# Reflected backward stochastic differential equations with rough drivers

Hanwu Li  Research Center for Mathematics and Interdisciplinary Sciences,
Shandong University, Qingdao 266237, China; Frontiers Science Center for
Nonlinear Expectations (Ministry of Education), Shandong University, Qingdao
266237, China<EMAIL_ADDRESS>,  Huilin Zhang  Research Center for
Mathematics and Interdisciplinary Sciences, Shandong University, Qingdao
266237, China; Institute of Mathematics, Humboldt University, Berlin, 10099,
Germany<EMAIL_ADDRESS>and  Kuan Zhang  Research Center for
Mathematics and Interdisciplinary Sciences, Shandong University, Qingdao
266237, China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study reflected backward stochastic differential equations
driven by rough paths (rough RBSDEs), which can be understood as a
probabilistic representation of nonlinear rough partial differential equations
(rough PDEs) or stochastic partial differential equations (SPDEs) with
obstacles. The well-posedness in the sense of [20] is proved via a variant of
Doss-Sussman transformation. Moreover, we show that our rough RBSDEs can be
approximated by a sequence of penalized BSDEs with rough drivers. As
applications, firstly we provide the equivalence between rough RBSDEs and the
obstacle problem of rough PDEs. Secondly, we show the solution of rough RBSDE
solves the corresponding optimal stopping problem.

###### Key words and phrases:

rough paths, reflected BSDEs, rough PDEs

###### 1991 Mathematics Subject Classification:

60L20, 60H10, 60L50

###### Contents

  1. 1 Introduction
    1. 1.1 Background and motivation
    2. 1.2 Structure of the paper
  2. 2 Preliminaries
    1. 2.1 Notations and basic definitions
    2. 2.2 Property of rough flows
  3. 3 Rough reflected BSDE
    1. 3.1 Well-posedness of rough RBSDE
    2. 3.2 Approximation via penalization
  4. 4 Applications
    1. 4.1 Obstacle problem for rough PDEs
    2. 4.2 Optimal stopping problem
  5. Appendix A Results on RBSDEs and rough BSDEs
  6. Appendix B Some facts about PDE with obstacles
  7. Appendix C Some elementary facts about increasing processes

##  1\. Introduction

###  1.1. Background and motivation

In this paper we focus on the reflected backward stochastic differential
equations (RBSDEs) involving an extra rough integral $d\mathbf{X}_{t}.$
Specifically, for $p>2,$ we are interested in the equation

(1.1) $\left\\{\begin{aligned}
&Y_{t}=\xi+\int_{t}^{T}f(r,S_{r},Y_{r},Z_{r})dr+\int_{t}^{T}H(S_{r},Y_{r})d\mathbf{X}_{r}+K_{T}-K_{t}-\int_{t}^{T}Z_{r}dW_{r},\\\
&Y_{t}\geq L_{t},\ t\in[0,T]\textrm{ and
}\int_{0}^{T}(Y_{r}-L_{r})dK_{r}=0,\end{aligned}\right.$

where

$\int_{t}^{T}H(S_{r},Y_{r})d\mathbf{X}_{r}=\sum_{i=1}^{l}\int_{t}^{T}H_{i}(S_{r},Y_{r})d\mathbf{X}^{i}_{r},$

the path $\mathbf{X}$ is a so called $p$-variation geometric rough path, and
the diffusion process $S$ satisfies

(1.2)
$S_{t}=x+\int_{0}^{t}b(r,S_{r})dr+\int_{0}^{t}\sigma(r,S_{r})dW_{r},\quad
t\in[0,T].$

In addition, we focus on the case when $f$ exhibits quadratic growth in $z.$
When $H\equiv 0,$ Eq. (1.1) degenerates to RBSDEs with quadratic growth, as
studied by, e.g., [10, 33, 35, 44].

The nonlinear backward stochastic differential equations (BSDEs), introduced
by Pardoux-Peng [40], has been a famous area of research in stochastic
analysis, with significant applications in various fields such as finance,
control theory, and mathematical physics. The BSDE has the following form

(1.3) $Y_{t}=\xi+\int_{t}^{T}f(r,Y_{r},Z_{r})dr-\int_{t}^{T}Z_{r}dW_{r},\quad
t\in[0,T],$

where $\xi$ is $\mathcal{F}_{T}$-measurable, and the solution of (1.3) is a
pair of processes $(Y,Z),$ with $Z$ derived from the martingale representation
theorem to ensure $Y$ is adapted. The concept of RBSDEs, first introduced by
El Karoui et al. [24], has expanded this theory to incorporate constraints
represented by the reflection term. Compared with classical BSDEs, the RBSDE
introduced by [24] requires an additional adapted increasing process
$K:\Omega\times[0,T]\rightarrow\mathbb{R}$ such that

(1.4) $\left\\{\begin{aligned}
&Y_{t}=\xi+\int_{t}^{T}f(r,Y_{r},Z_{r})dr-\int_{t}^{T}Z_{r}dW_{r}+K_{T}-K_{t},\\\
&Y_{t}\geq L_{t},\ t\in[0,T]\textrm{ and
}\int_{0}^{T}(Y_{r}-L_{r})dK_{r}=0.\end{aligned}\right.$

where the obstacle $L$ is continuous, adapted and $L_{T}\leq\xi.$

Our interest in RBSDE (1.1) is motivated by the following three related
problems.

(a) The first motivation is to solve the following obstacle problem of rough
partial differential equations (rough PDEs)

(1.5) $\left\\{\begin{aligned}
&du(t,x)+\mathcal{L}u(t,x)dt+H(x,u(t,x))d\mathbf{X}_{t}=0,\\\
&u(T,x)=g(x),\quad u(t,x)\geq l(t,x),\quad t\in[0,T],\end{aligned}\right.$

with

$\mathcal{L}u(t,x)=\frac{1}{2}\text{Tr}\left\\{\sigma\sigma^{\top}(t,x)D^{2}_{xx}u(t,x)\right\\}+b^{\top}(t,x)D_{x}u(t,x)+f\big{(}t,x,u(t,x),\sigma^{\top}(t,x)D_{x}u(t,x)\big{)}.$

Rough PDEs, which can be viewed both as an extension of classical PDEs with
singular drivers as well as a pathwise interpretation of SPDEs, have been
studied by various authors in different forms, see e.g., [12, 18, 19, 22, 30].
However, for the reflected or obstacle case, research on such rough
differential equations is very limited. Aida [1] and Deya et al. [17] study
the reflected rough differential equation in the deterministic case, but rough
PDEs with reflection in the infinite dimensional case remains a widely open
area. If the driver $\mathbf{X}$ is a smooth path, it is well-understood that
the solution $u(t,x)$ to (1.5) can be represented by a solution to RBSDE via
the so-called nonlinear Feynman-Kac formula. In the same spirit, when
$\mathbf{X}$ is rough, one expects to have the same equivalence between (1.5)
and rough RBSDEs (see Section 5.1). On the other hand, when $d\mathrm{X}_{t}$
term is a pathwise realization of Brownian integral, (1.5) corresponds to an
obstacle problem for SPDEs, which is well studied by Matoussi-Stoica [39].
Thus, the well-posedness of our rough RBSDE and thus (1.5) would nontrivially
generalize and solve the obstacle problem for SPDEs driven by rough stochastic
noises, e.g. fractional Brownian motion or strong Markov process which can be
lifted to rough signals (see e.g. [28, Chapter 15]).

(b) The second motivation comes from the reflected backward doubly stochastic
differential equations (RBDSDEs). The RBDSDEs have the form

(1.6) $\left\\{\begin{aligned}
&Y_{t}=\xi+\int_{t}^{T}f(r,S_{r},Y_{r},Z_{r})dr+\int_{t}^{T}H(S_{r},Y_{r})d\overset{\leftarrow}{\mathrm{B}}_{r}+K_{T}-K_{t}-\int_{t}^{T}Z_{r}dW_{r},\\\
&Y_{t}\geq L_{t},\ t\in[0,T]\textrm{ and
}\int_{0}^{T}(Y_{r}-L_{r})dK_{r}=0,\end{aligned}\right.$

where $d\overset{\leftarrow}{\mathrm{B}}_{r}$ is a so-called backward Itô
integral. We refer to, e.g., [3, 4, 5, 6, 8, 11, 36, 39, 42] for more
discussion on RBDSDEs, which plays an important role in SPDEs, infinite
dimensional stochastic control and nonlinear filtering. Our research on RBSDE
with a rough driver can provide a new perspective for the study of RBDSDE.
That is, letting $\mathbf{B}_{t}$ be the enhanced Brownian motion over
$\mathrm{B}_{t},$ each realization of the path of $\mathbf{B}_{\cdot}$ forms a
geometric rough path, and thus the RBDSDE can a.s. be regarded as our equation
driven by each sample path realization. Indeed, the Stratonovich integral
$H(S_{r},Y_{r})\circ d\mathrm{B}_{r}$ corresponds to the rough integral
$H(S_{r},Y_{r})d\mathbf{B}_{r},$ while the integral
$H(S_{r},Y_{r})d\overset{\leftarrow}{\mathrm{B}}_{r}$ in (1.6) equals to the
Stratonovich integral plus an extra $dr$ term when $H(\cdot,\cdot)$ is smooth
enough. One of the advantages of the rough path approach is the continuous
dependence of the equation solution on the trajectory, which may contribute to
the stability or numerical calculation of potential applications of RBDSDEs in
stochastic control or nonlinear filtering.

(c) The third motivation originates from the American option pricing under
stochastic or rough volatility. Consider a financial market model with
$P^{i},i=0,1,2,...,d$ being the price of $n+1$ assets. $P^{0}$ is the price of
a riskless bond satisfying

$dP^{0}_{t}=P^{0}_{t}r_{t}dt,$

where $r$ is the short rate. Others represent the price of stocks with
dynamics given by

(1.7)
$dP^{i}_{t}=P^{i}_{t}\left[b^{i}_{t}dt+\sum_{j=1}^{d}\left[\sigma^{i,j}_{t}dW^{j}_{t}+\lambda^{i,j}_{t}d\mathbf{X}^{j}_{t}\right]\right],$

where $b$ is the appreciation rate, $\sigma$ and $\lambda$ describe the
volatility, and $d\mathbf{X}_{t}$ term characterizes the undetected
information of the market which is represented as the integral of rough path.
Such rough volatility model (1.7) arises from the stochastic volatility model,
where the volatility of the stock price is a rough or diffusion process. In a
very recent study, Bank et al. [7] show that treating the asset price as a
solution to a rough stochastic differential equation (rough SDE) can bypass
difficulties coming from strict “controlledness assumptions” (such assumptions
can be seen in, e.g., [26, Definition 4.6]) central to rough paths and
regularity structures. However, [7] only deals with the pricing for European
options. When considering the American option pricing problem, let $L$ be its
payoff process. Then via a similar argument of El Karoui et al. [25] in a
rough setting, it is suggested that the super hedging price of this American
option under rough volatility should equal to the solution of the
corresponding rough RBSDE with obstacle $L$.

The theory of rough paths originates from Lyons’ seminal work [38]. It has
been developed by several excellent researchers in last two decades, see [26]
and references therein. The theory of rough SDE turns out to be a powerful
tool in areas of SPDE, nonlinear filtering and dynamical systems, see e.g.,
[2, 15, 21]. To our best knowledge, Diehl-Friz [20] firstly introduce the
rough BSDEs and show its well-posedness by applying a continuity argument.
Diehl-Zhang [23] consider a rough BSDE with a Young driver ($p$-variation with
$p\in[1,2)$ and prove its well-posedness in a strong sense. Very recently,
Liang-Tang [37] extend the result of [20] to a multidimensional case, and find
a strong unique solution to rough BSDE when the rough integral is linear. For
the reflected rough differential equation, except [1, 17] mentioned in our
motivation, we may refer to [29, 43] and the references therein for more
recent works. As far as we know, this is the first research on RBSDEs driven
by rough paths and thus rough PDEs with obstacles.

The contributions of this paper are threefold. First, we systematically
explore the characteristics and properties of rough RBSDEs, including the
stability, the comparison theorem, and moreover the construction and
approximation of solutions via penalization, which can be viewed as a
numerical approximation to nonlinear rough PDE or SPDE with obstacles. Second,
as applications, we relate rough RBSDEs with the obstacle problem within the
rough path framework, and then we derive a nonlinear Feynman-Kac formula for
the corresponding rough PDEs. Third, we delve into an optimal stopping problem
that involves RBSDEs and rough BSDEs. Roughly speaking, the solution to an
RBSDE driven by a rough path can be represented by the essential supremum over
a class of solutions to rough BSDEs with random terminal time $\tau$.

The main idea for the well-posedness part, developed from [20], is to
approximate the rough path via lifted smooth paths. Then solve the approximate
RBSDE via a modified Doss-Sussman transformation. However, by following such
idea, we will encounter fundamental problems when dealing with the
penalization approximation and applications of the rough RBSDE, since the
convergence of the solution via the prescribed approximation is rather weak.
More precisely, the modified Doss-Sussman transformation fails to provide the
continuous dependence of reflections $K_{t}$ on the rough driver, and thus the
convergence of $K_{t}$ fails to follow as the convergence of $(Y_{t},Z_{t}).$
In addition, it is not so clear if the continuity obtained by the
approximation argument is consistent with the penalized approximation or the
viscosity solution theory with obstacles. These two problems come down to the
fact that the rough integral $\int_{0}^{t}H(S_{r},Y^{m}_{r})d\mathbf{X}_{r}$
exists solely in a formal sense, while RBSDE theory relies heavily on Itô’s
theory to derive the convergence of
$\int_{0}^{t}H(S_{r},Y^{m}_{r})d\mathbf{X}_{r}$ when $Y^{m}\rightarrow Y$
uniformly. Moreover, for the obstacle problem involving rough PDEs, the mere
pointwise convergence of the approximated solution is not enough for the
existence of solutions to such rough PDEs. Indeed, the locally uniform
convergence of viscosity solutions to the limit should be considered.

We handle these problems separately. For the trouble caused by the definition
of solutions, we rely on the monotonic property of the diffeomorphism induced
by Doss-Sussman transformation, and the continuous property of the obstacle.
Indeed, these two properties make the convergence of reflections equivalent to
the weak convergence of corresponding measures via analyzing the uniform
convergence of cumulative distribution functions. For the similar trouble in
the penalization argument, inspired by Zhang [45, Section 6.3.3], we transform
the problem of convergence of penalization sequences into the study of
continuous dependence of rough RBSDEs on varying obstacles, and demonstrate
the convergence of these obstacles through the Doss-Sussman transformation.
That is, we treat the sequence of penalized rough BSDEs as a new kind of rough
RBSDEs, whose obstacle is the smaller value between the original obstacle and
the penalization solution. And then we employ the continuity and
differentiability of the Doss-Sussman transformation to study the trajectory
semi-continuity of the limits of penalization solutions. This semi-continuity
provides the a.s. uniform convergence of the new obstacles. Finally, the
desired convergence of penalization sequence follows by the stability of rough
RBSDEs. For the well-posedness of rough PDE with obstacles, motivated by [20],
to avoid the lack of uniform convergence, we employ a so-called “semi-relaxed
limits” technique, which is usually used to establish the invariance of
viscosity solution under limit. More specifically, this technique gives a tool
to establish a Dini type argument so that the uniform convergence can be
derived by the equivalence of the relaxed upper and lower limit. And the
equivalence of relaxed limits is a consequence of the comparison theorem for
viscosity solutions of transformed PDEs, as outlined in [14, Section 6].

###  1.2. Structure of the paper

The paper is organized as follows. In Section 2, we recall some notations and
facts on geometric rough path theory. In Section 3, we give the well-posedness
of rough RBSDEs and introduce some properties of our equations including the
continuity of solution map, the comparison result, and prove the penalization
approximation of our equations. In Section 4, we study the obstacle problem of
rough PDEs, and investigate the optimal stopping problem related to Eq. (1.1).
Finally in Appendix A we recall some concepts and results of RBSDEs and rough
BSDEs, and in Appendix B we give some facts of viscosity solutions for
obstacle problems and prove the comparison theorem that will be used in
Section 4.1.

##  2\. Preliminaries

###  2.1. Notations and basic definitions

Let $l,m\in\mathbb{N},$ and $(E,d_{E})$ be a complete metric space, and they
may vary in the context. Denote the Euclidean norm by $|\cdot|,$ of which the
dimension depends on the context. $T>0$ is a fixed time horizon. Let
$\left\\{W_{t}\right\\}_{t\in[0,T]}$ be a $d$-dimensional standard Brownian
motion in $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\in[0,T]},\mathbb{P})$ with
$(\mathcal{F}_{t})_{t\in[0,T]}$ being the augmented filtration of $W.$

Stochastic norms:  For any $k\in[1,\infty)$ and $[s,t]\subset[0,T],$ denote by

  * •

$\mathrm{H}^{k}_{[s,t]}(\mathbb{R}^{l})$ the space of progressively measurable
processes $X$ on $[s,t]$ with values in $\mathbb{R}^{l}$, equipped with norms

$\|X\|_{\mathrm{H}^{k};[s,t]}:=\left\\{\mathbb{E}\left[\left|\int_{s}^{t}|X_{r}|^{2}dr\right|^{\frac{k}{2}}\right]\right\\}^{\frac{1}{k}};$

  * •

$L^{k}(\mathcal{F}_{t},\mathbb{R}^{l})$ the space of random variable
$\xi\in\mathcal{F}_{t}$ with values in $\mathbb{R}^{l}$, equipped with norms

$\|\xi\|_{L^{k}}:=\left\\{\mathbb{E}\left[|\xi|^{k}\right]\right\\}^{\frac{1}{k}};$

  * •

$\mathrm{H}^{\infty}_{[s,t]}(\mathbb{R}^{l})$ the subspace of
$\mathrm{H}^{k}_{[s,t]}(\mathbb{R}^{l})$, which has finite essential supremum
norm

$\|Y\|_{\mathrm{H}^{\infty};[s,t]}:=\operatorname*{ess\,sup}\limits_{\omega\in\Omega,r\in[s,t]}|Y_{r}(\omega)|;$

  * •

$L^{\infty}(\mathcal{F}_{t},\mathbb{R}^{l})$ the space of all $\xi\in
L^{k}(\mathcal{F}_{t},\mathbb{R}^{l})$ such that
$\|\xi\|_{L^{\infty}}<\infty$, where

$\|\xi\|_{L^{\infty}}:=\operatorname*{ess\,sup}\limits_{\omega\in\Omega}|\xi(\omega)|;$

  * •

$\mathrm{I}^{k}_{[s,t]}$ the space of continuous increasing adapted process
$K$ on $[s,t]$ with $K_{s}=0$ and $K_{t}\in
L^{k}(\mathcal{F}_{t},\mathbb{R})$.

In the following we write $\mathrm{H}^{k}_{[s,t]}$ and
$L^{k}(\mathcal{F}_{t})$ short for $\mathrm{H}^{k}_{[s,t]}(\mathbb{R}^{l})$
and $L^{k}(\mathcal{F}_{t},\mathbb{R}^{l})$ if $l=1.$

Spaces of functions:  For any open set $U\subset\mathbb{R}^{m},$
$k,k_{1},k_{2}\in\mathbb{N},$ and $\lambda>1,$ denote by

  * •

$C(U,E)$ (resp. $C_{b}(U,E)$) the space of continuous (bounded and continuous)
function $g:U\rightarrow E$;

  * •

$\|\cdot\|_{\infty;U}$ the uniform norm for $g\in
C_{b}(U,\mathbb{R}^{l}):U\rightarrow\mathbb{R}^{l},$ which is defined by

$\|g\|_{\infty;U}:=\sup\limits_{r\in U}|g(r)|;$

  * •

$C^{k}(\mathbb{R}^{m},\mathbb{R}^{l})$ the space of $k$-th continuously
differentiable function $g\in C(\mathbb{R}^{m},\mathbb{R}^{l})$.

  * •

$C^{k_{1},k_{2}}(\mathbb{R}\times\mathbb{R}^{m},\mathbb{R}^{l})$ the space of
function $g\in C(\mathbb{R}\times\mathbb{R}^{m},\mathbb{R}^{l})$ such that for
each $i=1,2,...,k_{1}$ and $|\alpha|_{\text{index}}\leq k_{2},$
$\partial^{i}_{t}\partial^{\alpha}_{x}g(t,x)$ exists and is continuous, where
$\alpha\in\mathbb{N}^{m},$
$|\alpha|_{\text{index}}:=\sum\limits_{i=1}^{m}\alpha_{i},$ and
$\partial^{\alpha}_{x}g:=\frac{\partial^{|\alpha|_{\text{index}}}g}{\partial
x_{1}^{\alpha_{1}}\partial x_{2}^{\alpha_{2}}...\partial x_{m}^{\alpha_{m}}}$;

  * •

$C^{k_{1},k_{2}}([0,T]\times\mathbb{R}^{m},\mathbb{R}^{l})$ the space of
function $g\in C([0,T]\times\mathbb{R}^{m},\mathbb{R}^{l}),$ which is defined
in the same manners of
$C^{k_{1},k_{2}}(\mathbb{R}\times\mathbb{R}^{m},\mathbb{R}^{l}),$ requiring
that the derivatives continuous on $[0,T]\times\mathbb{R}^{m};$

  * •

$D_{x}g$ and $D^{2}_{xx}g$ the gradient and Hessian matrix of $g\in
C^{1}(\mathbb{R}^{m},\mathbb{R})$ and $g\in C^{2}(\mathbb{R}^{m},\mathbb{R})$
respectively, and denote by $D^{2}_{xy}g:=D_{x}(D_{y}g)$ and
$D^{3}_{xxy}g:=D^{2}_{xx}(D_{y}g)$ for $g\in
C^{1,1}(\mathbb{R}^{m}\times\mathbb{R},\mathbb{R})$ and $g\in
C^{2,1}(\mathbb{R}^{m}\times\mathbb{R},\mathbb{R})$ respectively;

  * •

$\text{Lip}^{1}(U,\mathbb{R}^{l})$ the space of function $V\in
C(U,\mathbb{R}^{l})$ such that
$\|V\|_{\text{Lip}^{1}(U,\mathbb{R}^{l})}<\infty,$ where

$\|V\|_{\text{Lip}^{1}(U,\mathbb{R}^{l})}:=\sup\limits_{x^{1},x^{2}\in
U:x^{1}\neq
x^{2}}\frac{|V(x^{1})-V(x^{2})|}{|x^{1}-x^{2}|}+\|V(\cdot)\|_{\infty;U};$

  * •

$\text{Lip}^{\lambda}(U,\mathbb{R}^{l})$ the space of function $V\in
C(U,\mathbb{R}^{l})$ such that
$\|V\|_{\text{Lip}^{\lambda}(U,\mathbb{R}^{l})}<\infty,$ where

$\|V\|_{\text{Lip}^{\lambda}(U,\mathbb{R}^{l})}:=\sum\limits_{|\alpha|_{\text{index}}=\lfloor\lambda\rfloor}\sup\limits_{x^{1},x^{2}\in
U:x^{1}\neq
x^{2}}\frac{|\partial^{\alpha}_{x}V(x^{1})-\partial^{\alpha}_{x}V(x^{2})|}{|x^{1}-x^{2}|^{\lambda-\lfloor\lambda\rfloor}}+\sum\limits_{|\beta|_{\text{index}}\leq\lfloor\lambda\rfloor}\|\partial^{\beta}_{x}V(\cdot)\|_{\infty;U},$

$\lfloor\lambda\rfloor$ represent the largest integer strictly smaller than
$\lambda,$ and $\alpha,\beta\in\mathbb{N}^{m}$;

  * •

$\text{Lip}^{1}([0,T],\mathbb{R}^{l})$ and
$\text{Lip}^{\lambda}([0,T],\mathbb{R}^{l})$ the space of $V\in
C([0,T],\mathbb{R}^{l}),$ which is defined in the same manners of
$\text{Lip}^{1}(U,\mathbb{R}^{l})$ and
$\text{Lip}^{\lambda}(U,\mathbb{R}^{l})$ respectively;

  * •

$\text{Lip}_{\text{loc}}^{1}(\mathbb{R}^{m},\mathbb{R}^{l})$ (resp.
$\text{Lip}_{\text{loc}}^{\lambda}(\mathbb{R}^{m},\mathbb{R}^{l})$) the local
$1$-Lipschitz ($\lambda$-Lipschitz) continuous space, which is the set of all
functions $V\in C(\mathbb{R}^{m},\mathbb{R}^{l})$ such that
$V\in\text{Lip}^{1}(U,\mathbb{R}^{l})$
($\text{Lip}^{\lambda}(U,\mathbb{R}^{l})$) for every bounded open subset
$U\subset\mathbb{R}^{m}$.

