# Backward Stochastic Differential Equations and Backward Stochastic Volterra
Integral Equations with Anticipating Generators

Hanxiao Wang, Jiongmin Yong, Chao Zhou College of Mathematics and Statistics,
Shenzhen University, Shenzhen 518060, China (Email: hxwang@szu.edu.cn). This
author would like to thank Chenchen Mou (of City University of Hong Kong) for
some useful discussions.Department of Mathematics, University of Central
Florida, Orlando, FL 32816, USA (Email: jiongmin.yong@ucf.edu). This author is
supported in part by NSF Grant DMS-1812921. Department of Mathematics and Risk
Management Institute, National University of Singapore, Singapore 119076,
Singapore (Email: matzc@nus.edu.sg). This author is supported by NSFC Grant
11871364 as well as Singapore MOE AcRF Grants R-146-000-271-112 and
R-146-000-284-114.

(This paper is dedicated to Professor Alain Bensoussan on the occasion of his
80th birthday)

Abstract. For a backward stochastic differential equation (BSDE, for short),
when the generator is not progressively measurable, it might not admit adapted
solutions, shown by an example. However, for backward stochastic Volterra
integral equations (BSVIEs, for short), the generators are allowed to be
anticipating. This gives, among other things, an essential difference between
BSDEs and BSVIEs. Under some proper conditions, the well-posedness of such
kind of BSVIEs is established. Further, the results are extended to path-
dependent BSVIEs, in which the generators can depend on the future paths of
unknown processes. An additional finding is that for path-dependent BSVIEs, in
general, the situation of anticipating generators is not avoidable and the
adaptedness condition similar to that imposed for anticipated BSDEs by
Peng–Yang [22] is not necessary.

Keywords. backward stochastic Volterra integral equation, backward stochastic
differential equation, anticipating generator, path-dependence.

AMS subject classifications. 60H20, 60H10, 45D05, 45G10.

##  1 Introduction

Let $(\Omega,{\cal F},\mathbb{P})$ be a complete probability space on which a
standard one-dimensional Brownian motion $W=\\{W(t);0\leqslant t<\infty\\}$ is
defined. The augmented natural filtration of $W$ is denoted by
$\mathbb{F}=\\{{\cal F}_{t}\\}_{t\geqslant 0}$. Let $T>0$ be a fixed time
horizon. We denote

$L^{p}_{{\cal
F}_{T}}(\Omega;\mathbb{R}^{n})=\big{\\{}\xi:\Omega\to\mathbb{R}^{n}~{}|~{}\xi\hbox{
is ${\cal F}_{T}$-measurable, }\mathbb{E}|\xi|^{p}<\infty\big{\\}}.$

For any given $\xi\in L^{p}_{{\cal F}_{T}}(\Omega;\mathbb{R}^{n})$ with $p>1$,
let us consider the following backward stochastic differential equation (BSDE,
for short) in its integral form:

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

where
$g:[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\Omega\to\mathbb{R}^{n}$
is a given ${\cal
B}([0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n})\otimes{\cal
F}_{T}$-measurable map, which is called the generator. Any pair
$(Y(\cdot),Z(\cdot))$ of $\mathbb{F}$-adapted processes satisfying (1.1) in
the Itô’s sense is called an adapted solution. To guarantee the well-posedness
of (1.1), the following standard conditions are normally assumed.

(H1)p The process $s\mapsto g(s,y,z)$ is $\mathbb{F}$-progressively measurable
on $[0,T]$, for all $(y,z)\in\mathbb{R}^{n}\times\mathbb{R}^{n}$. The map
$(y,z)\mapsto g(s,y,z)$ is uniformly Lipschitz continuous, and

$\mathbb{E}\Big{(}\int_{0}^{T}|g(s,0,0)|ds\Big{)}^{p}<\infty.$

In the above, the uniformly Lipschitz condition can be relaxed a little. For
convenience, if $s\mapsto g(s,y,z)$ is $\mathbb{F}$-progressively measurable,
we refer to $g(\cdot)$ as an adapted generator. If $s\mapsto g(s,y,z)$ is
merely $\mathscr{F}_{T}$-measurable, we refer to it as an anticipating
generator. The following result is by now standard (see Pardoux–Peng [21] and
El Karoui–Peng–Quenez [6], for example).

######  Theorem 1.1.

Let (H1)p hold with $p>1$. Then, for any $\xi\in L^{p}_{{\cal
F}_{T}}(\Omega;\mathbb{R}^{n})$, BSDE (1.1) admits a unique adapted solution
$(Y(\cdot),Z(\cdot))$ and the following holds:

$\mathbb{E}\Big{[}\sup_{t\in[0,T]}|Y(t)|^{p}+\Big{(}\int_{0}^{T}|Z(s)|^{2}ds\Big{)}^{p\over
2}\Big{]}\leqslant
K\mathbb{E}\Big{[}|\xi|^{p}+\Big{(}\int_{0}^{T}|g(s,0,0)|ds\Big{)}^{p}\Big{]}.$
(1.2)

Hereafter, $K>0$ stands for a generic constant which could be different from
line to line. Moreover, if $\bar{g}(\cdot)$ is another generator satisfying
(H1)p as well, and $\bar{\xi}\in
L^{p}_{\mathscr{F}_{T}}(\Omega;\mathbb{R}^{n})$, then by letting
$(\bar{Y}(\cdot),\bar{Z}(\cdot))$ be the corresponding adapted solution, one
has the following stability estimate:

$\displaystyle\mathbb{E}\Big{[}\sup_{t\in[0,T]}|Y(t)-\bar{Y}(t)|^{p}+\Big{(}\int_{0}^{T}|Z(s)-\bar{Z}(s)|^{2}ds\Big{)}^{p\over
2}\Big{]}$ $\displaystyle\quad\leqslant
K\mathbb{E}\Big{[}|\xi-\bar{\xi}|^{p}+\Big{(}\int_{0}^{T}\big{|}g(s,Y(s),Z(s))-\bar{g}(s,Y(s),Z(s))\big{|}ds\Big{)}^{p}\Big{]}.$
(1.3)

We point out that under (H1)p, the generator of BSDE (1.1) is adapted.
Therefore, it is natural to ask: _Can the generator $g(\cdot)$ of BSDEs be
anticipating?_ From (1.1), one intuitively sees that although the sum of the
first two terms on the right-hand side is $\mathscr{F}_{T}$-measurable, but
the last term is an Itô’s integral over $[t,T]$, which might make the whole
sum $\mathbb{F}$-adapted. However, this is not that simple. Let us present the
following simple example.

######  Example 1.2.

Consider the following BSDE:

$Y(t)=\int_{t}^{T}W(T)ds-\int_{t}^{T}Z(s)dW(s),\qquad t\in[0,T].$ (1.4)

We see that the above BSDE has the zero terminal state and has the generator:

$g(s,y,z)=W(T),\qquad(s,y,z)\in[0,T]\times\mathbb{R}\times\mathbb{R},$

which is anticipating, and independent of $(y,z)$. We claim that BSDE (1.4)
does not have an adapted solution $(Y(\cdot),Z(\cdot))$. To see this, let us
denote

$\psi(t)=\int_{t}^{T}W(T)ds=W(T)(T-t),\qquad t\in[0,T].$

Clearly, $\psi(\cdot)$ is an ${\cal F}_{T}$-measurable process (not
$\mathbb{F}$-adapted). Then, (1.4) can be rewritten as the following

$Y(t)=\psi(t)-\int_{t}^{T}Z(s)dW(s),\qquad t\in[0,T].$ (1.5)

Now, we introduce the following equation:

${\cal Y}(t)=\psi(t)-\int_{t}^{T}{\cal Z}(t,s)dW(s),\qquad t\in[0,T],$ (1.6)

which is a special form of backward stochastic Volterra integral equations
(BSVIEs, for short; see the next section for details). From Yong [42], it is
known that such a BSVIE admits a unique adapted solution $({\cal
Y}(\cdot),{\cal Z}(\cdot,\cdot))$, which has the explicit form:

$({\cal Y}(s),\,{\cal Z}(t,s))=(W(s)(T-s),\,T-t),\qquad 0\leqslant t\leqslant
s\leqslant T.$

Thus, if (1.4) (i.e., (1.5)) has an adapted solution $(Y(\cdot),Z(\cdot))$,
then we must have

$Z(s)={\cal Z}(t,s)=T-t,\qquad 0\leqslant t\leqslant s\leqslant T,$ (1.7)

because the adapted solution of (1.4) is also an adapted solution of BSVIE
(1.6). Clearly, (1.7) cannot be true.

The above example tells us the following: (i) BSDEs with anticipating
generators might not have adapted solutions, and they could have adapted
solutions within the framework of BSVIEs; (ii) BSVIEs with anticipating
generators might have adapted solutions, under some additional conditions. The
purpose of this paper is to explore the results of BSVIEs with anticipating
generators. We will establish a general theory for such kind of BSVIEs, and
beyond.

Besides the mathematical considerations, one might be wondering if there are
some practical motivations for BSVIEs with anticipating generators. We now
describe some possible situations.

Consider a position process $\psi(\cdot)$ on the time horizon $[0,T]$, which
is $\mathscr{F}_{T}$-measurable (not necessarily $\mathbb{F}$-adapted). As in
Yong [41] and Wang–Sun–Yong [29], one may use a BSVIE to describe the
equilibrium dynamic risk measure of $\psi(\cdot)$111One may also call it the
equilibrium utility/disutility of $\psi(\cdot)$.. More precisely, if
$(Y(\cdot),Z(\cdot\,,\cdot))$ is an adapted solution to the following BSVIE:

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

for some suitable generator $g(\cdot)$, then

$\rho(t,\psi(\cdot))=-Y(t),\qquad t\in[0,T],$

is an equilibrium dynamic risk measure of position process $\psi(\cdot)$. It
is not hard to understand that the generator $g(\cdot)$ could be a
deterministic function, or an $\mathbb{F}$-progressively measurable random
field. These cases were covered in the literature; see [42, 29], for examples.
We now describe some situations that the generator $g(\cdot)$ is anticipating
(or $\mathscr{F}_{T}$-measurable). Suppose the environment (in which a BSVIE
is considered) is depending on some random events, represented by a random
variable, say, $\theta$. These random events could include something like
default payments, bankruptcy of some relevant companies/institutes, the
approval/disapproval of FDA for some medicines, election results, pass/not
pass some tax bills, possible in-coming monetary policies, some possible
regional wars, to mention a few. The common feature of all these is that they
are not $\mathbb{F}$-adapted, and even not ${\cal F}_{T}$-measurable. However,
by the time $T$, it will be known whether such an event has happened.
Therefore, we may accept that the generator of the BSVIE is of form
$g(t,s,\mathbb{E}[\theta\,|\,{\cal F}_{T}],y,z)$, or simply allow $g(\cdot)$
to be $\mathscr{F}_{T}$-measurable222Of course, one could pursuit a more
general framework. However, at this stage, we are satisfied with such a
formulation..

In this paper, we will establish the well-posedness of BSVIEs with
anticipating generators, under proper conditions, which includes the situation
of adapted generators. The results will be extended to the path-dependent
BSVIE, which is an essential extension of the so-called anticipated BSDEs
introduced by Peng–Yang [22]. Let us mention that in studying path-dependent
BSVIEs, one will see that the situation of anticipating generators is not
avoidable, unless some restrictive (or special) cases are discussed. We will
explain this in more details in a later section.

The rest of the paper is organized as follows. In subsection 1.1, we provide a
literature review on the theory and applications of BSVIEs. In Section 2, we
give the preliminaries and recall the approach developed by Yong [42] to the
well-posedness of BSVIEs with adapted generators. In Section 3, we establish
the well-posedness of BSVIEs with anticipating generators under proper
conditions. The results of path-dependent BSVIEs are presented in Section 4.
In Section 5, we establish a connection between BSVIEs and coupled FBSDEs.
Some concluding remarks are presented in Section 6 and some technical results
are collected in Appendix.

###  1.1 Literature Review on BSVIEs

As a natural extension of BSDEs, BSVIE was initially studied by Lin [19] in a
special form of the following:

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

Motivated by the study of optimal control for forward SVIEs, Yong [42]
introduced the notion of adapted M-solution and the following more general
form of BSVIEs:

$Y(t)=\psi(t)+\int_{t}^{T}g(t,s,Y(s),Z(t,s),Z(s,t))ds-\int_{t}^{T}Z(t,s)dW(s),\qquad
t\in[0,T].$ (1.9)

The difference is that $Z(s,t)$ appears in (1.9), not in (1.8). Naturally, we
call (1.8) a Type-I BSVIE and (1.9) a Type-II BSVIE, respectively. This
important development has triggered extensive research on BSVIEs and their
applications. For instance, the well-posedness of BSVIEs was investigated by
Wang–Zhang [38], Ren [23], Anh–Grecksch–Yong [4], Shi–Wang–Yong [24],
Shi–Wen–Xiong [26], Wang–Sun–Yong [29], and Hamaguchi [8] in different
settings; Wang–Yong [34] established the comparison theorems of BSVIEs; the
variation of constants formula for linear BSVIEs was developed by Hu–Øksendal
[14], Wang–Yong–Zhang [31] and Hamaguchi [9]; the Feynman–Kac formula for
BSVIEs was established by Wang–Yong [35], Wang [28], and Wang–Yong–Zhang [31]
within Markovian and non-Markovian frameworks, respectively. It is noteworthy
that in [31], the forward and backward equations are both of Volterra type,
and then the associated PDE is non-local and path-dependent, due to which the
corresponding viscosity solution theory is challenging. A fundamental tool is
the functional Itô’s formula established by Viens–Zhang [27]. Some
applications of BSVIEs on optimal controls and mathematical finance can be
found in [2, 25, 37, 33, 36] and [41, 5, 17, 3, 29], respectively. In
particular, we remark that using BSVIEs, Wang–Yong [36] recently developed a
spike variation method for controlled forward SVIEs.

Besides the above works, another remarkable development of BSVIEs is the
finding of its connection with time-inconsistent control problems. Indeed,
BSVIE has become a popular tool for studying time-inconsistent problems
recently. See, for example, Wang–Sun–Yong [29], Wang–Yong [30],
Hernández–Possamaï [12, 13], Hamaguchi [7], Agram–Djehiche [1], Lei–Pun [18]
and Hernández [11]. From Yong [43, 44], we know that one of the main reasons
of causing time-consistency is the so-called subjective time-preferences,
which, mathematically, can be described by the non-exponential discounting. It
is shown [29, 30] that the most suitable dynamic recursive cost functional
allowing non-exponential discounting should be described by a Type-I BSVIE. As
a Feynman–Kac representation of the equilibrium HJB equation (or called an
extended HJB equation) associated with time-inconsistent problems, Wang–Yong
[30] introduced and initially investigated the following BSVIE:

$Y(t)=\psi(t)+\int_{t}^{T}g(t,s,Y(s),Z(s,s))ds-\int_{t}^{T}Z(t,s)dW(s),\qquad
t\in[0,T].$ (1.10)

The main feature of BSVIE (1.10) is that the diagonal value $Z(s,s)$ of
$Z(\cdot\,,\cdot)$ appears in the generator. In the Markovian case, Wang–Yong
[30] proved that (1.10) admits an adapted solution by an analytic approach,
and the uniqueness of adapted solutions was also discussed. The general well-
posedness of such kind of BSVIEs was recently studied by Hernández–Possamaï
[13]. A key step before [13] is that Hamaguchi [7] gave a proper definition
for $Z(s,s)$, using the derivatives of $Z(t,s)$ with respect to $t$. In [32],
we coined the name of Type-III BSVIEs for (1.10) to distinguish it from the
other two types of BSVIEs. Moreover, in [32] we found that Type-III BSVIEs can
be used to represent the M-solutions of linear Type-II BSVIEs, which serves as
a bridge to the decoupling approach of stochastic Hamiltonian systems of
Volterra type.

We emphasize that the generators of BSVIEs in all the above works are assumed
to be adapted. In the current paper, we would like to study the results of
BSVIEs with anticipating generators. More precisely, we shall focus on the
Type-I and Type-II BSVIEs, and the results of Type-III BSVIEs will be reported
in our future publications.

##  2 Preliminaries

In this section, we will recall a standard approach in establishing the well-
posedness of BSVIEs with adapted generators, from which, we will see why such
a standard method does not apply to the case of anticipating generators.

Recall that in (1.8) and (1.9), $g(\cdot)$ and $\psi(\cdot)$ are called the
generator and the free term of the BSVIE. The solution that we are looking for
is a pair $(Y(\cdot),Z(\cdot\,,\cdot))$ of processes satisfying certain
conditions. In order to make the above more precise, let us denote

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

which is an upper triangle domain in $[0,T]^{2}$. Note that $\Delta[0,T]$ does
not contain the diagonal segment $\\{(s,s)\bigm{|}s\in[0,T]\\}$. This will
allow our BSVIEs to have some singularities along the diagonal segment. We now
introduce some spaces. Let ${\cal H}^{p}_{\Delta}[0,T]$ be the set of all
pairs $(Y(\cdot),Z(\cdot\,,\cdot))$ of processes having the following
properties: $Y:[0,T]\times\Omega\to\mathbb{R}^{n}$ is $\mathbb{F}$-adapted,
$Z:\Delta[0,T]\times\Omega\to\mathbb{R}^{n}$ is ${\cal
B}(\Delta[0,T])\otimes{\cal F}_{T}$-measurable, for each $t\in[0,T]$,
$s\mapsto Z(t,s)$ is $\mathbb{F}$-adapted on $[t,T]$, and

$\|(Y(\cdot),Z(\cdot\,,\cdot))\|_{{\cal
H}^{p}_{\Delta}[0,T]}\equiv\Big{\\{}\mathbb{E}\Big{[}\int_{0}^{T}|Y(t)|^{p}dt+\int_{0}^{T}\Big{(}\int_{t}^{T}|Z(t,s)|^{2}ds\Big{)}^{p\over
2}dt\Big{]}\Big{\\}}^{1\over p}<\infty.$

Clearly, the above is a norm under which ${\cal H}^{p}_{\Delta}[0,T]$ is a
Banach space. Let ${\cal H}^{p}[0,T]$ be the set of all pairs
$(Y(\cdot),Z(\cdot\,,\cdot))$ of processes having the following properties:
$Y:[0,T]\times\Omega\to\mathbb{R}^{n}$ is $\mathbb{F}$-adapted,
$Z:[0,T]^{2}\times\Omega\to\mathbb{R}^{n}$ is ${\cal B}([0,T]^{2})\otimes{\cal
F}_{T}$-measurable, for each $t\in[0,T]$, $s\mapsto Z(t,s)$ is
$\mathbb{F}$-adapted on $[0,T]$, and

$\|(Y(\cdot),Z(\cdot\,,\cdot))\|_{{\cal
H}^{p}[0,T]}\equiv\Big{\\{}\mathbb{E}\Big{[}\int_{0}^{T}|Y(t)|^{p}dt+\int_{0}^{T}\Big{(}\int_{0}^{T}|Z(t,s)|^{2}ds\Big{)}^{p\over
2}dt\Big{]}\Big{\\}}^{1\over p}<\infty.$

