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\title{Reflected BSDEs with default time and irregular obstacles}
\alttitle{EDSRs rÃ©flÃ©chies avec temps de dÃ©faut et obstacles irrÃ©guliers}

\author{\firstname{Badr} \lastname{Elmansouri}\CDRorcid{0000-0003-2603-894X}}
\address{Laboratory of Analysis and Applied Mathematics (LAMA), Faculty of Sciences Agadir, Ibn Zohr University, BP 8106-CitÃ© Dakhla, 80000, Agadir, Morocco}
\email{badr.elmansouri@edu.uiz.ac.ma}

\subjclass{60H05, 60H30, 60G40, 91G80}

\keywords{\kwd{Reflected BSDEs} \kwd{irregular obstacle} \kwd{stochastic Lipschitz coefficient} \kwd{Mertens decomposition}}
\altkeywords{\kwd{EDSRs rÃ©flÃ©chies} \kwd{obstacle irrÃ©gulier} \kwd{coefficient de Lipschitz stochastique} \kwd{dÃ©composition de Mertens}}

\thanks{This research was supported by the National Center for Scientific and Technical Research (CNRST) in Morocco}

\begin{abstract}
In this note, we study reflected backward stochastic differential equations with a default time, where the reflecting obstacle is not necessarily right-continuous. We establish the existence and uniqueness of a solution for such equations under a stochastic Lipschitz condition on the coefficient.
\end{abstract}

\begin{altabstract}
Dans cette note, nous Ã©tudions les Ã©quations diffÃ©rentielles stochastiques rÃ©trogrades rÃ©flÃ©chies avec un temps de dÃ©faut, oÃ¹ l'obstacle de rÃ©flexion n'est pas nÃ©cessairement continu Ã&nbsp; droite. Nous Ã©tablissons l'existence et l'unicitÃ© d'une solution pour de telles Ã©quations sous une condition de Lipschitz stochastique sur le coefficient.
\end{altabstract}

\begin{document}

\maketitle

\section{Introduction}

In credit risk analysis, default risk is the most commonly studied type of financial risk, which refers to the risk that a counterparty in a financial transaction may fail to meet its contractual obligations. In fact, many papers have addressed this topic (see, e.g., \cite{el2010happens, jarrow2001counterparty, RTW, RTWJ, kusuoka1999remark}).

The notion of backward stochastic differential equations (BSDEs), first introduced by Pardoux and Peng~\cite{pardoux1990adapted} in a Brownian setting, has proven to be a powerful tool for solving various mathematical problems, particularly those arising in finance, such as the pricing and hedging of derivative securities, utility maximization, and other applications in different models, including the defaultable setting (see, e.g., \cite{el1997backward, rouge2000pricing, barrieu2005inf, peng2009bsdes, peng2010bsdes, DumitrescuR}).

In contrast, solving the pricing problem of American options using new stochastic approaches has been an active area of research over the past few years. Within this approach, reflected BSDEs (RBSDEs) have emerged as one of the most efficient tools for addressing this financial problem. In simple terms, RBSDEs are a class of BSDEs in which the state process of the solution is required to remain above a specified stochastic process, known as a barrier or obstacle. This concept was introduced by El Karoui et al.~\cite{el1997reflected} for continuous barriers and Lipschitz drivers in a Brownian setting. This framework was subsequently extended by several authors to more general settings, including right-continuous with left limits (RCLL) barriers in both discontinuous and defaultable frameworks (see, e.g., \cite{hamadene2002reflected, lepeltier2005penalization, dumitrescu2018american}). Additionally, Grigorova et al.~\cite{GriBasic} proposed a novel extension of RBSDEs to cases where the obstacle is only right-upper semi-continuous, rather than necessarily RCLL. These authors established the existence and uniqueness of this extension of RBSDEs under a Lipschitz coefficient in a filtration that supports both an independent Poisson random measure and a Brownian motion. Recently, building upon this development, several researchers have focused on advancing this extension (see, e.g., \cite{akdim2020strong, Bouhadou, baadi2017reflected, klimsiak2019reflected, marzougue2020predictable}).

This paper aims to address the existence and uniqueness problem for reflected BSDEs in a filtration that supports both a Brownian motion and a risk event occurrence modeled by a default time, where the barrier is only right-upper semi-continuous (not necessarily right-continuous) and the generator is stochastically Lipschitz. This last requirement is significant in finance, for instance, when examining a Black--Scholes model with unbounded stochastic parameters, including the risk-free rate process, the stock volatility process, and other related factors. In such cases, the pricing problem of an American option, expressed in terms of an RCLL barrier, requires the use of the stochastic Lipschitz constraint on the coefficient. Therefore, our work can be applied to solve this pricing problem for RCLL obstacles in the same market models considered in~\cite{dumitrescu2018american, bielecki2005pde}, without the need for bounded parameters. Additionally, this work can be applied to address a dynamic risk problem similar to that considered in~\cite{GriBasic} within a defaultable framework.

This note is organized as follows: Section~\ref{Sec2} introduces the necessary notations and definitions. In Section~\ref{Sec3}, we address the existence and uniqueness of a solution for RBSDEs with a default time, an irregular obstacle, and a stochastic Lipschitz driver.

\section{Preliminaries}\label{Sec2}

Let $(\Omega, \mathcal{G}, \mathbb{P})$ be a complete probability space equipped with two stochastic processes:
\begin{itemize}
\item A one-dimensional standard Brownian motion $W \coloneqq (W_t)_{t \geq 0}$, and we denote by $\mathbb{F} \coloneqq (\mathcal{F}_t)_{t \geq 0}$ its natural filtration.
\item A jump process $H = (H_t)_{t \geq 0}$, defined by $H_t = \mathds{1}_{\{\tau \leq t\}}$ for all $t \geq 0$, where $\tau$ is a $(0, +\infty)$-valued $\mathcal{G}$-measurable random variable that models a \emph{default time}. We refer to the process $H$ as the default process. We assume that $\mathbb{P}(\tau &gt; 0) = 1$ and $\mathbb{P}(\tau \geq t) &gt; 0$ for all $t &gt; 0$, allowing for the possibility of the default occurring at or after any fixed time.
\end{itemize}

Two types of information are available: the first comes from the public flow (represented by $\mathbb{F}$), and the second from the default time $\tau$. Therefore, to model the global flow of information, we consider $\mathbb{G}\coloneqq(\mathcal{G}_t)_{t \geq 0}$ to be the smallest right-continuous filtration that contains $\mathbb{F}$ and makes $\tau$ a $\mathbb{G}$-stopping time (but not necessarily an $\mathbb{F}$-stopping time). In other words, $\mathbb{G}$ is defined as:
\begin{equation*}
\mathcal{G}_t\coloneqq\bigcap_{s &gt;t}\bigl(\mathcal{F}_s \vee \sigma(H_u; u\leq s)\bigr),\quad t \geq 0.
\end{equation*}
Additionally, we assume that $\mathcal{G}_0=\mathcal{N}$, where $\mathcal{N}\coloneqq\bigl\{A \in \mathcal{G}:\mathbb{P}(A)=0 \;\text{ or }\; 1\bigr\}$, so that $\mathbb{G}$ satisfies the usual conditions of right continuity and completeness (see, e.g., \cite[Chapter~IV.48]{D2}).

