#  REFLECTED BSDEs AND DOUBLY REFLECTED BSDEs
DRIVEN BY RCLL MARTINGALES

Tianyang Nie$\,{}^{a}$111The research of T. Nie and M. Rutkowski was supported
by the DVC Research Bridging Support Grant Pricing of American and game
options in market with frictions. The work of T. Nie was supported by the
National Natural Science Foundation of China (No.
12022108,11971267,11831010,61977043). and Marek Rutkowski$\,{}^{b,c}$

${}^{a\,}$School of Mathematics, Shandong University,
Jinan, Shandong 250100, China

${}^{b\,}$School of Mathematics and Statistics, University of Sydney
Sydney, NSW 2006, Australia

${}^{c\,}$Faculty of Mathematics and Information Science, Warsaw University of
Technology,
00-661 Warszawa, Poland

###### Abstract

We prove some new results on reflected BSDEs and doubly reflected BSDEs driven
by a multi-dimensional RCLL martingale. The goal is to develop a general
multi-asset framework encompassing a wide spectrum of nonlinear financial
models, including as particular cases the setups studied by Peng and Xu [55]
and Dumitrescu et al. [13] who dealt with BSDEs driven by a one-dimensional
Brownian motion and a purely discontinuous martingale with a single jump. Our
results are not covered by existing literature on reflected and doubly
reflected BSDEs driven by a Brownian motion and a Poisson random measure.

Keywords: backward stochastic differential equation, RCLL martingale,
reflected BSDE AMS Subject Classification: 60H10, 60H30, 91G30, 91G40

**footnotetext: E-mail<EMAIL_ADDRESS>(Tianyang NIE);
<EMAIL_ADDRESS>(Marek RUTKOWSKI).

##  1 Introduction

Reflected backward stochastic differential equations (BSDEs) were introduced
in seminal papers by El Karoui et al. [20, 21] where they were applied to
solutions of optimal stopping problems and pricing of American options. They
were subsequently studied by several authors who dealt with various
frameworks, to mention a few: Aazizi and Ouknine [1], Baadi and Ouknine [2,
3], Bayraktar and Yao [5], Crépey and Matoussi [10], Essaky [24], Hamadène
[29], Hamadène and Ouknine [31], Grigorova et al. [26, 27], Klimsiak [41, 42],
Klimsiak et al. [43] Lepeltier and Xu [49], Peng and Xu [54], and Quenez and
Sulem [59].

Doubly reflected BSDEs were first introduced and studied by Cvitanić and
Karatzas [11] who also showed that classical Dynkin games can be solved using
the theory of doubly reflected BSDEs. Their studies were continued by, among
others, Bayraktar and Yao [6], Crépey and Matoussi [10], Dumitrescu et al.
[14], Essaky and Hassani [25], Grigorova et al. [27], Grigorova and Quenez
[28], Hamadène and Lepeltier [30], Hamadène and Ouknine [31], Hamadène and
Wang [32], Kobylanski et al. [46], and Lepeltier and San Martín [48]. For a
survey of results on Dynkin games and their applications to the valuation and
hedging of game options, the reader is referred to Kifer [38].

Our goal is to study the class of reflected BSDEs and doubly reflected BSDEs,
which are not covered by the existing literatures, but appear in a natural way
in applications of reflected BSDEs and doubly reflected BSDEs to optimal
stopping problems and financial mathematics. This work can be seen as a direct
continuation of our previous work [52] where a particular class of BSDEs
driven by discontinuous martingales was studied. The main motivation for this
research is to provide a theoretical underpinning for stochastic models of
nonlinear financial markets with credit risk where discontinuous martingales
appear in a natural way. In particular, our results encompass market models
with a single extraneous event, usually representing some credit event, which
were previously studied by Peng and Xu [55, 56] and, more recently, by
Dumitrescu et al. [13]. The interested reader is also referred to Dumitrescu
et al. [14, 15, 16] for applications of results from [13] on reflected and
doubly reflected BSDEs with a single jump to the nonlinear valuation and
superhedging of American and game options in financial models with the
reference credit risk.

It should be stressed that in related papers [13, 55, 56], the authors examine
a special class of BSDEs driven by a one-dimensional Brownian motion and a
purely discontinuous martingale with a single jump of a unit size, and the
reflected BSDEs and doubly reflected BSDEs were studied by Dumitrescu et al.
[14, 15, 16] in the same framework. We have extended the results of [13, 55,
56] to the BSDEs driven by a fairly general class of multi-dimensional RCLL
martingales in [52]. It should be acknowledged that several classes of BSDEs
driven by a Brownian motion, and possibly also a Poisson random measure, were
studied by, among others, Barles et al. [4], Crépey and Matoussi [10], El
Karoui et al. [19], Hamadène and Ouknine [31], Hamadène and Wang [32],
Jeanblanc et al. [36, 37], Papapantoleon et al. [53], Quenez and Sulem [58],
Royer [60] and Tang and Li [61]. However, the results obtained in the above-
mentioned papers do not cover the class of BSDEs driven by RCLL martingales
considered in [52].

In this paper, we will extend the work [52] to study the reflected BSDEs and
doubly reflected BSDEs driven by a fairly general martingale satisfying
Assumption 2.1 and thus results from [14, 15, 16] are covered by the present
work but our setup is manifestly more encompassing. Our motivation of studying
this kind of general reflected BSDEs and doubly reflected BSDEs comes from the
pricing of American and game options in general nonlinear financial market
models, for instance, market models with the credit risk where several default
times appears.

As in [52], we postulate here that the predictable representation property of
the driving martingale (see Assumption 2.2), which is known to hold in some
circumstances (see, e.g., Kusuoka [47] or Jeanblanc and Le Cam [35], in
particular it also holds in the framework of [14, 15, 16]). We stress that
this assumption can be readily relaxed in theorems yielding the existence and
uniqueness of solutions, provided that an additional orthogonal martingale
term is introduced in the definition of a solution to a reflected or doubly
reflected BSDE, as was done in Carbone et al. [8] and El Karoui and Huang [18]
(see Remark 2.2 of current paper and Remark 2.1 in [52] for more details). For
conciseness, we do not elaborate here on that fairly natural extension.

Our results are capable of covering a much broader spectrum of stochastic
models, for instance, the financial market models with the credit risk of
multiple entities where several default times appear in a natural way. More
generally, the results obtained in this work can be employed to derive the
results of the valuation, hedging and exercising of American and game options
in any market model where the price processes of primary assets are governed
by stochastic differential equations driven by discontinuous noise processes.
Some applications of our results to the American and game options are
presented in follow-up works by Kim et al. [39, 40].

The paper is organized as follows. We start by presenting in Section 2 some
definitions and assumptions from [8, 18, 51, 52] pertaining to BSDEs driven by
RCLL martingales. In Section 3, we study the reflected BSDEs driven by RCLL
matingales and we obtain the a priori estimates (Propositions 3.1–3.3), the
existence and uniqueness result (Theorem 3.1), as well as the convergence of
Picard’s iterations for solutions to reflected BSDEs (Proposition 3.4).
Analogous results for doubly reflected BSDEs driven by RCLL martingales are
established in Section 4.

##  2 Preliminaries

We start by describing the setup and assumptions introduced in [52] where
BSDEs driven by RCLL martingales are studied. Let
$(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$ be a filtered probability space
satisfying the usual conditions of right-continuity and completeness. The
initial $\sigma$-field $\mathcal{F}_{0}$ is assumed to be trivial. For any two
processes, say $X$ and $Y$, we denote the equality $X=Y$ as
$\mathbb{P}(X_{t}=Y_{t},\,\forall\,t\in[0,T])=1$. For any process $X$, we set
$X_{0-}=0$ by convention, and $\Delta X_{0}:=X_{0}-X_{0-}=X_{0}$ is the jump
of $X$ at time 0. Moreover, the integral $\int_{s}^{t}X_{u}\,dY_{u}$ is
interpreted as $\int_{]s,t]}X_{u}\,dY_{u}$ where $]s,t]=\\{u\in[0,T]:s<u\leq
t\\}$.

We assume that $M=((M^{1}_{t},M^{2}_{t},\ldots,M^{n}_{t})^{\ast},\,t\in[0,T])$
is an $n$-dimensional, square-integrable martingale on
$(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$. Since the filtration
$\mathbb{F}$ is right-continuous, an RCLL modification of any
$\mathbb{F}$-martingale is known to exist and thus it is assumed throughout
that $M$ is an RCLL process. We denote by $\langle M\rangle$ (resp. $[M]$) the
predictable covariation process (resp. the covariation process ) of $M$. As in
[8, 18, 51, 52], we work under the following assumption regarding the
martingale $M$.

######  Assumption 2.1.

There exists an $\mathbb{R}^{n\times n}$-valued, $\mathbb{F}$-predictable
process $m$ and an $\mathbb{F}$-adapted, continuous, nondecreasing process $Q$
with $Q_{0}=0$ such that, for all $t\in[0,T]$,

$\langle M\rangle_{t}=\int_{0}^{t}m_{u}m_{u}^{\ast}\,dQ_{u}.$ (2.1)

In addition, we assume that $Q$ is bounded i.e. $\exists$ a constant $C_{Q}$
s.t. $0\leq Q_{t}\leq C_{Q}$ for all $t\in[0,T]$.

######  Remark 2.1.

Assumption 2.1 holds naturally under some suitable assumptions, see [52] for
more details. Without loss of generality, we may assume that the process $m$
appearing in (2.1) takes values in the space of symmetric matrices so that
$m_{u}=m^{\ast}_{u}=(m_{u}m_{u}^{\ast})^{\frac{1}{2}}$, see e.g. [51, 52].

Now we introduce some spaces which will be used in the sequel of the paper. As
usual, $L^{2}({\mathcal{F}}_{T})$ stands for the space of all real-valued,
$\mathcal{F}_{T}$-measurable random variables $\eta$ such that
$\|\eta\|_{L^{2}({\mathcal{F}}_{T})}^{2}:={\mathbb{E}}(\eta^{2})<\infty$. Let
${\mathcal{L}}^{2}(M)$ be the space of all $\mathbb{R}^{n}$-valued,
$\mathbb{F}$-predictable processes $Z$ with the pseudo-norm

$\displaystyle\|Z\|_{{\mathcal{L}}^{2}(M)}^{2}:={\mathbb{E}}\bigg{[}\int_{0}^{T}\|m_{t}Z_{t}\|^{2}\,dQ_{t}\bigg{]}<\infty,$

where $\|\cdot\|$ denotes the Euclidean norm in $\mathbb{R}^{n}$. For any
fixed $\beta\geq 0$, we denote by ${\mathcal{L}}^{2}_{\beta}(M)$ the class of
all $\mathbb{R}^{n}$-valued, $\mathbb{F}$-predictable processes $Z$ with the
pseudo-norm

$\displaystyle\|Z\|_{{\mathcal{L}}^{2}_{\beta}(M)}^{2}:={\mathbb{E}}\bigg{[}\int_{0}^{T}e^{\beta
Q_{t}}\|m_{t}Z_{t}\|^{2}\,dQ_{t}\bigg{]}<\infty.$

Hence ${\mathcal{L}}^{2}(M)=\mathcal{L}^{2}_{0}(M)$ and the pseudo-norms
$\|\cdot\|_{{\mathcal{L}}^{2}(M)}$ and
$\|\cdot\|_{{\mathcal{L}}^{2}_{\beta}(M)}$ are equivalent for every $\beta>0$
since the nondecreasing process $Q$ is nonnegative and bounded.

Let ${\mathcal{S}}^{2}$ stand for the space of all real-valued, RCLL,
$\mathbb{F}$-adapted processes $X$ with the norm

$\displaystyle\|X\|_{{\mathcal{S}}^{2}}^{2}:={\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}X^{2}_{t}\bigg{]}<\infty.$

Similarly, we denote by ${\mathcal{S}}^{2}_{\beta}$ the space of all real-
valued, RCLL, $\mathbb{F}$-adapted processes $X$ with the norm

$\displaystyle\|X\|_{{\mathcal{S}}^{2}_{\beta}}^{2}:={\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}\big{(}e^{\beta
Q_{t}}X^{2}_{t}\big{)}\bigg{]}<\infty.$

It is clear that ${\mathcal{S}}^{2}=\mathcal{S}_{0}^{2}$ and the norms
$\|\cdot\|_{{\mathcal{S}}^{2}}$ and $\|\cdot\|_{{\mathcal{S}}^{2}_{\beta}}$
are equivalent for every $\beta>0$.

We denote by ${\mathcal{H}}^{2}(Q)$ the space of all real-valued,
$\mathbb{F}$-progressively measurable processes $X$ with the pseudo-norm

$\displaystyle\|X\|_{{\mathcal{H}}^{2}(Q)}^{2}:={\mathbb{E}}\bigg{[}\int_{0}^{T}X^{2}_{t}\,dQ_{t}\bigg{]}<\infty.$

Similarly, we denote by ${\mathcal{H}}^{2}_{\beta}(Q)$ the space of all real-
valued, $\mathbb{F}$-progressively measurable processes $X$ with the pseudo-
norm

$\displaystyle\|X\|_{{\mathcal{H}}^{2}_{\beta}(Q)}^{2}:={\mathbb{E}}\bigg{[}\int_{0}^{T}e^{\beta
Q_{t}}X^{2}_{t}\,dQ_{t}\bigg{]}<\infty.$

We observe that ${\mathcal{S}}^{2}\subset{\mathcal{H}}^{2}(Q)$ and, for
brevity, we denote by $\mathbb{S}^{2}$ the product space
${\mathcal{S}}^{2}\times{\mathcal{L}}^{2}(M)$ with the following pseudo-norm,
for every $(X,Z)\in{\mathcal{S}}^{2}\times{\mathcal{L}}^{2}(M)$,

$\displaystyle\|(X,Z)\|_{\mathbb{S}^{2}}=\|X\|_{{\mathcal{S}}^{2}}+\|Z\|_{{\mathcal{L}}^{2}(M)}.$

Similarly, we introduce the space
$\mathbb{H}^{2}:={\mathcal{H}}^{2}(Q)\times{\mathcal{L}}^{2}(M)$ endowed with
the natural pseudo-norm

$\displaystyle\|(X,Z)\|_{\mathbb{H}^{2}}=\|X\|_{{\mathcal{H}}^{2}(Q)}+\|Z\|_{{\mathcal{L}}^{2}(M)}.$

For a fixed $\beta\geq 0$, we denote by $\mathbb{S}^{2}_{\beta}$ the Banach
space
$({\mathcal{S}}^{2}_{\beta}\times{\mathcal{L}}^{2}_{\beta}(M),\|\cdot\|_{\beta})$
with the pseudo-norm

$\displaystyle\|(X,Z)\|^{2}_{\beta}:=\|Z\|^{2}_{{\mathcal{S}}^{2}_{\beta}}+\|Z\|^{2}_{{\mathcal{L}}^{2}_{\beta}(M)}$

and we note that $\mathbb{S}^{2}=\mathbb{S}^{2}_{0}$. In view of Assumption
2.1, the set $\mathbb{S}^{2}_{\beta}$ can be formally identified with
$\mathbb{S}^{2}$.

We introduce the following martingale representation assumption, which
provides a natural underpinning for the existence of a solution of the
following BSDE (2.2).

######  Assumption 2.2.

The martingale $M$ has the predictable representation property under
$\mathbb{P}$ with respect to the filtration $\mathbb{F}$, i.e. for any real-
valued, square-integrable $\mathbb{F}$-martingale $N$ with $N_{0}=0$ there
exists a process $Z\in{\mathcal{L}}^{2}(M)$ s.t.
$N_{t}=\int_{0}^{t}Z^{*}_{u}\,dM_{u}$. Moreover, the uniqueness of $Z$ in
$\|\cdot\|_{{\mathcal{L}}^{2}(M)}$ holds, that is, if
$N_{t}=\int_{0}^{t}Z^{*}_{u}\,dM_{u}=\int_{0}^{t}\widetilde{Z}^{*}_{u}\,dM_{u}$,
then $\|Z-\widetilde{Z}\|_{{\mathcal{L}}^{2}(M)}=0$.

######  Remark 2.2.

Assumption 2.2 is easily to be satisfied, the readers are referred to [52] for
more comments on Assumption 2.2. We just mention here that, similar to Remark
2.1 in [52], Assumption 2.2 can be relaxed to the case that the generalized
martingale representation property of $M$ is valid, in the sense that any
real-valued, square-integrable $\mathbb{F}$-martingale $N$ can be represented
as $N_{t}=\int_{0}^{t}Z^{*}_{u}\,dM_{u}+L_{t}$ where $L$ is a square-
integrable, $\mathbb{F}$-martingale orthogonal to $M$. Then the results of the
existence and uniqueness of solutions in this paper still hold, provided that
an additional orthogonal martingale term is introduced in the definition of a
solution to a reflected or doubly reflected BSDE.

For a fixed horizon date $T>0$, we consider the following BSDE on $[0,T]$ with
the data $(g,\eta,D)$

$\left\\{\begin{array}[c]{ll}dY_{t}=-g(t,Y_{t},Z_{t})\,dQ_{t}+Z_{t}^{\ast}\,dM_{t}+dD_{t},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ Y_{T}=\eta,\end{array}\right.$ (2.2)

where $\eta$ is a given the random variable in $L^{2}({\mathcal{F}}_{T})$, and
$D$ is a given processes in ${\mathcal{H}}^{2}(Q)$ s.t. $D_{T}\in
L^{2}({\mathcal{F}}_{T})$. Equation (2.2) can be written more explicitly in
the following manner, for every $t\in[0,T]$,

$\displaystyle
Y_{t}=\eta+\int_{t}^{T}g(t,Y_{u},Z_{u})\,dQ_{u}-\int_{t}^{T}Z_{u}^{\ast}\,dM_{u}-(D_{T}-D_{t}).$
(2.3)

Note that here $D$ is not necessary to be of finite variation. The next
definition states the measurability properties of the generator $g$ in (2.2).

######  Definition 2.1.

A generator
$g:\Omega\times[0,T]\times\mathbb{R}\times\mathbb{R}^{n}\rightarrow\mathbb{R}$
is a
$\mathcal{F}\otimes\mathcal{B}([0,T])\otimes\mathcal{B}(\mathbb{R})\otimes\mathcal{B}(\mathbb{R}^{n})$-measurable
function such that the process $g(\cdot,\cdot,y,z)$ is
$\mathbb{F}$-progressively measurable, for every
$(y,z)\in\mathbb{R}\times\mathbb{R}^{n}$.

Let us define
$\mu_{Q}(A):={\mathbb{E}}\big{[}\int_{0}^{T}\mathds{1}_{A}(\omega,t)\,dQ_{t}(\omega)\big{]}$
for every $A\in\mathcal{F}_{T}\otimes\mathcal{B}([0,T])$, which is the Doléans
measure of the process $Q$ on the space $\Omega\times[0,T]$ endowed with the
product $\sigma$-algebra $\mathcal{F}_{T}\otimes\mathcal{B}([0,T])$. We first
recall the uniform Lipschitz condition for the generator, which is frequently
employed in the study of classical BSDEs, reflected BSDEs and doubly reflected
BSDEs driven by a Brownian motion (also with jumps generated by a Poisson
random measure).