The above is also a norm under which ${\cal H}^{p}[0,T]$ is a Banach space.
Note that the difference between ${\cal H}^{p}_{\Delta}[0,T]$ and ${\cal
H}^{p}[0,T]$ is that for any $(Y(\cdot),Z(\cdot\,,\cdot))$, if it is in the
former, then $Z(\cdot\,,\cdot)$ is defined only on the upper triangle domain
$\Delta[0,T]$, whereas if it is in the latter, $Z(\cdot\,,\cdot)$ is defined
on the square domain $[0,T]^{2}$. Next, let ${\cal M}^{p}[0,T]$ be the
subspace of ${\cal H}^{p}[0,T]$, consisting of pairs
$(Y(\cdot),Z(\cdot\,,\cdot))$ satisfying the following additional condition:

$Y(t)=\mathbb{E}[Y(t)]+\int_{0}^{t}Z(t,s)dW(s),\qquad\hbox{\rm
a.e.}\,t\in[0,T],~{}\hbox{\rm a.s.}$ (2.1)

Note that for any $(Y(\cdot),Z(\cdot\,,\cdot))\in{\cal M}^{p}[0,T]$, one has

$\mathbb{E}\Big{(}\int_{0}^{t}|Z(t,s)|^{2}ds\Big{)}^{p\over 2}\leqslant
K_{p}\mathbb{E}\Big{|}\int_{0}^{t}Z(t,s)dW(s)\Big{|}^{p}=K_{p}\mathbb{E}|Y(t)-\mathbb{E}[Y(t)]|^{p}\leqslant
2^{p-1}K_{p}\mathbb{E}|Y(t)|^{p}.$

In the above, the first inequality is called a martingale moment inequality
(see 7.1 in Appendix), with $K_{p}$ only depending on $p>1$, and $K_{2}=1$.
Thus, for any $(Y(\cdot),Z(\cdot\,,\cdot))\in{\cal M}^{p}[0,T]$,

$\displaystyle\|(Y(\cdot),Z(\cdot\,,\cdot))\|_{{\cal
H}^{p}[0,T]}^{p}=\mathbb{E}\Big{[}\int_{0}^{T}|Y(t)|^{p}dt+\int_{0}^{T}\Big{(}\int_{0}^{T}|Z(t,s)|^{2}ds\Big{)}^{p\over
2}dt\Big{]}$ $\displaystyle\quad\leqslant 2^{(p-2)^{+}\over
2}\|(Y(\cdot),Z(\cdot\,,\cdot))\|_{{\cal
H}_{\Delta}^{p}[0,T]}^{p}+2^{(p-2)^{+}\over
2}\mathbb{E}\int_{0}^{T}\Big{(}\int_{0}^{t}|Z(t,s)|^{2}ds\Big{)}^{p\over 2}dt$
$\displaystyle\quad\leqslant 2^{(p-2)^{+}\over
2}\|(Y(\cdot),Z(\cdot\,,\cdot))\|_{{\cal
H}_{\Delta}^{p}[0,T]}^{p}+2^{p-1+{(p-2)^{+}\over
2}}K_{p}\mathbb{E}\int_{0}^{T}|Y(t)|^{p}dt$ $\displaystyle\quad\leqslant
K\|(Y(\cdot),Z(\cdot\,,\cdot))\|_{{\cal H}_{\Delta}^{p}[0,T]}^{p}\leqslant
K\|(Y(\cdot),Z(\cdot\,,\cdot))\|_{{\cal H}^{p}[0,T]}^{p}.$ (2.2)

This shows that in ${\cal M}^{p}[0,T]$, the norms $\|\cdot\|_{{\cal
H}^{p}[0,T]}$ and $\|\cdot\|_{{\cal H}^{p}_{\Delta}[0,T]}$ are equivalent. We
will use $\|\cdot\|_{{\cal M}^{p}[0,T]}$ to denote the induced norm in ${\cal
M}^{p}[0,T]$ (as a subspace of ${\cal H}^{p}[0,T]$). For any $0\leqslant
R<S\leqslant T$, $\Delta[R,S]$, ${\cal H}^{p}_{\Delta}[S,T]$, ${\cal
H}^{p}[S,T]$, and $\mathscr{M}^{p}[R,S]$ can be defined similarly. In the
following, we collect some other spaces for the convenience of the reader.

$\displaystyle C([R,S];\mathbb{R}^{n})$
$\displaystyle\textstyle=\Big{\\{}\varphi:[R,S]\to\mathbb{R}^{n}~{}|~{}\hbox{$\varphi(\cdot)$
is continuous, }\sup_{s\in[R,S]}|\varphi(s)|<\infty\Big{\\}},$ $\displaystyle
L^{p}_{{\cal F}_{T}}(R,S;\mathbb{R}^{n})$
$\displaystyle\textstyle=\Big{\\{}\varphi:[R,S]\times\Omega\to\mathbb{R}^{n}~{}|~{}\hbox{$\varphi(\cdot)$
is ${\cal B}([R,S])\otimes{\cal F}_{T}$-measurable, }$
$\displaystyle\qquad\mathbb{E}\int_{R}^{S}|\varphi(s)|^{p}ds<\infty\Big{\\}},$
$\displaystyle L^{p}_{\mathbb{F}}(\Omega;L^{q}(R,S;\mathbb{R}^{n}))$
$\displaystyle\textstyle=\Big{\\{}\varphi:[R,S]\times\Omega\to\mathbb{R}^{n}~{}|~{}\hbox{$\varphi(\cdot)$
is $\mathbb{F}$-progressively measurable, }$
$\displaystyle\qquad\mathbb{E}\Big{(}\int_{R}^{S}|\varphi(s)|^{q}ds\Big{)}^{p\over
q}<\infty\Big{\\}},$ $\displaystyle L^{p}_{\mathbb{F}}(R,S;\mathbb{R}^{n})$
$\displaystyle\textstyle=L^{p}_{\mathbb{F}}(\Omega;L^{p}(R,S;\mathbb{R}^{n})),$
$\displaystyle L_{\mathbb{F}}^{p}(\Omega;C([R,S];\mathbb{R}^{n}))$
$\displaystyle\textstyle=\Big{\\{}\varphi(\cdot)\in
L^{p}_{\mathbb{F}}(R,S;\mathbb{R}^{n})~{}|~{}\varphi(\cdot)~{}\hbox{is
continuous, }\mathbb{E}\big{[}\displaystyle\sup_{R\leqslant s\leqslant
S}|\varphi(s)|^{p}\big{]}<\infty\Big{\\}},$ $\displaystyle
L^{p}_{\mathbb{F}}(\Omega;L^{q}(\Delta[R,S];\mathbb{R}^{n}))$
$\displaystyle\textstyle=\Big{\\{}\varphi:\negthinspace\Delta[R,S]\negthinspace\times\negthinspace\Omega\negthinspace\to\negthinspace\mathbb{R}^{n}~{}|~{}\varphi(t,\cdot)\in
L^{p}_{\mathbb{F}}(\Omega;L^{q}(t,S;\mathbb{R}^{n})),\,\hbox{\rm
a.e.}~{}t\negthinspace\in\negthinspace[R,S],$
$\displaystyle\textstyle\hphantom{=\Big{\\{}\
}\qquad\qquad\qquad\qquad\qquad\sup_{t\in[R,S]}\mathbb{E}\Big{(}\int_{t}^{S}|\varphi(t,s)|^{q}ds\Big{)}^{p\over
q}<\infty\Big{\\}}.$

Next, we recall the following definition.

######  Definition 2.1.

(i) $(Y(\cdot),Z(\cdot\,,\cdot))\in{\cal H}^{p}_{\Delta}[0,T]$ is called an
adapted solution of Type-I BSVIE (1.8) if (1.8) is satisfied in the usual
Itô’s sense.

(ii) $(Y(\cdot),Z(\cdot\,,\cdot))\in{\cal M}^{p}[0,T]$ is called an adapted
M-solution of Type-II BSVIE (1.9) if (1.9) is satisfied in the usual Itô’s
sense.

Note that for an adapted solution $(Y(\cdot),Z(\cdot\,,\cdot))$,
$Z(\cdot\,,\cdot)$ is defined on the upper triangle domain $\Delta[0,T]$ only.
Whereas, for an adapted M-solution $(Y(\cdot),Z(\cdot\,,\cdot))$,
$Z(\cdot\,,\cdot)$ is defined on the square domain $[0,T]^{2}$ and (2.1) is
satisfied.

Now, we recall the standard approach to BSVIEs with adapted generators, and
will reveal why the method is not directly applicable to the case of
anticipating generators. For the simplicity, we only consider the Type-I
BSVIEs with $p=2$. Let us now introduce the following hypothesis.

(H2)2 The map
$g:\Delta[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\Omega\to\mathbb{R}^{n}$
is
$\mathscr{B}(\Delta[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n})\otimes\mathscr{F}_{T}$-measurable,
and for each $(t,y,z)\in[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}$, the
process $s\mapsto g(t,s,y,z)$ is $\mathbb{F}$-progressively measurable.
Further, there are deterministic measurable functions
$L_{y},L_{z}:\Delta[0,T]\to[0,\infty)$ having properties that for some
$\varepsilon>0$,

$\int_{0}^{T}\Big{(}\int_{t}^{T}L_{y}(t,s)^{2}ds\Big{)}^{1+\varepsilon}dt+\sup_{t\in[0,T]}\int_{t}^{T}L_{z}(t,s)^{2+\varepsilon}ds<\infty,$

such that

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle|g(t,s,y,z)|\leqslant|g(t,s,0,0)|+L_{y}(t,s)|y|+L_{z}(t,s)|z|,\quad(t,s)\in\Delta[0,T],~{}y,z\in\mathbb{R}^{n},\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle|g(t,s,y_{1},z_{1})\negthinspace-\negthinspace
g(t,s,y_{2},z_{2})|\negthinspace\leqslant\negthinspace
L_{y}(t,s)|y_{1}\negthinspace-\negthinspace y_{2}|\negthinspace+\negthinspace
L_{z}(t,s)|z_{1}\negthinspace-\negthinspace
z_{2}|,\quad(t,s)\in\Delta[0,T],~{}y_{i},z_{i}\in\mathbb{R}^{n},~{}i=1,2.\end{array}$

Also,

$\mathbb{E}\int_{0}^{T}\Big{(}\int_{t}^{T}|g(t,s,0,0,0)|ds\Big{)}^{2}dt<\infty.$

Below, we will denote

$L_{0}(t,s)=|g(t,s,0,0,0)|,\qquad\forall(t,s)\in\Delta[0,T].$

We now consider the well-posedness of Type-I BSVIE (1.8), under (H2)2.

Let a pair $(y(\cdot),z(\cdot\,,\cdot))\in\mathscr{H}_{\Delta}^{2}[0,T]$ and
$\psi(\cdot)\in L^{2}_{\mathscr{F}_{T}}(0,T;\mathbb{R}^{n})$ be given. For
almost every $t\in[0,T]$ with $\psi(t)$ and $g(t,\cdot\,,\cdot\,,\cdot)$ being
fixed, we consider the following BSDE for $(\eta(t,\cdot),\zeta(t,\cdot))$:

$\eta(t,r)=\psi(t)+\int_{r}^{T}g(t,s,y(s),\zeta(t,s))ds-\int_{r}^{T}\zeta(t,s)dW(s),\quad
r\in[t,T],$ (2.3)

with $t\in[0,T)$ being a parameter. Clearly, the above is a BSDE (on $[t,T]$)
with adapted generator $(s,\eta,\zeta)\mapsto\widetilde{g}(s,\eta,\zeta)\equiv
g(t,s,y(s),\zeta)$ (independent of $\eta$) satisfying

$\negthinspace\negthinspace\begin{array}[]{ll}\displaystyle|\widetilde{g}(s,0,0)|=|g(t,s,y(s),0)|\leqslant
L_{0}(t,s)+L_{y}(t,s)|y(s)|,\qquad s\in(t,T],\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle|\widetilde{g}(s,\eta_{1},\zeta_{1})-\widetilde{g}(s,\eta_{2},\zeta_{2})|\leqslant
L_{z}(t,s)|\zeta_{1}-\zeta_{2}|,\qquad
s\in(t,T],~{}\eta_{i},\zeta_{i}\in\mathbb{R}^{n},~{}i=1,2.\end{array}$

Thus,

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\mathbb{E}\Big{(}\int_{t}^{T}|\widetilde{g}(s,0,0)|ds\Big{)}^{2}\leqslant
2\mathbb{E}\Big{[}\Big{(}\int_{t}^{T}L_{0}(t,s)ds\Big{)}^{2}+\Big{(}\int_{t}^{T}L_{y}(t,s)|y(s)|ds\Big{)}^{2}\Big{]}\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle\quad\leqslant
2\mathbb{E}\Big{[}\Big{(}\int_{t}^{T}\negthinspace\negthinspace\negthinspace
L_{0}(t,s)ds\Big{)}^{2}\negthinspace\negthinspace\negthinspace+\negthinspace\Big{(}\int_{t}^{T}\negthinspace\negthinspace\negthinspace
L_{y}(t,s)^{2}ds\Big{)}\Big{(}\int_{t}^{T}\negthinspace\negthinspace\negthinspace|y(s)|^{2}ds\Big{)}\Big{]}<\infty.\end{array}$

Then, BSDE (2.3) admits a unique adapted solution
$(\eta(t,\cdot),\zeta(t,\cdot))\in
L^{2}_{\mathbb{F}}(\Omega;C([t,T];\mathbb{R}^{n}))\times
L^{2}_{\mathbb{F}}(t,T;\mathbb{R}^{n})$ and the following estimate holds:

$\mathbb{E}\Big{[}\sup_{r\in[t,T]}|\eta(t,r)|^{2}+\int_{t}^{T}|\zeta(t,s)|^{2}ds\Big{]}\leqslant
K\mathbb{E}\Big{[}|\psi(t)|^{2}+\Big{(}\int_{t}^{T}|g(t,s,y(s),0)|ds\Big{)}^{2}\Big{]}.$

Set

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

Then

$Y(t)=\psi(t)+\int_{t}^{T}g(t,s,y(s),Z(t,s))ds-\int_{t}^{T}Z(t,s)dW(s),\qquad\hbox{\rm
a.e.}\;t\in[0,T],$

and

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\mathbb{E}\Big{[}|Y(t)|^{2}+\int_{t}^{T}|Z(t,s)|^{2}ds\Big{]}\leqslant
K\mathbb{E}\Big{[}|\psi(t)|^{2}+\Big{(}\int_{t}^{T}|g(t,s,y(s),0)|ds\Big{)}^{2}\Big{]}\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle\quad\leqslant
K\mathbb{E}\Big{\\{}|\psi(t)|^{2}+\Big{[}\int_{t}^{T}\Big{(}L_{0}(t,s)+L_{y}(t,s)|y(s)|\Big{)}ds\Big{]}^{2}\Big{\\}},\qquad\hbox{\rm
a.e.}\;t\in[0,T].\end{array}$

Consequently,

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\mathbb{E}\Big{[}\int_{0}^{T}|Y(t)|^{2}dt+\int_{0}^{T}\int_{t}^{T}|Z(t,s)|^{2}dsdt\Big{]}\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle\quad\leqslant
K\mathbb{E}\Big{[}\int_{0}^{T}|\psi(t)|^{2}dt+\int_{0}^{T}\Big{(}\int_{t}^{T}L_{0}(t,s)ds\Big{)}^{2}dt+\int_{0}^{T}\int_{t}^{T}L_{y}(t,s)^{2}ds\int_{t}^{T}|y(s)|^{2}dsdt\Big{]}.\end{array}$

This means that the map
$(y(\cdot),z(\cdot\,,\cdot))\mapsto(Y(\cdot),Z(\cdot\,,\cdot))$ is well-
defined from $\mathscr{H}_{\Delta}^{2}[0,T]$ to itself.

Now, for given $(y_{i}(\cdot),z_{i}(\cdot\,,\cdot))\in{\cal
H}_{\Delta}^{2}[0,T]$, let $(\eta_{i}(t,\cdot),\zeta_{i}(t,\cdot))$ be the
corresponding adapted solution to the BSDE (2.3). Then the following stability
estimate holds:

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\mathbb{E}\Big{[}\sup_{r\in[t,T]}|\eta_{1}(t,r)-\eta_{2}(t,r)|^{2}+\int_{t}^{T}|\zeta_{1}(t,s)-\zeta_{2}(t,s)|^{2}ds\Big{]}\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle\quad\leqslant
K\mathbb{E}\Big{(}\int_{t}^{T}|g(t,s,y_{1}(s),\zeta_{1}(t,s))-g(t,s,y_{2}(s),\zeta_{1}(t,s))|ds\Big{)}^{2}.\end{array}$

Hence, by setting

$Y_{i}(t)=\eta_{i}(t,t),\quad
Z_{i}(t,s)=\zeta_{i}(t,s),\quad(t,s)\in\Delta[0,T],$

one has

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\mathbb{E}\Big{[}|Y_{1}(t)-Y_{2}(t)|^{2}+\int_{t}^{T}|Z_{1}(t,s)-Z_{2}(t,s)|^{2}ds\Big{]}\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle\quad\leqslant
K\mathbb{E}\Big{(}\int_{t}^{T}|g(t,s,y_{1}(s),\zeta_{1}(t,s))-g(t,s,y_{2}(s),\zeta_{1}(t,s))|ds\Big{)}^{2}\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle\quad\leqslant
K\mathbb{E}\Big{(}\int_{t}^{T}L_{y}(t,s)|y_{1}(s)-y_{2}(s)|ds\Big{)}^{2},\qquad\hbox{\rm
a.e.}\;t\in[0,T].\end{array}$

Then for $\delta\in(0,T)$, one has

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\mathbb{E}\Big{[}\int_{T-\delta}^{T}|Y_{1}(t)-Y_{2}(t)|^{2}dt+\int_{T-\delta}^{T}\int_{t}^{T}|Z_{1}(t,s)-Z_{2}(t,s)|^{2}dsdt\Big{]}\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle\quad\leqslant
K\mathbb{E}\Big{[}\int_{T-\delta}^{T}\Big{(}\int_{t}^{T}L_{y}(t,s)^{2}ds\Big{)}\Big{(}\int_{t}^{T}|y_{1}(s)-y_{2}(s)|^{2}ds\Big{)}dt\Big{]}.\end{array}$

This gives that
$(y(\cdot),z(\cdot\,,\cdot))\mapsto(Y(\cdot),Z(\cdot\,,\cdot))$ is a
contraction on $\mathscr{H}_{\Delta}^{2}[T-\delta,T]$, for $\delta>0$ small
enogh. Hence, there exists a unique fixed point on $\Delta[T-\delta,T]$. This
determines the values of $(Y(\cdot),Z(\cdot\,,\cdot))$ as follows:

$Y(t),\quad t\in[T-\delta,T],\qquad Z(t,s),\quad T-\delta\leqslant
t<s\leqslant T,$ (2.4)

where the domain for $Z(\cdot\,,\cdot)$ is the region $①$ indicated in the
following figure.