Let $(\Lambda_t)_{t \geq 0}$ be the $\mathbb{G}$-predictable compensator (or the $\mathbb{G}$-dual predictable projection) of the non-decreasing default process $(H_t)_{t \geq 0}$ (see, e.g., \cite[Definition~V.21]{bookHe}). As noted in~\cite{kusuoka1999remark}, since we are working in a financial context and following the defaultable setup of~\cite{DumitrescuR, dumitrescu2018american}, we assume that $(\Lambda_t)_{t \geq 0}$ is absolutely continuous with respect to the Lebesgue measure. Then, from~\cite[Theorem~V.14]{bookHe}, there exists a non-negative $\mathbb{G}$-predictable process $(\gamma_t)_{t \geq 0}$, called the \emph{intensity process}, such that $\Lambda_t = \int_{0}^{t} \gamma_s \dd s$ for $t \geq 0$. Moreover, it is well known that the process
\begin{equation}
M_t = H_t - \int_{0}^{t} (1 - H_s) \gamma_s \dd s = H_t - \int_{0}^{t \wedge \tau} \gamma_s \dd s = H_t - \int_{0}^{t} \gamma_s \dd s, \quad t \geq 0,\label{Martingale M}
\end{equation}
is a $(\mathbb{G}, \mathbb{P})$-martingale with integrable variation (i.e., the $\mathbb{G}$-compensated martingale; see, e.g., \cite[Corollary~V.13]{bookHe}). Note that the last equality in~\eqref{Martingale M} follows from the fact that $H_{t \wedge \tau} = H_t$ and the uniqueness of the $\mathbb{G}$-predictable compensator of the default process $H$ (see, e.g., \cite[Theorem~V.20]{bookHe} and~\cite[Definition~V.21]{bookHe}).

Throughout the paper, we assume the following:
%\paragraph{(H)-Hypothesis} Any $(\mathbb{F}, \mathbb{P})$-martingale is a $(\mathbb{G}, \mathbb{P})$-martingale.
%\begin{enonce*}{(H)-Hypothesis}
%Any $(\mathbb{F}, \mathbb{P})$-martingale is a $(\mathbb{G}, \mathbb{P})$-martingale.
%\end{enonce*}
\begin{equation}\label{Hhyp}
\tag{H}
\text{``Any $(\mathbb{F}, \mathbb{P})$-martingale is a $(\mathbb{G}, \mathbb{P})$-martingale.''}
\end{equation}

\begin{remark}
Under the \eqref{Hhyp}-assumption, the process $(W_t)_{t \geq 0}$ remains a Brownian motion in the enlarged filtration $\mathbb{G}$. This assumption, known as the \emph{immersion property} in credit risk modeling, is a highly general and important hypothesis in the field of filtration enlargement (see, e.g., \cite{mansuy2006random}).
\end{remark}

Let $T &gt; 0$ be a deterministic, finite-horizon time. Throughout the rest of the paper, we work within the defaultable setting $\bigl(\Omega, \mathcal{G}, \mathbb{G} = (\mathcal{G}_t)_{t \leq T}, \mathbb{P}\bigr)$.

We denote by $\mathcal{T}_{0,T}$ and $\mathcal{T}^p_{0,T}$ the set of $[0,T]$-valued $\mathbb{G}$-stopping times and $\mathbb{G}$-predictable times, respectively, with $\mathbb{G}$-predictable times defined as in~\cite[Chapter~IV.69]{D2} or~\cite[Definition~III.25]{bookHe}.

Let $\beta &gt; 0$ and $(\alpha_t)_{t \leq T}$ be a non-negative $\mathbb{G}$-adapted process. We define the increasing continuous process $(A_t)_{t \leq T}$ as $A_t \coloneqq \int_{0}^{t} \alpha^2_s \dd s$ and introduce the following spaces:
\begin{itemize}
\item $\mathcal{S}^{2}_{\beta}$: the space of $\mathbb{R}$-valued $\mathbb{G}$-optional (not necessarily RCLL) processes $(Y_t)_{t \leq T}$ such that
\[
\norm{Y}^2_{\mathcal{S}^{2}_{\beta}}\coloneqq\mathbb{E}\Biggl[\esssup_{\sigma \in \mathcal{T}_{0,T}}e^{\beta A_\sigma}\abs{Y_\sigma}^2\Biggr]&lt;+\infty,
\]
with the convention $\mathcal{S}^2_{}\coloneqq\mathcal{S}^2_{0}$.
\item $\mathcal{H}^{2,\alpha}_{\beta}$: the space of $\mathbb{R}$-valued $\mathbb{G}$-optional processes $(Y_t)_{t \leq T}$ such that
\[
\norm{Y}^2_{\mathcal{H}^{2,\alpha}_{\beta}}\coloneqq\mathbb{E}\int_{0}^{T}e^{\beta A_s}\abs{\alpha_s Y_s}^2 \dd s &lt;+\infty.
\]
\item $\mathcal{H}^2_{\beta}$: the space of $\mathbb{R}$-valued $\mathbb{G}$-predictable processes $(Z_t)_{t \leq T}$ such that
\[
\norm{Z}^2_{\mathcal{H}^2_{\beta}}\coloneqq\mathbb{E}\int_{0}^{T}e^{\beta A_s}\abs{Z_s}^2 \dd s &lt;+\infty,
\]
with the convention $\mathcal{H}^2_{}\coloneqq\mathcal{H}^2_{0}$.
\item $\mathcal{U}^2_{\gamma,\beta}$: the space of $\mathbb{R}$-valued $\mathbb{G}$-predictable process $(U_t)_{t \leq T}$ such that
\[
\norm{U}^2_{\mathcal{U}^{2}_{\gamma,\beta}}\coloneqq\mathbb{E}\int_{0}^{T}e^{\beta A_s} \abs{U_s}^2 \gamma_s \dd s&lt;+\infty,
\]
with the convention $\mathcal{U}^2_{\gamma}\coloneqq\mathcal{U}^2_{\gamma,0}$.
\end{itemize}

Recall that, under the immersion property~\eqref{Hhyp}, the martingale representation theorem holds for the filtration $\mathbb{G}$ with respect to the $\mathbb{G}$-martingales $W$ and $M$, where $M$ is defined by~\eqref{Martingale M}.
\begin{theorem}[\cite{kusuoka1999remark}]\label{theorem of representation}
For any $(\mathbb{G},\mathbb{P})$-square integrable martingale $(\mathcal{M}_t)_{t \leq T}$, there exists a unique pair of\/ $\mathbb{G}$-predictable processes $(\mathcal{Z}_t)_{t \leq T}$ and $(\mathcal{V}_t)_{t \leq T}$, such that $\mathcal{Z} \in \mathcal{H}^2$ and $\mathcal{V} \in \mathcal{U}^2_{\gamma}$ and
\[
\mathcal{M}_t=\mathcal{M}_0+\int_{0}^{t}\mathcal{Z}_s \dd W_s+\int_{0}^{t}\mathcal{V}_s \dd M_s,\quad \forall t \in [0,T],\quad \text{$\mathbb{P}$-a.s.}
\]
\end{theorem}