For $g\in C^{1}([0,T],\mathbb{R})$ ($g\in C^{1}(\mathbb{R},\mathbb{R}).$
resp.) we also write $\dot{g}_{t}:=D_{t}g(t)$ ($g^{\prime}(y):=D_{y}g(y)$
resp.).

The $p$-variation metric space and geometric rough path space:  For $p\geq 1,$
let $[p]$ be the integer part of $p$ and denote by

  * •

$C^{p\text{-var}}([0,T],E)$ the space of function $f\in C([0,T],E)$ with
$\|f\|_{p\text{-var};[s,t]}<\infty,$ where

$\|f\|_{p\text{-var};[s,t]}:=\sup\limits_{\pi\in\mathcal{P}[s,t]}\left\\{\sum\limits_{[t_{i},t_{i+1}]\in\pi}|d_{E}(f_{t_{i}},f_{t_{i+1}})|^{p}\right\\}^{\frac{1}{p}},$

$\mathcal{P}[s,t]$ is the set of all partitions of the interval $[s,t],$ and
$|\pi|$ is the mesh size of a partition $\pi\in\mathcal{P}[s,t]$;

  * •

$G^{[p]}(\mathbb{R}^{l})\subset\oplus_{k=0}^{[p]}(\mathbb{R}^{l})^{\otimes k}$
the free nilpotent group of step $[p]$ over $R^{l},$ which is equipped with
the product $*$ defined in [28, Chapter 7.3.2] ($\otimes$ in the literature)
and the Carnot-Caratheodory metric $d(\cdot,\cdot)$ defined in [28, Chapter
7.5.2];

  * •

$C^{p\text{-var}}([0,T],G^{[p]}(\mathbb{R}^{l}))$ the $p$-variation weak
geometric rough path space, equipped with the $p$-variation metric
$d_{p\text{-var};[0,T]}(\mathbf{X}^{1},\mathbf{X}^{2}):=\sup\limits_{\pi\in\mathcal{P}[0,T]}\left\\{\sum\limits_{[t_{i},t_{i+1}]\in\pi}|d(\mathbf{X}^{1}_{t_{i},t_{i+1}},\mathbf{X}^{2}_{t_{i},t_{i+1}})|\right\\}^{\frac{1}{p}},$
where $\mathbf{X}_{s,t}:=\mathbf{X}^{-1}_{s}*\mathbf{X}_{t}$;

  * •

$C^{0,p\text{-var}}([0,T],G^{[p]}(\mathbb{R}^{l}))$ the $p$-variation
geometric rough path space, which is the closure of
$C^{1\text{-var}}([0,T],G^{[p]}(\mathbb{R}^{l}))$ under
$d_{p\text{-var};[0,T]}.$

######  Remark 2.1.

Referring to [28, Theorem 8.13],
$\left(C^{p\text{-var}}([0,T],G^{[p]}(\mathbb{R}^{l})),d_{p\text{-var};[0,T]}\right)$
forms a complete, non-separable metric space. Comparatively, [28, Proposition
8.25] shows that the metric space
$\left(C^{0,p\text{-var}}([0,T],G^{[p]}(\mathbb{R}^{l})),d_{p\text{-var};[0,T]}\right)$
forms a polish space.

Recall the definition of the lift of bounded variation path given by [28,
Definition 7.2].

######  Definition 2.1.

The step-$[p]$ signature of $\mathrm{X}\in
C^{1\text{-var}}([0,T],\mathbb{R}^{l})$ is given by

$\displaystyle S_{[p]}(\mathrm{X})_{0,T}$
$\displaystyle:=\left(1,\int_{0<u<T}d\mathrm{X}_{u},\int_{0<u_{1}<u_{2}<T}d\mathrm{X}_{u_{1}}\otimes
d\mathrm{X}_{u_{2}},\cdots,\int_{0<u_{1}<...<u_{[p]}<T}d\mathrm{X}_{u_{1}}\otimes\cdots\otimes\mathrm{X}_{u_{[p]}}\right)$
$\displaystyle\in G^{[p]}(\mathbb{R}^{l}).$

The path $\mathbf{X}_{t}:=S_{[p]}(\mathrm{X})_{0,t}$ is called the
(step-$[p]$) lift of $\mathrm{X}.$

######  Remark 2.2.

For $\mathbf{X}\in C^{0,p\text{-var}}([0,T],G^{[p]}(\mathbb{R}^{l})),$
according to [28, Lemma 8.21] we can find
$\mathrm{X}^{n}\in\bigcap\limits_{k\geq
1}\text{Lip}^{k}([0,T],\mathbb{R}^{l})$ (i.e., $\mathrm{X}^{n}$ are smooth in
time) such that the lifts of $\mathrm{X}^{n},$ denoted by $\mathbf{X}^{n},$
converges to $\mathbf{X}$ in
$C^{p\text{-var}}([0,T],G^{[p]}(\mathbb{R}^{l})).$

###  2.2. Property of rough flows

Throughout this article, we make the following assumption:

######  Assumption $(H_{p,\gamma})$.

Given a natural number $l\geq 1$ and real numbers $\gamma>p\geq 2$, let
$H(x,\cdot):=(H_{1}(x,\cdot),...,H_{l}(x,\cdot))$ be a collection of vector
fields on $\mathbb{R},$ parameterized by $x\in\mathbb{R}^{d}.$ There exists a
constant $C_{H}>0$ such that

$|H|_{\rm{Lip}^{\gamma+2}(\mathbb{R}^{d+1},\mathbb{R}^{l})}\leq C_{H}.$

Under Assumption ($(H_{p,\gamma})$), according to [28, Chapter 11], given any
geometric rough path $\mathbf{X}\in
C^{0,p\text{-var}}([0,T],G^{[p]}(\mathbb{R}^{l}))$ and terminal time
$T^{\prime}\in[0,T],$ the following RDE admits a unique solution
$\phi^{T^{\prime}}(t,x,y).$

(2.1)
$\phi^{T^{\prime}}(t,x,y)=y+\int_{t}^{T^{\prime}}H(x,\phi^{T^{\prime}}(r,x,y))d\mathbf{X}_{r},\quad
t\in[0,T^{\prime}].$

In addition,
$\phi^{T^{\prime}}:[0,T^{\prime}]\times\mathbb{R}^{d}\times\mathbb{R}\rightarrow\mathbb{R}$
is a continuous $C^{3}$-diffeomorphism flow on $\mathbb{R}$. We denote the
space of such functions by
$\mathcal{D}_{3}([0,T^{\prime}]\times\mathbb{R}^{d};\mathbb{R}),$ i.e.

$\displaystyle\mathcal{D}_{3}([0,T^{\prime}]\times\mathbb{R}^{d};\mathbb{R}):=\Big{\\{}\phi\Big{|}\
\forall(t,x)\in[0,T^{\prime}]\times\mathbb{R}^{d},$ $\displaystyle\ \alpha\leq
3:D^{\alpha}_{y}(t,x,y),D^{\alpha}_{y}\phi^{-1}(t,x,y)$ $\displaystyle\text{
are continuous on }[0,T^{\prime}]\times\mathbb{R}^{d+1}\Big{\\}}.$

Moreover, the inverse of the map $y\mapsto\phi^{T^{\prime}}(t,x,y)$ satisfies
the following RDE (or ODE when $p=1$)

(2.2)
$(\phi^{T^{\prime}})^{-1}(t,x,y)=y-\int_{t}^{T^{\prime}}D_{y}(\phi^{T^{\prime}})^{-1}(r,x,y)H(x,y)d\mathbf{X}_{r},\quad
t\in[0,T^{\prime}].$

In view of [20, Appendix B], we obtain the following estimates for
$\phi^{T^{\prime}}$ and its derivatives.

######  Lemma 2.2.

Suppose $(H_{p,\gamma})$ holds and $\mathbf{X}\in
C^{0,p\text{-var}}([0,T],G^{[p]}(\mathbb{R}^{l}))$ with

$\|\mathbf{X}\|_{p\text{-var};[0,T]}\leq C_{\mathbf{X}},$

for some $C_{\mathbf{X}}>0$. Then the solution to (2.1) $\phi^{T^{\prime}}$
belongs to $\mathcal{D}_{3}([0,T^{\prime}]\times\mathbb{R}^{d};\mathbb{R}).$
Besides, there exists a positive constant $C=C(C_{H},C_{\mathbf{X}},T)$
independent of $T^{\prime}$ such that, uniformly over $x\in\mathbb{R}^{d},\
y\in\mathbb{R}$ and $t\in[0,T^{\prime}]$

$\displaystyle\max\Bigg{\\{}\left|D_{x}\phi^{T^{\prime}}\right|,\left|D_{y}\phi^{T^{\prime}}\right|,$
$\displaystyle\left|\frac{1}{D_{y}\phi^{T^{\prime}}}\right|,\left|D^{2}_{yy}\phi^{T^{\prime}}\right|,\left|D^{2}_{xy}\phi^{T^{\prime}}\right|,\left|D^{2}_{xx}\phi^{T^{\prime}}\right|,$
$\displaystyle\left|D^{3}_{yyy}\phi^{T^{\prime}}\right|,\left|D^{3}_{xyy}\phi^{T^{\prime}}\right|,\left|D^{3}_{xxy}\phi^{T^{\prime}}\right|\Bigg{\\}}\leq
C.$

where the derivatives of $\phi^{T^{\prime}}$ always take values at $(t,x,y).$
Moreover, for every $\varepsilon>0,$ there exists a positive constant
$\delta=\delta(\varepsilon,C_{H},C_{\mathbf{X}})$ independent of $T^{\prime}$
such that whenever $\|\mathbf{X}\|_{p\text{-var};[s,T^{\prime}]}\leq\delta,$
we have for all $t\in[s,T^{\prime}],$ $x\in\mathbb{R}^{d},$ and
$y\in\mathbb{R},$ the following inequality holds

$\max\bigg{\\{}\left|D_{x}\phi^{T^{\prime}}\right|,\left|D_{y}\phi^{T^{\prime}}-1\right|,\left|D^{2}_{yy}\phi^{T^{\prime}}\right|,\left|D^{2}_{xy}\phi^{T^{\prime}}\right|,\left|D^{2}_{xx}\phi^{T^{\prime}}\right|,\left|D^{3}_{yyy}\phi^{T^{\prime}}\right|,\left|D^{3}_{xyy}\phi^{T^{\prime}}\right|,\left|D^{3}_{xxy}\phi^{T^{\prime}}\right|\bigg{\\}}\leq\varepsilon.$

######  Remark 2.3.

According to [28, Proposition 11.11] we also have for
$(t,x,y)\in[0,T^{\prime}]\times\mathbb{R}^{d}\times\mathbb{R}$

$\displaystyle\max\bigg{\\{}\left|D_{x}(\phi^{T^{\prime}})^{-1}\right|,\left|D_{y}(\phi^{T^{\prime}})^{-1}\right|,$
$\displaystyle\left|D^{2}_{yy}(\phi^{T^{\prime}})^{-1}\right|,\left|D^{2}_{xy}(\phi^{T^{\prime}})^{-1}\right|,\left|D^{2}_{xx}(\phi^{T^{\prime}})^{-1}\right|,\left|D^{3}_{yyy}(\phi^{T^{\prime}})^{-1}\right|,$
$\displaystyle\left|D^{3}_{xyy}(\phi^{T^{\prime}})^{-1}\right|,\left|D^{3}_{xxy}(\phi^{T^{\prime}})^{-1}\right|\bigg{\\}}\leq
C(C_{H},C_{\mathbf{X}},T),$

where the derivatives of $(\phi^{T^{\prime}})^{-1}$ take values at $(t,x,y).$

We also have the following stability result of Eq. (2.1), which is referred to
[20, Lemma B.2] and [28, Theorem 11.13].

######  Lemma 2.3.

Under the same setting as Lemma 2.2, let $\left\\{\mathbf{X}^{n},\ n\geq
1\right\\}$ be a sequence of $p$-variation geometric rough paths that
converges to $\mathbf{X}$ in $p$-variation norm. Denote by
$\phi^{n,T^{\prime}}$ the solution of Eq. (2.1) driven by $\mathbf{X}^{n}.$
Then locally uniformly on
$[0,T^{\prime}]\times\mathbb{R}^{d}\times\mathbb{R}$, i.e. for any compact set
$U\subset\mathbb{R}^{d+1},$ the following convergence holds uniformly on
$[0,T^{\prime}]\times U,$

$\displaystyle\left(\phi^{n,T^{\prime}},D_{x}\phi^{n,T^{\prime}},D_{y}\phi^{n,T^{\prime}},\frac{1}{D_{y}\phi^{n,T^{\prime}}},D^{2}_{yy}\phi^{n,T^{\prime}},D^{2}_{xy}\phi^{n,T^{\prime}},D^{2}_{xx}\phi^{n,T^{\prime}}\right)$
$\displaystyle\ \ \ \ \ \
\longrightarrow\left(\phi^{T^{\prime}},D_{x}\phi^{T^{\prime}},D_{y}\phi^{T^{\prime}},\frac{1}{D_{y}\phi^{T^{\prime}}},D^{2}_{yy}\phi^{T^{\prime}},D^{2}_{xy}\phi^{T^{\prime}},D^{2}_{xx}\phi^{T^{\prime}}\right).$

In addition, let
$\psi^{n,T^{\prime}}(t,x,y):=(\phi^{n,T^{\prime}})^{-1}(t,x,y)$ and
$\psi^{T^{\prime}}(t,x,y):=(\phi^{T^{\prime}})^{-1}(t,x,y).$ Then the above
locally uniform convergence also holds with
$(\phi^{n,T^{\prime}},\phi^{T^{\prime}})$ replaced by
$(\psi^{n,T^{\prime}},\psi^{T^{\prime}})$.

The following lemma will be useful to do induction from subintervals to
subintervals of $[0,T].$

######  Lemma 2.4.

Let $X\in C^{\text{p\text{-var}}}([0,T],E).$ For any fixed $\delta>0,$ set
$N:=\left(\left[\frac{\|X\|_{p\text{-var};[0,T]}}{\delta}\right]+1\right)^{p}.$
Then, there exists a set of positive real numbers
$\left\\{h^{(i)},i=0,...,N\right\\}$ satisfying

$h^{(i)}\leq h^{(i+1)},\quad i=0,\dots,N-1,\quad h^{(N)}=T,\quad h^{(0)}=0,$

and

$\|X\|_{p\text{-var};[T-h^{(i)},T-h^{(i-1)}]}\leq\delta,\quad\text{for all
}i=1,2,...,N.$

###### Proof.

Fix $\delta>0.$ Let $h^{(0)}:=0$ and

$h^{(i)}:=\sup\left\\{h\in[h^{(i-1)},T];\|X\|_{p\text{-var};[T-h,T-h^{(i-1)}]}\leq\delta\right\\},\quad
i=1,2,...,N.$

The reader may easily prove that the set $\left\\{h^{(i)},i=0,...,N\right\\}$
satisfies the desired conditions of the lemma by the superadditivity
$\|X\|^{p}_{p\text{-var};[0,T]}\geq\sum\limits_{i=1}^{N}\|X\|^{p}_{p\text{-var};[T-h^{(i)},T-h^{(i-1)}]}.$
∎

##  3\. Rough reflected BSDE

In this section, we mainly consider the following rough RBSDE driven by
$\mathbf{X},$

(3.1) $\left\\{\begin{aligned}
&Y_{t}=\xi+\int_{t}^{T}f(r,S_{r},Y_{r},Z_{r})dr+\int_{t}^{T}H(S_{r},Y_{r})d\mathbf{X}_{r}+K_{T}-K_{t}-\int_{t}^{T}Z_{r}dW_{r},\\\
&Y_{t}\geq L_{t},\ t\in[0,T]\textrm{ and
}\int_{0}^{T}(Y_{r}-L_{r})dK_{r}=0,\end{aligned}\right.$

where the process $S$ satisfies

(3.2) $S=x+\int_{0}^{t}b(r,S_{r})dr+\int_{0}^{t}\sigma(r,S_{r})dW_{r},\
t\in[0,T].$

In the rest of our paper, we always assume that $Z$ takes values in the
$d$-dimensional row vector.

Throughout this section, we assume that
$f:\Omega\times[0,T]\times\mathbb{R}^{d}\times\mathbb{R}\times\mathbb{R}^{d}\rightarrow\mathbb{R}$
is
$\mathcal{P}\otimes\mathcal{B}(\mathbb{R}^{d})\otimes\mathcal{B}(\mathbb{R})\otimes\mathcal{B}(\mathbb{R}^{d})$-measurable,
where $\mathcal{P}$ denotes the predicable $\sigma$-algebra and $\mathcal{B}$
denotes the Borel $\sigma$-algebra. Let $C_{0}>0$ be a global constant and
$c(\cdot)\in C(\mathbb{R}^{+},\mathbb{R}^{+})$ be a strictly positive and
increasing continuous function with
$\int_{0}^{\infty}\frac{1}{c(y)}dy=\infty.$ We propose the following
assumption for the function $f$.

######  Assumption ($H_{f}$).

  * (A1)

$\mathbb{P}$-a.s., for all
$(t,x,y,z)\in[0,T]\times\mathbb{R}^{d}\times\mathbb{R}\times\mathbb{R}^{d}$,

$|f(t,x,y,z)|\leq c(|y|)+C_{0}|z|^{2}.$

  * (A2)

For all $M>0,$ there exists a constant $C_{M}>0$ such that $\mathbb{P}$-a.s,

$|D_{z}f(t,x,y,z)|\leq C_{M}(1+|z|),\text{ for all
}(t,x,y,z)\in[0,T]\times\mathbb{R}^{d}\times[-M,M]\times\mathbb{R}^{d}.$

  * (A3)

For all $M>0$ and $\varepsilon>0,$ there exists a constant
$C_{\varepsilon,M}>0$ such that $\mathbb{P}$-a.s.,

$D_{y}f(t,x,y,z)\leq C_{\varepsilon,M}+\varepsilon|z|^{2},\text{ for all
}(t,x,y,z)\in[0,T]\times\mathbb{R}^{d}\times[-M,M]\times\mathbb{R}^{d}.$

  * (A4)

For each $m\geq 1,$ there exists a random variable $C^{\prime}_{m}(\omega)>0$
such that $\mathbb{P}$-a.s.,

$|D_{y}f(t,x,y,z)|\leq C^{\prime}_{m},\text{ for all }|(t,x,y,z)|\leq m.$

######  Remark 3.1.

Concerning the above conditions, note that

  * (i).

Conditions (A1) and (A3) are slightly more general comparing with the
corresponding ones in [20]. For example, let
$f(t,x,y,z)=y\log(1+|y|)+|z|^{2}$, and $f$ satisfies (A1). Indeed, by choosing
$c(y)=C+(1+y)\log(1+y)$ with some $C>0$, it holds by applying Young’s
inequality. For Assumption (A3), $D_{y}f$ here is assumed to have quadratic
growth in $z$ while in [20, (F2)] they assume that $D_{y}f$ has a constant
upper bound.

  * (ii).

Condition (A3) is slightly different from [33, (H3)]. Here $c(\cdot)$ is
required to be increasing while (H3) in [33] only requires it to be strictly
positive. The additional monotonic of $c(\cdot)$ is to ensure that there
exists a constant $C>0$ such that

$c(|\phi^{T^{\prime}}(t,x,y)|)\leq C\cdot c(C|y|),$

which will provide upper bound convenient for the proof of Lemma 3.3.

  * (iii).

Here we assume that the partial derivatives of $f$ exist. However, as stated
in [20], the existence of partial derivatives is not necessary. This
requirement can be replaced by the Lipschitz continuous assumption for $f$.

  * (iv).

Compared with [33], we requires an additional condition $(A4)$, which assume
the local boundedness of $D_{y}f.$ This technical condition will be used in
Lemma 3.4 to ensure that a.s.,

$f(t,x,y^{n},z)\rightarrow f(t,x,y,z),\text{ locally uniformly in
}(t,x,z)\text{ as }y_{n}\rightarrow y,$

which will play an essential role in the proof of well-posedness of rough
RBSDEs.

Then, we introduce the assumptions of $b(\cdot)$ and $\sigma(\cdot)$ appearing
in Eq. (3.2).

######  Assumption ($H_{b,\sigma}$).

Assume that $b:\Omega\times[0,T]\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$
and
$\sigma:\Omega\times[0,T]\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d\times
d}$ are progressively measurable and satisfy

(3.3) $|D_{x}b(t,x)|\vee|b(t,x)|\vee|D_{x}\sigma(t,x)|\vee|\sigma(t,x)|\leq
C_{S},\text{ for all }(\omega,t,x)\in\Omega\times[0,T]\times\mathbb{R}^{d},$

where $C_{S}>0$ is a constant.

For the terminal value $\xi$ and the obstacle process $L,$ we assume the
following assumption.

######  Assumption ($H_{L,\xi}$).

Assume that $\xi\in L^{\infty}(\mathcal{F}_{T}),$ and
$L:\Omega\times[0,T]\rightarrow\mathbb{R}$ is a bounded, continuous, and
adapted process with $L_{T}\leq\xi.$ Assume further that

$\|\xi\|_{L^{\infty}}\leq C_{\xi},\quad\|L\|_{\mathrm{H}^{\infty};[0,T]}\leq
C_{L},$

where $C_{\xi},C_{L}>0$ are constants.

###  3.1. Well-posedness of rough RBSDE

Firstly, we consider the case when the driver is $d\mathrm{X}_{r},$ with
$\mathrm{X}\in\bigcap_{\lambda\geq
1}\text{Lip}^{\lambda}([0,T],\mathbb{R}^{l}).$ And we call such $\mathrm{X}$ a
smooth path. Then the equation is written as follows

(3.4)
$\left\\{\begin{array}[]{l}Y_{t}=\xi+\int_{t}^{T}f(r,S_{r},Y_{r},Z_{r})dr+\int_{t}^{T}H(S_{r},Y_{r})\dot{\mathrm{X}}_{r}dr+K_{T}-K_{t}-\int_{t}^{T}Z_{r}dW_{r},\\\
\\\ Y_{t}\geq L_{t},\ t\in[0,T]\textrm{ and
}\int_{0}^{T}(Y_{r}-L_{r})dK_{r}=0,\end{array}\right.$

where the diffusing process $S$ is the solution of Eq. (3.2). Here $H$
satisfies Assumption $(H_{p,\gamma})$, $(L,\xi)$ satisfies Assumption
($H_{L,\xi}$), $f,$ $b$ and $\sigma$ satisfy Assumption ($H_{f}$) and
($H_{b,\sigma}$) respectively.

######  Definition 3.1.

A triplet of processes
$(Y,Z,K)\in\mathrm{H}^{\infty}_{[0,T]}\times\mathrm{H}^{2}_{[0,T]}(\mathbb{R}^{d})\times\mathrm{I}^{1}_{[0,T]}$
is called a solution of reflected BSDE with parameter $(\xi,f,L,H,\mathrm{X})$
on $[0,T]$ if (3.4) holds.

######  Remark 3.2.

Suppose Assumption $(H_{p,\gamma})$, ($H_{f}$), ($H_{b,\sigma}$) and
($H_{L,\xi}$) hold. Assume $\mathrm{X}$ is a smooth path. Then Eq. (3.4) with
parameter $(\xi,f,L,H,\mathrm{X})$ admits a unique solution in the sense of
Definition 3.1, which is a consequence of [33, Theorem 1] and [33, Corollary
1].