$①$$②$$T$$T$$T-\delta$$T-\delta$$t$$s$

For $t\in[0,T-\delta]$, we consider

$Y(t)=\psi^{\delta}(t)+\int_{t}^{T-\delta}g(t,s,Y(s),Z(t,s))ds-\int_{t}^{T-\delta}Z(t,s)dW(s),\quad
t\in[0,T-\delta],$ (2.5)

where

$\psi^{\delta}(t)=\psi(t)+\int_{T-\delta}^{T}g(t,s,\bar{Y}(s),Z(t,s))ds-\int_{T-\delta}^{T}Z(t,s)dW(s),\qquad
t\in[0,T-\delta].$ (2.6)

Here, we use $\bar{Y}(\cdot)$ to indicate the already determined $Y(\cdot)$
(as in (2.4)), and (in the region $②$ of the above figure) $Z(\cdot\,,\cdot)$
is still unknown yet. Thus, (2.6) is actually a stochastic Fredholm integral
equation for $(\psi^{\delta}(\cdot),Z(\cdot\,,\cdot))$ and we need to find
$\psi^{\delta}(\cdot)$ which is $\mathscr{F}_{T-\delta}$-measurable so that
(2.5) is a BSVIE on $[0,T-\delta]$. To solve (2.6), we introduce the following
BSDE:

$\eta(t,r)=\psi(t)+\int_{r}^{T}g(t,s,\bar{Y}(s),\zeta(t,s))ds-\int_{r}^{T}\zeta(t,s)dW(s),\qquad
r\in[t\vee(T-\delta),T].$ (2.7)

This is very similar to (2.3), and it admits a unique adapted solution
$(\eta(t,\cdot),\zeta(t,\cdot))$. Now, we set

$\psi^{\delta}(t)=\eta(t,T-\delta),\quad
Z(t,s)=\zeta(t,s),\qquad(t,s)\in[0,T-\delta]\times[T-\delta,T].$

Then, $\psi^{\delta}(t)$ is $\mathscr{F}_{T-\delta}$-measurable and $Z(t,s)$
is determined for $(t,s)\in[0,T-\delta]\times[T-\delta,T]$ (see region $②$
indicated in the above figure). Since $g(\cdot)$ is $\mathbb{F}$-adapted,
(2.5) is a BSVIE on $[0,T-\delta]$ with an $\mathbb{F}$-adapted generator.
Then by induction, we can solve (1.9) over $[T-\delta,T]$, over
$[T-2\delta,T-\delta]$, etc., and eventually over $[0,T]$, with the adapted
solution $(Y(\cdot),Z(\cdot\,,\cdot))$ in the space
$\mathscr{H}_{\Delta}^{2}[0,T]$.

However, such an approach does not work for the case that $g(\cdot)$ is merely
$\mathscr{F}_{T}$-measurable, not $\mathbb{F}$-adapted. For such a case,
$s\mapsto g(t,s,y,z)$ is not necessarily $\mathscr{F}_{T-\delta}$-measurable
and thus (2.5) is not a BSVIE on $[0,T-\delta]$. As a matter of fact, since
the sum of the first two terms on the right-hand of (2.5) is
$\mathscr{F}_{T}$-measurable in general, the last term on the right-hand side
which is an Itô’s integral over $[0,T-\delta]$ will not be enough to make the
sum of the three terms on the right-hand side $\mathbb{F}$-adapted. Hence, the
approach for BSVIEs with $\mathbb{F}$-adapted generators does not work
directly for BSVISs with $\mathscr{F}_{T}$-measurable generators; in
particular, unlike the adapted generator cases, solving stochastic Fredholm
integral equation (2.6) is not helpful.

##  3 BSVIEs with Anticipating Generators

We now would like to improve the above approach and introduce some additional
conditions so that the well-posedness of BSVIEs with suitable anticipating
generators can be established. It turns out that the main part of this
approach for Type-I and Type-II BSVIEs are essentially the same. Therefore, we
directly discuss the case of Type-II BSVIEs, indicating some differences in
the process.

Let
$g:\Delta[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\Omega\to\mathbb{R}^{n}$
be a generator of a Type-II BSVIE, which is $\mathscr{F}_{T}$-measurable,
i.e., the map $s\mapsto g(t,s,y,z,\widehat{z})$ is ${\cal
B}([t,T])\otimes{\cal F}_{T}$-measurable. We introduce the following:

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle
g_{1}(t,s,y,z,\widehat{z})=\mathbb{E}_{s}[g(t,s,y,z,\widehat{z})],\\\ \vskip
3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle
g_{0}(t,s,y,z,\widehat{z})=g(t,s,y,z,\widehat{z})-g_{1}(t,s,y,z,\widehat{z}),\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\qquad\quad\forall(t,s,y,z,\widehat{z})\in\Delta[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^{n}.\end{array}$
(3.1)

We can show, by using Lemma 7.4 in the appendix, that one can take
$g_{1}:\Delta[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\Omega\to\mathbb{R}^{n}$
as an adapted generator. In other words, from now on, we may assume that the
following decomposition holds:

$g(t,s,y,z,\widehat{z})=g_{0}(t,s,y,z,\widehat{z})+g_{1}(t,s,y,z,\widehat{z}),\quad\forall(t,s,y,z,\widehat{z})\in\Delta[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^{n},~{}\hbox{\rm
a.s.},$ (3.2)

where $g_{1}(\cdot)$ is $\mathbb{F}$-adapted and $g_{0}(\cdot)$ is
$\mathscr{F}_{T}$-measurable. In the case that the generator is
$\mathbb{F}$-adapted, $g_{0}(\cdot)$ vanishes.

Let us rewrite our Type-II BSVIE as follows:

$\displaystyle Y(t)$
$\displaystyle=\psi(t)+\int_{t}^{T}g_{0}(t,s,Y(s),Z(t,s),Z(s,t))ds+\int_{t}^{T}g_{1}(t,s,Y(s),Z(t,s),Z(s,t))ds$
$\displaystyle\quad-\int_{t}^{T}Z(t,s)dW(s),\qquad t\in[0,T].$ (3.3)

The main idea is that we treat the sum of the first two terms on the right-
hand side of (3.3) as the free term. Then by some improved estimates for
BSVIEs with adapted generators, we establish the well-posedness of BSVIEs with
anticipating generators.

We now introduce the following hypothesis which is needed for Type-II BSVIEs.

(H3)p Let
$g_{0},g_{1}:\Delta[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\Omega\to\mathbb{R}^{n}$
be
$\mathscr{B}(\Delta[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n})\otimes\mathscr{F}_{T}$-measurable.
For any
$(t,y,z,\widehat{z})\in[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^{n}$,
$s\mapsto g_{1}(t,s,y,z,\widehat{z})$ is $\mathbb{F}$-progressively
measurable. There are deterministic functions
$L^{i}_{y},L^{i}_{z}:\Delta[0,T]\to[0,\infty)$, $i=0,1$, with the properties
that

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\int_{0}^{T}\Big{(}\int_{t}^{T}L^{i}_{y}(t,s)^{(p\land
2)(1+\varepsilon)\over(p\land 2)-1}ds\Big{)}^{(p\land
2)-1}dt+\sup_{t\in[0,T]}\Big{(}\int_{t}^{T}L^{i}_{z}(t,s)^{2}ds\Big{)}\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\qquad\qquad+\sup_{t\in[0,T]}\Big{(}\int_{t}^{T}L^{i}_{\widehat{z}}(t,s)^{(p\land
2)(1+\varepsilon)\over(p\land 2)-1}ds\Big{)}<\infty,\qquad i=0,1,\end{array}$
(3.4)

for some $\varepsilon>0$, and with the constant $K_{p}^{0}>0$ appearing in
(7.14),

$K^{0}_{p}\sup_{t\in[0,T]}\Big{(}\int_{t}^{T}L^{0}_{z}(t,s)^{2}ds\Big{)}^{p\over
2}<1,$ (3.5)

such that

$\displaystyle|g_{i}(t,s,y_{0},z_{0},\widehat{z}_{0})-g_{i}(t,s,y_{1},z_{1},\widehat{z}_{1})|\leqslant
L^{i}_{y}(t,s)|y_{0}-y_{1}|\negthinspace+\negthinspace
L^{i}_{z}(t,s)|z_{0}-z_{1}|\negthinspace+\negthinspace
L_{\widehat{z}}^{i}(t,s)|\widehat{z}_{0}-\widehat{z}_{1}|,$
$\displaystyle(t,s)\in\Delta[0,T],~{}y_{j},z_{j},\widehat{z}_{j}\in\mathbb{R}^{n},~{}i,j=0,1,$
(3.6)

and

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle|g_{i}(t,s,y,z,\widehat{z})|\negthinspace\leqslant\negthinspace
L^{i}_{0}(t,s)\negthinspace+\negthinspace
L_{y}^{i}(t,s)|y|\negthinspace+\negthinspace
L_{z}^{i}(t,s)|z|\negthinspace+\negthinspace
L^{i}_{\widehat{z}}(t,s)|\widehat{z}|^{1\wedge{2\over
p}},\quad(t,s)\in\Delta[0,T],~{}y,z,\widehat{z}\in\mathbb{R}^{n},\end{array}$
(3.7)

where

$L^{i}_{0}(\cdot\,,\cdot)=|g_{i}(\cdot\,,\cdot\,,0,0,0)|,\qquad\mathbb{E}\int_{0}^{T}\Big{(}\int_{t}^{T}L^{i}_{0}(t,s)ds\Big{)}^{p}dt<\infty,\qquad
i=0,1.$ (3.8)

Note that condition (3.5) means that the dependence $z\mapsto
g_{0}(t,s,y,z,\widehat{z})$ is well-controlled. Since in the appendix, we have
precise estimate for the constant $K_{p}^{0}$, this condition just gives a
size restriction of $L_{z}^{0}(\cdot\,,\cdot)$, which need not be sufficiently
small. This condition will play a crucial role below. We believe that a
restriction on the dependence of $z\mapsto g_{0}(t,s,y,z,\widehat{z})$ is
somewhat necessary; also see 5.2 for a comment on this condition from the
viewpoint of FBSDEs. However, at the moment, we can not provide satisfying
evidence. Hopefully, we can address this point in our future research.

For Type-I BSVIEs, we impose the following hypothesis.

(H3)${}_{p}^{\prime}$ In (H3)p, $L_{\widehat{z}}^{i}(\cdot\,,\cdot)=0$ and
(3.4) is replaced by the following:

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\int_{0}^{T}\Big{(}\int_{t}^{T}L^{i}_{y}(t,s)^{p(1+\varepsilon)\over
p-1}ds\Big{)}^{p-1}dt+\sup_{t\in[0,T]}\Big{(}\int_{t}^{T}L^{i}_{z}(t,s)^{2}ds\Big{)}<\infty,\quad
i=0,1.\end{array}$ (3.9)

Note that for $p>2$, one has ${p\over p-1}<2$. Thus, for such a case,
condition (3.4) is stronger than (3.9) for $L_{y}(\cdot\,,\cdot)$. We also
note that in the case $g_{0}(\cdot)$ being independent of $z$, condition (3.5)
is automatically true. In particular, such a condition holds true if
$g_{0}(\cdot)$ is identically equal to zero (or the generator of BSVIE is
adapted). In what follows, we denote

$\left\\{\negthinspace\negthinspace\begin{array}[]{ll}\displaystyle
L_{0}(t,s)=|g_{0}(t,s,0,0,0)|+|g_{1}(t,s,0,0,0)|,\\\ \vskip 3.0pt plus 1.0pt
minus 1.0pt\cr\displaystyle L_{y}(t,s)=L_{y}^{0}(t,s)+L^{1}_{y}(t,s),\quad
L_{\widehat{z}}(t,s)=L^{0}_{\widehat{z}}(t,s)+L^{0}_{\widehat{z}}(t,s),\end{array}\qquad(t,s)\in\Delta[0,T].\right.$

######  Theorem 3.1.

Let (H3)p hold. Then, for any $\psi(\cdot)\in
L^{p}_{\mathscr{F}_{T}}(\Omega;\mathbb{R}^{n})$, Type-II BSVIE (3.3) admits a
unique adapted M-solution
$(Y(\cdot),Z(\cdot\,,\cdot))\in\mathscr{M}^{p}[0,T]$. Moreover,

$\mathbb{E}\Big{[}\int_{0}^{T}|Y(t)|^{p}dt+\int_{0}^{T}\Big{(}\int_{t}^{T}|Z(t,s)|^{2}ds\Big{)}^{p\over
2}dt\Big{]}\leqslant
K\mathbb{E}\Big{[}\int_{0}^{T}|\psi(t)|^{p}dt+\int_{0}^{T}\Big{(}\int_{t}^{T}L_{0}(t,s)ds\Big{)}^{p}dt\Big{]}.$
(3.10)

If $\bar{g}(\cdot\,,\cdot\,,\cdot\,,\cdot\,,\cdot)$ is another generator
satisfying (H3)p, $\bar{\psi}(\cdot)\in L^{p}_{{\cal
F}_{T}}(\Omega;\mathbb{R}^{n})$, and
$\bar{Y}(\cdot),\bar{Z}(\cdot\,,\cdot))\in{\cal M}^{p}[0,T]$ is the unique
adapted M-solution to the corresponding BSVIE, then

$\displaystyle\mathbb{E}\Big{[}\int_{0}^{T}|Y(t)-\bar{Y}(t)|^{p}dt+\int_{0}^{T}\Big{(}\int_{t}^{T}|Z(t,s)-\bar{Z}(t,s)|^{2}ds\Big{)}^{p\over
2}dt\leqslant\negthinspace
K\mathbb{E}\Big{[}\negthinspace\int_{0}^{T}\negthinspace\negthinspace\negthinspace|\psi(t)\negthinspace-\negthinspace\bar{\psi}(t)|^{p}dt$
$\displaystyle\qquad+\negthinspace\negthinspace\int_{0}^{T}\negthinspace\negthinspace\negthinspace\negthinspace\Big{(}\negthinspace\int_{t}^{T}\negthinspace\negthinspace\negthinspace|g(t,s,Y(s),Z(t,s),Z(s,t))\negthinspace-\negthinspace\bar{g}(t,s,Y(s),Z(t,s),Z(s,t))|ds\Big{)}^{p}\negthinspace
dt\Big{]}.$ (3.11)

###### Proof.

Let $(y(\cdot),z(\cdot\,,\cdot))\in\mathscr{M}^{p}[0,T]$. We consider the
following Type-I BSVIE:

$\displaystyle Y(t)$
$\displaystyle=\psi(t)+\int_{t}^{T}\Big{[}g_{0}(t,s,y(s),z(t,s),z(s,t))+g_{1}(t,s,y(s),0,z(s,t))\Big{]}ds$
$\displaystyle\quad+\int_{t}^{T}\Big{[}g_{1}(t,s,y(s),Z(t,s),z(s,t))-g_{1}(t,s,y(s),0,z(s,t))\Big{]}ds$
$\displaystyle\quad-\int_{t}^{T}Z(t,s)dW(s),\qquad t\in[0,T].$ (3.12)

The generator is

$\widetilde{g}(t,s,\zeta)=g_{1}(t,s,y(s),\zeta,z(s,t))-g_{1}(t,s,y(s),0,z(s,t)),\qquad(t,s,\zeta)\in\Delta[0,T]\times\mathbb{R}^{n},$

with the free term

$\widetilde{\psi}(t)=\psi(t)+\int_{t}^{T}\Big{[}g_{0}(t,s,y(s),z(t,s),z(s,t))+g_{1}(t,s,y(s),0,z(s,t))\Big{]}ds.$

Clearly,

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\widetilde{g}(t,s,0)=0,\qquad(t,s)\in\Delta[0,T],\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle|\widetilde{g}(t,s,\zeta_{1})-\widetilde{g}(t,s,\zeta_{2})|\leqslant
L_{z}^{1}(t,s)|\zeta_{1}-\zeta_{2}|,\quad(t,s)\in\Delta[0,T],~{}\zeta_{1},\zeta_{2}\in\mathbb{R}^{n}.\end{array}$

Thus, BSVIE (3.12) admits a unique adapted solution
$(Y(\cdot),Z(\cdot\,,\cdot))\in{\cal H}^{p}_{\Delta}[0,T]$. We naturally
extend $Z(\cdot\,,\cdot)$ from $\Delta[0,T]$ to $[0,T]^{2}$ via the following

$Y(t)=\mathbb{E}[Y(t)]+\int_{0}^{t}Z(t,s)dW(s),\qquad t\in[0,T].$ (3.13)

Then, by (2.2) we have $(Y(\cdot),Z(\cdot\,,\cdot))\in{\cal M}^{p}[0,T]$. This
means a solution map
$(y(\cdot),z(\cdot\,,\cdot))\mapsto(Y(\cdot),Z(\cdot\,,\cdot))$ is well-
defined from ${\cal M}^{p}[0,T]$ to itself.

Now, let $(y_{i}(\cdot),z_{i}(\cdot\,,\cdot))\in{\cal M}^{p}[0,T]$ and let
$(Y_{i}(\cdot),Z_{i}(\cdot\,,\cdot))\in{\cal M}^{p}[0,T]$ be the adapted
M-solution to BSVIE (3.12) with $(y(\cdot),z(\cdot\,,\cdot))$ replaced by
$(y_{i}(\cdot),z_{i}(\cdot\,,\cdot))$. Then

$\displaystyle Y_{1}(t)-Y_{2}(t)$
$\displaystyle=\int_{t}^{T}\Big{[}g_{0}(t,s,y_{1}(s),z_{1}(t,s),z_{1}(s,t))-g_{0}(t,s,y_{2}(s),z_{2}(t,s),z_{2}(s,t))$
$\displaystyle\qquad\qquad+g_{1}(t,s,y_{1}(s),Z_{1}(t,s),z_{1}(s,t))-g_{1}(t,s,y_{2}(s),Z_{1}(t,s),z_{2}(s,t))\Big{]}ds$
$\displaystyle\quad+\int_{t}^{T}G(t,s)[Z_{1}(t,s)-Z_{2}(t,s)]ds-\int_{t}^{T}[Z_{1}(t,s)-Z_{2}(t,s)]dW(s),\quad
t\in[0,T],$ (3.14)

where

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle
G(t,s)={1\over|Z_{1}(t,s)-Z_{2}(t,s)|^{2}}\Big{[}g_{1}(t,s,y_{2}(s),Z_{1}(t,s),z_{2}(s,t))\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle\qquad\qquad-
g_{1}(t,s,y_{2}(s),Z_{2}(t,s),z_{2}(s,t))\Big{]}[Z_{1}(t,s)-Z_{2}(t,s)]^{\top}{\bf
1}_{\\{Z_{1}(t,s)\neq Z_{2}(t,s)\\}}.\end{array}$

Clearly,

$|G(t,s)|\leqslant L_{z}^{1}(t,s),\qquad(t,s)\in\Delta[0,T].$

By 7.3, we see that BSVIE (3.14) admits a unique adapted solution
$(Y_{1}(\cdot)-Y_{2}(\cdot),Z_{1}(\cdot\,,\cdot)-Z_{2}(\cdot\,,\cdot))\in{\cal
M}^{p}[0,T]$. Moreover,

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\mathbb{E}|Y_{1}(t)-Y_{2}(t)|^{p}+\mathbb{E}\Big{(}\int_{t}^{T}|Z_{1}(t,s)-Z_{2}(t,s)|^{2}ds\Big{)}^{p\over
2}\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle\quad\leqslant
K_{p}^{0}\mathbb{E}\Big{|}\int_{t}^{T}\big{[}g_{0}(t,s,y_{1}(s),z_{1}(t,s),z_{1}(s,t))-g_{0}(t,s,y_{2}(s),z_{2}(t,s),z_{2}(s,t))\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\qquad\qquad\qquad+g_{1}(t,s,y_{1}(s),Z_{1}(t,s),z_{1}(s,t))-g_{1}(t,s,y_{2}(s),Z_{1}(t,s),z_{2}(s,t))\big{]}ds\Big{|}^{p}\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle\quad\leqslant
K_{p}^{0}\mathbb{E}\Big{(}\int_{t}^{T}\big{[}L_{y}(t,s)|y_{1}(s)-y_{2}(s)|+L_{z}^{0}(t,s)|z_{1}(t,s)-z_{2}(t,s)|+L_{\widehat{z}}(t,s)|z_{1}(s,t)-z_{2}(s,t)|\big{]}ds\Big{)}^{p}\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\quad\leqslant(1+\varepsilon_{0})K_{p}^{0}\mathbb{E}\Big{(}\int_{t}^{T}L^{0}_{z}(t,s)|z_{1}(t,s)-z_{2}(t,s)|ds\Big{)}^{p}+K\mathbb{E}\Big{(}\int_{t}^{T}L_{y}(t,s)|y_{1}(s)-y_{2}(s)|ds\Big{)}^{p}\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\qquad+K\mathbb{E}\Big{(}\int_{t}^{T}L_{\widehat{z}}(t,s)|z_{1}(t,s)-z_{2}(t,s)|ds\Big{)}^{p},\qquad\hbox{\rm
a.e.}\;t\in[0.T].\end{array}$