######  Definition 2.2.

A generator $g$ is uniformly Lipschitz continuous if there exists a constant
$L$ such that, for $\mu_{Q}$-almost every $(\omega,t)$, for all
$y_{1},y_{2}\in\mathbb{R},\,z_{1},z_{2}\in\mathbb{R}^{n}$,

$|g(t,y_{1},z_{1})-g(t,y_{2},z_{2})|\leq
L\big{(}|y_{1}-y_{2}|+\|z_{1}-z_{2}\|\big{)}.$ (2.4)

We emphasis that condition (2.4) is no longer suitable when dealing with
BSDEs, reflected BSDEs and doubly reflected BSDEs with a single jump, which
may be used to represent the occurrence of some extraneous event $E$, and thus
condition (2.4) has been modified to explicitly account for the intensity
process $\lambda$ of $E$. For instance, in [13, 15, 16, 55, 56], the authors
postulate that a generator satisfies the following condition

$|g(t,y_{1},z_{1},k_{1})-g(t,y_{2},z_{2},k_{2})|\leq
L\big{(}|y_{1}-y_{2}|+\|z_{1}-z_{2}\|+\sqrt{\lambda_{t}}|k_{1}-k_{2}|\big{)}$
(2.5)

where the variables $k_{1},k_{2}$ correspond to the compensated martingale of
the indicator process of the event $E$. More generally, in existing papers on
BSDEs driven by a multi-dimensional martingale (see, e.g., [8, 18, 51]), a
generator is typically postulated to satisfy some kind of the $m$-Lipschitz
condition with $m$ being the process from Assumption 2.1. We refer the reader
to [51] for further references and comments on alternative Lipschitz-type
conditions, which can be used when dealing with some particular classes of
generators appearing in financial applications. In the present work, we focus
on the generator satisfying the following regularity condition.

######  Definition 2.3.

A generator $g$ is uniformly $m$-Lipschitz continuous if there exists a
constant $\widehat{L}>0$ such that, for $\mu_{Q}$-almost every $(\omega,t)$,
and all $y_{1},y_{2}\in\mathbb{R},\,z_{1},z_{2}\in\mathbb{R}^{n}$,

$|g(t,y_{1},z_{1})-g(t,y_{2},z_{2})|\leq\widehat{L}\big{(}|y_{1}-y_{2}|+\|m_{t}(z_{1}-z_{2})\|\big{)}.$
(2.6)

######  Remark 2.3.

We note that condition (2.5) employed in Peng and Xu [55, 56] and Dumitrescu
et al. [13, 16], can be seen as a special case of (2.6) (for more details, see
Remark 2.2 in [52]). The readers are also referred to [52] for the existence
and uniqueness theorem for solutions to the BSDE (2.2) with generator
satisfying the uniform $m$-Lipschitz condition.

##  3 Reflected BSDEs Driven by RCLL Martingales

Our aim in this section is to establish the existence and uniqueness result
for the reflected backward stochastic differential equation (RBSDE) driven by
RCLL martingales. Due to various kinds of the so-called minimality conditions,
which are also referred to as the Skorokhod conditions, governing the behavior
of a solution when it hits the obstacle, solutions to RBSDEs studied in
various papers are not always directly related to the nonlinear optimal
stopping problem associated with the obstacle process playing the role of
reward process. Specifically, the component $Y$ of a solution to the RBSDE is
not necessarily equal to the nonlinear Snell envelope of the obstacle process,
although, obviously, this is indeed the case when the proof of the existence
of a solution to the RBSDE hinges on the properties of the classical Snell
envelope for the standard optimal stopping problem under the linear
expectation, which was studied, in particular, by El Karoui [17], Kobylanski
and Quenez [44], and Maingueneau [50].

In view of our further applications to the study of American options in
nonlinear markets (see [39]), the equivalence between the component $Y$ in a
solution $(Y,Z,K)$ to the RBSDE and the value process of the corresponding
nonlinear optimal stopping problem is a highly desirable feature and thus it
will be ensured by a particular choice of minimality conditions in (3.1).

To be more specific, in this section we consider the following RBSDE on
$[0,T]$ with data $(g,\eta,D,\xi)$ (which satisfies the following Assumption
3.1)

$\left\\{\begin{array}[c]{ll}dY_{t}=-g(t,Y_{t},Z_{t})\,dQ_{t}+Z^{\ast}_{t}\,dM_{t}+dD_{t}-dK_{t},\
Y_{T}=\eta,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\ Y_{t}\geq\xi_{t},\
\forall\,t\in[0,T],\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\int_{0}^{T}(Y_{t}-\xi_{t})\,dK^{c}_{t}=0\ \mbox{\rm and}\ \Delta
K^{d}_{\tau}=\Delta
K^{d}_{\tau}\mathds{1}_{\\{(Y-D)_{\tau-}=(\xi-D)_{\tau-}\\}},\
\forall\,\tau\in\mathcal{T}_{p},\end{array}\right.$ (3.1)

where $\mathcal{T}_{p}$ is the class of $\mathbb{F}$-predictable stopping
times taking values in $[0,T]$ and $K$ is a nondecreasing, RCLL,
$\mathbb{F}$-predictable process with $K_{0}=0$. The continuous and
discontinuous components of sample paths of $K$ are denoted by $K^{c}$ and
$K^{d}$, respectively.

######  Assumption 3.1.

The quadruplet $(g,\eta,D,\xi)$ is such that: (i) the generator $g$ is
uniformly $m$-Lipschitz continuous and the process $g(\cdot,0,0)$ belongs to
${\mathcal{H}}^{2}(Q)$, (ii) the process $D$ belongs to ${\mathcal{H}}^{2}(Q)$
and the random variable $\eta-D_{T}$ belongs to $L^{2}({\mathcal{F}}_{T})$,
(iii) the process $\xi-D$ belongs to ${\mathcal{S}}^{2}$ and
$\xi_{T}\leq\eta$.

Let ${\mathcal{A}}^{2}$ be the class of nondecreasing, RCLL,
$\mathbb{F}$-predictable processes such that $A_{0}=0$ and
${\mathbb{E}}(A^{2}_{T})<+\infty$ and let
$\mathbb{H}^{2}_{{\mathcal{A}}}:=\mathbb{H}^{2}\times{\mathcal{A}}^{2}={\mathcal{H}}^{2}(Q)\times{\mathcal{L}}^{2}(M)\times{\mathcal{A}}^{2}$
and
$\mathbb{S}^{2}_{{\mathcal{A}}}:=\mathbb{S}^{2}\times{\mathcal{A}}^{2}={\mathcal{S}}^{2}\times{\mathcal{L}}^{2}(M)\times{\mathcal{A}}^{2}$.

######  Definition 3.1.

A solution to the RBSDE (3.1) with data $(g,\eta,D,\xi)$ is a triplet of
stochastic processes $(Y,Z,K)\in\mathbb{H}^{2}_{{\mathcal{A}}}$ such that
conditions (3.1) are satisfied in the following sense

$\displaystyle\mathbb{P}\bigg{(}Y_{t}=\eta+\int_{t}^{T}g(t,Y_{u},Z_{u})\,dQ_{u}-\int_{t}^{T}Z_{u}^{\ast}\,dM_{u}-(D_{T}-D_{t})+(K_{T}-K_{t}),\
\forall\,t\in[0,T]\bigg{)}=1,$

$\mathbb{P}(Y_{t}\geq\xi_{t},\,\forall\,t\in[0,T])=1$ and the Skorokhod
(minimality) conditions hold: $\int_{0}^{T}(Y_{t}-\xi_{t})\,dK^{c}_{t}=0$ and
$\Delta K^{d}_{\tau}=\Delta
K^{d}_{\tau}\mathds{1}_{\\{(Y-D)_{\tau-}=(\xi-D)_{\tau-}\\}}$ for every
$\tau\in\mathcal{T}_{p}$. We say that the uniqueness of a solution to (3.1)
holds if for any two solutions $(Y,Z,K)$ and
$(Y^{\prime},Z^{\prime},K^{\prime})$ to (3.1) we have
$\|(Y,Z,K)-(Y^{\prime},Z^{\prime},K^{\prime})\|_{\mathbb{H}^{2}_{{\mathcal{A}}}}=0$.

We define
$\widetilde{Y}:=Y-D,\,\widetilde{\xi}:=\xi-D,\,\widetilde{\eta}:=\eta-D_{T}$
and $\widetilde{g}(t,y,z):=g(t,y+D_{t},z)$ and we note that $\widetilde{g}$ is
uniformly $m$-Lipschitz continuous and, in view of Assumptions 3.1(i)-(ii),
the process $\widetilde{g}(\cdot,0,0)$ belongs to ${\mathcal{H}}^{2}(Q)$. We
also set $\widetilde{Z}=Z$ and $\widetilde{K}=K$. Then RBSDE (3.1) can be
written equivalently as the transformed RBSDE with data
$(\widetilde{g},\widetilde{\eta},0,\widetilde{\xi})$

$\left\\{\begin{array}[c]{ll}d\widetilde{Y}_{t}=-\widetilde{g}(t,\widetilde{Y}_{t},\widetilde{Z}_{t})\,dQ_{t}+\widetilde{Z}^{*}_{t}\,dM_{t}-d\widetilde{K}_{t},\
\widetilde{Y}_{T}=\widetilde{\eta},\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\widetilde{Y}_{t}\geq\widetilde{\xi}_{t},\ \forall\,t\in[0,T],\vskip 6.0pt
plus 2.0pt minus 2.0pt\\\
\int_{0}^{T}(\widetilde{Y}_{t}-\widetilde{\xi}_{t})\,d\widetilde{K}^{c}_{t}=0\
\mbox{\rm and}\
\Delta\widetilde{K}^{d}_{\tau}=\Delta\widetilde{K}^{d}_{\tau}\mathds{1}_{\\{\widetilde{Y}_{\tau-}=\widetilde{\xi}_{\tau-}\\}},\
\forall\,\tau\in\mathcal{T}_{p}.\end{array}\right.$ (3.2)

It is rather straightforward to check that the following lemma is valid.

######  Lemma 3.1.

Under Assumption 3.1, RBSDE (3.1) has a unique solution
$(Y,Z,K)\in\mathbb{S}^{2}_{{\mathcal{A}}}$ if and only if RBSDE (3.2) has a
unique solution
$(\widetilde{Y},\widetilde{Z},\widetilde{K})\in\mathbb{S}^{2}_{{\mathcal{A}}}$
and the relationships $Y=\widetilde{Y}+D,\,Z=\widetilde{Z},\,K=\widetilde{K}$
hold.

Without loss of generality, we henceforth assume that $D=0$, which means that
we will consider the following RBSDE on $[0,T]$ with data $(g,\eta,0,\xi)$

$\left\\{\begin{array}[c]{ll}dY_{t}=-g(t,Y_{t},Z_{t})\,dQ_{t}+Z^{\ast}_{t}\,dM_{t}-dK_{t},\
Y_{T}=\eta,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\ Y_{t}\geq\xi_{t},\
\forall\,t\in[0,T],\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\int_{0}^{T}(Y_{t}-\xi_{t})\,dK^{c}_{t}=0\ \mbox{\rm and}\ \Delta
K^{d}_{\tau}=\Delta K^{d}_{\tau}\mathds{1}_{\\{Y_{\tau-}=\xi_{\tau-}\\}},\
\forall\,\tau\in\mathcal{T}_{p}.\end{array}\right.$ (3.3)

###  3.1 A Priori Estimates for RBSDEs

We will first obtain some a priori estimates for solutions to the RBSDE.
Recall that we work under the natural assumption that the obstacle process
$\xi$ satisfies $\xi_{T}\leq\eta$. If we denote $\xi^{+}_{t}=\max(\xi_{t},0)$,
then it follows easily from Assumption 3.1(iii) that the process $\xi^{+}$
belongs to ${\mathcal{S}}^{2}$. The following result can be seen as a
generalization of Proposition 3.5 in El Karoui et al. [20].

######  Proposition 3.1.

Let Assumptions 2.1, 2.2 and 3.1 be valid and let $(Y,Z,K)$ be a solution to
the RBSDE (3.3). Then there exists a constant $C\geq 0$ such that

$\displaystyle\|Y\|_{{\mathcal{S}}^{2}}^{2}+\|Y\|_{{\mathcal{H}}^{2}(Q)}^{2}+\|Z\|_{{\mathcal{L}}^{2}(M)}^{2}+\|K\|_{{\mathcal{S}}^{2}}^{2}\leq
C\big{(}\|\eta\|_{L^{2}({\mathcal{F}}_{T})}^{2}+\|\xi^{+}\|_{{\mathcal{S}}^{2}}^{2}+\|g(\cdot,0,0)\|_{{\mathcal{H}}^{2}(Q)}^{2}\big{)}.$

Proof. The Itô formula yields

$\displaystyle
dY^{2}_{t}=2Y_{t-}\,dY_{t}+d[Y]_{t}=2Y_{t-}\,dY_{t}+dN_{t}+d\langle
Y\rangle_{t}$

where the process $N_{t}:=[Y]_{t}-\langle Y\rangle_{t}=\sum_{0\leq s\leq
t}|\Delta Y_{u}|^{2}-\langle Y^{d}\rangle_{t}$ is an $\mathbb{F}$-martingale.
Thus

$\displaystyle de^{\beta Q_{t}}Y^{2}_{t}$ $\displaystyle=\beta e^{\beta
Q_{t}}Y^{2}_{t}\,dQ_{t}+e^{\beta Q_{t}}\,dY^{2}_{t}$ $\displaystyle=e^{\beta
Q_{t}}\left[\beta
Y^{2}_{t}\,dQ_{t}-2Y_{t}g(t,Y_{t},Z_{t})\,dQ_{t}+2Y_{t-}Z^{*}_{t}\,dM_{t}-2Y_{t-}\,dK_{t}\right]$
(3.4) $\displaystyle+e^{\beta
Q_{t}}\left[dN_{t}+\|m_{t}Z_{t}\|^{2}\,dQ_{t}\right].$

Step 1. We will first show that

$\begin{array}[c]{ll}&\displaystyle\sup_{t\in[0,T]}{\mathbb{E}}(Y^{2}_{t})+{\mathbb{E}}\bigg{[}\int_{0}^{T}|Y_{u}|^{2}\,dQ_{u}+\int_{0}^{T}\|m_{u}Z_{u}\|^{2}\,dQ_{u}\bigg{]}\\\
&\displaystyle\leq
C\,{\mathbb{E}}\bigg{[}\eta^{2}+\int_{0}^{T}h_{u}\,dQ_{u}+2\int_{0}^{T}\xi^{+}_{u-}\,dK_{u}\bigg{]}\end{array}$
(3.5)

where we denote $h_{t}:=(g(t,0,0))^{2}$. By integrating (3.1) from $t$ to $T$,
taking the expectation and using the equalities
$\int_{0}^{T}(Y_{u}-\xi_{u})\,dK^{c}_{u}=0$ and $(Y_{t-}-\xi_{t-})\Delta
K^{d}_{t}=0$, we obtain

$\displaystyle{\mathbb{E}}(e^{\beta
Q_{t}}Y^{2}_{t})+{\mathbb{E}}\bigg{[}\int_{t}^{T}\beta e^{\beta
Q_{u}}Y_{u}^{2}\,dQ_{u}\bigg{]}+{\mathbb{E}}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}\|m_{u}Z_{u}\|^{2}\,dQ_{u}\bigg{]}\vskip 6.0pt plus 2.0pt minus 2.0pt$
$\displaystyle={\mathbb{E}}\bigg{[}e^{\beta
Q_{T}}\eta^{2}+2\int_{t}^{T}e^{\beta
Q_{u}}Y_{u}g(u,Y_{u},Z_{u})\,dQ_{u}+2\int_{t}^{T}e^{\beta
Q_{u}}\xi_{u-}\,dK_{u}\bigg{]}.$

Since the mapping $g$ is uniformly $m$-Lipschitz continuous, we have

$\displaystyle|2Y_{u}g(u,Y_{u},Z_{u})|\leq$ $\displaystyle
2\widehat{L}|Y_{u}|\big{(}g(u,0,0)+|Y_{u}|+\|m_{u}Z_{u}\|\big{)}\leq
h_{u}+(3\widehat{L}^{2}+2\widehat{L})Y^{2}_{u}+\frac{1}{2}\|m_{u}Z_{u}\|^{2}$

and thus

$\displaystyle{\mathbb{E}}(e^{\beta
Q_{t}}Y^{2}_{t})+{\mathbb{E}}\bigg{[}\int_{t}^{T}\beta e^{\beta
Q_{u}}Y_{u}^{2}\,dQ_{u}\bigg{]}+\frac{1}{2}\,{\mathbb{E}}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}\|m_{u}Z_{u}\|^{2}\,dQ_{u}\bigg{]}$
$\displaystyle\leq{\mathbb{E}}\bigg{[}e^{\beta
Q_{T}}\eta^{2}+\int_{t}^{T}e^{\beta
Q_{u}}h_{u}\,dQ_{u}+\int_{t}^{T}(3\widehat{L}^{2}+2\widehat{L})e^{\beta
Q_{u}}Y^{2}_{u}\,dQ_{u}+2\int_{t}^{T}e^{\beta
Q_{u}}\xi^{+}_{u-}\,dK_{u}\bigg{]}.$

If we take $\beta=3\widehat{L}^{2}+2\widehat{L}+1$, then we obtain the
following inequality, which holds for all $t\in[0,T]$,

$\displaystyle{\mathbb{E}}(e^{\beta
Q_{t}}Y^{2}_{t})+{\mathbb{E}}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}Y_{u}^{2}\,dQ_{u}\bigg{]}+\frac{1}{2}\,{\mathbb{E}}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}\|m_{u}Z_{u}\|^{2}\,dQ_{u}\bigg{]}$
$\displaystyle\leq{\mathbb{E}}\bigg{[}e^{\beta
Q_{T}}\eta^{2}+\int_{t}^{T}e^{\beta Q_{u}}h_{u}\,dQ_{u}+2\int_{t}^{T}e^{\beta
Q_{u}}\xi^{+}_{u-}\,dK_{u}\bigg{]}.$

Since $0\leq Q_{t}\leq C_{Q}$ for all $t\in[0,T]$, we have that $1\leq
e^{\beta Q_{t}}\leq e^{\beta Q_{T}}\leq e^{\beta C_{Q}}$, and thus it is easy
to deduce from the inequality above that

$\displaystyle\sup_{t\in[0,T]}{\mathbb{E}}(Y^{2}_{t})+{\mathbb{E}}\bigg{[}\int_{0}^{T}Y_{u}^{2}\,dQ_{u}+\int_{0}^{T}\|m_{u}Z_{u}\|^{2}\,dQ_{u}\bigg{]}\leq
e^{\beta
C_{Q}}\,{\mathbb{E}}\bigg{[}\eta^{2}+\int_{0}^{T}h_{u}\,dQ_{u}+2\int_{0}^{T}\xi^{+}_{u-}\,dK_{u}\bigg{]},$

which shows that (3.5) is valid for some constant $C\geq 0$. Note that the
value of a constant $C$ will vary from place to place in the remainder of the
proof.