######  Lemma 3.2 (Smooth case).

Suppose Assumption $(H_{p,\gamma})$, ($H_{f}$), ($H_{b,\sigma}$) and
($H_{L,\xi}$) hold. Assume $\mathrm{X}$ is smooth. Let $\phi^{T}(t,x,y)$ be
the solution of Eq. (2.1) with $T^{\prime}=T$ and $(Y,Z,K)$ be the unique
solution to the reflected BSDE with parameter $(\xi,f,L,H,\mathrm{X}).$ Then,
the triplet $(\widetilde{Y},\widetilde{Z},\widetilde{K})$ defined by

(3.5) $\begin{split}&\widetilde{Y}_{t}:=(\phi^{T})^{-1}(t,S_{t},Y_{t}),\
\widetilde{K}_{t}:=\int_{0}^{t}\frac{1}{D_{y}\phi^{T}(s,S_{s},\widetilde{Y}_{s})}dK_{s},\\\
&\widetilde{Z}_{t}:=-\frac{(D_{x}\phi^{T}(t,S_{t},\widetilde{Y}_{t}))^{\top}}{D_{y}\phi^{T}(t,S_{t},\widetilde{Y}_{t})}\sigma_{t}+\frac{1}{D_{y}\phi^{T}(t,S_{t},\widetilde{Y}_{t})}Z_{t},\
t\in[0,T],\end{split}$

is the solution of reflected BSDE with parameter
$(\xi,\widetilde{f},\widetilde{L},0,0),$ where (throughout, $\phi^{T}$ and all
its derivatives will always be evaluated at $(t,x,\widetilde{y}),$ and
$b,\sigma$ will always be evaluated at $(t,x)$)
$\widetilde{L}_{t}:=(\phi^{T})^{-1}(t,S_{t},L_{t}),$ and

$\displaystyle\widetilde{f}(t,x,\widetilde{y},\widetilde{z}):=\frac{1}{D_{y}\phi^{T}}\Big{\\{}f(t,x,\phi^{T},D_{y}\phi^{T}\widetilde{z}$
$\displaystyle+\sigma^{\top}D_{x}\phi^{T})+(D_{x}\phi^{T})^{\top}b+\frac{1}{2}\text{Tr}\left\\{D^{2}_{xx}\phi^{T}\sigma\sigma^{\top}\right\\}$
$\displaystyle+(D^{2}_{xy}\phi^{T})^{\top}\sigma\widetilde{z}+\frac{1}{2}D^{2}_{yy}\phi^{T}|\widetilde{z}|^{2}\Big{\\}}.$

###### Proof.

Applying Itô’s formula to $\psi(t,S_{t},Y_{t}),$ where $\psi:=(\phi^{T})^{-1}$
is the $y$-inverse of $\phi^{T}(t,x,y)$ satisfying Eq. (2.2), we have

$\displaystyle\psi(t,S_{t},Y_{t})=$
$\displaystyle\xi+\int_{t}^{T}\bigg{[}D_{y}\psi(\theta_{r})f(r,Y_{r},Z_{r})-(D_{x}\psi(\theta_{r}))^{\top}b_{r}-\frac{1}{2}\text{Tr}\left\\{D^{2}_{xx}\psi(\theta_{r})\sigma_{r}\sigma^{\top}_{r}\right\\}$
$\displaystyle\ \ \ \ \ \
-\frac{1}{2}D^{2}_{yy}\psi(\theta_{r})|Z_{r}|^{2}-(D^{2}_{xy}\psi(\theta_{r}))^{\top}\sigma_{r}Z^{\top}_{r}\bigg{]}dr$
$\displaystyle\ \ \ \ \ \
-\int_{t}^{T}\left[(D_{x}\psi(\theta_{r}))^{\top}\sigma_{r}+D_{y}\psi(\theta_{r})Z_{r}\right]dW_{r}+\int_{t}^{T}D_{y}\psi(\theta_{r})dK_{r},$

where $\theta_{r}:=(r,S_{r},Y_{r}),$ $b_{r}:=b(r,S_{r}),$ and
$\sigma_{r}:=\sigma(r,S_{r}).$ Let
$(\widetilde{Y},\widetilde{Z},\widetilde{K})$ be given by (3.5). Noting that
$\psi(t,x,y)$ is continuously strictly increasing in $y$ and $(Y,Z,K)$ solves
(3.4), we have

  * (i)

$\widetilde{L}_{T}=L_{T}\leq\xi,$

  * (ii)

$\widetilde{K}_{t}$ is increasing,

  * (iii)

$\widetilde{Y}_{t}\geq\widetilde{L}_{t}$, a.s. and
$Y_{t}=L_{t}\Leftrightarrow\widetilde{Y}_{t}=\tilde{L}_{t}.$

It follows that

(3.6)
$\int_{0}^{T}1_{\\{\widetilde{Y}_{t}=\widetilde{L}_{t}\\}}(t)d\widetilde{K}_{t}=\int_{0}^{T}\frac{1_{\\{Y_{t}=L_{t}\\}}(t)}{D_{y}\phi(s,S_{s},\widetilde{Y}_{s})}dK_{t}=0,$

which implies that
$\int_{0}^{T}(\widetilde{Y}_{t}-\widetilde{L}_{t})d\widetilde{K}_{t}=0.$
Therefore, $(\widetilde{Y},\widetilde{Z},\widetilde{K})$ is the solution of
reflected BSDE with parameter $(\xi,\widetilde{f},\widetilde{L},0,0).$ ∎

The objective is to give the definition of the solution to Eq. (3.1) driven by
a geometric rough path $\mathbf{X}.$ Our strategy is to study the transformed
RBSDEs with parameter $(\xi,\widetilde{f},\widetilde{L},0,0)$ suggested by the
Doss-Sussman transformation. Therefore, we need the following property of
$\widetilde{f}(\cdot)$ to ensure that $\widetilde{f}(\cdot)$ fulfills
Assumption (H1) for the well-posedness of classical RBSDEs.

######  Lemma 3.3.

Let $\mathbf{X}\in C^{0,p\text{-var}}([0,T],G^{[p]}(\mathbb{R}^{l})).$ Suppose
Assumption $(H_{p,\gamma})$, ($H_{f}$), and ($H_{b,\sigma}$) hold. Let
$\phi^{T^{\prime}}(t,x,y)$ be the unique solution to Eq. (2.1). Then, for

$\displaystyle\widetilde{f}^{T^{\prime}}(t,x,\widetilde{y},\widetilde{z}):=\frac{1}{D_{y}\phi^{T^{\prime}}}\Bigg{\\{}f\big{(}t,x,\phi^{T^{\prime}},D_{y}\phi^{T^{\prime}}\widetilde{z}$
$\displaystyle+\sigma^{\top}D_{x}\phi^{T^{\prime}}\big{)}+(D_{x}\phi^{T^{\prime}})^{\top}b+\frac{1}{2}\text{Tr}\left\\{D^{2}_{xx}\phi^{T^{\prime}}\sigma\sigma^{\top}\right\\}$
$\displaystyle+(D^{2}_{xy}\phi^{T^{\prime}})^{\top}\sigma\widetilde{z}+\frac{1}{2}D^{2}_{yy}\phi^{T^{\prime}}|\widetilde{z}|^{2}\Bigg{\\}},$

there exists an increasing positive continuous function
$\widetilde{c}(\cdot):\mathbb{R}\rightarrow\mathbb{R}$ depending on
$\|\mathbf{X}\|_{p\text{-var};[0,T^{\prime}]}$ such that
$\int_{0}^{\infty}\frac{dy}{\widetilde{c}(y)}=\infty$ and

  * (i).

there exists a constant $\widetilde{C}_{0}>0$ depending on
$\|\mathbf{X}\|_{p\text{-var};[0,T^{\prime}]}$ such that a.s.,

$\displaystyle|\widetilde{f}^{T^{\prime}}(t,x,\widetilde{y},\widetilde{z})|\leq\widetilde{c}(|\widetilde{y}|)+\widetilde{C}_{0}|\widetilde{z}|^{2},\text{
for all
}(t,x,\widetilde{y},\widetilde{z})\in[0,T^{\prime}]\times\mathbb{R}^{d}\times\mathbb{R}\times\mathbb{R}^{d};$

  * (ii).

for every $M>0,$ there exists a constant $\widetilde{C}_{M}>0$ depending on
$\|\mathbf{X}\|_{p\text{-var};[0,T^{\prime}]}$ such that a.s.,

$|D_{\widetilde{z}}\widetilde{f}^{T^{\prime}}(t,x,\widetilde{y},\widetilde{z})|\leq\widetilde{C}_{M}(1+|\widetilde{z}|),\text{
for all
}(t,x,\widetilde{y},\widetilde{z})\in[0,T^{\prime}]\times\mathbb{R}^{d}\times[-M,M]\times\mathbb{R}^{d};$

  * (iii).

for every $\varepsilon,M>0,$ there exist constants
$\delta_{\varepsilon,M},\widetilde{C}_{\varepsilon,M}>0$ depending on
$\|\mathbf{X}\|_{p\text{-var};[0,T^{\prime}]}$ such that whenever
$s\in[0,T^{\prime})$ and
$\|\mathbf{X}\|_{p\text{-var};[s,T^{\prime}]}\leq\delta_{\varepsilon,M}$, we
have $a.s.,$

$D_{\widetilde{y}}\widetilde{f}^{T^{\prime}}(t,x,\widetilde{y},\widetilde{z})\leq\widetilde{C}_{\varepsilon,M}+\varepsilon|\widetilde{z}|^{2},\text{
for all
}(t,x,\widetilde{y},\widetilde{z})\in[s,T^{\prime}]\times\mathbb{R}^{d}\times[-M,M]\times\mathbb{R}^{d}.$

###### Proof.

In view of [20, Lemma 7], it remains to show the first inequality in $(i)$. By
Assumption ($H_{f}$) we have

$\displaystyle\frac{1}{D_{y}\phi^{T^{\prime}}}\left|f\big{(}t,x,\phi^{T^{\prime}},D_{y}\phi^{T^{\prime}}\widetilde{z}+\sigma^{\top}D_{x}\phi^{T^{\prime}}\big{)}\right|$
$\displaystyle\leq
C^{\prime}c(|\phi^{T^{\prime}}|)+C^{\prime}C_{0}\left|D_{y}\phi^{T^{\prime}}\widetilde{z}+\sigma^{\top}D_{x}\phi^{T^{\prime}}\right|^{2}$
$\displaystyle\leq
C^{\prime}c(C^{\prime\prime}|\widetilde{y}|+C^{\prime\prime\prime})+C^{\prime}C_{0}\left|D_{y}\phi^{T^{\prime}}\widetilde{z}+\sigma^{\top}D_{x}\phi^{T^{\prime}}\right|^{2},$

where (the $\sup_{(t,x,\widetilde{y})}$ takes over
$[0,T^{\prime}]\times\mathbb{R}^{d}\times\mathbb{R}$)

$C^{\prime}:=\sup_{(t,x,\widetilde{y})}\left|\frac{1}{D_{y}\phi^{T^{\prime}}(t,x,\widetilde{y})}\right|,\
C^{\prime\prime}:=\sup_{(t,x,\widetilde{y})}|D_{y}\phi^{T^{\prime}}(t,x,\widetilde{y})|,\
C^{\prime\prime\prime}:=\sup\limits_{(t,x)}|\phi^{T^{\prime}}(t,x,0)|.$

By Lemma 2.2, $C^{\prime}$ and $C^{\prime\prime}$ are finite and depending on
$\|\mathbf{X}\|_{p\text{-var};[0,T^{\prime}]}$. In view of [28, Theorem 10.14]
and Assumption $(H_{p,\gamma})$, $C^{\prime\prime\prime}$ is finite as well.
Again by Lemma 2.2 and Assumption ($H_{b,\sigma}$), there exist constants
$\hat{C}$ and $\widetilde{C}_{0}$ such that

$\displaystyle
C^{\prime}C_{0}\left|D_{y}\phi^{T^{\prime}}\widetilde{z}+\sigma^{\top}D_{x}\phi^{T^{\prime}}\right|^{2}$
$\displaystyle+\frac{1}{D_{y}\phi^{T^{\prime}}}\Big{|}(D_{x}\phi^{T^{\prime}})^{\top}b+\frac{1}{2}\text{Tr}\left\\{D^{2}_{xx}\phi^{T^{\prime}}\sigma\sigma^{\top}\right\\}$
$\displaystyle+(D^{2}_{xy}\phi^{T^{\prime}})^{\top}\sigma\widetilde{z}+\frac{1}{2}D^{2}_{yy}\phi^{T^{\prime}}|\widetilde{z}|^{2}\Big{|}$
$\displaystyle\leq$
$\displaystyle\hat{C}+\widetilde{C}_{0}|\widetilde{z}|^{2}.$

Thus, letting
$\widetilde{c}(\widetilde{y}):=C^{\prime}c(C^{\prime\prime}|\widetilde{y}|+C^{\prime\prime\prime})+\hat{C},$
the desired bound follows. ∎

The following lemma shows the continuity property of
$\widetilde{f}^{T^{\prime}}(t,x,\widetilde{y},\widetilde{z})$ defined above
with respect to $\mathbf{X}\in
C^{0,p\text{-var}}([0,T],G^{[p]}(\mathbb{R}^{l})).$ In view of the continuity
of derivatives of $\phi^{T^{\prime}}$ (Lemma 2.3), it is sufficient to discuss
the continuity of

$\mathbf{X}\mapsto
f^{T^{\prime}}(t,x,\phi^{T^{\prime}}(t,x,\widetilde{y}),D_{y}\phi^{T^{\prime}}(t,x,\widetilde{y})\widetilde{z}+\sigma^{\top}(t,x)D_{x}\phi^{T^{\prime}}(t,x,\widetilde{y})).$

######  Lemma 3.4.

Let
$F:[0,T]\times\mathbb{R}^{d}\times\mathbb{R}\times\mathbb{R}^{d}\rightarrow\mathbb{R}.$
Assume that $\sigma:[0,T]\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d\times
d}$ satisfies ($H_{b,\sigma}$), and $\mathbf{X}^{n}\rightarrow\mathbf{X}^{0}$
in $C^{0,p\text{-var}}([0,T],G^{[p]}(\mathbb{R}^{l})).$ For
$T^{\prime}\in[0,T]$ let $\phi^{n,T^{\prime}}(t,x,y)$ be the solution of Eq.
(2.1) driven by $\mathbf{X}^{n}.$ For $m\geq 1$ set

$O_{m}:=\left\\{(t,x,y,z)\in[0,T]\times\mathbb{R}^{d}\times\mathbb{R}\times\mathbb{R}^{d},\
|(t,x,y,z)|<m\right\\}.$

Assume further that for each $m\geq 1,$ there exists $C_{m}>0$ such that for
all $(t,x,y,z)\in O_{m}$,

$|D_{y}F(t,x,y,z)|\vee|D_{z}F(t,x,y,z)|\leq C_{m}.$

For $n\in\mathbb{N}$, set

$F^{n}(t,x,y,z):=F(t,x,\phi^{n,T^{\prime}}(t,x,y),D_{y}\phi^{n,T^{\prime}}(t,x,y)z+\sigma^{\top}(t,x)D_{x}\phi^{n,T^{\prime}}(t,x,y)).$

Then, we have that locally uniformly on
$[0,T^{\prime}]\times\mathbb{R}^{d}\times\mathbb{R}\times\mathbb{R}^{d}$,

$F^{n}(t,x,y,z)\rightarrow F^{0}(t,x,y,z).$

###### Proof.

Note that by Lemma 2.2 and the boundedness of $\sigma$ by (3.3),

$\Theta:=(t,x,\phi^{n,T^{\prime}}(t,x,y),D_{y}\phi^{n,T^{\prime}}(t,x,y)z+\sigma^{\top}(t,x)D_{x}\phi^{n,T^{\prime}}(t,x,y))$

is bounded uniformly in $n\in\mathbb{N}$ whenever $(t,x,y,z)$ is bounded.
Suppose that both $(t,x,y,z),\Theta\in O_{m},$ then there exists a constant
$C^{\prime}_{m}>0$ such that

$\displaystyle|F^{n}(t,x,y,z)-F^{0}(t,x,y,z)|$ $\displaystyle\leq
C^{\prime}_{m}\Big{(}|\phi^{n,T^{\prime}}(t,x,y)-\phi^{0,T^{\prime}}(t,x,y)|+|D_{y}\phi^{n,T^{\prime}}(t,x,y)-D_{y}\phi^{0,T^{\prime}}(t,x,y)||z|$
$\displaystyle\quad\quad\quad\quad+|D_{x}\phi^{n,T^{\prime}}(t,x,y)-D_{x}\phi^{0,T^{\prime}}(t,x,y)||\sigma(t,x)|\Big{)}.$

Thus the locally uniform convergence of $F^{n}$ follows from Lemma 2.3 and the
boundedness of $\sigma.$

∎

In the following we will give a helpful property of increasing functions,
which implies the pointwise convergence of continuous increasing functions can
be strengthened to the uniform convergence. This property will play an
essential role in the proof of well-posedness of rough RBSDEs.

######  Lemma 3.5.

Let $K^{n}:[0,T]\rightarrow\mathbb{R}$, $n\geq 1,$ and $K\in
C([0,T],\mathbb{R})$ be positive increasing functions such that
$K^{n}_{0}=K_{0}=0.$ Assume that $\lim_{n}K^{n}_{t}=K_{t}$ for every
$t\in[0,T].$ Let $y_{t}\in C([0,T],\mathbb{R})$ be a positive function. Then
it follows that

$\int_{0}^{t}y_{s}dK^{n}_{s}\rightarrow\int_{0}^{t}y_{s}dK_{s}\text{ uniformly
in }t\in[0,T].$

###### Proof.

Denote by $t_{0}:=\sup\\{t\in[0,T];K_{t}=0\\}.$ Since the equality
$K_{t_{0}}-K_{0}=0$ leads to that

$\left|\int_{0}^{t}y_{s}d(K^{n}-K)_{s}\right|=\left|\int_{0}^{t}y_{s}dK^{n}_{s}\right|\leq\|y\|_{\infty;[0,t_{0}]}|K^{n}_{t_{0}}-K^{n}_{0}|,$

we have as $n\rightarrow\infty$

(3.7) $\int_{0}^{t}y_{s}dK^{n}_{s}\rightarrow\int_{0}^{t}y_{s}dK_{s}\text{
uniformly in }t\in[0,t_{0}].$

For $t\in(t_{0},T],$ from the convergence $\lim_{n}K^{n}_{\cdot}=K_{\cdot}$ it
follows that for all $r\in[t_{0},t],$

$(K^{n}_{r}-K^{n}_{t_{0}})\rightarrow(K_{r}-K_{t_{0}})>0\text{ and
}\int_{t_{0}}^{r}\frac{dK^{n}_{s}}{K^{n}_{t}-K^{n}_{t_{0}}}\rightarrow\int_{t_{0}}^{r}\frac{dK_{s}}{K_{t}-K_{t_{0}}}.$

Thus, we have that

(3.8) $dK^{n}_{s}/(K^{n}_{t}-K^{n}_{t_{0}})\rightarrow
dK_{s}/(K_{t}-K_{t_{0}}),\ \text{weakly in probability measure on }[t_{0},t].$

Since $y$ is continuous, by the definition of weak convergence we have

(3.9)
$\int_{t_{0}}^{t}\frac{y_{s}dK^{n}_{s}}{K^{n}_{t}-K^{n}_{t_{0}}}\rightarrow\int_{t_{0}}^{t}\frac{y_{s}dK_{s}}{K_{t}-K_{t_{0}}},\text{
for every }t\in(t_{0},T].$

Let

$\hat{K}^{n}_{t}:=\int_{t_{0}}^{t}y_{s}dK^{n}_{s},\
\hat{K}_{t}:=\int_{t_{0}}^{t}y_{s}dK_{s},\ t\in[t_{0},T].$

According to the convergence
$\lim_{n}(K^{n}_{t}-K^{n}_{t_{0}})=K_{t}-K_{t_{0}}$ and (3.9), we have

(3.10) $\hat{K}^{n}_{t}\rightarrow\hat{K}_{t},\text{ for every
}t\in(t_{0},T].$

Furthermore, by the positivity of $y,$ $\hat{K}^{n},\hat{K}$ are positive
increasing functions with $\hat{K}^{n}_{t_{0}}=\hat{K}_{t_{0}}=0.$ And by the
continuity of $K,$ $\hat{K}$ is also continuous. Set
$t_{0}^{\prime}:=\sup\\{t\in[t_{0},T];\hat{K}_{t}=0\\}.$ Then by repeating the
proof of (3.7) with $(K^{n},K)$ replaced by $(\hat{K}^{n},\hat{K}),$ and
letting the integrand be $1,$ we have that

(3.11)
$\int_{t_{0}}^{t}y_{s}dK^{n}_{s}\rightarrow\int_{t_{0}}^{t}y_{s}dK_{s}\text{
uniformly in }t\in[t_{0},t^{\prime}_{0}].$

By repeating the proof of (3.8) with $t=T,$ we have

(3.12)
$d\hat{K}^{n}_{s}/(\hat{K}^{n}_{T}-\hat{K}^{n}_{t^{\prime}_{0}})\rightarrow
d\hat{K}_{s}/(\hat{K}_{T}-\hat{K}_{t^{\prime}_{0}}),\text{ weakly in
probability measure on }[t^{\prime}_{0},T].$

Note that the cumulative distribution function (CDF) of
$d\hat{K}_{s}/(\hat{K}_{T}-\hat{K}_{t^{\prime}_{0}})$ is continuous. By
applying [41, Chapter 10, Exercise 5.2] to (3.12), the weak convergence of
$d\hat{K}^{n}_{s}/(\hat{K}^{n}_{T}-\hat{K}^{n}_{t^{\prime}_{0}})$ implies the
uniform convergence of CDFs, i.e.,

(3.13)
$\int_{t^{\prime}_{0}}^{t}\frac{y_{s}dK^{n}_{s}}{\hat{K}^{n}_{T}-\hat{K}^{n}_{t^{\prime}_{0}}}\rightarrow\int_{t^{\prime}_{0}}^{t}\frac{y_{s}dK_{s}}{\hat{K}_{T}-\hat{K}_{t^{\prime}_{0}}},\text{
uniformly in }t\in[t^{\prime}_{0},T].$

According to (3.10),
$\hat{K}^{n}_{T}-\hat{K}^{n}_{t_{0}^{\prime}}\rightarrow\hat{K}_{T}-\hat{K}_{t^{\prime}_{0}},$
and then by using (3.13) we have

(3.14)
$\int_{t^{\prime}_{0}}^{t}y_{s}dK^{n}_{s}\rightarrow\int_{t^{\prime}_{0}}^{t}y_{s}dK_{s},\text{
uniformly in }t\in[t^{\prime}_{0},T].$

By (3.7),(3.11), and (3.14), the proof is completed.

∎

Now we are ready to give the proof of our first main result.