Now, for any $\beta>0$, from the above, we have

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\mathbb{E}\Big{[}\int_{0}^{T}e^{\beta
pt}|Y_{1}(t)-Y_{2}(t)|^{p}dt+\int_{0}^{T}e^{\beta
pt}\Big{(}\int_{t}^{T}|Z_{1}(t,s)-Z_{2}(t,s)|^{2}ds\Big{)}^{p\over
2}dt\Big{]}\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\quad\leqslant(1+\varepsilon_{0})K_{p}^{0}\mathbb{E}\int_{0}^{T}e^{\beta
pt}\Big{(}\int_{t}^{T}L^{0}_{z}(t,s)|z_{1}(t,s)-z_{2}(t,s)|ds\Big{)}^{p}dt\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\qquad+K\mathbb{E}\int_{0}^{T}e^{\beta
pt}\Big{(}\int_{t}^{T}L_{y}(t,s)|y_{1}(s)-y_{2}(s)|ds\Big{)}^{p}dt\\\ \vskip
3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\qquad+K\mathbb{E}\int_{0}^{T}e^{\beta
pt}\Big{(}\int_{t}^{T}L_{\widehat{z}}(t,s)|z_{1}(s,t)-z_{2}(s,t)|ds\Big{)}^{p}dt.\end{array}$

Note that

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\mathbb{E}\int_{0}^{T}e^{\beta
pt}\Big{(}\int_{t}^{T}L^{0}_{z}(t,s)|z_{1}(t,s)-z_{2}(t,s)|ds\Big{)}^{p}dt\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\quad=\mathbb{E}\int_{0}^{T}\Big{(}\int_{t}^{T}L^{0}_{z}(t,s)^{2}ds\Big{)}^{p\over
2}e^{\beta pt}\Big{(}\int_{t}^{T}|z_{1}(t,s)-z_{2}(t,s)|^{2}ds\Big{)}^{p\over
2}dt\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\quad\leqslant\Big{[}\sup_{t\in[0,T]}\Big{(}\int_{t}^{T}L^{0}_{z}(t,s)^{2}ds\Big{)}^{p\over
2}\Big{]}\mathbb{E}\int_{0}^{T}e^{\beta
pt}\Big{(}\int_{t}^{T}|z_{1}(t,s)-z_{2}(t,s)|^{2}ds\Big{)}^{p\over
2}dt,\end{array}$

and

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\mathbb{E}\int_{0}^{T}e^{\beta
pt}\Big{(}\int_{t}^{T}L_{y}(t,s)|y_{1}(s)-y_{2}(s)|ds\Big{)}^{p}dt\\\ \vskip
3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\quad\leqslant\mathbb{E}\int_{0}^{T}\Big{(}\int_{t}^{T}e^{-{\beta
p(s-t)\over p-1}}L_{y}(t,s)^{p\over
p-1}ds\Big{)}^{p-1}\Big{(}\int_{t}^{T}e^{\beta
ps}|y_{1}(s)-y_{2}(s)|^{p}ds\Big{)}dt\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\quad\leqslant\int_{0}^{T}\Big{(}\int_{t}^{T}e^{-{\beta
p(1+\varepsilon)(s-t)\over(p-1)\varepsilon}}ds\Big{)}^{(p-1)\varepsilon\over
1+\varepsilon}\Big{(}\int_{t}^{T}L_{y}(t,s)^{p(1+\varepsilon)\over
p-1}ds\Big{)}^{p-1\over 1+\varepsilon}dt\,\mathbb{E}\int_{0}^{T}e^{\beta
ps}|y_{1}(s)-y_{2}(s)|^{p}ds\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\quad\leqslant\Big{(}{(p-1)\varepsilon\over\beta
p(1+\varepsilon)}\Big{)}^{(p-1)\varepsilon\over
1+\varepsilon}\int_{0}^{T}\Big{(}\int_{t}^{T}L_{y}(t,s)^{p(1+\varepsilon)\over
p-1}ds\Big{)}^{p-1\over 1+\varepsilon}dt\,\mathbb{E}\int_{0}^{T}e^{\beta
ps}|y_{1}(s)-y_{2}(s)|^{p}ds.\end{array}$

Also, since

$y_{1}(t)-y_{2}(t)=\mathbb{E}[y_{1}(t)-y_{2}(t)]+\int_{0}^{t}[z_{1}(t,s)-z_{2}(t,s)]dW(s),\quad
t\in[0,T],$

one has

$\mathbb{E}\Big{(}\int_{0}^{t}|z_{1}(t,s)-z_{2}(t,s)|^{2}ds\Big{)}^{p\over
2}\leqslant
K_{p}\mathbb{E}\Big{|}\int_{0}^{t}[z_{1}(t,s)-z_{2}(t,s)]dW(s)\Big{|}^{p}\leqslant
2^{p-1}K_{p}\mathbb{E}|y_{1}(t)-y_{2}(t)|^{p}.$

Therefore, when $p\in(1,2]$,

$\displaystyle\mathbb{E}\int_{0}^{T}e^{\beta
pt}\Big{(}\int_{t}^{T}L_{\widehat{z}}(t,s)|z_{1}(s,t)-z_{2}(s,t)|ds\Big{)}^{p}dt$
$\displaystyle\quad=\mathbb{E}\int_{0}^{T}\Big{(}\int_{t}^{T}e^{\beta(t-s)}L_{\widehat{z}}(t,s)e^{\beta
s}|z_{1}(s,t)-z_{2}(s,t)|ds\Big{)}^{p}dt$
$\displaystyle\quad\leqslant\mathbb{E}\int_{0}^{T}\Big{(}\int_{t}^{T}e^{{\beta
p\over p-1}(t-s)}L_{\widehat{z}}(t,s)^{p\over
p-1}ds\Big{)}^{p-1}\Big{(}\int_{t}^{T}e^{\beta
ps}|z_{1}(s,t)-z_{2}(s,t)|^{p}ds\Big{)}dt$
$\displaystyle\quad\leqslant\mathbb{E}\int_{0}^{T}\Big{(}\int_{t}^{T}e^{{\beta
p(1+\varepsilon)\over(p-1)\varepsilon}(t-s)}ds\Big{)}^{(p-1)\varepsilon\over
1+\varepsilon}\Big{(}\int_{t}^{T}L_{\widehat{z}}(t,s)^{p(1+\varepsilon)\over(p-1)}ds\Big{)}^{(p-1)\over
1+\varepsilon}\Big{(}\int_{t}^{T}e^{\beta
ps}|z_{1}(s,t)-z_{2}(s,t)|^{p}ds\Big{)}dt$
$\displaystyle\quad\leqslant\Big{(}{(p-1)\varepsilon\over\beta
p(1+\varepsilon)}\Big{)}^{(p-1)\varepsilon\over
1+\varepsilon}\Big{[}\sup_{t\in[0,T]}\Big{(}\int_{t}^{T}L_{\widehat{z}}(t,s)^{p(1+\varepsilon)\over(p-1)}ds\Big{)}^{(p-1)\over
1+\varepsilon}\Big{]}\mathbb{E}\int_{0}^{T}\int_{t}^{T}e^{\beta
ps}|z_{1}(s,t)-z_{2}(s,t)|^{p}dsdt$
$\displaystyle\quad\leqslant\Big{(}{(p-1)\varepsilon\over\beta
p(1+\varepsilon)}\Big{)}^{(p-1)\varepsilon\over
1+\varepsilon}\Big{[}\sup_{t\in[0,T]}\Big{(}\int_{t}^{T}L_{\widehat{z}}(t,s)^{p(1+\varepsilon)\over(p-1)}ds\Big{)}^{(p-1)\over
1+\varepsilon}\Big{]}\mathbb{E}\int_{0}^{T}e^{\beta
pt}\int_{0}^{t}|z_{1}(t,s)-z_{2}(t,s)|^{p}dsdt$
$\displaystyle\quad\leqslant\Big{(}{(p-1)\varepsilon\over\beta
p(1+\varepsilon)}\Big{)}^{(p-1)\varepsilon\over 1+\varepsilon}T^{1-{p\over
2}}\Big{[}\sup_{t\in[0,T]}\Big{(}\int_{t}^{T}L_{\widehat{z}}(t,s)^{p(1+\varepsilon)\over(p-1)}ds\Big{)}^{(p-1)\over
1+\varepsilon}\Big{]}\mathbb{E}\int_{0}^{T}e^{\beta
pt}\Big{(}\int_{0}^{t}|z_{1}(t,s)-z_{2}(t,s)|^{2}ds\Big{)}^{p\over 2}dt$
$\displaystyle\quad\leqslant\Big{(}{(p-1)\varepsilon\over\beta
p(1+\varepsilon)}\Big{)}^{(p-1)\varepsilon\over 1+\varepsilon}T^{1-{p\over
2}}2^{p-1}K_{p}\Big{[}\sup_{t\in[0,T]}\Big{(}\int_{t}^{T}L_{\widehat{z}}(t,s)^{p(1+\varepsilon)\over(p-1)}ds\Big{)}^{(p-1)\over
1+\varepsilon}\Big{]}\mathbb{E}\int_{0}^{T}e^{\beta
pt}|y_{1}(t)-y_{2}(t)|^{p}dt.$

Summarizing the above, we obtain that when $p\in(1,2]$, the following holds:

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\mathbb{E}\Big{[}\int_{0}^{T}e^{\beta
pt}|Y_{1}(t)-Y_{2}(t)|^{p}dt+\int_{0}^{T}e^{\beta
pt}\Big{(}\int_{t}^{T}|Z_{1}(t,s)-Z_{2}(t,s)|^{2}ds\Big{)}^{p\over
2}dt\Big{]}\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\quad\leqslant(1+\varepsilon_{0})K_{p}^{0}\Big{[}\sup_{t\in[0,T]}\Big{(}\int_{t}^{T}L^{0}_{z}(t,s)^{2}ds\Big{)}^{p\over
2}\Big{]}\mathbb{E}\int_{0}^{T}e^{\b{p}t}\Big{(}\int_{t}^{T}|z_{1}(t,s)-z_{2}(t,s)|^{2}ds\Big{)}^{p\over
2}dt\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\qquad+\Big{(}{(p-1)\varepsilon\over\beta
p(1+\varepsilon)}\Big{)}^{(p-1)\varepsilon\over
1+\varepsilon}K\Big{[}\int_{0}^{T}\Big{(}\int_{t}^{T}L_{y}(t,s)^{p(1+\varepsilon)\over
p-1}ds\Big{)}^{p-1\over 1+\varepsilon}dt\Big{]}\mathbb{E}\int_{0}^{T}e^{\beta
ps}|y_{1}(s)-y_{2}(s)|^{p}ds\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\qquad+\Big{(}{(p-1)\varepsilon\over\beta
p(1+\varepsilon)}\Big{)}^{(p-1)\varepsilon\over 1+\varepsilon}T^{1-{p\over
2}}K\Big{[}\sup_{t\in[0,T]}\Big{(}\int_{t}^{T}L_{\widehat{z}}(t,s)^{p(1+\varepsilon)\over(p-1)}ds\Big{)}^{(p-1)\over
1+\varepsilon}\Big{]}\mathbb{E}\int_{0}^{T}e^{\beta
pt}|y_{1}(t)-y_{2}(t)|^{p}dt.\end{array}$

If $L_{\widehat{z}}(\cdot\,,\cdot)=0$, i.e., our BSVIE is Type-I, then the
above holds for any $p\in(1,\infty)$. In this two cases, under the crucial
condition (3.5), for $\beta>0$ large enough, the map
$(y(\cdot),z(\cdot\,,\cdot))\mapsto(Y(\cdot),Z(\cdot\,,\cdot))$ is a
contraction on ${\cal H}^{p,\beta}_{\Delta}[0,T]$ whose norm is given by

$\|(Y(\cdot),Z(\cdot\,,\cdot)\|_{{\cal
H}^{p,\beta}_{\Delta}[0,T]}\equiv\Big{\\{}\mathbb{E}\Big{[}\int_{0}^{T}e^{\beta
pt}|Y(t)|^{p}dt+\int_{0}^{T}e^{\beta
pt}\Big{(}\int_{t}^{T}|Z(t,s)|^{2}ds\Big{)}^{p\over
2}dt\Big{]}\Big{\\}}^{1\over p}.$

Clearly, this norm is equivalent to that of ${\cal H}^{p}_{\Delta}[0,T]$.
Hence, BSVIE (3.3) admits a unique adapted solution
$(Y(\cdot),Z(\cdot\,,\cdot))\in{\cal H}^{p}_{\Delta}[0,T]$; and by naturally
extending $Z(\cdot\,,\cdot)$ from $\Delta[0,T]$ to $[0,T]^{2}$ as in (3.13),
one has $(Y(\cdot),Z(\cdot\,,\cdot))\in{\cal M}^{p}[0,T]$.

Finally, if our BSVIE is of Type-II, and $p\in[2,\infty)$. For $\psi(\cdot)\in
L^{p}_{{\cal F}_{T}}(0,T;\mathbb{R}^{m})\subseteq L^{2}_{{\cal
F}_{T}}(0,T;\mathbb{R}^{m})$, thanks to (3.4), there exists a unique adapted
M-solution $(Y(\cdot),Z(\cdot\,,\cdot))\in{\cal M}^{2}[0,T]$. We want to show
that in the current case, $(Y(\cdot),Z(\cdot\,,\cdot))$ is actually in ${\cal
M}^{p}[0,T]$. To show this, for the obtained adapted M-solution
$(Y(\cdot),Z(\cdot\,,\cdot))$, let us consider the following Type-I BSVIE for
$(\widetilde{Y}(\cdot),\widetilde{Z}(\cdot\,,\cdot))$:

$\widetilde{Y}(t)=\psi(t)+\int_{t}^{T}g\big{(}t,s,\widetilde{Y}(s),\widetilde{Z}(t,s),Z(s,t)\big{)}ds-\int_{t}^{T}\widetilde{Z}(t,s)dW(s),\qquad
t\in[0,T].$ (3.15)

Note that

$\begin{array}[]{ll}\displaystyle\mathbb{E}\int_{0}^{T}\Big{(}\int_{t}^{T}|g(t,s,0,0,Z(s,t))|ds\Big{)}^{p}dt\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle\quad\leqslant
K\mathbb{E}\int_{0}^{T}\Big{(}\int_{t}^{T}\big{[}L_{0}(t,s)+L_{\widehat{z}}(t,s)|Z(s,t)|^{2\over
p}\big{]}ds\Big{)}^{p}dt\\\ \displaystyle\quad\leqslant
K\mathbb{E}\Big{[}\int_{0}^{T}\Big{(}\int_{t}^{T}L_{0}(t,s)ds\Big{)}^{p}dt+\int_{0}^{T}\Big{(}\int_{t}^{T}L_{\widehat{z}}(t,s)^{p\over
p-1}ds\Big{)}^{p-1}\Big{(}\int_{t}^{T}|Z(s,t)|^{2}\Big{)}dsdt\Big{]}\\\ \vskip
3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle\quad\leqslant
K\mathbb{E}\Big{[}\int_{0}^{T}\Big{(}\int_{t}^{T}L_{0}(t,s)ds\Big{)}^{p}dt+\int_{0}^{T}\int_{0}^{t}|Z(t,s)|^{2}dsdt\Big{]}<\infty.\end{array}$

Thus, by the results on Type-I BSVIEs, (3.15) admits a unique adapted solution
$(\widetilde{Y}(\cdot),\widetilde{Z}(\cdot\,,\cdot))\in{\cal
H}^{p}_{\Delta}[0,T]$. Now, we extend $\widetilde{Z}(t,s)$ from $\Delta[0,T]$
to $[0,T]^{2}$ similarly to (3.13). It leads to
$(\widetilde{Y}(\cdot),\widetilde{Z}(\cdot\,,\cdot))\in{\cal
M}^{p}[0,T]\subseteq{\cal M}^{2}[0,T]$, which is an adapted M-solution to
(3.15). But, $(Y(\cdot),Z(\cdot\,,\cdot))$ is already the unique adapted
M-solution to (3.15) in ${\cal M}^{2}[0,T]$. Hence, it is necessary that

$(Y(\cdot),Z(\cdot\,,\cdot))=(\widetilde{Y}(\cdot),\widetilde{Z}(\cdot\,,\cdot))\in{\cal
M}^{p}[0,T].$

The stability estimates can be obtained similarly. ∎

######  Remark 3.2.

Let us revisit 1.2, in which the following BSDE is considered:

$Y(t)=\int_{t}^{T}W(T)ds-\int_{t}^{T}Z(s)dW(s),\quad t\in[0,T].$ (3.16)

Denote

$\psi(t)=\int_{t}^{T}W(T)ds,\quad t\in[0,T].$

Then we can rewrite (3.16) as follows:

$Y(t)=\psi(t)-\int_{t}^{T}Z(s)dW(s),\quad t\in[0,T].$ (3.17)

Note that the new free term $\psi(\cdot)$ depends on $t$, though the generator
of (3.16) is independent of $t$ and the free term of (3.16) equals zero.
Because of this feature, we should allow the unknown process $Z(t,s)$ to
depend on the parameter $t$; that is

$Y(t)=\int_{t}^{T}W(T)ds-\int_{t}^{T}Z(t,s)dW(s),\quad t\in[0,T].$ (3.18)

Naturally, the backward equation with the anticipating generator $W(T)$ should
be studied in the framework of BSVIEs, rather than BSDEs.