\begin{definition}[$\gamma$-stochastic Lipschitz driver]
A random function $f\colon [0,T] \times \Omega \times \mathbb{R} \times \mathbb{R} \times \mathbb{R} \to \mathbb{R}$, $(\omega,t,y,z,u) \mapsto f(t,\omega,y,z,u)$ is called a $\gamma$-stochastic Lipschitz driver (coefficient or generator) if:
\begin{itemize}
\item for all $(y,z)$, the stochastic process $f(\,\cdot\,,y,z)$ is\/ $\,\mathbb{G}$-progressively measurable;
\item there exist three non-negative\/ $\,\mathbb{G}$-adapted processes $(\mu_t)_{t \leq T}$, $(\rho_t)_{t \leq T}$, and $(\upsilon_t)_{t \leq T}$ such that:
\begin{itemize}
\item $\dd\mathbb{P} \otimes \dd t$-a.s., for each $(y_1, z_1, u_1)$, $(y_2, z_2, u_2) \in \mathbb{R}^3$,
\[
\bigl\lvert f(t,y_1,z_1,u_1)-f(t,y_2,z_2,u_2) \bigr\rvert \leq \mu_t \bigl\lvert y_1-y_2 \bigr\rvert+\rho_t \bigl\lvert z_1-z_2 \bigr\rvert+\upsilon_t \gamma_t \bigl\lvert u_1-u_2 \bigr\rvert;
\]
\item there exists $\epsilon &gt;0$ such that $\alpha_t^2\coloneqq\mu_t+\rho^2_t +\upsilon^2_t \gamma_t\geq \epsilon$;
\end{itemize}
\item the process $\biggl(\frac{f(t,0,0,0)}{\alpha_{t}}\biggr)_{t \leq T}$ belongs to $\mathcal{H}^2_{\beta}$.
\end{itemize}
\end{definition}
\begin{definition}[Barrier]
A process $(\xi_t)_{t \leq T}$ is said to be an obstacle (or barrier) if $\xi$ is $\mathbb{G}$-optional, left-limited, right-upper semi-continuous, and satisfies $\xi \in \mathcal{S}^2_{2\beta}$.
\end{definition}

\section{Reflected BSDE with default time and irregular barrier}\label{Sec3}

We consider RBSDEs with default time $\tau$ and irregular obstacle $\xi$ of the following general form:
\begin{definition}\label{def:RBSDE}
A process $(Y_t, Z_t, U_t, K_t, C_t)_{t \leq T}$ is said to be a solution to the reflected BSDE (RBSDE) with parameters $(\xi,f)$, if:
\begin{enumerate}\romanenumi
\item \label{def:RBSDE_1} $(Y_t, Z_t, U_t, K_t, C_t)_{t \leq T} \in \bigl(\mathcal{S}^2_{\beta} \cap \mathcal{H}^{2,\alpha}_{\beta} \bigr) \times \mathcal{H}^2_{\beta} \times \mathcal{U}^2_{\gamma,\beta}\times \mathcal{S}^2 \times \mathcal{S}^2$;\\[-0.5em]
\item \label{def:RBSDE_2}
\label{basic equation}
$\displaystyle
Y_t=\xi_T+\int_{t}^{T}f(s,Y_s,Z_s,U_s)\dd s-\int_{t}^{T}Z_s \dd W_s-\int_{t}^{T} U_s \dd M_s+(K_T-K_t)+(C_{T-}-C_{t-});$\\[-0.7em]
\item \label{def:RBSDE_3} $Y_t\geq \xi_t$ for all $t \in [0,T]$, a.s.;\\[-1em]
\item \label{def:RBSDE_4} $K$ is a non-decreasing right-continuous predictable process with
\begin{equation*}
K_0=0,\quad \int_{0}^{T}\mathds{1}_{\{Y_t&gt;\xi_t\}}\dd K^c_t=0\;\text{ a.s.\ \quad and }\quad (Y_{\sigma-}-\xi_{\sigma-})(K^d_{\sigma}-K^d_{\sigma-})=0\;\text{ a.s.\ for any }\; \sigma \in \mathcal{T}^p_{0,T};
\end{equation*}
\item \label{def:RBSDE_5} $C$ is a non-decreasing right-continuous adapted purely discontinuous process with
\begin{equation*}
C_{0-}=0,\quad \text{ and }\quad (Y_{\sigma}-\xi_{\sigma})(C_{\sigma}-C_{\sigma-})=0\;\text{ a.s.\ for any } \sigma \in \mathcal{T}_{0,T}.
\end{equation*}
\end{enumerate}
Here, $K^c$ denotes the continuous part of the non-decreasing process $K$ and $K^d$ its discontinuous part.
\end{definition}
\begin{remarks}\label{rema:RSBDE}
\mbox{}
\begin{enumerate}
\item By a classical result of the general theory of processes (see, e.g., \cite[Theorem~IV.84]{D2}), a process $(Y_t, Z_t, U_t, K_t, C_t)_{t \leq T} \in \bigl(\mathcal{S}^2_{\beta} \cap \mathcal{H}^{2,\alpha}_{\beta}\bigr) \times \mathcal{H}^2_{\beta} \times \mathcal{U}^2_{\gamma,\beta}\times \mathcal{S}^2 \times \mathcal{S}^2$ satisfies~\basicequation{} if and only if, a.s.\ for all $\sigma \in \mathcal{T}_{0,T}$,
\[
Y_{\sigma}=\xi_T+\int_{\sigma}^{T}f(s,Y_s,Z_s,U_s)\dd s-\int_{\sigma}^{T}Z_s \dd W_s-\int_{\sigma}^{T} U_s \dd M_s+(K_T-K_\sigma)+(C_{T-}-C_{\sigma-}).
\]

\item \label{Do not jump} Due to Theorem~\ref{theorem of representation} and the fact that the dual predictable projection $\Lambda$ of the default process $H$ is continuous, the enlarged filtration $\mathbb{G}$ is quasi-left continuous (see, e.g., \cite[Chapter~IV.82]{D2}, \cite[Definition~III.39]{bookHe}, or~\cite[Chapter~10, p.~218]{kallenberg2021foundations}). Indeed, from Theorem~\ref{theorem of representation}, we derive that any martingale $(\mathcal{M}_t)_{t \leq T}$ can be decomposed into two stochastic integrals: a continuous martingale $\bigl(\int_{0}^{t} \mathcal{Z}_s \dd W_s\bigr)_{t \leq T}$ and an RCLL finite-variation martingale $\bigl(\int_{0}^{t} \mathcal{V}_s \dd M_s\bigr)_{t \leq T}$ for some predictable processes $(\mathcal{Z}_t, \mathcal{V}_t)_{t \leq T}$ such that the representation theorem~\ref{theorem of representation} for the martingale $\mathcal{M}$ is well defined.

Now, for any predictable time $\sigma \in \mathcal{T}^p_{0,T}$, using~\cite[Theorem~13, p.~60]{bookProtter} and~\eqref{Martingale M}, we have $\Delta \bigl(\int_{0}^{\sigma} \mathcal{V}_s \dd M_s\bigr) = \mathcal{V}_\sigma \Delta M_\sigma = \mathcal{V}_\sigma \Delta H_\sigma$. Then, from~\cite[Theorem~V.27]{bookHe} we have $\mathbb{E}[\Delta H_\sigma] = \mathbb{E}[\Delta \Lambda_\sigma] = 0$. Since $H$ is a non-negative increasing process, we derive that $\Delta H_\sigma = 0$ a.s. Thus, $\Delta \bigl(\int_{0}^{\sigma} \mathcal{V}_s \dd M_s\bigr) = 0$ a.s.\ for any $\sigma \in \mathcal{T}^p_{0,T}$. Finally, from~\cite[Proposition~10.19]{kallenberg2021foundations}, we derive that the filtration $\mathbb{G}$ is quasi-left continuous.

