Maximum Principle for Quasi-linear Reflected Backward SPDEs
Guanxing Fu$^1$
Ulrich Horst$^1$
Jinniao Qiu$^2$
This paper establishes a maximum principle for quasi-linear reflected backward stochastic partial differential equations (RBSPDEs for short). We prove the existence and uniqueness of the weak solution to RBSPDEs allowing for non-zero Dirichlet boundary conditions and, using a stochastic version of De Giorgi's iteration, establish the maximum principle for RBSPDEs on a general domain. The maximum principle for RBSPDEs on a bounded domain and the maximum principle for backward stochastic partial differential equations (BSPDEs for short) on a general domain can be obtained as byproducts. Finally, the local behavior of the weak solutions is considered.
1604.02425.tex
%------------------------------------------------------------------------------
% Beginning of journal.tex
%------------------------------------------------------------------------------
%
% AMS-LaTeX version 2 sample file for journals, based on amsart.cls.
%
%        ***     DO NOT USE THIS FILE AS A STARTER.      ***
%        ***  USE THE JOURNAL-SPECIFIC *.TEMPLATE FILE.  ***
%
% Replace amsart by the documentclass for the target journal, e.g., tran-l.
%
%\documentclass{amsart}



\documentclass[10pt]{article}
%\usepackage{}
%\usepackage{mathbbold}

\usepackage{amssymb}   %basic
%\usepackage{amsfonts}
\usepackage{amsthm}    %theoremstyle
\usepackage{amsmath}   %\mathcal
%\usepackage{bbm}
%\usepackage[dvips]{graphics}
%\usepackage{newlfont}
\usepackage{stmaryrd}  %\interleave
\usepackage{titletoc}  %\ttl
\usepackage{mathrsfs}  %\mathscr
%\usepackage{hyperref}
\usepackage{graphicx}
\usepackage{xcolor}
\usepackage[toc,page,title,titletoc,header]{appendix}
\usepackage{ulem}



% Fuzz -------------------------------------------------------------------
\vfuzz2pt % Don't report over-full v-boxes if over-edge is small
\hfuzz2pt % Don't bother to report over-full boxes if over-edge is < 2pt

% Line spacing -----------------------------------------------------------
\newlength{\defbaselineskip}
\setlength{\defbaselineskip}{\baselineskip}
\newcommand{\setlinespacing}[1]%
           {\setlength{\baselineskip}{#1 \defbaselineskip}}
\newcommand{\doublespacing}{\setlength{\baselineskip}%
                           {2.0 \defbaselineskip}}
\newcommand{\singlespacing}{\setlength{\baselineskip}{\defbaselineskip}}

% THEOREMS ---------------------------------------------------------------
%\theoremstyle{plain}
%\newtheorem{thm}{Theorem}[section]
%\newtheorem{cor}[thm]{Corollary}
%\newtheorem{lem}[thm]{Lemma}
%\newtheorem{prop}[thm]{Proposition}
%\newtheorem{exam}[thm]{Example}
%\newtheorem{ques}[thm]{Question}
%\theoremstyle{definition}
%\newtheorem{defn}{Definition}[section]
\newtheorem*{assumption}{Standing Assumption}
%\newtheorem{rmk}{Remark}[section]

% \newcommand --------------------------------------------------------------
\newcommand{\eps}{\varepsilon}
\newcommand{\vf}{\varphi}
\DeclareMathOperator*{\esssup}{ess\,sup}
\DeclareMathOperator*{\essinf}{ess\,inf}
\newcommand{\cP}{\mathcal{P}}
\newcommand{\cO}{\mathcal{O}}
\newcommand{\cH}{\mathcal{H}}
\newcommand{\cL}{\mathcal{L}}
\newcommand{\cM}{\mathcal{M}}
\newcommand{\cB}{\mathcal{B}}
\newcommand{\cS}{\mathcal{S}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\cV}{\mathcal{V}}
\newcommand{\fD}{\mathfrak{D}}
\newcommand{\fX}{\mathfrak{X}}
\newcommand{\fS}{\mathfrak{S}}
\newcommand{\bH}{\mathbb{H}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bN}{\mathbb{N}}
\newcommand{\bD}{\mathbb{D}}
\newcommand{\bZ}{\mathbb{Z}}
\newcommand{\sF}{\mathscr{F}}
\newcommand{\sP}{\mathscr{P}}
\newcommand{\sD}{\mathscr{D}}
\newcommand{\sH}{\mathscr{H}}
\newcommand{\sS}{\mathscr{S}}
\newcommand{\sU}{\mathscr{U}}
\newcommand{\sV}{\mathscr{V}}
\newcommand{\la}{\langle}
\newcommand{\ra}{\rangle}
\newcommand{\intl}{\interleave}
\newcommand{\ptl}{\partial}
\newcommand{\rrow}{\rightarrow}
\newcommand{\tbf}{\textbf}

%the width and height of the text--------------------------------------------
\textwidth =16cm \topmargin =-18mm \textheight =23.5cm \oddsidemargin=5pt
\evensidemargin=0pt
%\baselineskip=5mm
\renewcommand{\baselinestretch}{1.1}

\renewcommand{\theequation}{\thesection.\arabic{equation}}
\makeatletter\@addtoreset{equation}{section} \makeatother
% \allowdisplaybreaks
%\setcounter{page}{1}
%===============================Add==========================================
 \allowdisplaybreaks
%=========================================================================



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\usepackage[all]{xy}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{xca}[theorem]{Exercise}

\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

\numberwithin{equation}{section}

%    Absolute value notation
\newcommand{\abs}[1]{\lvert#1\rvert}

%%    Blank box placeholder for figures (to avoid requiring any
%%    particular graphics capabilities for printing this document).
%\newcommand{\blankbox}[2]{%
%  \parbox{\columnwidth}{\centering
%%    Set fboxsep to 0 so that the actual size of the box will match the
%%    given measurements more closely.
%    \setlength{\fboxsep}{0pt}%
%    \fbox{\raisebox{0pt}[#2]{\hspace{#1}}}%
%  }%
%}




%\geometry{verbose,letterpaper,tmargin=20mm,bmargin=20mm,lmargin=20mm,rmargin=20mm}
\parindent 0em
\parskip 2ex





\begin{document}

\title{Maximum Principle for Quasi-linear Reflected Backward SPDEs}

%    Information for first author
\author{ Guanxing Fu$^1$ \and Ulrich Horst$^1$ \and Jinniao Qiu$^2$}
%    Address of record for the research reported here
%\address{Department of Mathematics, Humboldt-Universit\"at zu Berlin
        % Unter den Linden 6, 10099 Berlin, Germany.}

%\email{ fuguanxing725@gmail.com (Guanxing Fu), horst@math.hu-berlin.de (Ulrich Horst), qiujinn@gmail.com (Jinniao Qiu)}
%    \thanks will become a 1st page footnote.
%\thanks{}

\footnotetext[1]{Department of Mathematics, Humboldt-Universit\"{a}t zu Berlin, Unter den Linden 6, 10099 Berlin, Germany.
 Financial support from the Berlin Mathematical School and the SFB 649 ``Economic Risk'' is gratefully acknowledged.
  \textit{E-mail}: fuguanxing725@gmail.com (Guanxing Fu); horst@math.hu-berlin.de (Ulrich Horst).}

  \footnotetext[2]{  Department of Mathematics, University of Michigan, East Hall, 530 Church Street, Ann Arbor, MI 48109-1043, USA.
  \textit{Email}: qiujinn@gmail.com (Jinniao Qiu).}

%    General info
%\subjclass[2000]{60H15, 35R16}

%\date{Dec 16, 2014.}

%\dedicatory{This paper is dedicated to our advisors.}

%\keywords{backward stochastic partial differential equations, maximum principle, De Giorgi iteration.}

\maketitle

\begin{abstract}
This paper establishes a maximum principle for quasi-linear reflected backward stochastic partial differential equations (RBSPDEs for short). We prove the existence and uniqueness of the weak solution to RBSPDEs allowing for non-zero Dirichlet boundary conditions and, using a stochastic version of De Giorgi's iteration, establish the maximum principle for RBSPDEs on a general domain. The maximum principle for RBSPDEs on a bounded domain and the maximum principle for backward stochastic partial differential equations (BSPDEs for short) on a general domain can be obtained as byproducts. Finally, the local behavior of the weak solutions is considered.
\end{abstract}

{\bf AMS Subject Classification:} 60H15;  35R60

{\bf Keywords:} reflected backward stochastic partial differential equation, backward stochastic partial differential equation, maximum principle, De Giorgi's iteration

\section{Introduction}
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a complete probability space carrying a standard $m$-dimensional Brownian motion $W=\{W_t,t\geq 0\}$. Let $(\mathcal{F}_t)_{t \geq 0}$ be the natural filtration generated by $W,$ augmented by the $\mathbb{P}$-null sets in $\mathcal{F}$. In this paper, we establish a maximum principle for weak solutions to the reflected backward stochastic partial differential equation (RBSPDE)

\begin{equation}\label{equation1.1}
  \left\{\begin{array}{l}
  \begin{aligned}
-du(t,x)&=[\partial _j(a^{ij}\partial _iu(t,x)+\sigma ^{jr}v^r(t,x))
%\\
%&\quad
+f(t,x,u(t,x),\nabla u(t,x),v(t,x))\\
&\quad+\nabla \cdot g(t,x,u(t,x),\nabla u(t,x),v(t,x))]\,dt+\mu (dt,x)-v^r(t,x)dW^r_t,\\
&\quad~~~(t,x)\in Q:=[0,T]\times \mathcal{O},\\
u(T,x)&=G(x),~~~x\in \mathcal{O},\\
u(t,x)&\geq \xi (t,x) ~dt\times dx\times d\mathbb{P}-a.e.,\\
\int_Q(u(t,x)&-\xi (t,x))\mu (dt,dx)=0,
    \end{aligned}
  \end{array}\right.
\end{equation}
with general Dirichlet boundary conditions. Here and in what follows, the usual summation convention is applied, $\xi$ is a given stochastic process defined on $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq 0},\mathbb{P})$, called the {\sl obstacle process}, $T \in (0,\infty)$ is a deterministic {\sl terminal time}, $\mathcal{O}\subset \mathbb{R}^n$ is a possibly unbounded domain,  $\partial_ju=\frac{\partial u}{\partial x_j}$ and $\nabla=(\partial_{1},\cdots,\partial_{d})$ denotes the gradient operator. A {\sl solution} to the RBSPDE is a random triple $(u,v,\mu)$ defined on $\Omega\times[0,T]\times\bR^n$ such that (\ref{equation1.1}) holds in a suitable sense.

Since the introduction by Bensoussan \cite{Bensousan_83} backward stochastic partial differential equations (BSPDEs) have been extensively investigated in the probability and stochastic control literature. They naturally arise in many applications, for instance as stochastic Hamilton-Jacobi-Bellman equations associated with non-Markovian control problems \cite{peng}, as adjoint equations of the Duncan-Mortensen-Zakai equaltion in nonlinear filtering \cite{xhou} and as adjoint equations in stochastic control problems when formulating stochastic maximum principles \cite{Bensousan_83}. BSPDEs with singular terminal conditions arise in non-Markovian models for financial mathematics to describe optimal trading in illiquid financial markets \cite{paulwinqiuhorst}.

%in the optimal control of processes with incomplete information or more generally of systems driven by stochastic parabolic PDEs,
%We refer to \cite{duqiutang,englezoskaratzas,Hu_Ma_Yong02,tangwei} and references therein for an extensive analysis of BSPDEs.
Reflected BSPDEs arise as the Hamilton-Jacobi-Bellman equation for the optimal stopping problem of stochastic differential equations with random coefficients \cite{changpangyong,qiuwei}, and as the adjoint equations for the maximum principle of Pontryagin type in singular control problems of stochastic partial differential equations in, e.g. \cite{oksendalsulemzhang}
%{\color{blue}in which authors} derived RBSPDEs as the adjoint equations for the maximum principle of Pontryagin type.

Existence and uniqueness of solutions results for reflected PDEs and SPDEs have been established by many authors.
%including \cite{denismzhang,karouikapapeng}. In particular,
Pierre \cite{pierre,pierre1980} has studied parabolic PDEs with obstacles using parabolic potentials. Using methods and techniques from parabolic potential theory Denis, Matoussi and Zhang \cite{denismzhang} proved existence and uniqueness of solutions results for quasi-linear SPDEs driven by infinite dimensional Brownian motion. More recently, Qiu and Wei \cite{qiuwei} established a general theory of existence and uniqueness of solution for a class of quasi-linear RBSPDEs, which includes the classical results on obstacle problems for deterministic parabolic PDEs as special cases.

Adapting Moser's iteration scheme to the nonlinear case Aronson and Serrin \cite{AronsonSerrin1967} proved the maximum principle and local bounds of weak solutions for deterministic quasi-linear parabolic equations on bounded domains. Their method was extended by Denis, Matoussi, and Stoica \cite{DenisMatoussiStoica2005} to the stochastic case, obtaining an $L^p$ a priori estimate for the uniform norm of the solution of the stochastic quasi-linear parabolic equation with null Dirichlet condition, and further adapted by Denis, Matoussi, and Stoica \cite{denis-matoussi-stoica-2009} to local solutions. Later, Denis, Matoussi, and Zhang \cite{denismatoussizhang} established $L^p$ estimates for the uniform norm in time and space of weak solutions to reflected quasi-linear SPDEs along with a maximum principle for local solutions using a stochastic version of Moser's iteration scheme. Recently, Qiu and Tang \cite{qiutang} used the De Giorgi's iteration scheme, a technique that also works for degenerate parabolic equations, to establish a local and global maximum principle for weak solutions of BSPDEs without reflection. To the best of our knowledge a maximum principle for reflected BSPDEs has not yet been established in the literature.

In this paper we establish a maximum principle for reflected BSPDEs on possibly unbounded domains; a maximum principle and a comparison principle for BSPDEs on general domains, a maximum principle for RBSPDEs on bounded domains and a local maximum principle for RBSPDEs are obtained as well. Due to the obstacle, the maximum principle for RBSPDE is not a direct extension of that for BSPDE in \cite{qiutang}. Our proofs rely on a stochastic version of De Giorgi's iteration scheme that does not depend on the Lebesgue measure of the domain; this extends the scheme in \cite{qiutang} that only applies to bounded domains. Our iteration scheme requires an almost sure representation of the $L^2$ norm of the positive part of the of the weak solution of RBSPDEs. This, in turn requires generalizing the It\^o's formula for weak solutions to BSPDEs established in \cite{qiutang} and \cite{qiuwei} to the positive part of weak solutions.

It is worth pointing out that by contrast to $L^p$ estimates ($p\in (2,\infty)$) for the time and space maximal norm of weak solutions to \textit{forward} SPDEs or related obstacle problems as established in \cite{DenisMatoussiStoica2005,denismatoussizhang,qiu-2015-DSPDE}, our estimate for weak  solutions is uniform with respect to $w\in \Omega$ and hence establishes an $L^{\infty}$ estimate. This distinction comes from the essential difference between BSPDEs and \textit{forward} SPDEs: the noise term in the former endogenously originates from martingale representation and is hence governed by the coefficients, while the latter is fully exogenous, which prevents any $L^\infty$ estimate  for \textit{forward} SPDEs.

%
%Specifically, we prove the following result.
%
%\begin{theorem}
%Under suitable condition on the coefficients and the obstacle process the RBSPDE \eqref{equation1.1} has a unique bounded solution $(u,v,\mu)$ that satisfies the zero Dirichlet condition $u|_{\partial\mathcal{O}}=0$.
%\begin{itemize}
%	\item[i)] For a general domain $\mathcal{O}$,
%\begin{align*}
%\textrm{esssup}_{\Omega\times Q}u^{\pm}\leq C\left(\textrm{esssup}_{\Omega\times\partial_pQ}u^{\pm}+\Xi_1(f,g,\xi)\right),
%\end{align*}
%  where the constant $C$ depends only on $T$ and the structure parameters in (\ref{equation1.1}) and $\Xi_1(f,g,\xi)$  is expressed in terms of some quantities related to the coefficients $f,g$ and the obstacle $\xi$.
%  \item[ii)] For a bounded domain $\mathcal{O}$,
%\begin{align*}
%&\textrm{esssup}_{(\omega,t,x)\in\Omega \times Q}u^{\pm}(\omega,t,x) \leq  C\left(
%\textrm{esssup}_{(\omega,t,x)\in\Omega \times \partial _pQ}u^{\pm}
%+ \Xi_2(f,g,\xi) \right),
%\end{align*}
%where the parabolic boundary of $Q$ and the constant $C$ depends only on $T,|\cO|$ and the structure terms of RBSPDE \eqref{equation1.1}, while $\Xi_2(f,g,\xi)$ is expressed in terms of some quantities related to the coefficients $f,g$ and $\xi$.
%	\item[iii)] Let $Q_{2\rho}:=[t_0-4\rho^2,t_0)\times B_{2\rho}(x_0)\subset Q$ with $\rho \in (0,1)$. Then,
%\begin{align*}
%\textrm{esssup}_{(\omega,s,x)\in \Omega\times Q_{\frac{\rho}{2}}}u^{\pm}
%&\leq C \left(\rho^{-\frac{n+2}{2}}\left(\|u^{\pm}\|_{0,2;Q_{\rho}}+\|{\hat{\xi}}^{\pm}\|_{0,2;Q_{2\rho}}\right)+\rho ^{\theta} \Xi_2(f,g,\xi) \right),
%\end{align*}
% where the constant $C$ depends only on the structure terms of RBSPDE \eqref{equation1.1}, $\theta\in (0,1)$ is one of the structure terms of  RBSPDE \eqref{equation1.1} and $\Xi_2(f,g,\xi)$ are expressed in terms of some quantities related to the coefficients $f,g$ and obstacle $\xi$. In particular, solutions of RBSPDE \eqref{equation1.1} must be locally essentially bounded.
%\end{itemize}
%\end{theorem}
%
%Classical maximum principles for deterministic quasi-linear parabolic equations without obstacles were established by Aronson and Serrin \cite{AronsonSerrin1967} using Moser's iteration. Their results were later generalized by
%%for the deterministic versions of Theorems \ref{thm-intro-1} and \ref{thm-intro-2} and
%%Their proofs are mainly based on Moser's iteration which was generalized by
%Denis, Matoussi and Stoica~\cite{DenisMatoussiStoica2005} to SPDEs and further used by Denis, Matoussi and Zhang \cite{denismatoussizhang} to obtain a maximum principle for quasi-linear SPDEs with obstacles. The latter claims an $L^p$ estimate for the time-space uniform norm of weak solutions. An alternative way to obtain maximum principles is based on De Giorgi's iteration; this approach even works for degenerate parabolic cases as shown by, e.g. \cite{qiu-2015-DSPDE}. We shall, therefore, we shall adopt the method of De Giorgi's iteration to establish our maximum principle.
%
%By contrast to $L^p$ estimates ($p\in (2,\infty)$) for the time and space maximal norm of weak solutions to \textit{forward} SPDEs or related obstacle problems (see \cite{DenisMatoussiStoica2005,denismatoussizhang,qiu-2015-DSPDE}), our estimate for weak  solutions is uniform with respect to $w\in \Omega$ and hence establishes an $L^{\infty}$ estimate. This distinction comes from the essential difference between BSPDEs and \textit{forward} SPDEs: the noise term in the former comes from the martingale representation and is governed by the coefficients, and is thus endogenous, while in the latter it is exogenous, which prevents any $L^\infty$ estimate  for \textit{forward} SPDEs.

%For the maximum principle of quasi-linear RBSPDE \eqref{equation1.1}, the main difficulty arises from the appearance of the obstacle process $\xi$ which is rough and dominates the regularity and boundedness of weak solutions. This not only makes necessary a stochastic generalization of the parabolic potential and capacity theory by Pierre \cite{pierre} but also requires a new framework of spaces under which certain a priori estimates are derived (see \cite{denismatoussizhang, qiuwei} for more details). Moreover, due to the necessary adaptedness of the integrand of It\^o's stochastic integral, the technique of Steklov time average (see~\cite[page 100]{Lieberman}) finds difficulty in the stochastic situation and we have to establish some It\^o formulas to get around the difficulty.

%This paper is concerned with maximum principle for RBSPDE (\ref{equation1.1}). As for the maximum principle for RBSPDE on a general domain, we apply a new scheme, which is different from \cite{qiutang}. Our scheme dose not depend on the Lebesgue measure of the domain so that it can be applied to the general domain scenario. As corollaries, the maximum principle for RBSPDE on a bounded domain and the maximum principle for BSPDE on a general domain are obtained. As far as we know, there is still no result about them.

%In the recent work, \cite{qiuwei} established the uniqueness and existence of a class of quasi-linear RBSPDE. In order to give a rigorous meaning to the solution , they are motivated by Michael Pierre's seminal work \cite{pierre}, \cite{pierre1980} and \cite{pierre1983}. Since the measure $\mu$ in (1.1) in general is not absolutely continuous with respect to Lebesgue measure, so integral such as $\int(u-\xi)=0$ might make no sense. So \cite{pierre}, \cite{pierre1980} and \cite{pierre1983} introduced the notions capacity and quasi-continuous in order to characterize the solution $\mu$ a.s. such that the integral make sense. It should be mentioned that \cite{denismatoussizhang} and \cite{denismzhang} also applied Pierre's work into reflected SPDE theory. But \cite{qiuwei} extend the results to the backward and stochastic sense.

%As far as we know, no one considers the maximum principle of RBSPDE before, which is a powerful tool to study the regularity of solutions. Due to the existence of the obstacle, the maximum principle issue for RBSPDE is not a direct extension of the issue for BSPDE in \cite{qiutang}. Indeed, we have to generalize the It\^o formula established in \cite{qiutang} and \cite{qiuwei} and apply De Giorgi Iteration (see \cite{chen}) to the RBSPDE.

The paper is organized as follows: in Section 2, we list some notations and the standing assumptions on the parameters of the RBSPDE \eqref{equation1.1}. The existence and uniqueness of weak solution to the RBSPDE \eqref{equation1.1} is presented in Section 3. In Sections 4, we establish the maximum principle for the RBSPDE (\ref{equation1.1}) on a general domain as well as the maximum principles for RBSPDEs on a bounded domain and BSPDEs on a general domain. The local behavior of the weak solutions to (\ref{equation1.1}) is also considered. Finally, we list in the appendix some useful lemmas, the frequently used It\^o formulas and some definitions related to the stochastic regular measure.


\section{Preliminaries and standing assumptions}

%\subsection{Notations}
%
%Suppose $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq 0},\mathbb{P})$ is a complete filtered probability space with the filtration $(\mathcal{F}_t)_{t\geq 0}$ being generated by the Brownian motion $\{W_t,t\geq 0\}$ and augmented by all the $\mathbb{P}$ null sets in $\mathcal{F}$. Denote by $\mathbb{F}$ the $\sigma$-algebra generated by all predictable sets on $\Omega\times[0,T]$ associated with $(\mathcal{F}_t)_{t\geq 0}$.

 For an arbitrary domain $\Pi$ in some Euclidean space, let $\mathcal{C}^\infty_0(\Pi)$ be the class of infinitely differentiable functions with compact support in $\Pi$, and $L^2(\Pi)$ be the usual square integrable space on $\Pi$ with the scalar product $\langle u,v\rangle_{\Pi}=\int_{\Pi}u(x)v(x)dx$ and the norm $\|u\|_{L^2(\Pi)}=\langle u,u\rangle^{\frac{1}{2}}_{\Pi}$ for each pair $u,v\in L^2(\Pi)$. For $(k,p)\in\bZ\times [1,\infty)$ where $\bZ$ is the set of all the integers, let $H^{k,p}(\Pi)$ be the usual $k$-th order Sobolev space. For convenience, when $\Pi=\mathcal{O}$, we write $\langle\cdot,\cdot\rangle$ and $\|\cdot\|$ for $\langle \cdot,\cdot\rangle_{\mathcal{O}}$ and $\|\cdot\|_{L^2(\mathcal{O})}$ respectively. We recall that $Q = [0,T] \times {\cal O}$.
%We recall some spaces in \cite{qiutang}.
%\begin{definition}

For $t\in[0,T]$ and $\Pi\subseteq\mathbb{R}^n$, we put $\Pi _t :=[t,T]\times \Pi$. Denote by $H^{k,p}_{\mathcal{F}}(\Pi _t)$ the class of $H^{k,p}(\Pi)$-valued predictable processes on $[t,T]$ such that for each $u\in H^{k,p}_{\mathcal{F}}(\Pi _t)$ we have that
\begin{equation*}
\|u\|_{H^{k,p}_{\mathcal{F}}(\Pi _t)}:=\left(E\left[\int_t^T\|u(s,\cdot)\|^p_{H^{k,p}(\Pi)}ds\right]\right)^{1/p}<\infty.
\end{equation*}
Let $\mathcal{M}^{k,p}(\Pi _t)$ be the subspace of $H^{k,p}_{\mathcal{F}}(\Pi _t)$ such that
\begin{equation*}
\|u\|_{k,p;\Pi _t}:=\left(\textrm{esssup}_{\omega\in \Omega}\sup_{s\in[t,T]}E\left[\int_s^T\|u(\omega ,\tau , \cdot)\|^p_{H^{k,p}_{\mathcal{F}}(\Pi)}d\tau | \mathcal{F}_s\right]\right)^{1/p}<\infty
\end{equation*}
and  $\mathcal{L}^{\infty}(\Pi _t)$ be the subspace of $H^{0,p}_{\mathcal{F}}(\Pi _t)$ such that
\begin{equation*}
\|u\|_{\infty;\Pi _t}:=\textrm{esssup}_{(\omega ,s,x)\in \Omega \times \Pi _t}|u(\omega ,s,x)|<\infty.
\end{equation*}
Denote by $\mathcal{L}^{\infty,p}(\Pi _t)$ the subspace of $H^{0,p}_{\mathcal{F}}(\Pi _t)$ such that
\begin{equation*}
\|u\|_{\infty,p;\Pi _t}:=\textrm{esssup}_{(\omega , s)\in \Omega \times [t,T]}\|u(\omega,s,\cdot)\|_{L_p(\Pi)}<\infty.
\end{equation*}
Let $\mathcal{V}_2(\Pi _t)$ be the class of all $u\in H^{1,2}_{\mathcal{F}}(\Pi_t)$ such that
\begin{equation*}
\|u\|_{\mathcal{V}_2(\Pi _t)}:=\left(\|u\|_{\infty,2;\Pi_t}^2+\|\nabla u\|_{0,2;\Pi _t}^2\right)^{1/2}<\infty
\end{equation*}
and let $\mathcal{V}_{2,0}(\Pi _t)$ be the subspace of $\mathcal{V}_2(\Pi _t)$ for which
\begin{equation*}
\lim\limits_{r\rightarrow 0}||u(s+r,\cdot)-u(s,\cdot)||_{L^2(\Pi)}=0 ~~~\textrm{for}~\textrm{all}~s,s+r\in[t,T],\quad \text{a.s.}
\end{equation*}
%\end{definition}