##  4 Path-Dependent Type-I BSVIEs

In this section, we shall consider the following path-dependent Type-I BSVIE:

$Y(t)=\psi(t)+\int_{t}^{T}g(t,s,Y_{s})ds-\int_{t}^{T}Z(t,s)dW(s),\quad
t\in[0,T],$ (4.1)

where $Y_{s}$ denotes the path of $Y(\cdot)$ from the current time $s$ to the
terminal time $T$, that is

$Y_{s}\equiv\\{Y(r)\bigm{|}s\leqslant r\leqslant T\\},\quad\hbox{\rm a.s.}$

Denote $\mathbb{Y}\triangleq\bigcup_{s\in[0,T]}\mathbb{Y}_{s}$, where
$\mathbb{Y}_{s}\triangleq C([s,T];\mathbb{R}^{n})$, equipped with the uniform
norm:

$\|{\bf y}_{s}\|_{\mathbb{Y}_{s}}=\sup_{r\in[s,T]}|{\bf
y}_{s}(r)|,\quad\forall{\bf y}_{s}\in\mathbb{Y}_{s}.$

(H4)p The map $\psi:[0,T]\times\Omega\to\mathbb{R}^{n}$ is ${\cal
F}_{T}$-measurable and the map
$g:\Delta[0,T]\times\mathbb{Y}\times\Omega\to\mathbb{R}^{n}$ is ${\cal
B}(\Delta[0,T]\times\mathbb{Y})\otimes{\cal F}_{T}$-measurable satisfying:

$g(t,s,{\bf y})=g(t,s,{\bf y}_{s}),\quad\forall(t,s,{\bf
y})\in\Delta[0,T]\times\mathbb{Y}.$ (4.2)

There exist a constant $L>0$, a modulus of continuous function $\rho(\cdot)$
and a positive random variable $\xi\in L^{p}_{{\cal
F}_{T}}(\Omega;\mathbb{R}_{+})$ such that:

$\displaystyle|g(t,s,{\bf y}_{s})-g(t,s,{\bf y}^{\prime}_{s})|\leqslant
L\|{\bf y}_{s}-{\bf
y}^{\prime}_{s}\|_{\mathbb{Y}_{s}},\quad\int_{t}^{t+\delta}|g(t,s,{\bf
0})|ds\leqslant|\xi|\rho(\delta),$ $\displaystyle|g(t-\delta,s,{\bf
y}_{s})-g(t,s,{\bf y}_{s})|+|\psi(t-\delta)-\psi(t)|\leqslant(|\xi|+\|{\bf
y}_{s}\|_{\mathbb{Y}_{s}})\rho(\delta),$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\forall(t,s)\in\Delta[0,T],\,{\bf
y}_{s},{\bf y}_{s}^{\prime}\in\mathbb{Y}_{s},\,\delta\in[0,t].$ (4.3)

Moreover,

$\mathbb{E}\Big{[}\sup_{t\in[0,T]}\Big{|}\int_{t}^{T}g(t,s,{\bf
0})ds\Big{|}^{p}+\sup_{t\in[0,T]}|\psi(t)|^{p}\Big{]}<\infty.$ (4.4)

######  Theorem 4.1.

Let (H4)p hold with $p>1$. Then BSVIE (4.1) admits a unique adapted solution
$(Y(\cdot),Z(\cdot\,,\cdot))$ such that $Y(\cdot)$ is continuous in $t$ and
the following estimate holds true:

$\mathbb{E}\Big{[}\sup_{0\leqslant t\leqslant
T}|Y(t)|^{p}\Big{]}\negthinspace+\negthinspace\negthinspace\sup_{0\leqslant
t\leqslant
T}\mathbb{E}\Big{(}\int_{t}^{T}\negthinspace\negthinspace\negthinspace|Z(t,s)|^{2}ds\Big{)}^{p\over
2}\negthinspace\leqslant\negthinspace
K\mathbb{E}\Big{[}\sup_{t\in[0,T]}\Big{|}\int_{t}^{T}\negthinspace\negthinspace\negthinspace
g(t,s,{\bf
0})ds\Big{|}^{p}\negthinspace\negthinspace+\negthinspace\negthinspace\sup_{t\in[0,T]}|\psi(t)|^{p}\Big{]}.$
(4.5)

Note that the generator $g(\cdot)$ of BSVIE (4.1) could depend on the whole
pathes of the unknown process $Y(\cdot)$ in the future, due to which we would
like to call (4.1) a path-dependent BSVIE. To our best knowledge, it is the
first time to study the backward stochastic equations in a path-dependent
framework. It is noteworthy that the difficulty caused by the ${\cal
F}_{T}$-measurable generator is not avoidable here, because for any
$Y(\cdot)\in L^{p}_{\mathbb{F}}(\Omega;C([0,T];\mathbb{R}^{n}))$,
$g(t,s,Y_{s})$ is still ${\cal F}_{T}$-measurable even if $g(t,s,{\bf y}_{s})$
is assumed to be ${\cal F}_{s}$-measurable for any deterministic function
${\bf y}_{s}\in C([s,T];\mathbb{R}^{n})$.

A closely related work of path-dependent BSVIEs is the following so-called
anticipated BSDE (ABSDE, for short), introduced by Peng–Yang [22]:

$\left\\{\begin{aligned}
dY(t)&=-f(t,Y(t),Y(t+\delta),Z(t),Z(t+\delta))dt+Z(t)dW(t),\quad t\in[0,T],\\\
Y(t)&=\eta(t),\quad Z(t)=\zeta(t),\quad t\in[T,T+\delta],\end{aligned}\right.$
(4.6)

where $0<\delta<T$ is a fixed constant and $(\eta(\cdot),\zeta(\cdot))\in
L^{2}_{\mathbb{F}}(\Omega;C([T,T+\delta];\mathbb{R}^{n}))\times
L^{2}_{\mathbb{F}}(T,T+\delta;\mathbb{R}^{n})$ is the given terminal state.
Along this line, some extensions of ABSDEs to BSVIEs (called anticipated
BSVIEs) can be found in [39, 9]. The main feature of (4.6) is that the
generator $f(\cdot)$ at time $t$ depends on the values of the unknown
processes at the future time $t+\delta$. Note from Peng–Yang [22] that in
studying the well-posedness of ABSDEs, the following adaptedness condition is
assumed:

$f(s,y(s),y(s+\delta),z(s),z(s+\delta))\in{\cal
F}_{s},\quad\forall(y(\cdot),z(\cdot))\in
L^{2}_{\mathbb{F}}(\Omega;C([0,T+\delta];\mathbb{R}^{n}))\times
L^{2}_{\mathbb{F}}(0,T+\delta;\mathbb{R}^{n}).$ (4.7)

To satisfy the adaptedness condition (4.7), one normally needs to impose the
following assumption:

$\displaystyle
f(s,y(s),y(s+\delta),z(s),z(s+\delta))=\tilde{f}(s,y(s),\mathbb{E}_{s}[y(s+\delta)],z(s),\mathbb{E}_{s}[z(s+\delta)]),$
$\displaystyle\forall(y(\cdot),z(\cdot))\in
L^{2}_{\mathbb{F}}(\Omega;C([0,T+\delta];\mathbb{R}^{n}))\times
L^{2}_{\mathbb{F}}(0,T+\delta;\mathbb{R}^{n}),$ (4.8)

for some progressively function $\tilde{f}(\cdot)$, from which we see that the
anticipated terms $y(s+\delta)$ and $z(s+\delta)$ have been taken conditional
expectation. Mathematically, it is not satisfying, and then naturally one
hopes to remove this adaptedness condition of ABSDEs. However, the following
example shows that it cannot be achieved within the framework of ABSDEs. The
feasible approach (see Theorem 4.1) is to generalize the framework to BSVIEs.
The main reason is that the generator of BSVIEs is allowed to be anticipating,
whereas for BSDEs, the generator must be adapted.

######  Example 4.2.

Consider the ABSDE:

$Y(t)=W(2)+\int_{t}^{2}Y([s+1]\wedge 2)ds-\int_{t}^{2}Z(s)dW(s),\quad
t\in[0,2].$ (4.9)

Note that

$Y([t+1]\wedge 2)=Y(2)=W(2),\quad t\in[1,2].$

Then on $[1,2]$,

$Y(t)=(3-t)W(2)-\int_{t}^{2}Z(s)dW(s),\quad t\in[1,2].$

Obviously,

${\cal Y}(t)=(3-t)W(t),\quad{\cal Z}(t,s)=(3-t),\quad(t,s)\in\Delta[1,2],$

is the unique solution to the following BSVIE:

${\cal Y}(t)=(3-t)W(2)-\int_{t}^{2}{\cal Z}(t,s)dW(s),\quad t\in[1,2].$

Then by the argument in 1.2, we obtain that ABSDE (4.9) dose not have an
adapted solution on $[1,2]$.

Next, we are going to prove Theorem 4.1. For any given $y(\cdot)\in
L_{\mathbb{F}}^{p}(\Omega;C([0,T];\mathbb{R}^{n}))$, we consider the following
BSVIE:

$Y(t)=\psi(t)+\int_{t}^{T}g(t,s,y_{s})ds-\int_{t}^{T}Z(t,s)dW(s),\quad
t\in[0,T].$ (4.10)

By modifying the argument employed in [31, Proposition 2.4], we have the
following $L_{\mathbb{F}}^{p}(\Omega;C([0,T];\mathbb{R}^{n}))$-norm estimate
for $Y(\cdot)$.

######  Proposition 4.3.

Let (H4)p hold with $p>1$. Then BSVIE (4.10) admits a unique adapted solution
$(Y(\cdot),Z(\cdot,\cdot))$ such that $Y(\cdot)$ is continuous in $t$ and the
following estimate holds true:

$\displaystyle\mathbb{E}\Big{[}\sup_{0\leqslant t\leqslant
T}|Y(t)|^{p}\Big{]}+\sup_{0\leqslant t\leqslant
T}\mathbb{E}\Big{(}\int_{t}^{T}|Z(t,s)|^{2}ds\Big{)}^{p\over 2}$
$\displaystyle\quad\leqslant
K\mathbb{E}\Big{[}\sup_{t\in[0,T]}\Big{|}\int_{t}^{T}\negthinspace\negthinspace\negthinspace
g(t,s,y_{s})ds\Big{|}^{p}\negthinspace\negthinspace+\negthinspace\negthinspace\sup_{t\in[0,T]}|\psi(t)|^{p}\Big{]}.$
(4.11)

###### Proof.

By Theorem 3.1, BSVIE (4.10) admits a unique solution
$(Y(\cdot),Z(\cdot,\cdot))$. Thus, we only need to prove the pathwise
continuity of $Y(\cdot)$ and the estimate (4.5). Consider the following BSDE
parameterized by $t\in[0,T]$ on $[0,T]$:

$\eta(t,r)=\psi(t)+\int_{t}^{T}g(t,s,y_{s})ds-\int_{r}^{T}\zeta(t,s)dW(s),\quad
r\in[0,T].$ (4.12)

We emphasize that the initial value of the Lebesgue integral in the above is
given by the parameter $t$ (rather than $r$). By comparing (4.10) and (4.12),
it is obvious to see that

$Y(t)=\eta(t,t),\quad Z(t,s)=\zeta(t,s),\quad(t,s)\in\Delta[0,T].$ (4.13)

Then by Theorem 1.1, we get

$\sup_{t\in[0,T]}\mathbb{E}\Big{(}\int_{t}^{T}|Z(t,s)|^{2}ds\Big{)}^{p\over
2}\leqslant
K\mathbb{E}\Big{[}\sup_{t\in[0,T]}\Big{|}\int_{t}^{T}\negthinspace\negthinspace\negthinspace
g(t,s,y_{s})ds\Big{|}^{p}\negthinspace\negthinspace+\negthinspace\negthinspace\sup_{t\in[0,T]}|\psi(t)|^{p}\Big{]}.$
(4.14)

For a fixed $1<p^{\prime}<p$, by Theorem 1.1 again we have

$\displaystyle|\eta(t,r)-\eta(t^{\prime},r)|^{p^{\prime}}$
$\displaystyle\leqslant
K\mathbb{E}_{r}\Big{[}\Big{(}\int_{t^{\prime}}^{T}\big{|}g(t,s,y_{s})-g(t^{\prime},s,y_{s})\big{|}ds\Big{)}^{p^{\prime}}+\Big{(}\int^{t^{\prime}}_{t}|g(t,s,y_{s})|ds\Big{)}^{p^{\prime}}+|\psi(t)-\psi(t^{\prime})|^{p^{\prime}}\Big{]}$
$\displaystyle\leqslant
K\Big{\\{}\mathbb{E}_{r}\big{[}|\xi|^{p^{\prime}}+\|y(\cdot)\|^{p^{\prime}}_{\mathbb{Y}}\big{]}\rho(|t-t^{\prime}|)^{p^{\prime}}+\mathbb{E}_{r}[|\xi|^{p^{\prime}}+\|y(\cdot)\|^{p^{\prime}}_{\mathbb{Y}}]|t-t^{\prime}|^{p^{\prime}}\Big{\\}}$
$\displaystyle\leqslant
K\Big{\\{}\mathbb{E}_{r}\big{[}|\xi|^{p^{\prime}}+\|y(\cdot)\|^{p^{\prime}}_{\mathbb{Y}}\big{]}\Big{\\}}\rho(|t-t^{\prime}|)^{p^{\prime}},\qquad
0\leqslant t\leqslant t^{\prime}\leqslant T,\quad r\in[0,T].$ (4.15)

Note that $1<p^{\prime}<p$, by Doob’s maximum inequality, we get

$\displaystyle\mathbb{E}\Big{[}\sup_{r\in[0,T]}\mathbb{E}_{r}\big{[}|\xi|^{p^{\prime}}+\|y(\cdot)\|^{p^{\prime}}_{\mathbb{Y}}\big{]}\Big{]}\leqslant
K\Big{(}\mathbb{E}\big{[}|\xi|^{p}+\|y(\cdot)\|^{p}_{\mathbb{Y}}\big{]}\Big{)}^{p^{\prime}\over
p}<\infty.$ (4.16)

Let $\\{t_{i}\\}_{i\geqslant 1}$ be the rational numbers in $[0,T]$. There
exists an $\bar{\Omega}\subset\Omega$ such that $\mathbb{P}(\bar{\Omega})=1$,
$\eta(t_{i},s,\omega)$ is continuous in $s$, and

$\displaystyle\sup_{r\in[0,T]}|\eta(t_{i},r)-\eta(t_{j},r)|(\omega)\leqslant
C_{p}(\omega)\rho(|t_{i}-t_{j}|),\quad\forall(i,j),~{}\forall\omega\in\bar{\Omega},$
$\displaystyle\mbox{with}\quad
C_{p}(\omega)\triangleq\sup_{r\in[0,T]}\Big{(}K\mathbb{E}_{r}\big{[}|\xi|^{p^{\prime}}+\|y(\cdot)\|^{p^{\prime}}_{\mathbb{Y}}\big{]}\Big{)}^{1\over
p^{\prime}}(\omega)<\infty,\quad\forall\omega\in\bar{\Omega},$ (4.17)

where the inequality (4.17) is due to (4.15) and (4.16). For any $t\in[0,T]$,
by (4.15), there exists an $\Omega^{t}\subset\Omega$ such that
$\mathbb{P}(\Omega^{t})=1$ and

$\displaystyle\sup_{r\in[0,T]}|\eta(t,r)-\eta(t_{j},r)|(\omega)\leqslant
C_{p}(\omega)\rho(|t-t_{j}|),\quad\forall
j,~{}\forall\omega\in\Omega^{t}\cap\bar{\Omega}.$ (4.18)

Define

$\bar{\eta}(t,r,\omega)\triangleq\limsup_{t_{j}\to
t}\eta(t_{j},r,\omega),\quad\forall(t,r,\omega)\in[0,T]^{2}\times\Omega.$

From (4.17), note that for $\omega\in\bar{\Omega}$, the above $\limsup$ is
actually a limit. Then for any $\omega\in\bar{\Omega}$,
$\bar{\eta}(t,r,\omega)$ is continuous in $r$, and

$\sup_{r\in[0,T]}|\bar{\eta}(t,r)-\bar{\eta}(t^{\prime},r)|(\omega)\leqslant
C_{p}(\omega)\rho(|t-t^{\prime}|),\quad\forall t,t^{\prime}\in[0,T].$

Thus, for any $\omega\in\bar{\Omega}$, the function
$\bar{\eta}(\cdot,\cdot,\omega)$ is continuous in $(t,r)\in[0,T]^{2}$.
Moreover, by (4.18) we have

$\eta(t,r,\omega)=\bar{\eta}(t,r,\omega),\quad\forall
r\in[0,T],~{}\forall\omega\in\Omega^{t}\cap\bar{\Omega}.$

Since $\mathbb{P}(\Omega^{t}\cap\bar{\Omega})=1$, so $\bar{\eta}(\cdot,\cdot)$
is a desired version of $\eta(\cdot,\cdot)$. Then by always considering this
version, $\eta(\cdot,\cdot)$ is jointly continuous in $(t,r)$, a.s. In
particular, the process $Y(\cdot)$, defined by $Y(t)=\eta(t,t);t\in[0,T]$, is
continuous in $t$, a.s.

By Theorem 1.1, we have

$|\eta(r,r)|^{p^{\prime}}\leqslant
K\mathbb{E}_{r}\Big{[}\Big{|}\int_{r}^{T}g(r,s,y_{s})ds\Big{|}^{p^{\prime}}+|\psi(r)|^{p^{\prime}}\Big{]},$

which, together with $Y(r)=\eta(r,r)$, implies that

$|Y(r)|^{p^{\prime}}\leqslant
K\mathbb{E}_{r}\Big{[}\sup_{t\in[0,T]}\Big{|}\int_{t}^{T}g(t,s,y_{s})ds\Big{|}^{p^{\prime}}+\sup_{t\in[0,T]}|\psi(t)|^{p^{\prime}}\Big{]}.$

Note that both the processes $Y(\cdot)$ and
$\mathbb{E}_{\cdot}\big{[}\sup_{t\in[0,T]}\big{|}\int_{t}^{T}g(t,s,y_{s})ds\big{|}^{p^{\prime}}+\sup_{t\in[0,T]}|\psi(t)|^{p^{\prime}}\big{]}$
are pathwise continuous. Then by Doob’s maximum inequality again (noting
$p>p^{\prime}$), we get

$\mathbb{E}\Big{[}\sup_{t\in[0,T]}|Y(t)|^{p}\Big{]}\leqslant
K\mathbb{E}\Big{[}\sup_{t\in[0,T]}\Big{|}\int_{t}^{T}g(t,s,y_{s})ds\Big{|}^{p}+\sup_{t\in[0,T]}|\psi(t)|^{p}\Big{]}.$

∎

Proof of Theorem 4.1. By 4.3, the map

$\Gamma(y(\cdot),z(\cdot,\cdot))\triangleq(Y(\cdot),Z(\cdot,\cdot)),\quad\forall(y(\cdot),z(\cdot,\cdot))\in
L_{\mathbb{F}}^{p}(\Omega;C([0,T];\mathbb{R}^{n}))\times
L^{p}_{\mathbb{F}}(\Omega;L^{2}(\Delta[0,T];\mathbb{R}^{n})),$

is well-defined on the space
$L_{\mathbb{F}}^{p}(\Omega;C([0,T];\mathbb{R}^{n}))\times
L^{p}_{\mathbb{F}}(\Omega;L^{2}(\Delta[0,T];\mathbb{R}^{n}))$, where
$(Y(\cdot),Z(\cdot,\cdot))$ is the unique solution to (4.10). Let
$(\tilde{y}(\cdot),\tilde{z}(\cdot,\cdot))\in
L_{\mathbb{F}}^{p}(\Omega;C([0,T];\mathbb{R}^{n}))\times
L^{p}_{\mathbb{F}}(\Omega;L^{2}(\Delta[0,T];\mathbb{R}^{n}))$ and denote
$(\tilde{Y}(\cdot),\tilde{Z}(\cdot,\cdot))=\Gamma(\tilde{y}(\cdot),\tilde{z}(\cdot,\cdot))$.
Note that for any $\delta\in[0,T)$, by 4.3,

$\displaystyle\mathbb{E}\Big{[}\sup_{T-\delta\leqslant t\leqslant
T}|Y(t)-\tilde{Y}(t)|^{p}\Big{]}+\sup_{T-\delta\leqslant t\leqslant
T}\mathbb{E}\Big{(}\int_{t}^{T}|Z(t,s)-\tilde{Z}(t,s)|^{2}ds\Big{)}^{p\over
2}$ $\displaystyle\quad\leqslant
K\delta^{p}\mathbb{E}\Big{[}\sup_{T-\delta\leqslant t\leqslant
T}|y(t)-\tilde{y}(t)|^{p}\Big{]}.$ (4.19)

Then by the fixed point theorem, we can get that BSVIE (4.1) admits a unique
solution $(\bar{Y}(\cdot),\bar{Z}(\cdot,\cdot))\in
L_{\mathbb{F}}^{2}(\Omega;C([T-\delta,T];\mathbb{R}^{n}))\times
L^{p}_{\mathbb{F}}(\Omega;L^{2}(\Delta[T-\delta,T];\mathbb{R}^{n}))$ for some
small enough $\delta>0$.

