Furthermore, define

(4.17) $\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\displaystyle\Psi_{\lambda}(\nu):=\mathop{\rm
esssup}_{t\in[0,T]}\Phi_{\lambda}\Big{(}|\psi(t)|+\int_{t}^{T}\alpha(t,s)ds;\nu\Big{)},\
\ \nu\in[0,+\infty).\end{array}$

In view of (4.10)–(4.13), by using Lemma 2.7 we can deduce that for each
$i=1,\cdots,n$, almost every $t\in[0,T]$ and each $p>1$,

$\begin{array}[]{ll}\displaystyle\Phi_{\lambda}\Big{(}|Y^{i}(t)|;1\Big{)}=\displaystyle\Phi_{\lambda}\Big{(}|\eta^{i}(t,t)|;1\Big{)}\leqslant
K\mathbb{E}_{t}\Big{[}\Phi_{\lambda}\Big{(}|\psi^{i}(t)|+\int_{t}^{T}\big{(}\alpha(t,s)+\beta|y(s)|\big{)}ds;K\Big{)}\Big{]}\\\
\hskip 64.01869pt\leqslant\displaystyle
K\mathbb{E}_{t}\Big{[}\Psi_{\lambda}(K)\Phi_{\lambda}\Big{(}\int_{t}^{T}|y(s)|ds;K\Big{)}\Big{]}\end{array}$

and

$\displaystyle\mathbb{E}\Big{[}\Phi_{\lambda}\Big{(}\sup_{s\in[t,T]}|\eta^{i}(t,s)|;p\Big{)}\Big{]}+\mathbb{E}\Big{[}\Big{(}\int_{t}^{T}|Z^{i}(t,s)|^{2}ds\Big{)}^{p\over
2}\Big{]}\leqslant\displaystyle
K\mathbb{E}\Big{[}\Psi_{\lambda}(K_{p})\Phi_{\lambda}\Big{(}\int_{t}^{T}|y(s)|ds;K_{p}\Big{)}\Big{]},$

and then, by virtue of Jensen’s inequality, Doob’s martingale inequality and
Hölder’s inequality together with (4.14), (4.7) and (4.16), we have

(4.18)
$\begin{array}[]{ll}\displaystyle\Phi_{\lambda}\Big{(}|Y(t)|;1\Big{)}\leqslant\Phi_{\lambda}\Big{(}\sum_{i=1}^{n}|Y^{i}(t)|;1\Big{)}\displaystyle=\prod_{i=1}^{n}\Phi_{\lambda}\Big{(}|Y^{i}(t)|;1\Big{)}\\\
\ \
\displaystyle\leqslant\Big{\\{}K\mathbb{E}_{t}\Big{[}\Psi_{\lambda}(K)\Phi_{\lambda}\Big{(}\int_{t}^{T}|y(s)|ds;K\Big{)}\Big{]}\Big{\\}}^{n}\leqslant
K\mathbb{E}_{t}\Big{[}\Psi_{\lambda}(K)\Phi_{\lambda}\Big{(}\int_{t}^{T}|y(s)|ds;K\Big{)}\Big{]},\end{array}$
(4.19)
$\begin{array}[]{ll}\displaystyle\mathbb{E}\Big{[}\Phi_{\lambda}\Big{(}\mathop{\rm
esssup}_{t\in[0,T]}|Y(t)|;p\Big{)}\Big{]}=\displaystyle\mathbb{E}\Big{[}\mathop{\rm
esssup}_{t\in[0,T]}\Phi_{\lambda}\Big{(}|Y(t)|;p\Big{)}\Big{]}\\\ \vskip 3.0pt
plus 1.0pt minus 1.0pt\cr\displaystyle\leqslant
K_{p}\mathbb{E}\Big{\\{}\mathop{\rm