# A new type of reflected backward doubly stochastic differential equations
††thanks: The work of Auguste Aman is support by TWAS Research Grants to
individuals (No. 09-100 RG/MATHS/AF/AC-I- UNESCO FR: 3240230311) and it been
dedicated to all the dead of the post-election crisis in his country.
The work of Yong Ren is supported by the National Natural Science Foundation
of China (No 10901003), the Key Project of Chinese Ministry of Education (No
211077) and the Anhui Provincial Natural Science Foundation (No 10040606Q30).

Auguste Aman1 and Yong Ren 2
1\. U.F.R Mathématiques et Informatique, Université de Cocody,
582 Abidjan 22, Côte d’Ivoire
2\. Department of Mathematics, Anhui Normal University,
Wuhu 241000, China<EMAIL_ADDRESS>corresponding<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
###### Abstract

In this paper, we introduce a new kind of "variant" reflected backward doubly
stochastic differential equations (VRBDSDEs in short), where the drift is the
nonlinear function of the barrier process. In the one stochastic case, this
type of equations have been already studied by Ma and Wang [25]. They called
it as "variant" reflected BSDEs (VRBSDEs in short) based on the general
version of the Skorohod problem recently studied by Bank and El Karoui [6].
Among others, Ma and Wang [25] showed that VRBSDEs is a novel tool for some
problems in finance and optimal stopping problems where no existing methods
can be easily applicable. Since more of those models have their stochastic
counterpart, it is very useful to transpose the work of Ma and Wang [25] to
doubly stochastic version. In doing so, we firstly establish the stochastic
variant Skorohod problem based on the stochastic representation theorem, which
extends the work of Bank and El Karoui [6]. We prove the existence and
uniqueness of the solution for VRBDSDEs by means of the contraction mapping
theorem. By the way, we show the comparison theorem and stability result for
the solutions of VRBDSDEs.

AMS Subject Classification: 60H15; 60H20

Keywords: Reflected backward doubly stochastic differential equation,
stochastic Skorohod problem, stochastic representation theorem.

##  1 Introduction

The theory of backward stochastic differential equations (BSDEs in short) was
developed by Pardoux and Peng [31]. Precisely, given a data $(\xi,f)$
consisting of a progressively measurable process $f$, so-called the generator,
and a square integrable random variable $\xi$, they proved the existence and
uniqueness of an adapted process $(Y,Z)$ solution to the following BSDEs:

$Y_{t}=\xi+\int_{t}^{T}f(s,Y_{s},Z_{s})\,{\rm d}s-\int_{t}^{T}Z_{s}\,{\rm
d}B_{s},\quad 0\leq t\leq T.$

These equations have attracted great interest due to their connections with
mathematical finance [16, 17], stochastic control and stochastic games [20,
21, 22]. Furthermore, it was shown in various papers that BSDEs give the
probabilistic representation for the solution (at least in the viscosity
sense) of a large class of systems of semi-linear parabolic partial
differential equations (PDEs in short) [29, 30, 32, 34].

Further, other settings of BSDEs have been proposed. Especially, El-Karoui et
al. [15] have introduced the notion of reflected BSDEs (RBSDEs in short),
which is a BSDE but the solution is forced to stay above a lower barrier. In
details, a solution of such equations is a triple of processes $(Y,Z,K)$
satisfying that

$Y_{t}=\xi+\int_{t}^{T}f(s,Y_{s},Z_{s})\,{\rm
d}s+K_{T}-K_{t}-\int_{t}^{T}Z_{s}\,{\rm d}B_{s},\ Y_{t}\geq S_{t},$ (1.1)

where $S$, so-called the barrier, is a given stochastic process. The role of
the continuous increasing process $K$ is to push the state process upward with
the minimal energy, in order to keep it above $S$; in this sense, it satisfies
$\int_{0}^{T}(Y_{t}-S_{t})\,{\rm d}K_{t}=0.$ RBSDEs have been proven to be
powerful tools in mathematical finance [13, 19], the mixed game problems [12,
23], providing a probabilistic formula for the viscosity solution of an
obstacle problem for a class of parabolic PDEs ([14, 15, 37]) and so on. On
other interesting results on RBSDEs driven by a Brownian motion with different
barrier conditions, one can see Hamadène [18], Lepeltier and Xu [24] and Peng
and Xu [35].

Very recently, Ma and Wang [25] introduced the so-called Variant Reflected
Backward Stochastic Differential Equations (VRBSDEs in short) associated with
the notion of variant Skorohod problem studied by Bank and El Karoui [6], that
is

$Y_{t}=E\left\\{X_{T}+\int_{t}^{T}f(s,Y_{s},A_{s})ds|\mathcal{F}_{t}\right\\},0\leq
t\leq T,$ (1.2)

where $X=\\{X_{t}\\}_{t\geq 0}$ is an optional process of class (D) and the
solution $(Y,A)$ satisfies that

  1. (i)

$Y_{t}\leq X_{t},0\leq t\leq T,Y_{T}=X_{T};$

  2. (ii)

$A=\\{A_{t}\\}$ is an adapted, increasing process such that $A_{0-}=-\infty$,
and the following flat-off condition holds

$E\int_{0}^{T}|Y_{t}-X_{t}|dA_{t}=0.$ (1.3)

In addition, if the filtration $\mathcal{F}$ is generated by a Brownian motion
$B$, then (1.2) has the following extension form

$dY_{t}=-f(t,Y_{t},Z_{t},A_{t})dt+Z_{t}dB_{t},\ Y_{t}\leq X_{t},0\leq t\leq
T,\ Y_{T}=X_{T}.$ (1.4)

Unlike the role of $K$ in (1.1) as stated previously, the process $A$ here
could be regarded as a density of a reflecting force, which acts through the
drift $f$ in a nonlinear manner. From the above statements, we can see that
there is great difference between the frameworks of RBSDEs and VRBSDEs. Also,
even the fundamental well-posedness property of the VRBSDE cannot be obtained
by means of the usual ways used in BSDE and RBSDE. This brand new kind of
BSDEs has some important applications in finance and optimal stopping problems
([25]).

In [33], Pardoux and Peng proposed another class of BSDEs, named backward
doubly stochastic differential equations (BDSDEs in short) with the form:

$Y_{t}=\xi+\int_{t}^{T}f(s,Y_{s},Z_{s})ds+\int_{t}^{T}g(s,Y_{s},Z_{s})dB_{s}-\int_{t}^{T}Z_{t}dW_{t},0\leq
t\leq T,$ (1.5)

where the integral with respect to $\\{B_{t}\\}$ is a backward Itô integral
and the integral with respect to $\\{W_{t}\\}$ is a standard forward Itô
integral. Those two types of integrals are particular cases of the It
-Skorohod integral, see Nualart and Pardoux [28]. Following it, some well-
known works have been done in the probabilistic representation of certain
quasi-linear stochastic partial differential equations by means of BDSDEs from
different aspects, one can see Bally and Matoussi [5], Boufoussi et al. [7,
8], Buckdahn and Ma [9, 11, 10], Matoussi and Scheutzow [26], Zhang and Zhao
[38] and the references therein. Based on the reflected framework of El-Karoui
et al. [15], Bahlali et al. [4], Aman [2] and Ren [36] respectively proved the
existence and uniqueness of the solution for a class of reflected BDSDEs
(RBDSDEs in short) driven by Brownian motions and Lévy processes. Especially,
very recently, Matoussi and Stoica [27] proved the existence and uniqueness
result for the obstacle problem of quasi-linear parabolic stochastic PDEs by
means of the RBDSDEs.

Motivated by the aforementioned works, in this paper, we study a class of
variant reflected backward doubly stochastic differential equations (VRBDSDEs
in short). In doing so, we firstly establish the stochastic variant Skorohod
problem based on the stochastic representation theorem, which extends the work
of Bank and El Karoui [6]. We prove the existence and uniqueness of the
solution for VRBDSDEs by means of the contraction mapping theorem. In
addition, we show the comparison theorem and the stability result for the
solutions of VRBDSDEs.

Let us describe our plan. First, the formulation of the problems is proposed
in Section 2. The main results are presented in Section 3.

##  2 Formulation of the problems

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $T>0$ be
fixed throughout this paper. Let $\\{W_{t},\,0\leq t\leq T\\}$ and
$\\{B_{t},\,0\leq t\leq T\\}$ be two mutually independent standard Brownian
motion processes, with values respectively in $\mathbb{R}^{d}$ and in
$\mathbb{R}^{\ell}$, define on $(\Omega,\mathcal{F},\mathbb{P})$. Let
$\mathcal{N}$ denote the class of $\mathbb{P}$-null sets of $\mathcal{F}$. For
each $t\in[0,T]$, let us define

$\mathcal{F}_{t}=\mathcal{F}_{t}^{W}\vee\mathcal{F}_{t,T}^{B},$

where for any process $\\{\eta_{t};0\leq t\leq
T\\},\;\mathcal{F}_{s,t}^{\eta}=\sigma\\{\eta_{r}-\eta_{s};s\leq r\leq
t\\}\vee\mathcal{N}$ and, $\mathcal{F}_{t}^{\eta}=\mathcal{F}_{0,t}^{\eta}$.
Knowing that $\\{\mathcal{F}_{t}^{W},t\in[0,T]\\}$ is an increasing filtration
and $\\{\mathcal{F}_{t,T}^{B},t\in[0,T]\\}$ is a decreasing filtration, the
collection $\\{\mathcal{F}_{t},t\in[0,T]\\}$ is neither increasing nor
decreasing so it does not constitute a filtration.

Let us describe the following spaces will frequently used in the sequel.

$\bullet$ For any $n\in\mathbb{N}$, $\mathcal{M}^{2}(0,T,\mathbb{R}^{n})$
denotes the set of (class of $d\mathbb{P}\otimes dt$ a.e.) $n$-dimensional
jointly measurable random processes $\\{\varphi_{t};0\leq t\leq T\\}$ such
that

$\|\varphi\|_{\mathcal{M}^{2}}^{2}=\mathbb{E}\left(\int_{0}^{T}\mid\varphi_{t}\mid^{2}dt\right)<+\infty.$

$\bullet$ $\mathcal{S}^{\infty}([0,T],\mathbb{R})$ denotes the set of one
dimensional continuous $\mathcal{F}_{t}$-measurable bounded random processes.

$\bullet$ $\mathbb{L}^{\infty}(\mathbb{R})$ denotes the space of all
$\mathcal{F}_{T}$-measurable bounded random variables.

$\bullet$ $\mathcal{M}_{0,T}$ denotes the space of all stopping times taking
values in $[0,T]$.

$\bullet$ The process $X$ is said to belong to Class (D) on $[0,T]$ if the
family of random variables $\\{X_{\tau},\,\tau\in\mathcal{M}_{0,T}\\}$ is
uniformly integrable.