%From now on, we assume that $C$ is a positive constant and $C(\cdot)$ is a positive constant depending on $\cdot$, which may vary from line to line. Before investigating (1.1) further, we borrow an embedding lemma from \cite{qiutang}, which will be frequently used later.
%\begin{lemma}\label{lemma2.6}
%If for each $t\in[0,T]$, $u\in\mathcal{V}_2(\mathcal{O}_t)$, then we have
%$$\|u\|_{0,\frac{2(n+2)}{n};\mathcal{O}_t}\leq C\|\nabla u\|^{\frac{n}{n+2}}_{0,2;\mathcal{O}_t}\textrm{esssup}_{(\omega,s)\in\Omega\times[t,T]}\|u(\omega,s)\|^{\frac{n}{n+2}}\leq C\|u\|_{\mathcal{V}_2(\mathcal{O}_t)},$$
%where $C$ only depends on $n$.
%\end{lemma}


%\subsection{Standing Assumptions}


\begin{assumption}
We assume throughout that the coefficients and the obstacle process of the RBSPDE (1.1) satisfy the following conditions. Denote by $\mathbb{F}$ the $\sigma$-algebra generated by all predictable sets on $\Omega\times[0,T]$ associated with $(\mathcal{F}_t)_{t\geq 0}$.

\begin{itemize}
	\item [($\mathcal{A}_1$)] The random functions
\[
	g(\cdot,\cdot,\cdot,X,Y,Z):\Omega \times [0,T]\times \mathcal{O}\rightarrow \mathbb{R}^n \quad  \text{and}\quad f(\cdot,\cdot,	\cdot,X,Y,Z):\Omega \times [0,T]\times \mathcal{O}\rightarrow \mathbb{R}
\]
are $\mathbb{F}\otimes \mathcal{B}(\mathcal{O})$-measurable for any $(X,Y,Z)\in \mathbb{R}\times\mathbb{R}^n\times\mathbb{R}^m$ and there exist positive constants $L$, $\kappa$ and $\beta$ such that for each $(X_i,Y_i,Z_i)\in \mathbb{R}\times\mathbb{R}^n\times\mathbb{R}^m$, $i=1,2$,
$$|g(\cdot,\cdot,\cdot,X_1,Y_1,Z_1)-g(\cdot,\cdot,\cdot,X_2,Y_2,Z_2)|\leq L|X_1-X_2|+\frac{\kappa}{2}|Y_1-Y_2|+\sqrt{\beta}|Z_1-Z_2|$$
and
$$|f(\cdot,\cdot,\cdot,X_1,Y_1,Z_1)-f(\cdot,\cdot,\cdot,X_2,Y_2,Z_2)|\leq L(|X_1-X_2|+|Y_1-Y_2|+|Z_1-Z_2|).$$

	\item[($\mathcal{A}_2$)] The coefficients $a$ and $\sigma$ are $\mathbb{F}\otimes \mathcal{B}(\mathcal{O})$-measurable and there exist positive constants $\varrho >1$, $\lambda$ and $\Lambda$ such that for each $\eta \in \mathbb{R}^n$ and $(\omega,t,x)\in\Omega\times[0,T]\times\mathcal{O}$,
\begin{align*}
	\lambda |\eta|^2\leq (2a^{ij}(\omega ,t,x)-\varrho \sigma ^{ir}\sigma ^{jr}(\omega,t,x))\eta ^i\eta ^j &\leq \Lambda |\eta|^2\\
	|a(\omega ,t,x)|+|\sigma (\omega ,t,x)| &\leq \Lambda,
\end{align*}
and
$$\lambda -\kappa -\varrho '\beta >0 ~with~\varrho ':=\frac{\varrho}{\varrho -1}.$$

	\item[($\mathcal{A}_3$)]
The terminal value satisfies $G\in L^{\infty}(\Omega , \mathcal{F}_T, L^2(\mathcal{O}))\cap L^{\infty}(\Omega,\mathcal{O})$ and for some $p>\max\{n+2,2+4/n\}$, one has
\begin{align*}
	g_0 & :=g(\cdot,\cdot,\cdot,0,0,0)\in \mathcal{M}^{0,p}(Q)\cap \mathcal{M}^{0,2}(Q)\\
	f_0 & :=f(\cdot,\cdot,\cdot,0,0,0)\in \mathcal{M}^{0,\frac{p(n+2)}{p+n+2}}(Q)\cap\mathcal{M}^{0,2}(Q).
\end{align*}

	\item[($\mathcal{A}_4$)]	The obstacle process $\xi$ is almost surely quasi-continuous (see Appendix for the definition) on $Q$ and there exists a process $\hat{\xi}$ such that $\xi \leq \hat{\xi}$ $ds\times dx\times d\mathbb{P}$-a.e., where $\hat{\xi}\in \mathcal{V}_{2,0}(Q)$ together with some $\hat{v}\in\mathcal{M}^{0,2}(Q)$ is a solution to BSPDE
\begin{equation}\label{xi-hat}
\left\{\begin{array}{ll}
\begin{split}
-d\hat{\xi}(t,x)&=[\partial _j(a^{ij}\partial _i\hat{\xi}(t,x)+\sigma ^{jr}\hat{v}^r(t,x))
+\hat{f}(t,x)+\nabla \cdot \hat{g}(t,x)]dt
-\hat{v}^r(t,x)dW^r_t,\\
&~~~\quad (t,x)\in Q,\\
\hat{\xi}(T,x)&=\hat{G}(x),~~~x\in \mathcal{O},
\end{split}
\end{array}
\right.
\end{equation}
with the random functions $\hat{f}$, $\hat {g}$ and $\hat{G}$ satisfying
\begin{align*}
	 &\hat{G} \in L^{\infty}(\Omega , \mathcal{F}_T, L^2(\mathcal{O}))\cap L^{\infty}(\Omega,\mathcal{O}),
	 &\hat{f} \in \mathcal{M}^{0,\frac{p(n+2)}{p+n+2}}(Q)\cap\mathcal{M}^{0,2}(Q),
	 &~~\hat{g} \in \mathcal{M}^{0,p}(Q)\cap\mathcal{M}^{0,2}(Q).
\end{align*}

\item[($\mathcal{A}_5$)] The function $x \mapsto g(\cdot,\cdot,\cdot,x,0,0)$ is uniformly Lipschitz continuous in norm:
\begin{align*}
	\|g(\cdot,\cdot,\cdot,X_1,0,0)-g(\cdot,\cdot,\cdot,X_2,0,0)\|_{0,p;Q}\leq L|X_1-X_2|;\\
    \|g(\cdot,\cdot,\cdot,X_1,0,0)-g(\cdot,\cdot,\cdot,X_2,0,0)\|_{0,2;Q}\leq L|X_1-X_2|.
\end{align*}
\end{itemize}
\end{assumption}

\begin{remark}

While the assumptions $(\mathcal{A}_1-\mathcal{A}_4)$ are standard for the existence and uniqueness of solution, the assumption $\mathcal{A}_5$ is required for the iteration scheme for proof of the maximum principle in Theorem \ref{Theorem_MP_GD} below, which follows easily from $(\mathcal{A}_1)$ when the domain is bounded.

\end{remark}

For the index $p$ specified in $(\mathcal{A}_3)$ and $t\in[0,T]$, define the functional $A_p$ and $B_2$ as follows:
\begin{equation*}
A_p(l,h;\mathcal{O}_t):=\|l\|_{0,\frac{p(n+2)}{p+n+2};\mathcal{O}_t}+\|h\|_{0,p;\mathcal{O}_t}, ~~(l,h)\in \mathcal{M}^{0,\frac{p(n+2)}{p+n+2}}(\mathcal{O}_t)\times \mathcal{M}^{0,p}(\mathcal{O}_t)
\end{equation*}
and
\begin{equation*}
B_2(l,h;\mathcal{O}_t):=\|l\|_{0,2;\mathcal{O}_t}+\|h\|_{0,2;\mathcal{O}_t}, ~~(l,h)\in \mathcal{M}^{0,2}(\mathcal{O}_t)\times \mathcal{M}^{0,2}(\mathcal{O}_t).
\end{equation*}

In Sections 3 and 4, we will repeatedly use the Young inequality of the form
\begin{equation}\label{repeatedestimate}
\langle f,g\rangle=\langle \sqrt{\epsilon}f,\frac{1}{\sqrt{\epsilon}}g\rangle \leq \frac{1}{2}\left[\epsilon \|f\|^2+\frac{1}{\epsilon}\|g\|^2\right].
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Existence and uniqueness of weak solution to RBSPDE \eqref{equation1.1}}
In this section we prove an existence and uniqueness of weak solutions result for the RBSPDE \eqref{equation1.1} along with a strong norm estimate. The difficulty in defining weak solutions to the RBSPDE \eqref{equation1.1} is the random measure $\mu$. It is typically a local time so the Skorokhod condition $\int_Q(u-\xi)\,\mu(dt,dx)=0$ might not make sense. To give a rigorous meaning to the integral condition, the theory of parabolic potential and capacity introduced by \cite{pierre,pierre1980} was generalized by \cite{qiuwei} to a backward stochastic framework. We recall the definition of quasi continuity and stochastic regular measures in Appendix \ref{appendix-measure}.

\begin{definition}\label{definition3.1}
The triple $(u,v,\mu)$ is called a weak solution to the RBSPDE \eqref{equation1.1} if:
\begin{itemize}
	\item[(1)] $(u,v)\in \mathcal{V}_{2,0}(Q)\times \mathcal{M}^{0,2}(Q)$ and $\mu$ is a stochastic regular measure;
	\item[(2)] the RBSPDE \eqref{equation1.1} holds in the weak sense, i.e., for each $\varphi \in \mathcal{C}_0^{\infty}(\mathbb{R}^+)\otimes \mathcal{C}_0^{\infty}(\mathcal{O})$, we have
\begin{align*}
& \langle u(t,\cdot),\varphi (t,\cdot)\rangle \\
= &
\langle G(\cdot),\varphi(T,\cdot)\rangle
 -\int_t^T\left\{\langle u(s,\cdot),\partial _s\varphi (s,\cdot)\rangle+\langle\partial _j\varphi (s,\cdot),a^{ij}(s,\cdot)\partial _iu(s,\cdot) +\sigma ^{jr}v^r(s,\cdot)\rangle\right\}ds \\
&+\int_t^T\left[\langle f(s,\cdot,u(s,\cdot),\nabla u(s,\cdot),v(s,\cdot)),\varphi (s,\cdot)\rangle
-\langle g^j(s,\cdot,u(s,\cdot),\nabla u(s,\cdot),v(s,\cdot)),\partial_j \varphi (s,\cdot)\rangle\right]ds\\
&+\int_{[t,T]\times\mathcal{O}}\varphi (s,x)\mu (ds,dx)-\int_t^T\langle\varphi (s,\cdot),v^r(s,\cdot)dW_s^r\rangle,\quad\text{a.s.};
\end{align*}
	\item[(3)] $u$ admits a quasi-continuous version $\tilde{u}$ such that $\tilde{u}\geq \xi~ds\times dx\times d\mathbb{P}~a.e.$ and
\begin{equation}
\int_Q(\tilde{u}(t,x)-\xi (t,x))\mu (dt,dx)=0~~\mathbb{P}\text{-a.s.}
\end{equation}
\end{itemize}
\end{definition}

We denote by $\mathcal{U}(\xi,f,g,G)$ the set of all the weak solutions of the RBSPDE (\ref{equation1.1}) associated with the obstacle process $\xi$, the terminal condition $G$, and the coefficients $f$ and $g$. Further, $\mathcal{U}(-\infty,f,g,G)$ is the set of solutions  when there is no obstacle, i.e., $\mathcal{U}(-\infty,f,g,G)$ is the set of solution pairs $(u,v)$ to the associated BSPDE with terminal condition $G$ and coefficients $f$ and $g$. %Denote by $\mathcal{U}(-\infty)$ the totality of $(u,v)\in \mathcal{U}(-\infty,f,g,G)$ for some associated $f,g$ and $G$.

The following theorem guarantees the existence and uniqueness of weak solutions in the sense of Definition \ref{definition3.1}. The arguments for the norm estimate also apply to Lemma \ref{estimate-u-k} below, which is needed for the proof of our maximum principle.


\begin{theorem}\label{theorem3.2}
Suppose that Assumptions $(\mathcal{A}_1)$-$(\mathcal{A}_4)$ hold and that $\hat{\xi}|_{\partial\mathcal{O}}=0$. Then the RBSPDE (1.1) admits a unique solution $(u,v,\mu)$ that satisfies the zero Dirichlet condition $u|_{\partial\mathcal{O}}=0$. Moreover, for each $t\in [0,T]$, one has
\begin{equation}\label{equation3.2}
\begin{split}
\|u\|_{\mathcal{V}_2(\mathcal{O}_t)}+\|v\|_{0,2;\mathcal{O}_t}\leq &
C\left( \textrm{esssup}_{\omega \in \Omega}\|G(\omega,\cdot)\|_{L^2(\mathcal{O})}+\textrm{esssup}_{\omega \in \Omega}\|\hat{G}(\omega,\cdot)\|_{L^2(\mathcal{O})}\right. \\
&\left.+B_2(f_0,g_0;\mathcal{O}_t)+B_2(\hat{f},\hat{g};\mathcal{O}_t)\right),\\
\end{split}
\end{equation}
where the positive constant $C$ only depends on the constants $\lambda$, $\varrho$, $\kappa$, $\beta$, $L$  and $T$.
\end{theorem}

\begin{proof}
It has been shown in \cite[Theorem 4.12]{qiuwei} that the RBSPDE (1.1) admits a unique solution $(u,v,\mu)$ satisfying the zero Dirichlet condition $u|_{\partial\mathcal{O}}=0$ and that this solutions satisfies the integrability condition
\begin{equation*}
E\left[\sup_{t\in[0.T]}\|u(t)\|^2\right]+E\left[\int_0^T\|\nabla u(t)\|^2\,dt\right]+E\left[\int_0^T\|v(t)\|^2\,dt\right]<\infty.
\end{equation*}
Hence, we only need to prove the estimate (\ref{equation3.2}). To this end, notice first that
\begin{align*}
&\int_t^T\int_{\mathcal{O}}(u(s,x)-\hat{\xi}(s,x))\,\mu(dsdx)\\
=& \int_t^T\int_{\mathcal{O}}(u(s,x)-\xi(s,x)+\xi(s,x)-\hat{\xi}(s,x))\,\mu(dsdx)\\
\leq& 0.
\end{align*}
Thus for each $t\in [0,T]$, Proposition \ref{proposition6.1} yields almost surely,
\begin{equation} \label{estimate_begin}
\begin{split}
&\|u(t)-\hat{\xi}(t)\|^2+\int_t^T\|v(s)-\hat{v}(s)\|^2ds
\\
=&\|G-\hat{G}\|^2-\int_t^T\langle u(s)-\hat{\xi}(s),v^r(s) -\hat{v}^r(s)\rangle\, dW_s^r \\
& - \int_t^T\langle 2\partial _j(u-\hat{\xi}(s)),a^{ij}\partial _i(u-\hat{\xi})(s)  +\sigma ^{jr}(v^r-\hat{v}^r)\rangle\, ds \\
& -\int_t^T\langle 2\partial _j(u-\hat{\xi}(s)),g^j(s,u(s),\nabla u(s),v(s))-\hat{g}^j(s)\rangle\, ds \\
& +\int_t^T\langle 2(u-\hat{\xi}(s)),f(s,u(s),\nabla u(s),v(s))-\hat{f}(s)\rangle\, ds \\
& +\int_{\mathcal{O}_t}2(u(s,x)-\hat{\xi}(s,x))\mu (ds,dx)\\
\leq & \|G-\hat{G}\|^2-\int_t^T\langle u(s)-\hat{\xi}(s),v^r(s)-\hat{v}^r(s)\rangle\, dW_s^r \\
& -\int_t^T\langle 2\partial _j(u-\hat{\xi}(s)),a^{ij}\partial _i(u-\hat{\xi})(s)+\sigma ^{jr}(v^r-\hat{v}^r)\rangle \,ds \\
&-\int_t^T\langle 2\partial _j(u-\hat{\xi}(s)),g^j(s,u(s),\nabla u(s),v(s))-\hat{g}^j(s)\rangle \,ds \\
& +\int_t^T\langle 2(u-\hat{\xi}(s)),f(s,u(s),\nabla u(s),v(s))-\hat{f}(s)\rangle \,ds.
\end{split}
\end{equation}
Applying assumption $(\mathcal{A}_2)$ and (\ref{repeatedestimate}), one has
\begin{align*}
I_1:&
=-E\left[\int_t^T\langle 2\partial _j(u-\hat{\xi}(s)),a^{ij}\partial _i(u-\hat{\xi})(s)+\sigma ^{jr}(v_r-\hat{v}^r)\rangle \,ds\Big|\mathcal{F}_t\right]\\
=& -E\left[\int_t^T\langle\partial _j(u-\hat{\xi}(s)),(2a^{ij}-\sigma^{ir}\sigma^{jr}\varrho)\partial _i(u-\hat{\xi})(s)+\sigma^{ir}\sigma^{jr}\varrho\partial _i(u-\hat{\xi})(s)+2\sigma ^{jr}(v^r-\hat{v}^r)\rangle\, ds|\mathcal{F}_t\right]\\
\leq & -\lambda E\left[\int_t^T\|\nabla (u(s)-\hat{\xi}(s))\|^2\,ds|\mathcal{F}_t\right]+\frac{1}{\varrho}E\left[\int_t^T\|v(s)-\hat{v}(s)\|^2\,ds|\mathcal{F}_t\right].
\end{align*}
By assumptions $(\mathcal{A}_1)$ and $(\mathcal{A}_3)$ and the estimate (\ref{repeatedestimate}) it holds for each $\epsilon>0$ and $\theta>0$ that:
\begin{align*}
I_2:&
= -E\left[\int_t^T\langle 2\partial _j(u-\hat{\xi}(s)),g^j(s,u(s),\nabla u(s),v(s))-\hat{g}^j(s)\rangle\, ds|\mathcal{F}_t\right]\\
   \leq & E\left[\int_t^T\langle 2|\nabla(u(s)-\hat{\xi}(s))|,L|u(s)|+\frac{\kappa}{2}|\nabla u(s)|+\sqrt{\beta}|v(s)|\rangle\, ds|\mathcal{F}_t\right]\\
   &+E\left[\int_t^T\langle 2|\nabla(u(s)-\hat{\xi}(s))|,|\hat{g}(s)|+|g_0(s)|\rangle \,ds|\mathcal{F}_t\right]\\
\leq & 2\epsilon E\left[\int_t^T\|\nabla (u(s)-\hat{\xi}(s))\|^2\,ds|\mathcal{F}_t\right]+C(\epsilon)E\left[\int_t^T\|g_0(s)\|^2\,ds|\mathcal{F}_t\right]+C(\epsilon)E\left[\int_t^T\|\hat{g}(s)\|^2\,ds|\mathcal{F}_t\right]\\
&+E\left[\int_t^T\langle 2|\nabla(u(s)-\hat{\xi}(s))|,L|u(s)-\hat{\xi}(s)|+L|\hat{\xi}(s)|\rangle\, ds|\mathcal{F}_t\right]\\
&+E\left[\int_t^T\langle 2|\nabla(u(s)-\hat{\xi}(s))|,\frac{\kappa}{2}|\nabla(u(s)-\hat{\xi}(s))|+\frac{\kappa}{2}|\nabla\hat{\xi}(s)|\rangle\, ds|\mathcal{F}_t\right]\\
&+E\left[\int_t^T\langle 2|\nabla(u(s)-\hat{\xi}(s))|,\sqrt{\beta}|v(s)-\hat{v}(s)|+\sqrt{\beta}|\hat{v}(s)|\rangle\, ds|\mathcal{F}_t\right]\\
\leq & 2\epsilon E\left[\int_t^T\|\nabla(u(s)-\hat{\xi}(s))\|^2\,ds|\mathcal{F}_t\right]+C(\epsilon)E\left[\int_t^T\|\hat{g}(s)\|\,ds|\mathcal{F}_t\right]+C(\epsilon)E\left[\int_t^T\|g_0(s)\|\,ds|\mathcal{F}_t\right]\\
&+2\epsilon E\left[\int_t^T\|\nabla(u(s)-\hat{\xi}(s))\|^2\, ds|\mathcal{F}_t\right]+C(\epsilon,L)E\left[\int_t^T\|u(s)-\hat{\xi}(s)\|^2\, ds|\mathcal{F}_t\right]+C(\epsilon,L)E\left[\int_t^T\|\hat{\xi}(s)\|^2\, ds|\mathcal{F}_t\right]\\
&+\kappa E\left[\int_t^T\|\nabla(u(s)-\hat{\xi}(s))\|^2\,ds|\mathcal{F}_t\right]+\epsilon E\left[\int_t^T\|\nabla(u(s)-\hat{\xi}(s))\|^2\,ds|\mathcal{F}_t\right]+C(\epsilon,\kappa)E\left[\int_t^T\|\nabla\hat{\xi}(s))\|^2\,ds|\mathcal{F}_t\right]\\
&+\beta\theta E\left[\int_t^T\|\nabla(u(s)-\hat{\xi}(s))\|^2\,ds|\mathcal{F}_t\right]+\frac{1}{\theta}E\left[\int_t^T\|v(s)-\hat{v}(s)\|^2\,ds|\mathcal{F}_t\right]\\
&+\epsilon E\left[\int_t^T\|\nabla(u(s)-\hat{\xi}(s))\|^2\,ds|\mathcal{F}_t\right]+C(\epsilon,\beta)E\left[\int_t^T\|\hat{v}(s)\|^2\,ds|\mathcal{F}_t\right]\\
  % \end{align*}
%For each $\epsilon >0$, by the estimate (\ref{repeatedestimate}) there exist constants $C(\epsilon)$ and $C(\epsilon,|\mathcal{O}|,T,p)$ such that
%\begin{align*}
%&E\left[\int_t^T\langle 2|\partial _j(u-\hat{\xi}(s))|,|\hat{g}^j(s)|+|g^j_0(s)|\rangle ds|\mathcal{F}_t\right]\\
%&\leq  \epsilon E\left[\int_t^T\|\nabla (u-\hat{\xi})(s)\|^2ds|\mathcal{F}_t\right]+C(\epsilon)E\left[\int_t^T\left(\|g_0(s)\|^2+\|\hat{g}(s)\|^2\right)ds|\mathcal{F}_t\right]\\
%&\leq \epsilon E\left[\int_t^T\|\nabla (u-\hat{\xi})(s)\|^2ds|\mathcal{F}_t\right]+C(\epsilon,|\mathcal{O}|,T,p)\left(E\left[\int_t^T\|g_0(s)\|^pds|\mathcal{F}_t\right]\right)^{\frac{2}{p}}\\
%&+C(\epsilon,|\mathcal{O}|,T,p)\left(E\left[\int_t^T\|\hat{g}(s)\|^pds|\mathcal{F}_t\right]\right)^{\frac{2}{p}}.
%\end{align*}
%Moreover,
%\begin{align*}
%&E\left[\int_t^T\langle|\nabla(u(s)-\hat{\xi}(s))|,\kappa|\nabla u(s)|\rangle ds|\mathcal{F}_t\right]\\
%&\leq  \kappa E\left[\int_t^T\|\nabla(u(s)-\hat{\xi}(s))\|^2ds|\mathcal{F}_t\right]+\kappa E\left[\int_t^T|\langle \nabla(u(s)-\hat{\xi}(s)),|\nabla \hat{\xi}|\rangle|ds|\mathcal{F}_t\right]\\
%&\leq  (\kappa+\epsilon)E\left(\int_t^T\|\nabla (u(s)-\hat{\xi})\|^2|\mathcal{F}_t\right)+\frac{\kappa}{4\epsilon}E\left(\int_t^T\|\hat{\xi}(s)\|^2ds|\mathcal{F}_t\right).
%\end{align*}
%Finally,
%\begin{align*}
%&E\left[\int_t^T\langle 2|\nabla(u-\hat{\xi}(s))|,L|u(s)|\rangle ds|\mathcal{F}_t\right]\\
%&\leq E\left[\int_t^T\langle 2|\nabla(u-\hat{\xi}(s))|,L|u(s)-\hat{\xi}(s)|+L|\hat{\xi}(s)|\rangle ds|\mathcal{F}_t\right]\\
%&\leq  \epsilon E\left[\int_t^T\|\nabla (u-\hat{\xi})(s)\|^2ds|\mathcal{F}_t\right]+C(\epsilon,L)E\left[\int_t^T\left(\|u(s)-\hat{\xi}(s)\|^2+\|\hat{\xi}(s)\|^2\right)ds|\mathcal{F}_t\right]
%\end{align*}
%and by assumption $(\mathcal{A}_2)$, there exists $\theta>\varrho'$ and $\delta>0$ such that $\lambda-(\kappa+\delta)-\beta\theta>0$.
%\begin{align*}
%&E\left[\int_t^T\langle 2|\partial _j(u-\hat{\xi}(s))|,\sqrt{\beta}|v(s)|\rangle ds|\mathcal{F}_t\right]\\
%&\leq  \theta\beta E\left[\int_t^T\|\nabla (u-\hat{\xi})(s)\|^2ds|\mathcal{F}_t\right]+\frac{1}{\theta}E\left[\int_t^T\|v(s)\|^2ds|\mathcal{F}_t\right]\\
%&\leq \theta\beta E\left[\int_t^T\|\nabla %(u-\hat{\xi})(s)\|^2ds|\mathcal{F}_t\right]+\frac{2}{\theta}E\left[\int_t^T(\|\hat{v}(s)\|^2+\|v(s)-\hat{v}(s)\|^2)ds|\mathcal{F}_t\right],
%\end{align*}
%Combining the five estimates above, we have
%\begin{align*}
\leq & \left(6\epsilon +\kappa+\beta\theta\right)E\left[\int_t^T\|\nabla(u-\hat{\xi})(s)\|^2\,ds|\mathcal{F}_t\right]+\frac{1}{\theta}E\left[\int_t^T\|v(s)-\hat{v}(s)\|^2\,ds|\mathcal{F}_t\right]\\   &+C(\epsilon)E\left[\int_t^T\|g_0(s)\|^2\,ds|\mathcal{F}_t\right]+C(\epsilon)E\left[\int_t^T\|\hat{g}(s)\|^2\,ds|\mathcal{F}_t\right]\\
   & + C(\epsilon,L)E\left[\int_t^T\|u(s)-\hat{\xi}(s)\|^2\,ds|\mathcal{F}_t\right]+C(\epsilon ,L)E\left[\int_t^T\|\hat{\xi}(s)\|^2\,ds|\mathcal{F}_t\right]\\
   &+C(\epsilon, \kappa)E\left[\int_t^T\|\nabla\hat{\xi}(s)\|^2\,ds|\mathcal{F}_t\right]+C(\epsilon,\beta)E\left[\int_t^T\|\hat{v}(s)\|^2\,ds|\mathcal{F}_t\right].
\end{align*}