Step 2. We will now demonstrate that there exists a constant $C\geq 0$ such
that

$\displaystyle\|K\|_{{\mathcal{S}}^{2}}^{2}={\mathbb{E}}(K^{2}_{T})\leq CJ$
(3.6)

where

$\displaystyle
J:={\mathbb{E}}\bigg{[}\eta^{2}+\sup_{t\in[0,T]}(\xi^{+}_{t})^{2}+\int_{0}^{T}h_{t}\,dQ_{t}\bigg{]}=\|\eta\|_{L^{2}({\mathcal{F}}_{T})}^{2}+\|\xi^{+}\|_{{\mathcal{S}}^{2}}^{2}+\|g(\cdot,0,0)\|_{{\mathcal{H}}^{2}(Q)}^{2}.$

From the equality

$\displaystyle
K_{T}=Y_{0}-\eta-\int_{0}^{T}g(t,Y_{t},Z_{t})\,dQ_{t}+\int_{0}^{T}Z^{\ast}_{t}\,dM_{t}$

we obtain

$\displaystyle K^{2}_{T}\leq
C\bigg{[}Y^{2}_{0}+\eta^{2}+\bigg{(}\int_{0}^{T}g(t,Y_{t},Z_{t})\,dQ_{t}\bigg{)}^{2}+\bigg{(}\int_{0}^{T}Z^{\ast}_{t}\,dM_{t}\bigg{)}^{2}\bigg{]}.$

Since the process $Q$ is nondecreasing and bounded, there exists a constant
$C\geq 0$ (independent of $t$ and $\omega$) such that, for all $t\in[0,T]$,

$\displaystyle\bigg{(}\int_{t}^{T}g(u,Y_{u},Z_{u})\,dQ_{u}\bigg{)}^{2}\leq
C\,\int_{t}^{T}g^{2}(u,Y_{u},Z_{u})\,dQ_{u}.$

Using also the inequality $g^{2}(u,Y_{u},Z_{u})\leq
C(h_{u}+Y^{2}_{u}+\|m_{u}Z_{u}\|^{2})$, estimate (3.5) and the inequality

$\displaystyle 2C\,\int_{0}^{T}\xi^{+}_{t-}\,dK_{t}\leq
2C^{2}\sup_{t\in[0,T]}(\xi^{+}_{t})^{2}+\frac{1}{2}\,K^{2}_{T},$ (3.7)

we find that

$\displaystyle{\mathbb{E}}(K^{2}_{T})$ $\displaystyle\leq
C\,{\mathbb{E}}\bigg{[}\eta^{2}+\int_{0}^{T}h_{t}\,dQ_{t}+2\int_{0}^{T}\xi^{+}_{t-}\,dK_{t}\bigg{]}$
$\displaystyle\leq
C\,{\mathbb{E}}\bigg{[}\eta^{2}+\int_{0}^{T}h_{t}\,dQ_{t}\bigg{]}+2C^{2}\,{\mathbb{E}}\Big{[}\sup_{t\in[0,T]}(\xi^{+}_{t})^{2}\Big{]}+\frac{1}{2}\,{\mathbb{E}}(K^{2}_{T}),$

which implies that (3.6) holds. By combining (3.5), (3.6) and (3.7), we also
get

$\displaystyle\sup_{t\in[0,T]}{\mathbb{E}}(Y^{2}_{t})+{\mathbb{E}}\bigg{[}\int_{0}^{T}Y_{u}^{2}\,dQ_{u}+\int_{0}^{T}\|m_{u}Z_{u}\|^{2}\,dQ_{u}+K_{T}^{2}\bigg{]}\leq
CJ.$ (3.8)

Step 3. In view of (3.8), to complete the proof of the proposition, it remains
to show that

$\displaystyle\|Y\|^{2}_{{\mathcal{S}}^{2}}\leq CJ$ (3.9)

for some constant $C\geq 0$. Since

$\displaystyle
Y_{t}=\eta+\int_{t}^{T}g(u,Y_{u},Z_{u})\,dQ_{u}-\int_{t}^{T}Z^{\ast}_{u}\,dM_{u}+K_{T}-K_{t},$

we have, for every $t\in[0,T]$,

$\displaystyle Y^{2}_{t}\leq
C\bigg{[}\eta^{2}+\bigg{(}\int_{t}^{T}g(u,Y_{u},Z_{u})\,dQ_{u}\bigg{)}^{2}+\bigg{(}\int_{t}^{T}Z^{\ast}_{u}\,dM_{u}\bigg{)}^{2}+K^{2}_{T}\bigg{]}.$

The Burkholder-Davis-Gundy inequality with $p=2$ applied to the
$\mathbb{F}$-martingale $\widehat{M}_{t}:=\int_{0}^{t}Z^{\ast}_{u}\,dM_{u}$
gives

$\displaystyle\|\widehat{M}\|^{2}_{{\mathcal{S}}^{2}}\leq
C\,{\mathbb{E}}\big{(}[\widehat{M}]_{T}\big{)}=C\,{\mathbb{E}}\bigg{[}\int_{0}^{T}\|m_{u}Z_{u}\|^{2}\,dQ_{u}\bigg{]}$

and thus

$\displaystyle\|Y\|^{2}_{{\mathcal{S}}^{2}}\leq
C\,{\mathbb{E}}\bigg{[}\eta^{2}+\int_{0}^{T}\big{(}h_{t}+Y^{2}_{t}+\|m_{t}Z_{t}\|^{2}\big{)}\,dQ_{t}+K^{2}_{T}\bigg{]}.$

Therefore, using (3.8), we obtain (3.9) and thus the proof is completed.
$\Box$

For simplicity, we write $y:=Y^{1}-Y^{2},\,z:=Z^{1}-Z^{2}$ and
$k:=K^{1}-K^{2}$. The next result furnishes another a priori estimate for
solutions to the RBSDE (3.3).

######  Proposition 3.2.

Let Assumptions 2.1, 2.2 and 3.1(ii)-(iii) be valid. For $l=1,2$, assume that
$(Y^{l},Z^{l},K^{l})$ is a solution to the RBSDE (3.3) with generator $g^{l}$.
If the mapping $g^{1}$ is uniformly $m$-Lipschitz continuous with the constant
$\widehat{L}_{1}>0$, then for any $\gamma>0$ and $\beta>0$

$\displaystyle\big{(}\beta-2\widehat{L}_{1}-\gamma^{-1}\big{)}\|y\|^{2}_{{\mathcal{H}}^{2}_{\beta}(Q)}+(1-2\widehat{L}_{1}^{2}\gamma)\|z\|^{2}_{{\mathcal{L}}^{2}_{\beta}(M)}\leq
2\gamma\|\overline{g}\|^{2}_{{\mathcal{H}}^{2}_{\beta}(Q)}$

where
$\overline{g}_{t}:=g^{1}(t,Y^{2}_{t},Z^{2}_{t})-g^{2}(t,Y^{2}_{t},Z^{2}_{t})$
for all $t\in[0,T]$.

Proof. By taking the difference of the two RBSDEs, we obtain

$\left\\{\begin{array}[c]{ll}dy_{t}=-\delta_{t}\,dQ_{t}+z^{*}_{t}\,dM_{t}-dk_{t},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ y_{T}=0,\end{array}\right.$

where we denote
$\delta_{t}:=g^{1}(t,Y^{1}_{t},Z^{1}_{t})-g^{2}(t,Y^{2}_{t},Z^{2}_{t})$. The
Itô integration by parts formula gives

$\displaystyle d(e^{\beta Q_{t}}y^{2}_{t})=y^{2}_{t}\,de^{\beta
Q_{t}}+e^{\beta Q_{t}}\,dy^{2}_{t}=y^{2}_{t}\beta e^{\beta
Q_{t}}\,dQ_{t}+e^{\beta Q_{t}}\,dy^{2}_{t}$

where

$\displaystyle
dy^{2}_{t}=2y_{t-}\,dy_{t}+d[y]_{t}=2y_{t-}\,dy_{t}+dN_{t}+d\langle
y\rangle_{t}$

where the process $N:=[y]-\langle y\rangle$ is an $\mathbb{F}$-martingale,
$d\langle y\rangle_{t}=\|m_{t}z_{t}\|^{2}\,dQ_{t}$, and

$\displaystyle
y_{t-}\,dy_{t}=-y_{t}\delta_{t}\,dQ_{t}+y_{t-}z^{*}_{t}\,dM_{t}-y_{t-}\,dK^{1}_{t}+y_{t-}\,dK^{2}_{t}.$

By integrating from $t$ to $T$ and taking the conditional expectation with
respect to $\mathcal{F}_{t}$, we obtain

$\begin{split}&e^{\beta
Q_{t}}y^{2}_{t}+\beta\,{\mathbb{E}}_{t}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}y^{2}_{u}\,dQ_{u}\bigg{]}+{\mathbb{E}}_{t}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}\,\|m_{u}z_{u}\|^{2}\,dQ_{u}\bigg{]}\\\
&=2\,{\mathbb{E}}_{t}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}y_{u}\delta_{u}\,dQ_{u}\bigg{]}+2\,{\mathbb{E}}_{t}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}y_{u-}\,dK^{1}_{u}-2\int_{t}^{T}e^{\beta
Q_{u}}y_{u-}\,dK^{2}_{u}\bigg{]}.\end{split}$ (3.10)

Furthermore, we observe that

$\displaystyle
y_{u-}\,dK^{1,c}_{u}=y_{u}\,dK^{1,c}_{u}=\big{[}(Y^{1}_{u}-\xi_{u})-(Y^{2}_{u}-\xi_{u})\big{]}\,dK^{1,c}_{u}=-(Y^{2}_{u}-\xi_{u})\,dK^{1,c}_{u}\leq
0$

and

$\displaystyle y_{u-}\,\Delta
K^{1,d}_{u}=\big{[}(Y^{1}_{u-}-\xi_{u-})-(Y^{2}_{u-}-\xi_{u-})\big{]}\,\Delta
K^{1,d}_{u}\leq 0$

so that $y_{u-}\,dK^{1}_{u}\leq 0$. Analogous arguments show that
$y_{u}\,dK^{2,c}_{u}\geq 0$ and $y_{u-}\,\Delta K^{2,d}_{u}\geq 0$ and thus
the inequality $-y_{u-}\,dK^{2}_{u}\leq 0$ is valid as well.

In the next step, we use these inequalities, the fact that
$|\delta_{t}|\leq\widehat{L}_{1}(|y_{t}|+\|m_{t}z_{t}\|)+|\overline{g}_{t}|$
and the elementary inequality
$2a(\widehat{L}_{1}b+c)\leq\gamma^{-1}a^{2}+2\gamma(\widehat{L}_{1}^{2}b^{2}+c^{2})$,
which holds for arbitrary real numbers $a,b,c$ and $\gamma>0$. We deduce from
(3.10) that

$\displaystyle e^{\beta Q_{t}}y^{2}_{t}$
$\displaystyle+\beta\,{\mathbb{E}}_{t}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}y^{2}_{u}\,dQ_{u}\bigg{]}+{\mathbb{E}}_{t}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}\|m_{u}z_{u}\|^{2}\,dQ_{u}\bigg{]}$ $\displaystyle\leq
2\widehat{L}_{1}\,{\mathbb{E}}_{t}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}y^{2}_{u}\,dQ_{u}\bigg{]}+2\,{\mathbb{E}}_{t}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}|y_{u}|\big{(}\widehat{L}_{1}\|m_{u}z_{u}\|+|\overline{g}_{u}|\big{)}\,dQ_{u}\bigg{]}$
$\displaystyle\leq
2\widehat{L}_{1}\,{\mathbb{E}}_{t}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}|y_{u}|^{2}\,dQ_{u}\bigg{]}+{\mathbb{E}}_{t}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}\Big{(}\gamma^{-1}y^{2}_{u}+2\gamma\big{(}\widehat{L}_{1}^{2}\|m_{u}z_{u}\|^{2}+|\overline{g}_{u}|^{2}\big{)}\Big{)}\,dQ_{u}\bigg{]}.$

Rearranging the above equation and taking $t=0$, we obtain the desired
inequality. $\Box$

For any $(w,v)\in\mathbb{S}^{2}_{\beta}$, we suppose that the reflected BSDE
(3.3) associated with generator $g_{t}:=g(t,w_{t},v_{t})$ has a unique
solution
$(Y^{w,v},Z^{w,v},K^{w,v})\in\mathbb{S}^{2}_{\beta}\times{\mathcal{A}}^{2}$.
Then we have the following result, which will be used in the next section to
establish the existence and uniqueness of a solution to the reflected BSDE
(3.3).

######  Proposition 3.3.

Let Assumptions 2.1, 2.2 and 3.1(ii)-(iii) be valid. For $l=1,2$, we assume
that $(Y^{l},Z^{l},K^{l})$ is a solution to the RBSDE (3.3) with generator
$g^{l}_{t}=g(t,w^{l}_{t},v^{l}_{t})$. If the mapping $g$ is uniformly
$m$-Lipschitz continuous with the constant $\widehat{L}>0$, then for any
$\beta>0$

$\displaystyle\|y\|^{2}_{{\mathcal{S}}^{2}_{\beta}}+\|z\|^{2}_{{\mathcal{L}}^{2}_{\beta}(M)}\leq
1158\beta^{-1}\widehat{L}^{2}(C_{Q}+1)\big{(}\|w\|_{{\mathcal{S}}^{2}_{\beta}}^{2}+\|v\|^{2}_{{\mathcal{L}}^{2}_{\beta}(M)}\big{)}$

where $y=Y^{1}-Y^{2},\,z=Z^{1}-Z^{2},\,w=w^{1}-w^{2}$ and $v=v^{1}-v^{2}$.

Proof. By taking the difference of the two RBSDEs, we obtain

$\left\\{\begin{array}[c]{ll}dy_{t}=-\delta_{t}\,dQ_{t}+z^{*}_{t}\,dM_{t}-dk_{t},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ y_{T}=0,\end{array}\right.$

where we denote
$\delta_{t}:=g(t,w^{1}_{t},v^{1}_{t})-g(t,w^{2}_{t},v^{2}_{t})$. The Itô
integration by parts formula gives

$\displaystyle d(e^{\beta Q_{t}}y^{2}_{t})=y^{2}_{t}\,de^{\beta
Q_{t}}+e^{\beta Q_{t}}\,dy^{2}_{t}=y^{2}_{t}\beta e^{\beta
Q_{t}}\,dQ_{t}+e^{\beta Q_{t}}\,dy^{2}_{t}.$

Since

$\displaystyle
dy^{2}_{t}=2y_{t-}\,dy_{t}+d[y]_{t}=2y_{t-}\,dy_{t}+dN_{t}+d\langle
y\rangle_{t}$

where the process $N_{t}:=[y]_{t}-\langle y\rangle_{t}$ is an
$\mathbb{F}$-martingale and $d\langle
y\rangle_{t}=\|m_{t}z_{t}\|^{2}\,dQ_{t}$, we have

$\begin{split}d(e^{\beta Q_{t}}y^{2}_{t})&=e^{\beta Q_{t}}\big{(}\beta
y^{2}_{t}\,dQ_{t}-2y_{t-}\delta_{t}\,dQ_{t}+2y_{t-}z^{*}_{t}\,dM_{t}-2y_{t-}\,dk_{t}+d[y]_{t}\big{)}\\\
&=e^{\beta Q_{t}}\big{(}\beta
y^{2}_{t}\,dQ_{t}-2y_{t-}\delta_{t}\,dQ_{t}+2y_{t-}z^{*}_{t}\,dM_{t}-2y_{t-}\,dk_{t}+dN_{t}+d\langle
y\rangle_{t}\big{)}.\end{split}$ (3.11)

Similarly to Proposition 3.2, by integrating from $t$ to $T$, taking the
expectation, using the properties that $\int_{t}^{T}e^{\beta
Q_{u}}y_{u-}\,dk_{u}\leq 0$ and $N$ is an $\mathbb{F}$-martingale, we obtain

$\begin{split}&{\mathbb{E}}\bigg{[}e^{\beta
Q_{t}}y^{2}_{t}+\beta\,\int_{t}^{T}e^{\beta
Q_{u}}y^{2}_{u}\,dQ_{u}+\int_{t}^{T}e^{\beta
Q_{u}}\,\|m_{u}z_{u}\|^{2}\,dQ_{u}\bigg{]}\leq
2\,{\mathbb{E}}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}y_{u}\delta_{u}\,dQ_{u}\bigg{]}\\\ &\leq
2\widehat{L}\,{\mathbb{E}}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}|y_{u}|(|w_{u}|+\|m_{u}v_{u}\|)\,dQ_{u}\bigg{]}\\\ &\leq
2\lambda^{-1}\widehat{L}^{2}\,{\mathbb{E}}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}y^{2}_{u}\,dQ_{u}\bigg{]}+\lambda\,{\mathbb{E}}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}|w_{u}|^{2}\,dQ_{u}\bigg{]}+\lambda\,{\mathbb{E}}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}\|m_{u}v_{u}\|^{2}\,dQ_{u}\bigg{]}.\end{split}$

By taking $\lambda=2\beta^{-1}\widehat{L}^{2}$ and recalling that the process
$Q$ is bounded by a constant $C_{Q}$, we get

$\begin{split}&{\mathbb{E}}\bigg{[}e^{\beta
Q_{t}}y^{2}_{t}+\int_{t}^{T}e^{\beta
Q_{u}}\,\|m_{u}z_{u}\|^{2}\,dQ_{u}\bigg{]}\leq
2\,{\mathbb{E}}\,\int_{t}^{T}e^{\beta Q_{u}}y_{u}\delta_{u}\,dQ_{u}\\\
&\leq\lambda\,{\mathbb{E}}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}|w_{u}|^{2}\,dQ_{u}\bigg{]}+\lambda\,{\mathbb{E}}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}\|m_{u}v_{u}\|^{2}\,dQ_{u}\bigg{]}\\\
&\leq\lambda(C_{Q}+1)\big{(}\|w\|_{{\mathcal{S}}^{2}_{\beta}}^{2}+\|v\|^{2}_{{\mathcal{L}}^{2}_{\beta}(M)}\big{)}.\end{split}$
(3.12)

Using (3.11) and the noting that $[y]$ is nondecreasing and
$\int_{t}^{T}e^{\beta Q_{u}}y_{u-}\,dk_{u}\leq 0$, we get