######  Theorem 3.6.

Assume $(H_{p,\gamma})$, ($H_{f}$), ($H_{b,\sigma}$), and ($H_{L,\xi}$) hold.
Let $\mathbf{X}\in C^{0,p\text{-var}}([0,T],G^{[p]}(\mathbb{R}^{l}))$ and let
$\mathrm{X}^{n}\in\bigcap_{\lambda\geq
1}\text{Lip}^{\lambda}([0,T],\mathbb{R}^{l})$ for each $n\geq 1.$ Let
$\mathbf{X}^{n}$ be the lift of $\mathrm{X}^{n}$, and $(Y^{n},Z^{n},K^{n})$ be
the unique solutions to Eq. (3.1) with parameter $(\xi,f,L,H,\mathrm{X}^{n}).$
If $\mathbf{X}^{n}\rightarrow\mathbf{X}$ in
$C^{0,p\text{-var}}([0,T],G^{[p]}(\mathbb{R}^{l})),$ then there exists a
triplet
$(Y,Z,K)\in\mathrm{H}^{\infty}_{[0,T]}\times\mathrm{H}^{2}_{[0,T]}(\mathbb{R}^{d})\times\mathrm{I}^{1}_{[0,T]}$
such that

$(Y^{n}_{\cdot},K^{n}_{\cdot})\rightarrow(Y_{\cdot},K_{\cdot})\ \text{
uniformly in }\ [0,T]\ \text{ a.s., }\ Z^{n}\rightarrow Z\ \text{ in }\
\mathrm{H}^{2}_{[0,T]}(\mathbb{R}^{d}),$

and

$\int_{0}^{T}(Y_{r}-L_{r})dK_{r}=0.$

In addition, the triplet $(Y,Z,K)$ is unique in the sense that it does not
depend on the choice of sequence $\\{\mathrm{X}^{n},n\geq 1\\}.$ In this case
we call this $(Y,Z,K)$ the unique solution of Eq. (3.1) with parameter
$(\xi,f,L,H,\mathbf{X}).$

###### Proof.

Step 1. We first show that there exists a constant $M$ independent of $n$,
such that

$\displaystyle\operatorname*{ess\,sup}_{(t,\omega)\in[0,T]\times\Omega}|Y^{n}_{t}|\leq
M.$

In the following, we will denote by $(Y^{0},Z^{0},K^{0})$ and $\mathbf{X}^{0}$
the solution $(Y,Z,K)$ and the rough path $\mathbf{X},$ respectively. For each
$n\geq 0,$ let $\phi^{n,T^{\prime}}(t,x,y)$ be the following solution flow
parameterized by $x\in\mathbb{R}^{d}$ with terminal time $T^{\prime}\in[0,T]$

(3.15)
$\phi^{n,T^{\prime}}(t,x,y)=y+\int_{t}^{T^{\prime}}H(x,\phi^{n}(r,x,y))d\mathrm{X}^{n}_{r},\quad
t\in[0,T^{\prime}].$

Due to the smoothness of $\mathrm{X}^{n}$ for $n\geq 1,$ the above equation is
an ordinary differential equation (ODE) when $n\geq 1$; and it is an RDE when
$n=0.$ By Lemma 2.2, for every $x\in\mathbb{R}$
$\phi^{n,T^{\prime}}(t,x,\cdot)$ is a flow of $C^{3}$-diffeomorphisms in
$\mathbb{R}.$ In addition, Lemma 2.2 shows that
$\phi^{n,T^{\prime}}(t,\cdot,\cdot)$ and its derivatives up to order three are
bounded, and the same holds true for the inverse $y$-inverse
$(\phi^{n,T^{\prime}})^{-1}(t,\cdot,\cdot).$ Furthermore, using Lemma 2.3 we
have the following locally uniformly convergence in
$[0,T^{\prime}]\times\mathbb{R}^{d}\times\mathbb{R}$

(3.16)
$\displaystyle(\phi^{n,T^{\prime}},\frac{1}{D_{y}\phi^{n,T^{\prime}}},D_{y}\phi^{n,T^{\prime}},D^{2}_{yy}\phi^{n,T^{\prime}},D_{x}\phi^{n,T^{\prime}},D^{2}_{xx}\phi^{n,T^{\prime}},D^{2}_{xy}\phi^{n,T^{\prime}})$
$\displaystyle\longrightarrow(\phi^{0,T^{\prime}},\frac{1}{D_{y}\phi^{0,T^{\prime}}},D_{y}\phi^{0,T^{\prime}},D^{2}_{yy}\phi^{0,T^{\prime}},D_{x}\phi^{0,T^{\prime}},D^{2}_{xx}\phi^{0,T^{\prime}},D^{2}_{xy}\phi^{0,T^{\prime}}).$

For $n\geq 0,$ set

(3.17)
$\displaystyle\widetilde{f}^{n,T^{\prime}}(t,x,\widetilde{y},\widetilde{z}):=\frac{1}{D_{y}\phi^{n,T^{\prime}}}\Bigg{\\{}f(t,x,\phi^{n,T^{\prime}},D_{y}\phi^{n,T^{\prime}}\widetilde{z}+\sigma^{\top}D_{x}\phi^{n,T^{\prime}})+(D_{x}\phi^{n,T^{\prime}})^{\top}b$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\frac{1}{2}\text{Tr}\left\\{D^{2}_{xx}\phi^{n,T^{\prime}}\sigma\sigma^{\top}\right\\}+(D^{2}_{xy}\phi^{n,T^{\prime}})^{\top}\sigma\widetilde{z}+\frac{1}{2}D^{2}_{yy}\phi^{n,T^{\prime}}|\widetilde{z}|^{2}\Bigg{\\}},$
$\displaystyle\widetilde{L}^{n,T^{\prime}}_{t}:=(\phi^{n,T^{\prime}})^{-1}(t,S_{t},L_{t}),\quad(t,x,\widetilde{y},\widetilde{z})\in[0,T^{\prime}]\times\mathbb{R}^{d}\times\mathbb{R}\times\mathbb{R}^{d},$

where $\phi^{n,T^{\prime}}$ and all its derivatives are evaluated at
$(t,x,\widetilde{y}),$ and $b,\sigma$ are evaluated at $(t,x).$ For $n\geq 1,$
let $(Y^{n},Z^{n},K^{n})$ be the solution of Eq. (3.4) with terminal time $T$
and parameter $(\xi,f,L,H,\mathrm{X}^{n}).$ By Lemma 3.2, the triplet
$(\widetilde{Y}^{n,T^{\prime}},\widetilde{Z}^{n,T^{\prime}},\widetilde{K}^{n,T^{\prime}})$
solves Eq. (3.4) with terminal time $T^{\prime}$ and parameter
$(Y^{n}_{T^{\prime}},\widetilde{f}^{n,T^{\prime}},\widetilde{L}^{n,T^{\prime}},0,0),$
where

(3.18)
$\displaystyle\widetilde{Y}^{n,T^{\prime}}_{t}:=(\phi^{n,T^{\prime}})^{-1}(t,S_{t},Y^{n}_{t}),\quad\widetilde{K}^{n,T^{\prime}}_{t}:=\int_{0}^{t}\frac{1}{D_{y}\phi^{n,T^{\prime}}(s,S_{s},\widetilde{Y}^{n,T^{\prime}}_{s})}dK^{n}_{s},$
$\displaystyle\widetilde{Z}^{n,T^{\prime}}_{t}:=-\frac{(D_{x}\phi^{n,T^{\prime}}(t,S_{t},\widetilde{Y}^{n,T^{\prime}}_{t}))^{\top}}{D_{y}\phi^{n,T^{\prime}}(t,S_{t},\widetilde{Y}^{n,T^{\prime}}_{t})}\sigma_{t}+\frac{1}{D_{y}\phi^{n,T^{\prime}}(t,S_{t},\widetilde{Y}^{n,T^{\prime}}_{t})}Z^{n}_{t},\quad
t\in[0,T^{\prime}].$

By Lemma 3.3, there exist $\widetilde{C}_{0}>0$ and an increasing positive
continuous function $\widetilde{c}(\cdot)$ such that a.s.

(3.19)
$|\widetilde{f}^{n,T^{\prime}}(t,x,\widetilde{y},\widetilde{z})|\leq\widetilde{c}(|\widetilde{y}|)+\widetilde{C}_{0}|\widetilde{z}|^{2},\text{
for all
}(t,x,\widetilde{y},\widetilde{z})\in[0,T^{\prime}]\times\mathbb{R}^{d}\times\mathbb{R}\times\mathbb{R}^{d}.$

In view of (3.19) and Lemma A.2,

(3.20)
$\displaystyle\operatorname*{ess\,sup}_{(t,\omega)\in[0,T^{\prime}]\times\Omega}|\widetilde{Y}^{n,T^{\prime}}_{t}|$
$\displaystyle\leq\widetilde{M},\text{ and then
}\operatorname*{ess\,sup}_{(t,\omega)\in[0,T]\times\Omega}|Y^{n}_{t}|\leq M,$

where $\widetilde{M}$ is given by estimate (A.2) with $c(\cdot)$ replaced by
$\widetilde{c}(\cdot)$, and

$M:=\sup\limits_{n\geq
1}\left\\{\widetilde{M}\|D_{y}\phi^{n,T}(\cdot,\cdot,\cdot)\|_{\infty;[0,T]\times\mathbb{R}^{d}\times\mathbb{R}}+\|\phi^{n,T}(\cdot,\cdot,0)\|_{\infty;[0,T]\times\mathbb{R}^{d}}\right\\}.$

By Lemma 2.2 and the uniform boundedness of
$\|\mathbf{X}^{n}\|_{p\text{-var};[0,T]},$ $M$ defined above is finite.

Step 2. We show that $(Y^{n},Z^{n},K^{n})$ converges to some
$(Y^{0},Z^{0},K^{0})$ on small interval. Note that
$\sup_{n}\|\mathbf{X}^{n}\|_{p\text{-var};[0,T]}<\infty$. By Lemma 3.3, for
every $\varepsilon,M>0$ there exist $\delta,\widetilde{C}>0$ independent of
$n$ and $T^{\prime}$, such that

  * •

for every $M>0$ there exits $\widetilde{C}_{M}>0$ such that a.s.

$|D_{\widetilde{z}}\widetilde{f}^{n,T^{\prime}}(t,x,\widetilde{y},\widetilde{z})|\leq\widetilde{C}_{M}(1+|\widetilde{z}|),\text{
for all
}(t,x,\widetilde{y},\widetilde{z})\in[0,T^{\prime}]\times\mathbb{R}^{d}\times[-M,M]\times\mathbb{R}^{d};$

  * •

for every $\varepsilon,M>0,$ there exist
$\delta_{\varepsilon,M},\widetilde{C}_{\varepsilon,M}>0$ such that whenever
$s\in[0,T^{\prime})$ and
$\|\mathbf{X}^{n}\|_{p\text{-var};[s,T^{\prime}]}\leq\delta_{\varepsilon,M}$,
we have a.s.

$|D_{\widetilde{y}}\widetilde{f}^{n,T^{\prime}}(t,x,\widetilde{y},\widetilde{z})|\leq\widetilde{C}_{\varepsilon,M}+\varepsilon|\widetilde{z}|^{2}\text{
for all
}(t,x,\widetilde{y},\widetilde{z})\in[s,T^{\prime}]\times\mathbb{R}^{d}\times[-M,M]\times\mathbb{R}^{d}.$

Then for each $T^{\prime}\in(0,T]$ and $n\geq 1,$ in view of (3.19) and the
above facts, $\widetilde{f}^{n,T^{\prime}}$ satisfies the requirement of
Proposition A.5. Thus, by Proposition A.5, there exists $\varepsilon>0$ such
that for every $s\in[0,T^{\prime})$ satisfying
$\|\mathbf{X}^{n}\|_{p\text{-var};[s,T^{\prime}]}\leq\delta_{\varepsilon,\widetilde{M}},$
the Eq. (3.4) with terminal time $T^{\prime}$ and parameter
$(Y^{n}_{T^{\prime}},\widetilde{f}^{n,T^{\prime}},\widetilde{L}^{n,T^{\prime}},0,0)$
admits a unique solution
$(\widetilde{Y}^{n,T^{\prime}}_{\cdot},\widetilde{Z}^{n,T^{\prime}}_{\cdot},\widetilde{K}^{n,T^{\prime}}_{\cdot}-\widetilde{K}^{n,T^{\prime}}_{s})$
on $[s,T^{\prime}]$. Now fix $\varepsilon$ mentioned above, and denote by
$\delta:=\delta_{\varepsilon,\widetilde{M}}$. Since
$\lim_{n}d_{p\text{-var};[0,T]}(\mathbf{X}^{n},\mathbf{X}^{0})=0,$ we can
assume $\sup_{n\geq
1}d_{p\text{-var};[0,T]}(\mathbf{X}^{n},\mathbf{X}^{0})\leq\frac{\delta}{2}$
without loss of generality. By Lemma 2.4, there exists a set of increasing
real numbers $\left\\{h^{(i)},i=0,1,...,N\right\\}$ such that
$[T-h^{(N)},T-h^{(0)}]=[0,T]$ and

$\|\mathbf{X}^{0}\|_{p\text{-var};[T-h^{(i+1)},T-h^{(i)}]}\leq\frac{\delta}{2},$

where $N$ only depends on $\|\mathbf{X}^{0}\|_{p\text{-var};[0,T]}$ and
$\delta.$ Then for $i=0,1,2,...,N-1$ it follows that

$\sup_{n\geq
0}\|\mathbf{X}^{n}\|_{p\text{-var};[T-h^{(i+1)},T-h^{(i)}]}\leq\delta.$

Thus, the Eq. (3.4) with terminal time $T-h^{(i)}$ and parameter
$(Y^{n}_{T-h^{(i)}},\widetilde{f}^{n,T-h^{(i)}},\widetilde{L}^{n,T-h^{(i)}},0,0)$
admits a unique solution
$(\widetilde{Y}^{n,T-h^{(i)}}_{\cdot},\widetilde{Z}^{n,T-h^{(i)}}_{\cdot},\widetilde{K}^{n,T-h^{(i)}}_{\cdot}-\widetilde{K}^{n,T-h^{(i)}}_{T-h^{(i+1)}})$
on the interval $[T-h^{(i+1)},T-h^{(i)}].$

On the other hand, by Lemma 3.4 with $F(t,x,y,z):=f(\omega,t,x,y,z)$ and Lemma
2.3, $\widetilde{f}^{n,T}$ converges to $\widetilde{f}^{0,T}$ locally
uniformly in $(t,x,\widetilde{y},\widetilde{z})$ a.s., and
$\widetilde{L}^{n,T}$ converges to $\widetilde{L}^{0,T}$ uniformly in $t$ a.s.
as $n\rightarrow\infty.$ In view of Lemma A.6 we have

(3.21)
$\begin{split}&\widetilde{Y}^{n,T}_{\cdot}\rightarrow\widetilde{Y}^{0,T},\
\widetilde{K}^{n,T}_{\cdot}-\widetilde{K}^{n,T}_{T-h^{(1)}}\rightarrow\widetilde{K}^{0,T}_{\cdot}-\widetilde{K}^{0,T}_{T-h^{(1)}}\textrm{
uniformly in }[T-h^{(1)},T]\textrm{ a.s.,}\\\
&\widetilde{Z}^{n,T}\rightarrow\widetilde{Z}^{0,T}\textrm{ in
}\mathrm{H}^{2}_{[T-h^{(1)},T]},\end{split}$

where the triplet
$(\widetilde{Y}^{0,T},\widetilde{Z}^{0,T},\widetilde{K}^{0,T})$ uniquely
solves Eq. (3.4) with parameter
$(\xi,\widetilde{f}^{0,T},\widetilde{L}^{0,T},0,0)$ on interval
$[T-h^{(1)},T].$ For $t\in[T-h^{(1)},T]$, we define

(3.22) $\displaystyle Y^{0}_{t}:=\phi^{0,T}(t,S_{t},\widetilde{Y}^{0,T}_{t}),\
K^{0,T}_{t}:=\int_{T-h^{(1)}}^{t}D_{y}\phi^{0,T}(s,S_{s},\widetilde{Y}^{0,T}_{s})d\widetilde{K}^{0,T}_{s},$
$\displaystyle
Z^{0}_{t}:=D_{y}\phi^{0,T}(t,S_{t},\widetilde{Y}^{0,T}_{t})\left[\widetilde{Z}^{0,T}_{t}+\frac{(D_{x}\phi^{0,T}(t,S_{t},\widetilde{Y}^{0,T}_{t}))^{\top}}{D_{y}\phi^{0,T}(t,S_{t},\widetilde{Y}^{0,T}_{t})}\sigma_{t}\right].$

Since $\widetilde{Y}^{0,T}_{t}\geq\widetilde{L}^{0,T}_{t},$
$t\in[T-h^{(1)},T]$ and
$\int_{T-h^{(1)}}^{T}(\widetilde{Y}^{0,T}_{t}-\widetilde{L}^{0,T}_{t})d\widetilde{K}^{0,T}_{t}=0,$
we have for $t\in[T-h^{(1)},T]$ $Y^{0}_{t}\geq L_{t},$ and then
$\int_{T-h^{(1)}}^{T}(Y^{0}_{t}-L_{t})dK^{0,T}_{t}=0$ by a same calculation of
(3.6). Meanwhile, by Eq. (3.18) we have for $t\in[T-h^{(1)},T]$

(3.23) $\displaystyle Y^{n}_{t}:=\phi^{n,T}(t,S_{t},\widetilde{Y}^{n,T}_{t}),\
K^{n}_{t}:=\int_{0}^{t}D_{y}\phi^{n,T}(s,S_{s},\widetilde{Y}^{n,T}_{s})d\widetilde{K}^{n,T}_{s},$
$\displaystyle
Z^{n}_{t}:=D_{y}\phi^{n,T}(t,S_{t},\widetilde{Y}^{n,T}_{t})\left[\widetilde{Z}^{n,T}_{t}+\frac{(D_{x}\phi^{n,T}(t,S_{t},\widetilde{Y}^{n,T}_{t}))^{\top}}{D_{y}\phi^{n,T}(t,S_{t},\widetilde{Y}^{n,T}_{t})}\sigma_{t}\right].$

Fix a sample point $\omega,$ and note that the difference between
$K^{n}_{\cdot}-K^{n}_{T-h^{(1)}}$ and $K^{0,T}_{\cdot}$ is

(3.24)
$\displaystyle\int_{T-h^{(1)}}^{\cdot}\left(D_{y}\phi^{n,T}(s,S_{s},\widetilde{Y}^{n,T}_{s})-D_{y}\phi^{0,T}(s,S_{s},\widetilde{Y}^{0,T}_{s})\right)d\widetilde{K}^{n,T}_{s}$
$\displaystyle+\int_{T-h^{(1)}}^{\cdot}D_{y}\phi^{0,T}(s,S_{s},\widetilde{Y}^{0,T}_{s})d(\widetilde{K}^{n,T}-\widetilde{K}^{0,T})_{s}.$

By the uniform convergence of
$D_{y}\phi^{n,T}(t,S_{t},\widetilde{Y}^{n,T}_{t})\rightarrow
D_{y}\phi^{0,T}(t,S_{t},\widetilde{Y}^{0,T}_{t})$ and the uniform boundedness
of $\widetilde{K}^{n,T}_{T},$ the sequence of the first term of (3.24)
uniformly converges to $0$. In addition, note that $t\mapsto
D_{y}\phi^{0,T}(t,S_{t},\widetilde{Y}^{0,T}_{t})$ is continuous and positive,
and note that
$\widetilde{K}^{n,T}_{t}-\widetilde{K}^{n,T}_{T-h^{(1)}}\rightarrow\widetilde{K}^{0,T}_{t}-\widetilde{K}^{0,T}_{T-h^{(1)}}$
for every $t\in[T-h^{(1)},T].$ By Lemma 3.5 with
$K^{n}_{t}=\widetilde{K}^{n,T}_{t}-\widetilde{K}^{n,T}_{T-h^{(1)}},$
$K_{t}=\widetilde{K}^{0,T}_{t}-\widetilde{K}^{0,T}_{T-h^{(1)}},$ and
$y_{t}=D_{y}\phi^{0,T}(t,S_{t},\widetilde{Y}^{0,T}_{t}),$ we have

$\int_{T-h^{(1)}}^{t}D_{y}\phi^{0,T}(s,S_{s},\widetilde{Y}^{0,T}_{s})d\widetilde{K}^{n,T}_{s}\rightarrow\int_{T-h^{(1)}}^{t}D_{y}\phi^{0,T}(s,S_{s},\widetilde{Y}^{0,T}_{s})d\widetilde{K}^{0,T}_{s}\text{
uniformly in }t\in[T-h^{(1)},T].$

Hence, (3.24) converges to $0$ uniformly in $[T-h^{(1)},T].$ Combining (3.16)
and (LABEL:65) we obtain that

(3.25) $\begin{split}&{Y}^{n}_{\cdot}\rightarrow{Y}^{0}_{\cdot},\
{K}^{n}_{\cdot}-K^{n}_{T-h^{(1)}}\rightarrow{K}^{0,T}\textrm{ uniformly in
}[T-h^{(1)},T]\textrm{ a.s.,}\\\ &{Z}^{n}\rightarrow{Z}^{0}\textrm{ in
}\mathrm{H}^{2}_{[T-h^{(1)},T]}.\end{split}$

Step 3. We show that $(Y^{n},Z^{n},K^{n})$ converges to $(Y,Z,K)$ on the whole
interval and the Skorokhod condition holds by induction. Fix an
$i\in\left\\{0,1,...,N-1\right\\}$ and assume that we already have

${Y}^{n}_{\cdot}\rightarrow{Y}^{0}_{\cdot},\textrm{ uniformly in
}[T-h^{(i)},T]\textrm{ a.s..}$

First we show the above convergence holds on $[T-h^{(i+1)},T]$. Denote by
$(\widetilde{Y}^{0,T-h^{(i)}},\widetilde{Z}^{0,T-h^{(i)}},\widetilde{K}^{0,T-h^{(i)}})$
the unique solution of Eq. (3.4) on interval $[T-h^{(i+1)},T-h^{(i)}]$ with
terminal time $T-h^{(i)}$ and parameter
$(Y^{0}_{T-h^{(i)}},\widetilde{f}^{0,T-h^{(i)}},\widetilde{L}^{0,T-h^{(i)}},0,0).$
Note that $Y^{n}_{T-h^{(i)}}$ converges to $Y^{0}_{T-h^{(i)}}$ a.s. and
$(\widetilde{f}^{n,T-h^{(i)}},\widetilde{L}^{n,T-h^{(i)}})$ converges to
$(\widetilde{f}^{0,T-h^{(i)}},\widetilde{L}^{0,T-h^{(i)}})$ locally uniformly
in $(t,x,\widetilde{y},\widetilde{z}).$ Thus, by Lemma A.6, it follows that

$\displaystyle\widetilde{Y}^{n,T-h^{(i)}}_{\cdot}\rightarrow\widetilde{Y}^{0,T-h^{(i)}}_{\cdot},\
\widetilde{K}^{n,T-h^{(i)}}_{\cdot}-\widetilde{K}^{n,T-h^{(i)}}_{T-h^{(i+1)}}\rightarrow\widetilde{K}^{0,T-h^{(i)}}_{\cdot}-\widetilde{K}^{0,T-h^{(i)}}_{T-h^{(i+1)}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\textrm{ uniformly in
}[T-h^{(i+1)},T-h^{(i)}]\textrm{ a.s.,}$
$\displaystyle\widetilde{Z}^{n,T-h^{(i)}}\rightarrow\widetilde{Z}^{0,T-h^{(i)}}\textrm{
in }\mathrm{H}^{2}_{[T-h^{(i+1)},T-h^{(i)}]}.$

For $t\in[T-h^{(i+1)},T-h^{(i)}]$, set

$\displaystyle
Y^{0}_{t}:=\phi^{0,T-h^{(i)}}(t,S_{t},\widetilde{Y}^{0,T-h^{(i)}}_{t}),\
K^{0,T-h^{(i)}}_{t}:=\int_{T-h^{(i+1)}}^{t}D_{y}\phi^{0,T-h^{(i)}}(s,S_{s},\widetilde{Y}^{0,T-h^{(i)}}_{s})d\widetilde{K}^{0,T-h^{(i)}}_{s},$
$\displaystyle
Z^{0}_{t}:=D_{y}\phi^{0,T-h^{(i)}}(t,S_{t},\widetilde{Y}^{0,T-h^{(i)}}_{t})\left[\widetilde{Z}^{0,T-h^{(i)}}_{t}+\frac{(D_{x}\phi^{0,T-h^{(i)}}(t,S_{t},\widetilde{Y}^{0,T-h^{(i)}}_{t}))^{\top}}{D_{y}\phi^{0,T-h^{(i)}}(t,S_{t},\widetilde{Y}^{0,T-h^{(i)}}_{t})}\sigma_{t}\right].$

Repeating the same argument as $i=0,$ we have the following convergence in the
form of (LABEL:66)

(3.26) $\displaystyle{Y}^{n}_{\cdot}\rightarrow{Y}^{0}_{\cdot},\
{K}^{n}_{\cdot}-K^{n}_{T-h^{(i+1)}}\rightarrow{K}^{0,T-h^{(i)}}_{\cdot}\textrm{
uniformly in }[T-h^{(i+1)},T-h^{(i)}]\textrm{ a.s.,}$
$\displaystyle{Z}^{n}\rightarrow{Z}^{0}\textrm{ in
}\mathrm{H}^{2}_{[T-h^{(i+1)},T-h^{(i)}]},$

and then

$Y^{n}_{\cdot}\rightarrow Y^{0}_{\cdot},\text{ uniformly in
}[T-h^{(i+1)},T]\text{ a.s..}$

Thus, the induction is valid, and then (3.26) holds for every $i=0,1,...,N-1.$
Let $K^{0}_{0}:=0,$ and for each $i=0,1,...,N-1$,
$t\in(T-h^{(i+1)},T-h^{(i)}]$,

$K^{0}_{t}:=\sum_{j=1}^{N-i-1}K^{0,T-h^{(N-j)}}_{T-h^{(N-j)}}+K^{0,T-h^{(i)}}_{t},$

where we stipulate that
$\sum\limits_{j=1}^{0}K^{0,T-h^{(N-j)}}_{T-h^{(N-j)}}\equiv 0.$ Clearly for
$t\in(T-h^{(i+1)},T-h^{(i)}]$ we have

$\left|K^{n}_{t}-K^{0}_{t}\right|=\left|\sum_{j=1}^{N-i-1}\left[K^{n}_{T-h^{(N-j)}}-K^{n}_{T-h^{(N-j+1)}}-K^{0,T-h^{(N-j)}}_{T-h^{(N-j)}}\right]+K^{n}_{t}-K^{n}_{T-h^{(i+1)}}-K^{0,T-h^{(i)}}_{t}\right|.$

Thus, as $n\rightarrow\infty$ we have

(3.27)
$\displaystyle\sup_{t\in[0,T]}\left|Y^{n}_{t}-Y^{0}_{t}\right|\leq\sum_{i=0}^{N-1}\sup_{T-h^{(i+1)}<t\leq
T-h^{(i)}}\left|Y^{n}_{t}-Y^{0}_{t}\right|\rightarrow 0,\quad\text{a.s.,}$
$\displaystyle\sup_{t\in[0,T]}\left|K^{n}_{t}-K^{0}_{t}\right|\leq\sum_{i=0}^{N-1}\sup_{T-h^{(i+1)}<t\leq
T-h^{(i)}}\left|K^{n}_{t}-K^{n}_{T-h^{(i+1)}}-K^{0,T-h^{(i)}}_{t}\right|\rightarrow
0,\quad\text{a.s.,}$
$\displaystyle\mathbb{E}\left[\int_{0}^{T}\left|Z^{n}_{t}-Z^{0}_{t}\right|^{2}dt\right]\leq\sum_{i=0}^{N-1}\mathbb{E}\left[\int_{T-h^{(i+1)}}^{T-h^{(i)}}\left|Z^{n}_{t}-Z^{0}_{t}\right|^{2}dt\right]\rightarrow
0.$

Furthermore, by the same calculation of (3.6), we have for $t\in[0,T],$

$Y^{0}_{t}\geq L_{t},\ \text{and }\int_{0}^{T}(Y^{0}_{r}-L_{r})dK^{0}_{t}=0.$

Finally, $(Y^{0},Z^{0},K^{0})$ is independent of the approximation sequence
since the sequence of $\mathrm{X}^{n}$ is arbitrarily chosen in the beginning.
The proof is completed. ∎

In the above theorem, we have shown that
$\|Z\|_{\mathrm{H}^{2};[0,T]}<\infty.$ In the following we improve the
integrability via proving the boundedness of the BMO norm of $Z$, where

$\|Z\|_{\text{BMO}}:=\operatorname*{ess\,sup}_{(t,\omega)\in[0,T]\times\Omega}\left|\mathbb{E}_{t}\left[\int_{t}^{T}|Z_{r}|^{2}dr\right]\right|^{\frac{1}{2}}.$

Consequently, we have the $L^{2}$-boundedness of $K,$ which is essential to
the penalization method introduced in Subsection 3.2.