It follows from $(\mathcal{A}_3)$ that:
\begin{align*}
I_3:&=E\left[\int_t^T\langle 2(u(s)-\hat{\xi}(s)),f(s,u(s),\nabla u(s),v(s))-\hat{f}(s)\rangle \,ds|\mathcal{F}_t\right]\\
&\leq  E\left[\int_t^T\langle 2|u(s)-\hat{\xi}(s)|,|f_0|+L|u(s)|+L|\nabla u(s)|+L|v(s)|+|\hat{f}(s)|\rangle \,ds|\mathcal{F}_t\right].
\end{align*}
In view of (\ref{repeatedestimate}) it further holds for each $\epsilon_1>0$ that:
\begin{align*}
& E\left[\int_t^T\langle 2|u(s)-\hat{\xi}(s)|,L|u(s)|+L|\nabla u(s)|+L|v(s)|\rangle \,ds|\mathcal{F}_t\right]\\
\leq & \epsilon_1 E\left[\int_t^T\|\nabla u(s)\|^2\,ds|\mathcal{F}_t\right]+\epsilon_1 E\left[\int_t^T\|v(s)\|^2\,ds|\mathcal{F}_t\right]+C(\epsilon_1,L)E\left[\int_t^T\|u(s)-\hat{\xi}(s)\|^2\,ds|\mathcal{F}_t\right]\\
&+E\left[\int_t^T\langle 2|(u-\hat{\xi})(s)|,L|(u-\hat{\xi})(s)|+L|\hat{\xi}(s)|\rangle \,ds|\mathcal{F}_t\right]\\
\leq&  2\epsilon_1 E\left[\int_t^T\|\nabla (u(s)-\hat{\xi}(s))\|^2\,ds|\mathcal{F}_t\right]+2\epsilon _1 E\left[\int_t^T\|v(s)-\hat{v}(s)\|^2ds|\mathcal{F}_t\right]+2\epsilon _1E\left[\int_t^T\|\hat{v}(s)\|^2\,ds|\mathcal{F}_t\right]\\
&+2\epsilon _1E\left[\int_t^T\|\nabla\hat{\xi}(s)\|^2\,ds|\mathcal{F}_t\right]
+C(\epsilon_1,L)E\left[\int_t^T\|u(s)-\hat{\xi}(s)\|^2\,ds|\mathcal{F}_t\right]+E\left[\int_t^T\|\hat{\xi}(s)\|^2\,ds|\mathcal{F}_t\right],
\end{align*}
and by the H\"older inequality one has that:
\begin{align*}
& E\left[\int_t^T\langle 2|u(s)-\hat{\xi}(s)|,|f_0|+|\hat{f}(s)|\rangle \,ds|\mathcal{F}_t\right]\\
&\leq 2E\left[\int_t^T\|(u-\hat{\xi})(s)\|^2\,ds|\mathcal{F}_t\right]+E\left[\int_t^T\|f_0(s)\|^2\,ds|\mathcal{F}_t\right]+E\left[\int_t^T\|\hat{f}(s)\|^2\,ds|\mathcal{F}_t\right]
\end{align*}
In addition,
$$I_4:=E\left[\|G-\hat{G}\|^2|\mathcal{F}_t\right]\leq  \textrm{esssup}_{\omega \in \Omega}\|G-\hat{G}\|^2.$$

%\begin{align*}
%I_5:&=E\left[\int_t^T\int_{\mathcal{O}}2(u(s,x)-\hat{\xi}(s,x))\mu(dsdx)|\mathcal{F}_t\right]\\
%&=  E\left[\int_t^T\int_{\mathcal{O}}2(u(s,x)-\xi(s,x)+\xi(s,x)-\hat{\xi}(s,x))\mu(dsdx)|\mathcal{F}_t\right]\\
%&\leq 0.
%\end{align*}
Summing up the estimates $I_1$-$I_4$ and taking the supremum w.r.t. $(\omega,s)\in \Omega\times[t,T]$ on both sides we arrive at:
\begin{align*}
&\|u-\hat{\xi}\|^2_{\infty,2;\mathcal{O}_t}+\|v-\hat{v}\|^2_{0,2;\mathcal{O}_t}+\left(\lambda -\kappa-\beta\theta-6\epsilon-2\epsilon_1\right)\|\nabla(u-\hat{\xi})\|^2_{0,2;\mathcal{O}_t}\\
\leq \,& \left(\frac{1}{\varrho}+\frac{1}{\theta}+2\epsilon_1\right)\|v-\hat{v}\|^2_{0,2;\mathcal{O}_t}
+C(\epsilon,\epsilon_1,L)\int_t^T\|u-\hat{\xi}\|^2_{\infty,2;\mathcal{O}_s}\,ds+C(\epsilon,\epsilon_1,\beta)\|\hat{v}\|^2_{0,2;\mathcal{O}_t}\\
&+C(\epsilon,\epsilon_1,\kappa,L)\|\hat{\xi}\|^2_{\mathcal{V}_2(\mathcal{O}_t)}+\|f_0\|^2_{0,2;\mathcal{O}_t}+\|\hat{f}\|^2_{0,2;\mathcal{O}_t}
+C(\epsilon)\left(\|g_0\|^2_{0,2;\mathcal{O}_t}+\|\hat{g}\|^2_{0,2;\mathcal{O}_t}\right)\\
&+\textrm{esssup}_{\omega\in\Omega}\|G-\hat{G}\|^2.
\end{align*}
By assumption ($\mathcal{A}_2$) we can choose $\theta>\varrho'$ such that $\lambda-\kappa-\beta\theta>0$, and $\theta>\varrho'$ also implies $\frac{1}{\varrho}+\frac{1}{\theta}<1$. Now taking $\epsilon$ and $\epsilon_1$ small enough such that $\lambda -\kappa-\beta\theta-6\epsilon-2\epsilon_1>0$ and $\frac{1}{\varrho}+\frac{1}{\theta}+2\epsilon_1<1$, we have
\begin{equation}\label{equation3.7}
\begin{split}
&\|u-\hat{\xi}\|^2_{\mathcal{V}^2(\mathcal{O}_t)}+\|v-\hat{v}\|^2_{0,2;\mathcal{O}_t}\\
\leq &\, C(\epsilon,\epsilon_1,\lambda,\beta,\kappa,L,\varrho)\left(\int_t^T\|u-\hat{\xi}\|^2_{\infty,2;\mathcal{O}_s}\,ds+\|\hat{v}\|^2_{0,2;\mathcal{O}_t}+\|\hat{\xi}\|^2_{\mathcal{V}_2(\mathcal{O}_t)}\right.\\
&+B_2(f_0,g_0;\mathcal{O}_t)^2+B_2(\hat{f},\hat{g};\mathcal{O}_t)^2+\textrm{esssup}_{\omega\in\Omega}\|G-\hat{G}\|^2_{L^2(\mathcal{O})}\bigg).
\end{split}
\end{equation}
By Gronwall's inequality,
\begin{equation}\label{equation3.8}
\begin{split}
&\|u-\hat{\xi}\|^2_{\mathcal{V}_2(\mathcal{O}_t)}+\|v-\hat{v}\|^2_{0,2;\mathcal{O}_t}\\
\leq &C(\epsilon,\epsilon_1,\lambda,\beta,\kappa,L,\varrho,T)\left(\|\hat{v}\|^2_{0,2;\mathcal{O}_t}
+\|\hat{\xi}\|^2_{\mathcal{V}_2(\mathcal{O}_t)}+\textrm{esssup}_{\omega\in\Omega}\|G-\hat{G}\|^2_{L^2(\mathcal{O}_t)}\right.\\
&\left.+B_2(f_0,g_0;\mathcal{O}_t)^2+B_2(\hat{f},\hat{g};\mathcal{O}_t)^2\right).
\end{split}
\end{equation}
Since $\hat{\xi}|_{\partial\mathcal{O}}=0$, we can apply Proposition \ref{proposition6.1} to $\|\hat{\xi}(t)\|^2$. Starting from (\ref{estimate_begin}), using similar estimates,
 \begin{equation}\label{estimate_xi}
 \|\hat{\xi}\|^2_{\mathcal{V}_2(\mathcal{O}_t)}+\|\hat{v}\|^2_{0,2;\mathcal{O}_t}
\leq  C\left(B_2(\hat{f},\hat{g};\mathcal{O}_t)^2+\textrm{esssup}_{\omega\in\Omega}\|\hat{G}\|^2\right),
 \end{equation}
 where $C$ only depends on $\lambda$, $\beta$, $\kappa$, $\varrho$, $L$ and $T$. The estimate
 (\ref{equation3.8}) together with (\ref{estimate_xi}) yields (\ref{equation3.2}).
\end{proof}

%\begin{remark}
%In fact, by the similar approach to the proof of Theorem \ref{theorem3.2}, we can get a more precise estimate,
%\begin{equation}\label{preciseestimate}
%\begin{split}
%\|u\|_{\mathcal{V}_2(\mathcal{O}_t)}^2+\|v\|^2_{0,2;\mathcal{O}_t}\leq C\left(ess\sup_{\omega \in \Omega}\|G(\omega,\cdot)\|_{L^2(\mathcal{O})}^2+ess\sup_{\omega \in \Omega}\|\hat{G}(\omega,\cdot)\|_{L^2(\mathcal{O})}^2\right.\\
%\left.+\|g_0\|_{0,2;\mathcal{O}_t}^2+\|\hat{g}\|_{{0,2};\mathcal{O}_t}^2+\|f_0\|_{0,2;\mathcal{O}_t}^2+\|\hat{f}\|_{0,2;\mathcal{O}_t}\right).
%\end{split}
%\end{equation}
%But the estimate (\ref{equation3.2}) is needed during the proof of maximum principle in subsequent sections, so we still keep it.
%\end{remark}

With the same notation as in Theorem \ref{theorem3.2}, we can relax the zero Dirichlet boundary condition in Theorem \ref{theorem3.2} by assuming $u|_{\partial\mathcal{O}}=\tilde{u}|_{\partial\mathcal{O}}$ for some
$(\tilde{u},\tilde{v})\in\mathcal{U}(-\infty,\tilde{f},\tilde{g},\tilde{G})$
% $\tilde{u}$ is a solution to a RBSPDE,
%\begin{equation}\label{equation3.9}
%\left\{\begin{array}{ll}
%-d\tilde{u}=\partial_j(a^{ji}(t,x)\partial_i\tilde{u}(t,x)+\sigma^{jr}(t,x)\tilde{v}^r(t,x))dt+(\tilde{f}(t,x)+\nabla\tilde{g}(t,x))dt-\tilde{v}^rdW_t^r,\\\\
%\tilde{u}(T)=\tilde{G},
%\end{array}
%\right.
%\end{equation}
where the coefficients $a$, $\sigma$, $\tilde{f}$, $\tilde{g}$ and $\tilde{G}$ satisfy $(\mathcal{A}_2)$ and $(\mathcal{A}_3)$ respectively, and $\tilde{f}$ and $\tilde{g}$ do not depend on $\tilde{u}$, $\nabla \tilde{u}$ and $\tilde{v}$.
Assume further that $\hat{\xi}|_{\partial\mathcal{O}}\leq \tilde{u}|_{\partial{\mathcal{O}}}$ and put $\bar{\xi}:=\hat{\xi}-\tilde{u}$. Then, $(\bar{\xi},\bar{v})\in\mathcal{U}(-\infty,\bar{f},\bar{g},\bar{G})$,
%\begin{equation}
%\left\{\begin{array}{ll}
%-d\bar{\xi}=\partial_j(a^{ji}(t,x)\partial_i\bar{\xi}(t,x)+\sigma^{jr}(t,x)\bar{v}^r(t,x))dt+(\bar{f}(t,x)+\nabla\bar{g}(t,x))dt-\bar{v}^rdW_t^r,\\\\
%\bar{\xi}(T)=\bar{G},
%\end{array}
%\right.
%\end{equation}
where $\bar{v}=\hat{v}-\tilde{v}$, $\bar{f}=\hat{f}-\tilde{f}$, $\bar{g}=\hat{g}-\tilde{g}$ and $\bar{G}=\hat{G}-\tilde{G}$. Suppose now that $(\breve{\xi},\breve{v})\in\mathcal{U}(-\infty,\bar{f},\bar{g},\bar{G})$ with $\breve{\xi}|_{\partial_{\mathcal{O}}}=0$.
%\begin{equation}
%\left\{\begin{array}{ll}
%-d\breve{\xi}=\partial_j(a^{ji}(t,x)\partial_i\breve{\xi}(t,x)+\sigma^{jr}(t,x)\breve{v}^r(t,x))dt+(\breve{f}(t,x)+\nabla\breve{g}(t,x))dt-\breve{v}^rdW_t^r,\\
%\breve{\xi}(T)=\bar{G},\\
%\breve{\xi}|_{\partial\mathcal{O}}=0.
%\end{array}
%\right.
%\end{equation}
Then, $\breve{\xi}|_{\partial\mathcal{O}}=0\geq (\hat{\xi}-\tilde{u})|_{\partial\mathcal{O}}=\bar{\xi}|_{\partial\mathcal{O}}$ and the maximum principle in Lemma \ref{MP-general-domain-quasilinear-BSPDE} yields $\breve{\xi}\geq \hat{\xi}-\tilde{u}\geq \xi-\tilde{u}$. Therefore, our RBSPDE \eqref{equation1.1} is equivalent to the following one but with zero-Dirichlet condition:
\begin{equation}\label{equivalentequation}
\left\{\begin{array}{ll}
\begin{split}
-d\breve{u}(t,x)&=\left[\partial _j(a^{ij}\partial _i\breve{u}+\sigma ^{jr}\breve{v}^r)(t,x)
+(f+\nabla \cdot g)(t,x,\breve{u}+\tilde u,\nabla (\breve{u} + \tilde u),\breve{v}+\tilde v) \right. \\
&	\left. -(\tilde{f}+\nabla \cdot \tilde{g})(t,x)\right]\,dt +\mu (dt,x)-\breve{v}^r(t,x)dW^r_t,~~~(t,x)\in Q;\\
\breve{u}(T,x)&=G(x)-\tilde{G}(x),~~~x\in \mathcal{O};\\
\breve{u} &\geq \xi-\tilde{u},\quad \mathbb{P}\otimes dt\otimes dx\text{-a.e.};\\
\int_Q(\breve{u}&-(\xi -\tilde{u}))(t,x)\,\mu (dt,dx)=0.
\end{split}
\end{array}
\right.
\end{equation}
By Theorem \ref{theorem3.2}, there is a unique solution $u-\tilde{u}$ to the RBSPDE (\ref{equivalentequation}) satisfying zero-Dirichlet condition. In this way, Theorem \ref{theorem3.2} extends to RBSPDEs with general Dirichlet conditions.
%\end{remark}


%\begin{remark}
%Since the estimate of $u-\hat{\xi}$ and $\hat{\xi}$ are essentially based on the deduction starting from (\ref{estimate_begin}). If %$u^+|_{\partial\mathcal{O}}=0$, Theorem \ref{theorem6.2} can be applied. With similar approach as that starting from (\ref{estimate_begin}), we have
%\begin{equation}\label{estimate_u+_1}
%\|u^+\|_{\mathcal{V}_2(\mathcal{O}_t)}\leq %C\left(\textrm{esssup}_{\omega\in\Omega}\|G\|+\|f_0^+\|_{0,2;\mathcal{O}_t}+\|g_0\|_{0,2;\mathcal{O}_t}\right),
%\end{equation}
%where the positive constant $C$ only depends on $\lambda$, $\kappa$, $\varrho$, $\beta$, $L$ and $T$.
%If the zero Dirichlet condition of $u^+$ is violated, for each $t\in[0,T]$, we can choose $k=\textrm{esssup}_{\Omega\times\partial_p\mathcal{O}_t}u^+$ then we have $(u-k)^+|_{\partial\mathcal{O}\times[t,T]}=0$. Thus Theorem \ref{theorem6.2} and the estimate starting from (\ref{estimate_begin}) can be applied. Now we have
%\begin{equation}
%\|(u-k)^+\|_{\mathcal{V}_2(\mathcal{O}_t)}\leq C\left(\|f_0^+\|_{0,2;\mathcal{O}_t}+\|g_0\|_{0,2;\mathcal{O}_t}+k\right),
%\end{equation}
%where $C$ only depends on $\lambda$, $\kappa$, $\varrho$, $\beta$, $L$ and $T$.
%\end{remark}




\section{Maximum Principle for RBSPDE}

In this section we state and prove our maximum principles for RBSPDEs. We start with a global maximum principle on general domains, which states that the weak solution $u$ is bounded on the whole domain if it is bounded on the parabolic boundary. Subsequently we analyze the local behavior of $u^{\pm}$ when $u$ is not necessarily bounded on the parabolic boundary.

\subsection{Global Case}

This section establishes a maximum principle for the RBSPDE (\ref{equation1.1}) on a general domain $\mathcal{O}$. Since the Lebesgue measure of $\mathcal{O}$ might not be bounded, the scheme in \cite{qiutang} cannot be applied. Instead, motivated by \cite{qiu-2015-DSPDE}, we use a stochastic De Girogi's scheme that is independent of the measure of the domain. In what follows $\partial_pQ=(\{T\}\times\mathcal{O})\cup([0,T]\times\partial\mathcal{O})$ denotes the parabolic boundary of $Q$.

%we apply an iteration scheme, which has nothing to do with the domain itself. Our scheme is motivated by \cite{qiu-2015-DSPDE}, where stochastic De Girogi scheme is used to get an $L^p$ estimate for forward SPDEs. %In this section, we modify it to a proper version for our RBSPDE (\ref{equation1.1}).

%We establish the maximum principle for RBSPDE (\ref{equation1.1}) within a general domain as follows.
\begin{theorem}\label{Theorem_MP_GD}
\begin{itemize}
\item[(1)] Assume that $(\mathcal{A}_1)$-$(\mathcal{A}_5)$ hold. If the triplet $(u,v,\mu)$ is a solution to the RBSPDE (\ref{equation1.1}), then
\begin{equation*} %\label{MP-general-domain-1}
\begin{split}
&\textrm{esssup}_{(\omega,t,x)\in\Omega \times Q}u^{\pm} \\
\leq &  C\left(\textrm{esssup}_{(\omega,t,x)\in\Omega \times \partial _pQ}u^{\pm}+\textrm{esssup}_{(\omega,t,x)\in\Omega \times \partial _pQ}\hat{\xi}^{\pm}\right. \\
&\quad \left.+A_p(f_0^{\pm},g_0;Q)+B_2(f_0^{\pm},g_0;Q)+A_p(\hat{f}^{\pm},\hat{g};Q)+B_2(\hat{f}^{\pm},\hat{g};Q)
 \right),
\end{split}
\end{equation*}
where the constant $C$ depends only on $\lambda$, $\kappa$, $\beta$, $L$, $\varrho$, $T$, $p$ and $n$.
\item[(2)] If all conditions in (1) hold except assumption $(\mathcal{A}_5)$ is changed to
\begin{align}\label{additional-assumption-f-g}
g(t,x,r,0,0)=g(t,x,0,0)~\textrm{and}~f(t,x,r,0,0)~ \textrm{is}~ \textrm{non-increasing} ~w.r.t.~ \textrm{r},
\end{align}
then
\begin{equation*}%\label{MP-general-domain-2}
\begin{split}
&\textrm{esssup}_{(\omega,t,x)\in\Omega\times Q} u^{\pm} \\
\leq & \textrm{esssup}_{(\omega,t,x)\in\Omega\times\partial_p Q}u^{\pm}+\textrm{esssup}_{(\omega,t,x)\in\Omega\times\partial_p Q}\hat{\xi}^{\pm} \\
&+C\left(A_p(f_0^{\pm},g_0;Q)^{\frac{np}{np+2(p-n-2)}}B_2(f_0^{\pm},g_0;Q)^{\frac{2(p-n-2)}{np+2(p-n-2)}}\right. \\
&\left.+A_p(\hat{f}^{\pm},\hat{g};Q)^{\frac{np}{np+2(p-n-2)}}B_2(\hat{f}^{\pm},\hat{g};Q)^{\frac{2(p-n-2)}{np+2(p-n-2)}}\right),
\end{split}
\end{equation*}
where $C$ depends on $\lambda$, $\kappa$, $\beta$, $L$, $\varrho$, $n$, $p$ and $T$.
\end{itemize}
\end{theorem}
\begin{proof}
We only consider the estimate for the positive part $u^+$. The one for the negative part $u^-$ follows analogously. Further, we may w.l.o.g.~assume that $f(t,x,r,0,0)$ is non-increasing in $r$. Otherwise, the desired maximum principle can be derived from the maximum principle for the RBSPDE
\begin{equation*}
\left\{
\begin{array}{l}
\begin{aligned}
\label{u-bar}
-d\bar{u}(t,x)&=\left[\partial_j(a^{ij}(t,x)\partial_i\bar{u}(t,x)+\sigma^{jr}(t,x)\bar{v}^r(t,x))+\bar{f}(t,x,\bar{u}(t,x),\nabla\bar{u}(t,x),\bar{v}(t,x))\right.\\
              &\left.+\nabla\cdot\bar{g}(t,x,\bar{u}(t,x),\nabla\bar{u}(t,x),\bar{v}(t,x))\right]\,dt+\bar{\mu}(dt,x)-\bar{v}^r(t,x)\,dW^r_t,\\
\bar{u}(T,x)&=\bar{G}(x),\\
\bar{u}(t,x)&\geq\bar{\xi}(t,x)~dt\times dx\times d\mathbb{P}-a.e.,\\
\int_Q(\bar{u}(t,x)&-\bar{\xi}(t,x))\,\bar{\mu}(dt,dx)=0,
\end{aligned}
\end{array}
\right.
\end{equation*}
where $\bar{u}(t,x)=e^{Lt}u(t,x)$, $\bar{v}(t,x)=e^{Lt}v(t,x)$, $\bar{\mu}(dt,dx)=e^{Lt}\mu(dt,dx)$, $\bar{G}(x)=e^{LT}G(x)$, $\bar{\xi}(t,x) = e^{Lt}\xi(t,x)$ and
\begin{align*}
	\bar{f}(t,x,\bar{u}(t,x),\nabla\bar{u}(t,x),\bar{v}(t,x)) & =e^{Lt}f(t,x,e^{-Lt}\bar{u}(t,x),e^{-Lt}\nabla\bar{u}(t,x),e^{-Lt}\bar{v}(t,x))-L\bar{u}(t,x)\\
	\bar{g}(t,x,\bar{u}(t,x),\nabla\bar{u}(t,x),\bar{v}(t,x)) & =e^{Lt}g(t,x,e^{-Lt}\bar{u}(t,x),e^{-Lt}\nabla\bar{u}(t,x),e^{-Lt}\bar{v}(t,x)).
\end{align*}
%Moreover, $\bar{\xi}$ is dominated by $\bar{\hat{\xi}}$, which is a solution to the BSPDE,
%\begin{equation*}
%\left\{
%\begin{split}
%&-d\bar{\hat{\xi}}(t,x)=\left[\partial_j(a^{ij}(t,x)\partial_i\bar{\hat{\xi}}(t,x)+\sigma^{jr}(t,x)\bar{\hat{v}}^r(t,x))+ %e^{Lt}\hat{f}(t,x)-L\bar{\hat{\xi}}(t,x)+e^{Lt}\nabla\hat{g}(t,x)\right]\,dt-\bar{\hat{v}}^r\,dW^r_t,\\
%&\bar{\hat{\xi}}(T)=\bar{\hat{G}},
%\end{split}
%\right.
%\end{equation*}
%where $\bar{\hat{v}}(t,x)=e^{Lt}\hat{v}(t,x)$, $\bar{\hat{G}}=e^{LT}\hat{G}$.

%It is easy to check that $\bar{\mu}$ is still a stochastic regular measure and
%\begin{align}
%&|\bar{f}(t,x,X_1,Y_1,Z_1)-\bar{f}(t,x,X_2,Y_2,Z_2)|\leq 2L(|X_1-X_2|+|Y_1-Y_2|+|Z_1-Z_2|),\\
%&|\bar{g}(t,x,X_1,Y_1,Z_1)-\bar{g}(t,x,X_2,Y_2,Z_2)|\leq L|X_1-X_2|+\frac{\kappa}{2}|Y_1-Y_2|+\sqrt{\beta}|Z_1-Z_2|,\\
%&\bar{f}_0:=\bar{f}(\cdot,\cdot,\cdot,0,0,0)\in \mathcal{M}^{0,\frac{p(n+2)}{p+n+2}}(Q)\cap\mathcal{M}^{0,2}(Q),~\bar{g}_0:=\bar{g}(\cdot,\cdot,\cdot,0,0,0)\in \mathcal{M}^{0,p}(Q)\cap\mathcal{M}^{0,2}(Q).
%\end{align}
%By establishing the maximum principle for $\bar{u}$, we can get the one for $u$. So in the proof of $(1)$ we assume that $f(t,x,r,0,0)$ is non-increasing w.r.t. $r$.