Next, let us give the standing assumptions relative to VRBDSDE.

(A1) The boundary processes $X=\\{X_{t},0\leq t\leq T\\}$ is assumed to be an
optional process of class (D) and is lower-semi-continuous in expectation,

(A2) The coefficients
$f:[0,T]\times\Omega\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ and
$g:[0,T]\times\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ satisfy the
following assumptions:

  1. (i)

for fixed $(\omega,t,y)\in\Omega\times[0,T]\times\mathbb{R}$, the function
$f(t,\omega,y,\cdot)$ is continuous and strictly decreasing from $+\infty$ to
$-\infty$;

  2. (ii)

for fixed $(y,l)\in\mathbb{R}^{2}$, the processes $f(\cdot,\cdot,y,l)$ and
$g(\cdot,\cdot,y)$ are jointly measurable with

$\displaystyle\mathbb{E}\int_{0}^{T}[|f(t,\omega,y,l)|+|g(t,\omega,y)|^{2}]dt<+\infty;$

  3. (iii)

there exists a constant $L>0$, such that for all fixed $t,\omega,l,$ it holds
that

$\displaystyle|f(t,\omega,y,l)-f(t,\omega,y^{\prime},l)|$ $\displaystyle\leq$
$\displaystyle L|y-y^{\prime}|,$
$\displaystyle|g(t,\omega,y)-g(t,\omega,y^{\prime})|$ $\displaystyle\leq$
$\displaystyle L|y-y^{\prime}|,\;\;\forall\,y,\,y^{\prime}\in\mathbb{R};$

  4. (iv)

there exist two constants $k>0$ and $K>0$, such that for all fixed
$t,\omega,y,$ it holds that

$\displaystyle
k|l-l^{\prime}|\leq|f(t,\omega,y,l)-f(t,\omega,y,l^{\prime})|\leq
K|l-l^{\prime}|,\;\;\forall\,l,\,l^{\prime}\in\mathbb{R}.$

Given $\xi\in\mathbb{L}^{2}(\mathbb{R})$ and the boundary process $X$, we
consider the following VRBDSDE.

(i)

$\displaystyle
Y_{t}=\xi+\int_{t}^{T}f(s,Y_{s},A_{s})ds+\int_{t}^{T}g(s,Y_{s})dB_{s}-\int_{t}^{T}Z_{s}dW_{s},\;0\leq
t\leq T;$ (2.1)

(ii)

$Y_{t}\leq X_{t},\;0\leq t\leq T,\;\;Y_{T}=X_{T}=\xi;$

(iii)

the process $(A_{t})_{t}$ is $\mathcal{F}_{t}$-measurable, increasing, càdlàg
(right continuous with left limits), and $A_{0^{-}}=-\infty$, such that
$\displaystyle{\mathbb{E}\int_{0}^{T}|Y_{t}-X_{t}|dA_{t}=0}$.

The study of this new type of BDSDEs is based on the extension of Stochastic
Representation Theorem initiated by Bank and El Karoui [6]. To do this, let us
consider the following filtration $(\mathcal{G}_{t})_{t\geq 0}$ defined by

$\displaystyle\mathcal{G}_{t}=\mathcal{F}_{t}^{W}\vee\mathcal{F}_{T}^{B}.$

######  Theorem 2.1.

Assume (A2)–(i), (ii). Then, every optional process $X$ of class (D) which is
lower semi-continuous in expectation admits a representation of the form

$\displaystyle X_{S}=\mathbb{E}\left\\{X_{T}+\int_{S}^{T}f\left(u,\sup_{S\leq
v\leq
u}L_{v}\right)du+\int_{S}^{T}g\left(u\right)dB_{u}|\mathcal{F}_{S}\right\\}$
(2.2)

for any stopping times $S\in\mathcal{M}_{0,T}$, where $L$ is an optional
process taking values in $\mathbb{R}\cup\\{-\infty,+\infty\\}$, and it can be
characterized as follows

$(i)$

$\,f\left(u,\sup_{S\leq v\leq u}L_{v}\right)\in L^{1}(\mathbb{P}\otimes
dt),\,g\left(u\right)\in L^{2}(\mathbb{P}\otimes dt)$ for any stopping times
$S$,

$(ii)$

$\,L_{S}=\mbox{ess}\inf_{\tau>S}l_{S,\tau}$, where the "$\mbox{ess}\inf$" is
taken over all stopping times $S\in\mathcal{M}_{0,T}$ such that $S<T$, a.s.;
and $l_{S,\tau}$ is the unique $\mathcal{F}_{S}$-measurable random variable
satisfying that

$\displaystyle\mathbb{E}\\{X_{S}-X_{\tau}|\mathcal{F}_{S}\\}=\mathbb{E}\left\\{\int_{S}^{\tau}f\left(u,l_{S,\tau}\right)du+\int_{S}^{\tau}g\left(u\right)dB_{u}|\mathcal{F}_{S}\right\\},$

$(iii)$

if $\;V(t,l)=\mbox{ess}\inf_{\tau\geq
t}\mathbb{E}\left\\{X_{\tau}+\int_{t}^{\tau}f\left(u,l\right)du+\int_{t}^{\tau}g(u)dB_{u}|\mathcal{F}_{t}\right\\},\,t\in[0,T]$,
is the value functions of a family of optimal stopping problems indexed by
$l\in\mathbb{R}$, then

$\displaystyle L_{t}=\sup\\{l:V(t,l)=X_{t}\\},\;\;\;t\in[0,T].$

###### Proof.

Let $X$ be a optional process $X$ of class (D) which is lower semi-continuous
in expectation and $g$ be a function given above. Setting

$\displaystyle\widetilde{X}_{t}=X_{t}+\int^{t}_{0}g(u)dB_{u},$

according to assumption $({\bf A1})$ and $({\bf A2})$, it is clear that
$\widetilde{X}$ is a optional process of class (D) and is lower semi-
continuous. Therefore, it follows from Theorem 3 in [6] that there exists an
optional process $L$ taking values in $\mathbb{R}\cup\\{-\infty,+\infty\\}$
such that for any stopping times $S\in\mathcal{M}_{0,T}$,

$\displaystyle\widetilde{X}_{S}=\mathbb{E}\left\\{\widetilde{X}_{T}+\int_{S}^{T}f\left(u,\sup_{S\leq
v\leq u}L_{v}\right)du|\mathcal{G}_{S}\right\\}.$ (2.3)

Moreover, $L$ is characterized as follows:

  * •

$\,f\left(u,\sup_{S\leq v\leq u}L_{v}\right)\in L^{1}(\mathbb{P}\otimes dt)$
for any stopping times $S$, which satisfy $(i)$.

  * •

$\,L_{S}=\mbox{ess}\inf_{\tau>S}l_{S,\tau}$, where the "$\mbox{ess}\inf$" is
taken over all stopping times $S\in\mathcal{M}_{0,T}$ such that $S<T$, a.s.;
and $l_{S,\tau}$ is the unique $\mathcal{G}_{S}$-measurable random variable
satisfying that

$\displaystyle\mathbb{E}\\{\widetilde{X}_{S}-\widetilde{X}_{\tau}|\mathcal{G}_{S}\\}=\mathbb{E}\left\\{\int_{S}^{\tau}f\left(u,l_{S,\tau}\right)du|\mathcal{G}_{S}\right\\}.$
(2.4)

  * •

If $\widetilde{V}(t,l)=\mbox{ess}\inf_{\tau\geq
t}\mathbb{E}\left\\{\widetilde{X}_{\tau}+\int_{t}^{\tau}f\left(u,l\right)du|\mathcal{G}_{t}\right\\},\,t\in[0,T]$,
is the value functions of a family of optimal stopping problems indexed by
$l\in\mathbb{R}$, then

$\displaystyle
L_{t}=\sup\\{l:\widetilde{V}(t,l)=\widetilde{X}_{t}\\},\;\;\;t\in[0,T].$

Since $\widetilde{X}_{S}$ is $\mathcal{F}_{S}$-measurable and
$\mathcal{F}_{S}\subset\mathcal{G}_{S}$, and according to the definition of
$\widetilde{X}$, it follows from equalities (2.3) and (2.4) that

$\displaystyle X_{S}=\mathbb{E}\left\\{X_{T}+\int_{S}^{T}f\left(u,\sup_{S\leq
v\leq u}L_{v}\right)du+\int^{T}_{S}g(u)dB_{u}|\mathcal{F}_{S}\right\\}$

and

$\displaystyle\mathbb{E}\\{X_{S}-X_{\tau}|\mathcal{F}_{S}\\}=\mathbb{E}\left\\{\int_{S}^{\tau}f\left(u,l_{S,\tau}\right)du+\int^{\tau}_{S}g(u)dB_{u}|\mathcal{F}_{S}\right\\},$
(2.5)

respectively. To prove $(ii)$, it remains to show that $l_{S,\tau}$ is a
$\mathcal{F}_{S}$-measurable random variable, which is clear by (2.4). To end
the proof let us show $(iii)$. In fact, equalities (2.4) and (2.5) provide

$\displaystyle\mathbb{E}\left\\{X_{\tau}+\int_{S}^{\tau}f\left(u,l_{S,\tau}\right)du+\int^{\tau}_{S}g(u)dB_{u}|\mathcal{G}_{S}\right\\}$

$=\mathbb{E}\left\\{X_{\tau}+\int_{S}^{\tau}f\left(u,l_{S,\tau}\right)du+\int^{\tau}_{S}g(u)dB_{u}|\mathcal{F}_{S}\right\\}.$

Hence, denoting

$\displaystyle V(t,l)=\widetilde{V}(t,l)-\int^{\tau}_{t}g(u)dB_{u},$

we have

$\displaystyle V(t,l)=\mbox{ess}\inf_{\tau\geq
t}\mathbb{E}\left\\{X_{\tau}+\int_{t}^{\tau}f\left(u,l\right)du+\int^{\tau}_{S}g(u)dB_{u}|\mathcal{F}_{t}\right\\}$

and

$\displaystyle L_{t}=\sup\\{l:\,V(t,l)=X_{t}\\},\;\;\;t\in[0,T],$

which prove $(iii)$. ∎

A direct consequence of the previous stochastic representation theorem is the
following stochastic variant Skorohod problem.

######  Theorem 2.2.

Assume (A2)–(i), (ii). Then, for every optional process $X$ of class (D) which
is lower semi-continuous in expectation, there exists a unique pair of
$\mathcal{F}_{t}$-measurable processes $(Y,A)$, where $Y$ is continuous and
$A$ is increasing such that

$\displaystyle
Y_{t}=\mathbb{E}\left\\{X_{T}+\int_{t}^{T}f\left(u,A_{u}\right)du+\int_{t}^{T}g\left(u\right)dB_{u}|\mathcal{F}_{t}\right\\},\,\,t\in[0,T].$

Furthermore, the process $A$ can be expressed as $A_{t}=\sup_{0\leq s\leq
t^{+}}L_{s}$, where $L$ is the process in Theorem 2.1.