Next, we consider the following BSVIE

$Y(t)=\psi(t)+\int_{t}^{T}\Big{[}g(t,s,Y_{s})\textbf{1}_{[0,T-\delta)}(s)+g(t,s,\bar{Y}_{s})\textbf{1}_{[T-\delta,T]}(s)\Big{]}ds-\int_{t}^{T}Z(t,s)dW(s),\quad
t\in[0,T].$ (4.20)

The map

$\widehat{\Gamma}(y(\cdot),z(\cdot,\cdot))\triangleq(Y(\cdot),Z(\cdot,\cdot)),\quad\forall(y(\cdot),z(\cdot,\cdot))\in
L_{\mathbb{F}}^{p}(\Omega;C([0,T];\mathbb{R}^{n}))\times
L^{p}_{\mathbb{F}}(\Omega;L^{2}(\Delta[0,T];\mathbb{R}^{n})),$

is still well-defined on the space
$L_{\mathbb{F}}^{p}(\Omega;C([0,T];\mathbb{R}^{n}))\times
L^{p}_{\mathbb{F}}(\Omega;L^{2}(\Delta[0,T];\mathbb{R}^{n}))$, where
$(Y(\cdot),Z(\cdot,\cdot))$ is the unique solution to the following BSVIE:

$Y(t)=\psi(t)+\int_{t}^{T}\Big{[}g(t,s,y_{s})\textbf{1}_{[0,T-\delta)}(s)+g(t,s,\bar{Y}_{s})\textbf{1}_{[T-\delta,T]}(s)\Big{]}ds-\int_{t}^{T}Z(t,s)dW(s),\quad
t\in[0,T].$

We emphasize that in the above, the generator is independent of
$y(s),s\in[T-\delta,T]$. Let $(\tilde{y}(\cdot),\tilde{z}(\cdot,\cdot))\in
L_{\mathbb{F}}^{p}(\Omega;C([0,T];\mathbb{R}^{n}))\times
L^{p}_{\mathbb{F}}(\Omega;L^{2}(\Delta[0,T];\mathbb{R}^{n}))$ and denote
$(\tilde{Y}(\cdot),\tilde{Z}(\cdot,\cdot))=\widehat{\Gamma}(\tilde{y}(\cdot),\tilde{z}(\cdot,\cdot))$.
Then we have

$\displaystyle\mathbb{E}\Big{[}\sup_{T-2\delta\leqslant t\leqslant
T}|Y(t)-\tilde{Y}(t)|^{p}\Big{]}+\sup_{T-2\delta\leqslant t\leqslant
T}\mathbb{E}\Big{(}\int_{t}^{T}|Z(t,s)-\tilde{Z}(t,s)|^{2}ds\Big{)}^{p\over
2}$ $\displaystyle\quad\leqslant
K\delta^{p}\mathbb{E}\Big{[}\sup_{T-2\delta\leqslant t\leqslant
T-\delta}|y(t)-\tilde{y}(t)|^{p}\Big{]}.$

By the fixed point theorem again, we can get that BSVIE (4.20) admits a unique
solution $(\widehat{Y}(\cdot),\widehat{Z}(\cdot,\cdot))\in
L_{\mathbb{F}}^{p}(\Omega;C([T-2\delta,T];\mathbb{R}^{n}))\times
L^{p}_{\mathbb{F}}(\Omega;L^{2}(\Delta[T-2\delta,T];\mathbb{R}^{n}))$.
Clearly,

$\widehat{Y}(t)=\bar{Y}(t),\quad\widehat{Z}(t,s)=\bar{Z}(t,s),\quad
T-\delta\leqslant t\leqslant s\leqslant T.$

Thus, BSVIE (4.1) admits a unique solution on $[T-2\delta,T]$. By continuing
such a procedure, we obtain the existence and uniqueness of a solution to
BSVIE (4.1). Moreover, by 4.3, we have

$\displaystyle\mathbb{E}\Big{[}\sup_{0\leqslant t\leqslant
T}|Y(t)|^{p}\Big{]}+\sup_{0\leqslant t\leqslant
T}\mathbb{E}\Big{(}\int_{t}^{T}|Z(t,s)|^{2}ds\Big{)}^{p\over 2}$
$\displaystyle\quad\leqslant
K\mathbb{E}\int_{0}^{T}\|Y_{s}\|^{p}_{\mathbb{Y}_{s}}ds+K\mathbb{E}\Big{[}\sup_{t\in[0,T]}\Big{|}\int_{t}^{T}g(t,s,y_{s})ds\Big{|}^{p}+\sup_{t\in[0,T]}|\psi(t)|^{p}\Big{]}.$

Then by Grönwall’s inequality, we get the estimate (4.5) immediately.

###  4.1 Path-Dependent Type-I BSVIE with Adaptedness Conditions

Consider the following path-dependent BSVIE with progressively measurable
generators:

$Y(t)=\psi(t)+\int_{t}^{T}g(t,s,Y_{s},Z(t,s))ds-\int_{t}^{T}Z(t,s)dW(s),\quad
t\in[0,T].$ (4.21)

(H4)${}_{p}^{\prime}$ The map $\psi:[0,T]\to\mathbb{R}^{n}$ is ${\cal
F}_{T}$-measurable and the map
$g:\Delta[0,T]\times\mathbb{Y}\times\mathbb{R}^{n}\times\Omega\to\mathbb{R}^{n}$
is ${\cal B}(\Delta[0,T]\times\mathbb{Y}\times\mathbb{R}^{n})\otimes{\cal
F}_{T}$-measurable. For any $(t,s)\in\Delta[0,T]$, the geneator
$g(t,s,\cdot,\cdot)$ satisfies:

$\displaystyle g(t,s,{\bf y},z)=g(t,s,{\bf y_{s}},z),\quad\forall({\bf
y},z)\in\mathbb{Y}\times\mathbb{R}^{n},$ $\displaystyle g(t,s,y_{s},z)\in{\cal
F}_{s},\quad\forall(y(\cdot),z)\in
L^{p}_{\mathbb{F}}(\Omega;C([0,T];\mathbb{R}^{n}))\times\mathbb{R}^{n}.$
(4.22)

There exist a constants $L>0$ and a modulus of continuous function
$\rho(\cdot)$ and a positive random variable $\xi\in L^{p}_{{\cal
F}_{T}}(\Omega;\mathbb{R}_{+})$ such that:

$\displaystyle|g(t,s,{\bf y}_{s},z)-g(t,s,{\bf
y}^{\prime}_{s},z^{\prime})|\leqslant L\big{[}\|{\bf y}_{s}-{\bf
y}^{\prime}_{s}\|_{\mathbb{Y}_{s}}+|z-z^{\prime}|\big{]},$
$\displaystyle|g(t-\delta,s,{\bf y}_{s},z)-g(t,s,{\bf
y}_{s},z)|+|\psi(t-\delta)-\psi(t)|\leqslant(|\xi|+\|{\bf
y}\|_{\mathbb{Y}_{s}}+|z|)\rho(\delta),$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad\forall(t,s)\in\Delta[0,T],\,{\bf
y}_{s},{\bf
y}_{s}^{\prime}\in\mathbb{Y}_{s},\,(z,z^{\prime})\in\mathbb{R}^{n},\,\delta\in[0,t].$
(4.23)

Moreover,

$\mathbb{E}\Big{[}\sup_{t\in[0,T]}\Big{|}\int_{t}^{T}g(t,s,{\bf
0},0)ds\Big{|}^{p}+\sup_{t\in[0,T]}|\psi(t)|^{p}\Big{]}<\infty.$ (4.24)

######  Remark 4.4.

Note that the adaptedness condition (4.22) is much stronger than the following
progressively measurable condition:

$g(t,s,{\bf y_{s}},z)\in{\cal F}_{s},\quad\forall({\bf
y},z)\in\mathbb{Y}\times\mathbb{R}^{n}.$

Combining the arguments employed in 4.3 and [42, Theorem 3.7] together, the
following result can be obtained easily. We omit it here.

######  Theorem 4.5.

Let (H4)${}_{p}^{\prime}$ hold with $p>1$. Then BSVIE (4.21) admits a unique
adapted solution $(Y(\cdot),Z(\cdot,\cdot))$ such that $Y(\cdot)$ is
continuous in $t$ and the following estimate holds true:

$\displaystyle\mathbb{E}\Big{[}\sup_{0\leqslant t\leqslant
T}|Y(t)|^{p}\Big{]}+\sup_{0\leqslant t\leqslant
T}\mathbb{E}\Big{(}\int_{t}^{T}|Z(t,s)|^{2}ds\Big{)}^{p\over 2}$
$\displaystyle\quad\leqslant
K\mathbb{E}\Big{[}\sup_{t\in[0,T]}\Big{|}\int_{t}^{T}g(t,s,{\bf
0},0)ds\Big{|}^{p}+\sup_{t\in[0,T]}|\psi(t)|^{p}\Big{]}.$ (4.25)

##  5 Connections with FBSDEs

In this section, we shall show that if the generator of BSVIEs is allowed to
be anticipating, then it has an interesting connection with (coupled) FBSDEs.

Let us consider the following coupled FBSDE parameterized by $t\in[0,T)$:

$\left\\{\begin{aligned} X^{t}(s)&=x^{t}+\int_{t}^{s}b^{t}(r,Z^{t}(r))dr,\\\
Y^{t}(s)&=\xi^{t}X^{t}(T)+\int_{s}^{T}g^{t}(r,Z^{t}(r))dr-\int_{s}^{T}Z^{t}(r)dW(r),\end{aligned}\right.\quad
s\in[t,T].$ (5.1)

In the above, for any fixed $t\in[0,T)$,
$b^{t},\,g^{t}:[t,T]\times\mathbb{R}^{n}\times\Omega\to\mathbb{R}^{n}$ are
$\mathbb{F}$-progressively measurable and Lipschitz continuous with respect to
$z$, $x^{t}$ is an ${\cal F}_{t}$-measurable and square-integrable random
variable, and $\xi^{t}$ is an ${\cal F}_{T}$-measurable and essentially
bounded random variable.

######  Proposition 5.1.

333Hanxiao Wang would like to thank Chenchen Mou (City University of Hong
Kong) for suggesting him to study the connection between BSVIEs with
anticipating generators and coupled FBSDEs.

The well-posedness of FBSDE (5.1) on any time horizon $[t,T],\,t\in[0,T]$ is
equivalent to the well-posedness of the following BSVIE:

$\displaystyle Y(t)$
$\displaystyle=\xi^{t}x^{t}+\int_{t}^{T}\big{[}\xi^{t}b^{t}(r,Z(t,r))+g^{t}(r,Z(t,r))\big{]}dr-\int_{t}^{T}Z(t,r)dW(r),\qquad
t\in[0,T].$ (5.2)

###### Proof.

The proof is divided into two steps.

Step 1. Suppose that BSVIE (5.2) has a solution $(Y(\cdot),Z(\cdot,\cdot))$.
For any fixed $t$, we consider the following BSDE with unknown processes
$({\cal Y}(t,\cdot),{\cal Z}(t,\cdot))$:

${\cal
Y}(t,s)=\xi^{t}x^{t}+\int_{t}^{T}\big{[}\xi^{t}b^{t}(r,Z(t,r))+g^{t}(r,Z(t,r))\big{]}dr-\int_{s}^{T}{\cal
Z}(t,r)dW(r),\qquad s\in[t,T].$ (5.3)

Taking
$\xi^{t}x^{t}+\int_{t}^{T}\big{[}\xi^{t}b^{t}(r,Z(t,r))+g^{t}(r,Z(t,r))\big{]}dr$
as a terminal term, by the standard results of BSDEs, we get that BSDE (5.3)
admits a unique solution $({\cal Y}(t,\cdot),{\cal Z}(t,\cdot))$. Letting
$s=t$ and ${\cal Y}(t)={\cal Y}(t,t)$ in (5.3), we get

$\displaystyle{\cal Y}(t)$
$\displaystyle=\xi^{t}x_{t}+\int_{t}^{T}\big{[}\xi^{t}b^{t}(r,Z(t,r))+g^{t}(r,Z(t,r))\big{]}dr-\int_{t}^{T}{\cal
Z}(t,r)dW(r),\qquad t\in[0,T].$ (5.4)

Then by the uniqueness of the solution to BSVIE (5.4) (with $({\cal
Y}(\cdot),{\cal Z}(\cdot,\cdot))$ being unknown processes), we have

$Y(t)={\cal Y}(t),\quad Z(t,s)={\cal Z}(t,s),\qquad(t,s)\in\Delta[0,T].$

Substituting the above into (5.3) yields that

${\cal Y}(t,s)=\xi^{t}x^{t}+\int_{t}^{T}\big{[}\xi^{t}b^{t}(r,{\cal
Z}(t,r))+g^{t}(r,{\cal Z}(t,r))\big{]}dr-\int_{s}^{T}{\cal Z}(t,r)dW(r),\qquad
s\in[t,T].$

Let

$X^{t}(s)=x^{t}+\int_{t}^{s}b^{t}(r,{\cal Z}(t,r))dr,\quad Y^{t}(s)={\cal
Y}(t,s),\qquad Z^{t}(s)={\cal Z}(t,s),\qquad(t,s)\in\Delta[0,T].$

Then it is easily checked that for any $t\in[0,T)$,
$(X^{t}(\cdot),Y^{t}(\cdot),Z^{t}(\cdot))$ is a solution of FBSDE (5.1) on
$[t,T]$.

Step 2. Suppose that for any $t\in[0,T)$, FBSDE (5.1) has a solution
$(X^{t}(\cdot),Y^{t}(\cdot),Z^{t}(\cdot))$. Then from the fact

$X^{t}(T)=x^{t}+\int_{t}^{T}b^{t}(r,Z^{t}(r))dr,$

we have

$Y^{t}(s)=\xi^{t}x^{t}+\int_{t}^{T}\xi^{t}b^{t}(r,Z^{t}(r))dr+\int_{s}^{T}g^{t}(r,Z^{t}(r))dr-\int_{s}^{T}Z^{t}(r)dW(r),\qquad
s\in[t,T].$

By taking $s=t$ in the above, we get

$Y^{t}(t)=\xi^{t}x^{t}+\int_{t}^{T}\xi^{t}b^{t}(r,Z^{t}(r))dr+\int_{t}^{T}g^{t}(r,Z^{t}(r))dr-\int_{t}^{T}Z^{t}(r)dW(r),\qquad
t\in[0,T].$

Thus, the process $(Y(\cdot),Z(\cdot,\cdot))$, defined by

$Y(t)\triangleq Y^{t}(t),\quad Z(t,s)\triangleq
Z^{t}(s),\qquad(t,s)\in\Delta[0,T],$

is a solution to BSVIE (5.2). ∎

######  Remark 5.2.

Note that (5.1) is a multi-dimensional coupled FBSDE and the forms of
$b^{t}(\cdot),g^{t}(\cdot)$ are very general. Thus, the coefficients
$\xi^{t}$, $b^{t}(\cdot)$ and $g^{t}(\cdot)$ might not satisfy the monotone
condition imposed for coupled FBSDEs in the literature; see, [15, 40, 20, 45],
for example. To ensure that (5.1) admits a unique solution for any given time
horizon $[t,T]\subseteq[0,T]$, we can impose the small-time assumption, that
is, $\|\xi^{t}b^{t}_{z}(\cdot)\|_{L^{\infty}}$ is controlled by some constant
$K_{T}>0$ depending on $T$. On the other hand, by 5.1, we know that the well-
posedness of FBSDE (5.1) is equivalent to that of the BSVIE (5.2) with the
following generator:

$g(t,s,z)\equiv\xi^{t}b^{t}(s,z)+g^{t}(s,z),\quad(t,s,z)\in\Delta[0,T]\times\mathbb{R}^{n}.$

Clearly, $g(\cdot)$ is anticipating, because $\xi^{t}$ is ${\cal
F}_{T}$-measurable. The corresponding decomposition (see (3.1)) is given by

$g_{1}(t,s,y,z,\widehat{z})=\mathbb{E}_{s}[\xi^{t}]b^{t}(s,z)+g^{t}(s,z),\quad
g_{0}(t,s,y,z,\widehat{z})=\\{\xi^{t}-\mathbb{E}_{s}[\xi^{t}]\\}b^{t}(s,z).$

The corresponding $L_{z}^{0}(\cdot,\cdot)$ can be given by
$\|\\{\xi^{t}-\mathbb{E}_{s}[\xi^{t}]\\}b^{t}_{z}(\cdot)\|_{L^{\infty}}$.
Thus, (3.5) is in some sense equivalent to the small-time assumption for
coupled FBSDEs.

##  6 Conclusion

In conclusion, the BSVIEs with anticipating generators are explored for the
first time. The path-dependent BSVIEs are also introduced and studied. Under
proper conditions, the well-posedness of these kinds of BSVIEs are
established. Some essential differences between BSDEs and BSVIEs are revealed:
(i) If the generator is allowed to be anticipating, BSDEs are ill-posed in
general, but they can still have unique adapted solutions within the framework
of BSVIEs; (ii) For anticipated BSDEs, the adaptedness condition is not
avoidable, but it is no longer necessary within the framework of (path-
dependent) BSVIEs. In addition, we find that if the generator of BSVIEs is
allowed to be anticipating, it has an interesting connection with coupled
FBSDEs.

The following provides a clear path of how a BSVIE is reduced to a BSDE by
changing the measurable condition of the generator.

(I) Consider the following BSVIE with the anticipating generator $g(\cdot)$:

$Y(t)=\psi(t)+\int_{t}^{T}g(t,s,Y(s),Z(t,s))ds-\int_{t}^{T}Z(t,s)dW(s),\quad
t\in[0,T].$ (6.1)

If $g(t,s,y,z)$ and $\psi(t)$ are independent of $t$, (6.1) is still a BSVIE,
with the form:

$Y(t)=\xi+\int_{t}^{T}g(s,Y(s),Z(t,s))ds-\int_{t}^{T}Z(t,s)dW(s),\quad
t\in[0,T].$ (6.2)

If the generator $g(\cdot)$ is adapted, (6.2) is reduced to a BSDE:

$Y(t)=\xi+\int_{t}^{T}g(s,Y(s),Z(s))ds-\int_{t}^{T}Z(s)dW(s),\quad t\in[0,T].$
(6.3)

(II) Consider the following path-dependent BSVIE with the anticipating
generator $g(\cdot)$:

$Y(t)=\psi(t)+\int_{t}^{T}g(t,s,Y_{s})ds-\int_{t}^{T}Z(t,s)dW(s),\quad
t\in[0,T].$ (6.4)

If $g(t,s,{\bf y})$ and $\psi(t)$ are independent of $t$, (6.4) is still a
BSVIE, with the form:

$Y(t)=\xi+\int_{t}^{T}g(s,Y_{s})ds-\int_{t}^{T}Z(t,s)dW(s),\quad t\in[0,T].$
(6.5)

If the generator $g(s,y_{s})$ is ${\cal F}_{T}$-measurable for any
$y(\cdot)\in L_{\mathbb{F}}^{p}(\Omega;C([0,T];\mathbb{R}^{n}))$, in which
case $g(s,{\bf y}_{s})$ can be ${\cal F}_{s}$-measurable for any ${\bf
y}(\cdot)\in C([0,T];\mathbb{R}^{n})$, (6.5) is still a BSVIE, with the form:

$Y(t)=\xi+\int_{t}^{T}g(s,Y_{s})ds-\int_{t}^{T}Z(t,s)dW(s),\quad t\in[0,T].$
(6.6)

If the generator $g(s,y_{s})$ is ${\cal F}_{s}$-measurable for any
$y(\cdot)\in L_{\mathbb{F}}^{p}(\Omega;C([0,T];\mathbb{R}^{n}))$, (6.6) is
reduced to an anticipated BSDE:

$Y(t)=\xi+\int_{t}^{T}g(s,Y_{s})ds-\int_{t}^{T}Z(s)dW(s),\quad t\in[0,T].$
(6.7)

##  7 Appendix: BSDEs with Progressively Measurable Generators

First, let us recall the martingale moment inequalities (see [16, page 163]).