Now, for $t\in[0,T]$ define
\[
	\bar{k}=\textrm{esssup}_{(\omega,t,x)\in\Omega\times\partial_{p}\mathcal{O}_t}u^++\textrm{esssup}_{(\omega,t,x)\in\Omega\times\mathcal{O}_t}\hat{\xi}^+.
\]
For a positive constant $k$ to be determined later and each $m\in \mathbb{N}_0$, let $\bar{k}_m=k(1-2^{-m})$ and $k_m=\bar{k}_m+\bar{k}$.
By Theorem \ref{Ito-u+}, for $m\geq 1$,
\begin{equation}\label{Ito-for-u-k_m}
\begin{split}
&\|(u-k_m)^+(t)\|^2+\int_t^T\|v^{k_m}(s)\|^2ds
\\
=&-2\int_t^T\langle\partial_j(u-k_m)^+(s),a^{ij}\partial_i(u-k_m)^+(s)+\sigma ^{jr}(s)v^{k_m,r}(s)\rangle\, ds\\
&-2\int_t^T\langle\partial_j(u-k_m)^+(s),g^{j,k_m}\left(s,(u-k_m)^+(s),\nabla u(s),v^{k_m}(s)\right)\rangle\, ds\\
&+2\int_t^T\langle (u-k_m)^+(s),f^{k_m}\left(s,(u-k_m)^+(s),\nabla u(s),v^{k_m}(s)\right)\rangle \,ds\\
&+2\int_{\mathcal{O}_t}(u-k_m)^+(s,x)\,\mu(ds,dx)-2\int_t^T\langle (u-k_m)^+(s),v^{r,k_m}(s)\,dW^r_s\rangle,
\end{split}
\end{equation}
where $v^{r,k_m}:=v^r1_{\{u>k_m\}}$, $f^{k_m}(\cdot,\cdot,\cdot,X,\cdot,\cdot):=f(\cdot,\cdot,\cdot,X+k_m,\cdot,\cdot)$, $g^{j,k_m}(\cdot,\cdot,\cdot,X,\cdot,\cdot):=g^j(\cdot,\cdot,\cdot,X+k_m,\cdot,\cdot)$. All terms in (\ref{Ito-for-u-k_m}) are well defined. In particular, the stochastic integral is in fact a martingale.
 Taking conditional expectations on both sides w.r.t. $\mathcal{F}_t$ yields the following estimates for the remaining terms. Similar estimates as for $I_1$ in the proof of Theorem \ref{theorem3.2} yield,
\begin{equation}\label{J_1}
\begin{split}
J_1:=&-2E\left[\int_t^T\langle\partial_j(u-k_m)^+(s),a^{ij}\partial_i(u-k_m)^+(s)+\sigma ^{jr}(s)v^{r,k_m}(s)\rangle \, ds|\mathcal{F}_t\right]\\
\leq &-\lambda E\left[\int_t^T\|\nabla(u-k_m)^+(s)\|^2\,ds|\mathcal{F}_t\right]+\frac{1}{\varrho}E\left[\int_t^T\|v^{k_m}(s)\|^2\,ds|\mathcal{F}_t\right].
\end{split}
\end{equation}
By analogy to the estimate of $I_2$, for each $\epsilon>0$ and $\theta>0$, we have that
\begin{equation}\label{J_2-1}
\begin{split}
J_2:=&-2E\left[\int_t^T\langle\partial_j(u-k_m)^+(s),g^{j,k_m}\left(s,(u-k_m)^+(s),\nabla u(s),v^{k_m}(s)\right)\rangle \, ds|\mathcal{F}_t\right]  \\
\leq &2E\left[\int_t^T\langle|\nabla(u-k_m)^+(s)|,|g^{k_m}_0|+L|(u-k_m)^+(s)|+\frac{\kappa}{2}|\nabla (u-k_m)^+(s)| \right. \\
& \left. +\sqrt{\beta}|v^{k_m}(s)|\rangle \,ds|\mathcal{F}_t\right]\\
\leq &(\kappa+\beta\theta+\epsilon)E\left[\int_t^T\|\nabla(u-k_m)^+(s)\|^2\,ds|\mathcal{F}_t\right]+\frac{1}{\theta}E\left[\int_t^T\|v^{k_m}(s)\|^2\,ds|\mathcal{F}_t\right]\\
 &+\frac{L^2}{\epsilon}E\left[\int_t^T\|(u-k_m)^+(s)\|^2\,ds|\mathcal{F}_t\right]+2E\left[\int_t^T\langle|\nabla(u-k_m)^+(s)|,|g_0^{k_m}(s)|\rangle \,ds|\mathcal{F}_t\right].
%\leq & (\kappa+\beta\theta+2\epsilon)E\left[\int_t^T\|\nabla(u-k_m)^+(s)\|^2ds|\mathcal{F}_t\right]+\frac{1}{\theta}E\left[\int_t^T\|v^{k_m}(s)\|^2ds|\mathcal{F}_t\right]\nonumber\\
% &+C(\epsilon,L)E\left[\int_t^T\|(u-k_m)^+(s)\|^2ds|\mathcal{F}_t\right]\nonumber\\
% &+\left(\frac{2^m}{k}\right)^{1+\frac{2(p-n-2)}{np}}\|\nabla (u-k_{m-1})^+\|_{0,2;\mathcal{O}_t}\|g_0^{k_m}\|_{0,p;\mathcal{O}_t}\|(u-k_{m-1})^+\|_{0,\frac{2(n+2)}{n};\mathcal{O}_t}^{1+\frac{2(p-n-2)}{np}}.\nonumber\\
%\end{split}
\end{split}
\end{equation}

From
\begin{align*}
\left[(u-k_{m-1})^+-(u-k_m)^+\right]1_{\{u>k_m\}}=(k_m-k_{m-1})1_{\{u>k_m\}}=2^{-m}k1_{\{u>k_m\}}
\end{align*}
we get
\begin{align}\label{estimate_indicator}
1_{\{u>k_m\}}\leq\frac{2^m(u-k_{m-1})^+}{k}1_{\{u>k_m\}}\leq\frac{2^m(u-k_{m-1})^+}{k}.
\end{align}
By (\ref{estimate_indicator}) and  ($\mathcal{A}_5$) it holds that:
\begin{equation}\label{J_2-2}
\begin{split}
&E\left[\int_t^T\langle|\nabla(u-k_m)^+(s)|,|g^{k_m}_0(s)|\rangle\, ds|\mathcal{F}_t\right] \\
\leq & \left(E\left[\int_t^T\|\nabla(u-k_m)^+(s)\|^2\,ds|\mathcal{F}_t\right]\right)^{\frac{1}{2}}\left(E\left[\int_t^T\int_{\mathcal{O}}|g^{k_m}_0(s,x)1_{\{u>k_m\}}|^2\,dxds|\mathcal{F}_t\right]\right)^{\frac{1}{2}} \\
\leq & \|\nabla (u-k_m)^+\|_{0,2;\mathcal{O}_t}\|g^{k_m}_0\|_{0,p;\mathcal{O}_t}\left(E\left[\int_t^T\int_{\mathcal{O}}1_{\{u>k_m\}}\,dxds|\mathcal{F}_t\right]\right)^{\frac{1}{2}-\frac{1}{p}} \\
\leq & \|\nabla (u-k_m)^+\|_{0,2;\mathcal{O}_t}\|g^{k_m}_0\|_{0,p;\mathcal{O}_t}\left(E\left[\int_t^T\int_{\mathcal{O}}\left(\frac{2^m(u-k_{m-1})^+}{k}\right)^{\frac{2(n+2)}{n}}\,dxds|\mathcal{F}_t\right]\right)^{\frac{1}{2}-\frac{1}{p}} \\
\leq & \left(\frac{2^m}{k}\right)^{1+\frac{2(p-n-2)}{np}}\|\nabla (u-k_m)^+\|_{0,2;\mathcal{O}_t}\|g^{k_m}_0\|_{0,p;\mathcal{O}_t}\|(u-k_{m-1})^+\|_{0,\frac{2(n+2)}{n};\mathcal{O}_t}^{1+\frac{2(p-n-2)}{np}} \\
\leq & \left(\frac{2^m}{k}\right)^{1+\frac{2(p-n-2)}{np}}\|\nabla (u-k_{m-1})^+\|_{0,2;\mathcal{O}_t}\|g^{k_m}_0\|_{0,p;\mathcal{O}_t}\|(u-k_{m-1})^+\|_{0,\frac{2(n+2)}{n};\mathcal{O}_t}^{1+\frac{2(p-n-2)}{np}} \\
\leq & \left(\frac{2^m}{k}\right)^{1+\frac{2(p-n-2)}{np}}\|\nabla (u-k_{m-1})^+\|_{0,2;\mathcal{O}_t}(\|g_0\|_{0,p;\mathcal{O}_t}+Lk_m)\|(u-k_{m-1})^+\|_{0,\frac{2(n+2)}{n};\mathcal{O}_t}^{1+\frac{2(p-n-2)}{np}}.\\
\end{split}
\end{equation}
Combining (\ref{J_2-1}) and (\ref{J_2-2}), we see that
\begin{equation}
\begin{split}
J_2\leq & (\kappa+\beta\theta+\epsilon)E\left[\int_t^T\|\nabla(u-k_m)^+(s)\|^2\,ds|\mathcal{F}_t\right]+\frac{1}{\theta}E\left[\int_t^T\|v^{k_m}(s)\|^2\,ds|\mathcal{F}_t\right]\\
 &+\frac{L^2}{\epsilon}E\left[\int_t^T\|(u-k_m)^+(s)\|^2\,ds|\mathcal{F}_t\right]\\
 &+2\left(\frac{2^m}{k}\right)^{1+\frac{2(p-n-2)}{np}}\|\nabla (u-k_{m-1})^+\|_{0,2;\mathcal{O}_t}(\|g_0\|_{0,p;\mathcal{O}_t}+Lk_m)\|(u-k_{m-1})^+\|_{0,\frac{2(n+2)}{n};\mathcal{O}_t}^{1+\frac{2(p-n-2)}{np}}.\\
\end{split}
\end{equation}

For each $\epsilon_1>0$, by (\ref{repeatedestimate}) the monotonicity of $f(t,x,r,0,0)$ yields:
\begin{equation}\label{J_3-1}
\begin{split}
J_3:=& 2E\left[\int_t^T\langle (u-k_m)^+(s),f^{k_m}\left(s,(u-k_m)^+(s),\nabla u(s),v^{k_m}(s)\right)\rangle \, ds|\mathcal{F}_t\right]
\\
\leq & 2E\left[\int_t^T\langle(u-k_m)^+(s),f_0^{k_m}(s)+L(u-k_m)^+(s)+L\nabla(u-k_m)^+(s)+L|v^{k_m}(s)|\rangle \, ds|\mathcal{F}_t\right] \\
%=& 2E\left(\int_t^T\langle(\bar{u}-k_m)^+(s),e^{Ls}f(s,e^{-Ls}k_m,0,0)-Lk_m+2L(\bar{u}-k_m)^+(s)+2L\nabla(\bar{u}-k_m)^+(s)+2L|\bar{v}^{k_m}(s)|\rangle \, ds|\mathcal{F}_t\right)\nonumber\\
\leq & 2E\left[\int_t^T\langle(u-k_m)^+(s),f_0(s)+L(u-k_m)^+(s)+L\nabla(u-k_m)^+(s)+L|v^{k_m}(s)|\rangle \, ds|\mathcal{F}_t\right]\\
\leq & \left(2L+\frac{2L^2}{\epsilon_1}\right)E\left[\int_t^T\|(u-k_m)^+(s)\|^2\,ds|\mathcal{F}_t\right]+\epsilon_1E\left[\int_t^T\|\nabla(u-k_m)^+(s)\|^2\,ds|\mathcal{F}_t\right] \\
&+\epsilon_1E\left[\int_t^T\|v^{k_m}(s)\|^2\,ds|\mathcal{F}_t\right]
+2E\left[\int_t^T\langle(u-k_m)^+(s),f_0(s)\rangle \,ds|\mathcal{F}_t\right].
\end{split}
\end{equation}

By (\ref{estimate_indicator}) again, we have
\begin{equation}\label{J_3-2}
\begin{split}
&E\left[\int_t^T\langle (u-k_m)^+(s),f_0(s)\rangle \,ds|\mathcal{F}_t\right] \\
\leq & E\left[\int_t^T\langle (u-k_m)^+(s),f_0^+(s)\rangle \,ds|\mathcal{F}_t\right] \\
\leq & \left(E\left[\int_t^T\int_{\mathcal{O}}|(u-k_m)^+|^{\frac{2(n+2)}{n}}\,dxds|\mathcal{F}_t\right]\right)^{\frac{n}{2(n+2)}}\left(E\left[\int_t^T\int_{\mathcal{O}}|f_0^+(s,x)|^{\frac{p(n+2)}{p+n+2}}\,dxds|\mathcal{F}_t\right]\right)^{\frac{p+n+2}{p(n+2)}} \\
&\times\left(E\left[\int_t^T\int_{\mathcal{O}}1_{\{u>k_m\}}\,dxds|\mathcal{F}_t\right]\right)^{\frac{1}{2}-\frac{1}{p}} \\
\leq & \|(u-k_m)^+\|_{0,\frac{2(n+2)}{n};\mathcal{O}_t}\|f_0^+\|_{0,\frac{p(n+2)}{p+n+2};\mathcal{O}_t}\left(E\left[\int_t^T\int_{\mathcal{O}}\left(\frac{2^m(u-k_{m-1})^+}{k}\right)^{\frac{2(n+2)}{n}}\,dxds|\mathcal{F}_t\right]\right)^{\frac{1}{2}-\frac{1}{p}} \\
\leq & \left(\frac{2^m}{k}\right)^{1+\frac{2(p-n-2)}{np}}\|(u-k_m)^+\|_{0,\frac{2(n+2)}{n};\mathcal{O}_t}\|(u-k_{m-1})^+\|_{0,\frac{2(n+2)}{n};\mathcal{O}_t}^{1+\frac{2(p-n-2)}{np}}\|f_0^+\|_{0,\frac{p(n+2)}{p+n+2};\mathcal{O}_t} \\
\leq & \left(\frac{2^m}{k}\right)^{1+\frac{2(p-n-2)}{np}}\|(u-k_{m-1})^+\|_{0,\frac{2(n+2)}{n};\mathcal{O}_t}^{2+\frac{2(p-n-2)}{np}}\|f^+_0\|_{0,\frac{p(n+2)}{p+n+2};\mathcal{O}_t}.
\end{split}
\end{equation}
Therefore, by (\ref{J_3-1}) and (\ref{J_3-2}) we conclude
\begin{equation}
\begin{split}
J_3\leq & \left(2L+\frac{2L^2}{\epsilon_1}\right)E\left[\int_t^T\|(u-k_m)^+(s)\|^2\,ds|\mathcal{F}_t\right]+\epsilon_1E\left[\int_t^T\|\nabla(u-k_m)^+(s)\|^2\,ds|\mathcal{F}_t\right] \\
&+\epsilon_1E\left[\int_t^T\|v^{k_m}(s)\|^2\,ds|\mathcal{F}_t\right]
+2\left(\frac{2^m}{k}\right)^{1+\frac{2(p-n-2)}{np}}\|(u-k_{m-1})^+\|_{0,\frac{2(n+2)}{n};\mathcal{O}_t}^{2+\frac{2(p-n-2)}{np}}\|f^+_0\|_{0,\frac{p(n+2)}{p+n+2};\mathcal{O}_t}.
\end{split}
\end{equation}

%By Lemma \ref{MP_GD_L}, there must be a positive constant $C$ depending on $\lambda$, $\varrho$, $p$ and $n$ such that
%\begin{equation}
%\textrm{esssup}_{\Omega\times\mathcal{O}_t}\hat{\xi}^+\leq %C\left(\textrm{esssup}_{\Omega\times\partial_p\mathcal{O}_t}\hat{\xi}^++\|\hat{f}^+\|_{0,2;\mathcal{O}_t}+\|\hat{g}\|_{0,2;\mathcal{O}_t}+\|\hat{f}^+\|_{0,\frac{p(n+2)}{p+n+2};\mathcal{O}_t}+\|\hat{g}\|_{0,p;\mathcal{O}_t}\right).
%\end{equation}
%So if we take $k\geq %C\left(\textrm{esssup}_{\Omega\times\partial_p\mathcal{O}_t}\hat{\xi}^++\|\hat{f}^+\|_{0,2;\mathcal{O}_t}+\|\hat{g}\|_{0,2;\mathcal{O}_t}+\|\hat{f}^+\|_{0,\frac{p(n+2)}{p+n+2};\mathcal{O}_t}+\|\hat{g}\|_{0,p;\mathcal{O}_t}\right)$, %we have
Finally, note that
\begin{align*}%\label{J_4}
\int_t^T\int_{\mathcal{O}}(u-k_m)^+\,\mu(dxds) \leq \int_t^T\int_{\mathcal{O}}(u-\xi)^+\,\mu(dxds)+\int_t^T\int_{\mathcal{O}}(\xi-\hat{\xi}^+)^+\,\mu(dxds)=0.
\end{align*}
Combining the above estimates, we get
\begin{align*}
&\|(u-k_m)^+(t)\|^2+E\left[\int_t^T\|v^{k_m}(s)\|^2\,ds|\mathcal{F}_t\right]\nonumber\\
\leq & (-\lambda+\kappa+\beta\theta+\epsilon+\epsilon_1) E\left[\int_t^T\|\nabla(u-k_m)^+\|^2\,ds|\mathcal{F}_t\right]\nonumber\\
&+\left(\frac{1}{\theta}+\frac{1}{\varrho}+\epsilon_1\right)E\left[\int_t^T\|v^{k_m}\|^2\,ds|\mathcal{F}_t\right]
+\left(2L+\frac{L^2}{\epsilon}+\frac{2L^2}{\epsilon_1}\right)E\left[\int_t^T\|(u-k_m)^+\|^2\,ds|\mathcal{F}_t\right]\nonumber\\
&+2\left(\frac{2^m}{k}\right)^{1+\frac{2(p-n-2)}{np}}\|\nabla (u-k_{m-1})^+\|_{0,2;\mathcal{O}_t}\|(u-k_{m-1})^+\|_{0,\frac{2(n+2)}{n};\mathcal{O}_t}^{1+\frac{2(p-n-2)}{np}}\left(\|g_0\|_{0,p;\mathcal{O}_t}+Lk_m\right)\nonumber\\
&+2\left(\frac{2^m}{k}\right)^{1+\frac{2(p-n-2)}{np}}\|(u-k_{m-1})^+\|_{0,\frac{2(n+2)}{n};\mathcal{O}_t}^{2+\frac{2(p-n-2)}{np}}\|f_0^{+}\|_{0,\frac{p(n+2)}{p+n+2};\mathcal{O}_t}.\nonumber\\
\end{align*}
From this it is straightforward to see that
\begin{align*}
&\min\{1,\lambda-\kappa-\beta\theta-\epsilon-\epsilon_1\}\left\{\|(u-k_m)^+(t)\|^2+E\left[\int_t^T\|\nabla(u-k_m)^+\|^2\,ds|\mathcal{F}_t\right]\right\}\nonumber\\
&+\left(1-\frac{1}{\theta}-\frac{1}{\varrho}-\epsilon_1\right)E\left[\int_t^T\|v^{k_m}\|^2\,ds|\mathcal{F}_t\right]\nonumber\\
\leq
&
\left(2L+\frac{L^2}{\epsilon}+\frac{2L^2}{\epsilon_1}\right)E\left[\int_t^T\|(u-k_m)^+\|^2\,ds|\mathcal{F}_t\right]\nonumber\\
&+2\left(\frac{2^m}{k}\right)^{1+\frac{2(p-n-2)}{np}}\|\nabla (u-k_{m-1})^+\|_{0,2;\mathcal{O}_t}\|(u-k_{m-1})^+\|_{0,\frac{2(n+2)}{n};\mathcal{O}_t}^{1+\frac{2(p-n-2)}{np}}\left(\|g_0\|_{0,p;\mathcal{O}_t}+Lk_m\right)\nonumber\\
&+2\left(\frac{2^m}{k}\right)^{1+\frac{2(p-n-2)}{np}}\|(u-k_{m-1})^+\|_{0,\frac{2(n+2)}{n};\mathcal{O}_t}^{2+\frac{2(p-n-2)}{np}}\|f_0^{+}\|_{0,\frac{p(n+2)}{p+n+2};\mathcal{O}_t}.\nonumber\\
\end{align*}
By assumption ($\mathcal{A}_2$), there exits $\theta>\varrho'$ such that $\lambda-\kappa-\theta\beta>0$ and $\frac{1}{\theta}+\frac{1}{\varrho}<1$. So, we can take $\epsilon$ and $\epsilon_1$ small enough such that $\lambda-\kappa-\beta\theta-\epsilon-\epsilon_1>0$ and $1-\frac{1}{\theta}-\frac{1}{\varrho}-\epsilon_1>0$. Taking the supremum on both sides, Lemma \ref{lemma2.6} yields,
\begin{align*}
&\|(u-k_m)^+\|^2_{\mathcal{V}_2(\mathcal{O}_t)}+\|v^{k_m}\|^2_{0,2;\mathcal{O}_t} \\
\leq & C_1(\lambda,\kappa,\beta,L,\theta,\varrho,\epsilon,\epsilon_1)\int_t^T\|(u-k_m)^+\|^2_{\mathcal{V}_2(\mathcal{O}_s)}\,ds\\
&+C_1(\lambda,\kappa,\beta,L,\theta,\varrho,n,\epsilon,\epsilon_1)\left(\frac{2^m}{k}\right)^{1+\frac{2(p-n-2)}{np}}\|(u-k_{m-1})^+\|^{2+\frac{2(p-n-2)}{np}}_{\mathcal{V}_2(\mathcal{O}_t)}\left(A_p(f_0^{+},g_0;\mathcal{O}_t)+Lk_m\right).
\end{align*}
%where $C_1(L,\epsilon,\epsilon_1)=4L+\frac{L^2}{\epsilon}+\frac{4L^2}{\epsilon_1}$ and $C_2(\lambda,\kappa,\beta,\theta,,\varrho,\epsilon,\epsilon_1)=\min\{1,\lambda-\kappa-\beta\theta-2\epsilon-2\epsilon_1,1-\frac{1}{\theta}-\frac{1}{\varrho}-2\epsilon_1\}$.
Gronwall's inequality yields that
\begin{align}\label{gronwall}
&\|(u-k_m)^+\|_{\mathcal{V}_2(\mathcal{O}_t)}^2+\|v^{k_m}\|^2_{0,2;\mathcal{O}_t}\nonumber\\
\leq & C_2(\lambda,\kappa,\beta,L,\theta,\varrho,T,n,\epsilon,\epsilon_1)\left(\frac{2^m}{k}\right)^{1+\frac{2(p-n-2)}{np}}\|(u-k_{m-1})^+\|_{\mathcal{V}_2(\mathcal{O}_t)}^{2+\frac{2(p-n-2)}{np}}\left(A_p(f_0^+,g_0;\mathcal{O}_t)+Lk_m\right).
\end{align}
Letting $k\geq \bar{k}+\frac{A_p(f_0^+,g_0;\mathcal{O}_t)}{L}$, it follows from (\ref{gronwall}) that
\begin{align}\label{before-de-giorgi}
&\|(u-k_m)^+\|_{\mathcal{V}_2(\mathcal{O}_t)}^2+\|v^{k_m}\|^2_{0,2;\mathcal{O}_t}\nonumber\\
\leq & C_3(\lambda,\kappa,\beta,L,\theta,\varrho,T,n,\epsilon,\epsilon_1)\frac{2^{1+\frac{2(p-n-2)}{np}}}{k^{\frac{2(p-n-2)}{np}}}\left(2^{1+\frac{2(p-n-2)}{np}}\right)^{m-1}\|(u-k_{m-1})^+\|_{\mathcal{V}_2(\mathcal{O}_t)}^{2+\frac{2(p-n-2)}{np}}.
\end{align}
In terms of $a_m:=\|(u-k_m)^+\|_{\mathcal{V}_2(\mathcal{O}_t)}^2$, $C_0:=C_3(\lambda,\kappa,\beta,L,\theta,\varrho,T,\epsilon,\epsilon_1)\frac{2^{1+\frac{2(p-n-2)}{np}}}{k^{\frac{2(p-n-2)}{np}}}>0$, $b:=2^{1+\frac{2(p-n-2)}{np}}>1$ and $\delta:=\frac{(p-n-2)}{np}>0$, we get that
\begin{equation*} %\label{De Giorgi iteration}
a_m\leq C_0b^{m-1}a_{m-1}^{1+\delta}.
\end{equation*}
Now, let
\[
	k\geq C_3(\lambda,\kappa,\beta,L,\theta,\varrho,T,\epsilon,\epsilon_1)^{\frac{1}{2\delta}}2^{(1+2\delta)\left(\frac{1}{2\delta^2}+\frac{1}{2\delta}\right)}\|({u}-\bar{k})^+\|_{\mathcal{V}_2(\mathcal{O}_t)}.
\]
	Then $a_0 \leq C_0^{-\frac{1}{\delta}}b^{-\frac{1}{\delta^2}}$. Therefore, Lemma \ref{lemma4.1} can be applied to get $\lim\limits_{m\rightarrow\infty}a_m=0$. Along with the above estimates for $k$ this implies that
\begin{align} \label{ubar-kbar}
\textrm{esssup}_{(\omega,t,x)\in\Omega\times \mathcal{O}_t}(u-\bar{k})^+\leq C\left(\bar{k}+A_p(f_0^+,g_0;\mathcal{O}_t)+\|(u-\bar{k})^+\|_{\mathcal{V}_2(\mathcal{O}_t)}\right),
\end{align}
where $C$ depends on $\lambda$, $\kappa$, $\beta$, $L$, $\varrho$, $T$, $p$ and $n$. The estimates of terms $\|(u-\bar{k})^+\|_{\mathcal{V}_2(\mathcal{O}_t)}$ and $\textrm{esssup}_{(\omega,t,x)\in\Omega\times\mathcal{O}_t}\hat{\xi}^+$ are given in the following Lemma \ref{estimate-u-k}(1) and Lemma \ref{MP-general-domain-quasilinear-BSPDE}(1).
Finally we arrive at
\begin{align}\label{final-step-MP}
&\textrm{esssup}_{(\omega,t,x)\in\Omega\times \mathcal{O}_t}u^+\nonumber\\
\leq & C \left(\textrm{esssup}_{(\omega,t,x)\in\Omega\times \partial_p\mathcal{O}_t}u^++\textrm{esssup}_{(\omega,t,x)\in\Omega \times \partial _p\mathcal{O}_t}{\hat{\xi}}^{+}\right.\nonumber\\
&\left.+A_p(f_0^+,g_0;\mathcal{O}_t)+B_2(f_0^+,g_0;\mathcal{O}_t)+A_p(\hat{f}^{+},\hat{g};\mathcal{O}_t)+B_2(\hat{f}^{+},\hat{g};\mathcal{O}_t)
\right),
\end{align}
where $C$ depends on $\lambda$, $\kappa$, $\beta$, $L$, $\varrho$, $T$, $p$ and $n$.