Before give the proof of the above theorem, let us give a remark.

######  Remark 2.3.

The previous theorem can be enounced as follows: there exists a unique pair of
$\mathcal{F}_{t}$-measurable processes $(Y,Z,A)$, where $Y$ is continuous and
$A$ is increasing such that

$\displaystyle
Y_{t}=X_{T}+\int_{t}^{T}f\left(u,A_{u}\right)du+\int_{t}^{T}g\left(u\right)dB_{u}-\int^{T}_{t}Z_{u}dW_{u},\,\,t\in[0,T].$

###### Proof.

Let us define $A_{t}=\sup_{0\leq s\leq t^{+}}L_{s}$, where $L$ is the process
appears in (2.2) and the $\mathcal{G}_{t}$-square integrable martingale

$\displaystyle
M_{t}=\mathbb{E}\left\\{X_{T}+\int_{0}^{T}f\left(u,A_{u}\right)du+\int_{0}^{T}g\left(u\right)dB_{u}|\mathcal{G}_{t}\right\\},\;\;0\leq
t\leq T.$

An obvious extension of the Itô martingale representation theorem yields the
existence of a $\mathcal{G}_{t}$-progressively measurable process
$\\{Z_{t}\\}$ with values in $\mathbb{R}^{d}$ such that

$\displaystyle\mathbb{E}\left(\int^{T}_{0}|Z_{s}|^{2}ds\right)<+\infty,$
$\displaystyle M_{t}=M_{0}+\int_{0}^{t}Z_{s}dW_{s},\;\;\;0\leq t\leq T.$

Hence,

$\displaystyle M_{T}=M_{t}+\int_{t}^{T}Z_{s}dW_{s},\;\;\;0\leq t\leq T.$

Replacing $M_{T}$ and $M_{t}$, by their defining formulas and subtracting
$\int_{0}^{t}f\left(u,A_{u}\right)du+\int_{0}^{t}g\left(u\right)dB_{u}$ from
both sides of the equality yields that

$\displaystyle
Y_{t}=X_{T}+\int_{t}^{T}f\left(u,A_{u}\right)du+\int_{t}^{T}g\left(u\right)dB_{u}-\int^{T}_{t}Z_{u}dW_{u},$

where

$\displaystyle
Y_{t}=\mathbb{E}\left\\{X_{T}+\int_{t}^{T}f\left(u,A_{u}\right)du+\int_{t}^{T}g\left(u\right)dB_{u}|\mathcal{G}_{t}\right\\}.$
(2.6)

It remains to show that $\\{Y_{t}\\}$ and $\\{Z_{t}\\}$ are
$\mathcal{F}_{t}$-measurable. For $Y_{t}$, this is obvious since for each $t$,

$\displaystyle
Y_{t}=\mathbb{E}\left\\{\Theta|\mathcal{F}_{t}\vee\mathcal{F}^{B}_{t}\right\\}.$

where $\Theta$ is $\mathcal{F}^{W}_{T}\vee\mathcal{F}^{B}_{t,T}$-measurable.
Hence $\mathcal{F}^{B}_{t}$ is independent of
$\mathcal{F}_{t}\vee\sigma(\Theta)$, and

$\displaystyle Y_{t}=\mathbb{E}\left\\{\Theta|\mathcal{F}_{t}\right\\}.$

Now

$\displaystyle\int^{T}_{t}Z_{u}dW_{u}=X_{T}+\int_{t}^{T}f\left(u,A_{u}\right)du+\int_{t}^{T}g\left(u\right)dB_{u}-Y_{t},$

and the right side is
$\mathcal{F}^{W}_{T}\vee\mathcal{F}^{B}_{t,T}$-measurable. Hence, from the Itô
martingale representation theorem $\\{Z_{s},t<s<T\\}$ is
$\mathcal{F}^{W}_{s}\vee\mathcal{F}^{B}_{t,T}$-adapted. Consequently, $Z_{s}$
is $\mathcal{F}^{W}_{s}\vee\mathcal{F}^{B}_{t,T}$-measurable, for any $t<s$ so
it is $\mathcal{F}^{W}_{s}\vee\mathcal{F}^{B}_{s,T}$ measurable. Therefore,
the equality (2) becomes

$\displaystyle
Y_{t}=\mathbb{E}\left\\{X_{T}+\int_{t}^{T}f\left(u,A_{u}\right)du+\int_{t}^{T}g\left(u\right)dB_{u}|\mathcal{F}_{t}\right\\},$

which shows the desired result. ∎

##  3 Main results

The main objective of this section is to prove the existence and uniqueness
result to the new type of reflected BDSDEs. As mentioned in [25], we use the
well-known contraction mapping theorem, to provide the existence and
uniqueness of the solution. Next, like as in [25], we derive the comparison
theorem and a stability result of such equations.

###  3.1 Existence and uniqueness

Let us make the following extra assumptions on the boundary process $X$ and
the coefficients $f$ and $g$.

(A3) There exists a constant $\Gamma>0$, such that

  1. (i)

for any $\mu\in\mathcal{M}_{0,T}$, it holds that

$\displaystyle\mbox{ess}\sup_{\overset{\tau>\mu}{\tau\in\mathcal{M}_{0,T}}}\left\\{\left|\frac{\mathbb{E}\left\\{X_{\tau}-X_{\mu}|\mathcal{F}_{\mu}\right\\}}{\mathbb{E}\left\\{\tau-\mu|\mathcal{F}_{\mu}\right\\}}\right|+\left|\frac{\left[\left(\mathbb{E}\int_{\mu}^{\tau}|g(u,0)|^{2}du|\mathcal{F}_{\mu}\right)\right]^{1/2}}{\mathbb{E}\left\\{\tau-\mu|\mathcal{F}_{\mu}\right\\}}\right|\right\\}\leq\Gamma,\;\;a.s.;$

  2. (ii)

$|f(t,0,0)|\leq\Gamma,\;\;\;t\in[0,T]$.

Let us consider the following mapping $\Phi$ on
$\mathcal{S}^{\infty}([0,T],\mathbb{R})$: for a given process
$y\in\mathcal{S}^{2}([0,T],\mathbb{R})$, we define
$\Phi(y)_{t}=Y_{t},\,t\in[0,T]$, where $(Y,Z,A)$ is the unique solution of the
variant Skorohod problem:

$\displaystyle
Y_{t}=\xi+\int_{t}^{T}f\left(u,y_{u},A_{u}\right)du+\int_{t}^{T}g\left(u,y_{u}\right)dB_{u}-\int^{T}_{t}Z_{u}dW_{u},\,\,t\in[0,T],$
(3.1) $\displaystyle\mathbb{E}\int^{T}_{0}|Y_{t}-X_{t}|dA_{t}=0.$

It follows from Theorem 2.1 and Theorem 2.2 that the reflecting process $A$ is
exactly determined by $y$ in this sense: $A_{t}=\sup_{0\leq s\leq t^{+}}L_{s}$
and $L$ satisfies the stochastic representation:

$\displaystyle X_{t}=\xi+\int_{t}^{T}f\left(u,y_{u},\sup_{t\leq v\leq
u}L_{v}\right)du+\int_{t}^{T}g\left(u,y_{u}\right)dB_{u}-\int_{t}^{T}\overline{Z}dW_{u},\;\;t\in[0,T].$

Our goal is to prove that the mapping $\Phi$ is a contraction from
$\mathcal{S}^{2}([0,T],\mathbb{R})$ to itself. However, it should be noted
that the contraction can only show the existence and uniqueness of $Y$; the
uniqueness of $A$ must be established separately.

We now derive some priori estimates that will be useful in the sequel. To
begin with, let us consider the stochastic representation

$\displaystyle X_{t}=\xi+\int_{t}^{T}f\left(u,0,\sup_{t\leq v\leq
u}L^{0}_{v}\right)du+\int_{t}^{T}g\left(u,0\right)dB_{u}-\int^{T}_{t}Z^{0}_{u}dW_{u}.$

Let us denote $A^{0}_{t}=\sup_{0\leq s\leq t^{+}}L_{s}^{0}$. Then we have the
following result.

######  Lemma 3.1.

Assume $({\bf A1}),\,({\bf A2})$ and $({\bf A3})$ hold. Then, it holds that

$\displaystyle\|A^{0}\|_{\infty}\leq\frac{3\sqrt{3}\Gamma}{k},$ (3.2)

where $k$ and $\Gamma$ are the constants appearing in the previous
assumptions.

###### Proof.

For fixed $s\in[0,T]$ and any stopping times $\tau>s$, let us denote by
$l^{0}_{s,\tau}$ the $\mathcal{F}_{s}$-measurable random variable such that

$\displaystyle
X_{s}-X_{\tau}=\int_{s}^{\tau}f\left(u,0,l^{0}_{s,\tau}\right)du+\int_{s}^{\tau}g\left(u,0\right)dB_{u}-\int^{\tau}_{s}Z^{0}_{u}dW_{u}.$

Then, it follows from Theorem 2.1 that
$L^{0}_{s}=\mbox{ess}\inf_{\tau>s}l_{s,\tau}^{0}$ and $A_{t}^{0}=\sup_{0\leq
s\leq t^{+}}L^{0}_{s}$. On the other hand, we have

$\mathbb{E}(X_{s}-X_{\tau}|\mathcal{F}_{s})-\mathbb{E}\left(\int_{s}^{\tau}f\left(u,0,0\right)du|\mathcal{F}_{s}\right)-\mathbb{E}\left(\int_{s}^{\tau}g\left(u,0\right)dB_{u}|\mathcal{F}_{s}\right)$
$\displaystyle=\mathbb{E}\left(\int_{s}^{\tau}[f\left(u,0,l^{0}_{s,\tau}\right)-f\left(u,0,0\right)]du|\mathcal{F}_{s}\right).$
(3.3)

On the set $\\{\omega,\,l_{s,\tau}^{0}(\omega)<0\\}$, since $f(t,0,\cdot)$ is
decreasing and $l^{0}_{s,\tau}$ is $\mathcal{F}_{s}$-measurable, we have

$\displaystyle\mathbb{E}\left(\int_{s}^{\tau}[f\left(u,0,l^{0}_{s,\tau}\right)-f\left(u,0,0\right)]du|\mathcal{F}_{s}\right)$
$\displaystyle\geq$
$\displaystyle\mathbb{E}\left(\int_{s}^{\tau}k|l^{0}_{s,\tau}|du|\mathcal{F}_{s}\right)$
$\displaystyle\geq$ $\displaystyle
k|l^{0}_{s,\tau}|\mathbb{E}\left(\tau-s|\mathcal{F}_{s}\right).$