$\begin{split}&e^{\beta Q_{t}}y^{2}_{t}+\beta\,\int_{t}^{T}e^{\beta
Q_{u}}y^{2}_{u}\,dQ_{u}\leq 2\,\int_{t}^{T}e^{\beta
Q_{u}}y_{u}\delta_{u}\,dQ_{u}-2\int_{t}^{T}e^{\beta
Q_{u}}y_{u-}z^{*}_{u}\,dM_{u}\\\ &\leq
2\lambda^{-1}\widehat{L}^{2}\int_{t}^{T}e^{\beta
Q_{u}}y_{u}^{2}\,dQ_{u}+\lambda\int_{t}^{T}e^{\beta
Q_{u}}|w_{u}|^{2}\,dQ_{u}+\lambda\int_{t}^{T}e^{\beta
Q_{u}}\|m_{u}v_{u}\|^{2}\,dQ_{u}-2\int_{t}^{T}e^{\beta
Q_{u}}y_{u-}z^{*}_{u}\,dM_{u},\end{split}$

which yields

$\begin{split}e^{\beta Q_{t}}y^{2}_{t}&\leq\lambda\int_{t}^{T}e^{\beta
Q_{u}}|w_{u}|^{2}\,dQ_{u}+\lambda\int_{t}^{T}e^{\beta
Q_{u}}\|m_{u}v_{u}\|^{2}\,dQ_{u}-2\int_{t}^{T}e^{\beta
Q_{u}}y_{u-}z^{*}_{u}\,dM_{u}\\\
&\leq\lambda(C_{Q}+1)\left(\sup_{t\in[0,T]}\Big{(}e^{\beta
Q_{t}}w_{t}^{2}\Big{)}+\int_{0}^{T}e^{\beta
Q_{t}}\|m_{t}v_{t}\|^{2}\,dQ_{t}\right)-2\int_{t}^{T}e^{\beta
Q_{u}}y_{u-}z^{*}_{u}\,dM_{u}.\end{split}$ (3.13)

We denote $\widetilde{y}_{u}=y_{u}e^{\frac{\beta}{2}Q_{u}}$ and
$\widetilde{M}_{t}:=\int_{0}^{t}e^{\frac{\beta}{2}Q_{u}}z^{\ast}_{u}\,dM_{u}$
and we apply the Davis inequality (see, e.g., Theorem 10.24 in He et al. [33]
or Theorem 11.5.5 in Cohen and Elliott [9] with $p=1$)

$\displaystyle{\mathbb{E}}\left[\sup_{t\in[0,T]}\Big{|}\int_{0}^{t}\widetilde{y}_{u-}\,d\widetilde{M}_{u}\Big{|}\right]\leq
2\sqrt{6}\,{\mathbb{E}}\left[\bigg{(}\int_{0}^{T}|\widetilde{y}_{t-}|^{2}\,d[\widetilde{M}]_{t}\bigg{)}^{1/2}\right]\leq
6\,{\mathbb{E}}\left[\bigg{(}\int_{0}^{T}|\widetilde{y}_{t-}|^{2}\,d[\widetilde{M}]_{t}\bigg{)}^{1/2}\right].$

Consequently, using also the elementary inequalities

$\displaystyle{\mathbb{E}}\left[\bigg{(}\int_{0}^{T}|\widetilde{y}_{t-}|^{2}\,d[\widetilde{M}]_{t}\bigg{)}^{1/2}\right]$
$\displaystyle\leq{\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}|\widetilde{y}_{t}|\,[\widetilde{M}]^{\frac{1}{2}}_{T}\bigg{]}\leq\frac{1}{48}\,{\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}\widetilde{y}^{2}_{t}\bigg{]}+12\,{\mathbb{E}}\big{(}[\widetilde{M}]_{T}\big{)}$
$\displaystyle=\frac{1}{48}\,{\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}\widetilde{y}^{2}_{t}\bigg{]}+12\,{\mathbb{E}}\big{(}\langle\widetilde{M}\rangle_{T}\big{)},$

we obtain

$\displaystyle{\mathbb{E}}\left[\sup_{t\in[0,T]}\Big{|}\int_{0}^{t}\widetilde{y}_{u-}\,d\widetilde{M}_{u}\Big{|}\right]\leq\frac{1}{8}\,{\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}\widetilde{y}^{2}_{t}\bigg{]}+72\,{\mathbb{E}}\big{(}\langle\widetilde{M}\rangle_{T}\big{)},$

which is equivalent to

$\displaystyle{\mathbb{E}}\left[\sup_{t\in[0,T]}\Big{|}\int_{0}^{t}e^{\beta
Q_{u}}y_{u-}z^{*}_{u}\,dM_{u}\Big{|}\right]\leq\frac{1}{8}\,{\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}\Big{(}e^{\beta
Q_{t}}y^{2}_{t}\Big{)}\bigg{]}+72\,{\mathbb{E}}\bigg{[}\int_{0}^{T}e^{\beta
Q_{t}}\,\|m_{t}z_{t}\|^{2}\,dQ_{t}\bigg{]}.$

Thus, by using (3.12), we get

$\begin{split}&{\mathbb{E}}\left[\sup_{t\in[0,T]}\Big{|}\int_{t}^{T}e^{\beta
Q_{u}}y_{u-}z^{*}_{u}\,dM_{u}\Big{|}\right]\leq\frac{1}{4}\,{\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}\Big{(}e^{\beta
Q_{t}}y_{t}^{2}\Big{)}\bigg{]}+144\,{\mathbb{E}}\bigg{[}\int_{0}^{T}e^{\beta
Q_{t}}\,\|m_{t}z_{t}\|^{2}\,dQ_{t}\bigg{]}\\\
&\leq\frac{1}{4}\,{\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}\Big{(}e^{\beta
Q_{t}}y_{t}^{2}\Big{)}\bigg{]}+144\lambda(C_{Q}+1)\big{(}\|w\|_{{\mathcal{S}}^{2}_{\beta}}^{2}+\|v\|^{2}_{{\mathcal{L}}^{2}_{\beta}(M)}\big{)}.\end{split}$
(3.14)

From (3.13) and (3.14), we obtain

$\begin{split}{\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}\Big{(}e^{\beta
Q_{t}}y^{2}_{t}\Big{)}\bigg{]}&\leq\lambda(C_{Q}+1)\big{(}\|w\|_{{\mathcal{S}}^{2}_{\beta}}^{2}+\|v\|^{2}_{{\mathcal{L}}^{2}_{\beta}(M)}\big{)}+2\,{\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}\Big{|}\int_{t}^{T}e^{\beta
Q_{u}}y_{u-}z^{*}_{u}\,dM_{u}\bigg{]}\\\ &\leq
289\lambda(C_{Q}+1)\big{(}\|w\|_{{\mathcal{S}}^{2}_{\beta}}^{2}+\|v\|^{2}_{{\mathcal{L}}^{2}_{\beta}(M)}\big{)}+\frac{1}{2}\,{\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}\Big{(}e^{\beta
Q_{t}}y^{2}_{t}\Big{)}\bigg{]},\end{split}$

which gives

$\displaystyle{\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}\Big{(}e^{\beta
Q_{t}}y^{2}_{t}\Big{)}\bigg{]}\leq
578\lambda(C_{Q}+1)\big{(}\|w\|_{{\mathcal{S}}^{2}_{\beta}}^{2}+\|v\|^{2}_{{\mathcal{L}}^{2}_{\beta}(M)}\big{)}.$
(3.15)

By combining (3.12) and (3.15), we find that (recall that
$\lambda=2\beta^{-1}\widehat{L}^{2}$)

$\displaystyle\|y\|^{2}_{{\mathcal{S}}^{2}_{\beta}}+\|z\|^{2}_{{\mathcal{L}}^{2}_{\beta}(M)}\leq
1158\beta^{-1}\widehat{L}^{2}(C_{Q}+1)\big{(}\|w\|_{{\mathcal{S}}^{2}_{\beta}}^{2}+\|v\|^{2}_{{\mathcal{L}}^{2}_{\beta}(M)}\big{)},$

which ends the proof. $\Box$

###  3.2 Existence and Uniqueness Theorem for Reflected BSDEs

Our next goal is to study the existence and uniqueness of the solution to
RBSDE driven by RCLL martingales. From Lemma 3.1, we know that once we obtain
the existence and uniqueness of the solution to RBSDE (3.3), we also have
similar results for RBSDE (3.1). We henceforth assume that the obstacle $\xi$
in (3.3) (or the modified obstacle $\xi-D$ in (3.1)) belongs to the space
${\mathcal{S}}^{2}$. For the reader’s convenience, we give the detailed
demonstration of Lemma 3.2, which is adapted from the proof of Lemma 2.5 in
Quenez and Sulem [59] who in turn use, in particular, Proposition B.11 from
Kobylanski and Quenez [44] (see also [45]) where the case of the classical
optimal stopping problem was studied.

######  Lemma 3.2.

Let Assumptions 2.1, 2.2 and 3.1 be satisfied. If the generator $g$ does not
depend on $(y,z)$ so that the process $g_{t}:=g(t,y,z)=g(t,0,0)$ belongs to
${\mathcal{H}}^{2}(Q)$, then the RBSDE (3.3) has a unique solution
$(Y,Z,K)\in\mathbb{S}^{2}_{{\mathcal{A}}}$.

Proof. To alleviate the notation, we assume in the proof of Lemma 3.2 that
$\xi_{T}=\eta$. Otherwise, it suffices to introduce an auxiliary process
$\xi_{T}:=\xi_{t}\mathds{1}_{\\{t<T\\}}+\eta\mathds{1}_{\\{t=T\\}}$.

We denote by $\mathcal{T}_{[\sigma,T]}$ the class of all stopping times
$\sigma^{\prime}$ such that $\sigma\leq\sigma^{\prime}\leq T$ and
${\mathbb{E}}_{\sigma}[\,\cdot\,]={\mathbb{E}}[\,\cdot\,|\,\mathcal{F}_{\sigma}]$
and define the stochastic system $\overline{Y}$ by setting, for every
$\sigma\in\mathcal{T}$,

$\displaystyle\overline{Y}(\sigma):=\operatornamewithlimits{ess\,sup}_{\tau\in\mathcal{T}_{[\sigma,T]}}{\mathbb{E}}_{\sigma}\Big{[}\xi_{\tau}+\int_{\sigma}^{\tau}g_{t}\,dQ_{t}\Big{]}=\operatornamewithlimits{ess\,sup}_{\tau\in\mathcal{T}_{[\sigma,T]}}{\mathbb{E}}_{\sigma}\big{[}\xi_{\tau}+G_{\tau}-G_{\sigma}\big{]}$

where $G_{t}:=\int_{0}^{t}g_{s}\,dQ_{s}$. From the classical aggregation
theorems for stochastic systems due to Dellacherie and Lenglart [12] (see also
El Karoui [17], Kobylanski and Quenez [44] and the references therein), there
exists an RCLL, $\mathbb{F}$-adapted process $\overline{Y}$ such that the
equality $\overline{Y}_{\sigma}=\overline{Y}(\sigma)$ holds for every
$\sigma\in\mathcal{T}$. Furthermore, the process $V:=\overline{Y}+G$ is a
supermartingale of class (D) so it has the Doob-Meyer decomposition
$V=V_{0}+\overline{M}-D$ where $\overline{M}$ is a square-integrable
martingale and $D$ is a nondecreasing, RCLL, $\mathbb{F}$-predictable process
from ${\mathcal{A}}^{2}$. Let $D=D^{c}+D^{d}$ be the pathwise decomposition of
$D$ into its continuous and discontinuous components.

By Assumption 2.2, the $\mathbb{F}$-martingale $M$ has the predictable
representation property and thus there exists an $\mathbb{F}$-predictable
process $\overline{Z}\in{\mathcal{L}}^{2}(M)$ such that
$\overline{M}_{t}=\int_{0}^{t}\overline{Z}_{s}\,dM_{s}$. Moreover, the process
$\overline{Z}$ is unique in the pseudo-norm
$\|\cdot\|_{{\mathcal{L}}^{2}(M)}$. By an application of Proposition B.11 in
[44], we obtain the equality
$\int_{0}^{T}(\overline{Y}_{t}-\xi_{t})\,dD^{c}_{t}=0$. In addition, we have,
for every predictable stopping time $\tau\in\mathcal{T}_{p}$,

$\displaystyle\Delta D^{d}_{\tau}=\Delta
D^{d}_{\tau}\mathds{1}_{\\{\overline{Y}_{\tau-}=\widetilde{\xi}_{\tau-}\\}}.$

If we set $\overline{K}:=D^{c}+D^{d}$, then it is easy to check that
$(\overline{Y},\overline{Z},\overline{K})$ is a solution to the RBSDE (3.2).
We also note that the process $\overline{Y}$ is RCLL and, in view of
Proposition 3.1, the processes $\overline{Y}$ and $\overline{K}$ belong to
${\mathcal{S}}^{2}$. This completes the proof of the existence of a solution
$(Y,Z,K)$ to the RBSDE (3.3) with the desired properties.

To prove the uniqueness of a solution, we assume that
$(Y^{\prime},Z^{\prime},K^{\prime})$ is an arbitrary solution to the RBSDE
(3.2). We will show that $Y^{\prime}=\overline{Y}$, up to indistinguishability
of stochastic processes. Inspecting the RBSDE (3.3) with the fixed driver
$g_{t}$, we see that $Y^{\prime}+G$ is a supermartingale and thus, since
$Y^{\prime}\geq\xi$, we obtain, for every $\sigma\in\mathcal{T}$ and
$\tau\in\mathcal{T}_{[\sigma,T]}$,

$\displaystyle
Y^{\prime}_{\sigma}\geq{\mathbb{E}}_{\sigma}\big{[}Y^{\prime}_{\tau}+G_{\tau}-G_{\sigma}\big{]}\geq{\mathbb{E}}_{\sigma}\big{[}\xi_{\tau}+G_{\tau}-G_{\sigma}\big{]}.$

Since $\tau\in\mathcal{T}_{[\sigma,T]}$ is arbitrary, this in turn yields

$\displaystyle
Y^{\prime}_{\sigma}\geq\operatornamewithlimits{ess\,sup}_{\tau\in\mathcal{T}_{[\sigma,T]}}{\mathbb{E}}_{\sigma}\big{[}\xi_{\tau}+G_{\tau}-G_{\sigma}\big{]}=\overline{Y}_{\sigma}.$

We will now show that $Y^{\prime}_{\sigma}\leq\overline{Y}_{\sigma}$. To this
end, we fix $\varepsilon>0$ and we set
$\tau_{\sigma}^{\varepsilon}:=\inf\\{t\geq\sigma\,|\,Y^{\prime}_{t}\leq\xi_{t}+\varepsilon\\}$
so that $Y^{\prime}_{t}>\xi_{t}+\varepsilon$ on
$[\sigma,\tau_{\sigma}^{\varepsilon})$ and thus the process $K^{c}$ is
constant on $[\sigma,\tau_{\sigma}^{\varepsilon}]$. The process $K^{d}$ is
also manifestly constant on $[\sigma,\tau_{\sigma}^{\varepsilon})$ and, in
addition, $Y^{\prime}_{t}>\xi_{t}+\varepsilon$ on
$[\sigma,\tau_{\sigma}^{\varepsilon})$ gives

$\displaystyle
Y^{\prime}_{\tau_{\sigma}^{\varepsilon}-}\geq\xi_{\tau_{\sigma}^{\varepsilon}-}+\varepsilon>\xi_{\tau_{\sigma}^{\varepsilon}-},$

which implies that $\Delta K^{d}_{\tau_{\sigma}^{\varepsilon}}=0$ so that
$K^{d}$ is constant on $[\sigma,\tau_{\sigma}^{\varepsilon}]$. Therefore, the
process $Y^{\prime}+G$ is a martingale on the stochastic interval
$[\sigma,\tau_{\sigma}^{\varepsilon}]$. Furthermore, from the definition of
$\tau_{\sigma}^{\varepsilon}$ and the right-continuity of $Y^{\prime}$ and
$\widetilde{\xi}$, we have
$Y^{\prime}_{\tau_{\sigma}^{\varepsilon}}\leq\xi_{\tau_{\sigma}^{\varepsilon}}+\varepsilon$
and thus

$\displaystyle
Y^{\prime}_{\sigma}={\mathbb{E}}_{\sigma}\big{[}Y^{\prime}_{\tau_{\sigma}^{\varepsilon}}+G_{\tau_{\sigma}^{\varepsilon}}-G_{\sigma}\big{]}\leq{\mathbb{E}}_{\sigma}\big{[}\xi_{\tau_{\sigma}^{\varepsilon}}+G_{\tau_{\sigma}^{\varepsilon}}-G_{\sigma}\big{]}+\varepsilon\leq\overline{Y}_{\sigma}+\varepsilon.$

Since $\varepsilon>0$ was arbitrary, we conclude that
$Y^{\prime}_{\sigma}\leq\overline{Y}_{\sigma}$. The uniqueness of $Z$ and $K$
follows from the uniqueness of the Doob-Meyer decomposition and the postulated
uniqueness in the predictable representation property of $M$ (see Assumption
2.2). $\Box$

We now proceed to show the existence and uniqueness result for the solution of
the RBSDE (3.3). Recall that for a fixed $\beta\geq 0$, we denote by
$\mathbb{S}^{2}_{\beta}$ the Banach space
$({\mathcal{S}}^{2}_{\beta}\times{\mathcal{L}}^{2}(M),\|\,\cdot\,\|_{\beta})$
where the norm $\|\,\cdot\,\|_{\beta}$ is given by

$\displaystyle\|(w,v)\|^{2}_{\beta}:=\|w\|^{2}_{{\mathcal{S}}^{2}_{\beta}}+\|v\|^{2}_{{\mathcal{L}}^{2}_{\beta}(M)}.$

To set up the proof of Theorem 3.1, we need to define some mappings. We define
the mapping
$\Psi:\mathbb{S}^{2}_{\beta}\rightarrow\mathbb{S}^{2}_{{\mathcal{A}}}$ as
follows: for $(w,v)\in\mathbb{S}^{2}_{\beta}$ we set
$\Psi(w,v)=(Y^{w,v},Z^{w,v},K^{w,v})$ where
$(Y^{w,v},Z^{w,v},K^{w,v})\in\mathbb{S}^{2}_{{\mathcal{A}}}$ is the unique
solution of the RBSDE (3.3) associated with generator
$g_{t}(\omega):=g(\omega,t,w_{t}(\omega),v_{t}(\omega))$. In view of Lemma
3.2, the mapping $\Psi$ is well-defined. In addition, we define the projection
$\pi:\mathbb{S}^{2}_{{\mathcal{A}}}\rightarrow\mathbb{S}^{2}_{\beta}$, which
maps $(w,v,l)\in\mathbb{S}^{2}_{{\mathcal{A}}}$ to
$\pi(w,v,l):=(w,v)\in\mathbb{S}^{2}_{\beta}$. Finally, we define the mapping
$S:\mathbb{S}^{2}_{\beta}\rightarrow\mathbb{S}^{2}_{\beta}$ by
$S:=\pi\circ\Psi$. The following lemma shows that $S$ is a contraction on
$\mathbb{S}^{2}_{\beta}$, provided that $\beta$ is sufficiently large.