As a consequence of the preceding analysis, $\mathbb{G}$-martingales have only totally inaccessible jump times (see~\cite[Chapter~I, Definition~2.20]{bookJacod}), and the state process $Y$ has two types of left-jumps: one totally inaccessible jump, which stems from a stochastic integral with respect to $M$ (or $H$) defined by decomposition~\eqref{Martingale M}, and predictable jumps, which come from the irregular barrier $\xi$. Therefore, it is necessary to introduce the predictable process $K$ to control these left jumps. Moreover, the processes $K$ and $H$ do not have jumps in common, i.e., $\Delta H_t \, \Delta K_t = \Delta M_t \, \Delta K_t = 0$ for any $t \in [0,T]$.

\item If the obstacle $\xi$ is right-continuous, then the state process $Y$ of the RBSDE~\sbasicequation{} is also right-continuous, and we have $C \equiv 0$. Then, we recover the usual formulation of RBSDEs with a right-continuous barrier, which can be used to solve the pricing problem of an American option in the same complete market model considered in~\cite{dumitrescu2018american, bielecki2005pde}.
\end{enumerate}
\end{remarks}

The following proposition will be needed to obtain a priori estimates for the solution of the RBSDE~\sbasicequation{}.
\begin{proposition}\label{LacanellePropo}
Let $(Y_t, Z_t, U_t, K_t, C_t)_{t \leq T}$ be a solution of the RBSDE~\sbasicequation{}. Then,
\[
\mathbb{E}{\left[\esssup_{\sigma \in \mathcal{T}_0^T}\Biggl\lvert\int_{0}^{\sigma}e^{\beta A_s}Y_{s-}Z_s \dd W_s\Biggr\rvert\right]} \leq \frac{1}{8}\norm{Y}^2_{\mathcal{S}^2_{\beta}}+2c^2 \norm{Z}^2_{\mathcal{H}^2_{\beta}}
\]
and
\[
\mathbb{E}{\left[\esssup_{\sigma \in \mathcal{T}_0^T}\Biggl\lvert\int_{0}^{\sigma}e^{\beta A_s}Y_{s-}U_s \dd M_s\Biggr\rvert\right]} \leq \frac{1}{8}\norm{Y}^2_{\mathcal{S}^2_{\beta}}+2c^2 \norm{U}^2_{\mathcal{U}^2_{\gamma,\beta}},
\]
where $c&gt;0$ is the universal constant of the Burkholder--Davis--Gundy (B-D-G) inequality. In particular, the process
\[
\Biggl(\int_{0}^{t}e^{\beta A_s}\bigl\{Y_{s-} Z_s \dd W_s+Y_{s-}U_s \dd M_s\bigr\}\Biggr)_{t \leq T}
\]
is a uniformly integrable martingale.
\end{proposition}
\begin{proof}
Using the left-continuity of the trajectories of the process $(e^{\beta A_t}Y_{t-})_{t \leq T}$, we have
\[
\sup_{t \in [0,T]} \abs{Y_{t-}}^2= \sup_{t \in [0,T] \cap \mathbb{Q}} e^{\beta A_t}\abs{Y_{t-}}^2 \leq \esssup_{\sigma \in \mathcal{T}_0^T} e^{\beta A_{\sigma}} \abs{Y_{\sigma}}^2.
\]
Moreover, we have
\[
\bigl\lvert Y_{s-}(\omega)\bigr\rvert^2 \leq \sup_{t \in [0,T]} \abs{Y_{t-}}^2, \quad \forall (s,\omega) \in [0,T] \times \Omega.
\]
Then, for any $\nu \in \mathcal{T}_0^T$, we get
%\[
%\sqrt{\int_{0}^{\nu} e^{2\beta A_s} \abs{Y_{s-}}^2 \abs{Z_s}^2 \dd s} \leq \sqrt{\esssup_{\sigma \in \mathcal{T}_0^T} e^{\beta A_{\sigma}} \abs{Y_{\sigma}}^2} \, \sqrt{\int_{0}^{\nu} e^{\beta A_s} \abs{Z_s}^2 \dd s}.
%\]
\[
\sqrt{\int_{0}^{\nu} e^{2\beta A_s} \abs{Y_{s-}}^2 \abs{Z_s}^2 \dd s} \leq \sqrt{\vphantom{\int}\smash{\esssup_{\sigma \in \mathcal{T}_0^T} e^{\beta A_{\sigma}} \abs{Y_{\sigma}}^2}} \, \sqrt{\int_{0}^{\nu} e^{\beta A_s} \abs{Z_s}^2 \dd s}.
\]
It remains to apply the B-D-G inequality (see, e.g., \cite[Theorem~IV.48]{bookProtter}) to the continuous local martingale
\[
\Biggl(\int_{0}^{t} e^{\beta A_s} Y_{s-} Z_s \dd W_s\Biggr)_{t \leq T}
\]
and use the basic inequality $ ab \leq \frac{\varepsilon}{2} a^2 + \frac{1}{2\varepsilon} b^2 $ for any $a, b \in \mathbb{R}$ and $\varepsilon=8$. Similarly, we follow the same argument for the RCLL local martingale
\[
\Biggl(\int_{0}^{t} e^{\beta A_s} Y_{s-} U_s \dd M_s\Biggr)_{t \leq T}.\qedhere
\]
\end{proof}

\subsection{Existence and uniqueness of a solution} 

The proof of the existence and uniqueness result is divided into two main steps:
\begin{enumerate}
\item in the first step, we consider the case where the generator $f$ does not depend on the variables $(y,z,u)$, using some a priori estimates and results from optimal stopping theory;
\item in the second step, with the help of the result from the previous step, we prove the existence and uniqueness of the solution to the RBSDE~\sbasicequation{} in the case of a general $\gamma$-stochastic Lipschitz driver $f$ using a fixed-point theorem in an appropriate Banach space.
\end{enumerate}

\subsubsection{Case of a generator independent of the variables \texorpdfstring{$(y,z,u)$}{(y,z,u)}}

In this part, we assume that the generator $f$ of the RBSDE~\sbasicequation{} is of the form $f(t,\omega,y,z,u)=\mathfrak{f}(t,\omega)$, where $\bigl(\mathfrak{f}(t,\omega)\bigr)_{t \leq T}$ is a $\mathbb{G}$-progressively measurable process such that $\frac{\mathfrak{f}(\,\cdot\,)}{\alpha_{\cdot}} \in \mathcal{H}^2_{\beta}$. We maintain this condition on the generator $\mathfrak{f}$ throughout this part. The objective is to prove the existence and uniqueness result for the RBSDE~\sbasicequation{} associated with the data $(\xi,\mathfrak{f})$.