According to (3.3), we get

$\displaystyle\mathbb{E}(X_{s}-X_{\tau}|\mathcal{F}_{s})-\mathbb{E}\left(\int_{s}^{\tau}f\left(u,0,0\right)du|\mathcal{F}_{s}\right)-\mathbb{E}\left(\int_{s}^{\tau}g\left(u,0\right)dB_{u}|\mathcal{F}_{s}\right)\geq
k|l^{0}_{s,\tau}|\mathbb{E}\left(\tau-s|\mathcal{F}_{s}\right).$

In other words, on $\\{l_{s,\tau}^{0}<0\\}$, we have

$\displaystyle|l^{0}_{s,\tau}|$ $\displaystyle\leq$
$\displaystyle\frac{1}{k}\left\\{\frac{\mathbb{E}(X_{s}-X_{\tau}|\mathcal{F}_{s})}{\mathbb{E}\left(\tau-s|\mathcal{F}_{s}\right)}-\frac{\mathbb{E}\left(\int_{s}^{\tau}f\left(u,0,0\right)du|\mathcal{F}_{s}\right)}{\mathbb{E}\left(\tau-s|\mathcal{F}_{s}\right)}-\frac{\mathbb{E}\left(\int_{s}^{\tau}g\left(u,0\right)dB_{u}|\mathcal{F}_{s}\right)}{\mathbb{E}\left(\tau-s|\mathcal{F}_{s}\right)}\right\\}.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
(3.4)

We can show similarly that on the set $\\{l_{s,\tau}^{0}>0\\}$ the following
relation holds

$\displaystyle l^{0}_{s,\tau}$ $\displaystyle\leq$
$\displaystyle\frac{1}{k}\left\\{-\frac{\mathbb{E}(X_{s}-X_{\tau}|\mathcal{F}_{s})}{\mathbb{E}\left(\tau-s|\mathcal{F}_{s}\right)}+\frac{\mathbb{E}\left(\int_{s}^{\tau}f\left(u,0,0\right)du|\mathcal{F}_{s}\right)}{\mathbb{E}\left(\tau-s|\mathcal{F}_{s}\right)}+\frac{\mathbb{E}\left(\int_{s}^{\tau}g\left(u,0\right)dB_{u}|\mathcal{F}_{s}\right)}{\mathbb{E}\left(\tau-s|\mathcal{F}_{s}\right)}\right\\}.\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
(3.5)

Putting (3.4) and (3.5) together, we have

$\displaystyle|l^{0}_{s,\tau}|^{2}$ $\displaystyle\leq$
$\displaystyle\frac{3}{k^{2}}\Bigg{\\{}\left|\frac{\mathbb{E}(X_{s}-X_{\tau}|\mathcal{F}_{s})}{\mathbb{E}\left(\tau-s|\mathcal{F}_{s}\right)}\right|^{2}+\left|\frac{\mathbb{E}\left(\int_{s}^{\tau}|f\left(u,0,0\right)|du|\mathcal{F}_{s}\right)}{\mathbb{E}\left(\tau-s|\mathcal{F}_{s}\right)}\right|^{2}$
(3.6)
$\displaystyle+\frac{\mathbb{E}\left(\left|\int_{s}^{\tau}g\left(u,0\right)dB_{u}\right|^{2}|\mathcal{F}_{s}\right)}{\left|\mathbb{E}\left(\tau-s|\mathcal{F}_{s}\right)\right|^{2}}\Bigg{\\}}.\;\;\;\;\;\;\;\;\;\;\;\;\;\;$

Using conditional expectation version of isometry property, we get

$\displaystyle\mathbb{E}\left(\left|\int_{s}^{\tau}g\left(u,0\right)dB_{u}\right|^{2}|\mathcal{F}_{s}\right)$
$\displaystyle=$
$\displaystyle\mathbb{E}\left(\int_{s}^{\tau}|g\left(u,0\right)|^{2}du|\mathcal{F}_{s}\right)$

which together with (3.6) leads to

$\displaystyle|l^{0}_{s,\tau}|$ $\displaystyle\leq$
$\displaystyle\frac{\sqrt{3}}{k}\Bigg{\\{}\left|\frac{\mathbb{E}(X_{s}-X_{\tau}|\mathcal{F}_{s})}{\mathbb{E}\left(\tau-s|\mathcal{F}_{s}\right)}\right|+\frac{\mathbb{E}\left(\int_{s}^{\tau}|f\left(u,0,0\right)|du|\mathcal{F}_{s}\right)}{\mathbb{E}\left(\tau-s|\mathcal{F}_{s}\right)}$
(3.7)
$\displaystyle+\frac{\left[\mathbb{E}\left(\int_{s}^{\tau}|g\left(u,0\right)|^{2}du|\mathcal{F}_{s}\right)\right]^{1/2}}{\mathbb{E}\left(\tau-s|\mathcal{F}_{s}\right)}\Bigg{\\}}.\;\;\;\;\;\;\;\;\;\;\;\;\;\;$

Since

$\displaystyle|A^{0}_{t}|=\left|\sup_{0\leq s\leq
t^{+}}L^{0}_{s}\right|\leq\sup_{0\leq s\leq t^{+}}|L^{0}_{s}|=\sup_{0\leq
s\leq t^{+}}\left\\{\mbox{ess}\inf_{\tau>s}|l_{s,\tau}|\right\\},$

we derive from (3.6) and $({\bf A3})$ that

$\displaystyle|A^{0}_{t}|\leq\sup_{0\leq s\leq
t^{+}}\left\\{\mbox{ess}\inf_{\tau>s}|l_{s,\tau}|\right\\}\leq\frac{3\,\sqrt{3}\,\Gamma}{k}$

and ends the proof. ∎

######  Lemma 3.2.

Assume $({\bf A1}),\,({\bf A2})$ and $({\bf A3})$ hold. Then, for any
$t\in[0,T]$, it holds almost surely that

$\displaystyle|A_{t}-A^{\prime}_{t}|\leq\frac{\sqrt{2}\,L}{k}(1+\sqrt{T})\|y-y^{\prime}\|_{\infty}.$

###### Proof.

Again, we fix $s\in[0,T]$ and let $\tau\in\mathcal{M}(0,T)$ be such that
$\tau>s$ a.s. Let us consider, according to Theorem 2.1,
$l_{s,\tau},\,l^{\prime}_{s,\tau}$ two $\mathcal{F}_{s}$-measurable random
variables such that

$\displaystyle\mathbb{E}(X_{s}-X_{\tau}|\mathcal{F}_{s})=\mathbb{E}\left\\{\int_{s}^{\tau}f\left(u,y_{u},l_{s,\tau}\right)du+\int_{s}^{\tau}g\left(u,y_{u}\right)dB_{u}|\mathcal{F}_{s}\right\\}$

$\displaystyle=\mathbb{E}\left\\{\int_{s}^{\tau}f\left(u,y^{\prime}_{u},l^{\prime}_{s,\tau}\right)du+\int_{s}^{\tau}g\left(u,y^{\prime}_{u}\right)dB_{u}|\mathcal{F}_{s}\right\\}.$
(3.8)

Let us denote
$D^{\tau}_{s}=\left\\{\omega/l_{s,\tau}(\omega)>l^{\prime}_{s,\tau}(\omega)\right\\}$,
thus $D^{\tau}_{s}\in\mathcal{F}_{s}$, for any stopping times $\tau>s$. Since
${\bf 1}_{D^{\tau}_{s}}$ is $\mathcal{F}_{s}$-measurable, it follows from
(3.8) that
$\displaystyle\left[\mathbb{E}\left(\int_{s}^{\tau}|f\left(u,y_{u},l_{s,\tau}\right)-f\left(u,y_{u},l^{\prime}_{s,\tau}\right)|{\bf
1}_{D^{\tau}_{s}}du|\mathcal{F}_{s}\right)\right]^{2}$

$\displaystyle=\left[\mathbb{E}\left(\int_{s}^{\tau}[f\left(u,y^{\prime}_{u},l^{\prime}_{s,\tau}\right)-f\left(u,y_{u},l^{\prime}_{s,\tau}\right)]{\bf
1}_{D^{\tau}_{s}}du+\int_{s}^{\tau}(g\left(u,y^{\prime}_{u}\right)-g\left(u,y_{u}\right)){\bf
1}_{D^{\tau}_{s}}dB_{u}|\mathcal{F}_{s}\right)\right]^{2}.$ (3.9)

By assumption $({\bf A2})$-$(iv)$, we have

$\displaystyle\left[\mathbb{E}\left(\int_{s}^{\tau}|f\left(u,y_{u},l_{s,\tau}\right)-f\left(u,y_{u},l^{\prime}_{s,\tau}\right)|{\bf
1}_{D^{\tau}_{s}}du|\mathcal{F}_{s}\right)\right]^{2}\geq
k^{2}|l_{s,\tau}-l^{\prime}_{s,\tau}|^{2}[\mathbb{E}\left\\{\tau-s|\mathcal{F}_{s}\right\\}{\bf
1}_{D^{\tau}_{s}}]^{2}.$ (3.10)

Next, assumption (A2)-(iii) together with conditional expectation version of
isometry property lead to
$\displaystyle\left[\mathbb{E}\left(\int_{s}^{\tau}[f\left(u,y^{\prime}_{u},l^{\prime}_{s,\tau}\right)-f\left(u,y_{u},l^{\prime}_{s,\tau}\right)]{\bf
1}_{D^{\tau}_{s}}du+\int_{s}^{\tau}(g\left(u,y^{\prime}_{u}\right)-g\left(u,y_{u}\right)){\bf
1}_{D^{\tau}_{s}}dB_{u}|\mathcal{F}_{s}\right)\right]^{2}$

$\displaystyle\leq$ $\displaystyle
2\left[\mathbb{E}\left(\int_{s}^{\tau}|f\left(u,y_{u},l^{\prime}_{s,\tau}\right)-f\left(u,y^{\prime}_{u},l^{\prime}_{s,\tau}\right)|{\bf
1}_{D^{\tau}_{s}}du\right)\right]^{2}$ (3.11)
$\displaystyle+2\left[\mathbb{E}\int_{s}^{\tau}|g\left(u,y^{\prime}_{u}\right)-g\left(u,y_{u}\right)|^{2}{\bf
1}_{D^{\tau}_{s}}du|\mathcal{F}_{s}\right]$ $\displaystyle\leq$ $\displaystyle
2L^{2}\|y-y^{\prime}\|^{2}_{\infty}[\mathbb{E}\left\\{\tau-s|\mathcal{F}_{s}\right\\}{\bf
1}_{D^{\tau}_{s}}]^{2}+2L^{2}\|y-y^{\prime}\|^{2}_{\infty}\mathbb{E}\left(\tau-s|\mathcal{F}_{s}\right){\bf
1}_{D^{\tau}_{s}}.$