######  Lemma 3.3.

If Assumptions 2.1, 2.2 and 3.1 hold, then
$S:\mathbb{S}^{2}_{\beta}\rightarrow\mathbb{S}^{2}_{\beta}$ is a contraction
mapping for some $\beta\geq 0$ and thus $S$ has a unique fixed point, which is
denoted as $(\widehat{Y},\widehat{Z})$.

Proof. We consider two pairs $(w^{1},v^{1})$ and $(w^{2},v^{2})$ in
$\mathbb{S}^{2}_{\beta}$ and, for brevity, we denote their respective images
through $S$ by $(Y^{1},Z^{1})$ and $(Y^{2},Z^{2})$. Using the a priori
estimate from Proposition 3.3 and the fact that the generator $g$ is uniformly
$m$-Lipschitz continuous with a constant $\widehat{L}$, we obtain for
arbitrary $\beta\geq 0$ and $\gamma>0$

$\displaystyle\|Y\|^{2}_{{\mathcal{S}}^{2}_{\beta}}+\|Z\|^{2}_{{\mathcal{L}}^{2}_{\beta}(M)}\leq
1158\beta^{-1}\widehat{L}^{2}(C_{Q}+1)\big{(}\|w^{1}-w^{2}\|^{2}_{{\mathcal{S}}^{2}_{\beta}}+\|v^{1}-v^{2}\|^{2}_{{\mathcal{L}}^{2}_{\beta}(M)}\big{)}.$

If we set $\beta=2\times 1158\widehat{L}^{2}(C_{Q}+1)>0$ for $\beta\geq 0$,
then we obtain

$\displaystyle\|S(w^{1},v^{1})-S(w^{2},v^{2})\|_{\beta}^{2}=\|(Y^{1},Z^{1})-(Y^{2},Z^{2})\|_{\beta}^{2}\leq\frac{1}{2}\|(w^{1},v^{1})-(w^{2},v^{2})\|_{\beta}^{2}.$

By the Banach theorem, the mapping
$S:\mathbb{S}^{2}_{\beta}\rightarrow\mathbb{S}^{2}_{\beta}$ has a unique fixed
point $(\widehat{Y},\widehat{Z})$. $\Box$

The next result establishes the existence and uniqueness of a solution to the
RBSDE (3.3). Let the mapping
$T:\mathbb{S}^{2}_{{\mathcal{A}}}\rightarrow\mathbb{S}^{2}_{{\mathcal{A}}}$ be
defined by $T=\Psi\circ\pi$ so that $T(w,v,l)=(Y^{w,v},Z^{w,v},K^{w,v})$ for
all $(w,v,l)\in\mathbb{S}^{2}_{{\mathcal{A}}}$.

######  Theorem 3.1.

If Assumptions 2.1, 2.2 and 3.1 are satisfied, then the reflected BSDE (3.3)
has a unique solution $(Y,Z,K)$ in $\mathbb{H}^{2}_{{\mathcal{A}}}$. Moreover,
the triplet $(Y,Z,K)$ belongs to $\mathbb{S}^{2}_{{\mathcal{A}}}$ and the
process $K$ is in ${\mathcal{S}}^{2}$.

Proof. From the definitions of $S$ and $T$, it is clear that, for every
$(w,v,l)\in\mathbb{S}^{2}_{{\mathcal{A}}}$,

$\displaystyle T(w,v,l)=(S(w,v),K^{w,v}).$ (3.16)

From Lemma 3.3, we know that the mapping $S$ has a unique fixed point
$(\widehat{Y},\widehat{Z})\in\mathbb{S}^{2}_{\beta}$. We claim that
$(\widehat{Y},\widehat{Z},\widehat{K})$ where
$\widehat{K}:=K^{\widehat{Y},\widehat{Z}}\in{\mathcal{A}}^{2}$ is a unique
fixed point of $T$. The equality
$T(\widehat{Y},\widehat{Z},\widehat{K})=(\widehat{Y},\widehat{Z},\widehat{K})$
follows from the equality
$S(\widehat{Y},\widehat{Z})=(\widehat{Y},\widehat{Z})$ and (3.16) with
$l=\widehat{K}$. Furthermore, in view of Lemma 3.2, the process $\widehat{K}$
belongs to ${\mathcal{S}}^{2}$, since the triplet
$(\widehat{Y},\widehat{Z},\widehat{K})$ can also be interpreted as a unique
solution to the RBSDE with a fixed generator
$g_{t}=g(t,\widehat{Y},\widehat{Z})$.

To check that the fixed point of $T$ is unique, it suffices to recall that the
fixed point of $S$ is unique and use again (3.16). We have thus shown that the
RBSDE (3.3) has a unique solution $(\widehat{Y},\widehat{Z},\widehat{K})$
where $\widehat{K}=K^{\widehat{Y},\widehat{Z}}$. It is also easy to verify
that $(\widehat{Y},\widehat{Z},\widehat{K})$ is also a unique fixed point of
the mapping $T^{k}$ for every $k\geq 2$. To this end, we notice that the
equality $T^{k}(w,v,l)=(S^{k}(w,v),K^{S^{k-1}(w,v)})$ holds for every $k\geq
2$ and, by Lemma 3.3,
$S^{k}(\widehat{Y},\widehat{Z})=S^{k-1}(\widehat{Y},\widehat{Z})=(\widehat{Y},\widehat{Z})$.
The second assertion is now an immediate consequence of Proposition 3.1.
$\Box$

###  3.3 Picard’s Iterations for Solutions to Reflected BSDEs

In this section, we work under the assumptions of Theorem 3.1. Our next goal
is to construct Picard’s sequence and to show that it converges to the unique
solution $(Y,Z,K)\in\mathbb{S}^{2}_{{\mathcal{A}}}$ to the RBSDE (3.3). For
this purpose, we take the sequence $\\{(Y^{k},Z^{k})\\}_{k=1}^{\infty}$ where
$(Y^{k},Z^{k})=S^{k}(w,v)$ for some $(w,v)\in\mathbb{S}^{2}$ and we define the
sequence $\\{K^{k}\\}_{k=1}^{\infty}$ by postulating that, for every
$k\in\mathbb{N}$, the triplet
$(Y^{k},Z^{k},K^{k})\in\mathbb{S}^{2}_{{\mathcal{A}}}$ is the unique solution
to the RBSDE with generator $g^{k}_{t}:=g(t,Y_{t}^{k-1},Z_{t}^{k-1})$ (see
Lemma 3.2). Hence we have that, for all $k\in\mathbb{N}$,

$K^{k}_{t}=-\int_{0}^{t}g(u,Y_{u}^{k-1},Z_{u}^{k-1})\,dQ_{u}+\int_{0}^{t}Z^{k,\ast}_{u}\,dM_{u}-Y_{t}^{k}+Y_{0}^{k},$

$Y^{k}_{t}\geq\xi_{t}$ for all $t\in[0,T]$, and the continuous and
discontinuous components $K^{k,c}$ and $K^{k,d}$ of $K^{k}$ satisfy

$\displaystyle\int_{0}^{T}(Y^{k}_{t}-\xi_{t})\,dK_{t}^{k,c}=0,\quad\Delta
K_{\tau}^{k,d}=\Delta
K_{\tau}^{k,d}\mathds{1}_{\\{\widetilde{Y}^{k}_{\tau-}=\widetilde{\xi}_{\tau-}\\}}.$

From Lemma 3.3, we obtain the convergence
$\lim\limits_{n\rightarrow\infty}\|(Y^{k},Z^{k})-(Y,Z)\|_{\beta}^{2}=0$ where
$(Y,Z,K)$ is the unique solution to the RBSDE (3.3). The following result
establishes the convergence of Picard’s iterations
$(Y^{k},Z^{k},K^{k})_{k=1}^{\infty}$ to $(Y,Z,K)$ in the Banach space
$\mathbb{S}^{2}_{{\mathcal{A}}}$.

######  Proposition 3.4.

If Assumptions 2.1, 2.2 and 3.1 are valid, then the sequence
$(Y^{k},Z^{k},K^{k})_{k=1}^{\infty}$ converges in
$\mathbb{S}^{2}_{{\mathcal{A}}}$ to the unique solution $(Y,Z,K)$ of the
reflected BSDE (3.3), meaning that

$\displaystyle\lim_{k\rightarrow\infty}\|Y^{k}-Y\|_{{\mathcal{S}}^{2}}=0,\quad\lim_{k\rightarrow\infty}\|Z^{k}-Z\|_{{\mathcal{L}}^{2}(M)}=0,\quad\lim_{k\rightarrow\infty}\|K^{k}-K\|_{{\mathcal{S}}^{2}}=0.$
(3.17)

Proof. The properties
$\lim_{k\rightarrow\infty}\|Y^{k}-Y\|_{{\mathcal{S}}^{2}}=0$ and
$\lim_{k\rightarrow\infty}\|Z^{k}-Z\|_{{\mathcal{L}}^{2}(M)}=0$ are immediate
from the Banach fixed point theorem (see Lemma 3.3). It thus suffices to show
$\lim_{k\rightarrow\infty}\|K^{k}-K\|_{{\mathcal{S}}^{2}}=0$. If we denote
$\widetilde{Y}^{k}:=Y^{k}-Y$, $\widetilde{Z}^{k}:=Z^{k}-Z$ and
$\widetilde{K}^{k}:=K^{k}-K$, then we have

$\left\\{\begin{array}[c]{ll}d\widetilde{Y}^{k}_{t}=-\widetilde{g}^{k-1}_{t}\,dQ_{t}+\widetilde{Z}^{k,*}_{t}\,dM_{t}-d\widetilde{K}^{k}_{t},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ \widetilde{Y}^{k}_{T}=0,\end{array}\right.$

where
$\widetilde{g}^{k-1}_{t}:=g(t,Y^{k-1}_{t},Z^{k-1}_{t})-g(t,Y_{t},Z_{t})$. We
claim that that there exists a constant $C\geq 0$ such that

$\displaystyle\|\widetilde{Y}^{k}\|^{2}_{{\mathcal{S}}^{2}}+\|\widetilde{K}^{k}\|^{2}_{{\mathcal{S}}^{2}}\leq
CJ_{k}$ (3.18)

where

$\displaystyle J_{k}$
$\displaystyle:=\|\widetilde{Y}^{k-1}\|^{2}_{{\mathcal{H}}^{2}(Q)}+\|\widetilde{Y}^{k}\|^{2}_{{\mathcal{H}}^{2}(Q)}+\|\widetilde{Z}^{k-1}\|^{2}_{{\mathcal{L}}^{2}(M)}+\|\widetilde{Z}^{k}\|^{2}_{{\mathcal{L}}^{2}(M)}$

with $\widetilde{Y}^{k-1}:=Y^{k-1}-Y$ and $\widetilde{Z}^{k-1}:=Z^{k-1}-Z$.
Assuming that (3.18) holds, from the convergence
$\lim\limits_{k\rightarrow\infty}\|(Y^{k},Z^{k})-(Y,Z)\|_{\beta}^{2}=0$, we
deduce that $\lim_{k\rightarrow\infty}J_{k}=0$ and thus we see that (3.17) is
indeed a consequence of (3.18). It thus remains to show that (3.18) is
satisfied.

Step 1. We will first show that

$\displaystyle\|\widetilde{Y}^{k}\|^{2}_{{\mathcal{S}}^{2}}\leq CJ_{k}.$
(3.19)

The Itô formula gives

$\displaystyle
d|\widetilde{Y}^{k}_{t}|^{2}=2\widetilde{Y}^{k}_{t-}\,d\widetilde{Y}^{k}_{t}+d[\widetilde{Y}^{k}]_{t}$

and thus

$\displaystyle
d|\widetilde{Y}^{k}_{t}|^{2}=-2\widetilde{Y}^{k}_{t}\widetilde{g}^{k-1}_{t}\,dQ_{t}+2\widetilde{Y}^{k}_{t-}\widetilde{Z}^{k,*}_{t}\,dM_{t}-2\widetilde{Y}^{k}_{t-}\,d\widetilde{K}^{k}_{t}+d[\widetilde{Y}^{k}]_{t}$

where the process

$\displaystyle[\widetilde{Y}^{k}]_{t}=\sum_{0\leq u\leq
t}|\Delta\widetilde{Y}^{k}_{u}|^{2}+\langle\widetilde{Y}^{k,c}\rangle_{t}$

is nondecreasing. As in the proof of Proposition 3.2, we infer from the
Skorokhod conditions that
$\int_{t}^{T}\widetilde{Y}^{k}_{u-}\,d\widetilde{K}^{k}_{u}\leq 0$ for every
$t\in[0,T]$ and thus

$\displaystyle|\widetilde{Y}^{k}_{t}|^{2}\leq
2\int_{t}^{T}\widetilde{Y}^{k}_{u}\widetilde{g}^{k-1}_{u}\,dQ_{u}-2\int_{t}^{T}\widetilde{Y}^{k}_{u-}\widetilde{Z}^{k,*}_{u}\,dM_{u},$

which gives

$\sup_{t\in[0,T]}|\widetilde{Y}^{k}_{t}|^{2}\leq
2\,\int_{0}^{T}|\widetilde{Y}^{k}_{t}\widetilde{g}^{k-1}_{t}|\,dQ_{t}+2\sup_{t\in[0,T]}\,\Big{|}\int_{0}^{t}\widetilde{Y}^{k}_{u-}\,dM^{k}_{u}\Big{|}$
(3.20)

where the $\mathbb{F}$-martingale $M^{k}$ is given by
$M^{k}_{t}:=\int_{0}^{t}\widetilde{Z}^{k,\ast}_{u}\,dM_{u}$ so that $\langle
M^{k}\rangle_{t}=\int_{0}^{t}\|m_{u}\widetilde{Z}^{k}_{u}\|^{2}\,dQ_{u}$.
Using also the inequality
$|\widetilde{g}^{k-1}_{t}|\leq\widehat{L}\big{(}|\widetilde{Y}^{k-1}_{t}|+\|m_{t}\widetilde{Z}^{k-1}_{t}\|\big{)}$,
we deduce that there exists a positive constant $C$ (as usual, the value of
$C$ may vary from place to place in the remainder of the proof) such that, for
all $t\in[0,T]$,

$\displaystyle|\widetilde{Y}^{k}_{t}\widetilde{g}^{k-1}_{t}|\leq\widehat{L}|\widetilde{Y}^{k}_{t}|\big{(}|\widetilde{Y}^{k-1}_{t}|+\|m_{t}\widetilde{Z}^{k-1}_{t}\|\big{)}\leq
C\big{(}|\widetilde{Y}^{k-1}_{t}|^{2}+|\widetilde{Y}^{k}_{t}|^{2}+\|m_{t}\widetilde{Z}^{k-1}_{t}\|^{2}\big{)}$

and thus it is easy to see that

$\displaystyle{\mathbb{E}}\bigg{[}\int_{0}^{T}|\widetilde{Y}^{k}_{t}\widetilde{g}^{k-1}_{t}|\,dQ_{t}\bigg{]}\leq
CJ_{k}.$ (3.21)

Furthermore, an application of the Davis inequality gives

$\displaystyle{\mathbb{E}}\left[\sup_{t\in[0,T]}\Big{|}\int_{0}^{t}\widetilde{Y}^{k}_{u-}\,dM^{k}_{u}\Big{|}\right]\leq
2\sqrt{6}\,{\mathbb{E}}\left[\bigg{(}\int_{0}^{T}|\widetilde{Y}^{k}_{u-}|^{2}\,d[M^{k}]_{u}\bigg{)}^{1/2}\right]\leq
6\,{\mathbb{E}}\left[\bigg{(}\int_{0}^{T}|\widetilde{Y}^{k}_{u-}|^{2}\,d[M^{k}]_{u}\bigg{)}^{1/2}\right].$

Consequently, using also the elementary inequalities

$\displaystyle{\mathbb{E}}\left[\bigg{(}\int_{0}^{T}|\widetilde{Y}^{k}_{u-}|^{2}\,d[M^{k}]_{u}\bigg{)}^{1/2}\right]$
$\displaystyle\leq{\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}|\widetilde{Y}^{k}_{t}|\,[M^{k}]^{\frac{1}{2}}_{T}\bigg{]}\leq\frac{1}{16}\,{\mathbb{E}}\Big{[}\sup_{t\in[0,T]}|\widetilde{Y}^{k}_{t}|^{2}\Big{]}+4\,{\mathbb{E}}\big{(}[M^{k}]_{T}\big{)}$
$\displaystyle=\frac{1}{16}\,{\mathbb{E}}\Big{[}\sup_{t\in[0,T]}|\widetilde{Y}^{k}_{t}|^{2}\Big{]}+4\,{\mathbb{E}}\big{(}\langle
M^{k}\rangle_{T}\big{)},$

we obtain

$\displaystyle{\mathbb{E}}\left[\sup_{t\in[0,T]}\Big{|}\int_{0}^{t}\widetilde{Y}^{k}_{u-}\,dM^{k}_{u}\Big{|}\right]\leq\frac{3}{8}\,{\mathbb{E}}\Big{[}\sup_{t\in[0,T]}|\widetilde{Y}^{k}_{t}|^{2}\Big{]}+24\,{\mathbb{E}}\big{(}\langle
M^{k}\rangle_{T}\big{)}.$ (3.22)

It is now easy to deduce from (3.20), (3.21) and (3.22) that there exists a
constant $C$ such that

$\displaystyle\|\widetilde{Y}^{k}_{t}\|^{2}_{{\mathcal{S}}^{2}}={\mathbb{E}}\Big{[}\sup_{t\in[0,T]}|\widetilde{Y}^{k}_{t}|^{2}\Big{]}\leq
C\,{\mathbb{E}}\bigg{[}\int_{0}^{T}|\widetilde{Y}^{k}_{t}\widetilde{g}^{k-1}_{t}|\,dQ_{t}\bigg{]}+C\,{\mathbb{E}}\big{(}\langle
M^{k}\rangle_{T}\big{)}\leq CJ_{k}$

and thus (3.19) is valid.