######  Proposition 7.1.

For any $p>1$, there exists a constant $K_{p}>0$ such that for any
$z(\cdot)\in L^{p}_{\mathbb{F}}(\Omega;$ $L^{2}(0,T;\mathbb{R}^{n}))$,

$\mathbb{E}\Big{(}\int_{0}^{T}|z(s)|^{2}ds\Big{)}^{p\over 2}\leqslant
K_{p}\mathbb{E}\Big{|}\int_{0}^{T}z(s)dW(s)\Big{|}^{p}.$ (7.1)

Note that if $p=2$, then $K_{2}=1$ and the equality holds in the above.

Let us now consider the following BSDE:

$Y(t)=\xi+\int_{t}^{T}g_{1}(s,Z(s))ds-\int_{t}^{T}Z(s)dW(s),\qquad t\in[0,T].$
(7.2)

We introduce the following hypothesis.

(H0)p Let $g_{1}:[0,T]\times\mathbb{R}^{n}\times\Omega\to\mathbb{R}^{n}$
satisfy the following: For each $z\in\mathbb{R}^{n}$, $t\mapsto g_{1}(t,z)$ is
$\mathbb{F}$-progressively measurable. There exists a function
$L_{z}^{1}(\cdot)\in L^{2}(0,T)$ such that

$|g_{1}(t,z_{1})-g_{1}(t,z_{2})|\leqslant
L_{z}^{1}(t)|z_{1}-z_{2}|,\qquad\forall t\in[0,T],~{}z\in\mathbb{R}^{n}.$

Further, $g_{1}(\cdot\,,0)\in
L_{\mathbb{F}}^{p}(\Omega;L^{1}(0,T;\mathbb{R}^{n}))$.

In what follows, we will denote

$L_{0}^{1}(s)=|g_{1}(s,0)|,\qquad s\in[0,T].$

######  Proposition 7.2.

Let (H0)p hold. Then for any $\xi\in L^{p}_{{\cal
F}_{T}}(\Omega;\mathbb{R}^{n})$, BSDE (7.2) admits a unique adapted solution
$(Y(\cdot),Z(\cdot))$, and the following holds:

$\Big{[}\sup_{t\in[0,T]}\mathbb{E}|Y(t)|^{p}\Big{]}^{1\over
p}+\Big{[}\mathbb{E}\Big{(}\int_{0}^{T}|Z(s)|^{2}ds\Big{)}^{p\over
2}\Big{]}^{1\over
p}\leqslant\widehat{K}_{p}\Big{(}\mathbb{E}|\xi|^{p}\Big{)}^{1\over
p}+N_{p}\widehat{K}_{p}\Big{[}\mathbb{E}\Big{(}\int_{0}^{T}L_{0}^{1}(s)ds\Big{)}^{p}\Big{]}^{1\over
p},$ (7.3)

where

$\displaystyle\widehat{K}_{p}$
$\displaystyle=\min\Big{\\{}\Big{[}1+2\tilde{K}_{p}N\Big{(}{\sqrt{N}\over\sqrt{N}-\sqrt{\bar{K}_{p}}}\Big{)}\Big{]}\Big{(}{\sqrt{N}\over\sqrt{N}-\sqrt{\bar{K}_{p}}}\Big{)}^{N}\bigm{|}N\in\mathbb{N},~{}N>\bar{K}_{p}\Big{\\}}$
(7.4)
$\displaystyle\equiv\Big{[}1+2\tilde{K}_{p}N_{p}\Big{(}{\sqrt{N_{p}}\over\sqrt{N_{p}}-\sqrt{\bar{K}_{p}}}\Big{)}\Big{]}\Big{(}{\sqrt{N_{p}}\over\sqrt{N_{p}}-\sqrt{\bar{K}_{p}}}\Big{)}^{N_{p}},$

with

$\tilde{K}_{p}=K_{p}^{1\over
p},\qquad\bar{K}=4\tilde{K}^{2}_{p}\int_{0}^{T}L_{z}^{1}(s)^{2}ds.$ (7.5)

Further, if $g_{i}(\cdot\,,\cdot)$ satisfies (H0)p, $\xi_{i}\in L^{p}_{{\cal
F}_{T}}(\Omega;\mathbb{R}^{n})$, and $(Y_{i}(\cdot),Z_{i}(\cdot))$ is the
adapted solution to the corresponding BSDE, $i=1,2$, then

$\begin{array}[]{ll}\displaystyle\Big{[}\sup_{t\in[0,T]}\mathbb{E}|Y_{1}(t)-Y_{2}(t)|^{p}\Big{]}^{1\over
p}+\Big{[}\mathbb{E}\Big{(}\int_{0}^{T}|Z_{1}(s)-Z_{2}(s)|^{2}ds\Big{)}^{p\over
2}\Big{]}^{1\over p}\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\quad\leqslant\widehat{K}_{p}\Big{(}\mathbb{E}|\xi_{1}-\xi_{2}|^{p}\Big{)}^{1\over
p}+N_{p}\widehat{K}_{p}\Big{[}\mathbb{E}\Big{(}\int_{0}^{T}|g_{1}(s,Z_{1}(s))-g_{2}(s,Z_{1}(s))|ds\Big{)}^{p}\Big{]}^{1\over
p}.\end{array}$ (7.6)

Proof. First of all, BSDE (7.2) leads to

$Y(t)=\mathbb{E}_{t}\Big{[}\xi+\int_{t}^{T}g_{1}(s,Z_{1}(s))ds\Big{]}.$

Thus,

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\Big{(}\mathbb{E}|Y(t)|^{p}\Big{)}^{1\over
p}\leqslant\Big{(}\mathbb{E}|\xi|^{p}\Big{)}^{1\over
p}+\Big{[}\mathbb{E}\Big{(}\int_{t}^{T}L^{1}_{0}(s)ds\Big{)}^{p}\Big{]}^{1\over
p}+\Big{[}\mathbb{E}\Big{(}\int_{t}^{T}L_{z}^{1}(s)|Z(s)|ds\Big{)}^{p}\Big{]}^{1\over
p}\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\quad\leqslant\Big{(}\mathbb{E}|\xi|^{p}\Big{)}^{1\over
p}+\Big{[}\mathbb{E}\Big{(}\int_{t}^{T}L^{1}_{0}(s)ds\Big{)}^{p}\Big{]}^{1\over
p}+\Big{(}\int_{t}^{T}L_{z}^{1}(s)^{2}ds\Big{)}^{1\over
2}\Big{[}\mathbb{E}\Big{(}\int_{t}^{T}|Z(s)|^{2}ds\Big{)}^{p\over
2}\Big{]}^{1\over p}.\end{array}$

As a result,

$\sup_{s\in[t,T]}\Big{(}\mathbb{E}|Y(s)|^{p}\Big{)}^{1\over
p}\leqslant\Big{(}\mathbb{E}|\xi|^{p}\Big{)}^{1\over
p}+\Big{[}\mathbb{E}\Big{(}\int_{t}^{T}L^{1}_{0}(s)ds\Big{)}^{p}\Big{]}^{1\over
p}+\Big{(}\int_{t}^{T}L_{z}^{1}(s)^{2}ds\Big{)}^{1\over
2}\Big{[}\mathbb{E}\Big{(}\int_{t}^{T}|Z(s)|^{2}ds\Big{)}^{p\over
2}\Big{]}^{1\over p}.$

Also, making use of the martingale moment inequality (7.1), one has

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\Big{[}\mathbb{E}\Big{(}\int_{t}^{T}|Z(s)|^{2}ds\Big{)}^{p\over
2}\Big{]}^{1\over
p}\leqslant\tilde{K}_{p}\Big{[}\mathbb{E}\Big{|}\int_{t}^{T}Z(s)dW(s)\Big{|}^{p}\Big{]}^{1\over
p}\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\quad\leqslant\tilde{K}_{p}\Big{[}\mathbb{E}\Big{|}\xi+\int_{t}^{T}g_{1}(s,Z(s))ds-Y(t)\Big{|}^{p}\Big{]}^{1\over
p}\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle\quad\leqslant
2\tilde{K}_{p}\Big{\\{}\Big{(}\mathbb{E}|\xi|^{p}\Big{)}^{1\over
p}+\Big{[}\mathbb{E}\Big{(}\int_{t}^{T}L^{1}_{0}(s)ds\Big{)}^{p}\Big{]}^{1\over
p}+\Big{(}\int_{t}^{T}L^{1}_{z}(s)^{2}ds\Big{)}^{1\over
2}\Big{[}\mathbb{E}\Big{(}\int_{t}^{T}|Z(s)|^{2}ds\Big{)}^{p\over
2}\Big{]}^{1\over p}\Big{\\}},\end{array}$ (7.7)

where $\tilde{K}_{p}=K_{p}^{1\over p}$ with $K_{p}$ being given by 7.1. For
any $0\leqslant t\leqslant\tau\leqslant T$, denote

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\Theta_{y}(t,\tau)=\Big{(}\sup_{s\in[t,\tau]}\mathbb{E}|Y(s)|^{p}\Big{)}^{1\over
p},\qquad\Theta_{z}(t,\tau)=\Big{[}\mathbb{E}\Big{(}\int_{t}^{\tau}|Z(s)|^{2}ds\Big{)}^{p\over
2}\Big{]}^{1\over p},\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle{\cal
L}(t,\tau)=\Big{(}\int_{t}^{\tau}L^{1}_{z}(s)^{2}ds\Big{)}^{1\over
2},\qquad{\cal
L}_{0}(t,\tau)=\Big{[}\mathbb{E}\Big{(}\int_{t}^{\tau}L_{0}^{1}(s)ds\Big{)}^{p}\Big{]}^{1\over
p}.\end{array}$

Then the above procedure leads to

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\Theta_{y}(t,\tau)\leqslant\Theta_{y}(\tau,\tau)+{\cal
L}_{0}(t,\tau)+{\cal L}(t,\tau)\Theta_{z}(t,\tau),\\\ \vskip 3.0pt plus 1.0pt
minus 1.0pt\cr\displaystyle\Theta_{z}(t,\tau)\leqslant
2\tilde{K}_{p}\Big{[}\Theta_{y}(\tau,\tau)+{\cal L}_{0}(t,\tau)+{\cal
L}(t,\tau)\Theta_{z}(t,\tau)\Big{]}.\end{array}$

Thus, by letting $\tau-t>0$ small so that $2\tilde{K}_{p}{\cal L}(t,\tau)<1$,
one has

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\Theta_{z}(t,\tau)\leqslant{2\tilde{K}_{p}\Big{[}\Theta_{y}(\tau,\tau)+{\cal
L}_{0}(t,\tau)\Big{]}\over 1-2\tilde{K}_{p}{\cal
L}(t,\tau)},\qquad\Theta_{y}(t,\tau)\leqslant{\Theta_{y}(\tau,\tau)+{\cal
L}_{0}(t,\tau)\over 1-2\tilde{K}_{p}{\cal L}(t,\tau)}.\end{array}$

Pick $0=t_{0}<t_{1}<t_{2}<\cdots<t_{N}=T$ so that

$\alpha\equiv{1\over 1-2\tilde{K}_{p}{\cal
L}(t_{k-1},t_{k})}\equiv{1\over\displaystyle
1-2\tilde{K}_{p}\Big{(}\int_{t_{k-1}}^{t_{k}}L^{1}_{z}(s)^{2}ds\Big{)}^{1\over
2}}\in(1,\infty),\quad k=1,2,\cdots,N.$ (7.8)

Then we have

$\displaystyle\Theta_{y}(t_{k-1},t_{k})$
$\displaystyle\leqslant\alpha\Theta_{y}(t_{k},t_{k})+\alpha{\cal
L}_{0}(t_{k-1},t_{k})\leqslant\alpha\Theta_{y}(t_{k},t_{k+1})+\alpha{\cal
L}_{0}(t_{k-1},t_{k})$
$\displaystyle\leqslant\alpha^{2}\Theta_{y}(t_{k+1},t_{k+2})+\alpha^{2}{\cal
L}_{0}(t_{k},t_{k+1})+\alpha{\cal L}_{0}(t_{k-1},t_{k})\leqslant\cdots$
$\displaystyle\leqslant\alpha^{N-k+1}\Theta_{y}(T,T)+\sum_{i=k}^{N}\alpha^{i-k+1}{\cal
L}_{0}(t_{i-1},t_{i})\leqslant\alpha^{N}\Theta_{y}(T,T)+N\alpha^{N}{\cal
L}_{0}(0,T),$

and

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\Theta_{z}(0,T)\leqslant\sum_{k=1}^{N}\Theta_{z}(t_{k-1},t_{k})\leqslant\sum_{k=1}^{N}2\tilde{K}_{p}\alpha\Big{[}\Theta_{y}(t_{k},t_{k})+{\cal
L}_{0}(t_{k-1},t_{k})\Big{]}\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\qquad\qquad\leqslant
2\tilde{K}_{p}N\alpha^{N+1}\Theta_{y}(T,T)+2\tilde{K}_{p}N^{2}\alpha^{N+1}{\cal
L}_{0}(0,T).\end{array}$

That is

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\Big{[}\sup_{t\in[0,T]}\mathbb{E}|Y(t)|^{p}\Big{]}^{1\over
p}\leqslant\alpha^{N}\Big{(}\mathbb{E}|\xi|^{p}\Big{)}^{1\over
p}+N\alpha^{N}\Big{[}\mathbb{E}\Big{(}\int_{0}^{T}L_{0}(s)ds\Big{)}^{p}\Big{]}^{1\over
p},\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\Big{[}\mathbb{E}\Big{(}\int_{0}^{T}|Z(s)|^{2}ds\Big{)}^{p\over
2}\Big{]}^{1\over p}\leqslant
2\tilde{K}_{p}N\alpha^{N+1}\Big{(}\mathbb{E}|\xi|^{p}\Big{)}^{1\over
p}+2\tilde{K}_{p}N^{2}\alpha^{N+1}\Big{[}\mathbb{E}\Big{(}\int_{0}^{T}L_{0}(s)ds\Big{)}^{p}\Big{]}^{1\over
p}.\end{array}$

Hence,

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\Big{[}\sup_{t\in[0,T]}\mathbb{E}|Y(t)|^{p}\Big{]}^{1\over
p}+\Big{[}\mathbb{E}\Big{(}\int_{0}^{T}|Z(s)|^{2}ds\Big{)}^{p\over
2}\Big{]}^{1\over p}\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\quad\leqslant(1+2\tilde{K}_{p}N\alpha)\alpha^{N}\Big{(}\mathbb{E}|\xi|^{p}\Big{)}^{1\over
p}+(1+2\tilde{K}_{p}N\alpha)N\alpha^{N}\Big{[}\mathbb{E}\Big{(}\int_{0}^{T}L_{0}(s)ds\Big{)}^{p}\Big{]}^{1\over
p}.\end{array}$

We now determine $N$ and $\alpha$ in terms of the known conditions. From
(7.8), we have have the relation between $\alpha$ and $N$:

$N{(\alpha-1)^{2}\over\alpha^{2}}=4\tilde{K}^{2}_{p}\int_{0}^{T}L_{z}^{1}(s)^{2}ds\equiv\bar{K}_{p}.$

Thus, for given large $N\in\mathbb{N}$, we have

$0=N(\alpha-1)^{2}-\bar{K}_{p}\alpha^{2}=(N-\bar{K}_{p})\alpha^{2}-2N\alpha+N.$

This means that for any integer $N>\bar{K}$,

$1<\alpha={2N+\sqrt{4N^{2}-4(N-\bar{K}_{p})N}\over
2(N-\bar{K}_{p})}={N+\sqrt{N\bar{K}_{p}}\over
N-\bar{K}_{p}}={\sqrt{N}\over\sqrt{N}-\sqrt{\bar{K}_{p}}}.$

Let

$f(x)=\Big{[}1+2\tilde{K}_{p}x\Big{(}{\sqrt{x}\over\sqrt{x}-\sqrt{\bar{K}_{p}}}\Big{)}\Big{]}\Big{(}{\sqrt{x}\over\sqrt{x}-\sqrt{\bar{K}_{p}}}\Big{)}^{x},\qquad
x>\bar{K}_{p}.$

Clearly,

$\lim_{x\to\bar{K}_{p}}f(x)=\lim_{x\to\infty}f(x)=\infty.$

Hence, the following is well-defined:

$\displaystyle\widehat{K}_{p}$
$\displaystyle=\min\Big{\\{}\Big{[}1+2\tilde{K}_{p}N\Big{(}{\sqrt{N}\over\sqrt{N}-\sqrt{\bar{K}_{p}}}\Big{)}\Big{]}\Big{(}{\sqrt{N}\over\sqrt{N}-\sqrt{\bar{K}_{p}}}\Big{)}^{N}\bigm{|}N\in\mathbb{N},~{}N>\bar{K}_{p}\Big{\\}}$
(7.9)
$\displaystyle\equiv\Big{[}1+2\tilde{K}_{p}N_{p}\Big{(}{\sqrt{N_{p}}\over\sqrt{N_{p}}-\sqrt{\bar{K}_{p}}}\Big{)}\Big{]}\Big{(}{\sqrt{N_{p}}\over\sqrt{N_{p}}-\sqrt{\bar{K}_{p}}}\Big{)}^{N_{p}}.$

Consequently, we have

$\Big{[}\sup_{t\in[0,T]}\mathbb{E}|Y(t)|^{p}\Big{]}^{1\over
p}+\Big{[}\mathbb{E}\Big{(}\int_{0}^{T}|Z(s)|^{2}ds\Big{)}^{p\over
2}\Big{]}^{1\over
p}\leqslant\widehat{K}_{p}\Big{(}\mathbb{E}|\xi|^{p}\Big{)}^{1\over
p}+N_{p}\widehat{K}_{p}\Big{[}\mathbb{E}\Big{(}\int_{0}^{T}L_{0}^{1}(s)ds\Big{)}^{p}\Big{]}^{1\over
p}.$ (7.10)

Now, let $g_{i}(\cdot\,,\cdot)$ satisfy (H0)p, $\xi_{i}\in L^{p}_{{\cal
F}_{T}}(\Omega;\mathbb{R}^{n})$ and let $(Y_{i}(\cdot),Z_{i}(\cdot))$ be the
corresponding adapted solution:

$Y_{i}(t)=\xi_{i}+\int_{t}^{T}g_{i}(s,Z_{i}(s))ds-\int_{t}^{T}Z_{i}(s)dW(s),\qquad
t\in[0,T],~{}i=1,2.$

Then we have the following BSDE for
$(Y_{1}(\cdot)-Y_{2}(\cdot),Z_{1}(\cdot)-Z_{2}(\cdot))$:

$\displaystyle Y_{1}(t)-Y_{2}(t)$
$\displaystyle=\xi_{1}-\xi_{2}+\int_{t}^{T}\big{\\{}g_{1}(s,Z_{1}(s))-g_{2}(s,Z_{1}(s))+G(s)[Z_{1}(s)-Z_{2}(s)]\big{\\}}ds$
$\displaystyle\quad-\int_{t}^{T}[Z_{1}(s)-Z_{2}(s)]dW(s),$

where

$G(s)={[g_{2}(s,Z_{1}(s))-g_{2}(s,Z_{2}(s))][Z_{1}(s)-Z_{2}(s)]^{\top}\over|Z_{1}(s)-Z_{2}(s)|^{2}}{\bf
1}_{\\{Z_{1}(s)\neq Z_{2}(s)\\}}.$

Clearly,

$|G(s)|\leqslant L_{z}^{1}(s),\qquad s\in[0,T],~{}\hbox{\rm a.s.}$

By (7.10), we obtain

$\begin{array}[]{ll}\displaystyle\Big{[}\sup_{t\in[0,T]}\mathbb{E}|Y_{1}(t)-Y_{2}(t)|^{p}\Big{]}^{1\over
p}+\Big{[}\mathbb{E}\Big{(}\int_{0}^{T}|Z_{1}(s)-Z_{2}(s)|^{2}ds\Big{)}^{p\over
2}\Big{]}^{1\over p}\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\quad\leqslant\widehat{K}_{p}\Big{(}\mathbb{E}|\xi_{1}-\xi_{2}|^{p}\Big{)}^{1\over
p}+N_{p}\widehat{K}_{p}\Big{[}\mathbb{E}\Big{(}\int_{0}^{T}|g_{1}(s,Z_{1}(s))-g_{2}(s,Z_{1}(s))|ds\Big{)}^{p}\Big{]}^{1\over
p}.\end{array}$

This completes the proof.