Combining (3.10) and (3.11) with (3.9), we obtain

$\displaystyle
k|l_{s,\tau}-l^{\prime}_{s,\tau}|\mathbb{E}\left\\{\tau-s|\mathcal{F}_{s}\right\\}\leq\sqrt{2}L\|y-y^{\prime}\|_{\infty}\mathbb{E}\left\\{\tau-s|\mathcal{F}_{s}\right\\}+\sqrt{2}L\|y-y^{\prime}\|_{\infty}[\mathbb{E}\left(\tau-s|\mathcal{F}_{s}\right)]^{1/2},$

on $D^{\tau}_{s}$. Thus,

$\displaystyle|l_{s,\tau}-l^{\prime}_{s,\tau}|\leq\frac{\sqrt{2}\,L}{k}(1+[\mathbb{E}\left\\{(\tau-s)|\mathcal{F}_{s}\right\\}]^{-1/2})\|y-y^{\prime}\|_{\infty}$

on $D^{\tau}_{s}$, since $\tau>s$. Similarly, we can show that the inequality
holds on the complement of $D^{\tau}_{s}$ as well. Therefore, we have

$\displaystyle|l_{s,\tau}-l^{\prime}_{s,\tau}|\leq\frac{\sqrt{2}\,L}{k}(1+[\mathbb{E}\left\\{\tau-s|\mathcal{F}_{s}\right\\}]^{-1/2})\|y-y^{\prime}\|_{\infty}.$
(3.12)

Next, since
$L_{s}=\mbox{ess}\inf_{\tau>s}l_{s,\tau},\;L^{\prime}_{s}=\mbox{ess}\inf_{\tau>s}l^{\prime}_{s,\tau},\;A_{t}=\sup_{0\leq
s\leq t}L_{s}$ and $A^{\prime}_{t}=\sup_{0\leq s\leq t}L^{\prime}_{s}$, we
conclude from (3.12) that

$\displaystyle|A_{t}-A^{\prime}_{t}|=\left|\sup_{0\leq s\leq
t}L_{s}-\sup_{0\leq s\leq t}L^{\prime}_{s}\right|$ $\displaystyle\leq$
$\displaystyle\sup_{0\leq s\leq
t}\left|\mbox{ess}\inf_{\tau>s}l_{s,\tau}-\mbox{ess}\inf_{\tau>s}l^{\prime}_{s,\tau}\right|$
$\displaystyle\leq$ $\displaystyle\sup_{0\leq s\leq
t}\mbox{ess}\sup_{\tau>s}|l_{s,\tau}-l^{\prime}_{s,\tau}|$ $\displaystyle\leq$
$\displaystyle\sup_{0\leq s\leq
t}\mbox{ess}\sup_{\tau>s}\frac{\sqrt{2}L}{k}\left[1+\left(\mathbb{E}\left\\{(\tau-s)|\mathcal{F}_{s}\right\\}\right)^{-1/2}\right]\|y-y^{\prime}\|_{\infty}$
$\displaystyle\leq$
$\displaystyle\frac{\sqrt{2}L}{k}(1+\sqrt{T})\|y-y^{\prime}\|_{\infty}.$

∎

We are now ready to prove the main result of this section, the existence and
uniqueness of the solution to the VRBDSDE.

######  Theorem 3.3.

Assume $({\bf A1}),\,({\bf A2})$ and $({\bf A3})$ hold. Assume further that

${2TL\left(1+\sqrt{2}\frac{K}{k}\left(1+\sqrt{T}\right)\right)+L\sqrt{2T}<1},$

then the VRBDSDE (2.1) admits a unique solution $(Y,A)$.

###### Proof.

First, let us show that the mapping $\Phi$ defined by (3.4) is from
$\mathcal{S}^{\infty}$ to itself. To do this, we note that by using assumption
$({\bf A1})$ and Lemmas 3.1 and 3.2, we derive

$\displaystyle|Y_{t}|^{2}=|\Phi(y)_{t}|^{2}\leq
3\mathbb{E}\left\\{|\xi|+\left(\int^{T}_{t}|f(s,y_{s},A_{s})|ds\right)^{2}+\left(\int_{t}^{T}g(s,y_{s})dB_{s}\right)^{2}|\mathcal{F}_{t}\right\\}.$
(3.13)

We have

$\displaystyle\mathbb{E}\left\\{\left(\int^{T}_{t}f(s,y_{s},A_{s})ds\right)^{2}|\mathcal{F}_{t}\right\\}$

$\displaystyle\leq$ $\displaystyle
4T^{2}\left(K^{2}\|A-A^{0}\|^{2}_{\infty}+L^{2}\|y\|^{2}_{\infty}+K^{2}\|A^{0}\|^{2}_{\infty}+\Gamma^{2}\right)$
(3.14) $\displaystyle\leq$ $\displaystyle
4T^{2}L^{2}\left(1+2\frac{K^{2}}{k^{2}}\left(1+\sqrt{T}\right)\|y\|^{2}_{\infty}\right)+4T^{2}\left(1+27\frac{K^{2}}{k^{2}}\right)\Gamma^{2}$

and

$\displaystyle\mathbb{E}\left\\{\left(\int^{T}_{t}g(s,y_{s})|ds\right)^{2}|\mathcal{F}_{t}\right\\}$
$\displaystyle\leq$ $\displaystyle
2\mathbb{E}\left\\{\left(\int_{0}^{T}|g(s,0)|^{2}ds\right)|\mathcal{F}_{t}\right\\}+2L^{2}T\|y\|_{\infty}^{2}.$
(3.15)

It follows from (3.13), (3.14) and (3.15) that

$\displaystyle|Y_{t}|$ $\displaystyle\leq$
$\displaystyle\|\xi\|+\sqrt{2}\left[\mathbb{E}\left(\int_{0}^{T}|g(s,0)|^{2}ds|\mathcal{F}_{t}\right)\right]^{1/2}+L\left[2T\left(1+\sqrt{2}\frac{K}{k}\left(1+\sqrt{T}\right)\right)+\sqrt{2T}\right]\|y\|_{\infty}$
$\displaystyle+2T\left(1+3\sqrt{3}\frac{K}{k}\right)\Gamma.$

As it is known by assumption that $\xi$ belongs to $\mathbb{L}^{\infty}$, we
deduce from $({\bf A3})$-$(i)$ that $Y=\Phi(y)$ belongs to $S^{\infty}$. Now,
let us prove that $\Phi$ is a contraction. For $y,\,y^{\prime}\in S^{\infty}$,
we denote $Y=\Phi(y)$ and $Y^{\prime}=\Phi(y^{\prime})$. Then, for
$t\in[0,T]$, we have

$\displaystyle|\Phi(y)-\Phi(y^{\prime})|^{2}$ $\displaystyle=$
$\displaystyle\left|\mathbb{E}\left\\{\int_{t}^{T}[f(s,y_{s},A_{s})-f(s,y^{\prime}_{s},A^{\prime}_{s})]ds+\int_{t}^{T}[g(s,y_{s})-g(s,y^{\prime}_{s})]dB_{s}|\mathcal{F}_{t}\right\\}\right|^{2}$
(3.16) $\displaystyle\leq$ $\displaystyle
2\left|\mathbb{E}\left\\{\int_{t}^{T}|f(s,y_{s},A_{s})-f(s,y^{\prime}_{s},A^{\prime}_{s})|ds|\mathcal{F}_{t}\right\\}\right|^{2}$
$\displaystyle+2\mathbb{E}\left\\{\left|\int_{t}^{T}[g(s,y_{s})-g(s,y^{\prime}_{s})]dB_{s}\right|^{2}|\mathcal{F}_{t}\right\\}.$

Applying assumption on $f$ and Lemma 3.2, we derive that

$\displaystyle\left|\mathbb{E}\left(\int_{t}^{T}|f(s,y_{s},A_{s})-f(s,y^{\prime}_{s},A^{\prime}_{s})|ds|\mathcal{F}_{t}\right)\right|^{2}$
$\displaystyle\leq$ $\displaystyle
2T^{2}\left(L^{2}\|y^{\prime}-y\|^{2}_{\infty}+K^{2}\|A-A^{\prime}\|_{\infty}^{2}\right)$
$\displaystyle\leq$ $\displaystyle
2T^{2}\left[L^{2}+K^{2}\frac{2L^{2}}{k^{2}}\left(1+\sqrt{T}\right)^{2}\right]\|y^{\prime}-y\|_{\infty}.$

Moreover, it follows from conditional expectation version of isometry property
that

$\displaystyle\mathbb{E}\left\\{\left|\int_{t}^{T}[g(s,y_{s})-g(s,y^{\prime}_{s})]dB_{s}\right|^{2}|\mathcal{F}_{t}\right\\}$
$\displaystyle=$
$\displaystyle\mathbb{E}\left\\{\left(\int_{t}^{T}|g(s,y_{s})-g(s,y^{\prime}_{s})|^{2}ds\right)|\mathcal{F}_{t}\right\\}$
(3.18) $\displaystyle\leq$ $\displaystyle
L^{2}T\|y-y^{\prime}\|^{2}_{\infty}.$

Finally, putting (LABEL:eq9) and (3.18) into (3.16), we obtain

$\displaystyle|\Phi(y)-\Phi(y^{\prime})|\leq\left[2TL\left(1+\sqrt{2}\frac{K}{k}\left(1+\sqrt{T}\right)\right)+L\sqrt{2T}\right]\|y-y^{\prime}\|_{\infty}.$

Since we assume that
$2TL\left(1+\sqrt{2}\frac{K}{k}\left(1+\sqrt{T}\right)\right)+L\sqrt{2T}<1$,
it is not difficult to see that $\Phi$ is a contraction.