Step 2. In the second step, we will show that

$\displaystyle\|\widetilde{K}^{k}\|^{2}_{{\mathcal{S}}^{2}}\leq CJ_{k}.$
(3.23)

Using the equality

$\displaystyle\widetilde{K}^{k}_{t}=\widetilde{Y}_{0}-\widetilde{Y}_{t}^{k}-\int_{0}^{t}\widetilde{g}^{k-1}_{u}\,dQ_{u}+\int_{0}^{t}\widetilde{Z}^{k,\ast}_{u}\,dM_{u}=\widetilde{Y}_{0}-\widetilde{Y}_{t}^{k}-\int_{0}^{t}\widetilde{g}^{k-1}_{u}\,dQ_{u}+M^{k}_{t},$

we obtain

$\displaystyle\sup_{t\in[0,T]}|\widetilde{K}^{k}_{t}|\leq
2\,\sup_{t\in[0,T]}|\widetilde{Y}^{k}_{t}|+\int_{0}^{T}|\widetilde{g}^{k-1}_{t}|\,dQ_{t}+\sup_{t\in[0,T]}|M^{k}_{t}|$

and thus

$\displaystyle\|\widetilde{K}^{k}\|_{{\mathcal{S}}^{2}}={\mathbb{E}}\Big{[}\sup_{t\in[0,T]}|\widetilde{K}^{k}_{t}|^{2}\Big{]}$
$\displaystyle\leq
C\,{\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}|\widetilde{Y}^{k}_{t}|^{2}+\Big{(}\int_{0}^{T}|\widetilde{g}^{k-1}_{t}|\,dQ_{t}\Big{)}^{2}+\sup_{t\in[0,T]}|M^{k}_{t}|^{2}\bigg{]}$
$\displaystyle\leq
C\,{\mathbb{E}}\bigg{[}J_{k}+\int_{0}^{T}|\widetilde{g}^{k-1}_{t}|^{2}\,dQ_{t}+\int_{0}^{T}\|m_{t}\widetilde{Z}^{k}_{t}\|^{2}\,dQ_{t}\bigg{]}$

where we used (3.19), the boundedness of $Q$ and we observed that the
Burkholder-Davis-Gundy inequality with $p=2$ (see, e.g., Theorem 10.36 in He
et al. [33] or Theorem 11.5.5 in Cohen and Elliott [9]) yields (one may take
here $C=4$)

$\displaystyle\|M^{k}\|^{2}_{{\mathcal{S}}^{2}}={\mathbb{E}}\Big{[}\,\sup_{t\in[0,T]}\big{|}M^{k}_{t}\big{|}^{2}\Big{]}\leq
C\,{\mathbb{E}}([M^{k}]_{T})=C\,{\mathbb{E}}(\langle
M^{k}\rangle_{T})=C\,{\mathbb{E}}\bigg{[}\int_{0}^{T}\|m_{t}\widetilde{Z}^{k}_{t}\|^{2}\,dQ_{t}\bigg{]}=\|\widetilde{Z}^{k}\|^{2}_{{\mathcal{L}}^{2}(M)}.$

Using the inequality $|\widetilde{g}^{k-1}_{t}|^{2}\leq
C\big{(}|\widetilde{Y}^{k-1}_{t}|^{2}+\|m_{t}\widetilde{Z}^{k-1}_{t}\|^{2}\big{)}$,
we obtain (3.23). To conclude that inequality (3.18) is valid, it suffices to
combine (3.19) with (3.23). $\Box$

##  4 Doubly Reflected BSDEs with RCLL Martingales

The goal of this section is to study the doubly reflected backward stochastic
differential equation (DRBSDE) on $[0,T]$ with data $(g,\eta,D,\xi,\zeta)$

$\left\\{\begin{array}[c]{ll}dY_{t}=-g(t,Y_{t},Z_{t})\,dQ_{t}+Z^{*}_{t}\,dM_{t}+dD_{t}-dL_{t}+dU_{t},\
Y_{T}=\eta,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\ \xi_{t}\leq
Y_{t}\leq\zeta_{t},\ \forall\,t\in[0,T],\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\int_{0}^{T}(Y_{t}-\xi_{t})\,dL^{c}_{t}=\int_{0}^{T}(\zeta_{t}-Y_{t})\,dU^{c}_{t}=0,\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ \Delta L^{d}_{\tau}=\Delta
L^{d}_{\tau}\mathds{1}_{\\{(Y-D)_{\tau-}=(\xi-D)_{\tau-}\\}}\ \mbox{\rm and}\
\Delta U^{d}_{\tau}=\Delta
U^{d}_{\tau}\mathds{1}_{\\{(Y-D)_{\tau-}=(\zeta-D)_{\tau-}\\}},\
\forall\,\tau\in\mathcal{T}_{p},\end{array}\right.$ (4.1)

where the data $(g,\eta,D,\xi,\zeta)$ satisfies the following assumption

######  Assumption 4.1.

The quintuplet $(g,\eta,D,\xi,\zeta)$ is such that: (i) the generator $g$ is
uniformly $m$-Lipschitz continuous and the process $g(\cdot,0,0)$ belongs to
${\mathcal{H}}^{2}(Q)$, (ii) the process $D$ belongs to ${\mathcal{H}}^{2}(Q)$
and the random variable $\eta-D_{T}$ belongs to $L^{2}({\mathcal{F}}_{T})$,
(iii) the processes $\xi-D$ and $\zeta-D$ belong to ${\mathcal{S}}^{2}$ and
satisfy $\xi_{t}-D_{t}\leq\zeta_{t}-D_{t}$ for all $t\in[0,T]$; in addition,
$\xi_{T}\leq\eta\leq\zeta_{T}$.

In (4.1), $\mathcal{T}_{p}$ denotes all $\mathbb{F}$-predictable stopping
times taking values in $[0,T]$, $L$ and $U$ are non-decreasing, RCLL,
$\mathbb{F}$-predictable processes with $L_{0}=U_{0}=0$ and such that the
measures generated by $L$ and $U$ satisfy $dL_{t}\perp dU_{t}$ where the
notation $dL_{t}\perp dU_{t}$ means that the measures generated by
nondecreasing processes $L$ and $U$ are singular, in the sense that there
exists an $\mathbb{F}$-predictable set $F\subset\Omega\times[0,T]$ such that

$\displaystyle\int_{0}^{T}\mathds{1}_{F(t)}\,dL_{t}=\int_{0}^{T}\mathds{1}_{F^{c}(t)}\,dU_{t}=0$

where $F(t):=\\{\omega\in\Omega\,|\,(\omega,t)\in F\\}$ and $F^{c}(t)$ is its
complement. Note that here the equality $L=L^{c}+L^{d}$ (respectively,
$U=U^{c}+U^{d}$) gives the pathwise decomposition of $L$ (respectively, $U$)
into its continuous and jump components. The last two rows in (4.1) are
collectively referred to as the Skorokhod (or the minimality) conditions for
DRBSDEs.

To examine solutions to the RDBSDE (4.1), we introduce the spaces of
stochastic processes
$\widetilde{\mathbb{H}}^{2}_{{\mathcal{A}}}=\mathbb{H}^{2}_{{\mathcal{A}}}\times{\mathcal{A}}^{2}$
and
$\widetilde{\mathbb{S}}^{2}_{{\mathcal{A}}}=\mathbb{S}^{2}_{{\mathcal{A}}}\times{\mathcal{A}}^{2}$
(recall that the spaces
$\mathbb{H}^{2}_{{\mathcal{A}}},\mathbb{S}^{2}_{{\mathcal{A}}},{\mathcal{A}}^{2}$
are introduced in Section 3).

######  Definition 4.1.

A solution to the DRBSDE (4.1) with data $(g,\eta,D,\xi,\zeta)$ is a
quadruplet of stochastic processes
$(Y,Z,L,U)\in\widetilde{\mathbb{H}}^{2}_{{\mathcal{A}}}$ satisfying (4.1), in
the sense that:

$\displaystyle\mathbb{P}\bigg{(}Y_{t}=\eta+\int_{t}^{T}g(t,Y_{u},Z_{u})\,dQ_{u}-\int_{t}^{T}Z_{u}^{\ast}\,dM_{u}-(D_{T}-D_{t})+(L_{T}-L_{t})-(U_{T}-U_{t}),\
\forall\,t\in[0,T]\bigg{)}=1,$

$\mathbb{P}(\xi_{t}\leq Y_{t}\leq\zeta_{t},\,\forall\,t\in[0,T])=1$ and the
Skorokhod (minimality) conditions stated in (4.1) hold. We say that the
uniqueness of a solution to (4.1) holds if for any two solutions $(Y,Z,L,U)$
and $(Y^{\prime},Z^{\prime},L^{\prime},U^{\prime})$ to (4.1) we have
$\|(Y,Z,L,U)-(Y^{\prime},Z^{\prime},L^{\prime},U^{\prime})\|_{\widetilde{\mathbb{H}}^{2}_{{\mathcal{A}}}}=0$.

It can be observed that, without loss of generality, it suffices to consider
the modified obstacles

$\xi^{\prime}_{t}:=\xi_{t}\mathds{1}_{\\{t<T\\}}+\eta\mathds{1}_{\\{t=T\\}},\quad\zeta^{\prime}_{t}:=\zeta_{t}\mathds{1}_{\\{t<T\\}}+\eta\mathds{1}_{\\{t=T\\}},$

and thus the random variable $\eta$ can be omitted, provided that we postulate
that $\xi_{T}=\zeta_{T}$ in (4.1).

As in preceding sections, we make use of the following transformations:
$\widetilde{Y}:=Y-D$, $\widetilde{\xi}=\xi^{\prime}-D$,
$\widetilde{\zeta}=\zeta^{\prime}-D$ and
$\widetilde{g}(t,y,z):=g(t,y+D_{t},z)$ and we set
$\widetilde{Z}=Z,\,\widetilde{L}=L,\,\widetilde{U}=U$. Therefore, we
henceforth study a solution
$(\widetilde{Y},\widetilde{Z},\widetilde{L},\widetilde{U})\in\widetilde{\mathbb{S}}^{2}_{{\mathcal{A}}}$
to the transformed DRBSDE with data
$(\widetilde{g},\widetilde{\xi}_{T},0,\widetilde{\xi},\widetilde{\zeta})$

$\left\\{\begin{array}[c]{ll}d\widetilde{Y}_{t}=-\widetilde{g}(t,\widetilde{Y}_{t},\widetilde{Z}_{t})\,dQ_{t}+\widetilde{Z}^{*}_{t}\,dM_{t}-d\widetilde{L}_{t}+d\widetilde{U}_{t},\
\widetilde{Y}_{T}=\widetilde{\eta}_{T},\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\widetilde{\xi}_{t}\leq\widetilde{Y}_{t}\leq\widetilde{\zeta}_{t},\
\forall\,t\in[0,T],\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\int_{0}^{T}(\widetilde{Y}_{t}-\widetilde{\xi}_{t})\,dL^{c}_{t}=\int_{0}^{T}(\widetilde{\zeta}_{t}-\widetilde{Y}_{t})\,d\widetilde{U}^{c}_{t}=0,\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\Delta\widetilde{L}^{d}_{\tau}=\Delta\widetilde{L}^{d}_{\tau}\mathds{1}_{\\{\widetilde{Y}_{\tau-}=\widetilde{\xi}_{\tau-}\\}}\
\mbox{\rm and}\
\Delta\widetilde{U}^{d}_{\tau}=\Delta\widetilde{U}^{d}_{\tau}\mathds{1}_{\\{\widetilde{Y}_{\tau-}=\widetilde{\zeta}_{\tau-}\\}},\
\forall\,\tau\in\mathcal{T}_{p},\end{array}\right.$ (4.2)

and we argue that DRBSDEs (4.1) and (4.2) are equivalent under Assumption 4.1.
Notice that, in view of inequalities
$\widetilde{\xi}\leq\widetilde{Y}\leq\widetilde{\zeta}$ and Assumption
4.1(iii), it is readily seen that the process $\widetilde{Y}$ necessarily
belongs to the space ${\mathcal{S}}^{2}$.

###  4.1 A Priori Estimates for Doubly Reflected BSDEs

To address the case of a uniformly $m$-Lipschitz continuous generator $g$, we
will prove some auxiliary result furnishing the a priori estimates for
solutions to (4.2). As we argued before, we may and do assume, without loss of
generality, that $D=0$ and $\xi_{T}=\zeta_{T}=\eta$ and thus we henceforth
examine the DRBSDE with data $(g,\xi_{T},0,\xi,\zeta)$.

######  Proposition 4.1.

For $i=1,2$, let $(Y^{i},Z^{i},L^{i},U^{i})$ be a solution to the doubly
reflected BSDE with data $(g^{i},\xi^{i}_{T},0,\xi^{i},\zeta^{i})$ and a
uniformly $m$-Lipschitz continuous generator $g^{i}$. If we denote
$y:=Y^{1}-Y^{2}$, $z:=Z^{1}-Z^{2}$ and
$\overline{g}_{t}(\omega):=g^{1}(\omega,t,Y^{2}_{t},Z^{2}_{t})-g^{2}(\omega,t,Y^{2}_{t},Z^{2}_{t})$,
then for any constant $\gamma>0$

$\displaystyle(\beta-2\widehat{L}_{1}-\gamma^{-1})\|y\|^{2}_{{\mathcal{H}}^{2}_{\beta}(Q)}+(1-2\widehat{L}_{1}^{2}\gamma)\|z\|^{2}_{{\mathcal{L}}^{2}_{\beta}(M)}\leq
2\gamma\|\overline{g}\|^{2}_{{\mathcal{H}}^{2}_{\beta}(Q)}$

where $\widehat{L}_{1}$ is the Lipschitz constant associated with generator
$g^{1}$.

Proof. By taking the difference of the two DRBSDEs, we obtain

$\left\\{\begin{array}[c]{ll}y_{t}=-\delta_{t}\,dQ_{t}+z^{*}_{t}\,dM_{t}-(dL^{1}_{t}-dL^{2}_{t})+(dU^{1}_{t}-dU^{2}_{t}),\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ y_{T}=0,\end{array}\right.$

where $\delta_{t}:=g^{1}(t,Y^{1}_{t},Z^{1}_{t})-g^{2}(t,Y^{2}_{t},Z^{2}_{t})$.
As in the proof of Proposition 3.2, we obtain

$\begin{split}&e^{\beta
Q_{t}}y_{t}^{2}+\beta{\mathbb{E}}_{t}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}y^{2}_{u}\,dQ_{u}\bigg{]}+{\mathbb{E}}_{t}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}\|m_{u}z_{u}\|^{2}\,dQ_{u}\bigg{]}\\\
&=2\,{\mathbb{E}}_{t}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}y_{u}\delta_{u}\,dQ_{u}\bigg{]}+2{\mathbb{E}}_{t}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}y_{u-}\,dL^{1}_{u}-\int_{t}^{T}e^{\beta
Q_{u}}y_{u-}\,dL^{2}_{u}\bigg{]}\\\
&-2\,{\mathbb{E}}_{t}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}y_{u-}\,dU^{1}_{u}-\int_{t}^{T}e^{\beta
Q_{u}}y_{u-}\,dU^{2}_{u}\bigg{]}.\end{split}$

On the one hand, using the Skorokhod conditions, we observe that

$\displaystyle
y_{u-}\,dL^{1,c}_{u}=y_{u}\,dL^{1,c}_{u}=\big{[}(Y^{1}_{u}-\xi_{u})-(Y^{2}_{u}-\xi_{u})\big{]}\,dL^{1,c}_{u}=-(Y^{2}_{u}-\xi_{u})\,dL^{1,c}_{u}\leq
0$

and

$\displaystyle y_{u-}\,\Delta
L^{1,d}_{u}=\big{[}(Y^{1}_{u-}-\xi_{u-})-(Y^{2}_{u-}-\xi_{u-})\big{]}\,\Delta
L^{1,d}_{u}\leq 0$

so that $y_{u-}\,dL^{1}_{u}\leq 0$. Analogous arguments show that
$y_{u}\,dL^{2,c}_{u}\geq 0$ and $y_{u-}\,\Delta L^{2,d}_{u}\geq 0$ and thus
the inequality $-y_{u-}\,dL^{2}_{u}\leq 0$ is valid as well. On the other
hand, we observe that

$\displaystyle
y_{u-}\,dU^{1,c}_{u}=y_{u}\,dU^{1,c}_{u}=\big{[}(Y^{1}_{u}-\zeta_{u})-(Y^{2}_{u}-\zeta_{u})\big{]}\,dU^{1,c}_{u}=-(Y^{2}_{u}-\zeta_{u})\,dU^{1,c}_{u}\geq
0$

and

$\displaystyle y_{u-}\,\Delta
U^{1,d}_{u}=\big{[}(Y^{1}_{u-}-\zeta_{u-})-(Y^{2}_{u-}-\zeta_{u-})\big{]}\,\Delta
U^{1,d}_{u}\geq 0$

so that $y_{u-}\,dU^{1}_{u}\geq 0$. Similarly, one can show that
$y_{u}\,dU^{2,c}_{u}\leq 0$ and $y_{u-}\,\Delta U^{2,d}_{u}\leq 0$ and thus we
conclude that $-y_{u-}\,dU^{2}_{u}\geq 0$. Therefore, we obtain

$e^{\beta Q_{t}}y_{t}^{2}+\beta{\mathbb{E}}_{t}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}y^{2}_{u}\,dQ_{u}\bigg{]}+{\mathbb{E}}_{t}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}\|m_{u}z_{u}\|^{2}\,dQ_{u}\bigg{]}\leq
2\,{\mathbb{E}}_{t}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}y_{u}\delta_{u}\,dQ_{u}\bigg{]}.$

Then, arguing as in the proof of Proposition 3.2, we obtain the desired
inequality. $\Box$

The next result is a counterpart of Proposition 3.3.

######  Proposition 4.2.

Let Assumptions 2.1, 2.2 and 4.1(ii)-(iii) be valid. For $l=1,2$, assume that
$(Y^{l},Z^{l},K^{l})$ is a solution to the doubly reflected BSDE (4.1) with
generator $g^{l}_{t}=g(t,w^{l}_{t},v^{l}_{t})$. If the mapping $g$ is
uniformly $m$-Lipschitz continuous with the constant $\widehat{L}>0$, then for
any $\beta>0$

$\displaystyle\|y\|^{2}_{{\mathcal{S}}^{2}_{\beta}}+\|z\|^{2}_{{\mathcal{L}}^{2}_{\beta}(M)}\leq
1158\beta^{-1}\widehat{L}^{2}(C_{Q}+1)\big{(}\|w\|_{{\mathcal{S}}^{2}_{\beta}}^{2}+\|v\|^{2}_{{\mathcal{L}}^{2}_{\beta}(M)}\big{)}$
(4.3)

where $y=Y^{1}-Y^{2},\,z=Z^{1}-Z^{2},\,w=w^{1}-w^{2}$ and $v=v^{1}-v^{2}$.