(2) For each $t\in[0,T]$, let $k\geq \textrm{esssup}_{(\omega,t,x)\in\Omega\times\partial_p\mathcal{O}_t}u^++\textrm{esssup}_{(\omega,t,x)\in\Omega\times\mathcal{O}_t}\hat{\xi}^+$. By Theorem \ref{Ito-u+} we obtain
\begin{align*} %\label{startingpoint-mp-comparison}
&\|(u-k)^+(t)\|^2+\int_t^T\|v^{k}(s)\|^2\,ds\nonumber\\
&=-2\int_t^T\langle\partial_j(u-k)^+(s),a^{ij}\partial_i(u-k)^+(s)+\sigma ^{jr}(s)v^{k,r}{s}\rangle\, ds\nonumber\\
&-2\int_t^T\langle\partial_j(u-k)^+(s),g^{j,k}(s,(u-k)^+(s),\nabla u(s),v^k(s))\rangle\, ds\nonumber\\
&+2\int_t^T\langle (u-k)^+(s),f^{k}(s,(u-k)^+(s),\nabla u(s),v^{k}(s))\rangle \,ds+2\int_{\mathcal{O}_t}(u-k)^+(s,x)\,\mu(ds,dx)\nonumber\\
&-2\int_t^T\langle (u-k)^+(s),v^{r,k}(s)\,dW^r_s\rangle.
\end{align*}
For every $k>l\geq \textrm{esssup}_{(\omega,t,x)\in\Omega\times\partial_p\mathcal{O}_t}u^++\textrm{esssup}_{(\omega,t,x)\in\Omega\times\mathcal{O}_t}\hat{\xi}^+$, we get
\begin{align*}
\left((u-l)^+-(u-k)^+\right)1_{(u>k)}=(k-l)1_{(u>k)},
\end{align*}
which implies
\begin{align*} %\label{1_u-k}
1_{(u>k)}\leq \frac{(u-l)^+}{k-l}.
\end{align*}
By the assumptions on $g$ and $f$, and using the same arguments in (\ref{J_1}), (\ref{J_2-1}) and (\ref{J_2-2})-(\ref{gronwall}) we obtain that
\begin{align*}
\|(u-k)^+\|_{\mathcal{V}_2(\mathcal{O}_t)}^2+\|v^{k}\|^2_{0,2;\mathcal{O}_t}
\leq & C\frac{A_p(f_0^+,g_0;\mathcal{O}_t)}{(k-l)^{1+\frac{2(p-n-2)}{np}}}\|(u-l)^+\|_{\mathcal{V}_2(\mathcal{O}_t)}^{2+\frac{2(p-n-2)}{np}},
\end{align*}
where $C$ depends on $\lambda$, $\kappa$, $\beta$, $L$, $\varrho$, $T$ and $n$. By setting $\phi(k):=\|(u-k)^+\|_{\mathcal{V}_2(\mathcal{O}_t)}^2$, $\alpha:=1+\frac{2(p-n-2)}{np}>0$, $\zeta:=1+\frac{p-n-2}{np}$ and $C_1:=CA_p(f_0^+,g_0;\mathcal{O}_t)$, the following statement holds for each $k>l\geq \textrm{esssup}_{(\omega,t,x)\in\Omega\times\partial_p\mathcal{O}_t}u^++\textrm{esssup}_{(\omega,t,x)\in\Omega\times\mathcal{O}_t}\hat{\xi}^+$:
\begin{align*}
\phi(k)\leq \frac{C_1}{(k-l)^{\alpha}}\phi(l)^{\zeta}.
\end{align*}
If we define $d:=C_1^{\frac{1}{\alpha}}\left|\phi(\textrm{esssup}_{(\omega,t,x)\in\Omega\times\partial_p\mathcal{O}_t}u^++\textrm{esssup}_{(\omega,t,x)\in\Omega\times\mathcal{O}_t}\hat{\xi}^+)\right|^{\frac{\zeta-1}{\alpha}}2^{\frac{1+\alpha}{\alpha}}$, then by Corollary \ref{corollary-de-giorgi},
\begin{align*}
\|(u-d-\textrm{esssup}_{(\omega,t,x)\in\Omega\times\partial_p\mathcal{O}_t}u^+-\textrm{esssup}_{(\omega,t,x)\in\Omega\times\mathcal{O}_t}\hat{\xi}^+)^+\|_{\mathcal{V}_2(\mathcal{O}_t)}=0,
\end{align*}
and so Lemma \ref{estimate-u-k}(2) yields
\begin{align}\label{estimate-last-but-one-2}
\textrm{esssup}_{(\omega,t,x)\in\Omega\times \mathcal{O}_t}u^+\leq & \textrm{esssup}_{(\omega,t,x)\in\Omega\times\partial_p\mathcal{O}_t}u^++\textrm{esssup}_{(\omega,t,x)\in\Omega\times\mathcal{O}_t}\hat{\xi}^+\nonumber\\
&+CA_p(f_0^+,g_0;\mathcal{O}_t)^{\frac{1}{\alpha}}B_2(f_0^+,g_0;\mathcal{O}_t)^{\frac{2(\zeta-1)}{\alpha}},
\end{align}
where $C$ depends on $\lambda$, $\kappa$, $\beta$, $L$, $\varrho$, $n$, $p$ and $T$. Therefore
(\ref{estimate-last-but-one-2}) and (\ref{MP-general-domain-quasilinear-BSPDE-2}) yield
\begin{align*} %\label{finalpoint-mp-comparison}
\textrm{esssup}_{(\omega,t,x)\in\Omega\times \mathcal{O}_t}u^+\leq & \textrm{esssup}_{(\omega,t,x)\in\Omega\times\partial_p\mathcal{O}_t}u^++\textrm{esssup}_{(\omega,t,x)\in\Omega\times\partial_p\mathcal{O}_t}\hat{\xi}^+\nonumber\\
&+CA_p(\hat{f}^+,\hat{g};\mathcal{O}_t)^{\frac{1}{\alpha}}B_2(\hat{f}^+,\hat{g};\mathcal{O}_t)^{\frac{2(\zeta-1)}{\alpha}}\nonumber\\
&+CA_p(f_0^+,g_0;\mathcal{O}_t)^{\frac{1}{\alpha}}B_2(f_0^+,g_0;\mathcal{O}_t)^{\frac{2(\zeta-1)}{\alpha}}.
\end{align*}
\end{proof}


When the domain $\mathcal{O}$ is bounded, $\|\cdot\|_{0,2;Q}$ can be bounded by $\|\cdot\|_{0,p;Q}$ and $\|\cdot\|_{0,\frac{p(n+2)}{p+n+2};Q}$ and we have the following maximum principle for the RBSPDE (\ref{equation1.1}) on a bounded domain.

\begin{corollary}\label{MP-RBSPDE-bounded-domain}
\begin{itemize}
	\item[(1)] Assume $(\mathcal{A}_1)$-$(\mathcal{A}_5)$ hold and $\mathcal{O}$ is bounded. If the triplet $(u,v,\mu)$ is a solution to the RBSPDE (\ref{equation1.1}), then
\begin{align*} %\label{MP-bounded-domain-1}
&\textrm{esssup}_{(\omega,t,x)\in\Omega \times Q}u^{\pm} \\
\leq& \, C\left(\textrm{esssup}_{(\omega,t,x)\in\Omega \times \partial _pQ}u^{\pm}+\textrm{esssup}_{(\omega,t,x)\in\Omega \times \partial _pQ}\hat{\xi}^{\pm}\right.\nonumber\\
&\quad \left.+A_p(f_0^{\pm},g_0;Q)+A_p(\hat{f}^{\pm},\hat{g};Q)\nonumber
 \right),
\end{align*}
where the constant $C$ depends only on $\lambda$, $\kappa$, $\beta$, $L$, $\varrho$, $T$, $p$, $n$ and $|\mathcal{O}|$.
	\item[(2)] Suppose that $(\mathcal{A}_1)$-$(\mathcal{A}_4)$ and (\ref{additional-assumption-f-g}) hold. Then for each solution $(u,v,\mu)$ to the RBSPDE (\ref{equation1.1}), it holds true that
\begin{align*} %\label{MP-bounded-domain-2}
&\textrm{esssup}_{(\omega,t,x)\in\Omega\times Q}u^{\pm}\nonumber\\
\leq \,& \textrm{esssup}_{(\omega,t,x)\in\Omega\times\partial_pQ}u^{\pm}+\textrm{esssup}_{(\omega,t,x)\in\Omega\times\partial_pQ}\hat{\xi}^{\pm}\nonumber\\
&+C\left(A_p(f_0^{\pm},g_0;Q)+A_p(\hat{f}^{\pm},\hat{g};Q)\right),
\end{align*}
where $C$ depends on $\lambda$, $\kappa$, $\beta$, $L$, $\varrho$, $T$, $p$, $n$, and $|\mathcal{O}|$.
\end{itemize}
\end{corollary}

%\begin{corollary}
%Let $(u_i,v_i,\mu_i)$ be solutions to the RBSPDE (\ref{equation1.1}) with parameters $(f_i,g,G_i,a,\sigma,\xi_i)$, $i=1,2$, respectively. $\xi_i$ is dominated by $\hat{\xi}_i$, $i=1,2$, respectively, where $\hat{\xi}_i$ is the solution to the linear BSPDE (\ref{xi-hat}) with parameters $(\hat{f}_i,\hat{g},\hat{G}_i,a,\sigma)$, $i=1,2$. In addition to $(\mathcal{A}_1)$-$(\mathcal{A}_4)$, we suppose either $(\mathcal{A}_5)$ or (\ref{additional-assumption-f-g}) holds. Furthermore, assume $u_i$ and $\hat{\xi}_i$ satisfy zero Dirichlet conditions, $i=1,2$. Then if $f_1(t,x,u_2,\nabla u_2,v_2)\leq f_2(t,x,u_2,\nabla u_2,v_2)$ $dt\times dx\times d\mathbb{P}\textrm{-}a.e.$, $G_1\leq G_2$ $dx\times d\mathbb{P}\textrm{-}a.e.$, $\hat{f}_1\leq \hat{f}_2$ $dt\times dx\times d\mathbb{P}\textrm{-}a.e.$, $\hat{G}_1\leq \hat{G}_2$ $dx\times d\mathbb{P}\textrm{-}a.e.$, we have $u_1\leq u_2$ $dt\times dx\times d\mathbb{P}\textrm{-}a.e.$.
%\end{corollary}
%\begin{proof}
%Letting $(\underline{u},\underline{v})=(u_1-u_2,v_1-v_2)$, one has
%\begin{equation*}
 % \left\{\begin{array}{l}
 % \begin{split}
%-d\underline{u}(t,x)&=[\partial _j(a^{ij}\partial _i\underline{u}(t,x)+\sigma ^{jr}\underline{v}^r(t,x))
%\\
%&\quad
%+\underline{f}(t,x,\underline{u}(t,x),\nabla \underline{u}(t,x),\underline{v}(t,x))\\
%&\quad+\nabla \cdot \underline{g}(t,x,\underline{u}(t,x),\nabla \underline{u}(t,x),\underline{v}(t,x))]\,dt+\mu_1 (dt,x)-\mu_2(dt,x)-\underline{v}^r(t,x)dW^r_t,\\
%&\quad~~~(t,x)\in Q:=[0,T]\times \mathcal{O},\\
%\underline{u}(T,x)&=\underline{G}(x),~~~x\in \mathcal{O},\\
 %   \end{split}
 % \end{array}\right.
%\end{equation*}
%where
%\begin{align*}
%&\underline{f}(t,x,\cdot,\cdot,\cdot)=f_1(t,x,\cdot+u_2,\cdot+\nabla u_2,\cdot+v_2)-f_2(t,x,u_2,\nabla u_2,v_2)\nonumber\\
%&\underline{g}(t,x,\cdot,\cdot,\cdot)=g(t,x,\cdot+u_2,\cdot+\nabla u_2,\cdot+v_2)-g(t,x,u_2,\nabla u_2,v_2)\nonumber\\
%&\underline{G}=G_1-G_2.\nonumber
%\end{align*}
%By It\^o formula \ref{theorem6.2} and the arguments in Theorem \ref{Theorem_MP_GD}, we get that:
%If $(\mathcal{A}_1)$-$(\mathcal{A}_5)$ hold, %let $\bar{\underline{u}}(t,x)=e^{Lt}\underline{u}(t,x)$. Then we have
%\begin{equation*}
%\left\{
%\begin{split}\label{u-bar}
%&-d\bar{\underline{u}}(t,x)=\left[\partial_j(a^{ij}(t,x)\partial_i\bar{\underline{u}}(t,x)+\sigma^{jr}(t,x)\bar{\underline{v}}^r(t,x))+\bar{\underline{f}}(t,x,\bar{\underline{u}}(t,x),\nabla\bar{\underline{u}}(t,x),\bar{\underline{v}}(t,x))\right.\\
 %&              \left.+\nabla\cdot\bar{\underline{g}}(t,x,\bar{\underline{u}}(t,x),\nabla\bar{\underline{u}}(t,x),\bar{\underline{v}}(t,x))\right]\,dt+\bar{\mu}_1(dt,x)-\bar{\mu}_2(dt,x)-\bar{\underline{v}}^r(t,x)\,dW^r_t,\\
 % &\bar{\underline{u}}(T,x)=\bar{\underline{G}}(x),\\
%\end{split}
%\right.
%\end{equation*}
%where $\bar{\underline{v}}(t,x)=e^{Lt}\underline{v}(t,x)$, $\bar{\underline{G}}(x)=e^{LT}\underline{G}(x)$, $\bar{\mu}_i(dt,dx)=e^{Lt}\mu_i(dt,dx)$, $\bar{\xi}_i(t,x)=e^{Lt}\xi_i(t,x)$, $i=1,2$, $$\bar{\underline{f}}(t,x,\bar{\underline{u}}(t,x),\nabla\bar{\underline{u}}(t,x),\bar{\underline{v}}(t,x))=e^{Lt}\underline{f}(t,x,e^{-Lt}\bar{\underline{u}}(t,x),e^{-Lt}\nabla\bar{\underline{u}}(t,x),e^{-Lt}\bar{\underline{v}}(t,x))-L\bar{\underline{u}}(t,x)$$ and $$\bar{\underline{g}}(t,x,\bar{\underline{u}}(t,x),\nabla\bar{\underline{u}}(t,x),\bar{\underline{v}}(t,x))=e^{Lt}\underline{g}(t,x,e^{-Lt}\bar{\underline{u}}(t,x),e^{-Lt}\nabla\bar{\underline{u}}(t,x),e^{-Lt}\bar{\underline{v}}(t,x)).$$

%For $k=1,2$, $\bar{\xi_k}$ is dominated by $\bar{\hat{\xi}}_k$, which is a solution to the BSPDE,
%\begin{equation*}
%\left\{
%\begin{split}
%&-d\bar{\hat{\xi}}_k(t,x)=\left[\partial_j(a^{ij}(t,x)\partial_i\bar{\hat{\xi}}_k(t,x)+\sigma^{jr}(t,x)\bar{\hat{v}}_k^r(t,x))+ e^{Lt}\hat{f}_k(t,x)-L\bar{\hat{\xi}}_k(t,x)+e^{Lt}\nabla\hat{g}_k(t,x)\right]\,dt-\bar{\hat{v}}_k^r\,dW^r_t,\\
%&\bar{\hat{\xi}}_k(T)=\bar{\hat{G}}_k,
%\end{split}
%\right.
%\end{equation*}
%where $\bar{\hat{v}}_k(t,x)=e^{Lt}\hat{v}_k(t,x)$, $\bar{\hat{G}}_k=e^{LT}\hat{G}_k$.

%for each $t\in[0,T]$, we let $\bar{k}=\textrm{esssup}_{\Omega\times\partial_p\mathcal{O}_t}\bar{u}^++\textrm{esssup}_{\Omega\times\mathcal{O}_t}(\hat{\xi}_1-\hat{\xi}_2)^+$. For each $k$ to be determined and $m\in \mathbb{N}$, let $\bar{k}_m=k(1-2^{-m})$ and $k_m=\bar{k}_m+\bar{k}$. By It\^o formula \ref{theorem6.2}, %we have
%\begin{align}\label{equation5.26}
%&\|(\bar{\underline{u}}-k_m)^+(t)\|^2+\int_t^T\|\bar{\underline{v}}^{k_m}(s)\|^2\,ds
%\nonumber
%\\
%&=-2\int_t^T\langle\partial_j(\bar{\underline{u}}-k_m)^+(s),a^{ij}\partial_i(\bar{\underline{u}}-k_m)^+(s)+\sigma ^{jr}(s)\bar{\underline{v}}^{k_m,r}{s}\rangle\, ds\nonumber\\
%&-2\int_t^T\langle\partial_j(\bar{\underline{u}}-k_m)^+(s),\bar{\underline{g}}^{k_m,j}\left(s,(\bar{\underline{u}}-k_m)^+(s),\nabla \bar{\underline{u}}(s),\bar{\underline{v}}^{k_m}(s)\right)\rangle\, ds\nonumber\\
%&+2\int_t^T\langle (\bar{\underline{u}}-k_m)^+(s),\bar{\underline{f}}^{k_m}\left(s,(\bar{\underline{u}}-k_m)^+(s),\nabla \bar{\underline{u}}(s),\bar{\underline{v}}^{k_m}(s)\right)\rangle \,ds+2\int_{\mathcal{O}_t}(\bar{\underline{u}}-k_m)^+(s,x)\,\bar{\mu}_1(ds,dx)\nonumber\\
%&-2\int_{\mathcal{O}_t}(\bar{\underline{u}}-k_m)^+(s,x)\,\bar{\mu}_2(ds,dx)-2\int_t^T\langle (\bar{\underline{u}}-k_m)^+(s),\bar{\underline{v}}^{k_m,r}(s)\,dW^r_s\rangle,
%\end{align}
%where $\bar{\underline{v}}^{k_m}:=\bar{\underline{v}}1_{\{\bar{\underline{u}}>k_m\}}$, $\bar{\underline{f}}^{k_m}(\cdot,\cdot,\cdot,X,Y,Z):=\bar{\underline{f}}(\cdot,\cdot,\cdot,X+k_m,Y,Z)$ and $\bar{\underline{g}}^{k_m}(\cdot,\cdot,\cdot,X,Y,Z):=\bar{\underline{g}}(\cdot,\cdot,\cdot,X+k_m,Y,Z)$.
%By Corollary \ref{comparison-quasi-linear-BSPDE}, we have $\hat{\xi}_1\leq\hat{\xi}_2$ $dt\otimes dx\otimes d\mathbb{P}\textrm{-}a.e.$, which implies $\bar{\hat{\xi}}_1\leq\bar{\hat{\xi}}_2$ $dt\otimes dx\otimes d\mathbb{P}\textrm{-}a.e.$. So we have
%\begin{align}
%&\int_{\mathcal{O}_t}(\bar{\underline{u}}-k_m)^+(s,x)\,(\bar{\mu}_1(ds,dx)-\bar{\mu}_2(ds,dx))\nonumber\\
%\leq & \int_{\mathcal{O}_t}(\bar{u}_1-\bar{u}_2-\textrm{esssup}_{\Omega\times\mathcal{O}_t}(\bar{\hat{\xi}}_1-\bar{\hat{\xi}}_2)^+)^+(s,x)\,(\bar{\mu}_1(ds,dx)-\bar{\mu}_2(ds,dx))\nonumber\\
%\leq & \int_{\mathcal{O}_t}(\bar{u}_1-\bar{u}_2-\bar{\hat{\xi}}_1+\bar{\hat{\xi}}_2)^+(s,x)\,(\bar{\mu}_1(ds,dx)-\bar{\mu}_2(ds,dx))\nonumber\\
%\leq & 0.
%\end{align}
%Therefore, by the same arguments from (\ref{J_1}) to (\ref{final-step-MP}), we get










%\begin{align*}
%&\textrm{esssup}_{(\omega,t,x)\in\Omega \times Q}\underline{u}^{+} \\
%\leq &  C\left(\textrm{esssup}_{\Omega \times \partial _pQ}\underline{u}^{+}+\textrm{esssup}_{\Omega \times \partial _pQ}\hat{\xi}^{+}+\|\underline{f}_0^{\pm}\|_{0,2;Q}+\|\underline{f}_0^{+}\|_{0,\frac{p(n+2)}{p+n+2};Q}\right.\nonumber\\
%&\quad \left.+\|\hat{\underline{f}}^{+}\|_{0,2;Q}+\|\hat{\underline{f}}^{+}\|_{0,\frac{p(n+2)}{p+n+2};Q}+\|\underline{g}_0\|_{0,2;Q}+\|\underline{g}_0\|_{0,p;Q}+\|\hat{\underline{g}}\|_{0,2;Q}+\|\hat{\underline{g}}\|_{0,p;Q}\nonumber
% \right),
%\end{align*}
%which is zero by assumptions. So we have $\textrm{esssup}_{(\omega,t,x)\in\Omega \times Q}\underline{u}^{+}=0$.





%If $\mathcal{A}_1$-$\mathcal{A}_4$ and (\ref{additional-assumption-f-g}) hold, let $k\geq \textrm{esssup}_{\Omega\times\partial_p\mathcal{O}_t}\underline{u}^++\textrm{esssup}_{\Omega\times\mathcal{O}_t}(\hat{\xi}_1-\hat{\xi}_2)^+$, by the arguments from (\ref{startingpoint-mp-comparison}) to (\ref{finalpoint-mp-comparison}), we have
%\begin{align*}
%\textrm{esssup}_{\Omega\times Q} \underline{u}^{+}\leq & \textrm{esssup}_{\Omega\times\partial_p Q}\underline{u}^{+}+\textrm{esssup}_{\Omega\times\partial_p Q}\hat{\xi}^{+}\nonumber\\
%&+C\left(\|\underline{f}_0^{+}\|_{0,\frac{p(n+2)}{p+n+2};Q}+\|\underline{g}_0\|_{0,p;Q}\right)^{\frac{np}{np+2(p-n-2)}}\left(\|\underline{f}_0^{+}\|_{0,2;Q}+\|\underline{g}_0\|_{0,2;Q}\right)^{\frac{2(p-n-2)}{np+2(p-n-2)}}\nonumber\\
%&+C\left(\|\hat{\underline{f}}_0^{+}\|_{0,\frac{p(n+2)}{p+n+2};Q}+\|\hat{\underline{g}}_0\|_{0,p;Q}\right)^{\frac{np}{np+2(p-n-2)}}\left(\|\hat{\underline{f}}_0^{+}\|_{0,2;Q}+\|\hat{\underline{g}}_0\|_{0,2;Q}\right)^{\frac{2(p-n-2)}{np+2(p-n-2)}},
%\end{align*}
%which implies $\textrm{esssup}_{(\omega,t,x)\in\Omega \times Q}\underline{u}^{+}=0$.

%Anyway, we have $u_1\leq u_2$ $dt\otimes dx\otimes d\mathbb{P}\textrm{-}a.e.$.
%\end{proof}



\begin{lemma}\label{estimate-u-k}
\begin{itemize}
	\item[(1)] Under the same conditions as in Theorem \ref{Theorem_MP_GD}(1), for each $t\in[0,T]$ and each $k\geq \textrm{esssup}_{(\omega,t,x)\in\Omega\times\partial_p\mathcal{O}_t}u^++\textrm{esssup}_{(\omega,t,x)\in\Omega\times\mathcal{O}_t}\hat{\xi}^+$, we have
\begin{align*}
\|(u-k)^{+}\|_{\mathcal{V}_2(\mathcal{O}_t)}\leq C(B_2(f_0^{+},g_0;\mathcal{O}_t)+k),
\end{align*}
where $C$ depends on $\lambda$, $\kappa$, $\beta$, $L$, $\varrho$ and $T$.
	\item[(2)] Under the same conditions as in Theorem \ref{Theorem_MP_GD}(2), for each $t\in[0,T]$ and $k\geq \textrm{esssup}_{(\omega,t,x)\in\Omega\times\partial_p\mathcal{O}_t}u^++\textrm{esssup}_{(\omega,t,x)\in\Omega\times\mathcal{O}_t}\hat{\xi}^+$ and we have
\begin{align*}
\|(u-k)^+\|_{\mathcal{V}_2(\mathcal{O}_t)}\leq CB_2(f_0^+,g_0;\mathcal{O}_t),
\end{align*}
where $C$ depends on $\lambda$, $\kappa$, $\beta$, $L$, $\varrho$ and $T$.
\end{itemize}
\end{lemma}
\begin{proof}
\begin{itemize}
\item[(1)] As in the proof of Theorem \ref{Theorem_MP_GD}, we may assume w.l.o.g that $f(t,x,r,0,0)$ is non-increasing in $r$. For
\[	
    k\geq\textrm{esssup}_{(\omega,t,x)\in\Omega\times\partial_{p}\mathcal{O}_t}u^++\textrm{esssup}_{(\omega,t,x)\in\Omega\times\mathcal{O}_t}\hat{\xi}^+,
\]
we have
\begin{align*}
\int_t^T\int_{\mathcal{O}}(u-k)^+\,\mu(dxds) \leq \int_t^T\int_{\mathcal{O}}(u-\xi)^+\,\mu(dxds)+\int_t^T\int_{\mathcal{O}}(\xi-\hat{\xi}^+)^+\,\mu(dxds)=0.
\end{align*}
\end{itemize}
Applying Theorem \ref{Ito-u+}, we have
\begin{equation}
\begin{split}
&\|(u-k)^+(t)\|^2+E\left(\int_t^T\|v^{k}(s)\|^2\,ds|\mathcal{F}_t\right)
\\
\leq &-2E\left(\int_t^T\langle\partial_j(u-k)^+(s),a^{ij}\partial_i(u-k)^+(s)+\sigma ^{jr}(s)v^{k,r}(s)\rangle\, ds|\mathcal{F}_t\right)\\
&-2E\left(\int_t^T\langle\partial_j(u-k)^+(s),g^{j,k}\left(s,(u-k)^+(s),\nabla u(s),v^{k}(s)\right)\rangle\, ds|\mathcal{F}_t\right)\\
&+2E\left(\int_t^T\langle (u-k)^+(s),f^{k}\left(s,(u-k)^+(s),\nabla u(s),v^{k}(s)\right)\rangle \,ds|\mathcal{F}_t\right)\\
:=& K_1+K_2+K_3,
\end{split}
\end{equation}
where $v^{r,k}:=v^r1_{\{u>k\}}$, $g^{j,k}(\cdot,\cdot,\cdot,X,\cdot,\cdot):=g^j(\cdot,\cdot,\cdot,X+k,\cdot,\cdot)$, $f^{k}(\cdot,\cdot,\cdot,X,\cdot,\cdot):=f(\cdot,\cdot,\cdot,X+k,\cdot,\cdot)$. The quantities $K_i$ $(i=1,2,3)$ can now be estimated by analogy to the constants $I_i$ $(i=1,2,3)$ in the proof of Theorem \ref{theorem3.2}. Specifically, $K_1$ can be estimated as $I_1$, with $u-\hat{\xi}$ and $v-\hat{v}$ being replaced by $(u-k)^+$ and $v$, respectively; $K_2$ can be estimated as $I_2$, without $\hat{g}$ (because we now have no obstacle process involved in), and $u-\hat{\xi}$ and $g^j(s,x,u,\nabla u,v)$ being replaced by $(u-k)^+$ and $g^{j,k}\left(s,x,(u-k)^+(s),\nabla u(s),v^{k}(s)\right)$, respectively and the estimate for $K_3$ is similar to that for $I_3$, without $\hat{f}$ and $u-\hat{\xi}$ and $f(s,x,u,\nabla u,v)$ being replaced by $(u-k)^+$ and $f^{k}\left(s,x,(u-k)^+,\nabla u,v^{k}\right)$, respectively. Finally, by $(\mathcal{A}_5)$, $\|g^k_0\|_{0,2;\mathcal{O}_t}$ can be estimated by $\|g_0\|_{0,2;\mathcal{O}_t}+Lk$. This yields the desired result.
\item[(2)] The proof is the same as that of $(1)$ if we note that $\|g^k_0\|_{0,2;\mathcal{O}_t}=\|g_0\|_{0,2;\mathcal{O}_t}$ by assumption.
\end{proof}







The following lemma establishes the maximum principle for quasi-linear BSPDE on general domains.
\begin{lemma}\label{MP-general-domain-quasilinear-BSPDE}
Let $(u,v)$ be a weak solution to the following quasi-linear BSPDE
\begin{equation}\label{quasilinear-BSPDE}
\left\{\begin{array}{ll}
\begin{split}
-du(t,x)&=[\partial _j(a^{ij}\partial _iu(t,x)+\sigma ^{jr}v^r(t,x))
+f(t,x,u(t,x),\nabla u(t,x),v(t,x))\\
&+\nabla \cdot g(t,x,u(t,x),\nabla u(t,x),v(t,x))]\,dt
-v^r(t,x)\,dW^r_t,
\quad (t,x)\in Q,\\
u(T,x)&=G(x),~~~x\in \mathcal{O}.
\end{split}
\end{array}
\right.
\end{equation}
\begin{itemize}
	\item[(1)] If the coefficients satisfy assumptions $(\mathcal{A}_1)$, $(\mathcal{A}_2)$, $(\mathcal{A}_3)$ and $(\mathcal{A}_5)$, then for each $t\in[0,T]$ we have
\begin{equation}\label{MP-general-domain-quasilinear-BSPDE-1}
\begin{split}
&\textrm{esssup}_{(\omega,t,x)\in\Omega \times \mathcal{O}_t}u^{\pm} \\
\leq \, & C\left(\textrm{esssup}_{(\omega,t,x)\in\Omega \times \partial _p\mathcal{O}_t}u^{\pm}+A_p(f_0^{\pm},g_0;\mathcal{O}_t)+B_2(f_0^{\pm},g_0;\mathcal{O}_t)\right)
\end{split}
\end{equation}
where $C$ depends on $\lambda$, $\kappa$, $\beta$, $L$, $\varrho$, $T$, $p$ and $n$;
	\item[(2)] If $(\mathcal{A}_1)$, $(\mathcal{A}_2)$, $(\mathcal{A}_3)$ and (\ref{additional-assumption-f-g}) hold true,
then for each $t\in[0,T]$ we have
\begin{align}\label{MP-general-domain-quasilinear-BSPDE-2}
&\textrm{esssup}_{(\omega,t,x)\in\Omega\times \mathcal{O}_t}u^{\pm}\nonumber\\
\leq \, & \textrm{esssup}_{(\omega,t,x)\in\Omega\times\partial_p\mathcal{O}_t}u^{\pm}\nonumber\\
&+C A_p(f_0^{\pm},g_0;\mathcal{O}_t)^{\frac{np}{np+2(p-n-2)}}B_2({f}_0^{\pm},{g}_0;\mathcal{O}_t)^{\frac{2(p-n-2)}{np+2(p-n-2)}},
%&+C\left(\|f_0^{\pm}\|_{0,\frac{p(n+2)}{p+n+2};\mathcal{O}_t}+\|g_0\|_{0,p;\mathcal{O}_t}\right)^{\frac{np}{np+2(p-n-2)}}\left(\|f_0^{\pm}\|_{0,2;\mathcal{O}_t}+\|g_0\|_{0,2;\mathcal{O}_t}\right)^{\frac{2(p-n-2)}{np+2(p-n-2)}},
\end{align}
where $C$ depends on $\lambda$, $\kappa$, $\beta$, $L$, $\varrho$, $n$, $p$ and T.
\end{itemize}
\end{lemma}


\begin{proof}
    In terms of $\bar{k}=\textrm{esssup}_{(\omega,t,x)\in\Omega\times\partial_{p}\mathcal{O}_t}u^+$ the assertion follows by establishing estimates analogous to (\ref{J_1})-(\ref{ubar-kbar}) and Lemma \ref{estimate-u-k}(1).
\end{proof}



The proceeding lemmas allow us to establish the comparison principle for the quasi-linear BSPDE on a general domain.
\begin{corollary}\label{comparison-quasi-linear-BSPDE}
Let $(u_i,v_i)$ be solutions to the quasi-linear BSPDE (\ref{quasilinear-BSPDE}) with parameters $(f_i,g,G_i,a,\sigma)$ respectively, $i=1,2$. Suppose that assumptions in Lemma 4.4 hold and that  $(u_1-u_2)^+|_{\partial\mathcal{O}}=0$. Then if $f_1(t,x,u_2,\nabla u_2,v_2)\leq f_2(t,x,u_2,\nabla u_2,v_2)$ $dt\times dx\times d\mathbb{P}\textrm{-}a.e.$ and $G_1\leq G_2$ $dx\times d\mathbb{P}\textrm{-}a.e.$, we have $u_1\leq u_2$ $dt\times dx\times d\mathbb{P}\textrm{-}a.e.$.
\begin{proof}
Let $(\underline{u},\underline{v})=(u_1-u_2,v_1-v_2)$. Then $(\underline{u},\underline{v})$ is a solution to the quasi-linear BSPDE (\ref{quasilinear-BSPDE}) with parameters $(\underline{f},\underline{g},\underline{G},a,\sigma)$, where
\begin{align*}
&\underline{f}(t,x,\cdot,\cdot,\cdot)=f_1(t,x,\cdot+u_2,\cdot+\nabla u_2,\cdot+v_2)-f_2(t,x,u_2,\nabla u_2,v_2)\nonumber\\
&\underline{g}(t,x,\cdot,\cdot,\cdot)=g(t,x,\cdot+u_2,\cdot+\nabla u_2,\cdot+v_2)-g(t,x,u_2,\nabla u_2,v_2)\nonumber\\
&\underline{G}=G_1-G_2.\nonumber
\end{align*}
Then we have $\underline{f}_0:=\underline{f}(\cdot,\cdot,0,0,0)\leq 0$, $\underline{g}_0:=\underline{g}(\cdot,\cdot,0,0,0)=0$ and $\textrm{esssup}_{\Omega\times\partial_p Q}u^+=0$. Therefore by Lemma \ref{estimate-u-k} or Lemma \ref{MP-general-domain-quasilinear-BSPDE}, there holds that $u_1\leq u_2$ $dt\times dx\times d\mathbb{P}\textrm{-}a.e.$.
\end{proof}
\end{corollary}
%\begin{proof}
%\textcolor{blue}{the proof of (1) is the same as in last theorem (1) and (2) is the same as (2).}
%\end{proof}

%\begin{remark}
%Similar iteration sequence has been applied to obtain $L^p$ estimate for a linear forward degenerate SPDE in \cite{qiu-2015-DSPDE}. We believe this %sequence by also be applied to obtain the maximum principle for degenerate reflected quasi-linear backward SPDE, which is left for future work.
%\end{remark}
% in a similar way to ..