Let us denote by $Y\in\mathcal{S}^{\infty}$ the unique fixed point and by $A$
the associating reflecting process defined by $A_{t}=\sup_{0\leq v\leq
t^{+}}L_{v}$, where $L$ satisfies the representation

$\displaystyle
X_{t}=\mathbb{E}\left\\{\xi+\int_{t}^{T}f\left(s,Y_{s},\sup_{t\leq v\leq
s}L_{v}\right)ds+\int_{t}^{T}g\left(s,Y_{s}\right)dB_{u}|\mathcal{F}_{t}\right\\}.$
(3.19)

Let us now prove that $(Y,A)$ is the solution to the VRBDSDE (2.1). For this
instance, it follows from (3.19), the definition of $A$, and the monotonicity
of $f$ on the third variable that for all $t\in[0,T]$,

$\displaystyle Y_{t}$ $\displaystyle=$
$\displaystyle\mathbb{E}\left\\{\xi+\int_{t}^{T}f\left(s,Y_{s},A_{s}\right)ds+\int_{t}^{T}g\left(s,Y_{s}\right)dB_{s}|\mathcal{F}_{t}\right\\}$
$\displaystyle\leq$
$\displaystyle\mathbb{E}\left\\{\xi+\int_{t}^{T}f\left(s,Y_{s},\sup_{t\leq
v\leq
s}L_{v}\right)ds+\int_{t}^{T}g\left(s,Y_{s}\right)dB_{s}|\mathcal{F}_{s}\right\\}=X_{t}.$

To end the proof of existence, it remains to show that the flat-off conditions
holds. The properties of optional projection and definition of $A$ and $L$
lead to

$\displaystyle\mathbb{E}\int_{0}^{T}(Y_{t}-X_{t})dA_{t}$ $\displaystyle=$
$\displaystyle\mathbb{E}\int_{0}^{T}\left\\{\int_{t}^{T}\left[f\left(s,Y_{s},\sup_{0\leq
v\leq s^{+}}L_{v}\right)-f\left(s,Y_{s},\sup_{t\leq v\leq
s}L_{v}\right)\right]ds\right\\}dA_{t}.$

Next, using the Fubini theorem and the fact that Lebesgue measure does not
charge the discontinuities of the path $u\mapsto\sup_{t\leq v\leq u}L_{v}$,
which are only countably many, we have

$\displaystyle\mathbb{E}\int_{0}^{T}(Y_{t}-X_{t})dA_{t}$ $\displaystyle=$
$\displaystyle\mathbb{E}\int_{0}^{T}\left\\{\int_{0}^{s}\left[f\left(s,Y_{s},\sup_{0\leq
v\leq s^{+}}L_{v}\right)-f\left(s,Y_{s},\sup_{t\leq v\leq
s^{+}}L_{v}\right)\right]dA_{t}\right\\}ds,$

which provide by the same argument used in [25] that

$\displaystyle\mathbb{E}\int_{0}^{T}|Y_{t}-X_{t}|dA_{t}=0.$

For the uniqueness, let us suppose that there is another solution
$(Y^{\prime},A^{\prime})$ to the VRBDSDE such that $Y^{\prime}_{t}\leq
X_{t},\;t\in[0,T]$, and

$\displaystyle
Y^{\prime}_{t}=\mathbb{E}\left\\{\xi+\int_{t}^{T}f\left(s,Y^{\prime}_{s},A^{\prime}_{s}\right)ds+\int_{t}^{T}g\left(s,Y^{\prime}_{s}\right)dB_{s}|\mathcal{F}_{t}\right\\},\;\;\;\mathbb{E}\int_{0}^{T}|Y^{\prime}_{t}-X_{t}|dA_{t}=0.$

Since both $Y$ and $Y^{\prime}$ are the unique fixed points of the mapping
$\Phi$, it follows that $Y=Y^{\prime}$. Let us consider the stochastic variant
Skorohod problem

$\displaystyle\widetilde{Y}_{t}=\mathbb{E}\left\\{\xi+\int_{t}^{T}f^{Y}\left(s,\widetilde{A}_{s}\right)ds+\int_{t}^{T}g^{Y}\left(s\right)dB_{s}|\mathcal{F}_{t}\right\\},$
$\displaystyle\widetilde{Y}_{t}\leq X_{t},\;\;\;\;\;Y_{T}=X_{T}=\xi,$ (3.20)
$\displaystyle\mathbb{E}\int_{0}^{T}|\widetilde{Y}_{t}-X_{t}|d\widetilde{A}_{t}=0,$

where $f^{Y}\left(s,l\right)=f\left(s,Y_{s},l\right)$ and
$g^{Y}\left(s\right)=g\left(s,Y_{s}\right)$. Thanks to Theorem 2.2, there
exists a unique pair of process $(\widetilde{Y},\widetilde{A})$ that solves
the stochastic variant Skorohod problem. Moreover, since $(Y,A)$ and
$(Y^{\prime},A^{\prime})$ are the solutions to the variant BDSDE (3.20), it
follows that $Y_{t}=\widetilde{Y}_{t}$ and
$A_{t}=\widetilde{A}_{t}=A^{\prime}_{t},\;t\in[0,T]$, a.s., which proves the
uniqueness, whence the theorem. ∎

######  Corollary 3.4.

Suppose that $(Y,A)$ is a solution to VRBDSDE with generator $f$ and $g$ and
upper boundary $X$. Then $A_{0^{-}}=-\infty$ and $Y_{0}=X_{0}$.

###### Proof.

Since the existence and uniqueness proof depends heavily on the well-posedness
result of the extended stochastic representation theorem, we must require that
$A_{0^{-}}=-\infty$. On the other hand, since $Y$ is a fixed point of the
mapping $\Phi$ defined by (3.4), it not difficult to see that $Y_{0}$ and
$X_{0}$ satisfy the following equalities:

$\displaystyle X_{0}$ $\displaystyle=$
$\displaystyle\mathbb{E}\left\\{\xi+\int_{0}^{T}f\left(s,Y_{s},\sup_{0\leq
v\leq s}L_{v}\right)ds+\int_{0}^{T}g\left(s,Y_{s}\right)dB_{s}\right\\},$
$\displaystyle Y_{0}$ $\displaystyle=$
$\displaystyle\mathbb{E}\left\\{\xi+\int_{0}^{T}f\left(s,Y_{s},A_{s}\right)ds+\int_{0}^{T}g\left(s,Y_{s}\right)dB_{s}\right\\}$
$\displaystyle=$
$\displaystyle\mathbb{E}\left\\{\xi+\int_{0}^{T}f\left(s,Y_{s},\sup_{0\leq
v\leq s^{+}}L_{v}\right)ds+\int_{0}^{T}g\left(s,Y_{s}\right)dB_{s}\right\\}.$

Hence, by the same argument that the paths of the increasing process
$u\mapsto\sup_{t\leq v\leq u}L_{v}$ has only countably many discontinuities,
which are negligible under the Lebesgue measure, we prove that $Y_{0}=X_{0}$.
∎

###  3.2 Comparison theorems

This section is devoted to study the comparison theorem of the VRBDSDE, one of
the very important tools in the theory of BSDEs. Let us remark that our method
follows closely to one appeared in [25], which is quite different from all the
existing arguments in the BSDE literature.

To state, let us consider the following two VRBDSDEs for $i=1,2$,

$\displaystyle
Y^{i}_{t}=\mathbb{E}\left\\{\xi^{i}+\int_{t}^{T}f^{i}\left(s,Y^{i}_{s},A^{i}_{s}\right)ds+\int_{t}^{T}g\left(s,Y^{i}_{s}\right)dB_{s}|\mathcal{F}_{t}\right\\},$
$\displaystyle Y^{i}_{t}\leq X^{i}_{t},\;\;\;\;\;Y^{i}_{T}=X^{i}_{T}=\xi^{i},$
(3.21) $\displaystyle\mathbb{E}\int_{0}^{T}|Y^{i}_{t}-X^{i}_{t}|dA^{i}_{t}=0.$

In the sequel, we call $(f^{i},g,X^{i}),\;i=1,2$ as the "parameters" of the
VRBDSDE (3.21), $i=1,2$, respectively. We also define the two following
stopping times

$\displaystyle\mu$ $\displaystyle=$
$\displaystyle\inf\left\\{t\in[0,T),\;A_{t}^{2}>A^{1}_{t}+\varepsilon\right\\}\wedge
T;$ $\displaystyle\tau$ $\displaystyle=$
$\displaystyle\inf\left\\{t\in[\mu,T),\;A_{t}^{1}>A^{2}_{t}-\frac{\varepsilon}{2}\right\\}\wedge
T.$ (3.22)

We recall the following result appear in [25].

######  Lemma 3.5.

The stopping times $\mu$ and $\tau$ defined by (3.22) have the standing
properties:

(i)

$\;\mu$ and $\tau$ are points of increase for $A^{2}$ and $A^{1}$,
respectively. In other word, for any $\delta>0$, it holds that
$A^{2}_{\mu^{-}}<A^{2}_{\mu+\delta}$ and
$A^{1}_{\tau^{-}}<A^{1}_{\tau+\delta}$.

(ii)

$\;\mathbb{P}(\mu<\tau)=1$, and $A^{1}_{t}\leq
A^{2}_{t}-\frac{\varepsilon}{2}$, for all $t\in[\mu,\tau],\,\mathbb{P}$-a.s.,

(iiii)

it holds that $Y^{2}_{\mu}=X^{2}_{\mu}$ and
$Y^{1}_{\tau}=X^{1}_{\tau},\;\mathbb{P}$-a.s.

Before give the comparison theorem, in order to simplify the notations, let us
give the following. For $(Y^{i},A^{i}),\;i=1,2$ be the solution to two
VRBDSDEs with boundaries $X^{1}$ and $X^{2}$ respectively, we denote
$\Delta\Theta=\Theta^{1}-\Theta^{2},\;\Theta=X,Y,A$, and $\xi$. Furthermore,
recall

$\displaystyle\mathcal{G}_{t}=\mathcal{F}^{W}_{t}\vee\mathcal{F}^{B}_{T},$

we define two martingales

$\displaystyle
M^{i}_{t}=\mathbb{E}\left\\{\int_{0}^{T}f^{i}\left(s,Y^{i}_{s},A^{i}_{s}\right)ds+\int_{0}^{T}g\left(s,Y^{i}_{s}\right)dB_{s}|\mathcal{G}_{t}\right\\},\;t\in[0,T],\,i=1,2.$

######  Theorem 3.6.

Assume that the parameters of the VRBDSDEs (3.21) $(f^{i},g,X^{i}),\,i=1,2$,
satisfy $({\bf A1})$ and $({\bf A2})$. Assume further that

(i)

$\;f^{1}(t,y,l)\geq f^{2}(t,y,l),\;d\mathbb{P}\otimes dt$ a.s.,

(ii)

$\;X^{1}_{t}\leq X^{2}_{t},\;\;0\leq t\leq T$, a.s.,

(iii)

$\;\Delta X_{s}\leq\mathbb{E}\\{{\rm e}^{(L+\frac{1}{2}L^{2})(t-s)}\Delta
X_{t}|\mathcal{G}_{s}\\}$ a.s. for all $s$ and $t$ such that $s<t$.

Then, we have $A^{1}_{t}\geq A^{2}_{t},\;t\in[0,T],\ \mathbb{P}$-a.s.

######  Remark 3.7.

As it is explained in [25], the assumption $(iii)$ in above theorem
significate that the process ${\rm e}^{Ls}\Delta X_{s}$, is a submartingale
and it does not add restrictive on the regularity of the boundary processes
$X^{1}$ and $X^{2}$, which are only required to be the optional processes
satisfying $({\bf A3})$.