Proof. By taking the difference of the two DRBSDEs, we obtain

$\left\\{\begin{array}[c]{ll}y_{t}=-\delta_{t}\,dQ_{t}+z^{*}_{t}\,dM_{t}-(dL^{1}_{t}-dL^{2}_{t})+(dU^{1}_{t}-dU^{2}_{t}),\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ y_{T}=0,\end{array}\right.$

where $\delta_{t}:=g^{1}(t,w^{1}_{t},v^{1}_{t})-g^{2}(t,w^{2}_{t},v^{2}_{t})$.
As in the proof of Proposition 3.3, the Itô integration by parts formula gives

$\begin{split}d(e^{\beta Q_{t}}y^{2}_{t})&=e^{\beta Q_{t}}\big{(}\beta
y^{2}_{t}\,dQ_{t}-2y_{t-}\delta_{t}\,dQ_{t}+2y_{t-}z^{*}_{t}\,dM_{t}+dN_{t}+d\langle
y\rangle_{t}\big{)}\\\ &\qquad-2y_{t-}e^{\beta
Q_{t}}\,(dL^{1}_{t}-dL^{2}_{t}-dU^{1}_{t}+dU^{2}_{t})\\\ \end{split}$ (4.4)

where the process $N_{t}:=[y]_{t}-\langle y\rangle_{t}$ is an
$\mathbb{F}$-martingale. By integrating from $t$ to $T$, taking the
expectation, using the property that $\int_{t}^{T}e^{\beta
Q_{u}}y_{u-}\,(dL^{1}_{u}-dL^{2}_{u}-dU^{1}_{u}+dU^{2}_{u})\leq 0$, we obtain

$\begin{split}&{\mathbb{E}}\bigg{[}e^{\beta
Q_{t}}y^{2}_{t}+\beta\,\int_{t}^{T}e^{\beta
Q_{u}}y^{2}_{u}\,dQ_{u}+\int_{t}^{T}e^{\beta
Q_{u}}\,\|m_{u}z_{u}\|^{2}\,dQ_{u}\bigg{]}\leq
2{\mathbb{E}}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}y_{u}\delta_{u}\,dQ_{u}\bigg{]}\\\ &\leq
2\widehat{L}\,{\mathbb{E}}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}|y_{u}|(|w_{u}|+\|m_{u}v_{u}\|)\,dQ_{u}\bigg{]}\\\ &\leq
2\lambda^{-1}\widehat{L}^{2}\,{\mathbb{E}}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}y^{2}_{u}\,dQ_{u}\bigg{]}+\lambda\,{\mathbb{E}}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}w^{2}_{u}\,dQ_{u}\bigg{]}+\lambda\,{\mathbb{E}}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}\|m_{u}v_{u}\|^{2}\,dQ_{u}\bigg{]}.\end{split}$

By taking $\lambda=2\beta^{-1}\widehat{L}^{2}$ and recalling that the process
$Q$ is bounded by a constant $C_{Q}$, we get

$\begin{split}&{\mathbb{E}}\bigg{[}e^{\beta
Q_{t}}y^{2}_{t}+\int_{t}^{T}e^{\beta
Q_{u}}\,\|m_{u}z_{u}\|^{2}\,dQ_{u}\bigg{]}\leq
2\,{\mathbb{E}}\,\int_{t}^{T}e^{\beta Q_{u}}y_{u}\delta_{u}\,dQ_{u}\\\
&\leq\lambda\,{\mathbb{E}}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}|w_{u}|^{2}\,dQ_{u}\bigg{]}+\lambda\,{\mathbb{E}}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}\|m_{u}v_{u}\|^{2}\,dQ_{u}\bigg{]}\\\
&\leq\lambda(C_{Q}+1)\big{(}\|w\|_{{\mathcal{S}}^{2}_{\beta}}^{2}+\|v\|^{2}_{{\mathcal{L}}^{2}_{\beta}(M)}\big{)}.\end{split}$
(4.5)

Using (4.4) and noting that $[y]$ is nondecreasing and $\int_{t}^{T}e^{\beta
Q_{u}}y_{u-}\,(dL^{1}_{u}-dL^{2}_{u}-dU^{1}_{u}+dU^{2}_{u})\leq 0$, we obtain

$\begin{split}&e^{\beta Q_{t}}y^{2}_{t}+\beta\,\int_{t}^{T}e^{\beta
Q_{u}}y^{2}_{u}\,dQ_{u}\leq 2\,\int_{t}^{T}e^{\beta
Q_{u}}y_{u}\delta_{u}\,dQ_{u}-2\int_{t}^{T}e^{\beta
Q_{u}}y_{u-}z^{*}_{u}\,dM_{u}\\\ &\leq
2\lambda^{-1}\widehat{L}^{2}\int_{t}^{T}e^{\beta
Q_{u}}y_{u}^{2}\,dQ_{u}+\lambda\int_{t}^{T}e^{\beta
Q_{u}}|w_{u}|^{2}\,dQ_{u}+\lambda\int_{t}^{T}e^{\beta
Q_{u}}\|m_{u}v_{u}\|^{2}\,dQ_{u}-2\int_{t}^{T}e^{\beta
Q_{u}}y_{u-}z^{*}_{u}\,dM_{u},\end{split}$

Arguing as in the proof of Proposition 3.3 and using (4.5) and the Davis
inequality, we can show that (4.3) is valid. $\Box$

###  4.2 Existence and Uniqueness Theorem for Doubly Reflected BSDEs

Before stating the next result, we need to introduce some notations. For any
process $g\in{\mathcal{H}}^{2}(Q)$, we define

$\widetilde{\xi}^{g}_{t}:=\widetilde{\xi}_{t}-{\mathbb{E}}\bigg{[}\widetilde{\xi}_{T}+\int_{t}^{T}g_{u}\,dQ_{u}\,\Big{|}\,\mathcal{F}_{t}\bigg{]},\quad\widetilde{\zeta}^{g}_{t}:=\widetilde{\zeta}_{t}-{\mathbb{E}}\bigg{[}\widetilde{\zeta}_{T}+\int_{t}^{T}g_{u}\,dQ_{u}\,\Big{|}\,\mathcal{F}_{t}\bigg{]},$
(4.6)

so that $\widetilde{\xi}^{g}_{T}=\widetilde{\zeta}^{g}_{T}=0$. If
$\widetilde{\zeta}$ and $\widetilde{\xi}$ are in ${\mathcal{S}}^{2}$, then the
processes $\widetilde{\xi}^{g}$ and $\widetilde{\zeta}^{g}$ belong to
${\mathcal{S}}^{2}$ as well. We define the sequences
$\\{I^{g,k}\\}_{k=0}^{\infty}$ and $\\{J^{g,k}\\}_{k=0}^{\infty}$ by setting
$I^{g,0}=J^{g,0}=0$ and, by recursion, for every stopping time
$\sigma\in\mathcal{T}$,

$\displaystyle
I^{g,k+1}_{\sigma}:=\operatornamewithlimits{ess\,sup}_{\tau\in\mathcal{T}_{[\sigma,T]}}{\mathbb{E}}\big{[}J^{g,k}_{\tau}+\widetilde{\xi}^{g}_{\tau}\,|\,\mathcal{F}_{\sigma}\big{]},\quad
J^{g,k+1}_{\sigma}:=\operatornamewithlimits{ess\,sup}_{\tau\in\mathcal{T}_{[\sigma,T]}}{\mathbb{E}}\big{[}I^{g,k}_{\tau}-\widetilde{\zeta}^{g}_{\tau}\,|\,\mathcal{F}_{\sigma}\big{]}$

where $I^{g,k+1}$ and $J^{g,k+1}$ are taken to be RCLL. The following lemma
was proven in Dumitrescu et al. [14] in the special case where $Q_{t}=t$. The
following minor extension of Lemma 3.3 in [14] can be established by applying
to the processes $\widetilde{\xi}^{g}_{t}$ and $\widetilde{\zeta}^{g}_{t}$
given by (4.6) the method of the proof of Lemma 3.3 in [14].

######  Lemma 4.1.

The sequences of processes $\\{I^{g,k}\\}_{k=0}^{\infty}$ and
$\\{J^{g,k}\\}_{k=0}^{\infty}$ are non-decreasing. Moreover, the processes
$I^{g}$ and $J^{g}$, which are defined by, for all $t\in[0,T]$,

$\displaystyle I^{g}_{t}:=\lim_{k\rightarrow\infty}I^{g,k},\quad
J^{g}_{t}:=\lim_{k\rightarrow\infty}J^{g,k}$

are $\mathbb{R}\cup\\{+\infty\\}$-valued, strong supermartingales with
$I^{g}_{T}=J^{g}_{T}=0$ and, for every $\sigma\in\mathcal{T}$,

$I^{g}_{\sigma}=\operatornamewithlimits{ess\,sup}_{\tau\in\mathcal{T}_{[\sigma,T]}}{\mathbb{E}}\big{[}J^{g}_{\tau}+\widetilde{\xi}^{g}_{\tau}\,|\,\mathcal{F}_{\sigma}\big{]},\quad
J^{g}_{\sigma}=\operatornamewithlimits{ess\,sup}_{\tau\in\mathcal{T}_{[\sigma,T]}}{\mathbb{E}}\big{[}I^{g}_{\tau}-\widetilde{\zeta}^{g}_{\tau}\,|\,\mathcal{F}_{\sigma}\big{]}.$
(4.7)

If $I^{g}_{0}<+\infty$ and $J^{g}_{0}<+\infty$, then $I^{g}$ and $J^{g}$ are
real-valued, RCLL supermartingales.

The proof of Lemma 3.3 in [14] uses also some results on a general optimal
stopping problem, which was studied, in particular, by Maingueneau [50], El
Karoui [17], Kobylanski and Quenez [44] and Kobylanski et al. [46]. We observe
that some of their arguments can be applied to our setup. Let $\varphi$ be an
RCLL process of class (D) and let the RCLL $\mathbb{F}$-supermartingale $v$ be
obtained through the aggregation of the supermartingale family
$\bar{v}:\mathcal{T}\to\mathbb{R}$, which is given by

$\displaystyle\bar{v}(\sigma):=\operatornamewithlimits{ess\,sup}_{\tau\in\mathcal{T}_{[\sigma,T]}}{\mathbb{E}}[\varphi_{\tau}\,|\,\mathcal{F}_{\sigma}].$

Let $B$ be the $\mathbb{F}$-predictable, non-decreasing process in the Doob-
Meyer decomposition of $v$. If $B=B^{c}+B^{d}$ is the pathwise decomposition
of $B$ into its continuous and jump components, then it can be shown that
(see, for instance, Proposition B11 in [44])

$\displaystyle\int_{0}^{T}\mathds{1}_{\\{v_{t}>\varphi_{t}\\}}\,dB^{c}_{t}=0\
\ \mbox{\rm and}\ \ \Delta B^{d}_{t}=\Delta
B^{d}_{t}\mathds{1}_{\\{v_{t-}=\varphi_{t-}\\}}.$ (4.8)

Moreover, if the filtration $\mathbb{F}$ is quasi-left-continuous, then
$\Delta v_{\tau}=\Delta
B^{d}_{\tau}\mathds{1}_{\\{v_{\tau-}=\varphi_{\tau-}\\}}$ for any jump time
$\tau\in\mathcal{T}_{p}$ of $v$.

We first consider the case where the generator $g$ in (4.2) is independent of
$(y,z)$ and the process $g:\Omega\times[0,T]\rightarrow\mathbb{R}$ belongs to
${\mathcal{H}}^{2}(Q)$. To show the existence and uniqueness of a solution to
the RBSDE (4.2) under the assumption that $I^{g}$ and $J^{g}$ belong to
${\mathcal{S}}^{2}$, we may directly apply the method of the proof of Theorem
3.5 in Dumitrescu et al. [14] and thus the proof of Lemma 4.2 is omitted.

######  Lemma 4.2.

Let Assumptions 2.1–2.2 and 4.1 be satisfied and the generator $g$ do not
depend on $(y,z)$. Suppose that the processes $I^{g}$ and $J^{g}$ belong to
${\mathcal{S}}^{2}$ and let the RCLL process $\overline{Y}$ be given by, for
all $t\in[0,T]$,

$\displaystyle\overline{Y}_{t}:=I^{g}_{t}-J^{g}_{t}+{\mathbb{E}}\bigg{[}\xi_{T}+\int_{t}^{T}g_{u}\,dQ_{u}\,\Big{|}\,\mathcal{F}_{t}\bigg{]}.$
(4.9)

Then there exists a unique solution
$(\overline{Y},Z,U,L)\in\widetilde{\mathbb{S}}^{2}_{{\mathcal{A}}}$ to the
doubly reflected BSDE (4.2).

Lemma 4.2 is not completely satisfactory, since it requires that the processes
$I^{g}$ and $J^{g}$ belong to ${\mathcal{S}}^{2}$. However, it is known that
the following condition, which was used in several papers on the theory of
Dynkin games and DRBSDEs, ensures the existence of the value of a Dynkin game
and a solution to a DRBSDE.

######  Assumption 4.2.

Assume that the processes $\widetilde{\xi}$ and $\widetilde{\zeta}$ belong to
${\mathcal{S}}^{2}$, the inequality
$\widetilde{\xi}_{t}\leq\widetilde{\zeta}_{t}$ holds for all $t\in[0,T]$ and
$\widetilde{\xi}_{T}=\widetilde{\zeta}_{T}$. Let the _Mokobodzki condition_
holds, that is, there exist two nonnegative supermartingales $H$ and
$H^{\prime}$ in ${\mathcal{S}}^{2}$ such that $\widetilde{\xi}_{t}\leq
H_{t}-H^{\prime}_{t}\leq\widetilde{\zeta}_{t}$ for all $t\in[0,T]$.

It was shown in Proposition 3.10 in Dumitrescu et al. [14] that if
$g\in{\mathcal{H}}^{2}(Q)$, then the Mokobodzki condition holds if and only if
$I^{g}\in{\mathcal{S}}^{2}$ (or, equivalently, $J^{g}\in{\mathcal{S}}^{2}$).
Hence if Assumptions 2.1–2.2 are complemented by Assumption 4.2, then the
existence of a unique solution to the DRBSDE (4.2) with a fixed generator
$g\in{\mathcal{H}}^{2}(Q)$ is ensured by Lemma 4.2. We may thus state without
proof the following result.

######  Proposition 4.3.

Let Assumptions 2.1–2.2 and 4.1–4.2 be satisfied. If the generator $g$ does
not depend on $(y,z)$, then there exists a unique solution
$(\widetilde{Y},Z,U,L)\in\widetilde{\mathbb{S}}^{2}_{{\mathcal{A}}}$ to the
doubly reflected BSDE (4.2).

We are now ready to demonstrate the existence and uniqueness of a solution to
the DRBSDE (4.1). For a fixed $\beta\geq 0$, we denote by
$\mathbb{S}^{2}_{\beta}$ the Banach space
$({\mathcal{S}}^{2}_{\beta}\times{\mathcal{L}}^{2}_{\beta}(M),\|\cdot\|_{\beta})$
with the norm

$\displaystyle\|(w,v)\|^{2}_{\mathbb{S}^{2}_{\beta}}:=\|w\|^{2}_{{\mathcal{S}}^{2}_{\beta}}+\|v\|^{2}_{{\mathcal{L}}^{2}_{\beta}(M)}.$

We define the mapping
$\Psi:\mathbb{S}^{2}_{\beta}\rightarrow\widetilde{\mathbb{S}}^{2}_{{\mathcal{A}}}$
as follows: for an arbitrary $(w,v)\in\mathbb{S}^{2}_{\beta}$ we set
$\Psi(w,v):=(Y^{w,v},Z^{w,v},L^{w,v},U^{w,v})$ where
$(Y^{w,v},Z^{w,v},L^{w,v},U^{w,v})$ is the unique solution to the DRBSDE (4.2)
with generator $g_{t}:=g(t,w_{t},v_{t})$. In addition, we define the
projection
$\pi:\widetilde{\mathbb{S}}^{2}_{{\mathcal{A}}}\rightarrow{\mathcal{S}}^{2}_{\beta}$,
which maps $(w,v,l,u)\in\widetilde{\mathbb{S}}^{2}_{{\mathcal{A}}}$ to
$\pi(w,v,l,u):=(w,v)\in{\mathcal{S}}^{2}_{\beta}$. Finally, the mapping
$S:\mathbb{S}^{2}_{\beta}\rightarrow\mathbb{S}^{2}_{\beta}$ is given by
$S:=\pi\circ\Psi$. In view of Proposition 4.3, the mapping $\Psi$ (hence also
$S$) is well defined under Assumptions 2.1–2.2 and 4.1–4.2.

######  Lemma 4.3.

If Assumptions 2.1–2.2 and 4.1–4.2 are satisfied, then
$S:\mathbb{S}^{2}_{\beta}\rightarrow\mathbb{S}^{2}_{\beta}$ is a contraction
mapping for some $\beta\geq 0$ and thus $S$ has a unique fixed point
$(\widehat{Y},\widehat{Z})\in\mathbb{S}^{2}_{\beta}.$

Proof. We consider two pairs $(w^{1},v^{1})$ and $(w^{2},v^{2})$ in
$\mathbb{S}^{2}_{\beta}$ and, for brevity, we denote their respective images
through $S$ by $(Y^{1},Z^{1})$ and $(Y^{2},Z^{2})$. Using the a priori
estimate from Proposition 4.2 and the fact that the generator $g$ is uniformly
$m$-Lipschitz continuous with a constant $\widehat{L}$, we obtain for
arbitrary $\beta\geq 0$ and $\gamma>0$

$\displaystyle\|Y\|^{2}_{{\mathcal{S}}^{2}_{\beta}}+\|Z\|^{2}_{{\mathcal{L}}^{2}_{\beta}(M)}\leq
1158\beta^{-1}\widehat{L}^{2}(C_{Q}+1)\big{(}\|w^{1}-w^{2}\|^{2}_{{\mathcal{S}}^{2}_{\beta}}+\|v^{1}-v^{2}\|^{2}_{{\mathcal{L}}^{2}_{\beta}(M)}\big{)}.$

If we set $\beta=(2\times 1158)\widehat{L}^{2}(C_{Q}+1)>0$, then we obtain

$\displaystyle\|S(w^{1},v^{1})-S(w^{2},v^{2})\|_{\beta}^{2}=\|(Y^{1},Z^{1})-(Y^{2},Z^{2})\|_{\beta}^{2}\leq\frac{1}{2}\|(w^{1},v^{1})-(w^{2},v^{2})\|_{\beta}^{2}.$

By the Banach theorem, the mapping
$S:\mathbb{S}^{2}_{\beta}\rightarrow\mathbb{S}^{2}_{\beta}$ has a unique fixed
point $(\widehat{Y},\widehat{Z})$. $\Box$

We are now ready to prove the main existence and uniqueness result for the
DRBSDE (4.1).

######  Theorem 4.1.

If Assumptions 2.1–2.2 and 4.1–4.2 are satisfied, then the doubly reflected
BSDE (4.1) with data $(g,\eta,D,\xi,\zeta)$ has a unique solution $(Y,Z,L,U)$.