######  Proposition 3.7.

Under the same assumption of Theorem 3.6, let $(Y,Z,K)$ be the solution of Eq.
(3.4) with parameter $(\xi,f,L,H,\mathbf{X}).$ Then there exists a constant
$C>0$ depending on $\|\mathbf{X}\|_{p\text{-var};[0,T]}$ such that

$\|Z\|_{\text{BMO}}\leq C,\ \text{ and }\ \|K_{T}\|_{L^{2}}\leq C.$

###### Proof.

For each $n\geq 1$ let $(Y^{n},Z^{n},K^{n})$ be given by Theorem 3.6, and
$(\widetilde{Y}^{n,T},\widetilde{Z}^{n,T},\widetilde{K}^{n,T})$ be defined by
(3.18). We first claim that there exits a constant
$\widetilde{C}_{\text{BMO}}>0$ such that

(3.28) $\sup_{n\geq
1}\|\widetilde{Z}^{n,T}\|_{\text{BMO}}\leq\widetilde{C}_{\text{BMO}}.$

If so, by the representation (3.22), and boundedness of the inverse
transformation shown by Lemma 2.2, the above boundedness implies that
$\sup_{n\geq 1}\|Z^{n}\|_{\text{BMO}}\leq C_{\text{BMO}}$ with some constant
$C_{\text{BMO}}>0.$ In addition, recalling that $Z$ is the limit of $Z^{n}$ in
$\mathrm{H}^{2}_{[0,T]}(\mathbb{R}^{d}),$ it follows that
$\int_{t}^{T}|Z^{n}_{r}|^{2}dr\rightarrow\int_{t}^{T}|Z_{r}|^{2}dr$ in
probability. Then we obtain $\|Z\|_{\text{BMO }}\leq C_{\text{BMO}}$ by
Fatou’s Lemma for conditional expectations. Indeed, (3.28) can be proved
similarly as the proof of [44, Proposition 2.2] so we omit the
proof111Although in [44] parameters of the corresponding RBSDE are assumed to
be Markovian, the boundedness of BMO norm can be extended to the case
satisfying $(A1)$ of Assumption (H1)..

To complete the proof, it remains to show the boundedness of
$\|K\|_{\mathrm{I}^{2};[0,T]}.$ Firstly, according to [31, Corollary 2.1], the
estimate (3.28) implies that there exist a constant $C^{\prime}>0$ such that

$\sup_{n\geq 1}\|\widetilde{Z}^{n,T}\|_{\mathrm{H}^{4}_{[0,T]}}\leq
C^{\prime}.$

Secondly, by (3.20) we have $\sup_{n\geq
1}|\widetilde{Y}^{n,T}_{t}|\leq\widetilde{M}$ with some constant
$\widetilde{M}>0.$ Finally, noting that

$\widetilde{K}^{n,T}_{T}=\widetilde{Y}^{n,T}_{0}-\xi-\int_{0}^{T}\widetilde{f}^{n,T}(r,S_{r},\widetilde{Y}^{n,T}_{r},\widetilde{Z}^{n,T}_{r})dr+\int_{0}^{T}\widetilde{Z}^{n,T}_{r}dW_{r},$

we obtain the following inequality from (A1) in ($H_{f}$)

$\sup_{n\geq
1}\mathbb{E}\left[\left|\widetilde{K}^{n,T}_{T}\right|^{2}\right]\leq
C^{\prime\prime}+C^{\prime\prime}\sup_{n\geq
1}\mathbb{E}\left[\left|\int_{0}^{T}|\widetilde{Z}^{n,T}_{r}|^{2}dr\right|^{2}\right]$

with some constant $C^{\prime\prime}>0,$ and then the boundedness of
$\mathbb{E}\left[|K_{T}|^{2}\right]$ follows. ∎

The following comparison theorem is a direct consequence of the comparison
theorem for RBSDEs with quadratic growth (we refer it to [33, Proposition
3.2]) and Theorem 3.6.

######  Corollary 3.8.

Suppose that for each $j=1,2,$ the parameter
$(\xi^{j},f^{j},L^{j},H,\mathbf{X})$ satisfy the conditions in Theorem 3.6.
Assume further that $\xi^{1}\leq\xi^{2},$ $f^{1}\leq f^{2},$ and $L^{1}\leq
L^{2}.$ Let $(Y^{1},Z^{1},K^{1})$ and $(Y^{2},Z^{2},K^{2})$ be the solution of
Eq. (3.4) with parameters $(\xi^{1},f^{1},L^{1},H,\mathbf{X})$ and
$(\xi^{2},f^{2},L^{2},H,\mathbf{X})$ respectively, we have, a.s., for all
$t\in[0,T]$

$Y^{1}_{t}\leq Y^{2}_{t}.$

Now we give a very useful stability result of RBSDEs with rough drivers, which
is an essential tool to obtain results in the following sections.

######  Proposition 3.9.

Suppose that for each $n\geq 0,$
$(\xi^{(n)},f^{(n)},L^{(n)},H^{(n)},\mathbf{X}^{(n)})$ satisfies the
conditions in Theorem 3.6. Assume further that the sequence $f^{n}$ converges
to $f^{0}$ locally uniformly in
$(t,x,y,z)\in[0,T]\times\mathbb{R}^{d}\times\mathbb{R}\times\mathbb{R}^{d}$
a.s., $\xi^{n}$ converges to $\xi^{0}$ a.s., $L^{n}$ converges to $L^{0}$
uniformly in $t\in[0,T]$ a.s., $H^{(n)}$ converges to $H^{(0)}$ in
$\text{Lip}^{\gamma+2}(\mathbb{R}^{d+1},R^{l}),$ and $\mathbf{X}^{(n)}$
converges to $\mathbf{X}^{(0)}$ in
$C^{0,p\text{-var}}([0,T],G^{[p]}(\mathbb{R}^{l})).$ Let
$(Y^{(n)},Z^{(n)},K^{(n)})$ be the solution of Eq. (3.4) with parameter
$(\xi^{(n)},f^{(n)},L^{(n)},H^{(n)},\mathbf{X}^{(n)})$. Then, we have as
$n\rightarrow\infty$

$\displaystyle(Y^{(n)},K^{(n)})$
$\displaystyle\rightarrow(Y^{(0)},K^{(0)})\text{ uniformly on }[0,T],\
\mathbb{P}\text{-a.s.,}$ $\displaystyle Z^{(n)}$ $\displaystyle\rightarrow
Z^{(0)}\text{ in }\mathrm{H}^{2}_{[0,T]}(\mathbb{R}^{d}).$

###### Proof.

The idea is to apply Lemma A.6 to the transformed equations. For
$T^{\prime}\in[0,T)$ and $t\in[T^{\prime},T],$ for $n\geq 0$ denote by
$\phi^{(n),T^{\prime}}(t,x,\widetilde{y})$ the unique solution of the
following RDE on $t\in[0,T^{\prime}]$

(3.29)
$\phi^{(n),T^{\prime}}(t,x,\widetilde{y})=\widetilde{y}+\int_{t}^{T^{\prime}}H^{(n)}(x,\phi^{(n),T^{\prime}}(r,x,\widetilde{y}))d\mathbf{X}^{(n)}_{r}.$

In addition, set

$\displaystyle\widetilde{Y}^{(n),T^{\prime}}_{t}:=(\phi^{(n),T^{\prime}})^{-1}(t,S_{t},Y^{(n)}_{t}),\
\widetilde{K}^{(n),T^{\prime}}_{t}:=\int_{0}^{t}\frac{1}{D_{y}\phi^{(n),T^{\prime}}(s,S_{s},\widetilde{Y}^{(n),T^{\prime}}_{s})}dK^{(n)}_{s},$
$\displaystyle\widetilde{Z}^{(n),T^{\prime}}_{t}:=-\frac{(D_{x}\phi^{(n),T^{\prime}}(t,S_{t},\widetilde{Y}^{(n),T^{\prime}}_{t}))^{\top}}{D_{y}\phi^{(n),T^{\prime}}(t,S_{t},\widetilde{Y}^{(n),T^{\prime}}_{t})}\sigma_{t}+\frac{1}{D_{y}\phi^{(n),T^{\prime}}(t,S_{t},\widetilde{Y}^{(n),T^{\prime}}_{t})}Z^{(n)}_{t},$
$\displaystyle\widetilde{f}^{(n),T^{\prime}}\big{(}t,x,\widetilde{y},\widetilde{z}):=\frac{1}{D_{y}\phi^{(n),T^{\prime}}}\bigg{\\{}f^{(n)}(t,x,\phi^{(n),T^{\prime}},D_{y}\phi^{(n),T^{\prime}}\widetilde{z}+\sigma^{\top}D_{x}\phi^{(n),T^{\prime}}\big{)}$
$\displaystyle\qquad+(D_{x}\phi^{(n),T^{\prime}})^{\top}b+\frac{1}{2}\text{Tr}\left\\{D^{2}_{xx}\phi^{(n),T^{\prime}}\sigma\sigma^{\top}\right\\}+(D^{2}_{xy}\phi^{(n),T^{\prime}})^{\top}\sigma\widetilde{z}+\frac{1}{2}D^{2}_{yy}\phi^{(n),T^{\prime}}|\widetilde{z}|^{2}\bigg{\\}},$

and
$\widetilde{L}^{(n),T^{\prime}}:=(\phi^{(n),T^{\prime}})^{-1}(t,S_{t},L^{(n)}_{t}).$
Here $\phi^{(n),T^{\prime}}$ and all its derivatives are valued at
$(t,x,\widetilde{y}).$ By the same argument as the step 2 of the proof of
Theorem 3.6, there exists a sequence of increasing positive numbers
$\left\\{h^{(i)},i=0,1,2,...,N\right\\}$ with $[T-h^{(N)},T-h^{(0)}]=[0,T]$
such that, for each $n\geq 0$ and each $i\in\left\\{0,1,2,...,N-1\right\\},$

$(\widetilde{Y}^{(n),T-h^{(i)}}_{\cdot},\widetilde{Z}^{(n),T-h^{(i)}}_{\cdot},\widetilde{K}^{(n),T-h^{(i)}}_{\cdot}-\widetilde{K}^{(n),T-h^{(i)}}_{T-h^{(i+1)}})$

uniquely solves Eq. (3.4) on $[T-h^{(i+1)},T-h^{(i)}]$ with terminal time
$T-h^{(i)}$ and parameter
$(Y^{(n),T-h^{(i)}}_{T-h^{(i)}},\widetilde{f}^{(n),T-h^{(i)}},\widetilde{L}^{(n),T-h^{(i)}},0,0).$

Now we verify conditions to apply Lemma A.6 to
$(Y^{(n),T-h^{(i)}}_{T-h^{(i)}},\widetilde{f}^{(n),T-h^{i}},\widetilde{L}^{(n),T-h^{i}})$
on $[T-h^{(i+1)},T-h^{(i)}]$. Indeed, Lemma 2.2 implies that for each
$i\in\left\\{0,1,...,N-1\right\\},$ the terminal values $Y^{(n)}_{T-h^{(i)}}$
and obstacles $\widetilde{L}^{(n),T-h^{(i)}}$ satisfy (i) and (ii) in Lemma
A.6. In addition, set

$\left\\{\begin{aligned}
\hat{\phi}^{(n),T-h^{(i)}}_{t}&:=\Big{(}x,\phi^{(n),T-h^{(i)}},D_{x}\phi^{(n),T-h^{(i)}},D_{y}\phi^{(n),T-h^{(i)}},D^{2}_{xy}\phi^{(n),T-h^{(i)}},\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad
D^{2}_{xx}\phi^{(n),T-h^{(i)}},D^{2}_{yy}\phi^{(n),T-h^{(i)}}\Big{)},\\\
\hat{\phi}^{(n),T-h^{(i)}}_{T}&=(x,y,0,1,0,0,0),\end{aligned}\right.$

where $\phi^{(n),T-h^{(i)}}$ and its derivatives are valuated at $(t,x,y).$
Then according to the proof [28, Theorem 11.12], $\hat{\phi}^{(n),T-h^{(i)}}$
is the unique solution to an RDE along some
$\hat{H}^{(n)}\in\text{Lip}^{\gamma-1}(\mathbb{R}^{d^{2}+3d+3},\mathbb{R}^{(d^{2}+3d+3)\times
l})$ and driven by $d\mathbf{X}^{(n)}_{t}.$ By [28, Proposition 11.5], the
derivatives of $\phi^{(n),T-h^{(i)}}$ satisfies the RDE obtained by formal
differentiation. Then by this observation, each element of $\hat{H}^{(n)}$ is
multiplication of some derivatives of $H^{(n)}$ and some truncated linear
functions. Since $H^{(n)}\rightarrow H^{(0)}$ in
$\text{Lip}^{\gamma+1}(\mathbb{R}^{d}\times\mathbb{R},\mathbb{R}^{l})$ we have

(3.30) $\hat{H}^{(n)}\rightarrow\hat{H}^{(0)}\text{ in
}\text{Lip}^{\gamma-1}(\mathbb{R}^{d^{2}+3d+3},\mathbb{R}^{(d^{2}+3d+3)\times
l}).$

In view of (3.30) and the convergence of
$\mathbf{X}^{(n)}\rightarrow\mathbf{X}^{0}$ in
$C^{0,p\text{-var}}([0,T],G^{[p]}(\mathbb{R}^{l})),$ by [28, Corollary 10.27]

$\hat{\phi}^{(n),T-h^{(i)}}\rightarrow\hat{\phi}^{(0),T-h^{(i)}}\text{ locally
uniformly in }(t,x,y)\in[0,T]\times\mathbb{R}^{d}\times\mathbb{R}.$

Similarly, we also have the above locally convergence with
$\phi^{(n),T-h^{(i)}}$ replaced by $(\phi^{(n),T-h^{(i)}})^{-1}.$ Thus, in
view of the convergence of derivatives of $\phi^{(n),T-h^{(i)}}$ and its
inverse, and by adapting the proof of Lemma 3.4 with $\phi^{n,T^{\prime}}$
replaced by $\phi^{(n),T-h^{(i)}},$ the sequence of
$(\widetilde{f}^{(n),T-h^{(i)}},\widetilde{L}^{(n),T-h^{(i)}})$ satisfies the
requirements (iii) in Lemma A.6.

Finally, we finish the proof by Lemma A.6 and induction argument. Note that
$T=T-h^{(0)}$ and

$Y^{(n)}_{T}=\xi^{n}\rightarrow\xi=Y^{(0)}_{T}\ \text{ a.s..}$

For $i=0,1,2,...,N-1,$ assume that the terminal value
$Y^{(n)}_{T-h^{(i)}}\rightarrow Y^{(0)}_{T-h^{(i)}}$ a.s.. Then by Lemma A.6

$\displaystyle(\widetilde{Y}^{(n),T-h^{(i)}}_{\cdot},\widetilde{K}^{(n),T-h^{(i)}}_{\cdot}-\widetilde{K}^{(n),T-h^{(i)}}_{T-h^{(i+1)}})\rightarrow(\widetilde{Y}^{(0),T-h^{(i)}}_{\cdot},\widetilde{K}^{(0),T-h^{(i)}}_{\cdot}-\widetilde{K}^{(0),T-h^{(i)}}_{T-h^{(i+1)}})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\text{ uniformly in
}[T-h^{(i+1)},T-h^{(i)}]\ \text{ a.s.,}$
$\displaystyle\widetilde{Z}^{(n),T-h^{(i)}}\rightarrow\widetilde{Z}^{(0),T-h^{(i)}}\
\text{ in }\ \mathrm{H}^{2}_{[T-h^{(i+1)},T-h^{(i)}]}(\mathbb{R}^{d}).$

In view of (3.23), we can repeat the proof of (LABEL:66) to obtain that

$\displaystyle(Y^{(n)}_{\cdot},K^{(n)}_{\cdot}-K^{(n)}_{T-h^{(i+1)}})$
$\displaystyle\rightarrow(Y^{(0)}_{\cdot},K^{(0)}_{\cdot}-K^{(0)}_{T-h^{(i+1)}})\
\text{ uniformly in }\ [T-h^{(i+1)},T-h^{(i)}]\ \text{ a.s.,}$ $\displaystyle
Z^{(n)}$ $\displaystyle\rightarrow Z^{(0)}\ \text{ in }\
\mathrm{H}^{2}_{[T-h^{(i+1)},T-h^{(i)}]}(\mathbb{R}^{d}).$

And then we have

$Y^{(n)}_{T-h^{(i+1)}}\rightarrow Y^{(0)}_{T-h^{(i+1)}}\text{ a.s. .}$

Thus by induction, it follows that $Y^{(n)}_{T-h^{(i)}}\rightarrow
Y^{(0)}_{T-h^{(i)}}$ a.s. for all $i\in\\{0,1,...,N\\}$, which implies that

$\displaystyle(Y^{(n)}_{\cdot},K^{(n)}_{\cdot})$
$\displaystyle\rightarrow(Y^{(0)}_{\cdot},K^{(0)}_{\cdot})\ \text{ uniformly
in }\ [0,T]\ \text{ a.s.,}$ $\displaystyle Z^{(n)}$ $\displaystyle\rightarrow
Z^{(0)}\ \text{ in }\ \mathrm{H}^{2}_{[0,T]}(\mathbb{R}^{d}).$

∎

###  3.2. Approximation via penalization

In this subsection, we show that the solution of rough RBSDE can be
constructed by a penalization method. Recall the rough BSDE as follows

(3.31)
$Y_{t}=\xi+\int_{t}^{T}f(r,S_{r},Y_{r},Z_{r})dr+\int_{t}^{T}H(S_{r},Y_{r})d\mathbf{X}_{r}-\int_{t}^{T}Z_{r}dW_{r},\quad
t\in[0,T].$

Suppose that Assumptions $(H_{p,\gamma})$, ($H_{f}$), ($H_{b,\sigma}$) hold
and $\xi\in L^{\infty}(\mathcal{F}_{T})$. [20] shows that Eq. (3.31) admits a
unique solution
$(Y,Z)\in\mathrm{H}^{\infty}_{[0,T]}\times\mathrm{H}^{2}_{[0,T]}(\mathbb{R}^{d})$
(see Proposition A.7).