First, we state the following lemma, which gives an a priori estimate of the solution.
\begin{lemma}\label{Uniqueness}
Let $(Y^1_t, Z^1_t, U^1_t, K^1_t, C^1_t)_{t \leq T}$ and $(Y^2_t, Z^2_t, U^2_t, K^2_t, C^2_t)_{t \leq T}$ be two solutions to the RBSDE~\sbasicequation{} with parameters $\bigl(\mathfrak{f}^1(\,\cdot\,), \xi^1\bigr)$ and $\bigl(\mathfrak{f}^2(\,\cdot\,), \xi^2\bigr)$, respectively. We denote $\bar{\mathfrak{S}} \coloneqq \mathfrak{S}^1 - \mathfrak{S}^2$ for $\mathfrak{S} \in \bigl\{Y, Z, U, K, C, \mathfrak{f}(\,\cdot\,), \xi\bigr\}$. Then, there exists a constant $\mathfrak{c}_{\beta} &gt; 0$ depending on $\beta$ such that for all $\beta &gt; 1$
\begin{equation*}
\begin{split}
\norm{\bar{Y}}^2_{\mathcal{S}^2_{\beta}}+\norm{\bar{Y}}^2_{\mathcal{H}^{2,\alpha}_{\beta}}+\norm{\bar{Z}}^2_{\mathcal{H}^{2}_{\beta}}+\norm{\bar{U}}^2_{\mathcal{U}^{2}_{\gamma,\beta}}
\leq \mathfrak{c}_{\beta} {\left(\norm{\bar{\xi}}^2_{\mathcal{S}^{2}_{2\beta}}+\Biggl\lVert\frac{\bar{\mathfrak{f}}(\,\cdot\,)}{\alpha_{\cdot}}\Biggr\rVert^2_{\mathcal{H}^{2}_{\beta}}\right)}.
\end{split}
\end{equation*}
\end{lemma}

\begin{proof}
Using the Gal'{\v c}uk--Lenglart formula (see, e.g., \cite[Theorem~8.2]{Galuk1982}), we have
\begin{multline*}
e^{\beta A_t}\abs{\bar{Y}_t}^2+\beta\int_{t}^{T}e^{\beta A_s} \abs{\bar{Y}_s}^2 \dd A_s+\int_{t}^{T}e^{\beta A_s} \abs{\bar{Z}_s}^2 \dd s\\
=e^{\beta A_T}\abs{\xi_T}^2+2\int_{t}^{T}e^{\beta A_s}\bar{Y}_s \bar{\mathfrak{f}}(s)\dd s-2\int_{t}^{T}e^{\beta A_s}\bar{Y}_{s-} \bar{Z}_s \dd W_s-2\int_{t}^{T}e^{\beta A_s}\bar{Y}_{s-} \bar{U}_s \dd M_s\\
+2\int_{t}^{T}e^{\beta A_s}\bar{Y}_{s-} \dd\bar{K}_s+2\int_{t}^{T}e^{\beta A_s}\bar{Y}_s \dd\bar{C}_s-\sum_{t&lt;s \leq T} e^{\beta A_s}\bigl(\Delta \bar{Y}_s\bigr)^2-\sum_{t\leq s &lt; T} e^{\beta A_s}\bigl(\Delta_{+} \bar{Y}_s\bigr)^2.
\end{multline*}
Using~\basicequation{}, Remark~\ref{rema:RSBDE}\eqref{Do not jump}, and the fact that $H$ is a purely discontinuous process, we derive that
\[
\sum_{t &lt; s \leq T} e^{\beta A_s} \bigl(\Delta \bar{Y}_s\bigr)^2 = \int_{t}^{T} e^{\beta A_s} \bar{U}_{s} \dd H_s + \sum_{t &lt; s \leq T} e^{\beta A_s} \bigl(\Delta \bar{K}_s\bigr)^2.
\]
Then, we have
\[
\int_{t}^{T} e^{\beta A_s} \abs{\bar{U}_s}^2 \gamma_s \dd s - \sum_{t &lt; s \leq T} e^{\beta A_s} \bigl(\Delta \bar{Y}_s\bigr)^2 \leq -\int_{t}^{T} e^{\beta A_s} \abs{\bar{U}_s}^2 \dd M_s.
\]
This, together with the Skorokhod and minimality conditions on $\bar{K}$ and $\bar{C}$, implying
\[
\bar{Y}_{s-} \dd \bar{K}_s + \bar{Y}_{s} \dd \bar{C}_s \leq 0,
\]
the inequality
\[
2 \bar{Y}_s \bar{\mathfrak{f}}(s) \leq (\beta-1) \alpha_s^2 \abs{\bar{Y}_s}^2 + \frac{1}{1-\beta} \Biggl\lvert\frac{\bar{\mathfrak{f}}(s)}{\alpha_s}\Biggr\rvert^2
\]
for any $\beta &gt; 1$, and finally the result from Proposition~\ref{LacanellePropo}, allows us to derive the claim.
\end{proof}
\begin{remark}
The result of Lemma~\ref{Uniqueness} can be easily generalized to the case of a general $\gamma$-stochastic Lipschitz generator, where the term $\bar{\mathfrak{f}}(\,\cdot\,)$ is replaced by $\bar{\mathfrak{f}}(\,\cdot\,, Y^2_{\cdot}, Z^2_{\cdot}, U^2_{\cdot})$.
\end{remark}

\goodbreak

The main result of the current part is given in the following theorem.
\begin{theorem}\label{basic theorem for f independent of y,z,u}
The RBSDE~\sbasicequation{} admits a unique solution $(Y_t, Z_t, U_t, K_t, C_t)_{t \leq T} \in \bigl(\mathcal{S}^2_{\beta} \cap \mathcal{H}^{2,\alpha}_{\beta} \bigr) \times \mathcal{H}^2_{\beta} \times \mathcal{U}^2_{\gamma,\beta} \times \mathcal{S}^2 \times \mathcal{S}^2$ for all $\beta &gt; 1$. Moreover, for each $\nu \in \mathcal{T}_{0,T}$, we have
\begin{equation}\label{Representation of Y}
Y_{\nu}=\esssup_{\sigma \in \mathcal{T}_{\nu,T}}\mathbb{E}\Biggl[\xi_{\sigma}+\int_{\nu}^{\sigma}\mathfrak{f}(s)\dd s\,\Bigg\vert\, \mathcal{G}_{\nu}\Biggr]\;\text{ a.s.}
\end{equation}
\end{theorem}