%\subsection{Global Case}

%\begin{theorem}\label{theorem4.3}
%Assume $\mathcal{A}_1$-$\mathcal{A}_4$ hold. The triplet $(u,v,\mu)$ is a solution to RBSPDE (1.1). Then we have
%\begin{align}\label{equation4.1}
%&\textrm{esssup}_{(\omega,t,x)\in\Omega \times Q}u^{\pm}(\omega,t,x) \\
%&\leq  C\left( \|u^{\pm}\|_{0,2;Q} +   \textrm{esssup}_{(\omega,t,x)\in\Omega \times \partial _pQ}u^{\pm}+\textrm{esssup}_{(\omega,t,x)\in\Omega \times %\partial _pQ}\hat{\xi}^{\pm}+\|f_0^{\pm}\|_{0,\frac{p(n+2)}{p+n+2};Q}\right.\nonumber\\
%&\quad \left.+\|\hat{f}^{\pm}\|_{0,\frac{p(n+2)}{p+n+2};Q}+\|g_0\|_{0,p;Q}+\|\hat{g}\|_{0,p;Q}\nonumber
 %\right),
%\end{align}
%where $\partial _pQ:=\left([0,T]\times \partial \mathcal{O}\right)\cup \left(\{T\}\times \mathcal{O}\right)$ is the parabolic boundary of $Q$ and the %constant $C$ depends only on $\lambda$, $\kappa$, $\beta$, $L$, $p$, $\varrho$, $n$, $T$ and $|\mathcal{O}|$.
%\end{theorem}

%\begin{proof}
%We only prove the $"u^+"$ part and the other part is similar to obtain.\\
%Since from $u|_{\mathcal{O}}=0$ we cannot get $u^+|_{\mathcal{O}}=0$, the It\^o formula in Theorem 3.10 \cite{qiuwei} cannot be applied directly here. So first we should generalize the It\^o formula for the case $u^+|_{\mathcal{O}}=0$, which we put in the Appendix, see Theorem \ref{theorem6.2}.
%Choose a constant $k\geq \textrm{esssup}_{(\omega,t,x)\in\Omega \times \partial_pQ}u^{+}(\omega,t,x)$, then $(u-k)^+|_{\partial\mathcal{O}}=0$. Therefore, by Theorem \ref{theorem6.2}, for each $t\in[0,T]$ there holds almost surely
%\begin{align}\label{equation4.2}
%&\|(u-k)^+(t)\|^2+\int_t^T\|v^k(s)\|^2ds
%\nonumber
%\\
%&=-2\int_t^T\langle\partial_j(u-k)^+(s),a^{ij}\partial_i(u-k)^+(s)+\sigma ^{jr}(s)v^{k,r}{s}\rangle\, ds\nonumber\\
%&-2\int_t^T\langle\partial_j(u-k)^+(s),g^{k,j}(s,(u-k)^+(s),\nabla u(s),v^k(s))\rangle\, ds\nonumber\\
%&+2\int_t^T\langle (u-k)^+(s),f^k(s,(u-k)^+(s),\nabla u(s),v^k(s))\rangle \,ds+2\int_{\mathcal{O}_t}(u-k)^+(s,x)\mu(s,x)\nonumber\\
%&-2\int_t^T\langle (u-k)^+(s),v^{k,r}(s)dW^r_s\rangle,
%\end{align}
%where $v^k:=v1_{\{u>k\}}$, $f^k(\cdot,\cdot,\cdot,X,Y,Z):=f(\cdot,\cdot,\cdot,X+k,Y,Z)$ and $g^k(\cdot,\cdot,\cdot,X,Y,Z):=g(\cdot,\cdot,\cdot,X+k,Y,Z)$. %All terms in (\ref{equation4.2}) are well defined. In particular, the stochastic integral is in fact a martingale.

% Taking conditional expectation on both sides w.r.t. $\mathcal{F}_t$, with similar estimate for $I_1$ in the proof of Theorem \ref{theorem3.2}, we have
%\begin{align}\label{equation4.4}
%J_1:=&-2E\left[\int_t^T\langle\partial_j(u-k)^+(s),a^{ij}\partial_i(u-k)^+(s)+\sigma ^{jr}(s)v^{k,r}{s}\rangle ds|\mathcal{F}_t\right]\nonumber\\
%\leq &-\lambda E\left[\int_t^T\|\nabla(u-k)^+(s)\|^2ds|\mathcal{F}_t\right]+\frac{1}{\varrho}E\left[\int_t^T\|v^k(s)\|^2ds|\mathcal{F}_t\right].
%\end{align}
%With similar estimate for $I_2$ and $B_2$, for each $\epsilon>0$ we have
%\begin{align}\label{equation4.5}
%\begin{split}
%J_2:=&-2E\left[\int_t^T\langle\partial_j(u-k)^+(s),g^{k,j}(s,(u-k)^+(s),\nabla u(s),v^k(s))\rangle ds|\mathcal{F}_t\right] \nonumber \\
%\leq &2E\left[\int_t^T\langle|\nabla(u-k)^+(s)|,g^k_0+L|(u-k)^+(s)|+\frac{\kappa}{2}|\nabla (u-k)^+(s)|+\sqrt{\beta}|v^k(s)|\rangle ds|\mathcal{F}_t\right]\nonumber\\
%\leq &(\kappa+\beta\theta+2\epsilon)E\left[\int_t^T\|\nabla(u-k)^+(s)\|^2ds|\mathcal{F}_t\right]+\frac{1}{\theta}E\left[\int_t^T\|v^k(s)\|^2ds|\mathcal{F}_t\right]\nonumber\\
% &+C(\epsilon,L)E\left[\int_t^T\|(u-k)^+(s)\|^2ds|\mathcal{F}_t\right]+2E\left[\int_t^T\langle|\nabla(u-k)^+(s)|,|g_0^k|\rangle ds|\mathcal{F}_t\right]\nonumber\\
%\end{split}
%\end{align}
%where
% \begin{equation}\label{equation4.6}
%|{u>k}|_{\infty;\mathcal{O}_t}:=esssup_{(\omega,s)\in \Omega \times [t,T]}E\left[\mathcal{O}_s\cap \{u>k\}|\mathcal{F}_s\right],
%\end{equation}
%$\epsilon$ is a positive constant to be determined later and $\theta>\varrho'$ with $\lambda-\kappa-\theta\beta>0$.
%\newline For each $\epsilon_1>0$, one has
%\begin{align}\label{equation4.7}
%\begin{split}
%J_3:&= 2E\left[\int_t^T\langle (u-k)^+(s),f^k(s,(u-k)^+(s),\nabla u(s),v^k(s))\rangle ds|\mathcal{F}_t\right]\nonumber\\
%&\leq  2\|(u-k)^+\|_{0,\frac{2(n+2)}{n};\mathcal{O}_t}\|f_0^k\|_{0,\frac{p(n+2)}{n+p+2};\mathcal{O}_t}|{u>k}|_{\infty;\mathcal{O}_t}^{\frac{1}{2}-\frac{1}{p}}\nonumber\\
%&\leq  \epsilon _1\|(u-k)^+\|_{0,\frac{n(n+2)}{n};\mathcal{O}_t}^2+C(\epsilon _1)\left(\|f_0^+\|_{0,\frac{p(n+2)}{n+p+2};\mathcal{O}_t}^2+k^2\right)|{u>k}|_{\infty;\mathcal{O}_t}^{1-\frac{2}{p}}.
%\end{split}
%\end{align}
%By the virtue of \cite[Theorem 5.4]{qiutang}, $\hat{\xi}$ satisfies the maximum principle. So there must be some suitable constant $C$ which only depends on $p$, $\kappa$, $L$, $\beta$, $n$, $\lambda$, $|\mathcal{O}|$, $\varrho$ and $T$, such that
%\begin{equation}
%\textrm{esssup}_{(\omega,s,x)\in\Omega\times[t,T]\times\mathcal{O}}\hat{\xi}^+(\omega,s,x)\leq \textrm{esssup}_{\Omega \times \partial _pQ}\hat{\xi}^++C\left(\|\hat{f}^+\|_{0,\frac{p(n+2)}{p+n+2};Q}+\|\hat{g}\|_{0,p;Q}\right)|\mathcal{O}|^{\frac{1}{n+2}-\frac{1}{p}}.
%\end{equation}

%Taking $$k\geq \textrm{esssup}_{\Omega \times \partial _pQ}\hat{\xi}^++C\left(\|\hat{f}^+\|_{0,\frac{p(n+2)}{p+n+2};Q}+\|\hat{g}\|_{0,p;Q}\right)|\mathcal{O}|^{\frac{1}{n+2}-\frac{1}{p}},$$ we have \begin{equation}\label{equation4.9}
%\int_{\mathcal{O}_t}(u-k)^+(s,x)\mu(dsdx)\leq \int_{\mathcal{O}_t}(u-\xi+\hat{\xi}-k)^+\mu(dsdx)\leq 0.
%\end{equation}
%Summing up the estimates for $J_1$-$J_4$, taking supremum w.r.t. $(\omega, s)\in\Omega \times [t,T]$ and taking $\epsilon$ small enough, due to assumption $(\mathcal{A}_2)$, it follows from (\ref{equation4.4}), (\ref{equation4.5}), (\ref{equation4.7}), (\ref{equation4.9}) that
%\begin{align*}
%&\|(u-k)^2\|^2_{\mathcal{V}_2(\mathcal{O}_t)}+\|v^k\|^2_{0,2;\mathcal{O}_t}\\
%&\leq C\left\{\|(u-k)^+\|_{0,2;\mathcal{O}_t}^2+\epsilon _1 \|(u-k)^+\|_{0,\frac{2(n+2)}{n};\mathcal{O}_t}^2
%+\left(\|f_0^+\|_{0,\frac{p(n+2)}{p+n+2};Q}^2+\|g_0\|^2_{0,p;Q}+k^2\right)|{u>k}|_{\infty;\mathcal{O}_t}^{1-\frac{2}{p}}\right\}.
%\end{align*}
%Now take $\epsilon _1$ small enough. By the embedding Lemma \ref{lemma2.6}, we have
%\begin{align*}
%&\|(u-k)^+\|_{0,\frac{2(n+2)}{n};\mathcal{O}_t}^2\\
%&\leq \|(u-k)^2\|^2_{\mathcal{V}_2(\mathcal{O}_t)}+\|v^k\|^2_{0,2;\mathcal{O}_t}
%\\
%&\leq C\left\{\|(u-k)^+\|_{0,2;\mathcal{O}_t}^2+\left(\|f_0^+\|_{0,\frac{p(n+2)}{p+n+2};Q}^2+\|g_0\|^2_{0,p;Q}+k^2\right)|{u>k}|_{\infty;\mathcal{O}_t}^{1-\frac{2}{p}}\right\}\\
%&\leq C\left(|T-t||\mathcal{O}|\right)^{\frac{2}{n+2}}\|(u-k)^+\|^2_{0,\frac{2(n+2)}{n};\mathcal{O}_t}
%+C\left(\|f_0^+\|_{0,\frac{p(n+2)}{p+n+2};Q}^2+\|g_0\|^2_{0,p;Q}+k^2\right)|{u>k}|_{\infty;\mathcal{O}_t}^{1-\frac{2}{p}}.
%\end{align*}

%Taking $t_0$ such that $C\left(|T-t_0||\mathcal{O}|\right)^{\frac{2}{n+2}}=1/2$ and $k\geq \|f_0^+\|_{0,\frac{p(n+2)}{p+n+2};Q}+\|g_0\|_{0,p;Q}$, we have
%\begin{equation}\label{equation4.12}
%\|(u-k)^+\|^2_{0,\frac{2(n+2)}{n};\mathcal{O}_{t_0}} \leq Ck^2|{u>k}|_{\infty;\mathcal{O}_{t_0}}^{1-\frac{2}{p}}.
%\end{equation}
%If we define
%\begin{equation*}
%\psi (h)= |{u>h}|_{\infty;\mathcal{O}_{t_0}},~h\geq k,
%\end{equation*}
%then (\ref{equation4.12}) implies
%If we define
%\begin{equation*}
%|u>k|_{\infty;\mathcal{O}_t}=\textrm{esssup}_{(\omega,s)\in\Omega\times[t,T]}E\left(|\mathcal{O}_s\cap\{u>k\}|\,|\mathcal{F}_s\right),
%\end{equation*}
% thus starting from (\ref{equation4.2}) and in a similar way to Theorem 5.1 in \cite{qiutang}, we have
%\begin{equation}\label{De Giorgi Sequence_bounded domain_global}
%\phi(m)\leq Ck^{\frac{4}{n}}\phi (m-1)^{\frac{2(p-n-2)}{pn}+1},
%\end{equation}
%where $\phi (m)= |u>k(2-2^{-m})|_{\infty;\mathcal{O}_{t_0}}$ for some $t_0$ such that $C\left(|T-t_0||\mathcal{O}|\right)^{\frac{2}{n+2}}=1/2$ with $C$ large enough. Thus by (\ref{De Giorgi Sequence_bounded domain_global}) and Lemma \ref{lemma4.1} we get the desired result for $u^+$.
%Taking $k_m=k(2-2^{-m}),m\in\mathbb{N}$, we get the De Giorgi's sequence $\psi(k_m)$. By Lemma \ref{lemma4.1} we have: if there is constant $\theta$ s.t.
%\begin{equation}\label{equation4.15}
%\psi (k_0)\leq \theta ,
%\end{equation}
%then $\lim_{m\rightarrow \infty}\psi(k_m)=0$.
%To verify (\ref{equation4.15}), we only need to let
%\begin{equation*}
%k\geq \theta ^{-\frac{1}{2}}\|u^+\|^2_{0,2;\mathcal{O}_{t_0}}.
%\end{equation*}
%Now $\lim_{m\rightarrow \infty}\psi(k_m)=0$ implies
%\begin{equation*}
%\textrm{esssup}_{(\omega,s,x)\in \Omega \times [t_0,T]\times \mathcal{O}} u^+\leq 2k,
%\end{equation*}
%from which we obtain,
%\begin{align*}
%&\textrm{esssup}_{(\omega,t,x)\in\Omega \times \mathcal{O}_{t_0}}u^{+}(\omega,t,x) \\
%\leq & C\left(
%\|u^{+}\|_{0,2;Q}+
%\textrm{esssup}_{(\omega,t,x)\in\Omega \times \partial _pQ}u^{+}+\textrm{esssup}_{(\omega,t,x)\in\Omega \times \partial _pQ}\hat{\xi}^{+}+\|g_0\|_{0,p;Q}+\|\hat{g}\|_{0,p;Q}\right.\\
%&\left.+\|f_0^{+}\|_{0,\frac{p(n+2)}{p+n+2};Q}+\|\hat{f}^{+}\|_{0,\frac{p(n+2)}{p+n+2};Q}  \right).
%\end{align*}
%By iteration w.r.t. $t_0$, we arrive at (\ref{equation4.1}).

%The estimate on $u^-$ follows in a similar way.
%\end{proof}

%\begin{theorem}\label{MP_BD_Global}
%Assume $\mathcal{A}_1$-$\mathcal{A}_4$ hold and the triple $(u,v,\mu)$ is a solution to RBSPDE (1.1),
%as defined by Definition \ref{definition3.1}. If we further assume that $g(t,x,r,0,0)=g(t,x,0,0)$ and $f(t,x,r,0,0)$ is non-increasing w.r.t. $r$, then
%\begin{align*}
%\textrm{esssup}_{(\omega,t,x)\in \Omega\times Q}u^{\pm}(\omega,t,x)
%&\leq \textrm{esssup}_{(\omega,t,x)\in \Omega\times \partial_pQ}u^{\pm}(\omega,t,x)+\textrm{esssup}_{(\omega,t,x)\in \Omega\times \partial_pQ}\hat{\xi}^{\pm}(\omega,t,x)\\
%&+C\left(\|f_0^{\pm}\|_{0,\frac{p(n+2)}{n+p+2};Q}+\|g_0\|_{0,p;Q}+\|\hat{f}^{\pm}\|_{0,\frac{p(n+2)}{n+p+2};Q}+\|\hat{g}\|_{0,p;Q}\right)|\mathcal{O}|^{\frac{1}{n+2}-\frac{1}{p}},
%\end{align*}
%where the positive constant $C$ depends on the structure parameters $\lambda$, $\kappa$, $\beta$, $p$, $\varrho$, $L$, $n$, $T$, $\varrho$ and %$|\mathcal{O}|$.
%\end{theorem}
%\begin{proof}
%Thanks to the further assumptions in this theorem, we have%from (\ref{equation4.5}) and (\ref{equation4.7}) it follows that
%\begin{align*}
%J_2 &\leq  %(\kappa+\beta\theta+3\epsilon)E\left[\int_t^T\|\nabla(u-k)^+(s)\|^2ds|\mathcal{F}_t\right]+\frac{1}{\theta}E\left[\int_t^T\|v^k(s)\|^2ds|\mathcal{F}_t\right]\\
%&+C(\epsilon,L)E\left[\int_t^T\|(u-k)^+(s)\|^2ds|\mathcal{F}_t\right]+C(\epsilon)\|g_0\|_{0,p;Q}^2|{u>k}|_{\infty;\mathcal{O}_t}^{1-\frac{2}{p}}
%\end{align*}
%and
%\begin{align*}
%\begin{split}
%J_3\leq & \epsilon _1\|(u-k)^+\|_{0,\frac{n(n+2)}{n};\mathcal{O}_t}^2+C(\epsilon _1)\|f_0^+\|_{0,\frac{p(n+2)}{n+p+2};\mathcal{O}_t}^2|{u>k}|_{\infty;\mathcal{O}_t}^{1-\frac{2}{p}}.
%\end{split}
%\end{align*}
%With the similar procedure in the proof of Theorem \ref{theorem4.3}, we have
%\begin{equation}\label{equation4.22}
%\phi(m)\leq %\frac{C(\|f_0\|^2_{0,\frac{p(n+2)}{n+p+2};\mathcal{O}_{t_0}}+\|g_0\|^2_{0,p;\mathcal{O}_{t_0}})^{\frac{n+2}{n}}}{(k2^{-m})^{\frac{2(n+2)}{n}}}\phi^{1-\frac{2}{p}}(m-1),
%\end{equation}
%from which and Corollary \ref{corollary4.2}, we get
%\begin{align*}
%\textrm{esssup}_{(\omega,t,x)\in\Omega\times\mathcal{O}_t}u^+(\omega,t,x)
%&\leq \textrm{esssup}_{(\omega,t,x)\in \Omega \times \partial _pQ}u^+(\omega,t,x)+\textrm{esssup}_{(\omega,t,x)\in \Omega \times \partial %_pQ}\hat{\xi}^+(\omega,t,x)\\
%&+C\left(\|f_0^+\|_{0,\frac{p(n+2)}{n+p+2};\mathcal{O}_t}+\|g_0\|_{0,p;Q}+\|\hat{f}^+\|_{0,\frac{p(n+2)}{n+p+2};\mathcal{O}_t}+\|\hat{g}\|_{0,p;Q}\right).
%\end{align*}
%The estimate for $u^-$ follows in a similary way.
%\end{proof}

% following is comparison principle. be deleted as first due to some mistake.
%This section is completed with the following comparison theorem as a corollary of the maximum principle.
%\begin{corollary}
%Suppose $\mathcal{A}_1$ and $\mathcal{A}_2$ hold. Let $(f^1,g,G^1,\xi^1,\hat{\xi}^1)$ and $(f^2,g,G^2,\xi^2,\hat{\xi}^2)$ satisfy $\mathcal{A}_3$ and %$\mathcal{A}_4$ and further assume $f^1$, $f^2$ and $g$ satisfy the condition in Proposition \ref{theorem4.4}. Let $u_i$ be the solution to RBSPDE (1.1) with data $(f^i,g,G^i,\xi^i,\hat{\xi}^i)$, $i=1,2$. If $f^1 (\omega,t,x,u_2,\nabla u_2,v_2)\leq f^2(\omega,t,x,u_2,\nabla u_2,v_2)$ $dt\otimes dx\otimes d\mathbb{P}$ a.e., $\hat{f}^1\leq \hat{f}^2$ $dt\otimes dx\otimes d\mathbb{P}$ a.e., $(u_1-u_2)^+|_{\partial\mathcal{O}}=0$, $(\hat{\xi}^1-\hat{\xi}^2)^+|_{\partial\mathcal{O}}=0$ and $G^1\leq G^2$ $dx\otimes d\mathbb{P}$ a.e., then we have $u_1\leq u_2$ $dt\otimes dx\otimes d\mathbb{P}$ a.e..
%\end{corollary}
%\begin{proof}
%The assertion is obtained directly by applying the maximum principle in Proposition \ref{theorem4.4} to $u_1-u_2$.
%\end{proof}


\subsection{Local Behavior of the Random Field $u^{\pm}$}
The global maximum principle in Theorem \ref{Theorem_MP_GD} tells us that if the random field $u^{\pm}$ is bounded on the parabolic boundary, it must be bounded in the whole domain. This section studies the local behavior of $u^{\pm}$ when it is not necessarily bounded on the parabolic boundary.
\begin{definition}\label{cutofffunction}
A function $\zeta$ is called a cut-off function on the sub-domain $Q'\subset Q$ if it satisfies the following properties:
\begin{itemize}
	\item[(1)] there exits some smooth function sequence $\{\zeta_m\}\subset C_0^{\infty}(Q')$ such that $\zeta_m$, $\partial_s\zeta_m$ and $\nabla\zeta_m$ converge to $\zeta$, $\partial_s\zeta$ and $\nabla\zeta$ in $L^{\infty}(Q')$ respectively;
	\item[(2)] $\zeta\in [0,1]$;
	\item[(3)] there exits a domain $Q'' \subset\subset Q'$ and a nonempty domain $Q''' \subset\subset Q''$ such that
            $$  \zeta(t,x)=
   \begin{cases}
   0 &\mbox{if $(t,x)\in Q' \backslash Q''$}\\
   1 &\mbox{if $(t,x)\in Q'''$,}
   \end{cases}$$
   where by $A\subset\subset B$ we mean the closure $\bar{A}\subseteq B$.
\end{itemize}
\end{definition}

We modify the definition of backward stochastic parabolic De Giorgi class in \cite{qiutang} as follows.
\begin{definition}
We say a function $u\in\mathcal{V}_{2,0}(Q)$ belongs to a backward stochastic parabolic De Giorgi class $BSPDG^{\pm}(a_0,b_0,k_0,\eta;\delta,Q)$ with $$(a_0,b_0,k_0,\eta,\delta)\in[0,\infty)\times[0,\infty)\times[0,\infty)\times(n+2,\infty)\times(0,1),$$
 if for any $Q_{\rho,\tau}:=[t_0-\tau,t_0)\times B_{\rho}(x_0)\subset Q$ with $(\rho,\tau)\in (0,\delta]\times (0,\delta^2]$, each cut-off function $\zeta$ on $Q_{\rho,\tau}$ and for each $k\geq k_0$, we have
\begin{align}
\|\zeta (u-k)^{\pm}\|^2_{\mathcal{V}_2(Q_{\rho,\tau})}
&\leq b_0 \bigg\{
\|(u-k)^{\pm}\|^2_{0,2;Q_{\rho,\tau}}\left(1+\|\partial _t\zeta\|_{L^{\infty}(Q_{\rho,\tau})}+\|\nabla \zeta\|^2_{L^{\infty}(Q_{\rho,\tau})}\right)\nonumber\\
&\quad \left.+\left(k^2+a_0^2\right)|{(u-k)^{\pm}>0}|_{\infty;Q_{\rho,\tau}}^{1-\frac{2}{\eta}}\right\},
\label{relatn-DG}
\end{align}
where $|{(u-k)^{\pm}>0}|_{\infty;Q_{\rho,\tau}}:=\textrm{esssup}_{\omega\in \Omega}\sup_{s\in[t_0-\tau,t_0)}E\left[\int_{[s,t_0)\times B_{\rho}(x_0)}1_{\{(u(t,x)-k)^{\pm}>0\}}\,dxdt|\mathcal{F}_s\right]$.
\end{definition}