We now look at the following Type-I BSVIE:

$Y(t)=\psi(t)+\int_{t}^{T}g_{1}(t,s,Z(t,s))ds-\int_{t}^{T}Z(t,s)dW(s),\qquad
t\in[0,T].$ (7.11)

We introduce the following assumption.

(H0)${}_{p}^{\prime}$ Let
$g_{1}:\Delta[0.T]\times\mathbb{R}^{n}\times\Omega\to\mathbb{R}^{n}$ satisfy
the following: For each $(t,z)\in[0,T]\times\mathbb{R}^{n}$, $s\mapsto
g_{1}(t,s,z)$ is $\mathbb{F}$-progressively measurable on $[t,T]$. There
exists a deterministic function $L_{z}^{1}(\cdot\,,\cdot)\in
L^{2}(\Delta[0,T];\mathbb{R}_{+})$ such that

$|g_{1}(t,s,z_{1})-g_{1}(t,s,z_{2})|\leqslant
L_{z}^{1}(t,s)|z_{1}-z_{2}|,\qquad\forall(t,s)\in\Delta,~{}z\in\mathbb{R}^{n}.$

with the property

$K^{0}_{p}\sup_{t\in[0,T]}\Big{(}\int_{t}^{T}L^{1}_{z}(t,s)^{2}ds\Big{)}^{p\over
2}<1,$ (7.12)

where $K_{p}^{0}\equiv(\widehat{K}_{p})^{p}$ and $\widehat{K}_{p}$ is given by
(7.5) with $\bar{K}$ replaced by

$\bar{K}=4\tilde{K}^{2}_{p}\sup_{t\in[0,T]}\int_{t}^{T}L_{z}^{1}(t,s)^{2}ds.$
(7.13)

Further, $g_{1}(\cdot\,,\cdot\,,0)\in
L_{\mathbb{F}}^{p}(\Omega;L^{1}(\Delta[0,T];\mathbb{R}^{n}))$.

The following proposition is concerned with the above BSVIE.

######  Proposition 7.3.

Let (H0)${}_{p}^{\prime}$ hold. Then for any $\psi(\cdot)\in L^{p}_{{\cal
F}_{T}}(0,T;\mathbb{R}^{n})$, BSVIE (7.11) admits a unique adapted solution
$(Y(\cdot),Z(\cdot\,,\cdot))\in{\cal H}_{\Delta}^{p}[0,T]$ and with
$K_{p}^{0}=(\widehat{K}_{p})^{p}$, the following holds:

$\mathbb{E}|Y(t)|^{p}+\mathbb{E}\Big{(}\int_{t}^{T}|Z(t,s)|^{2}ds\Big{)}^{p\over
2}\leqslant
K_{p}^{0}\mathbb{E}\Big{|}\psi(t)+\int_{t}^{T}g(t,s,0)ds\Big{|}^{p},\quad\hbox{\rm
a.e.}\;t\in[0,T].$ (7.14)

If $g_{i}(\cdot\,,\cdot\,,\cdot)$ satisfies (H0)${}_{p}^{\prime}$,
$\psi_{i}(\cdot)\in L^{p}_{{\cal F}_{T}}(\Omega;\mathbb{R}^{n})$, and
$(Y_{i}(\cdot),Z_{i}(\cdot\,,\cdot))$ is adapted solution to the corresponding
BSVIE, $i=1,2$, then

$\begin{array}[]{ll}\displaystyle\mathbb{E}|Y_{1}(t)-Y_{2}(t)|^{p}+\mathbb{E}\Big{(}\int_{t}^{T}|Z_{1}(t,s)-Z_{2}(t,s)|^{2}ds\Big{)}^{p\over
2}\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle\quad\leqslant
K_{p}^{0}\mathbb{E}\Big{|}\psi_{1}(t)-\psi_{2}(t)+\int_{t}^{T}[g_{1}(t,s,Z_{1}(t,s))-g_{2}(t,s,Z_{1}(t,s))]ds\Big{|}^{p},\quad\hbox{\rm
a.e.}\;t\in[0,T].\end{array}$ (7.15)

###### Proof.

First, we write the equation as follows:

$Y(t)=\psi(t)+\int_{t}^{T}g_{1}(t,s,0)ds+\int_{t}^{T}[g_{1}(t,s,Z(t,s))-g_{1}(t,s,0)]ds-\int_{t}^{T}Z(t,s)dW(s).$

For almost all $t\in[0,T]$, consider BSDE:

$\eta(t,r)=\psi(t)+\int_{t}^{T}g_{1}(t,s,0)ds+\int_{r}^{T}[g_{1}(t,s,\zeta(t,s))-g_{1}(t,s,0)]ds-\int_{r}^{T}\zeta(t,s)dW(s),\quad
r\in[t,T].$ (7.16)

Then regarding the sum of the first two terms on the right-hand side as the
terminal state, and note that the generator $(s,\zeta)\mapsto
g_{1}(t,s,\zeta)-g_{1}(t,s,0)\equiv\widetilde{g}_{1}(t,s,\zeta)$ has the
property that $\widetilde{g}_{1}(t,s,0)=0$, satisfying (H0)p, one has

$\begin{array}[]{ll}\displaystyle\Big{[}\sup_{r\in[t,T]}\mathbb{E}|\eta(t,r)|^{p}\Big{]}^{1\over
p}+\Big{[}\mathbb{E}\Big{(}\int_{t}^{T}|\zeta(t,s)|^{2}ds\Big{)}^{p\over
2}\Big{]}^{1\over
p}\leqslant\widehat{K}_{p}\Big{(}\mathbb{E}\Big{|}\psi(t)+\int_{t}^{T}g_{1}(t,s,0)ds\Big{|}^{p}\Big{)}^{1\over
p}.\end{array}$ (7.17)

By setting

$Y(t)=\eta(t,t),\quad Z(t,s)=\zeta(t,s),$

we have that $(Y(\cdot),Z(\cdot\,,\cdot))$ is an adapted solution to BSVIE
(7.11), and

$\displaystyle\Big{[}\mathbb{E}|Y(t)|^{p}+\mathbb{E}\Big{(}\int_{t}^{T}|Z(t,s)|^{2}ds\Big{)}^{p\over
2}\Big{]}^{1\over p}\leqslant\Big{[}\mathbb{E}|Y(t)|^{p}\Big{]}^{1\over
p}+\Big{[}\mathbb{E}\Big{(}\int_{t}^{T}|Z(t,s)|^{2}ds\Big{)}^{p\over
2}\Big{]}^{1\over p}$
$\displaystyle\quad\leqslant\widehat{K}_{p}\Big{(}\mathbb{E}\Big{|}\psi(t)+\int_{t}^{T}g_{1}(t,s,0)ds\Big{|}^{p}\Big{)}^{1\over
p}.$

Then, (7.14) holds. The uniqueness follows from (7.12). Next, let
$g_{i}(\cdot\,,\cdot)$ satisfy (H0)${}_{p}^{\prime}$, $\psi_{i}(\cdot)\in
L^{p}_{{\cal F}_{T}}(\Omega;\mathbb{R}^{n})$, and let
$(Y_{i}(\cdot),Z_{i}(\cdot\,,\cdot))$ be the corresponding BSVIE. Then

$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle
Y_{1}(t)-Y_{2}(t)=\psi_{1}(t)-\psi_{2}(t)+\int_{t}^{T}\Big{(}g_{1}(t,s,Z_{1}(t,s))-g_{2}(t,s,Z_{1}(t,s))\Big{)}ds\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\qquad\qquad\qquad\quad+\int_{t}^{T}G(t,s)[Z_{1}(t,s)-Z_{2}(t,s)]ds-\int_{t}^{T}[Z_{1}(t,s)-Z_{2}(t,s)]dW(s),\qquad
t\in[0,T],\end{array}$

where,

$G(t,s)={[g_{2}(t,s,Z_{1}(t,s))-g_{2}(t,s,Z_{2}(t,s))][Z_{1}(t,s)-Z_{2}(t,s)]^{\top}\over|Z_{1}(t,s)-Z_{2}(t,s)|^{2}}{\bf
1}_{\\{Z_{1}(t,s)\neq Z_{2}(t,s)\\}}.$

It is clear that

$|G(t,s)|\leqslant L_{z}^{1}(t,s),\qquad\forall(t,s)\in\Delta[0,T].$

Then, from the above, one has

$\begin{array}[]{ll}\displaystyle\Big{[}\mathbb{E}|Y_{1}(t)-Y_{2}(t)|^{p}+\mathbb{E}\Big{(}\int_{t}^{T}|Z_{1}(t,s)-Z_{2}(t,s)|^{2}ds\Big{)}^{p\over
2}\Big{]}^{1\over p}\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\quad\leqslant\widehat{K}_{p}\Big{(}\mathbb{E}\Big{|}\psi_{1}(t)-\psi_{2}(t)+\int_{t}^{T}[g_{1}(t,s,Z_{1}(t,s))-g_{2}(t,s,Z_{1}(t,s))]ds\Big{|}^{p}\Big{)}^{1\over
p}.\end{array}$ (7.18)

Hence, (7.15) follows. ∎

We remark that 7.3 still holds true if (7.12) is replaced by

$\sup_{t\in[0,T]}\Big{(}\int_{t}^{T}L^{1}_{z}(t,s)^{2+\varepsilon}ds\Big{)}<\infty.$
(7.19)

with $\varepsilon>0$. The following is concerned with the property of
$g_{1}(\cdot)$ in the decomposition (3.2).

Let $h:[0,T]\times\mathbb{R}\times\Omega\to\mathbb{R}$ be ${\cal
B}([0,T]\times\mathbb{R}\times{\cal F}_{T})$-measurable and $h(s,y)$ be
continuous with respect to the variable $y$. Moreover, suppose that
$h(\cdot\,,0)\in L^{1}_{{\cal F}_{T}}(0,T;\mathbb{R})$ and there exists a
function $L(\cdot)\in L^{\infty}(0,T;\mathbb{R}_{+})$ such that

$|h(s,y)|\leqslant|h(s,0)|+L(s)|y|.$

Let

$\widehat{h}(s,y)=\mathbb{E}_{s}[h(s,y)],\quad\forall(s,y)\in[0,T]\times\mathbb{R}.$

Then the mapping $y\mapsto\widehat{h}(s,y,\omega)$ is continuous, and for any
fixed $y\in\mathbb{R}$, the process $\widehat{h}(\cdot\,,y)$ is
$\mathbb{F}$-adapted.

######  Lemma 7.4.

For any $Y(\cdot)\in L^{1}_{\mathbb{F}}(0,T;\mathbb{R})$, the process
$\widehat{h}(s,Y(s));s\in[0,T]$ is $\mathbb{F}$-adapted. 444This result is
obtained by a useful discussion with Chenchen Mou (City University of Hong
Kong).

###### Proof.

By the definition of $\mathbb{F}$-adapted processes, we need only to prove
that for any $s\in[0,T]$, $\widehat{h}(s,Y(s))\in{\cal F}_{s}$. Then it
suffices to show that for any $O=(a,b)\subseteq\mathbb{R}$,

$\\{\omega\in\Omega\,|\,\widehat{h}(s,Y(s,\omega),\omega)\in\bar{O}=[a,b]\\}\in{\cal
F}_{s},$

because $(a,\infty)=\bigcup_{k,n=1}^{\infty}[a+1/n,a+k]$. For the ease of
presentation, we write
$\\{\omega\in\Omega\,|\,\widehat{h}(s,Y(s,\omega),\omega)\in O\\}$ as
$\\{\widehat{h}(s,Y(s))\in O\\}$.

It is clearly seen that

$\\{\widehat{h}(s,Y(s))\in\bar{O}\\}=\bigcup_{y\in\mathbb{R}}\big{(}\\{\widehat{h}(s,y)\in\bar{O}\\}\cap\\{Y(s)=y\\}\big{)}.$
(7.20)

Let $A$ be a countable dense set of $R$. We claim that

$\bigcup_{y\in\mathbb{R}}\big{(}\\{\widehat{h}(s,y)\in\bar{O}\\}\cap\\{Y(s)=y\\}\big{)}=\bigcap_{m=1}^{\infty}\bigcap_{n=1}^{\infty}\bigcup_{y\in
A}\Big{(}\big{\\{}\widehat{h}(s,B_{1\over n}(y))\in
O_{m}\big{\\}}\cap\big{\\{}Y(s)\in B_{1\over n}(y)\big{\\}}\Big{)},$ (7.21)

where

$\displaystyle\big{\\{}\widehat{h}(s,B_{1\over n}(y))\in
O_{m}\big{\\}}=\bigcup_{x\in B_{1\over n}(y)}\big{\\{}\widehat{h}(s,x)\in
O_{m}\big{\\}}\quad\hbox{and}\quad O_{m}=(a-1/m,b+1/m).$

Since the mapping $y\mapsto h(s,y)$ is continuous and $O_{m}$ is an open set,
we have

$\big{\\{}\widehat{h}(s,B_{1\over n}(y))\in O_{m}\big{\\}}=\bigcup_{x\in
B_{1\over n}(y)\cap A}\big{\\{}\widehat{h}(s,x)\in O_{m}\big{\\}}.$

If the claim (7.21) holds, the above implies that

$\bigcup_{y\in\mathbb{R}}\big{(}\\{\widehat{h}(s,y)\in\bar{O}\\}\cap\\{Y(s)=y\\}\big{)}=\bigcap_{m=1}^{\infty}\bigcap_{n=1}^{\infty}\bigcup_{y\in
A}\Big{(}\bigcup_{x\in B_{1\over n}(y)\cap A}\big{\\{}\widehat{h}(s,x)\in
O_{m}\big{\\}}\bigcap\big{\\{}Y(s)\in B_{1\over n}(y)\big{\\}}\Big{)}.$

Then from the facts $\\{\widehat{h}(s,x)\in O_{m}\\}\in{\cal F}_{s}$,
$\\{Y(s)\in B_{1\over n}(y)\\}\in{\cal F}_{s}$, and that $A$ is a countable
set, we get

$\\{\widehat{h}(s,Y(s))\in\bar{O}\\}\in{\cal F}_{s}.$

Now let us prove that the claim (7.21) really holds.

_Step 1._
$\bigcup_{y\in\mathbb{R}}\big{(}\\{\widehat{h}(s,y)\in\bar{O}\\}\cap\\{Y(s)=y\\}\big{)}\subseteq\bigcap_{m=1}^{\infty}\bigcap_{n=1}^{\infty}\bigcup_{y\in
A}\big{(}\big{\\{}\widehat{h}(s,B_{1\over n}(y))\in
O_{m}\big{\\}}\cap\big{\\{}Y(s)\in B_{1\over n}(y)\big{\\}}\big{)}.$

For any
$\omega\in\bigcup_{y\in\mathbb{R}}\big{(}\\{\widehat{h}(s,y)\in\bar{O}\\}\cap\\{Y(s)=y\\}\big{)}$,
there exists a $y\in\mathbb{R}$ such that $\widehat{h}(s,y,\omega)\in\bar{O}$
and $Y(s,\omega)=y$. For any $m>0$ and $n>0$, $\bar{O}\subseteq O_{m}$ and
there exists a $y_{mn}\in A$ such that $y\in B_{1\over n}(y_{mn})$. Then

$\omega\in\big{\\{}\widehat{h}(s,B_{1\over n}(y_{mn}))\in
O_{m}\big{\\}}\cap\big{\\{}Y(s)\in B_{1\over n}(y_{mn})\big{\\}}.$

Thus, “$\subseteq$” holds.

_Step 2._
$\bigcup_{y\in\mathbb{R}}\big{(}\\{\widehat{h}(s,y)\in\bar{O}\\}\cap\\{Y(s)=y\\}\big{)}\supseteq\bigcap_{m=1}^{\infty}\bigcap_{n=1}^{\infty}\bigcup_{y\in
A}\big{(}\big{\\{}\widehat{h}(s,B_{1\over n}(y))\in
O_{m}\big{\\}}\cap\big{\\{}Y(s)\in B_{1\over n}(y)\big{\\}}\big{)}.$

Let $\omega\in\bigcap_{m=1}^{\infty}\bigcap_{n=1}^{\infty}\bigcup_{y\in
A}\big{(}\big{\\{}\widehat{h}(s,B_{1\over n}(y))\in
O_{m}\big{\\}}\cap\big{\\{}Y(s)\in B_{1\over n}(y)\big{\\}}\big{)}$. Then for
any $m$ and $n$, there exists a $y_{m,n}\in A$ such that

$Y(s,\omega)\in B_{1\over n}(y_{mn})\quad\hbox{and}\quad\exists z_{mn}\in
B_{1\over
n}(y_{mn})\hbox{~{}~{}such~{}that~{}~{}}\widehat{h}(s,z_{mn},\omega)\in
O_{m}.$

Note that

$|y_{mn}-Y(s,\omega)|\leqslant{1\over
n}\quad\hbox{and}\quad|y_{mn}-z_{mn}|\leqslant{1\over n}.$

By the continuity of $\widehat{h}(s,y)$ with respect to $y$, we get

$\widehat{h}(s,Y(s,\omega),\omega)\in\bigcap_{m=1}^{\infty}O_{m}=\bar{O}.$

∎

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