\begin{proof}
We define the value functions $\bigl\{\overline{\overline{\!Y}}(\nu) : \nu \in \mathcal{T}_{0,T}\bigr\}$ as
\[
\overline{\overline{\!Y}}(\nu) \coloneqq \esssup_{\sigma \in \mathcal{T}_{\nu,T}} \mathbb{E}\Biggl[\xi_{\sigma} + \int_{\nu}^{\sigma} \mathfrak{f}(s) \dd s \,\Bigg\vert\, \mathcal{G}_{\nu}\Biggr]
\]
and the family $\bigl\{\tilde{Y}(\nu) : \nu \in \mathcal{T}_{0,T}\bigr\}$ by
\[
\tilde{Y}(\nu) \coloneqq \overline{\overline{\!Y}}(\nu) + \int_{0}^{\nu} \mathfrak{f}(s) \dd s = \esssup_{\sigma \in \mathcal{T}_{\nu,T}} \mathbb{E}\Biggl[\xi_{\sigma} + \int_{0}^{\sigma} \mathfrak{f}(s) \dd s \,\Bigg\vert\, \mathcal{G}_{\nu}\Biggr].
\]
Using the fact that $\bigl(\xi_t + \int_{0}^{t} \mathfrak{f}(s) \dd s\bigr)_{t \leq T}$ is an optional process of class (D) (recall that a process $\mathcal{X}$ is of class (D) if the set $\bigl\{\mathcal{X}_{\sigma} : \sigma \in \mathcal{T}_0^T\bigr\}$ is uniformly integrable). Then, using~\cite[Theorem~22, p.~432]{Dellacherire} (see also~\cite[Remark~b, p.~435]{Dellacherire}), we can find a strong optional supermartingale that aggregates the family $\bigl\{\tilde{Y}(\nu) : \nu \in \mathcal{T}_{0,T}\bigr\}$. That is, there exists a strong optional supermartingale $(\tilde{Y}_t)_{t \leq T}$ such that $\tilde{Y}_{\sigma} = \tilde{Y}(\sigma)$ for all $\sigma \in \mathcal{T}_{0,T}$. Thus, by definition, we have $\overline{\overline{\!Y}}(\nu) = \tilde{Y}_{\sigma} - \int_{0}^{\sigma} \mathfrak{f}(s) \dd s$ for all $\sigma \in \mathcal{T}_{0,T}$. Therefore, the process $(\overline{\overline{\!Y}}_t)_{t \leq T} \coloneqq \bigl(\tilde{Y}_{t} - \int_{0}^{t} \mathfrak{f}(s) \dd s\bigr)_{t \leq T}$ aggregates the family $\bigl\{\overline{\overline{\!Y}}(\nu) : \nu \in \mathcal{T}_{0,T}\bigr\}$. Additionally, from~\cite[Theorem~4, p.~408]{Dellacherire}, we derive that almost all trajectories of the process $(\overline{\overline{\!Y}}_t )_{t \leq T}$ have finite left and right limits on $[0,T]$. Now, from an integration by parts formula (see, e.g., \cite[Theorem~II.31]{bookProtter}), we have $e^{\beta A_{\nu}} \int_{\nu}^{T} e^{-\beta A_s} \dd A_s = \frac{1}{\beta} \bigl(1 - e^{\beta (A_{\nu} - A_{T})}\bigr) \leq \frac{1}{\beta}$ for any $(\sigma, \nu) \in \mathcal{T}_{0,T} \times \mathcal{T}_{0,{\sigma}}$. Then, using Jensen's, Young's, and HÃ¶lder's inequalities, we obtain, for any $\nu \in \mathcal{T}_{0,T}$,
\begin{equation*}
\begin{split}
e^{\frac{\beta}{2}A_{\nu}}\abs{\overline{\overline{\!Y}}_{\nu}}&amp; \leq \sqrt{2}\mathbb{E}{\left[\left\{\esssup_{\sigma \in \mathcal{T}_\nu^T \vphantom{\mathcal{T}_0^T}}e^{\beta A_{\nu}}\abs{\xi_{\sigma}}^2+e^{\beta A_{\nu}}\Biggl(\int_{\nu}^{T}e^{-\beta A_s}\dd A_s\Biggr)\int_{\nu}^{T}e^{\beta A_s}\Biggl\lvert\frac{\mathfrak{f}(s)}{\alpha_s}\Biggr\rvert^2 \dd s\right\}^{\frac{1}{2}}\,\middle\vert\, \mathcal{G}_{\nu}\right]} \\
&amp; \leq \sqrt{2}\mathbb{E}{\left[\left\{\esssup_{\sigma \in \mathcal{T}_0^T}e^{2 \beta A_{\sigma}}\abs{\xi_{\sigma}}^2+\frac{1}{\beta}\int_{0}^{T}e^{\beta A_s}\Biggl\lvert\frac{\mathfrak{f}(s)}{\alpha_s}\Biggr\rvert^2 \dd s\right\}^{\frac{1}{2}}\,\middle\vert\, \mathcal{G}_{\nu}\right]}.
\end{split}
\end{equation*}
Taking the essential supremum above over all $\nu \in \mathcal{T}_{0,T}$ on both sides, using the fact that every martingale in $\mathbb{G}$ has an RCLL modification (see, e.g., \cite[Theorem~I.9]{bookProtter}), and Doob's maximal quadratic inequality (see, e.g., \cite[Theorem~I.1.43]{bookJacod}), we conclude that
\[
\norm{\overline{\overline{\!Y}}}^2_{\mathcal{S}^2_{\beta}} \leq 8 \Biggl(\norm{\xi}^2_{\mathcal{S}^{2}_{2\beta}} + \biggl\lVert\frac{\mathfrak{f}(\,\cdot\,)}{\alpha_{\cdot}}\biggr\rVert^2_{\mathcal{H}^{2}_{\beta}}\Biggr) &lt; +\infty,
\]
i.e., $\overline{\overline{\!Y}} \in \mathcal{S}^2_{\beta}$ for any $\beta &gt; 0$. Using this and $\frac{\mathfrak{f}(\,\cdot\,)}{\alpha} \in \mathcal{H}^2_{\beta}$, we deduce that the process $(\tilde{Y}_t)_{t \leq T}$ belongs to $\mathcal{S}^2_{\beta}$; in particular, it is a strong optional supermartingale of class (D). Applying Mertens' decomposition (see, e.g., \cite[Theorem~A.1]{GriBasic}), we get the existence of a uniformly integrable RCLL martingale $(\mathcal{N}_t)_{t \leq T}$, a non-decreasing right-continuous predictable process $(K_t)_{t \leq T}$ (with $K_0 = 0$) such that $\mathbb{E}[K_T] &lt; \infty$, and a non-decreasing right-continuous adapted purely discontinuous process $(C_t)_{t \leq T}$ (with $C_{0-} = 0$) such that $\mathbb{E}[C_T] &lt; \infty$, with the following decomposition $\tilde{Y}_{\sigma} = \mathcal{N}_{\sigma} - K_{\sigma} - C_{\sigma-}$, a.s., for each $\sigma \in \mathcal{T}_{0,T}$. Moreover, it is not hard to show that
\[
\mathbb{E}\bigl[(K_T + C_{T-})^2\bigr] \leq \mathfrak{c}_{\beta} \Biggl(\norm{\xi}^2_{\mathcal{S}^{2}_{2\beta}} + \biggl\lVert\frac{\mathfrak{f}(\,\cdot\,)}{\alpha_{\cdot}}\biggr\rVert^2_{\mathcal{H}^{2}_{\beta}}\Biggr) &lt; +\infty
\]