Here, we take $(k,\rho, \tau)\in [k_0,\infty)\times (0,\delta]\times (0,\delta^2]$ for given $(k_0,\delta)\in [0,\infty)\times(0,1)$ in the above definition, instead of $(k,\rho, \tau)\in \bR\times (0,1)\times (0,1)$ as in \cite[Definition 5.2]{qiutang}. However, a direct extension of \cite[Theorem 5.8]{qiutang} yields the following lemma.
%
%In fact, the proof of \cite[Theorem 5.8]{qiutang} is based on a stochastic version of  De Giorgi's iteration scheme starting from relation \eqref{relatn-DG},   Theorem 5.8 of \cite{qiutang}  holds with modifications due to $\delta$. In particular, an immediate consequence is the following lemma.
\begin{lemma}\label{lem-DG}
Given $k_0^{\pm}\geq 0$, if $u\in BSPDG^{\pm}(a^{\pm}_0,b^{\pm}_0,k_0^{\pm},\eta;\delta,Q)$, then
\begin{align*} %\label{equation5.16}
\textrm{esssup}_{(\omega,t,x)\in \Omega\times Q_{\frac{\rho}{2}}}u^{\pm}
&\leq 2k_0^{\pm} + C_{\pm} \left\{\rho^{-\frac{n+2}{2}}\|u^{\pm}\|_{0,2;Q_{\rho}}+a_0^{\pm}\rho^{1-\frac{2+n}{\eta}}\right\},
\end{align*}
where $Q_{\rho}:=[t_0-\rho ^2,t_0)\times B_{\rho}(x_0)\subset Q$ with $\rho \in (0,\delta]$ and the constants $C_{\pm}$ depend on $a^{\pm}_0$, $b^{\pm}_0$ and $n$.
\end{lemma}

%\begin{remark}
%The proof of \cite[Theorem 5.8]{qiutang} is just using a stochastic version of  De Giorgi's iteration scheme starting from relation \eqref{relatn-DG}.
%
%\end{remark}

For the solution to the RBSPDE (\ref{equation1.1}), we further have the following result.


\begin{lemma}\label{RBSPDE-DG}
Let assumptions $(\mathcal{A}_1)$-$(\mathcal{A}_4)$ hold. Suppose $(u,v,\mu)$ is a solution to the RBSPDE (1.1). Given $Q_{\delta}:=[t_0-\delta ^2,t_0)\times B_{\delta}(x_0)\subset Q$ with $\delta \in (0,1)$, let $k_0^{\pm}=\textrm{esssup}_{\Omega\times Q_{\delta}}\hat{\xi}^{\pm}$. Then we have $u\in BSPDG^{\pm}(a^{\pm}_0,b_0,k^{\pm}_0,\eta;\delta,Q_{\delta})$ with $\eta=p$, $a^{\pm}_0=A_p(f_0^{\pm},g_0;Q_{\delta})$ and $b_0$ depending on $\lambda$, $\kappa$, $\beta$, $\varrho$, $\Lambda$, $L$, $n$ and $p$.
%, i.e., for any $Q_{\rho,\tau}:=[t_0-\tau,t_0)\times B_{\rho}(x_0)\subset Q_{\delta}$ with $(\rho,\tau)\in (0,\delta)\times (0,\delta^2)$, any cut-off function $\zeta$ on $Q_{\rho,\tau}$ and each $k\geq k_0^{\pm}$, we have
%\begin{equation}\label{equation5.5}
%\begin{split}
%\|\zeta (u-k)^{\pm}\|^2_{\mathcal{V}_2(Q_{\rho,\tau})}
%&\leq b_0 \bigg\{
%\|(u-k)^{+}\|^2_{0,2;Q_{\rho,\tau}}\left(1+\|\partial _t\zeta\|_{L^{\infty}(Q_{\rho,\tau})}+\|\nabla \zeta\|^2_{L^{\infty}(Q_{\rho,\tau})}\right)\\
%& \left.+\left(k^2+\|f_0^{\pm}\|^2_{0,\frac{p(n+2)}{p+n+2};Q}+\|g_0\|^2_{0,p;Q}\right)|{(u-k)^{\pm}>0}|_{\infty;Q_{\rho,\tau}}^{1-\frac{2}{p}}\right\}.
%\end{split}
%\end{equation}
%where $b_0$ depends on $\lambda$, $\kappa$, $\beta$, $\varrho$, $\Lambda$, $L$, $n$ and $p$.
\end{lemma}

\begin{proof}
First we generalize the It\^o formula to a local case for the RBSPDE (\ref{equation1.1}).
For each cut-off function $\zeta$ on $Q_{\rho,\tau}$ with $(\rho,\tau)\in(0,\delta]\times(0,\delta^2]$, we can choose a sequence of smooth functions $\{\zeta _m\}\subset C_0^{\infty}(Q_{\rho,\tau})$ such that $\zeta_m$ and its gradients w.r.t. $s$ and $x$ converge uniformly to $\zeta$ and its gradient, respectively, as $m\rightarrow \infty$.

%It is easy to verify that for each $\varphi \in C_0^{\infty}(Q_{\rho,\tau})$,
%\begin{align*}
%&\langle u(t)\zeta_m(t),\varphi(t)\rangle_{B_{\rho}(x_0)}+\int_t^{t_0}\langle u(s)\zeta_m(s),\partial_s \varphi(s)\rangle_{B_{\rho}(x_0)}\,ds
%\\
%&+\int_t^{t_0}\langle\partial _j \varphi(s),a^{ij}(s)\partial _i(\zeta_m(s)u(s))+\sigma ^{jr}(s)v^r(s)\zeta_m(s)\rangle_{B_{\rho}(x_0)}\,ds\\
%=&\int_t^{t_0}\Big\langle f(s,u(s),\nabla u(s),v(s))\zeta_m(s)-u(s)\partial_s\zeta_m(s)-a^{ij}(s)\partial_j\zeta_m(s)\partial _iu(s)\\
%&-\sigma ^{jr}(s)\partial _j\zeta _m(s)v^r(s)-\partial_j\zeta_m(s)g^j(s,u(s),\nabla u(s),v(s)),\varphi(s)\Big\rangle_{B_{\rho}(x_0)}\,ds\\
%&-\int_t^{t_0}\Big\langle\zeta_m(s)g^{i}(s,u(s),\nabla u(s),v(s))-a^{ij}(s)\partial _j\zeta_m(s)u(s),\partial_i\varphi(s)\Big\rangle_{B_{\rho}(x_0)}\,ds\\
%&+\int_t^{t_0}\int_{B_{\rho}(x_0)}\varphi (s,x)\zeta_m(s,x)\,\mu(ds,dx)-\int_t^{t_0}\langle\varphi(s),\zeta_m(s)v^r(s)\rangle_{B_{\rho}(x_0)}\,dW^r_s.
%\end{align*}
%So, from the definition of the weak solution, $(\zeta _mu,\zeta_m v,\zeta_m \mu)$ satisfy the following equation in the weak sense,
%\begin{align*}
%-d\bar{u}(t,x)=&[\partial _j(a^{ij}\partial _i\bar{u}(t,x)+\sigma ^{jr}\bar{v}^r(t,x))+\bar{f}(t,x)+\nabla \cdot \bar{g}(t,x)]\,dt+\bar{\mu} (dt,x)-\bar{v}^r(t,x)\,dW^r_t,
%\end{align*}
%where $\bar{u}(t,x)=u(t,x)\zeta_m(t,x)$, $\bar{v}(t,x)=v(t,x)\zeta_m(t,x)$, $\bar{\mu}(t,x)=\mu(t,x)\zeta_m(t,x)$
%\begin{align*}
%\bar{f}(t,x)=&f(t,x,u(t,x),\nabla u(t,x),v(t,x))\zeta_m(t,x)-u(t,x)\partial_t\zeta_m(t,x)-a^{ij}(t,x)\partial_j\zeta_m(t,x)\partial _iu(t,x)\\
%&-\sigma ^{jr}(t,x)\partial _j\zeta _m(t,x)v^r(t,x)-\partial_j\zeta_m(t,x)g^j(t,x,u(t,x),\nabla u(t,x),v(t,x)),
%\end{align*}
%and $\bar{g}^i(t,x)=\zeta_m(t,x)g^{i}(t,x,u(t,x),\nabla u(t,x),v(t,x))-a^{ij}(t,x)\partial _j\zeta_m(t,x)u(t,x)$.


For $k \geq k_0^+$, Theorem \ref{Ito-u+} yields that
\begin{align*}
&\|(u-k)^+(t)\zeta_m(t)\|^2_{L^2(B_{\rho}(x_0))}+\int_t^{t_0}\|\zeta_m(s)v^k(s)\|^2_{L^2(B_{\rho}(x_0))}\,ds\\
=&-2\int_t^{t_0}\langle\zeta_m(s)\partial _s\zeta_m(s),|(u-k)^+(s)|^2\rangle_{B_{\rho}(x_0)}\,ds
+2\int_t^{t_0}\langle\zeta_m^2(s)(u-k)^+(s),f(s,u(s),\nabla u(s),v(s))\rangle_{B_{\rho}(x_0)}\,ds\\
&-2\int_t^{t_0}\langle\partial _j(\zeta_m^2(s)(u-k)^+(s)),a^{ij}(s)\partial_iu(s)+\sigma ^{jr}(s)v^{r}(s)+g^{j}(s,u(s),\nabla u(s),v(s))\rangle_{B_{\rho}(x_0)}\,ds\\
&-2\int_t^{t_0}\langle\zeta_m^2(s)(u-k)^+(s),v^{r,k}(s)\rangle_{B_{\rho}(x_0)}\,dW^r_s+2\int_t^{t_0}\int_{B_{\rho}(x_0)}(u-k)^+(s,x)\zeta^2_m\,\mu(ds,dx),
\end{align*}
where $v^{r,k}:=v^r1_{\{u>k\}}$.

Thus, by letting $m\rightarrow \infty$ and the dominated convergence theorem, we can get
\begin{align*}%\label{Ito formula_local}
&\|(u-k)^+(t)\zeta(t)\|^2_{L^2(B_{\rho}(x_0))}+\int_t^{t_0}\|\zeta(s)v^k(s)\|^2_{L^2(B_{\rho}(x_0))}\,ds\nonumber\\
&=-2\int_t^{t_0}\langle\zeta(s)\partial _s\zeta(s),|(u-k)^+(s)|^2\rangle_{B_{\rho}(x_0)}\,ds
+2\int_t^{t_0}\langle\zeta^2(s)(u-k)^+(s),f(s,u(s),\nabla u(s),v(s))\rangle_{B_{\rho}(x_0)}ds\nonumber\\
&-2\int_t^{t_0}\langle\partial _j(\zeta^2(s)(u-k)^+(s)),a^{ij}(s)\partial_iu(s)+\sigma ^{jr}(s)v^{r}(s)
+g^{j}(s,u(s),\nabla u(s),v(s))\rangle_{B_{\rho}(x_0)}\,ds\nonumber\\
&-2\int_t^{t_0}\langle\zeta^2(s)(u-k)^+(s),v^{r,k}(s)\rangle_{B_{\rho}(x_0)}\,dW^r_s
+2\int_t^{t_0}\int_{B_{\rho}(x_0)}(u-k)^+(s,x)\zeta^2\,\mu(ds,dx).
\end{align*}
Taking conditional expectation, we obtain
\begin{align}\label{startingpoint_local}
%\begin{split}
&\|((u-k)\zeta)^+(t)\|^2_{L^2(B_{\rho}(x_0))}+E\left[\int_t^{t_0}\|\zeta(s)v^k(s)\|^2_{L^2(B_{\rho}(x_0))}\,ds|\mathcal{F}_t\right]\nonumber\\
&=-2E\left[\int_t^{t_0}\langle\zeta(s)\partial _s\zeta(s),|(u-k)^+(s)|^2\rangle_{B_{\rho}(x_0)}\,ds|\mathcal{F}_t\right]\nonumber\\
&\quad+2E\left[\int_t^{t_0}\langle\zeta^2(s)(u-k)^+(s),f^k(s,(u(s)-k)^+,\nabla (u(s)-k)^+,v(s))\rangle_{B_{\rho}(x_0)}\,ds|\mathcal{F}_t\right]\nonumber\\
&\quad-2E\left[\int_t^{t_0}\langle\partial _j(\zeta^2(s)(u-k)^+(s)),a^{ji}(s)\partial_iu(s)+\sigma ^{jr}(s)v^{r}(s)\right.\nonumber\\
&\quad\quad\quad\quad\quad\quad +g^{j,k}(s,(u(s)-k)^+,\nabla (u(s)-k)^+,v(s))\rangle_{B_{\rho}(x_0)}\,ds|\mathcal{F}_t\bigg]\nonumber\\
&\quad +2E\left[\int_t^{t_0}\int_{B_{\rho}(x_0)}(u-k)^+(s,x)\zeta^2\,\mu(ds,dx)|\mathcal{F}_t\right],
%\end{split}
\end{align}
where $f^k(\cdot,\cdot,\cdot,X,Y,Z):=f(\cdot,\cdot,\cdot,X+k,Y,Z)$ and $g^{j,k}(\cdot,\cdot,\cdot,X,Y,Z):=g^j(\cdot,\cdot,\cdot,X+k,Y,Z)$. As $k \geq k_0^+$, the last term on the right hand side of \eqref{startingpoint_local} vanishes. Hence, starting from (\ref{startingpoint_local}), we derive the desired result in a similar way to \cite[Proposition 5.6]{qiutang}.
%\begin{equation*}
%K_1\leq 2\|\partial_s\zeta\|_{L^{\infty}(Q_{\rho,\tau})}E\left[\int_t^{t_0}\|(u-k)^+(s)\|^2ds|\mathcal{F}_t\right]
%\end{equation*}
%and
%\begin{align*}
%K_2\leq & CE\left[\int_t^{t_0}\|\zeta(s)(u(s)-k)^+\|^2ds|\mathcal{F}_t\right]+\epsilon E\left[\int_t^{t_0}\|\zeta(s)\nabla(u(s)-k)^+\|^2ds|\mathcal{F}_t\right]\\
%&+ \epsilon E\left[\int_t^{t_0}\|\zeta(s)v^k(s)\|^2ds|\mathcal{F}_t\right]+\epsilon\|\zeta(s)(u(s)-k)^+\|^2_{0,\frac{2(n+2)}{n};Q_{\rho,\tau}}\\
%&+C(\|f_0\|^2_{0,\frac{p(n+2)}{p+2+n};Q_{\rho,\tau}}+k^2)|u>k|^{1-\frac{2}{p}}_{\infty;Q_{\rho,\tau}}.
%\end{align*}
%Notice
%\begin{align*}
%&E\left[\int_t^{t_0}\Big\langle\zeta(s)((u(s)-k)^+\partial_j\zeta(s)),a^{ji}(s)\partial_i(u(s)-k)^+\right.\\
%&\quad\quad\quad\left.+\sigma^{jr}(s)v^r(s)+g(s,(u(s)-k)^+,\nabla (u(s)-k)^+,v(s))\Big\rangle ds|\mathcal{F}_t\right]\\
%&\leq  C\|(u-k)^+\|^2_{0,2;Q_{\rho,\tau}}\|\nabla \zeta\|^2_{L^{\infty}(Q_{\rho,\tau})}+(\|g_0\|^2_{0,p;Q_{\rho,\tau}}+k^2)|u>k|_{\infty;Q_{\rho,\tau}}^{1-\frac{2}{p}}+\epsilon E\left[\int_t^{t_0}\|\zeta(s)v^k(s)\|^2ds|\mathcal{F}_t\right]\\
%&+\epsilon E\left[\int_t^{t_0}\|\zeta(s)\nabla(u(s)-k)^+\|^2ds|\mathcal{F}_t\right]
%\end{align*}
%and
%\begin{align*}
%&-2E\left[\int_t^{t_0}\langle\zeta^2(s)\partial_j(u(s)-k)^+,a^{ji}(s)\partial_iu(s)+\sigma ^{jr}(s)v^{k,r}(s)\right.\\
%&\quad\quad\quad\quad \left.+g^{k,j}(s,(u(s)-k)^+,\nabla (u(s)-k)^+,v(s))\rangle_{B_{\rho}(x_0)}ds|\mathcal{F}_t\right]\\
%&\leq  -(\lambda-\kappa-\theta \beta-2\epsilon)E\left[\int_t^{t_0}\|\zeta(s)\nabla(u(s)-k)^+\|^2ds|\mathcal{F}_t\right]
%+C(\|g_0\|^2_{0,p;Q_{\rho,\tau}}+k^2)|u>k|^{1-\frac{2}{p}}_{\infty;Q_{\rho,\tau}} \\
%&\quad +CE\left[\int_t^{t_0}\|\zeta(s)(u(s)-k)^+\|^2ds|\mathcal{F}_t\right]
%+\left(\frac{1}{\varrho}+\frac{1}{\theta}\right)E\left[\int_t^{t_0}\|\zeta(s)v^k(s)\|^2ds|\mathcal{F}_t\right],
%&\quad
%+C(\|g_0\|^2_{0,p;Q_{\rho,\tau}}+k^2)|u>k|^{1-\frac{2}{p}}_{\infty;Q_{\rho,\tau}},
%\end{align*}
%where $\theta>\varrho'$ is such that $\lambda-\kappa-\theta\beta>0$ and $\epsilon$ is to be determined later.
%Therefore,
%\begin{align*}
%K_3\leq &-(\lambda-\beta\theta-\kappa-6\epsilon)E\left[\int_t^{t_0}\|\zeta(s)\nabla(u(s)-k)^+\|^2ds|\mathcal{F}_t\right]\\
%&+\left(\frac{1}{\theta}+\frac{1}{\varrho}+4\epsilon\right)E\left[\int_t^{t_0}\|\zeta(s)v^k(s)\|^2ds|\mathcal{F}_t\right]
%+C(\|g_0\|^2_{0,p;Q_{\rho,\tau}}+k^2)|u>k|^{1-\frac{2}{p}}_{\infty;Q_{\rho,\tau}}\\
%&+CE\left[\int_t^{t_0}\|\zeta(s)(u(s)-k)^+\|^2ds|\mathcal{F}_t\right]+C\|(u-k)^+\|^2_{0,2;Q_{\rho,\tau}}\|\nabla \zeta\|^2_{L^{\infty}(Q_{\rho,\tau})}.
%&.
%\end{align*}
%It is obvious that
%\begin{equation*}
%K_4\leq \int_{Q_{\rho,\tau}}(\hat{\xi}-k)^+\mu(ds,dx).
%\end{equation*}
%Combining the estimations above, taking supremum w.r.t. $(\omega,s)\in\Omega\times[t_0-\tau,t_0]$ and letting $\epsilon$ small enough, we get
%\begin{align}\label{equation5.13}
%&\|\zeta(u-k)^+\|^2_{\infty,2;Q_{\rho,\tau}}+\|\zeta\nabla(u-k)^+\|^2_{0,2;Q_{\rho,\tau}}+\|\zeta v^k\|^2_{0,2;Q_{\rho,\tau}}-\epsilon \|\zeta(u-k)^+\|^2_{0,\frac{2(n+2)}{n};Q_{\rho,\tau}}\nonumber\\
%&\leq  C\Bigg(\|(u-k)^+\|^2_{0,2;Q_{\rho,\tau}}(1+\|\partial_s\zeta\|_{L^{\infty}(Q_{\rho,\tau})}+\|\nabla\zeta\|^2_{L^{\infty}(Q_{\rho,\tau})})\nonumber\\
%&\quad +\textrm{esssup}_{(\omega,s)\in\Omega\times[t_0-\tau,t_0]}\int_{Q_{\rho,\tau}}(\hat{\xi}-k)^{+}\mu(ds,dx)\nonumber\\
%&\quad +\big(k^2+\|f_0\|^2_{0,\frac{p(n+2)}{p+n+2};Q}+\|g_0\|^2_{0,p;Q}\big)|{(u-k)^{+}>0}|_{\infty;Q}^{1-\frac{2}{p}}\Bigg),
%\end{align}
%where the constant $C$ depends only on $\kappa$, $\lambda$, $\beta$, $\varrho$, $L$, $n$ and $|\mathcal{O}|$. (\ref{equation5.13}) implies
%\begin{align*}
%&\|\zeta(u-k)^+\|^2_{\mathcal{V}_2(Q_{\rho,\tau})}+\|\zeta v^k\|^2_{0,2;Q_{\rho,\tau}}-\epsilon \|\zeta(u-k)^+\|^2_{0,\frac{2(n+2)}{n};Q_{\rho,\tau}}\\
%\leq & C\left(\|(u-k)^+\|^2_{0,2;Q_{\rho,\tau}}(1+\|\partial_s\zeta\|_{L^{\infty}(Q_{\rho,\tau})}+\|\nabla\zeta\|^2_{L^{\infty}(Q_{\rho,\tau})})\right.\\
%&+\textrm{esssup}_{(\omega,s)\in\Omega\times[t_0-\tau,t_0]}\int_{Q_{\rho,\tau}}(\hat{\xi}-k)^{+}\mu(ds,dx)\\
%&\left.+\left(k^2+\|f_0\|^2_{0,\frac{p(n+2)}{p+n+2};Q}+\|g_0\|^2_{0,p;Q}\right)|{(u-k)^{+}>0}|_{\infty;Q}^{1-\frac{2}{p}}\right)+\|\nabla\zeta(u-k)^+\|^2_{0,2;Q_{\rho,\tau}}\\
%\leq & C\left\{\|(u-k)^+\|^2_{0,2;Q_{\rho,\tau}}(1+\|\partial_s\zeta\|_{L^{\infty}(Q_{\rho,\tau})}+\|\nabla\zeta\|^2_{L^{\infty}(Q_{\rho,\tau})})\right.\\
%&+\textrm{esssup}_{(\omega,s)\in\Omega\times[t_0-\tau,t_0]}\int_{Q_{\rho,\tau}}(\hat{\xi}-k)^{+}\mu(ds,dx)\\
%&\left.+\left(k^2+\|f_0\|^2_{0,\frac{p(n+2)}{p+n+2};Q}+\|g_0\|^2_{0,p;Q}\right)|{(u-k)^{+}>0}|_{\infty;Q}^{1-\frac{2}{p}}\right\}.\\
%\end{align*}
%Finally the proof is finished by the embedding Lemma \ref{lemma2.6}.
\end{proof}

%\begin{remark}
%Through similar estimation, we have a simpler version of (\ref{equation5.5}) for $(u-k)^-$,
%\begin{equation}\label{equation5.15}
%\begin{split}
%\|\zeta (u-k)^{-}\|^2_{\mathcal{V}_2(Q_{\rho,\tau})}
%\leq& C \left\{\|(u-k)^{-}\|^2_{0,2;Q_{\rho,\tau}}(1+\|\partial _t\zeta\|_{L^{\infty}(Q_{\rho,\tau})}+\|\nabla \zeta\|^2_{L^{\infty}(Q_{\rho,\tau})})\right.\\
%&\left.+\left(k^2+\|f_0\|^2_{0,\frac{p(n+2)}{p+n+2};Q}+\|g_0\|^2_{0,p;Q}\right)|{(u-k)^{-}>0}|_{\infty;Q}^{1-\frac{2}{p}}\right\}.
%\end{split}
%\end{equation}
%\end{remark}


Given $Q_{2\rho}:=[t_0-4\rho ^2,t_0)\times B_{2\rho}(x_0)\subset Q$ with $\rho \in (0,1)$,
 let $k_0^{\pm}=\textrm{esssup}_{\Omega\times Q_{\rho}}\hat{\xi}^{\pm}$. Lemma \ref{RBSPDE-DG} shows that $u\in BSPDG^{\pm}(a^{\pm}_0,b_0,k_0^{\pm},\eta;\rho,Q_{\rho})$ with $\eta=p$. $a^{\pm}_0=A_p(f_0^{\pm},g_0;Q_{\rho})$ and $b_0$ given therein. On the other hand,
in view of the local boundedness of weak solutions for BSPDEs (\cite[Proposition 5.6 and Theorem 5.8]{qiutang}), we have
\begin{align*}
k_0^{\pm}
&\leq C \left\{\rho^{-\frac{n+2}{2}}\|\hat{\xi}^{\pm}\|_{0,2;Q_{2\rho}}+A_p(\hat{f}^{\pm},\hat{g};Q_{2\rho})\rho ^{1-\frac{2+n}{p}}\right\}
\end{align*}
with $C$ depending on $\lambda$, $\kappa$, $\varrho$, $\Lambda$, $n$ and $p$. Hence, further by Lemmas \ref{lem-DG} and \ref{RBSPDE-DG}, we obtain finally the local behavior of weak solutions to the RBSPDE (1.1).


\begin{theorem}\label{MP-RBSPDE-Local}
Let assumptions $(\mathcal{A}_1)$-$(\mathcal{A}_4)$ hold. Let $(u,v,\mu)$ be a weak solution to the RBSPDE (1.1). Given $Q_{2\rho}:=[t_0-4\rho ^2,t_0)\times B_{2\rho}(x_0)\subset Q$ with $\rho \in (0,1)$, we have
\begin{equation*}%\label{equation5.16}
\begin{split}
\textrm{esssup}_{(\omega,s,x)\in \Omega\times Q_{\frac{\rho}{2}}}u^{\pm}
\leq &\, C \left\{\rho^{-\frac{n+2}{2}}(\|u^{\pm}\|_{0,2;Q_{\rho}}+\|\hat{\xi}^{\pm}\|_{0,2;Q_{2\rho}})\right.\\
&\left.+\left(A_p(f_0^{\pm},g_0;Q_{\rho})+A_p(\hat{f}^{\pm},\hat{g};Q_{2\rho})\right)\rho ^{1-\frac{2+n}{p}}\right\},
\end{split}
\end{equation*}
where $C$ is a positive constant depending on $\lambda$, $\kappa$, $\beta$, $\varrho$, $\Lambda$, $L$, $n$ and $p$.
\end{theorem}