###### Proof of Theorem 3.6.

According to (3.21) and the previous notations, we can write, on the set
$\\{\mu<T\\}$

$\displaystyle\Delta Y_{\mu}$ $\displaystyle=$
$\displaystyle\mathbb{E}\left\\{\Delta
Y_{\tau}+\int_{\mu}^{\tau}\left[f^{1}\left(s,Y^{1}_{s},A^{1}_{s}\right)-f^{2}\left(s,Y^{2}_{s},A^{2}_{s}\right)\right]ds\right.$
(3.23)
$\displaystyle+\left.\int_{\mu}^{\tau}\left[g\left(s,Y^{1}_{s}\right)-g\left(s,Y^{2}_{s}\right)\right]dB_{s}+(\Delta
M_{\tau}-\Delta M_{\mu})|\mathcal{F}_{\mu}\right\\},$

where $\Delta M=M^{1}-M^{2}$, and

$\displaystyle\nabla_{y}f^{1}_{s}$ $\displaystyle=$
$\displaystyle\frac{f^{1}\left(s,Y^{1}_{s},A^{1}_{s}\right)-f^{1}\left(s,Y^{2}_{s},A^{1}_{s}\right)}{Y^{1}_{s}-Y^{2}_{s}}{\bf
1}_{\\{Y^{1}_{s}\neq Y^{2}_{s}\\}},$ $\displaystyle\nabla_{y}g_{s}$
$\displaystyle=$
$\displaystyle\frac{g\left(s,Y^{1}_{s}\right)-g\left(s,Y^{2}_{s}\right)}{Y^{1}_{s}-Y^{2}_{s}}{\bf
1}_{\\{Y^{1}_{s}\neq Y^{2}_{s}\\}},$ $\displaystyle\Delta_{l}f^{1}_{s}$
$\displaystyle=$ $\displaystyle
f^{1}\left(s,Y^{2}_{s},A^{1}_{s}\right)-f^{2}\left(s,Y^{2}_{s},A^{2}_{s}\right),$
$\displaystyle\Delta_{2}f_{s}$ $\displaystyle=$ $\displaystyle
f^{1}\left(s,Y^{2}_{s},A^{2}_{s}\right)-f^{2}\left(s,Y^{2}_{s},A^{2}_{s}\right).$

It is clear that $({\bf A2})$ implies that $\nabla_{y}f^{1}$ and $\nabla_{y}g$
are bounded progressively measurable processes, and by the definition of
$\mu,\,\tau$ and the monotonicity of $f$ on it variable $l$, we have
$\Delta_{l}f^{1}>0$ on the interval $[\mu,\tau]$. Hence, $\Delta Y$ is a
unique solution of the following linear BDSDE

$\displaystyle\Delta Y_{\mu}$ $\displaystyle=$
$\displaystyle\mathbb{E}\left\\{\Delta
Y_{\tau}+\int_{\mu}^{\tau}\nabla_{y}f^{1}_{s}\Delta
Y_{s}ds+\int_{\mu}^{\tau}[\Delta_{l}f^{1}_{s}+\Delta_{2}f_{s}]ds\right.$
$\displaystyle+\left.\int_{\mu}^{\tau}\nabla_{y}g_{s}\Delta
Y_{s}dB_{s}+(\Delta M_{\tau}-\Delta M_{\mu})|\mathcal{F}_{\mu}\right\\}$

Setting
$\Gamma_{t}=\exp\left(\int_{0}^{t}\nabla_{y}f^{1}_{s}ds+\int_{0}^{t}\nabla_{y}g_{s}dB_{s}-\frac{1}{2}\int_{0}^{t}|\nabla_{y}g_{s}|^{2}ds\right)$,
as it done in [3], one can derive

$\displaystyle\mathbb{E}\left\\{\Gamma_{\mu}\Delta Y_{\mu}-\Gamma_{\tau}\Delta
Y_{\tau}|\mathcal{F}_{\mu}\right\\}=\mathbb{E}\left\\{\int_{\mu}^{\tau}\Gamma_{s}[\Delta_{l}f^{1}_{s}+\Delta_{2}f_{s}]ds-\int_{\mu}^{\tau}\Gamma_{s}d(\Delta
M_{s})|\mathcal{F}_{\mu}\right\\}.$

Therefore, since $f^{1}\geq f^{2},\;\Delta_{2}f\geq 0,\,d\mathbb{P}\otimes
dt$-a.s., and consequently, since $M^{i},\;i=1,2$, is a martingale, we get

$\displaystyle\mathbb{E}\left\\{\Gamma_{\mu}\Delta Y_{\mu}-\Gamma_{\tau}\Delta
Y_{\tau}|\mathcal{F}_{\mu}\right\\}=\mathbb{E}\left\\{\int_{\mu}^{\tau}\Gamma_{s}[\Delta_{l}f^{1}_{s}+\Delta_{2}f_{s}]ds|\mathcal{F}_{\mu}\right\\}>0.$
(3.24)

On the other hand, by the flat-off condition and Lemma 3.5-$(iii)$, one can
check that $Y^{1}_{\mu}-Y^{2}_{\mu}\leq X^{1}_{\mu}-X^{2}_{\mu}$ and
$Y^{1}_{\tau}-Y^{2}_{\tau}\leq X^{1}_{\tau}-X^{2}_{\tau}$,

$\displaystyle\mathbb{E}\left\\{\Gamma_{\mu}\Delta Y_{\mu}-\Gamma_{\tau}\Delta
Y_{\tau}|\mathcal{F}_{\mu}\right\\}\leq\mathbb{E}\left\\{\Gamma_{\mu}\Delta
X_{\mu}-\Gamma_{\tau}\Delta X_{\tau}|\mathcal{F}_{\mu}\right\\}.$ (3.25)

It is now clear that if the right hand side of (3.25) is non-positive, then
(3.25) contradicts to (3.24), and therefore one must have
$\mathbb{P}(\mu<T)=0$. In other words, $A^{2}_{t}\leq A^{1}_{t}+\varepsilon$,
for all $t\in[0,T],\,\mathbb{P}$-a.s. Since $\varepsilon$ is taken arbitrary,
entails that

$\displaystyle A^{2}_{t}\leq
A^{1}_{t},\;\;\;\;t\in[0,T],\;\;\mathbb{P}\mbox{-a.s.}$

Now it remain to show that the right hand side of (3.25) is non-positive. To
do this, let us note that since by assumption $(ii)$ we have $\Delta
X_{\tau}\leq 0$, it follows from (3.25) and assmption $(iii)$ that

$\displaystyle\mathbb{E}\left\\{\Gamma_{\mu}\Delta Y_{\mu}-\Gamma_{\tau}\Delta
Y_{\tau}|\mathcal{F}_{\mu}\right\\}$ $\displaystyle\leq$
$\displaystyle\Gamma_{\mu}\mathbb{E}\left\\{\Delta X_{\mu}-{\rm
e}^{\int_{\mu}^{\tau}\nabla_{y}f^{1}_{s}ds+\int_{\mu}^{\tau}\nabla_{y}g_{s}dB_{s}-\frac{1}{2}\int_{\mu}^{\tau}|\nabla_{y}g_{s}|^{2}ds}\Delta
X_{\tau}|\mathcal{F}_{\mu}\right\\}$ $\displaystyle\leq$
$\displaystyle\Gamma_{\mu}\mathbb{E}\left\\{\Delta X_{\mu}-{\rm
e}^{(L+\frac{1}{2}L^{2})(\tau-\mu)}\Delta X_{\tau}|\mathcal{F}_{\mu}\right\\}$
$\displaystyle\leq$ $\displaystyle 0.$

∎

As it is emphasized in [25], Theorem 3.6 only gives the comparison between the
two reflecting processes $A^{1}$ and $A^{2}$. This is still one step away from
comparison between $Y^{1}$ and $Y^{2}$, which is much desirable for obvious
reason. But, the latter is not true in general, due do the "opposite"
monotonicity on $f^{i}$’s on the variable $l$. We nevertheless have the
following corollary of Theorem 3.6.

######  Corollary 3.8.

Assume all the assumptions of Theorem 3.6 hold and further $f^{1}=f^{2}$. Then
$Y^{1}_{t}\leq Y^{2}_{t}$, for all $t\in[0,T],\,\mathbb{P}$-a.s.

###### Proof.

Let us denote $f=f^{1}=f^{2}$ and define two random functions
$\widetilde{f}^{i}(t,\omega,y)=f(t,\omega,y,A^{i}_{t}(\omega))$, for
$(t,\omega,y)\in[0,T]\times\Omega\times\mathbb{R},i=1,2$. Then $Y^{1}$ and
$Y^{2}$ can be seen as the solution of BDSDEs

$\displaystyle
Y^{i}_{t}=\mathbb{E}\left\\{\xi^{i}+\int_{t}^{T}\widetilde{f}^{i}\left(s,Y^{i}_{s}\right)ds+\int_{t}^{T}g\left(s,Y^{i}_{s}\right)dB_{s}|\mathcal{F}_{t}\right\\},\;t\in[0,T],\;\;i=1,2.$

It follows from the fact $A^{1}\geq A^{2}$ that
$\widetilde{f}^{1}(t,\omega,y)=f(t,\omega,y,A^{1}_{t}(\omega))\leq
f(t,\omega,y,A^{2}_{t}(\omega))=\widetilde{f}^{2}(t,\omega,y)$. Therefore,
since $\xi^{1}=X^{1}_{T}\leq X^{2}_{T}=\xi^{2}$, and according to the
comparison theorem of BDSDEs, we have $Y^{1}_{t}\leq Y^{2}_{t}$, for all
$t\in[0,T],\ \mathbb{P}$-a.s. ∎

###  3.3 Stability results

In this section, we study another useful aspect of the well-posedness of the
VRBDSDE, which it is called the continuous dependence of the solution on the
boundary process whence the terminal process as well. For this instance, let
us introduce, for any optional process $X$ and any stopping time $\mu$ and
$\tau$ satisfy that $\mu<\tau$,

$\displaystyle
m_{\mu,\tau}(X)=\frac{\mathbb{E}\\{X_{\tau}-X_{\mu}|\mathcal{F}_{\mu}\\}}{\mathbb{E}\\{\tau-\mu|\mathcal{F}_{\mu}\\}}.$

Let us note that the random variable $m_{\mu,\tau}(X)$ measures the path
regularity of the "nonmartingale" part of the boundary process $X$. In the
sequel, we will show that this will be a major measurement for the "closeness"
of the boundary processes, as far as the continuous dependence is concerned.