######  Theorem 3.10.

Assume the same assumption as in Theorem 3.6. Let $(Y,Z,K)$ be the unique
solution of Eq. (3.1). For each $m\geq 1$ let $(Y^{m},Z^{m})$ be the solution
of the following rough BSDE on $[0,T]$

(3.32)
$Y^{m}_{t}=\xi+\int_{t}^{T}f(r,S_{r},Y^{m}_{r},Z^{m}_{r})dr+\int_{t}^{T}H(S_{r},Y^{m}_{r})d\mathbf{X}_{r}+m\int_{t}^{T}(Y^{m}_{r}-L_{r})^{-}dr-\int_{t}^{T}Z^{m}_{r}dW_{r}.$

Then $(Y^{m}_{\cdot},K^{m}_{\cdot})\rightarrow(Y_{\cdot},K_{\cdot})$ uniformly
in $[0,T]$ a.s., and $Z^{m}\rightarrow Z$ in
$\mathrm{H}^{2}_{[0,T]}(\mathbb{R}^{d}),$ where
$K^{m}_{t}:=m\int_{0}^{t}(Y^{m}_{r}-L_{r})^{-}dr.$

###### Proof.

Denote by $f^{m}(t,x,y,z):=f(t,x,y,z)+m(y-L_{t})^{-}.$ Recall that for each
$m\geq 1$, $(Y^{m},Z^{m})$ is the solution of Eq. (3.31) with parameter
$(\xi,f^{m},H,\mathbf{X}).$ Denote by $L^{m}_{t}:=Y^{m}_{t}\land L_{t}.$

Step 1. We show that for $m\geq 1$, the solution of BSDE (3.32) is also the
solution to the following RBSDE with rough driver $d\mathbf{X}_{r}$

(3.33) $\left\\{\begin{aligned}
&Y^{m}_{t}=\xi+\int_{t}^{T}f(r,S_{r},Y^{m}_{r},Z^{m}_{r})dr+\int_{t}^{T}H(S_{r},Y^{m}_{r})d\mathbf{X}_{r}+K^{m}_{T}-K^{m}_{t}-\int_{t}^{T}Z^{m}_{r}dW_{r},\\\
&Y^{m}_{t}\geq
L^{m}_{t},\quad\int_{0}^{T}(Y^{m}_{r}-L^{m}_{r})dK^{m}_{r}=0,\quad
t\in[0,T].\end{aligned}\right.$

Let $\left\\{\mathbf{X}^{n},n\geq 1\right\\}$ be the same lifts sequence of
the smooth paths given by Theorem 3.6 such that
$\mathbf{X}^{n}\rightarrow\mathbf{X}\in
C^{0,p\text{-var}}([0,T],G^{[p]}(\mathbb{R}^{l})).$ For each $m\geq 1,$ by
Proposition A.7 we have that $(Y^{m},Z^{m}),$ the solution of Eq. (3.32) with
parameter $(\xi,f^{m},H,\mathbf{X}),$ is the limit of solutions of Eq. (3.32)
with $(\xi,f^{m},H,\mathbf{X}^{n}),$ $n\geq 1.$ And we denote by
$(\bar{Y}^{m,n},\bar{Z}^{m,n})$ the solution of Eq. (3.32) with
$(\xi,f^{m},H,\mathbf{X}^{n}).$ For $m,n\geq 1,$ set
$\bar{K}^{m,n}_{t}:=m\int_{0}^{t}(\bar{Y}^{m,n}_{r}-L_{r})^{-}dr$ and
$\bar{L}^{m,n}_{t}:=\bar{Y}^{m,n}_{t}\land L_{t}.$ By Definition 3.1, for each
$m\geq 1$ and $n\geq 1$ $(\bar{Y}^{m,n},\bar{Z}^{m,n},\bar{K}^{m,n})$ also
solves RBSDE (3.4) with parameter
$(\xi,f^{m},\bar{L}^{m,n},H,\mathbf{X}^{n}).$ Note $\bar{L}^{m,n}_{\cdot}$
converges to $L^{m}_{\cdot}$ uniformly in $[0,T]$ a.s. as $n\rightarrow\infty$
by the same convergence of $\bar{Y}^{m,n}\rightarrow\bar{Y}^{m}.$ Then the
triplet $(\bar{Y}^{m,n},\bar{Z}^{m,n},\bar{K}^{m,n})$ converges to the
solution of Eq. (3.33) as $n\rightarrow\infty,$ which is a consequence of the
stability result shown in Proposition 3.9. Recalling that
$(\bar{Y}^{m,n},\bar{Z}^{m,n})\rightarrow(Y^{m},Z^{m})$ as
$n\rightarrow\infty,$ we also have

$\bar{K}^{m,n}_{\cdot}=m\int_{0}^{\cdot}(\bar{Y}^{m,n}_{r}-L_{r})^{-}dr\rightarrow
m\int_{0}^{\cdot}(Y^{m}_{r}-L_{r})^{-}dr=K^{m}_{\cdot},\ \text{ uniformly in
}\ [0,T]\ \text{ a.s., }\ n\rightarrow\infty.$

Thus, the triplet $(Y^{m},Z^{m},K^{m})$ solves Eq. (3.33), or equally, it
solves Eq. (3.4) with parameter $(\xi,f,L^{m},H,\mathbf{X}).$

Step 2. According Proposition 3.9, it is sufficient to show that
$L^{m}_{\cdot}\rightarrow L_{\cdot}$ uniformly on $[0,T]$ a.s. to prove the
convergence $(Y^{m},Z^{m},K^{m})\rightarrow(Y,Z,K).$ Noting that $Y^{m}$ is
increasing by Proposition A.8 and the limit process $L$ is continuous, Dini’s
Lemma implies the uniformly convergence of $L^{m}:=Y^{m}\land L$ is equal to
the convergence by point. To this end, let
$Y^{0}:=\lim_{m\rightarrow\infty}Y^{m}$. it suffices to prove

(3.34) $Y^{0}_{t}\geq L_{t},\text{ for all }t\in[0,T].$

We show the above inequality following three steps.

Step 3. We show that $(Y^{m},Z^{m})$ are uniformly bounded in $m\geq 1.$ Let
$(\hat{Y},\hat{Z})$ be the solution of Eq. (3.31) with parameter
$(\xi,f,H,\mathbf{X}).$ Recalling that $(Y^{m},Z^{m})$ is the solution of Eq.
(3.31) with parameter $(\xi,f^{m},H,\mathbf{X}).$ By Proposition A.8, we have
a.s.

$\hat{Y}_{t}\leq Y^{m}_{t},\ \text{ for all }\ t\in[0,T].$

In view of [20, Theorem 3] and Remark A.2, an estimate similar to (3.20) can
be established for $\hat{Y}$. Consequently, there exists a constant
$\hat{M}>0$ such that

$\operatorname*{ess\,sup}\limits_{(t,\omega)\in[0,T]\times\Omega}|\hat{Y}_{t}|\leq\hat{M}.$

Thus, we have $(-\hat{M})\land L_{t}\leq L^{m}_{t}\leq L_{t}.$ According to
Step 1, for each $m\geq 1$ the triplet $(Y^{m},Z^{m},K^{m})$ is also the
solution of Eq. (3.4) with parameter $(\xi,f,L^{m},H,\mathbf{X}).$ By the same
argument to obtain (3.20), there exists a constant $M>0$ such that

(3.35) $\sup_{m\geq
1}\operatorname*{ess\,sup}_{(t,\omega)\in[0,T]\times\Omega}|Y^{m}_{t}|\leq M.$

In addition, Proposition 3.7 shows that
$\|\int_{0}^{\cdot}Z^{m}_{r}dW_{r}\|_{\text{BMO}}$ are uniformly bounded in
$m\geq 1.$ By [31, Corollary 2.1], we have

(3.36) $\sup_{m\geq
1}\mathbb{E}\left[\left|\int_{0}^{T}|Z^{m}_{r}|^{2}dr\right|^{2}\right]<\infty.$

Step 4. We show that a.s., $Y^{0}_{t}\geq L_{t}$ $dt$-a.e.. For the same
reason of Proposition 3.7, we have

$\sup\limits_{m\geq
1}\mathbb{E}\left[\left|K^{m}_{T}\right|^{2}\right]<\infty.$

Thus, recalling that $|Y^{m}|$ is uniformly bounded in $m\geq 1,$ it follows
from the dominated convergence theorem that

$\mathbb{E}\left[\int_{0}^{T}(Y^{0}_{t}-L_{t})^{-}dt\right]=\lim_{m\rightarrow\infty}\mathbb{E}\left[\int_{0}^{T}(Y^{m}_{t}-L_{t})^{-}dt\right]=\lim_{m\rightarrow\infty}\frac{1}{m}\mathbb{E}\left[K^{m}_{T}\right]=0.$

This implies that $a.s.,$ $Y^{0}_{t}\geq L_{t}$ for a.e. $t\in[0,T].$

Step 5. We show that a.s. the right limit of $Y^{0}_{t}$ (denoted by
$Y^{0}_{t+},$) exists for all $t\in[0,T],$ and it satisfies a.s.,
$Y^{0}_{t}\geq Y^{0}_{t+}$ for all $t\in[0,T].$ If this claim is true,
combined with the fact that a.s., $Y^{0}_{t}\geq L_{t}$ for a.e. $t\in[0,T]$,
it follows that a.s., $Y^{0}_{t}\geq L_{t}$ for all $t\in[0,T],$ which is
exactly (3.34). For any fixed
$(T^{\prime},x,\widetilde{y})\in[0,T]\times\mathbb{R}^{d}\times\mathbb{R}$,
let $\phi^{T^{\prime}}(\cdot,x,\widetilde{y})$ be the unique solution of the
following RDE

$\phi^{T^{\prime}}(t,x,\widetilde{y})=\widetilde{y}+\int_{t}^{T^{\prime}}H(x,\phi^{T^{\prime}}(r,x,\widetilde{y}))d\mathbf{X}_{r},\quad
t\in[0,T^{\prime}].$

For each $m\geq 1,$ $T^{\prime}\in[0,T],$ and $t\in[0,T^{\prime}],$ denote by
$(\widetilde{Y}^{m,T^{\prime}}_{t},\widetilde{Z}^{m,T^{\prime}}_{t},\widetilde{K}^{m,T^{\prime}}_{t})$
as follows

$\displaystyle\widetilde{Y}^{m,T^{\prime}}_{t}$
$\displaystyle:=(\phi^{T^{\prime}})^{-1}(t,S_{t},Y^{m}_{t}),\quad\widetilde{K}^{m,T^{\prime}}_{t}:=\int_{0}^{t}\frac{1}{D_{y}\phi^{T^{\prime}}(s,S_{s},\widetilde{Y}^{m,T^{\prime}}_{s})}dK^{m}_{s},$
$\displaystyle\widetilde{Z}^{m,T^{\prime}}_{t}$
$\displaystyle:=-\frac{(D_{x}\phi^{T^{\prime}}(t,S_{t},\widetilde{Y}^{m,T^{\prime}}_{t}))^{\top}}{D_{y}\phi^{T^{\prime}}(t,S_{t},\widetilde{Y}^{m,T^{\prime}}_{t})}\sigma_{t}+\frac{1}{D_{y}\phi^{T^{\prime}}(t,S_{t},\widetilde{Y}^{m,T^{\prime}}_{t})}Z^{m}_{t}.$

And for
$(t,x,\widetilde{y},\widetilde{z})\in[0,T^{\prime}]\times\mathbb{R}^{d}\times\mathbb{R}\times\mathbb{R}^{d}$,
denote by

$\displaystyle\widetilde{f}^{T^{\prime}}(t,x,\widetilde{y},\widetilde{z}):=\frac{1}{D_{y}\phi^{T^{\prime}}}\bigg{\\{}f\big{(}t,x,\phi^{T^{\prime}},D_{y}\phi^{T^{\prime}}\widetilde{z}+\sigma^{\top}D_{x}\phi^{T^{\prime}}\big{)}+(D_{x}\phi^{T^{\prime}})^{\top}b$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\frac{1}{2}\text{Tr}\left\\{D^{2}_{xx}\phi^{T^{\prime}}\sigma\sigma^{\top}\right\\}+(D^{2}_{xy}\phi^{T^{\prime}})^{\top}\sigma\widetilde{z}+\frac{1}{2}D^{2}_{yy}\phi^{T^{\prime}}|\widetilde{z}|^{2}\bigg{\\}},$
$\displaystyle\widetilde{L}^{m,T^{\prime}}_{t}:=(\phi^{T^{\prime}})^{-1}(t,S_{t},L^{m}_{t}),$

where $\phi^{T^{\prime}}$ and its derivatives will always take values at
$(t,x,\widetilde{y}).$ By (3.35) and Lemma 2.2, there exists a constant
$\widetilde{M}>0$ such that

$\sup_{m\geq
1}\operatorname*{ess\,sup}_{(t,\omega)\in[0,T^{\prime}]\times\Omega}|\widetilde{Y}^{m,T^{\prime}}|\leq\widetilde{M}.$

Thus, by the same argument in the step 2 of the proof pf Theorem 3.6, there
exist an integer $N>0$ and a sequence of
$\left\\{h^{(i)},i=0,2,...,N\right\\}$ such that
$[T-h^{(N)},T-h^{(0)}]=[0,T],$ and that, in addition, the triplet
$(\widetilde{Y}^{m,T-h^{(i)}},\widetilde{Z}^{m,T-h^{(i)}},\widetilde{K}^{m,T-h^{(i)}})$
uniquely solves Eq. (3.4) on $[T-h^{(i+1)},T-h^{(i)}]$ with terminal time
$T-h^{(i)}$ and parameter
$(Y^{m}_{T-kh},\widetilde{f}^{T-h^{(i)}},L^{m,T-h^{(i)}},0,0).$ For any
$t\in[T-h^{(i+1)},T-h^{(i)}],$ let

(3.37)
$\widetilde{Y}^{0,T-h^{(i)}}_{t}:=(\phi^{T-h^{(i)}})^{-1}(t,S_{t},Y^{0}_{t}).$

By the definition of $Y^{0}$ we have a.s.
$\widetilde{Y}^{m,T-h^{(i)}}_{t}\rightarrow\widetilde{Y}^{0,T-h^{(i)}}_{t}$
for all $t\in[T-h^{(i+1)},T-h^{(i)}]$ as $m\rightarrow\infty.$ In addition,
from (3.35) and (3.36) we have both
$\|\widetilde{Z}^{m,T-h^{(i)}}_{t}\|_{\mathrm{H}^{2};[T-h^{(i+1)},T-h^{(i)}]}$
and
$\|\widetilde{f}^{T-h^{(i)}}(t,S_{t},\widetilde{Y}^{m,T-h^{(i)}}_{t},\widetilde{Z}^{m,T-h^{(i)}}_{t})\|_{\mathrm{H}^{2};[T-h^{(i+1)},T-h^{(i)}]}$
are uniformly bounded in $m\geq 1$ and $i=0,1,2,...,N-1,$ which implies that
there exists a pair of processes
$(\widetilde{Z}^{0,T-h^{(i)}},\widetilde{f}^{0,T-h^{(i)}})$ such that, by
passing to a subsequence if necessary,

$\displaystyle(\widetilde{Z}^{m,T-h^{(i)}}_{\cdot},\widetilde{f}^{m,T-h^{(i)}}_{\cdot}):=(\widetilde{Z}^{m,T-h^{(i)}}_{\cdot},\widetilde{f}^{T-h^{(i)}}(\cdot,S_{\cdot},\widetilde{Y}^{m,T-h^{(i)}}_{\cdot},\widetilde{Z}^{m,T-h^{(i)}}_{\cdot}))$
$\displaystyle\qquad\qquad\overset{\text{weakly}}{\longrightarrow}(\widetilde{Z}^{0,T-h^{(i)}}_{\cdot},\widetilde{f}^{0,T-h^{(i)}}_{\cdot})\
\text{ in }\
\mathrm{H}^{2}_{[T-h^{(i+1)},T-h^{(i)}]}(\mathbb{R}^{d})\times\mathrm{H}^{2}_{T-h^{(i+1)},T-h^{(i)}]}(\mathbb{R}).$

For $i=0,1,...,N-1$ and $t\in[T-h^{(i+1)},T-h^{(i)}],$ define
$\widetilde{K}^{0,T-h^{(i)}}_{t}$ by

(3.38)
$\widetilde{K}^{0,T-h^{(i)}}_{t}:=\widetilde{Y}^{0,T-h^{(i)}}_{T-h^{(i+1)}}-\widetilde{Y}^{0,T-h^{(i)}}_{t}-\int_{T-h^{(i+1)}}^{t}\widetilde{f}^{0,T-h^{(i)}}_{r}dr+\int_{T-h^{(i+1)}}^{t}\widetilde{Z}^{0,T-h^{(i)}}_{r}dW_{r}.$

Noting that
$\widetilde{Y}^{m,T-h^{(i)}}_{\tau}\rightarrow\widetilde{Y}^{0,T-h^{(i)}}_{\tau}$
a.s. as $m\rightarrow\infty,$ and for $\zeta\in L^{2}(\mathcal{F}_{T})$

$\mathbb{E}\left[\int_{T-h^{(i+1)}}^{\tau}\widetilde{f}^{m,T-h^{(i)}}_{r}dr\cdot\zeta\right]=\mathbb{E}\left[\int_{T-h^{(i+1)}}^{T}\widetilde{f}^{m,T-h^{(i)}}_{r}\cdot\mathbb{E}_{r}[\zeta
1_{\\{r<\tau}\\}]dr\right],$

we have for each stopping time $\tau$ taking values in
$[T-h^{(i+1)},T-h^{(i)}]$

$\widetilde{K}^{m,T-h^{(i)}}_{\tau}\overset{\text{weakly}}{\longrightarrow}\widetilde{K}^{0,T-h^{(i)}}_{\tau}\
\text{ in }\ L^{2}(\mathcal{F}_{T}).$

Since for each $m\geq 1$ and $i\in\\{0,1,...,N-1\\}$, the process
$\widetilde{K}^{m,T-h^{(i)}}_{t}$ is increasing in
$t\in[T-h^{(i+1)},T-h^{(i)}]$ a.s., Lemma C.2 implies that
$\widetilde{K}^{0,T-h^{(i)}}_{t}$ is also increasing in
$t\in[T-h^{(i+1)},T-h^{(i)}]$ a.s.. In view of the equality (3.38), for any
$i\in\\{0,1,...,N-1\\}$ and $t\in\left[T-h^{(i+1)},T-h^{(i)}\right)$, we
obtain that $\widetilde{Y}^{0,T-h^{(i)}}_{t+}$ exists and
$\widetilde{Y}^{0,T-h^{(i)}}_{t}\geq\widetilde{Y}^{0,T-h^{(i)}}_{t+}.$ Thus,
$Y^{0}_{t+}=\phi^{T-h^{(i)}}(t+,S_{t+},\widetilde{Y}^{0,T-h^{(i)}}_{t+})$
exists by the continuity of $\phi^{T-h^{(i)}}(t,x,\widetilde{y})$ in
$(t,x,\widetilde{y}).$ And then by the representation (3.37) and the
positivity of $\frac{1}{D_{y}\phi^{T-h^{(i)}}}$ from Lemma 2.2, it follows
that $Y^{0}_{t}\geq Y^{0}_{t+}.$ The proof is completed. ∎

##  4\. Applications

###  4.1. Obstacle problem for rough PDEs

In this subsection, we suppose that
$f:[0,T]\times\mathbb{R}^{d}\times\mathbb{R}\times\mathbb{R}^{d}\rightarrow\mathbb{R}$
is deterministic and jointly continuous. Given
$b:[0,T]\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d},$
$\sigma:[0,T]\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d\times d},$
$l:[0,T]\times\mathbb{R}^{d}\rightarrow\mathbb{R},$
$g:\mathbb{R}^{d}\rightarrow\mathbb{R}$ and $\mathbf{X}\in
C^{0,p\text{-var}}([0,T],G^{[p]}(\mathbb{R}^{l})),$ consider the following
system of forward-backward stochastic differential equations

(4.1) $\left\\{\begin{aligned}
&Y^{t,x}_{s}=g(S^{t,x}_{T})+\int_{s}^{T}f(r,S^{t,x}_{r},Y^{t,x}_{r},Z^{t,x}_{r})dr+\int_{s}^{T}H(S^{t,x}_{r},Y^{t,x}_{r})d\mathbf{X}_{r}\\\
&\quad\quad\quad\quad+K^{t,x}_{T}-K^{t,x}_{s}-\int_{s}^{T}Z^{t,x}_{r}dW_{r},\\\
&Y^{t,x}_{s}\geq
l(s,S^{t,x}_{s}),\quad\int_{t}^{T}\left(Y^{t,x}_{r}-l(r,S^{t,x}_{r})\right)dK^{t,x}_{r}=0,\quad
s\in[t,T],\\\
&S^{t,x}_{s}=x+\int_{t}^{s}b(r,S^{t,x}_{r})dr+\int_{t}^{s}\sigma(r,S^{t,x}_{r})dW_{r},\quad
s\in[t,T],\end{aligned}\right.$

and the corresponding rough PDE

(4.2) $\left\\{\begin{aligned}
&dv(t,x)+\mathcal{L}v(t,x)dt+H(x,v(t,x))d\mathbf{X}_{t}=0,\\\
&v(T,x)=g(x),\quad v(t,x)\geq l(t,x),\end{aligned}\right.$

where ($(\sigma,b)$ takes value at $(t,x)$)

$\mathcal{L}v(t,x):=\frac{1}{2}\text{Tr}\left\\{(\sigma\sigma^{\top}D^{2}_{xx})v(t,x)\right\\}+(b^{\top}D_{x})v(t,x)+f\big{(}t,x,v(t,x),\sigma^{\top}D_{x}v(t,x)\big{)}.$

Assume $(H_{p,\gamma})$, ($H_{f}$), ($H_{b,\sigma}$) hold for
$(H,f,b,\sigma).$ Assume further $(l,g)\in
C_{b}([0,T]\times\mathbb{R}^{d},\mathbb{R})\times
C_{b}(\mathbb{R}^{d},\mathbb{R})$ such that $g(x)\geq l(T,x).$ For a smooth
driver $d\mathrm{X}_{t}$ (i.e. $\mathrm{X}\in\bigcap_{\lambda\geq
1}\text{Lip}^{\lambda}([0,T],\mathbb{R}^{l})$),
$f^{\prime}(t,x,y,z):=f(t,x,y,z)+H(x,y)\dot{\mathrm{X}}_{t}$ also satisfies
($H_{f}$). Thus, by Proposition B.2, PDE. (4.2) driven by $d\mathrm{X}_{t}$
(i.e. with $d\mathbf{X}_{t}$ replaced by $d\mathrm{X}_{t}$) admits a unique
viscosity solution. On the other hand, according to Theorem 3.6, the system
(4.1) admits a unique solution $(Y^{t,x},Z^{t,x},K^{t,x})$. We aim to
establish the relation between $Y^{t,x}$ and the solution of rough PDE (4.2).
For this purpose, we first introduce the following definition inspired by [20,
Theorem 12].

######  Definition 4.1.

For any $\mathbf{X}\in C^{0,p\text{-var}}([0,T],G^{[p]}(\mathbb{R}^{l})),$ for
$n\geq 1$ let $\mathrm{X}^{n}$ be smooth such that their lifts
$\mathbf{X}^{n}\rightarrow\mathbf{X}$ in
$C^{0,p\text{-var}}([0,T],G^{[p]}(\mathbb{R}^{l})).$ Assume that $u^{n}(t,x)$
is the unique viscosity solution of obstacle PDE (4.2) driven by
$d\mathrm{X}^{n}_{t}$ (i.e. with $\mathbf{X}$ replaced by $\mathrm{X}^{n}$),
and there exists $u\in C([0,T]\times\mathbb{R}^{d},\mathbb{R})$ such that

$u^{n}\rightarrow u,\quad\text{locally uniformly.}$

Then we call $u(t,x)$ a solution of the rough PDE (4.2). Moreover, if $u(t,x)$
does not depend on the choice of approximation sequence
$\left\\{\mathrm{X}^{n},n\geq 1\right\\},$ we call $u(t,x)$ the unique
solution of Eq. (4.2).

######  Theorem 4.2.

Assume that $(H,f)$ satisfies Assumption $(H_{p,\gamma})$, ($H_{f}$),
$(l,g)\in C_{b}([0,T]\times\mathbb{R}^{d},\mathbb{R})\times
C_{b}(\mathbb{R}^{d},\mathbb{R})$ such that $g(x)\geq l(T,x),$ and
$(b,\sigma)$ satisfies ($H_{b,\sigma}$). In addition, suppose that $f$
satisfies

(4.3) $|D_{x}f(t,x,y,z)|\leq C^{\prime}(1+|z|^{2}),$

for some $C^{\prime}>0$. For any $\mathbf{X}\in
C^{0,\text{p-var}}([0,T],G^{[p]}(\mathbb{R}^{l}))$, let $u(t,x):=Y^{t,x}_{t},$
where $(Y^{t,x},Z^{t,x},K^{t,x})$ is the unique solution of Eq. (4.1). Then
$u(t,x)$ is the unique solution of the rough PDE (4.2).

###### Proof.

Denote by $\mathbf{X}^{0}:=\mathbf{X}$ and $(b,\sigma):=(b(t,x),\sigma(t,x))$.
For $T^{\prime}\in[0,T]$, let $\phi^{n,T^{\prime}},n\in\mathbb{N}$ be the
unique solution of Eq. (3.15), and $\mathrm{X}^{n}$ be smooth with their lifts
$\mathbf{X}^{n}\rightarrow\mathbf{X}^{0}$ in
$C^{0,p\text{-var}}([0,T],G^{[p]}(\mathbb{R}^{l})).$ In view of Proposition
B.2, for $n\geq 1,$ Eq. 4.2 driven by $d\mathrm{X}^{n}_{t}$ admits a unique
viscosity solution $u^{n}.$ On the other hand, by the monotonicity of
$\phi^{n,T^{\prime}}(t,x,y)$ and $(\phi^{n,T^{\prime}})^{-1}(t,x,y)$ in $y,$

(4.4)
$u^{n}(t,x)>l(t,x)\Leftrightarrow(\phi^{n,T^{\prime}})^{-1}(t,x,u^{n}(t,x))>(\phi^{n,T^{\prime}})^{-1}(t,x,l(t,x)).$

Let $\widetilde{f}^{n,T^{\prime}}(t,x,\widetilde{y},\widetilde{z})$ be defined
by (3.17), and
$\widetilde{l}^{n,T^{\prime}}(t,x):=(\phi^{n,T^{\prime}})^{-1}(t,x,l(t,x)).$
According to (4.4) and [27, Lemma 5], it follows that for $n\geq 1,$ $u^{n}$
is a viscosity solution to Eq. (4.2) driven by $d\mathbf{X}^{n}_{t}$ if and
only if $v^{n,T^{\prime}}(t,x):=(\phi^{n,T^{\prime}})^{-1}(t,x,u^{n}(t,x))$ is
a viscosity solution of the following equation

(4.5) $\left\\{\begin{aligned}
&dv(t,x)+\left[\frac{1}{2}\text{Tr}\Big{\\{}(\sigma\sigma^{\top}D^{2}_{xx}v(t,x))\right\\}+b^{\top}D_{x}v(t,x)\\\
&\ \ \ \ \ \
+\widetilde{f}^{n,T^{\prime}}(t,x,v(t,x),\sigma^{\top}D_{x}v(t,x))\Big{]}dt=0\\\
&v(T^{\prime},x)=u^{n}(T^{\prime},x),\quad
v(t,x)\geq\widetilde{l}^{n,T^{\prime}}(t,x),\quad(t,x)\in[0,T^{\prime}]\times\mathbb{R}^{d}.\end{aligned}\right.$

For $n=0,$ let
$\widetilde{l}^{0,T^{\prime}}(t,x):=(\phi^{0,T^{\prime}})^{-1}(t,x,l(t,x)),$
and $\widetilde{f}^{0,T^{\prime}}(t,x,\widetilde{y},\widetilde{z})$ be defined
by (3.17) with $n=0.$ We divide our proof into three parts.