for some constant $\mathfrak{c}_{\beta} &gt; 0$. In particular, $K_T$ and $C_{T-} (= C_T)$ are square-integrable random variables, which implies that $(K, C) \in \mathcal{S}^2 \times \mathcal{S}^2$. Consequently, the process $(\mathcal{N}_t)_{t \leq T}$ is a square-integrable martingale. In light of Theorem~\ref{theorem of representation}, there exist unique predictable processes $(Z, U) \in \mathcal{H}^2 \times \mathcal{U}^2_{\gamma}$ such that $\dd\mathcal{N}_t = Z_t \dd W_t + U_t \dd M_t$ for all $t \in [0,T]$. Henceforth, the process $(\overline{\overline{\!Y}}_t, Z_t, U_t, K_t, C_t)_{t \leq T}$ satisfies~\basicequation{}. Note that, as $\overline{\overline{\!Y}}_{\sigma} = \overline{\overline{\!Y}}({\sigma}) \geq \xi_{\sigma}$ for any $\sigma \in \mathcal{T}_{0,T}$, using the optional section theorem (see, e.g., \cite[Theorem~IV.4.5]{bookHe}), we derive that $\overline{\overline{\!Y}}$ verifies Definition~\ref{def:RBSDE}\eqref{def:RBSDE_3}. Let us now concentrate on the minimality and Skorokhod condition. From~\basicequation{} and~\cite[Remark~A.4]{GriBasic}, we have $\Delta C_{\sigma} = -\Delta_{+} \overline{\overline{\!Y}}_{\sigma}$, then $\Delta C_{\sigma} = \mathds{1}_{\{\overline{\overline{\!Y}}_{\sigma} = \xi_{\sigma}\}} \Delta C_{\sigma}$ a.s., for each $\sigma \in \mathcal{T}_{0,T}$. It follows that $C$ satisfies Definition~\ref{def:RBSDE}\eqref{def:RBSDE_5}. Further, due to~\cite[Proposition~B.11]{Kobylanski}, for each $\sigma \in \mathcal{T}_{0,T}^p$, we have
\[
\int_{0}^{T} \mathds{1}_{\{\overline{\overline{\!Y}}_s &gt; \xi_s\}} \dd K^c_s = 0 \;\text{ a.s.\ \quad and \quad}\Delta K^d_{\sigma} = \mathds{1}_{\{\overline{\overline{\!Y}}_{\sigma-} = \xi_{\sigma-}\}} \Delta K^d_{\sigma} \;\text{ a.s.}
\]
Then the process $(K_t)_{t \leq T}$ satisfies Definition~\ref{def:RBSDE}\eqref{def:RBSDE_4}. By setting $Y \coloneqq \overline{\overline{\!Y}}$, we derive that the process $(Y_t, Z_t, U_t, K_t, C_t)_{t \leq T}$ verifies Definition~\ref{def:RBSDE}\eqref{def:RBSDE_2}--\eqref{def:RBSDE_5}. As $(Y_t, K_t, C_t)_{t \leq T} \in \mathcal{S}^2_{\beta} \times \mathcal{S}^2 \times \mathcal{S}^2$, it remains to show that $(Y_t, Z_t, U_t)_{t \leq T} \in \mathcal{H}^{2,\alpha}_{\beta} \times \mathcal{H}^2_{\beta} \times \mathcal{U}^2_{\gamma,\beta}$ for any $\beta &gt; 1$. To this end, it suffices to prove the result in the case where $A_T$ is a bounded random variable. Then, following the same argument as in Lemma~\ref{Uniqueness} with the Skorokhod and minimality conditions on $(K_t)_{t \leq T}$ and $(C_t)_{t \leq T}$, and finally applying Fatou's Lemma, we obtain the following estimation:
\[
\norm{Y}^2_{\mathcal{H}^{2,\alpha}_{\beta}}+\norm{Z}^2_{\mathcal{H}^{2}_{\beta}}+\norm{U}^2_{\mathcal{U}^{2}_{\gamma,\beta}}
\leq3\norm{\xi}^2_{\mathcal{S}^{2}_{2\beta}}+\frac{1}{\beta-1}\Biggl\lVert\frac{\mathfrak{f}(\,\cdot\,)}{\alpha_{\cdot}}\Biggr\rVert^2_{\mathcal{H}^{2}_{\beta}}+\mathbb{E}\bigl[\abs{K_T}^2\bigr]+\mathbb{E}\bigl[\abs{C_T}^2\bigr]&lt;+\infty.
\]
The uniqueness of the solution follows from Lemma~\ref{Uniqueness} (for the state process $(Y_t)_{t \leq T}$), Mertens' decomposition (for the non-decreasing processes $(K_t, C_t)_{t \leq T}$), and the martingale representation property (for the control processes $(Z_t, U_t)_{t \leq T}$). As a conclusion, the process $(Y_t, Z_t, U_t, K_t, C_t)_{t \leq T}$ is the unique solution of the RBSDE~\sbasicequation{} associated with the data $\bigl(\xi, \mathfrak{f}(\,\cdot\,)\bigr)$ such that the state process $(Y_t)_{t \leq T}$ is characterized by~\eqref{Representation of Y}, completing the proof.
\end{proof}
\begin{corollary}\label{app well defined}
Let $\beta &gt; 1$ and $(y_t, z_t, u_t)_{t \leq T} \in \bigl(\mathcal{S}^2_{\beta} \cap \mathcal{H}^{2,\alpha}_{\beta} \bigr) \times \mathcal{H}^2_{\beta} \times \mathcal{U}^2_{\gamma,\beta}$. There exists a unique process $(Y_t, Z_t, U_t, K_t, C_t)_{t \leq T} \in \bigl(\mathcal{S}^2_{\beta} \cap \mathcal{H}^{2,\alpha}_{\beta} \bigr) \times \mathcal{H}^2_{\beta} \times \mathcal{U}^2_{\gamma,\beta} \times \mathcal{S}^2 \times \mathcal{S}^2$ that is the unique solution to the RBSDE~\sbasicequation{} associated with the data $\bigl(\xi, \hat{\mathfrak{f}}(\,\cdot\,)\bigr)$, where $\hat{\mathfrak{f}}(t,\omega) \coloneqq f\bigl(t,\omega,y_{t}(\omega), z_{t}(\omega), u_{t}(\omega)\bigr)$. Moreover, the state process $(Y_t)_{t \leq T}$ is characterized as the value function of the optimal stopping problem~\eqref{Representation of Y} expressed in terms of $\bigl(\xi, \hat{\mathfrak{f}}(\,\cdot\,)\bigr)$.
\end{corollary}
\begin{proof}
This is a direct consequence of Theorem~\ref{basic theorem for f independent of y,z,u} and the fact that
\[
\mathbb{E}{\left[\int_{0}^{T} e^{\beta A_s} \Biggl\lvert\frac{\hat{\mathfrak{f}}(s)}{\alpha_s}\Biggr\rvert^2 \dd s\right]} \leq 4 {\left(\norm{y}^2_{\mathcal{H}^{2,\alpha}_{\beta}} + \norm{z}^2_{\mathcal{H}^{2}_{\beta}} + \norm{u}^2_{\mathcal{U}^{2}_{\gamma,\beta}} + \Biggl\lVert\frac{f(\,\cdot\,,0,0,0)}{\alpha_{\cdot}}\Biggr\rVert^2_{\mathcal{H}^{2}_{\beta}}\right)} &lt; +\infty,
\]
using the $\gamma$-stochastic Lipschitz property of $f$.
\end{proof}