Let us consider $\\{X^{n}\\}_{n=1}^{\infty}$, a sequence of optional processes
satisfying that $({\bf A3})$. We suppose that $\\{X^{n}\\}_{n=1}^{\infty}$
converges to $X^{0}$ in $\mathcal{S}^{\infty}$, and that $X^{0}$ satisfies
$({\bf A3})$ as well. Let $(Y^{n},A^{n})$ be the unique solution to the
VRBDSDE’s with parameters $(f,g,X^{n})$, for
$n=0,1,2,\cdot\cdot\cdot\cdot\cdot$. Roughly speaking, for
$n=0,1,2,\cdot\cdot\cdot\cdot$, we have

$\displaystyle
X^{n}_{t}=\mathbb{E}\left\\{\xi^{n}+\int_{t}^{T}f\left(s,Y^{n}_{s},\sup_{t\leq
v\leq
s}L^{n}_{v}\right)ds+\int_{t}^{T}g\left(s,Y^{n}_{s}\right)dB_{s}|\mathcal{F}_{t}\right\\},$
$\displaystyle A^{n}_{s}=\sup_{0\leq v\leq s^{+}}L^{n}_{v},$ $\displaystyle
Y^{n}_{t}=\mathbb{E}\left\\{\xi^{n}+\int_{t}^{T}f\left(s,Y^{n}_{s},A^{n}_{s}\right)ds+\int_{t}^{T}g\left(s,Y^{n}_{s}\right)dB_{s}|\mathcal{F}_{t}\right\\}.$

Next, let us give the following lemma that provides the control of
$|A^{n}_{t}-A_{t}^{0}|$, which is needed in the sequel.

######  Lemma 3.9.

Assume $({\bf A2})$ and $({\bf A3})$ hold. Then, for all $t\in[0,T]$, it holds
that

$\displaystyle|A^{n}_{t}-A_{t}^{0}|\leq\frac{\sqrt{3}}{k}\sup_{0\leq s\leq
t}\mbox{ess}\sup_{\tau>s}\left|m^{n}_{\mu,\tau}-m^{0}_{\mu,\tau}\right|+\frac{\sqrt{3}L}{k}(1+\sqrt{T})\|Y^{n}-Y^{0}\|_{\infty}.$

###### Proof.

The proof follows the similar step as the proof of Lemma 3.2. Let us consider,
$l^{n}_{s,\tau},\;n=0,1,\cdot\cdot\cdot$ the $\mathcal{F}_{s}$-measurable
random variable such that

$\displaystyle\mathbb{E}(X^{n}_{s}-X^{n}_{\tau}|\mathcal{F}_{s})$
$\displaystyle=$
$\displaystyle\mathbb{E}\left\\{\int_{s}^{\tau}f\left(u,Y^{n}_{u},l^{n}_{s,\tau}\right)du+\int_{s}^{\tau}g\left(u,Y^{n}_{u}\right)dB_{u}|\mathcal{F}_{s}\right\\}.$
(3.26)

Therefore, for $n=1,\cdot\cdot\cdot$, we have

$\displaystyle\mathbb{E}(X^{n}_{s}-X^{n}_{\tau}|\mathcal{F}_{s})-\mathbb{E}(X^{0}_{s}-X^{0}_{\tau}|\mathcal{F}_{s})$
$\displaystyle=$
$\displaystyle\mathbb{E}\left\\{\int_{s}^{\tau}[f\left(u,Y^{n}_{u},l^{n}_{s,\tau}\right)-f\left(u,Y^{0}_{u},l^{0}_{s,\tau}\right)]du\right.$
$\displaystyle\left.+\int_{s}^{\tau}[g\left(u,Y^{n}_{u}\right)-g\left(u,Y^{0}_{u}\right)]dB_{u}|\mathcal{F}_{s}\right\\}.$

On the set
$D^{\tau}_{s}=\left\\{\omega/l^{n}_{s,\tau}(\omega)>l^{0}_{s,\tau}(\omega)\right\\}\in\mathcal{F}_{s}$,
we get

$\displaystyle{\bf
1}_{D^{\tau}_{s}}\mathbb{E}(X^{n}_{s}-X^{n}_{\tau}|\mathcal{F}_{s})-\mathbb{E}(X^{0}_{s}-X^{0}_{\tau}|\mathcal{F}_{s})$
$\displaystyle=$ $\displaystyle{\bf
1}_{D^{\tau}_{s}}\mathbb{E}\left\\{\int_{s}^{\tau}[f\left(u,Y^{n}_{u},l^{n}_{s,\tau}\right)-f\left(u,Y^{0}_{u},l^{n}_{s,\tau}\right)+f\left(u,Y^{0}_{u},l^{n}_{s,\tau}\right)-f\left(u,Y^{0}_{u},l^{0}_{s,\tau}\right)]du\right.$
$\displaystyle\left.+\int_{s}^{\tau}[g\left(u,Y^{n}_{u}\right)-g\left(u,Y^{0}_{u}\right)]dB_{u}|\mathcal{F}_{s}\right\\}.$

From $({\bf A2})$, it clear that on
$D^{\tau}_{s},\,f\left(u,Y^{0}_{u},l^{n}_{s,\tau}\right)-f\left(u,Y^{0}_{u},l^{0}_{s,\tau}\right)\geq
k|l^{n}_{s,\tau}-l^{0}_{s,\tau}|$ and hence

$\displaystyle
k^{2}[|l^{n}_{s,\tau}-l^{0}_{s,\tau}|\mathbb{E}\left\\{\tau-s|\mathcal{F}_{s}\right\\}]^{2}{\bf
1}_{D^{\tau}_{s}}$ $\displaystyle\leq$ $\displaystyle
3|\mathbb{E}(X^{n}_{s}-X^{n}_{\tau}|\mathcal{F}_{s})-\mathbb{E}(X^{0}_{s}-X^{0}_{\tau}|\mathcal{F}_{s})|^{2}{\bf
1}_{D^{\tau}_{s}}$
$\displaystyle+3\left|\mathbb{E}\left\\{\int_{s}^{\tau}L|Y^{n}_{u}-Y^{0}_{u}|du\mathcal{F}_{s}\right\\}\right|^{2}{\bf
1}_{D^{\tau}_{s}}$
$\displaystyle\left.+3\left|\int_{s}^{\tau}[g\left(u,Y^{n}_{u}\right)-g\left(u,Y^{0}_{u}\right)]dB_{u}|\mathcal{F}_{s}\right\\}\right|^{2}{\bf
1}_{D^{\tau}_{s}}.$

Next, assumption $({\bf A2})$-$(iii)$ together with conditional expectation
version of isometry
property lead to

$\displaystyle|l_{s,\tau}-l^{\prime}_{s,\tau}|\leq\frac{\sqrt{3}}{k}\left|m^{n}_{\mu,\tau}-m^{0}_{\mu,\tau}\right|+\frac{\sqrt{3}\,L}{k}\left(1+[\mathbb{E}\left\\{(\tau-s)|\mathcal{F}_{s}\right\\}]^{-1/2}\right)\|y-y^{\prime}\|_{\infty}$

on $D^{\tau}_{s}$. Similarly, we can show that the inequality holds on the
complement of $D^{\tau}_{s}$ as well. Therefore, we have

$\displaystyle|l_{s,\tau}-l^{\prime}_{s,\tau}|\leq\frac{\sqrt{3}}{k}\left|m^{n}_{\mu,\tau}-m^{0}_{\mu,\tau}\right|+\frac{\sqrt{3}\,L}{k}\left(1+[\mathbb{E}\left\\{(\tau-s)|\mathcal{F}_{s}\right\\}]^{-1/2}\right)\|y-y^{\prime}\|_{\infty}$

Finally, according to the definition of $A^{n},\,n=0,1,\cdot\cdot\cdot$, we
conclude that for $n=1,2,\cdot\cdot\cdot$,

$\displaystyle|A_{t}-A^{\prime}_{t}|$ $\displaystyle=$
$\displaystyle\left|\sup_{0\leq s\leq t}L^{n}_{s}-\sup_{0\leq s\leq
t}L^{0}_{s}\right|$ $\displaystyle\leq$ $\displaystyle\sup_{0\leq s\leq
t}\left|\mbox{ess}\inf_{\tau>s}l^{n}_{s,\tau}-\mbox{ess}\inf_{\tau>s}l^{0}_{s,\tau}\right|$
$\displaystyle\leq$ $\displaystyle\sup_{0\leq s\leq
t}\mbox{ess}\sup_{\tau>s}|l^{n}_{s,\tau}-l^{0}_{s,\tau}|$ $\displaystyle\leq$
$\displaystyle\sup_{0\leq s\leq
t}\mbox{ess}\sup_{\tau>s}\frac{\sqrt{3}}{k}\left[\left|m^{n}_{\mu,\tau}-m^{0}_{\mu,\tau}\right|+L\left(1+\left(\mathbb{E}\left\\{(\tau-s)|\mathcal{F}_{s}\right\\}\right)^{-1/2}\right)\right]\|Y^{n}-Y^{0}\|_{\infty}$
$\displaystyle\leq$ $\displaystyle\frac{\sqrt{3}}{k}\sup_{0\leq s\leq
t}\mbox{ess}\sup_{\tau>s}\left|m^{n}_{\mu,\tau}-m^{0}_{\mu,\tau}\right|+\frac{\sqrt{3}L}{k}(1+\sqrt{T})\|Y^{n}-Y^{0}\|_{\infty}.$

∎

Now, we are ready to derive the main result of this subsection.

######  Theorem 3.10.

Assume $({\bf A2})$ and $({\bf A3})$ hold. Further, assume that

${\sqrt{6}TL\left(1+\sqrt{3}\frac{K}{k}\left(1+\sqrt{T}\right)\right)+L\sqrt{3T}<1}.$

Then, it holds that

$\displaystyle\|Y^{n}-Y^{0}\|_{\infty}$ $\displaystyle\leq$
$\displaystyle\frac{\sqrt{3}}{1-\left[\sqrt{6}TL\left(1+\sqrt{3}\frac{K}{k}\left(1+\sqrt{T}\right)\right)+L\sqrt{3T}\right]}\times$
$\displaystyle\times\left\\{\left\|\xi^{n}-\xi^{0}\right\|_{\infty}+\frac{\sqrt{6}TK}{k}\left\|\sup_{\mu\in[0,T]}ess\sup_{\tau>\mu}\frac{1}{k}\left|m^{n}_{\mu,\tau}-m^{0}_{\mu,\tau}\right|\right\|_{\infty}\right\\}.$

###### Proof.

Using the similar arguments as Theorem 3.3, we obtain this estimation

$\displaystyle|Y^{n}_{t}-Y^{0}_{t}|\leq\sqrt{3}\|\xi^{n}-\xi^{0}\|_{\infty}+\sqrt{3}L(\sqrt{2}T+\sqrt{T})\|Y^{n}-Y^{0}\|_{\infty}+\sqrt{6}TK\|A^{n}-A^{0}\|_{\infty},$

which, together with Lemma 3.9, proves the desired result. ∎

######  Remark 3.11.

Let us emphasize that, since all the model study in section 6 of [25] have
their stochastic counterpart, we can with no more difficulty establish
respectively the stochastic version of recursive intertemporal utility
minimization, optimal stopping problems. It suffice to follows the similar
step as in [25] with some additional argument due to the presence of the
backward stochastic integral with respect the Brownian motion $B$.

Acknowledgement The work was partially done while the second author visited
Shandong University. He would like to thank Prof. Shige Peng for providing a
stimulating working environment.

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