Step 1. Assuming additional $u^{n}(T^{\prime},\cdot)\rightarrow
u(T^{\prime},\cdot),$ we use the method of semi-relaxed limits (refer to [14,
Section 6]) to show that the limits

$\overline{v}^{0,T^{\prime}}:=\limsup_{n\rightarrow\infty}{}^{*}v^{n,T},\quad(\text{resp.
}\underline{v}_{0,T^{\prime}}:=\liminf_{n\rightarrow\infty}{}_{*}v^{n,T^{\prime}}),$

is a viscosity sub (resp. super) solution of Eq. (4.5) with $n=0,$ where

(4.6) $\displaystyle\limsup_{n\rightarrow\infty}{}^{*}v^{n,T^{\prime}}(t,x)$
$\displaystyle:=\lim_{n\rightarrow\infty}\sup\left\\{v^{m,T^{\prime}}(s,y):|s-t|\vee|y-x|\leq\frac{1}{n},\
\text{and }m\geq n\right\\},$
$\displaystyle\liminf_{n\rightarrow\infty}{}_{*}v^{n,T^{\prime}}(t,x)$
$\displaystyle:=\lim_{n\rightarrow\infty}\inf\left\\{v^{m,T^{\prime}}(s,y):|s-t|\vee|y-x|\leq\frac{1}{n},\
\text{and }m\geq n\right\\}.$

To see the finiteness of above limits, for each $n\geq 1$, the Feynman–Kac
representation shown in Proposition B.2 and the prior bound of the
corresponding BSDEs given by Lemma A.2 imply that $u^{n}(t,x)$ are uniform
bounded in $(t,x)\in[0,T]\times\mathbb{R}^{d}$ and $n\geq 1.$ In addition, due
to the uniform bound of
$\|D_{y}(\phi^{n,T^{\prime}})^{-1}\|_{\infty;[0,T^{\prime}]\times\mathbb{R}^{d}\times\mathbb{R}}$
in $n\geq 1,$ the uniform boundedness of $u^{n}(t,x)$ implies that of
$v^{n,T},$ i.e.,

$\sup_{n\geq
1}\|v^{n,T^{\prime}}(\cdot,\cdot)\|_{\infty;[0,T^{\prime}]\times\mathbb{R}^{d}}<\infty.$

Now let us show $\overline{v}^{0,T^{\prime}}(\cdot,\cdot)$ (resp.
$\underline{v}^{0,T^{\prime}}(\cdot,\cdot)$) is a subsolution (resp.
supersolution). Firstly, by [9, Lemma 1.5 Chapter V] we have that
$\overline{v}^{0,T^{\prime}}(\cdot,\cdot)$ (resp.
$\underline{v}^{0,T^{\prime}}(\cdot,\cdot)$) is upper (resp. lower)
semicontinuous. Secondly, by Lemma 2.3
$\widetilde{l}^{n,T^{\prime}}(\cdot,\cdot)\rightarrow\widetilde{l}^{0,T^{\prime}}(\cdot,\cdot)$
locally uniformly, and by Lemma 3.4,
$\widetilde{f}^{n,T^{\prime}}\rightarrow\widetilde{f}^{0,T^{\prime}}$ locally
uniformly. In view of $u^{n}(T^{\prime},\cdot)\rightarrow u(T^{\prime},\cdot)$
and these two facts, Proposition B.5 applies and we have
$\overline{v}^{0,T^{\prime}}(T^{\prime},x)=\underline{v}^{0,T^{\prime}}(T^{\prime},x)=u(T^{\prime},x).$
Thirdly, assume that $(t,x)\in(0,T^{\prime})\times\mathbb{R}^{d}$ satisfies

(4.7) $\overline{v}^{0,T^{\prime}}(t,x)>\widetilde{l}^{0,T^{\prime}}(t,x),$

and let $(a,p,\Lambda)\in P^{2,+}\overline{v}^{0,T^{\prime}}(t,x)$, where the
semi-jets $P^{2,+}$ and $P^{2,-}$ are defined in Appendix B. Using the same
argument of [14, Lemma 6.1], there exist sequences

$n_{j}\rightarrow\infty,\ (t_{j},x_{j})\in(0,T^{\prime})\times\mathbb{R}^{d},\
(a_{j},p_{j},\Lambda_{j})\in P^{2,+}v^{n_{j},T^{\prime}}(t_{j},x_{j}),\quad
j\geq 1,$

such that

$(t_{j},x_{j},v^{n_{j},T^{\prime}}(t_{j},x_{j}),a_{j},p_{j},\Lambda_{j})\rightarrow(t,x,\overline{v}^{0,T^{\prime}}(t,x),a,p,\Lambda),\quad
j\rightarrow\infty.$

In view of the locally uniform convergence of
$\widetilde{l}^{n_{j},T^{\prime}}(\cdot,\cdot)\rightarrow\widetilde{l}^{0,T^{\prime}}(\cdot,\cdot),$
and inequality (4.7), we can assume that
$v^{n_{j},T^{\prime}}(t_{j},x_{j})>\widetilde{l}^{n_{j},T^{\prime}}(t_{j},x_{j})$
for sufficient large $j.$ Due to the viscosity property of $v^{n,T^{\prime}},$
we have

$a_{j}+(p_{j}^{\top}b(t_{j},x_{j}))+\frac{1}{2}\text{Tr}\left\\{\sigma\sigma^{\top}(t_{j},x_{j})\Lambda_{j}\right\\}+\widetilde{f}^{n_{j},T^{\prime}}(t_{j},x_{j},v^{n_{j},T^{\prime}}(t_{j},x_{j}),\sigma^{\top}(t_{j},x_{j})p_{j})\geq
0.$

According to the fact that
$\widetilde{f}^{n,T^{\prime}}\rightarrow\widetilde{f}^{0,T^{\prime}}$ locally
uniformly, by sending $j\rightarrow\infty$ we have

$a+(p^{\top}b(t,x))+\frac{1}{2}\text{Tr}\left\\{\sigma\sigma^{\top}(t,x)\Lambda\right\\}+\widetilde{f}^{0,T^{\prime}}(t,x,\overline{v}^{0}(t,x),\sigma^{\top}(t,x)p)\geq
0,$

which implies that $\overline{v}^{0,T^{\prime}}$ is a viscosity subsolution to
Eq. (4.5) with $n=0.$ In addition, the uniform continuity of $l(\cdot,\cdot)$
implies that
$\underline{v}^{0,T^{\prime}}(\cdot,\cdot)\geq\widetilde{l}^{0,T^{\prime}}(\cdot,\cdot),$
and then we can repeat the above procedure to show that
$\underline{v}^{0,T^{\prime}}$ is a viscosity supersolution to Eq. (4.5) with
$n=0.$

Step 2. Following the above step, we show that $v^{n,T^{\prime}}(\cdot,\cdot)$
locally uniformly converges to a limit. Note that
$\overline{v}^{0,T^{\prime}}$ ($\underline{v}^{0,T^{\prime}}$ resp.) is the
viscosity subsolution (supersolution resp.). In view of Lemma 3.3, we see that
comparison theorem Proposition B.4 applies. Thus, there exists $\delta>0$ such
that for whenever $s\in[0,T^{\prime})$ and
$\|\mathbf{X}^{n}\|_{p\text{-var};[s,T^{\prime}]}\leq\delta,$

$\overline{v}^{0,T^{\prime}}(t,x)\leq\underline{v}^{0,T^{\prime}}(t,x)\text{
for }(t,x)\in(s,T^{\prime}]\times\mathbb{R}^{d}.$

And then by definition we have
$\overline{v}^{0,T^{\prime}}=\underline{v}^{0,T^{\prime}}.$ Following the
Dini-type argument [14, Remark 6.4], for every $r\in(s,T^{\prime})$ we have

(4.8)
$v^{n,T^{\prime}}(\cdot,\cdot)\rightarrow\overline{v}^{0,T^{\prime}}(\cdot,\cdot)\text{
locally uniformly on }[r,T^{\prime}]\times\mathbb{R}^{d}.$

Step 3. Now we are back to prove the convergence of $u^{n}(\cdot,\cdot).$ Let
$\widetilde{u}^{0,T^{\prime}}(t,x):=\phi^{0,T^{\prime}}(t,x,\overline{v}^{0,T^{\prime}}(t,x)).$
Then Lemma 2.3 and (4.8) yield that

$u^{n}(\cdot,\cdot)\rightarrow\widetilde{u}^{0,T^{\prime}}(\cdot,\cdot)\text{
locally uniformly on }[r,T^{\prime}]\times\mathbb{R}^{d}.$

On the other hand, for $n\geq 1,$ denote by $Y^{n;t,x}$ the solution of RBSDE
(4.1) with $\mathbf{X}=\mathbf{X}^{n}.$ Recall that $u(t,x)=Y^{t,x}_{t}.$ Then
by Proposition B.2 and Theorem 3.6, for $n\geq 1,$ $u^{n}(t,x)=Y^{n;t,x}_{t}$
and $u^{n}(t,x)\rightarrow Y^{t,x}_{t}=u(t,x).$ Thus, assuming the locally
uniform convergence of $u^{n}(T^{\prime},\cdot)\rightarrow
u(T^{\prime},\cdot),$ we have $\widetilde{u}^{0,T^{\prime}}(t,x)=u(t,x)$ on
$[r,T^{\prime}]\times\mathbb{R}^{d}$ and then

(4.9) $u^{n}(\cdot,\cdot)\rightarrow u(\cdot,\cdot)\text{ locally uniformly on
}[r,T^{\prime}]\times\mathbb{R}^{d}.$

Let $\theta\in(0,T)$ satisfying
$\sup_{n}\|\mathbf{X}^{n}\|_{p\text{-var};[0,\theta]}\leq\delta/2.$ By the
fact that $\mathbf{X}^{n}\rightarrow\mathbf{X}^{0}$ in $p$-variation metric,
and Lemma 2.4 with $[0,T]$ replaced by $[\theta,T],$ there exists a sequence
of increasing numbers $\\{h^{(i)},i=0,1,...,N\\}$ satisfying

$h^{(i)}<h^{(i+1)},\ i=0,1,...,N-1,\ h^{(N)}=T-\theta,\ h^{(0)}=0,$

and (take $n$ large enough if necessary)

(4.10)
$\sup_{n}\|\mathbf{X}^{n}\|_{p\text{-var};[T-h^{(i)},T-h^{(i-1)}]}\leq\frac{\delta}{2},\text{
for all }i=1,2,...,N.$

By (4.10) and the fact that
$\|\mathbf{X}^{n}\|_{p\text{-var};[0,\theta]}\leq\delta/2,$ we have for $i<N$

$\|\mathbf{X}^{n}\|_{p\text{-var};[T-h^{(i+1)},T-h^{(i-1)}]}\leq\delta,\text{
and }\|\mathbf{X}^{n}\|_{p\text{-var};[0,T-h^{(N-1)}]}\leq\delta.$

Then in view of the above inequalities and the fact that

$T-h^{(i)}\in(T-h^{(i+1)},T-h^{(i-1)}),\ T-h^{(N)}\in(0,T-h^{(N-1)}),$

by (4.9) it follows that whenever $u^{n}(T-h^{(i-1)},\cdot)\rightarrow
u(T-h^{(i-1)},\cdot)$ locally uniformly,

(4.11) $u^{n}(\cdot,\cdot)\rightarrow u(\cdot,\cdot)\text{ locally uniformly
on }[T-h^{(i)},T-h^{(i-1)}]\times\mathbb{R}^{d}.$

Since $u^{n}(T,\cdot)=g(\cdot)=u(\cdot)$ for all $n,$ (4.11) holds for $i=1.$
Thus by induction, the uniform convergence of
$u^{n}(T-h^{(i-1)},\cdot)\rightarrow u(T-h^{(i-1)},\cdot)$ can be deduced, and
then (4.11) holds for every $i>1.$ Since $[T-h^{(N)},T-h^{(0)}]=[\theta,T],$
we have

$u^{n}(\cdot,\cdot)\rightarrow u(\cdot,\cdot)\text{ locally uniformly on
}[\theta,T]\times\mathbb{R}^{d}.$

And then the proof is completed by taking the terminal time $T$ replaced
$T+\theta,$ and doing the transition of time $t\mapsto t+\theta.$

∎

######  Remark 4.1.

In the above proof, by the same reason mentioned in [20, Theorem 12], the
convergence at time $t=0$ is not immediate, due to that the comparison result
in Proposition B.4 may not hold at the left endpoint. However, we can take the
transition of time: $t\mapsto t+\theta,$ and extend the coefficients as well
as the rough path up to time $0.$ Then the rough PDEs are considered on
$[0,T+\theta]$ to get the locally uniformly convergence of corresponding
$u^{n}(t-\theta,x)$ on the interval $[\theta,T+\theta].$ Then the convergence
of $u^{n}$ on $[0,T]$ follows by doing the reverse transition of time:
$t\mapsto t-\theta.$

######  Remark 4.2.

Assume $(H,f,l,g,b,\sigma)$ satisfies the requirements in Proposition B.2, and
$\mathrm{X}$ is smooth. Let $u$ be the unique viscosity solution of obstacle
PDE (4.2) driven by $d\mathrm{X}_{t}.$ Then from Theorem 4.2, it follows that
$u$ is also the unique solution in the sense of Definition 4.1.

###  4.2. Optimal stopping problem

In this subsection, we aim to show that the solution $Y_{t}$ of Eq. (3.4) with
parameter $(\xi,f,L,H,\mathbf{X})$ corresponds to an optimal stopping time
problem in the context of BSDEs with rough drivers $d\mathbf{X}_{t}$. In the
classical case, one of the tasks of optimal stopping problem is to
characterize the following value function

$\operatorname*{ess\,sup}_{\tau\in\mathcal{T}_{t}}\mathbb{E}_{t}\left[G_{\tau}\right],$

where $G$ is an $\mathcal{F}$-adapted process, $\mathcal{T}_{t}$ is the set of
all stopping times taking values in $[t,T].$ Assuming $(Y,Z,K)$ uniquely
solves RBSDE (1.4), it follows from El Karoui et al. [25, Theorem 5.2] that

(4.12)
$Y_{t}=\operatorname*{ess\,sup}_{\tau\in\mathcal{T}_{t}}B_{t}(\tau,L^{\prime}_{\tau}),$

where $B_{t}(\tau,L^{\prime}_{\tau})$ represents the solution to BSDE (1.3)
with terminal time $\tau$, terminal value $L^{\prime}_{\tau}$ and

(4.13) $L^{\prime}_{s}:=L_{s}1_{\\{s<T\\}}+\xi 1_{\\{s=T\\}}.$

Actually, $B_{t}(\tau,L^{\prime}_{\tau})$ can be interpreted as
$\mathbb{E}_{t}[G_{\tau}]$ with
$G_{s}=\int_{t}^{s}f(r,Y_{r},Z_{r})dr+L^{\prime}_{s}$. In the following, we
will establish the well-posedness of rough RBSDEs with random terminal times
and provide a characterization of the optimal stopping problem in the rough
path setting.

Fix $\tau\in\mathcal{T}_{0},$ for $\xi\in L^{\infty}(\mathcal{F}_{\tau})$
consider the following equation

(4.14) $\left\\{\begin{aligned}
Y_{t}&=\xi+\int_{t\land\tau}^{\tau}f(r,S_{r},Y_{r},Z_{r})dr+\int_{t\land\tau}^{\tau}H(S_{r},Y_{r})d\mathbf{X}_{r}+K_{\tau}-K_{t\land\tau}-\int_{t\land\tau}^{\tau}Z_{r}dW_{r}\\\
Y_{t}&\geq L_{t\land\tau},\quad\int_{0}^{\tau}(Y_{r}-L_{r})dK_{r}=0,\quad
t\in[0,T],\end{aligned}\right.$

where $L_{\tau}\leq\xi.$ Set

(4.15)
$\displaystyle\left(\hat{f}(t,x,y,z),\hat{L}_{t},\hat{H}(t,x,y)\right):=\left(f(t,x,y,z)1_{\\{t\leq\tau\\}},L_{t\land\tau},H(x,y)1_{\\{t\leq\tau\\}}\right).$

######  Definition 4.3.

Let $\\{\mathrm{X}^{n}\\}_{n\geq 1}$ be a sequence of smooth paths with lifts
$\mathbf{X}^{n}\rightarrow\mathbf{X}$ in
$C^{0,p\text{-var}}([0,T],G^{[p]}(\mathbb{R}^{l})).$ Assume that for each
$n\geq 1$ Eq. (3.4) with parameter
$(\xi,\hat{f},\hat{L},\hat{H},\mathrm{X}^{n})$ admits a unique solution
$(Y^{n},Z^{n},K^{n})\in\mathrm{H}^{\infty}_{[0,T]}\times\mathrm{H}^{2}_{[0,T]}(\mathbb{R}^{d})\times\mathrm{I}^{1}_{[0,T]}$.
We call a triple
$(Y,Z,K)\in\mathrm{H}^{\infty}_{[0,T]}\times\mathrm{H}^{2}_{[0,T]}(\mathbb{R}^{d})\times\mathrm{I}^{1}_{[0,T]}$
a solution of Eq. (4.14) with terminal time $\tau$ and parameter
$(\xi,f,L,H,\mathbf{X})$ if

$(Y^{n}_{\cdot},K^{n}_{\cdot})\rightarrow(Y_{\cdot},K_{\cdot})\ \text{
uniformly in }\ [0,T]\ \text{ a.s., }\ Z^{n}\rightarrow Z\ \text{ in }\
\mathrm{H}^{2}_{[0,T]}(\mathbb{R}^{d}),$

and

$\int_{0}^{T}(Y_{r}-L_{r})dK_{r}=0,$

If $(Y,Z,K)$ is independent of the choice of $\left\\{\mathrm{X}^{n},n\geq
1\right\\},$ we call $(Y,Z,K)$ the unique solution of Eq. (4.14) with terminal
time $\tau$ and parameter $(\xi,f,L,H,\mathbf{X}).$

Now we give the well-posedness of Eq. (4.14) by adapting the arguments of
Theorem 3.6 into a stopping time scenario.

######  Proposition 4.4.

Under the same assumptions of Theorem 3.6, and assume further that $\xi\in
L^{\infty}(\mathcal{F}_{\tau}),$ where $\tau\in\mathcal{T}_{0}.$ Then there
exists a unique solution $(Y,Z,K)$ of Eq. (4.14) with terminal time $\tau$ and
parameter $(\xi,f,L,H,\mathbf{X}).$

###### Proof.

Let $\left\\{\mathrm{X}^{n}:[0,T]\rightarrow\mathbb{R}^{l},n\geq 1\right\\}$
be a sequence of smooth paths such that their lifts
$\mathbf{X}^{n}\rightarrow\mathbf{X}$ in
$C^{0,p\text{-var}}([0,T],G^{[p]}(\mathbb{R}^{l})).$ Denoted by
$\mathbf{X}^{0}:=\mathbf{X}.$ For $n\geq 1$, let $(Y^{n},Z^{n},K^{n})$ be the
unique solution of Eq. (3.4) with parameter
$(\xi,\hat{f},\hat{L},\hat{H},\mathrm{X}^{n}),$ where
$(\hat{f},\hat{L},\hat{H})$ are defined in (4.15). For $n\geq 1,$
$T^{\prime}\in[0,T],$ and $t\in[0,T^{\prime}],$ we give the definitions of
$(\widetilde{Y}^{n,T^{\prime}},\widetilde{Z}^{n,T^{\prime}},\widetilde{K}^{n,T^{\prime}})$
by

(4.16)
$\displaystyle\widetilde{Y}^{n,T^{\prime}}_{t}:=(\phi^{n,T^{\prime}})^{-1}(t\land\tau,S_{t\land\tau},Y^{n}_{t\land\tau}),\
\widetilde{K}^{n,T^{\prime}}_{t}:=\int_{0}^{t\land\tau}\frac{1}{D_{y}\phi^{n,T^{\prime}}(s,S_{s},\widetilde{Y}^{n,T^{\prime}}_{s})}dK^{n}_{s},$
$\displaystyle\widetilde{Z}^{n,T^{\prime}}_{t}:=1_{\\{t\leq\tau\\}}\left(-\frac{(D_{x}\phi^{n,T^{\prime}}(t,S_{t},\widetilde{Y}^{n,T^{\prime}}_{t}))^{\top}}{D_{y}\phi^{n,T^{\prime}}(t,S_{t},\widetilde{Y}^{n,T^{\prime}}_{t})}\sigma_{t}+\frac{1}{D_{y}\phi^{n,T^{\prime}}(t,S_{t},\widetilde{Y}^{n,T^{\prime}}_{t})}Z^{n}_{t}\right),$

where $\phi^{n,T^{\prime}}$ satisfies Eq. (3.15). For $n\geq 0$ let
$(\widetilde{f}^{n,T^{\prime}},\widetilde{L}^{n,T^{\prime}})$ be given by
(3.17). For $n\geq 1$, applying Itô’s formula to
$(\phi^{n,T^{\prime}})^{-1}(t,S_{t},Y^{n}_{t})$ and using (3.6), we have for
$t\in[0,T^{\prime}]$

$\left\\{\begin{aligned}
\widetilde{Y}^{n,T^{\prime}}_{t\land\tau}&=(\phi^{n,T^{\prime}})^{-1}(T^{\prime}\land\tau,S_{T^{\prime}\land\tau},Y^{n}_{T^{\prime}\land\tau})+\int_{t\land\tau}^{T^{\prime}\land\tau}\widetilde{f}^{n,T^{\prime}}(r,S_{r},\widetilde{Y}^{n,T^{\prime}}_{r},\widetilde{Z}^{n,T^{\prime}}_{r})dr\\\
&\qquad\qquad\qquad+\widetilde{K}^{n,T^{\prime}}_{T^{\prime}\land\tau}-\widetilde{K}^{n,T^{\prime}}_{t\land\tau}-\int_{t\land\tau}^{T^{\prime}\land\tau}\widetilde{Z}^{n,T^{\prime}}_{r}dW_{r},\\\
\widetilde{Y}^{n,T^{\prime}}_{t\land\tau}&\geq\widetilde{L}^{n,T^{\prime}}_{t\land\tau},\quad\int_{0}^{T^{\prime}\land\tau}\left(\widetilde{Y}^{n,T^{\prime}}_{r}-\widetilde{L}^{n,T^{\prime}}_{r}\right)d\widetilde{K}^{n}_{r}=0,\end{aligned}\right.$

and then the fact
$(\widetilde{Y}^{n,T^{\prime}}_{t},\widetilde{Z}^{n,T^{\prime}}_{t},\widetilde{K}^{n,T^{\prime}}_{t})=(\widetilde{Y}^{n,T^{\prime}}_{t\land\tau},\widetilde{Z}^{n,T^{\prime}}_{t}1_{\\{t\leq\tau\\}},\widetilde{K}^{n,T^{\prime}}_{t\land\tau})$
yields that for $t\in[0,T^{\prime}]$