Proof. Let the mapping
$T:\widetilde{\mathbb{S}}^{2}_{{\mathcal{A}}}\rightarrow\widetilde{\mathbb{S}}^{2}_{{\mathcal{A}}}$
be defined by $T=\Psi\circ\pi$, which means that
$T(w,v,l,u)=(Y^{w,v},Z^{w,v},L^{w,v},U^{w,v})$ for all
$(w,v,l,u)\in\widetilde{\mathbb{S}}^{2}_{{\mathcal{A}}}$. From Lemma 4.3, we
know that the mapping $S$ has a unique fixed point
$(\widehat{Y},\widehat{Z})\in{\mathcal{H}}^{2}(Q)\times{\mathcal{L}}^{2}(M)$.
From the definitions of $S$ and $T$, it is clear that, for every
$(w,v,u,l)\in\widetilde{\mathbb{S}}^{2}_{{\mathcal{A}}}$,

$T(w,v,l,u)=(S(w,v),L^{w,v},U^{w,v}).$ (4.10)

We claim that $(\widehat{Y},\widehat{Z},\widehat{L},\widehat{U})$ where
$(\widehat{L},\widehat{U}):=(L^{\widehat{Y},\widehat{Z}},U^{\widehat{Y},\widehat{Z}})$
is a unique fixed point of $T$. To this end, we observe that the equality
$T(\widehat{Y},\widehat{Z},\widehat{L},\widehat{U})=(\widehat{Y},\widehat{Z},\widehat{L},\widehat{U})$
follows from the equality
$S(\widehat{Y},\widehat{Z})=(\widehat{Y},\widehat{Z})$ and (4.10) with
$(l,u)=(\widehat{L},\widehat{U})$. To check that a fixed point of $T$ is
unique, it suffices to recall that the fixed point of $S$ is unique and use
again (4.10). We have thus shown that the DRBSDE (4.2) has a unique solution
$(\widehat{Y},\widehat{Z},\widehat{L},\widehat{U})$ where
$(\widehat{L},\widehat{U}):=(L^{\widehat{Y},\widehat{Z}},U^{\widehat{Y},\widehat{Z}})$.
$\Box$

###  4.3 Picard’s Iterations for Solutions to Doubly Reflected BSDEs

To construct the Picard’s sequence converging to the solution of a
$(Y,Z,L,U)$, we consider the sequence $\\{(Y^{k},Z^{k})\\}_{k=1}^{\infty}$
where $(Y^{k},Z^{k})=S^{k}(w,v)$ for some $(w,v)\in\mathbb{H}^{2}_{\beta}$.
The associated sequence $\\{(L^{k},U^{k})\\}_{k=1}^{\infty}$ is obtained by
choosing for every $k\in\mathbb{N}$ the last two components of the unique
solution $(Y^{k},Z^{k},L^{k},U^{k})$ to the DRBSDE with generator
$g^{k-1}_{t}:=g(t,Y^{k-1}_{t},Z^{k-1}_{t})$. Notice that the existence of a
solution is guaranteed under the conditions of Proposition 4.3. The next
result can be demonstrated in a similar way to the proof of Proposition 3.4
and thus we merely sketch the main steps in the proof of Proposition 4.4.

######  Proposition 4.4.

If Assumptions 2.1–2.2 and 4.1–4.2 are valid, then

$\lim_{k\rightarrow\infty}\|Y^{k}-Y\|_{{\mathcal{S}}^{2}}=0,\quad\lim_{k\rightarrow\infty}\|(L^{k}-L)-(U^{k}-U)\|_{{\mathcal{S}}^{2}}=0,\quad\lim_{k\rightarrow\infty}\|Z^{k}-Z\|_{{\mathcal{L}}^{2}(M)}=0.$
(4.11)

Proof. We first observe that last convergence in (4.11) is clear from Lemma
4.3. If we denote $\widetilde{Y}^{k}:=Y^{k}-Y$, $\widetilde{Z}^{k}:=Z^{k}-Z$,
$\widetilde{L}^{k}:=L^{k}-L$ and $\widetilde{U}^{k}:=U^{k}-U$, then we have

$\left\\{\begin{array}[c]{ll}d\widetilde{Y}^{k}_{t}=-\widetilde{g}^{k-1}_{t}\,dQ_{t}+\widetilde{Z}^{k,*}_{t}\,dM_{t}-d\widetilde{L}^{k}_{t}+d\widetilde{U}^{k}_{t},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ \widetilde{Y}^{k}_{T}=0,\end{array}\right.$

where
$\widetilde{g}^{k-1}_{t}:=g(t,Y^{k-1}_{t},Z^{k-1}_{t})-g(t,Y_{t},Z_{t})$. We
claim that that there exists a constant $C\geq 0$ such that

$\displaystyle\|\widetilde{Y}^{k}\|^{2}_{{\mathcal{S}}^{2}}+\|\widetilde{L}^{k}-\widetilde{U}^{k}\|^{2}_{{\mathcal{S}}^{2}}\leq
CJ_{k}$ (4.12)

where

$\displaystyle
J_{k}:=\|\widetilde{Y}^{k-1}\|^{2}_{{\mathcal{H}}^{2}(Q)}+\|\widetilde{Y}^{k}\|^{2}_{{\mathcal{H}}^{2}(Q)}+\|\widetilde{Z}^{k-1}\|^{2}_{{\mathcal{L}}^{2}(M)}+\|\widetilde{Z}^{k}\|^{2}_{{\mathcal{L}}^{2}(M)}.$

Assuming (4.12) and using Lemma 4.3 and the Banach theorem, we deduce that
$\lim_{k\rightarrow\infty}J_{k}=0$ and thus (4.11) follows. It thus remains to
show that (4.12) is valid.

Step 1. To show that $\|\widetilde{Y}^{k}\|^{2}_{{\mathcal{S}}^{2}}\leq
CJ_{k}$ for some constant $C\geq 0$, it suffices to follow Step 1 in
Proposition 4.2. By applying the Itô formula to $|\widetilde{Y}^{k}_{t}|^{2}$
and using the Skorokhod conditions, we get
$\int_{t}^{T}\widetilde{Y}^{k}_{u-}\,d\widetilde{L}^{k}_{u}-\int_{t}^{T}\widetilde{Y}^{k}_{u-}\,d\widetilde{U}^{k}_{u}\leq
0$ for every $t\in[0,T]$, we obtain the inequality

$\displaystyle|\widetilde{Y}^{k}_{t}|^{2}$
$\displaystyle+\int_{t}^{T}\|m_{u}\widetilde{Z}^{k}_{u}\|^{2}\,dQ_{u}+\sum_{t<s\leq
T}|\Delta\widetilde{Y}^{k}_{u}|^{2}-\int_{t}^{T}\|m_{u}^{d}(\widetilde{Z}^{k}_{u})^{d}\|^{2}\,dQ_{u}$
$\displaystyle\leq
2\int_{t}^{T}\widetilde{Y}^{k}_{u}\widetilde{g}^{k-1}_{u}\,dQ_{u}-2\int_{t}^{T}\widetilde{Y}^{k}_{u-}\widetilde{Z}^{k,*}_{u}\,dM_{u}.$

The rest follows by repeating the arguments from Step 1 in Proposition 4.2.

Step 2. We claim that
$\|\widetilde{L}^{k}-\widetilde{U}^{k}\|^{2}_{{\mathcal{S}}^{2}}\leq CJ_{k}$.
To check this, we observe that

$\displaystyle\widetilde{L}^{k}_{t}-\widetilde{U}^{k}_{t}=\widetilde{Y}_{0}-\widetilde{Y}_{t}^{k}-\int_{0}^{t}\widetilde{g}^{k-1}_{u}\,dQ_{u}+\int_{0}^{t}\widetilde{Z}^{k,\ast}_{u}\,dM_{u}.$

and by using the arguments from Step 2 in Proposition 3.4, we can obtain the
desired inequality. $\Box$

## References

  * [1] Aazizi, S., and Ouknine, Y.: Strong envelope and strong supermartingale: Application to reflected backward stochastic differential equation. Working paper, 2016 (arXiv:1112.0255v2).
  * [2] Baadi, B., and Ouknine, Y.: Reflected BSDEs when the obstacle is not right-continuous in a general filtration. ALEA – Latin American Journal of Probability and Mathematical Statistics 14 (2017), 201–218.
  * [3] Baadi, B., and Ouknine, Y.: Reflected BSDEs with optional barrier in a general filtration. Afrika Matematika 29 (2018), 1049–1064.
  * [4] Barles, G., Buckdahn, R., and Pardoux, E.: Backward stochastic differential equations and integral-partial differential equations. Stochastics and Stochastics Reports 60 (1995), 57–83.
  * [5] Bayraktar, E., and Yao, S.: Quadratic reflected BSDEs with unbounded obstacles. Stochastic Processes and their Applications 122 (2012), 1155–1203.
  * [6] Bayraktar, E., and Yao, S.: Doubly reflected BSDEs with integrable parameters and related Dynkin games. Stochastic Processes and their Applications 125 (2015), 4489–4542.
  * [7] Bielecki, T. R., Jeanblanc, M., and Rutkowski, M.: Credit Risk Modeling. Osaka University Press, Osaka, 2009.
  * [8] Carbone, R., Ferrario, B., and Santacroce, M.: Backward stochastic differential equations driven by càdlàg martingales. Theory of Probability and its Applications 52(2) (2008), 304–314.
  * [9] Cohen, S. N., and Elliott, R.: Stochastic Calculus and Applications. Springer, New York, 2015. Bikrhäuser, Berlin,
  * [10] Crépey, S., and Matoussi, A.: Reflected and doubly reflected BSDEs with jumps: a priori estimates and comparison. The Annals of Probability 18(5) (2008), 2041–2069.
  * [11] Cvitanić, J., and Karatzas, I.: Backward stochastic differential equations with reflection and Dynkin games. The Annals of Probability 24(4) (1996), 2024–2056.
  * [12] Dellacherie, C., and Lenglart, E.: Sur des problèmes de régularisation, de recollement et d’interpolation en théorie des processus. In Lecture Notes in Mathematics 920, Séminaire de Probabilités XVI, Springer, Berlin, 1982, pp. 298–313.
  * [13] Dumitrescu, R., Grigorova, M., Quenez, M. C., and Sulem, A.: BSDEs with default jump. In Computation and Combinatorics in Dynamics, Stochastics and Control, Abel Symposia, Vol. 13, E. Celledoni, G. Di Nunno, K. Ebrahimi-Fard, H. Munthe-Kaas (Eds.), Springer, Cham, 2018, pp. 233–263.
  * [14] Dumitrescu, R., Quenez, M. C., and Sulem, A.: Generalized Dynkin games and doubly reflected BSDEs with jumps. Electronic Journal of Probability 21/64 (2016), 1–32.
  * [15] Dumitrescu, R., Quenez, M. C., and Sulem, A.: Game options in an imperfect market with default. SIAM Journal on Financial Mathematics 8 (2017), 532–559.
  * [16] Dumitrescu, R., Quenez, M. C., and Sulem, A.: American options in an imperfect complete market with default. ESAIM: Proceedings and Surveys 64 (2018), 93–110.
  * [17] El Karoui, N.: Les aspects probabilistes du contrôle stochastique. In Lecture Notes in Mathematics 876, Ecole d’Eté de Probabilités de Saint-Flour IX, 1979, P.-L. Hennequin (Ed.), Springer, Berlin, 1981, pp. 73–238.
  * [18] El Karoui, N., and Huang, S.: A general result of existence and uniqueness of backward stochastic differential equations. In Backward Stochastic Differential Equations, Pitman Research Notes in Mathematics Series 364, N. El Karoui and L. Mazliak (Eds.), Addison Wesley Longman Ltd, Harlow, Essex, 1997, pp. 27–36.
  * [19] El Karoui, N., Matoussi, A., and Ngoupeyou, A.: Quadratic exponential semimartingales and application to BSDEs with jumps. Working paper, 2016 (arXiv:1603.06191).
  * [20] El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, S., and Quenez, M. C.: Reflected solutions of backward SDE’s, and related obstacle problems for PDE’s. The Annals of Probability 25 (1997), 702–737.
  * [21] El Karoui, N., Pardoux, E., and Quenez, M. C.: Reflected backward SDEs and American options. In Numerical Methods in Finance, L. C. G. Rogers and D. Talay (Eds.), Cambridge University Press, Cambridge, 1997, pp. 215–231.
  * [22] El Karoui, N., Peng, S., and Quenez, M. C.: Backward stochastic differential equations in finance. Mathematical Finance 7 (1997), 1–71.
  * [23] El Karoui, N., and Quenez, M. C.: Non-linear pricing theory and backward stochastic differential equations. In Lecture Notes in Mathematics 1656, W. J. Runggaldier (Ed.), Springer, Berlin, 1997, pp. 191–246.
  * [24] Essaky, E. H.: Reflected backward stochastic differential equation with jumps and RCLL obstacle. Bulletin des Sciences Mathématiques 132(8) (2008), 690–710.
  * [25] Essaky, E. H., and Hassani, M.: Generalized BSDE with 2-reflecting barriers and stochastic quadratic growth. Journal of Differential Equations 254 (2013), 1500–1528.
  * [26] Grigorova, M., Imkeller, P., Offen, E., Ouknine, Y., and Quenez, M. C.: Reflected BSDEs when the obstacle is not right-continuous and optimal stopping. The Annals of Applied Probability 27 (2017), 3153–3188.
  * [27] Grigorova, M., Imkeller, P., Ouknine, Y., and Quenez, M. C.: Optimal stopping with $f$-expectations: the irregular case. Stochastic Processes and their Applications 130/3 (2020), 1258–1288.
  * [28] Grigorova, M., and Quenez, M. C.: Optimal stopping and a non-zero-sum Dynkin game in discrete time with risk measures induced by BSDEs. Stochastics: An International Journal of Probability and Stochastic Processes 89 (2017), 259–279.
  * [29] Hamadène, S.: Reflected BSDE’s with discontinuous barrier and application. Stochastics and Stochastics Reports 74(3-4) (2002), 571–596.
  * [30] Hamadène, S., and Lepeltier, J. P.: Reflected BSDEs and mixed game problem. Stochastic Processes and their Applications 85 (2000), 177–188.
  * [31] Hamadène, S., and Ouknine, Y.: Reflected backward SDEs with general jumps. Theory of Probability and its Applications 60 (2016), 263–280.
  * [32] Hamadène, S., and Wang, H.: BSDEs with two RCLL reflecting obstacles driven by Brownian motion and Poisson measure and a related mixed zero-sum game. Stochastic Processes and their Applications 119 (2009), 2881–2912.
  * [33] He, S., Wang, J., and Yan, J.: Semimartingale Theory and Stochastic Calculus. Science Press and CRS Press, Beijing and Boca Raton, 1992.
  * [34] Jacod, J., and Shiryaev, A. N.: Limit Theorems for Stochastic Processes. Second edition. Springer, Berlin, 2003.
  * [35] Jeanblanc, M., and Le Cam, Y.: Immersion property and credit risk modelling. In Optimality and Risk - Modern Trends in Mathematical Finance: The Kabanov Festschrift, F. Delbaen, M. Rasonyi and C. Stricker (Eds.), Springer, Berlin, 2009, pp. 99–131.
  * [36] Jeanblanc, M., Matoussi, A., and Ngoupeyou, A.: Robust utility maximization problem in model with jumps and unbounded claim. Working paper, 2012 (arXiv:1201.2690).
  * [37] Jeanblanc, M., Matoussi, A., and Ngoupeyou, A.: Robust utility maximization problem in a discontinuous filtration. Working paper, 2013 (arXiv:1201.2690v3).
  * [38] Kifer, Y.: Dynkin games and Israeli options. ISRN Probability and Statistics (2013), ID856458.
  * [39] Kim, E., Nie, T., and Rutkowski, M.: Arbitrage-free pricing of American options in nonlinear markets. Working paper, 2018 (arXiv: 1804.10753v2).
  * [40] Kim, E., Nie, T., and Rutkowski, M.: Arbitrage-free pricing of game options in nonlinear markets. Working paper, 2018 (arXiv: 1807.05448).
  * [41] Klimsiak, T.: Reflected BSDEs with monotone generator. Electronic Journal of Probability 17/107 (2012), 1–25.
  * [42] Klimsiak, T.: Reflected BSDEs on filtered probability spaces. Stochastic Processes and their Applications 125 (2015), 4204–4241.
  * [43] Klimsiak, T., Rzymowski, M., and Słomiński, L.: Reflected BSDEs with regulated trajectories. Stochastic Processes and their Applications 129 (2019), 1153–1184.
  * [44] Kobylanski, M., and Quenez, M. C.: Optimal stopping time problem in a general framework. Electronic Journal of Probability 17/72 (2012), 1–28.
  * [45] Kobylanski, M., and Quenez, M. C.: Erratum: Optimal stopping time problem in a general framework. Available at https://hal.archives-ouvertes.fr/hal-01328196. 2016.
  * [46] Kobylanski, M., Quenez, M. C., and de Campagnolle, M. R.: Dynkin games in a general framework. Stochastics: An International Journal of Probability and Stochastic Processes 86 (2014), 304–329.
  * [47] Kusuoka, S.: A remark on default risk models. Advances in Mathematical Economics 1 (1999), 69–82.
  * [48] Lepeltier, J. P., and San Martín, J.: Backward SDEs with two barriers and continuous coefficients. Journal of Applied Probability 41 (2004), 162–175.
  * [49] Lepeltier, J. P., and Xu, M.: Penalization method for reflected backward stochastic differential equations with one r.c.l.l. barrier. Statistics and Probability Letters 75 (2005), 58-66.
  * [50] Maingueneau, A. M.: Temps d’arrêt optimaux et théorie générale. In Lecture Notes in Mathematics 649, Séminaire de Probabilités XII, C. Dellacherie, P. A. Meyer and M. Weil (Eds.), Springer, Berlin, 1978, pp. 457–467.
  * [51] Nie, T., and Rutkowski, M.: BSDEs driven by multidimensional martingales and their applications to markets with funding costs. Theory of Probability and its Applications 60 (2016), 604–630.
  * [52] Nie, T., and Rutkowski, M.: Existence, uniqueness and strict comparison theorems for backward stochastic differential equations driven by RCLL martingales. Working paper, 2019.
  * [53] Papapantoleon, T., Possamaï, D., and Saplaouras, A.: Existence and uniqueness results for BSDEs with jumps: the whole nine yards. Electronic Journal of Probability 23 (2018), 1–68.
  * [54] Peng, S., and Xu, M.: The smallest $g$-supermartingale and reflected BSDE with single and double $L^{2}$ obstacles. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 41 (2005), 605–630.
  * [55] Peng, S., and Xu, X.: BSDEs with random default time and their applications to default risk. Working paper, 2009 (arXiv:0910.2091).
  * [56] Peng, S., and Xu, X.: BSDEs with random default time and related zero-sum stochastic differential games. C. R. Acad. Sci. Paris, Ser I. 348 (2010), 193–198.
  * [57] Protter, P. E.: Stochastic Integration and Differential Equations. Second Edition. Springer, Berlin, 2004.
  * [58] Quenez, M. C., and Sulem, A.: BSDEs with jumps, optimization and applications to dynamic risk measures. Stochastic Processes and their Applications 123 (2013), 3328–3357.
  * [59] Quenez, M. C., and Sulem, A.: Reflected BSDEs and robust optimal stopping time for dynamic risk measures with jumps. Stochastic Processes and their Applications 124 (2014), 3031–3054.
  * [60] Royer, M.: Backward stochastic differential equations with jumps and related non-linear expectations. Stochastic Processes and their Applications 116 (2006), 1358–1376.
  * [61] Tang, S., and Li, X.: Necessary conditions for optimal control of stochastic systems with random jumps. SIAM Journal on Control and Optimization 32 (1994) 1447–1475.
  * [62]