\subsubsection{Case of a general \texorpdfstring{$\gamma$}{gamma}-stochastic Lipschitz generator}

Now, we prove the main result of the paper.
\begin{theorem}
Let $f$ be a $\gamma$-stochastic Lipschitz driver and $\xi$ be a barrier. The RBSDE~\sbasicequation{} associated with the data $(\xi, f)$ has a unique solution $(Y_t, Z_t, U_t, K_t, C_t)_{t \leq T}$, which belongs to $\bigl(\mathcal{S}^2_{\beta} \cap \mathcal{H}^{2,\alpha}_{\beta} \bigr) \times \mathcal{H}^2_{\beta} \times \mathcal{U}^2_{\gamma,\beta} \times \mathcal{S}^2 \times \mathcal{S}^2$ for each $\beta &gt; 4$.
\end{theorem}
\begin{proof}
For any $\beta &gt; 1$, we denote by $\mathfrak{B}^2_{\beta}$ the space $\mathcal{H}^{2,\alpha}_{\beta} \times \mathcal{H}^2_{\beta} \times \mathcal{U}^2_{\gamma,\beta}$, which we equip with the norm $\norm{\,\cdot\,}_{\mathfrak{B}^2_{\beta}}$ defined by $\norm{(Y, Z, U)}_{\mathfrak{B}^2_{\beta}} \coloneqq \norm{Y}^2_{\mathcal{H}^{2,\alpha}_{\beta}} + \norm{Z}^2_{\mathcal{H}^{2}_{\beta}} + \norm{U}^2_{\mathcal{U}^{2}_{\gamma,\beta}}$, for $(Y_t, Z_t, U_t)_{t \leq T} \in \mathcal{H}^{2,\alpha}_{\beta} \times \mathcal{H}^2_{\beta} \times \mathcal{U}^2_{\gamma,\beta}$. Clearly, the space $\bigl(\mathfrak{B}^2_{\beta}, \norm{\,\cdot\,}_{\mathfrak{B}^2_{\beta}}\bigr)$ is a Banach space. Now, we define a mapping $\Psi \colon \mathfrak{B}^2_{\beta} \to \mathfrak{B}^2_{\beta}$ as follows: given $(y_t, z_t, u_t)_{t \leq T} \in \mathfrak{B}^2_{\beta}$, we set $\Psi(y, z, u) = (Y, Z, U)$, where $(Y_t, Z_t, U_t)_{t \leq T}$ are the first three components of the solution of the RBSDE~\sbasicequation{} associated with the data $\bigl(\xi, \hat{f}(\,\cdot\,)\bigr)$, where $\hat{f}(t, \omega) \coloneqq f\bigl(t, \omega, y_{t}(\omega), z_{t}(\omega), u_{t}(\omega)\bigr)$. From Corollary~\ref{app well defined}, the mapping $\Psi$ is well defined. We aim to prove that the functional $\Psi$ is a contraction on $\mathfrak{B}^2_{\beta}$ and hence, by Banach's fixed-point theorem, it admits a unique fixed point $(Y_t, Z_t, U_t)_{t \leq T} \in \mathfrak{B}^2_{\beta}$, i.e., $\Psi(Y, Z, U) = (Y, Z, U)$, which corresponds to the three components of the unique solution of the RBSDE~\sbasicequation{} associated with parameters $(\xi, f)$. The fact that $Y \in \mathcal{S}^2_{\beta}$ can be simply derived from the characterization of the process $Y$ given by~\eqref{Representation of Y} for a general $\gamma$-stochastic Lipschitz generator $f$ and the same argument as in the proof of Theorem~\ref{basic theorem for f independent of y,z,u} for $(\overline{\overline{\!Y}}_t)_{t \leq T}$, together with the fact that $\biggl((Y, Z, U), \xi, \frac{f(\,\cdot\,, 0, 0, 0)}{\alpha}\biggr) \in \mathfrak{B}^2_{\beta} \times \mathcal{S}^2_{2\beta} \times \mathcal{H}^2_{\beta}$. To show this result, we consider $(y'_t, z'_t, u'_t)_{t \leq T}$ to be another element of $\mathfrak{B}^2_{\beta}$ and define $\Psi(y', z', u') = (Y', Z', U')$, where $(Y'_t, Z'_t, U'_t, K'_t, C'_t)_{t \leq T}$ is the solution of the RBSDE~\sbasicequation{} associated with the data $\bigl(\xi, \hat{f}'(\,\cdot\,)\bigr)$, where $\hat{f}'(t, \omega) \coloneqq f\bigl(t, \omega, y'_{t}(\omega), z'_{t}(\omega), u'_{t}(\omega)\bigr)$. Set $\bar{\mathfrak{S}} \coloneqq \mathfrak{S} - \mathfrak{S}'$ for $\mathfrak{S} \in \bigl\{Y, Z, U, K, C, y, z, u\bigr\}$. Using the standard computations performed in Lemma~\ref{Uniqueness}, together with the $\gamma$-stochastic Lipschitz condition of $f$, the Skorokhod and minimality conditions on $(K_t)_{t \leq T}$, $(K'_t)_{t \leq T}$, $(C_t)_{t \leq T}$, and $(C'_t)_{t \leq T}$, we get
\begin{multline*}
\beta\mathbb{E}{\left[\int_{0}^{T}e^{\beta A_s} \abs{\bar{Y}_s}^2 \dd A_s\right]}+\mathbb{E}{\left[\int_{0}^{T}e^{\beta A_s} \abs{\bar{Z}_s}^2 \dd s\right]}+\mathbb{E}{\left[\int_{0}^{T}e^{\beta A_s} \abs{\bar{U}_s}^2 \gamma_s \dd s\right]}\\
\leq (\beta-1)\mathbb{E}{\left[\int_{0}^{T}e^{\beta A_s}\abs{\bar{Y}_s}^2\dd A_s\right]}+\frac{3}{\beta-1}\mathbb{E}{\left[\int_{0}^{T}e^{\beta A_s}\biggl\{\abs{\bar{y}_s}^2\dd A_s+\bigl(\abs{\bar{z}_s}^2+\abs{\bar{u}_s}^2\gamma_s\bigr)\dd s\biggr\}\right]}.
\end{multline*}
Choosing $\beta &gt; 4$, we derive that
\[
\bigl\lVert\Psi(y, z, u) - \Psi(y', z', u')\bigr\rVert_{\mathfrak{B}^2_{\beta}} \leq \mathfrak{c}_{\beta} \bigl\lVert(y, z, u) - (y', z', u')\bigr\rVert_{\mathfrak{B}^2_{\beta}}
\]
for a constant $0 &lt; \mathfrak{c}_{\beta} &lt; 1$ depending on $\beta$ (specifically, $\mathfrak{c}_{\beta} = 3 / (\beta - 1)$). Therefore, the mapping $\Psi$ is a strict contraction. By the Banach fixed-point theorem, we get that $\Psi$ has a unique fixed point in $\mathfrak{B}^2_{\beta}$, denoted by $(Y_t, Z_t, U_t)_{t \leq T}$, where the process $(Y_t, Z_t, U_t)_{t \leq T}$ is equal to the first three components of the solution $(Y_t, Z_t, U_t, K_t, C_t)_{t \leq T}$ to the RBSDE~\sbasicequation{} associated with the data $(\xi, f)$. Consequently, $(Y_t, Z_t, U_t, K_t, C_t)_{t \leq T}$ is the unique solution to the RBSDE~\sbasicequation{} with the data $(\xi, f)$.
\end{proof}

\section*{Acknowledgments}

The corresponding author expresses gratitude to Prof.\ Mohamed El Otmani for his insightful guidance and productive discussions. We would also like to thank the anonymous referee for their careful review and valuable suggestions, which helped improve this article.

\section*{Declaration of interests}

The author does not work for, advise, own shares in, or receive funds from any organization that could benefit from this article, and has declared no affiliations other than their research organizations.

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