$\left\\{\begin{aligned}
\widetilde{Y}^{n,T^{\prime}}_{t}&=(\phi^{n,T^{\prime}})^{-1}(T^{\prime}\land\tau,S_{T^{\prime}\land\tau},Y^{n}_{T^{\prime}\land\tau})+\int_{t}^{T}\widetilde{f}^{n,T^{\prime}}(r,S_{r},\widetilde{Y}^{n,T^{\prime}}_{r},\widetilde{Z}^{n,T^{\prime}}_{r})1_{\\{r\leq\tau\\}}dr\\\
&\qquad\qquad\qquad+\widetilde{K}^{n,T^{\prime}}_{T}-\widetilde{K}^{n,T^{\prime}}_{t}-\int_{t}^{T}\widetilde{Z}^{n,T^{\prime}}_{r}dW_{r},\\\
\widetilde{Y}^{n,T^{\prime}}_{t}&\geq\widetilde{L}^{n,T^{\prime}}_{t\land\tau},\quad\int_{0}^{T^{\prime}}\left(\widetilde{Y}^{n,T^{\prime}}_{r}-\widetilde{L}^{n,T^{\prime}}_{r\land\tau}\right)d\widetilde{K}^{n}_{r}=0,\end{aligned}\right.$

which implies that
$(\widetilde{Y}^{n,T^{\prime}},\widetilde{Z}^{n,T^{\prime}},\widetilde{K}^{n,T^{\prime}})$
is a solution of Eq. (3.4) with terminal time $T^{\prime}$ and parameter
$(\widetilde{\xi}^{n,T^{\prime}},\widetilde{f}^{n,T^{\prime}}(t,x,\widetilde{y},\widetilde{z})1_{\\{t\leq\tau\\}},\widetilde{L}^{n,T^{\prime}}_{t\land\tau},0,0),$
where

(4.17)
$\widetilde{\xi}^{n,T^{\prime}}:=(\phi^{n,T^{\prime}})^{-1}(T^{\prime}\land\tau,S_{T^{\prime}\land\tau},Y^{n}_{T^{\prime}\land\tau}).$

The rest of the proof is similar as the procedure from (3.20) to (3.27) in the
proof of Theorem 3.6, so we just illustrate it briefly. By the same argument
to obtain $\eqref{e:bounde of Yn},$ there exists a constant $\widetilde{M}>0$
such that for all $n\geq 1$ and $T^{\prime}\in[0,T]$

$\operatorname*{ess\,sup}_{(t,\omega)\in[0,T^{\prime}]\times\Omega}|\widetilde{Y}^{n,T^{\prime}}_{t}|\leq\widetilde{M}.$

In addition, by the same argument in the step 2 of the proof of Theorem 3.6,
there exist an integer $N>0$ and a set of increasing real numbers
$\left\\{h^{(i)},i=0,1,...,N\right\\}$ such that
$[T-h^{(N)},T-h^{(0)}]=[0,T],$ and that the Eq. (4.14) with terminal time
$T-h^{(i)}$ and parameter

$(\widetilde{\xi}^{0,T-h^{(i)}},\widetilde{f}^{0,T-h^{(i)}}(t,x,\widetilde{y},\widetilde{z})1_{\\{t\leq\tau\\}},\widetilde{L}^{0,T-h^{(i)}}_{t\land\tau},0,0)$

admits a unique solution on interval $[T-h^{(i+1)},T-h^{(i)}].$ For $n\geq 1$
the unique solution coincides with
$(\widetilde{Y}^{n,T-h^{(i)}},\widetilde{Z}^{n,T-h^{(i)}},\widetilde{K}^{n,T-h^{(i)}}).$
And for $n=0$ we denote the unique solution by
$(\widetilde{Y}^{0,T-h^{(i)}},\widetilde{Z}^{0,T-h^{(i)}},\widetilde{K}^{0,T-h^{(i)}}).$
For each $i=0,1,...,N-1$ and $t\in[T-h^{(i+1)},T-h^{(i)}],$ define by

(4.18) $\displaystyle
Y^{0}_{t}:=\phi^{0,T-h^{(i)}}(t\land\tau,S_{t\land\tau},\widetilde{Y}^{0,T-h^{(i)}}_{t}),$
$\displaystyle
K^{0,T-h^{(i)}}_{t}:=\int_{T-h^{(i+1)}}^{t\land\tau}D_{y}\phi^{0,T-h^{(i)}}(s,S_{s},\widetilde{Y}^{0,T-h^{(i)}}_{s})d\widetilde{K}^{0,T-h^{(i)}}_{s},$
$\displaystyle Z^{0}_{t}:=1_{\\{t\leq\tau\\}}\cdot
D_{y}\phi^{0,T-h^{(i)}}(t,S_{t},\widetilde{Y}^{0,T-h^{(i)}}_{t})\left[\widetilde{Z}^{0,T-h^{(i)}}_{t}+\frac{(D_{x}\phi^{0,T-h^{(i)}}(t,S_{t},\widetilde{Y}^{0,T-h^{(i)}}_{t}))^{\top}}{D_{y}\phi^{0,T-h^{(i)}}(t,S_{t},\widetilde{Y}^{0,T-h^{(i)}}_{t})}\sigma_{t}\right].$

Then using Lemma A.6, and doing an induction indexed by $i=0,1,...,N-1$
similar to the one as in the proof of Theorem 3.6, it follows that

$\displaystyle\widetilde{Y}^{n,T-h^{(i)}}_{\cdot}\rightarrow\widetilde{Y}^{0,T-h^{(i)}}_{\cdot},\
\widetilde{K}^{n,T-h^{(i)}}_{\cdot}-\widetilde{K}^{n,T-h^{(i)}}_{T-h^{(i+1)}}\rightarrow\widetilde{K}^{0,T-h^{(i)}}_{\cdot}-\widetilde{K}^{0,T-h^{(i)}}_{T-h^{(i+1)}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\textrm{ uniformly in
}[T-h^{(i+1)},T-h^{(i)}]\textrm{ a.s.,}$
$\displaystyle\widetilde{Z}^{n,T-h^{(i)}}\rightarrow\widetilde{Z}^{0,T-h^{(i)}}\textrm{
in }\mathrm{H}^{2}_{[T-h^{(i+1)},T-h^{(i)}]},$

and then

$\displaystyle{Y}^{n}_{\cdot}\rightarrow{Y}^{0}_{\cdot},\
{K}^{n}_{\cdot}\rightarrow{K}^{0}_{\cdot}\textrm{ uniformly in }[0,T]\textrm{
a.s.,}$ $\displaystyle{Z}^{n}\rightarrow{Z}^{0}\textrm{ in
}\mathrm{H}^{2}_{[0,T]},$

where $K^{0}_{0}:=0$ and

$K^{0}_{t}:=\sum_{j=1}^{N-j-1}K^{0,T-h^{(N-j)}}_{T-h^{(N-j)}}+K^{0,T-h^{(i)}}_{t},\quad
t\in(T-h^{(i+1)},T-h^{(i)}],\quad i=0,1,...,N-1.$

∎

######  Remark 4.3.

For the solutions obtained by Proposition 4.4, we claim that similar result as
in Corollary 3.8 and Proposition 3.9 (i.e., the comparison property and the
stability property) still hold.

Now for a stopping time $\tau\in\mathcal{T}_{0}$ and $\xi\in
L^{\infty}(\mathcal{F}_{\tau})$ consider a BSDE with random terminal time

(4.19)
$Y_{t}=\xi+\int_{t\land\tau}^{\tau}f(r,S_{r},Y_{r},Z_{r})dr+\int_{t\land\tau}^{\tau}H(S_{r},Y_{r})d\mathbf{X}_{r}-\int_{t\land\tau}^{\tau}Z_{r}dW_{r},\
t\in[0,T].$

Similar to Definition 4.3 and Proposition 4.4, we provide the definition for
the solution to a BSDE with random terminal time and its existence and
uniqueness result.

######  Definition 4.5.

Let the parameter $(\hat{f},\hat{H})$ be defined by (4.15), and
$\\{\mathrm{X}^{n}\\}_{n\geq 1}$ be a sequence of smooth paths with lifts
$\mathbf{X}^{n}\rightarrow\mathbf{X}$ in
$C^{0,p\text{-var}}([0,T],G^{[p]}(\mathbb{R}^{l})).$ Assume for each $n\geq 1$
Eq. (3.31) with parameter $(\xi,\hat{f},\hat{H},\mathrm{X}^{n})$ admits a
unique solution
$(Y^{n},Z^{n})\in\mathrm{H}^{\infty}_{[0,T]}\times\mathrm{H}^{2}_{[0,T]}(\mathbb{R}^{d}).$
We call
$(Y,Z)\in\mathrm{H}^{\infty}_{[0,T]}\times\mathrm{H}^{2}_{[0,T]}(\mathbb{R}^{d})$
a solution of Eq. (4.14) with terminal time $\tau$ and parameter
$(\xi,f,H,\mathbf{X})$ if

$Y^{n}_{\cdot}\rightarrow Y_{\cdot}\ \text{ uniformly in }\ [0,T]\ \text{
a.s., }\ Z^{n}\rightarrow Z\ \text{ in }\
\mathrm{H}^{2}_{[0,T]}(\mathbb{R}^{d}).$

If $(Y,Z)$ is independent of the choice of $\left\\{\mathrm{X}^{n},n\geq
1\right\\},$ we call $(Y,Z)$ the unique solution of Eq. (4.19).

######  Remark 4.4.

In our case (i.e. under the assumption of Proposition 4.4), Definition 4.3 is
a generalization of Definition 4.5 with $K=0$. By Remark A.2 and estimate
(3.20), a uniform prior estimate can be established for $Y^{n},$ and thus $Y$
is uniform bounded. Then a solution to (4.19) is a solution to (4.14) with
$L\equiv-M$ for some $M>0$ large enough and $K=0.$

According to Remark 4.4, we have the well-posedness of (4.19) by Proposition
4.4.

######  Proposition 4.6.

Assume $H$ and $f$ satisfy the conditions in Assumption $(H_{p,\gamma})$ and
($H_{f}$) respectively. Assume $\xi\in L^{\infty}(\mathcal{F}_{\tau}),$ and
$(\sigma,b)$ satisfies Assumption ($H_{b,\sigma}$). Then there exists a unique
solution of Eq. (4.19).

Now we are ready to show the main result of this subsection. For each
$\tau\in\mathcal{T}_{t},$ let the pair of processes
$(B_{s}(\tau,L^{\prime}_{\tau}),\pi_{s}(\tau,L^{\prime}_{\tau}))$ be the
unique solution of the BSDE (4.19) with terminal time $\tau$ and parameter
$(L^{\prime}_{\tau},f,H,\mathbf{X})$, where $L^{\prime}$ is defined in (4.13).

######  Theorem 4.7.

Assume $(\xi,f,L,H,\mathbf{X})$ satisfies the conditions in Theorem 3.6. Let
$(Y,Z,K)$ be the unique solution of the rough RBSDE Eq. (3.1) with parameter
$(\xi,f,L,H,\mathbf{X}).$ Then, for each $t\in[0,T],$

(4.20)
$Y_{t}=B_{t}(D_{t},L^{\prime}_{D_{t}})=\operatorname*{ess\,sup}_{\tau\in\mathcal{T}_{t}}B_{t}(\tau,L^{\prime}_{\tau}),$

where $D_{t}=\inf\\{u\geq t;Y_{u}=L^{\prime}_{u}\\}.$

###### Proof.

Denoted by $\mathbf{X}^{0}:=\mathbf{X}.$ Let
$\left\\{\mathrm{X}^{n}:[0,T]\rightarrow\mathbb{R}^{l},n\geq 1\right\\}$ be a
sequence of smooth paths with their lifts
$\mathbf{X}^{n}\rightarrow\mathbf{X}^{0}$ in
$C^{0,p\text{-var}}([0,T],G^{[p]}(\mathbb{R}^{l})).$ For each $n\geq 1,$
according to [34] and [32], the BSDE with stopping terminal time $\tau$ and
parameter $(L^{\prime}_{\tau},f,H,\mathrm{X}^{n})$ admits a unique solution,
and we denote it by
$(B^{n}_{t}(\tau,L^{\prime}_{\tau}),\pi^{n}_{t}(\tau,L^{\prime}_{\tau})).$
Using the same argument as in the proof of Proposition 4.4, an estimate
similar to (3.20) can be established, i.e., there exists a constant $M>0$ such
that, for all $\tau\in\mathcal{T}_{t}$

(4.21) $\sup_{n\geq
1}\operatorname*{ess\,sup}_{(t,\omega)\in[0,T]\times\Omega}|B^{n}_{t}(\tau,L^{\prime}_{\tau})|\leq
M.$

Let $\overline{L}_{t}:=(-M-1)\land L_{t}.$ By (4.21) we have
$B^{n}_{t}(\tau,L^{\prime}_{\tau})>\overline{L}_{t}.$ According to the
uniqueness result shown by [33, Corollary 1], the triplet
$(B^{n}_{t}(\tau,L^{\prime}_{\tau}),\pi^{n}_{t}(\tau,L^{\prime}_{\tau}),0)$
uniquely solves the RBSDE (4.14) with terminal time $\tau$ and parameter
$(L^{\prime}_{\tau},f,\overline{L},H,\mathrm{X}^{n}).$ Consequently, by
Definition 4.3, 4.5, and Proposition 4.6,
$(B_{t}(\tau,L^{\prime}_{\tau}),\pi_{t}(\tau,L^{\prime}_{\tau}),0)$ uniquely
solves the RBSDE (4.14) associated with terminal time $\tau$ and parameter
$(L^{\prime}_{\tau},f,\overline{L},H,\mathbf{X}).$ Since $Y_{\tau}\geq
L^{\prime}_{\tau}$ and $L_{\cdot}\geq\overline{L}_{\cdot}$ a.s., in view of
Remark 4.3, we have $Y_{t}\geq B_{t}(\tau,L^{\prime}_{\tau}).$ Thus, we have

$Y_{t}\geq\operatorname*{ess\,sup}_{\tau\in\mathcal{T}_{t}}B_{t}(\tau,L^{\prime}_{\tau}).$

It remains to show $Y_{t}=B_{t}(D_{t},L^{\prime}_{D_{t}}).$ Firstly, Let
$(Y^{0},Z^{0},K^{0})$ be the solution of Eq. (4.14) with terminal time
$\tau=D_{t}$ and parameter $(Y_{D_{t}},f,L,H,\mathbf{X}).$ By Definition 4.3,
$(Y^{0},Z^{0},K^{0})$ is the limit of $(Y^{n},Z^{n},K^{n})$, which is the
unique solution of Eq. (4.14) with terminal time $\tau=D_{t}$ and parameter
$(Y_{D_{t}},f,L,H,\mathrm{X}^{n}).$ Meanwhile, by Theorem 3.6, $(Y,Z,K)$ is
the limits of $(Y^{(n)},Z^{(n)},K^{(n)})$, which is the unique solution of Eq.
(3.1) with parameter $(\xi,f,L,H,\mathrm{X}^{n})$. Then $(Y_{\cdot\land
D_{t}},Z_{\cdot}1_{\\{\cdot\leq D_{t}\\}},K_{\cdot\land D_{t}})$ is the limits
of $(Y^{(n)}_{\cdot\land D_{t}},Z^{(n)}_{\cdot}1_{\\{\cdot\leq
D_{t}\\}},K^{(n)}_{\cdot\land D_{t}}),$ which is the unique solution of Eq.
(4.14) with terminal time $\tau=D_{t}$ and parameter
$(Y^{(n)}_{D_{t}},f,L,H,\mathrm{X}^{n}).$ Since $Y^{(n)}_{D_{t}}\rightarrow
Y_{D_{t}}$ a.s., by the stability of solutions implied by Remark 4.3, we have

(4.22) $(Y^{0}_{s},Z^{0}_{s},K^{0}_{s})=(Y_{s\land D_{t}},Z_{s}1_{\\{s\leq
D_{t}\\}},K_{s\land D_{t}}),\ \text{ for all }\ s\in[0,T].$

Secondly, we will show that $Y^{0}_{s}=B_{s}(D_{t},L^{\prime}_{D_{t}})$ for
$s\in[t,T],$ which implies our conclusion. For any $T^{\prime}\in[t,T]$ and
$s\in[t,T^{\prime}],$ let $\phi^{0,T^{\prime}}$ be the unique solution of Eq.
(3.15) with $n=0,$ and
$(\widetilde{f}^{0,T^{\prime}},\widetilde{L}^{0,T^{\prime}})$ be given by Eq.
(3.17) with $n=0.$ Let
$(\widetilde{Y}^{0,T^{\prime}},\widetilde{Z}^{0,T^{\prime}},\widetilde{K}^{0,T^{\prime}})$
be defined by (4.16) with $n=0$ and $\tau=D_{t}.$ By the proof of Proposition
4.4, we see the boundedness of $Y^{0}$, and thus there exists a constant
$\widetilde{M}>0$ such that for all $T^{\prime}\in[t,T]$ and
$s\in[t,T^{\prime}]$

$\left|\widetilde{Y}^{0,T^{\prime}}_{s}\right|\leq\widetilde{M}.$

Let $\widetilde{\xi}^{0,T^{\prime}}$ be defined by (4.17) with $n=0$ and
$\tau=D_{t}.$ By the proof of Proposition 4.4, there exist an integer $N>0$
and a set of increasing real numbers $\left\\{h^{(i)},i=0,1,...,N\right\\}$
such that $[T-h^{(N)},T-h^{(0)}]=[t,T],$ and that the triplet

$(\widetilde{Y}^{0,T-h^{(i)}}_{\cdot},\widetilde{Z}^{0,T-h^{(i)}}_{\cdot},\widetilde{K}^{0,T-h^{(i)}}_{\cdot}-\widetilde{K}^{0,T-h^{(i)}}_{T-h^{(i)}})$

is the unique solution of Eq. (4.14) on $[T-h^{(i+1)},T-h^{(i)}]$ with
terminal time $(T-h^{(i)})\land D_{t}$ and parameter
$(\widetilde{\xi}^{0,T-h^{(i)}},\widetilde{f}^{0,T-h^{(i)}},\widetilde{L}^{0,T-h^{(i)}},0,0).$
On each subinterval $[T-h^{(i+1)},T-h^{(i)}],$ since $Y_{s}>L_{s}$ for all
$s\in[t,D_{t}),$ we have
$\widetilde{Y}^{0,T-h^{(i)}}_{s}>\widetilde{L}^{0,T-h^{(i)}}_{s},$
$s\in[T-h^{(i+1)},T-h^{(i)}]\cap[t,D_{t}).$ Thus,
$\widetilde{K}^{0,T-h^{(i)}}_{\cdot}\equiv 0.$ Note that by [20, Theorem A.2],
we see that Eq. (4.19) with terminal time $D_{t}$ and parameter
$(\widetilde{\xi}^{0,T-h^{(i)}},\widetilde{f}^{0,T-h^{(i)}},0,0)$ has a unique
solution. Then it follows that
$(\widetilde{Y}^{0,T-h^{(i)}},\widetilde{Z}^{0,T-h^{(i)}})$ is the solution.
In view of (4.18), we have that $K^{0,T-h^{(i)}}=0$ and $(Y^{0},Z^{0})$ is
exactly the solution constructed in the proof of [20, Theorem 3].222The case
for rough BSDEs with stopping terminal times is not involved in the proof of
[20, Theorem 3]. However, according to Proposition 4.4, their proof can be
adapted to the stopping terminal case without difficulties. It follows that
$(Y^{0},Z^{0})$ is the unique solution of Eq. (4.19) with terminal time
$D_{t}$ and parameter $(L^{\prime}_{D_{t}},f,H,\mathbf{X})$ which implies our
claim and completes the proof.

∎

## Aknowlegdements

Li’s research was supported by the National Natural Science Foundation of
China (No. 12301178), the Natural Science Foundation of Shandong Province for
Excellent Young Scientists Fund Program (Overseas) (No. 2023HWYQ-049) and the
Natural Science Foundation of Shandong Province (No. ZR2023ZD35). HZ is
partially supported by NSF of China (Grant Numbers 12031009) and Young
Research Project of Tai-Shan (No.tsqn202306054). Li and HZ are supported by
NSF of Shandong (No. ZR2023MA026).

##  Appendix A. Results on RBSDEs and rough BSDEs

In this section, we recall some basic definitions and results concerning
reflected BSDEs with superlinear quadratic coefficient in [33] and results
concerning BSDEs with rough drivers in [20]. Assume the obstacle process $L$
and the terminal value $\xi$ satisfy Assumption ($H_{L,\xi}$). Consider the
following RBSDE

(A.1) $\left\\{\begin{aligned}
&Y_{t}=\xi+\int_{t}^{T}f(s,S_{s},Y_{s},Z_{s})ds+K_{T}-K_{t}-\int_{t}^{T}Z_{s}dW_{s},\
\ t\in[0,T],\\\ &Y_{t}\geq
L_{t},\quad\int_{0}^{T}(Y_{r}-L_{r})dK_{r}=0,\end{aligned}\right.$

where $S$ is given by (3.2). We assume

######  Assumption (H1).

Let
$f:\Omega\times[0,T]\times\mathbb{R}^{d}\times\mathbb{R}\times\mathbb{R}^{d}\rightarrow\mathbb{R}$
be
$\mathcal{P}\otimes\mathcal{B}(\mathbb{R}^{d})\otimes\mathcal{B}(\mathbb{R})\otimes\mathcal{B}(\mathbb{R}^{d})$-measurable.
Assume there is a strictly positive increasing continuous function $c$
satisfying $\int_{0}^{\infty}\frac{dy}{c(y)}=\infty$ so that

there exists a constant $C_{f}>0$ such that $\mathbb{P}$-a.s.,

$\displaystyle|f(t,x,y,z)|\leq c(|y|)+C_{f}|z|^{2},\text{ for all
}(t,x,y,z)\in[0,T]\times\mathbb{R}^{d}\times\mathbb{R}\times\mathbb{R}^{d};$

for every $M>0,$ there exists a constant $C_{M}>0$ such that
$\mathbb{P}$-a.s.,

$|D_{z}f(t,x,y,z)|\leq C_{M}(1+|z|),\text{ for all
}(t,x,y,z)\in[0,T]\times\mathbb{R}^{d}\times[-M,M]\times\mathbb{R}^{d};$

for every $\varepsilon,M>0,$ there exist two constants
$C_{\varepsilon,M},h_{\varepsilon,M}>0$ such that $\mathbb{P}$-a.s.,

$D_{y}f(t,x,y,z)\leq C_{\varepsilon,M}+\varepsilon|z|^{2},\text{ for all
}(t,x,y,z)\in[T-h_{\varepsilon,M},T]\times\mathbb{R}^{d}\times[-M,M]\times\mathbb{R}^{d}.$

######  Definition A.1.

We say that $(Y,Z,K)$ is a solution of the RBSDE (A.1) associated with
terminal value $\xi,$ obstacle $L$ and coefficient $f,$ which is denoted by
Eq. $(\xi,L,f),$ if

  * (a)

it satisfies Eq. (A.1) for every $t\in[0,T]$ a.s..

  * (b)

$Y\in\mathrm{H}^{\infty}_{[0,T]},$
$Z\in\mathrm{H}^{2}_{[0,T]}(\mathbb{R}^{d}),$ and
$K\in\mathrm{I}^{1}_{[0,T]}.$

Kobylanski et al. [33, Theorem 1] gives the following prior estimate of $Y.$

######  Lemma A.2.

Assume that $(b,\sigma)$ satisfies Assumption ($H_{b,\sigma}$), $f$ satisfies
(A1) in (H1), $(L,\xi)$ satisfy ($H_{L,\xi}$), and
$(Y,Z,K)\in\mathrm{H}_{[0,T]}^{\infty}(\mathbb{R})\times\mathrm{H}_{[0,T]}^{2}(\mathbb{R}^{d})\times\mathrm{I}_{[0,T]}^{1}(\mathbb{R})$
is a solution of the Eq. (A.1) with parameter $(\xi,f,L),$ we have

(A.2) $|Y_{t}|\leq U_{0}=:\bar{M},$

with $U:[0,T]\rightarrow\mathbb{R}$ solving the following backward ODE

$U_{t}=b+\int_{t}^{T}c(U_{r})dr,$

where