%\begin{proof}
%For each $l\in \mathbb{N}$, let $R_l=\frac{\rho}{2}+\frac{\rho}{2^{l+1}}$, $k_l=k(2-\frac{1}{2^l})$ and $Q_{R_l}=[t_0,t_0+R_l^2)\times B_{R_l}(x_0)$. Choose $\zeta_l$ to be a cut-off function on $Q_{R_l}$ such that
%\begin{equation*}
%\zeta_l(t,x)=
%\left\{\begin{array}{ll}
%1,~~~(t,x)\in Q_{R_{l+1}},\\\\
%0,~~~(t,x)\in Q_{R_l}\backslash Q_{\frac{R_l+R_{l+1}}{2}},
%$\end{array}
%\right.
%\end{equation*}
%and
%$$\|\nabla \zeta\|^2_{L^2(Q_{\rho})}+\|\partial_s\zeta\|_{L^2(Q_{\rho})}\leq \frac{C}{(R_{Q_l}-R_{Q_{l+1}})^2}.$$
%Recall $\hat{\xi}$ satisfies the maximum principle, if we take $k$ large enough, there must be a constant $C$ such that $$k\geq C(esssup_{(\omega,s,x)\in \Omega\times \partial_pQ_{\rho}}\hat{\xi}^{+}+\|\hat{f}^{+}\|^2_{0,\frac{p(n+2)}{p+n+2};Q}+\|\hat{g}\|^2_{0,p;Q})\geq\textrm{esssup}_{\Omega\times[t,T]\times\mathcal{O}}\hat{\xi}^+,$$
%then from (\ref{equation5.5}) and the embedding lemma \ref{lemma2.6} we have
%\begin{equation}\label{eq5.17}
%\begin{split}
%\|\zeta_l(u-k_{l+1})^+\|^2_{0,\frac{2(n+2)}{n};Q_{R_l}}
%&\leq C\left(\|(u-k_{l+1})^+\|^2_{0,2;Q_{R_l}}(1+\rho^{-2}4^{l+2})\right.\\
%&\quad+\left.(4k^2+\|f_0^{+}\|^2_{0,\frac{p(n+2)}{p+n+2};Q}+\|g_0\|^2_{0,p;Q})|u>k_{l+1}|^{1-\frac{2}{p}}_{\infty;Q_{R_l}}\right).
%\end{split}
%\end{equation}
%Take $l$ large enough, then (\ref{eq5.17}) implies
%\begin{align*}
%&\|\zeta_l(u-k_{l+1})^+\|^2_{0,\frac{2(n+2)}{n};Q_{R_l}}\\
%&\leq C\left(\|(u-k_{l+1})^+\|^2_{0,2;Q_{R_l}}\rho^{-2}4^{l+2}+(4k^2+\|f_0^{+}\|^2_{0,\frac{p(n+2)}{p+n+2};Q}+\|g_0\|^2_{0,p;Q})|u>k_{l+1}|^{1-\frac{2}{p}}_{\infty;Q_{R_l}}\right).
%\end{align*}
%Define $\phi _l=\|(u-k_{l})^+\|^2_{0,2;Q_{R_l}}$, then we have
%\begin{align}
%\phi _{l+1}
%&\leq\|\zeta_l(u-k_{l+1})^+\|^2_{0,2;Q_{R_{l}}}\nonumber\\
%&\leq |u>k_{l+1}|^{\frac{2}{n+2}}_{\infty;Q_{R_l}}\|\zeta_l(u-k_{l+1})^+\|^2_{0,\frac{2(n+2)}{n};Q_{R_{l}}}\nonumber\\
%&	\leq C|u>k_{l+1}|^{\frac{2}{n+2}}_{\infty;Q_{R_l}}\left(\phi_l\rho^{-2}4^{l+2}+(4k^2+\|f_0\|^2_{0,\frac{p(n+2)}{p+n+2};Q}+\|g_0\|^2_{0,p;Q})|u>k_{l+1}|^{1-\frac{2}{p}}_{\infty;Q_{R_l}}\right). \label{equation5.19}
%\end{align}
%Note that $$\phi _l \geq (k_{l+1}-k_l)^2|u>k_{l+1}|_{\infty;Q_{R_l}},$$
%then we have
%\begin{align*}
%\phi _{l+1} &\leq
%C4^{\frac{2l+2}{n+2}+(l+1)(1-\frac{2}{p})}k^{-\frac{4}{n+2}-2+\frac{4}{p}}\phi_l^{1+\frac{2}{n+2}-\frac{2}{p}}(4k^2+\|f_0\|^2_{0,\frac{p(n+2)}{p+n+2};Q}+\|g_0\|^2_{0,p;Q})\\
%&\quad + C4^{\frac{2l+2}{n+2}+l+2}k^{-\frac{4}{n+2}}\rho^{-2}\phi_l^{1+\frac{2}{n+2}}\\
%&\leq %C4^{\frac{2l+2}{n+2}+l+2}k^{-\frac{4}{n+2}}\phi_l^{1+\frac{2}{n+2}-\frac{2}{p}}\left(\rho^{-2}\phi_l^{\frac{2}{p}}+4^{-\frac{2l+2}{p}-1}k^{-2+\frac{4}{p}}(4k^2+\|f_0\|^2_{0,\frac{p(n+2)}{p+n+2};Q}+\|g_0\|^2_{0,p;Q})\right).
%\end{align*}
%Now let $$\|f_0\|^2_{0,\frac{p(n+2)}{p+n+2};Q}+\|g_0\|^2_{0,p;Q}\leq 4k^2\rho^{2(\frac{n+2}{p})-1}.$$
%Hence (\ref{equation5.19}) implies
%\begin{align*}
%\phi _{l+1} &\leq  C4^{\frac{2l+2}{n+2}+l+2}k^{-\frac{4}{n+2}}\phi_l^{1+\frac{2}{n+2}-\frac{2}{p}}(\rho^{-2}\phi_l^{\frac{2}{p}}+4^{-\frac{2l+2}{p}-1}k^{-2+\frac{4}{p}}(4k^2+4k^2\rho^{2(\frac{n+2}{p}-1)}))\\
%&\leq  C4^{\frac{2l+2}{n+2}+l+2}k^{-\frac{4}{n+2}}\phi_l^{1+\frac{2}{n+2}-\frac{2}{p}}(\rho^{-2}\phi_l^{\frac{2}{p}}+4^{-\frac{2l+2}{p}+\frac{1}{2}}k^{\frac{4}{p}}\rho^{2(\frac{n+2}{p}-1)})\\
%&\leq  C4^{\frac{2l+2}{n+2}+l+2}k^{\frac{4}{p}-\frac{4}{n+2}}\phi_l^{1+\frac{2}{n+2}-\frac{2}{p}}\rho^{2(\frac{n+2}{p}-1)}(\rho^{-\frac{2n+4}{p}}\phi_l^{\frac{2}{p}}k^{-\frac{4}{p}}+4^{-\frac{2l+2}{p}+\frac{1}{2}}).
%\end{align*}
%Further assume $k\geq 2^{-\frac{p}{4}}\rho^{-\frac{n+2}{2}}\|u^+\|_{0,2;Q_{\rho}}$, which implies $k\geq \rho^{-\frac{n+2}{2}}\phi_l^{\frac{1}{2}}4^{-\frac{p}{8}}$, hence we have
%\begin{equation*}
%\phi_{l+1}\leq C4^{\frac{2l+2}{n+2}+l+3}k^{\frac{4}{p}-\frac{4}{n+2}}\rho^{2(\frac{n+2}{p}-1)}\phi_l^{1+\frac{2}{n+2}-\frac{2}{p}}.
%\end{equation*}
%Define $\alpha _l=\rho^{-n-2}k^{-2}\phi_l$ as the De Giorgi sequence, then we have
%$$\alpha _{l+1}\leq C4^{l+3+\frac{2l+2}{n+2}}\alpha_l^{1-\frac{2}{p}+\frac{2}{n+2}}.$$
%If we let
%\begin{equation*}
%\rho^{-n-2}k^{-2}\|u^+\|^2_{0,2;Q_{\rho}}\leq (C4^{\frac{2}{n+2}+3})^{-\frac{(n+2)p}{2p-2(n+2)}}4^{-\frac{n+4}{n+2}\frac{(n+2)^2p^2}{(2p-2(n+2))^2}},
%\end{equation*}
%then by Lemma 3.1 we finally get $\|(u-2k)^+\|_{0,2;Q_{\frac{\rho}{2}}}=0$, together with the estimation of $k$ above, (\ref{equation5.16}) is obtained.

%Starting from (\ref{equation5.5}), by letting $k\geq C\left((2\rho)^{-\frac{n+2}{2}}\|\hat{\xi}^+\|_{0,2;Q_{2\rho}}+(2\rho)^{1-\frac{n+2}{\rho}}(\|\hat{f}\|_{0,\frac{p(n+2)}{p+n+2};Q}+\|\hat{g}\|_{0,p;Q})\right)$ for some $C$ large enough such that the integral term in (\ref{equation5.5}) vanish, and in the similar approach to Theorem 5.8 in \cite{qiutang}, we can get
%\begin{equation}\label{scheme_local}
%\phi(m+1)\leq C4^{m+3+\frac{2m+2}{n+2}}\phi^{1-\frac{2}{p}+\frac{2}{n+2}}(m),
%\end{equation}
%where $\phi(m)=\rho^{-n-2}k^{-2}\|(u-k(2-2^{-m}))^+\|^2_{0,2;[t_0,t_0+R_m^2)\times B_{R_m}(x_0)}$ is the scheme constructed in \cite{qiutang} with $R_m=\frac{\rho}{2}+\frac{\rho}{2^{m+1}}$.

%Therefore, applying Lemma \ref{lemma4.1} to (\ref{scheme_local}) we get (\ref{equation5.16}).
%\end{proof}


\begin{appendix}
\section{Auxiliary lemmas and It\^o's Formulas}

This subsection states some useful lemmas and It\^o formulas, which have been frequently used. The first lemma and corollary are from \cite{chen}.

\begin{lemma}\label{lemma4.1}
Let $\{a_k,k\in\mathbb{N}\}$ be a sequence of nonnegative numbers satisfying
$$a_{k+1}\leq C_0b^ka_k^{1+\delta},$$
where $b>1$, $\delta >0$ and $C_0$ is a positive constant. Then, if $a_0\leq \theta _0:=C_0^{-\frac{1}{\delta}}b^{-\frac{1}{\delta^2}}$, we have $\lim_{k\rightarrow \infty}a_k=0$.
\end{lemma}

\begin{corollary}\label{corollary-de-giorgi}
Let $\phi :[r_0,\infty]\rightarrow\mathbb{R}^+$ be a nonnegative and decreasing function. Assume there exist constants $C_1>0$, $\alpha >0$ and $\varsigma >1$ such that for any $r_0<r<l$,
$$\phi(l)\leq \frac{C_1}{(l-r)^{\alpha}}\phi (r)^{\varsigma}.$$
Then for any $d$ satisfying
$$d\geq C_1^{\frac{1}{\alpha}}|\phi(r_0)|^{\frac{\varsigma-1}{\alpha}}2^{\frac{\varsigma}{\varsigma-1}},$$
we have $\phi(r_0+d)=0$.
\end{corollary}

The following embedding lemma is from \cite{qiutang}.
\begin{lemma}\label{lemma2.6}
If for each $t\in[0,T]$, $u\in\mathcal{V}_2(\mathcal{O}_t)$, then we have
$$\|u\|_{0,\frac{2(n+2)}{n};\mathcal{O}_t}\leq C\|\nabla u\|^{\frac{n}{n+2}}_{0,2;\mathcal{O}_t}\textrm{esssup}_{(\omega,s)\in\Omega\times[t,T]}\|u(\omega,s)\|^{\frac{n}{n+2}}\leq C\|u\|_{\mathcal{V}_2(\mathcal{O}_t)},$$
where $C$ only depends on $n$.
\end{lemma}

 Now, we are going to present the It\^o formulas, which have been frequently used in the main text. We assume that $\Phi$ is a function that satisfies the following properties:
 \begin{itemize}
\item[(1)] $\Phi\in \mathcal{C}(\mathbb{R}^+\times\mathbb{R}^{n}\times\mathbb{R}\rightarrow \mathbb{R})$ and $\partial_t\Phi(t,x,u)$, $\Phi'(t,x,u)$, $\Phi''(t,x,u)$ and $\partial_j\Phi'(t,x,u)$, $j=1,2,\cdots,n$ exist and are continuous;
\item[(2)] $\Phi'(t,x,0)=0$ for any $(t,x)\in \mathbb{R}^+\times\mathbb{R}^n$;
\item[(3)] $\sup_{t\in\mathbb{R}^{+},x\in\mathbb{R}^n}|\partial_j\Phi'(t,x,u)|\leq C |u|$, $j=1,2,\cdots,n$;
\item[(4)]
\begin{align*}
\sup\limits_{t\in\mathbb{R}^{+},x\in\mathbb{R}^n,u\in\mathbb{R}/\{0\}} \left\{|\Phi''(t,x,u)|+\frac{1}{|u|^2}|\partial_t\Phi(t,x,u)-\partial_t\Phi(t,x,0)|\right\}<\infty,
\end{align*}
where $\partial_j\Phi(t,x,u)=\partial_{x_j}\Phi(t,x,u)$, $\Phi'(t,x,u)=\partial_u\Phi(t,x,u)$ and $\Phi''(t,x,u)=\partial^2_u\Phi(t,x,u)$.
\end{itemize}
Suppose that the following BSPDE
\begin{equation}\label{appeneq}
\left\{\begin{array}{ll}
\begin{split}
-du(t,x)&=[\partial _j(a^{ij}\partial _iu(t,x)+\sigma ^{jr}v^r(t,x))
+\bar{f}(t,x)+\nabla \cdot \bar{g}(t,x)]\,dt \\
& +\mu (dt,x)-v^r(t,x)\,dW^r_t,
~~~(t,x)\in Q,\\
u(T,x)&=G(x),~~~x\in \mathcal{O},\\
\end{split}
\end{array}
\right.
\end{equation}
holds in the weak sense where $(u,v)\in\mathcal{V}_2(Q)\times\mathcal{M}^{0,2}(Q)$, $\mu$ is a stochastic regular measure, $\bar{f}$, $\bar{g}$ and $G$ satisfy $(\mathcal{A}_3)$, $a$ and $\sigma$ satisfy $(\mathcal{A}_2)$.

When $\Phi$ is independent of $x$, i.e., $\Phi(t,x,u)=\Phi(t,u)$, the first It\^o formula is from \cite[Theorem 3.10]{qiuwei}.

\begin{proposition}\label{proposition6.1}
Let BSPDE (\ref{appeneq}) hold in the weak sense with $u|_{\partial \mathcal{O}}=0$. Then there holds almost surely that
\begin{align*}
&\int_{\mathcal{O}}\Phi(t,u(t,x))\,dx+\frac{1}{2}\int_t^T\langle\Phi''(s,u(s)),|v(s)|^2\rangle \,ds\\
=& \int_{\mathcal{O}}\Phi(T,G(x))\,dx-\int_t^T\int_{\mathcal{O}}\partial_s\Phi(s,u(s,x))\,dxds+\int_t^T\langle \Phi'(s,u(s)),\bar{f}(s)\rangle \,ds\\
&-\int_t^T\langle \Phi''(s,u(s))\partial_{j}u(s),a^{ij}(s)\partial_{i}u(s)+\sigma^{jr}(s)v^r(s)+\bar{g}^j(s)\rangle \,ds\\
&+\int_{[t,T]\times\mathcal{O}}\Phi'(s,u(s,x))\,\mu(ds,dx)-\int_t^T\langle\Phi'(s,u(s)),v^r(s)\rangle \,dW_s^r.
\end{align*}
\end{proposition}

The following It\^o formula extends the preceding one to the positive parts of the weak solutions to BSPDEs.

\begin{theorem}\label{Ito-u+}
Let BSPDE (\ref{appeneq}) hold in the weak sense but with $u^+|_{\partial \mathcal{O}}=0$. Then there holds almost surely that
\begin{equation}\label{equation6.5}
\begin{split}
        &\int_{\mathcal{O}}\Phi(t,x,u^+(t,x))\,dx+\frac{1}{2}\int_t^T\langle \Phi''(s,u^+(s)),|v^u(s)|^2\rangle \,ds\\
        =& \int_{\mathcal{O}}\Phi(T,x,G^+(x))\,dx-\int_t^T\int_{\mathcal{O}}\partial_s\Phi(s,x,u^+(s,x))\,dxds\\
        &+\int_t^T\langle \Phi'(s,u^+(s)),\bar{f}^u(s)\rangle\,ds+\int_t^T\int_{\mathcal{O}}\Phi'(s,x,u^+(s,x))\,\mu(dsdx)\\
        &-\int_t^T\langle \Phi''(s,u^+(s))\partial_j u^+(s)+\partial_j \Phi'(s,u^+(s)),a^{ij}(s)\partial_i u^+(s)+\sigma^{j,r}(s)v^{r,u}(s)+\bar{g}^{j,u}(s)\rangle \,ds\\
        &-\int_t^T\langle\Phi'(s,u^+(s)),v^{r,u}(s)\rangle \,dW^r_s,
\end{split}
\end{equation}
where
$$v^{r,u}=1_{\{u>0\}}v^r,~~~~\bar{f}^u=1_{\{u>0\}}\bar{f},~~~~\bar{g}^{j,u}=1_{\{u>0\}}\bar{g}^j.$$
\end{theorem}



\begin{proof}
Note that in general we cannot get $u|_{\partial\mathcal{O}}=0$ from $u^+|_{\partial\mathcal{O}}=0$, so Proposition \ref{proposition6.1} is not applicable. Insread, we shall apply an approximation scheme similar to that for \cite[Theorem 3.10]{qiuwei}
%We apply an approximation scheme that allows us to deduce the desired results from Proposition \ref{proposition6.1}.

Let $\check{u}$ be the stochastic regular parabolic potential (see next subsection for the definition) associated with $\mu$. Now define
\begin{equation*}
\left\{\begin{array}{ll}
\begin{split}
-d\hat{u}(t,x)&=\left(-\Delta \hat{u}(t,x)+\bar{f}(t,x)+\nabla\cdot\hat{g}(t,x)\right)dt
-v^r(t,x)dW^r_t,\\
&~~~(t,x)\in Q,\\
\hat{u}(0,x)&=u(0,x),~~~x\in \mathcal{O},\\
\end{split}
\end{array}
\right.
\end{equation*}
where $\hat{g}^j(t,x)=\partial_j u(t,x)+a^{ij}\partial _iu(t,x)+\sigma ^{jr}v^r(t,x)+\bar{g}^j(t,x)$. Then, $u=\hat{u}-\check{u}$ and the zero Dirichlet conditions of $u^+$ and $\check{u}$ imply $\hat{u}^+|_{\partial\mathcal{O}}=0$. By \cite[Proposition 3.9(i)]{qiuwei} $u$ is almost surely quasi-continuous. So, the integral w.r.t. $\mu$ in (\ref{equation6.5}) is well defined. We can also check that all the other terms in (\ref{equation6.5}) are well defined.

%Our approximation scheme uses the following sequence of functions: for any $k=1,2, ...$ we put
%\begin{equation}
%\psi_k(z)=
%\left\{\begin{array}{ll}
%0,& z\in(-\infty,1/k];\\
%\frac{k^2}{10}(z-\frac{1}{k})^3, & z\in(1/k,2/k];\\
%-\frac{k^3}{5}(z-\frac{2}{k})^4+\frac{3k^2}{10}(z-\frac{2}{k})^3+\frac{3k}{10}(z-\frac{2}{k})^2+\frac{3}{10}z-\frac{1}{2k},&z\in (2/k,3/k];\\
%z-\frac{11}{5k},& z\in(3/k,\infty).\\
%\end{array}
%\right.
%\end{equation}
%The approximating functions $\psi_k$ have the following properties:
%\begin{itemize}
%	\item[(P-1)]: $|\psi_k(z)- z^+|\leq \frac{3}{k}$;
%	\item[(P-2)]: $|\psi'_k(z)- 1_{[0,+\infty)}(z)|\leq 1_{[0,\frac{3}{k}]}(z)$;
 %	\item[(P-3)]: $u^+|_{\partial\mathcal{O}}=0$ yields $\psi_k(u)|_{\partial\mathcal{O}}=0$ for $k$ large enough;
 %	\item[(P-4)]: $\psi_k(z)$ is convex on $[\frac{1}{k},\frac{3}{k}]$. So $\psi_k(\frac{1}{k})=0$ and $\psi_k(\frac{3}{k})<\frac{3}{k}$ imply that we always have $\psi_k(z)\leq |z|$ on $[\frac{1}{k},\frac{3}{k}]$ and thus on the whole real line.
	%\item[(P-5)]: $\psi_k'(z)$ is bounded.
%\end{itemize}

%Since (\ref{appeneq}) holds in the weak sense we have for each $(\varphi, \varsigma)\in\mathcal{C}_{0}^{\infty}(\mathcal{O})\times \mathcal{C}_{0}^{\infty}(\mathcal{O})$, $\varphi\geq 0$ that
%\begin{align*}
%\begin{split}
%&\langle u(t,\cdot)\varphi ,\varsigma\rangle+\int_t^T\langle \partial_j\varsigma,a^{ji}(s)\partial_i(\varphi u(s))+v^r(s)\sigma^{jr}(s)\varphi\rangle \,ds\\
%=&\langle G\varphi,\varsigma\rangle+\int_t^T\langle\bar{f}(s)\varphi,\varsigma\rangle \,ds-\int_t^T\langle \varphi\bar{g}(s),\nabla \cdot\varsigma\rangle ds-\int_t^T\langle \bar{g}(s),\varsigma\nabla\cdot\varphi\rangle \,ds\\
%&+\int_t^T\langle \partial_j\varsigma,a^{ij}(s)u(s)\partial_i\varphi\rangle \,ds-\int_t^T\langle \varsigma \partial_j\varphi,a^{ij}(s)\partial_iu(s)+\sigma^{jr}(s)v^r(s)\rangle \,ds\\
%&+\int_t^T\int_{\mathcal{O}}\varsigma(x)\varphi(x)\,\mu(dsdx)-\int_t^T\langle \varsigma,\varphi v^r(s)\rangle \,dW^r_s.
%\end{split}
%\end{align*}
%So
%\[
%	-d\bar{u}=[\partial_j(a^{ij}\partial_i\bar{u}+\sigma^{jr}\bar{v}^r)+\tilde{f}+\nabla \cdot\tilde{g}]\,dt+\bar{\mu}(dt,x)-\bar{v}^r\,dW^r_t
%\]
%holds in the weak sense, where $\bar{u}=u\varphi$, $\bar{v}=v\varphi$ and $\bar{\mu}=\varphi\mu$,
%\begin{equation*}
%\begin{split}
%\tilde{f}(t,x)=& \bar{f}(t,x)\varphi(x)-a^{ij}(t,x)\partial_j\varphi(x)\partial_iu(t,x)-\sigma^{jr}(t,x)\partial_j\varphi(x)v^r(t,x)-\partial_j\varphi(x)\bar{g}^j(t,x)
%\end{split}
%\end{equation*}
%and $\tilde{g}^i(t,x)=\varphi(x)g^i(t,x)-a^{ij}(t,x)\partial_j\varphi(x)u(t,x)$. {To apply Proposition \ref{proposition6.1} we need to} verify that $\bar{\mu}$ is still a stochastic regular measure. To this end, let $\tilde{u}\in\mathcal{V}_{2,0}(Q)$ be the unique solution to the following SPDE,
%\begin{equation}\label{checkstochasticregularmeasure1}
%\left\{\begin{array}{ll}
%d\tilde{u}(t,x)=(\Delta \tilde{u}-\tilde{f}-\nabla\cdot\breve{g})(t,x)\,dt+\bar{v}^r(t,x)\,dW^r_s,\\\\
%\tilde{u}(0,x)=\bar{u}(0,x),
%\end{array}
%\right.
%\end{equation}
%where $\breve{g}^i=\partial_i\bar{u}+a^{ij}\partial_j\bar{u}+\sigma^{ir}\bar{v}^r+\tilde{g}^i$. Then we have
%\begin{equation}\label{checkstochasticregularmeasure2}
%\bar{\mu}=\partial_t(\tilde{u}-\bar{u})-\Delta (\tilde{u}-\bar{u}),
%\end{equation}
 %from which and the fact $\tilde{u}(0,x)-\bar{u}(0,x)=0$ it is easy to check that $\bar{\mu}$ is still a stochastic regular measure.

%Since $\bar{u}|_{\partial \mathcal{O}}=0$, it follows from Proposition \ref{proposition6.1} that for any $\tilde{\phi}\in\mathcal{C}_0^{\infty}(\mathcal{O})$ and any $k\in \mathbb{N}^+$ we have almost surely that,
%\begin{align*}
%&\int_{\mathcal{O}}\psi_k(\bar{u}(t,x))\tilde{\phi}(x)\,dx+\frac{1}{2}\int_t^T\langle \psi_k''(\bar{u}(s))\tilde{\phi},|\bar{v}|^2(s)\rangle \,ds\\
%=& \int_{\mathcal{O}}\psi_k(\varphi(x)G(x))\tilde{\phi}(x)\,dx+\int_t^T\langle \psi_k'(\bar{u}(s))\tilde{\phi},\bar{f}(s)\rangle \,ds
%+\int_{[t,T]\times\mathcal{O}}\psi_k'(\bar{u}(s,x))\tilde{\phi}(x)\,\bar{\mu}(dsdx)\\
%& -\int_t^T\!\!\!\langle\psi_k''(\bar{u}(s))\tilde{\phi}\partial_{j}\bar{u}+\psi_k'(\bar{u}(s))\partial_{j}\tilde{\phi},a^{ij}(s)\partial_{x_i}\bar{u}(s)+\sigma^{jr}(s)\bar{v}^r(s)+\bar{g}^j(s)\rangle \,ds-\int_t^T\!\!\!\langle \psi_k'(\bar{u}(s))\tilde{\phi},\bar{v}^r(s)\rangle \,dW^r_s.
%\end{align*}
%Now for any $\mathcal{O}'\subset\subset\mathcal{O}$, taking $\varphi\equiv 1$ on $\mathcal{O}'$ and $supp(\tilde{\phi})\subset\mathcal{O}'$, it holds almost surely that
%\begin{align*}
%&\int_{\mathcal{O}}\psi_k(u(t,x))\tilde{\phi}(x)\,dx+\frac{1}{2}\int_t^T\langle \psi_k''(u(s))\tilde{\phi},|\bar{v}|^2(s)\rangle \,ds\\
%=& \int_{\mathcal{O}}\psi_k(G(x))\tilde{\phi}(x)\,dx+\int_t^T\langle \psi_k'(u(s))\tilde{\phi},\bar{f}(s)\rangle
%+\int_{[t,T]\times\mathcal{O}}\psi_k'(u(s,x))\tilde{\phi}(x)\,\mu(dsdx)
%\\
%& -\int_t^T\langle\psi_k''(u(s))\tilde{\phi}\partial_{x_j}u+\psi_k'(u(s))\partial_{j}\tilde{\phi},a^{ij}(s)\partial_{i}u(s)+\sigma^{jr}(s)v^r(s)+\bar{g}^j(s)\rangle \,ds\\
%&-\int_t^T\langle \psi_k'(u(s))\tilde{\phi},v^r(s)\rangle \,dW^r_s.
%\end{align*}
%Since $\mathcal{O}'$ and $\tilde{\phi}$ are arbitrary, there holds almost surely that
%\begin{align*}
%&\psi_k(u(t,x))+\frac{1}{2}\int_t^T\psi_k''(u(s,x))|v|^2(s,x)\,ds\\
%= & \psi_k(G(x))+\int_t^T\psi_k'(u(s,x))\bar{f}(s,x)\,ds+\int_t^T\psi_k'(u(s,x))\,\mu(ds,x)(ds,x)\\
%&+\int_t^T\psi_k'(u(s,x))\partial_{x_i}(a^{ij}(s,x)\partial_{x_j}u(s,x)+\sigma^{ir}(s,x)v^r(s,x)+g^i(s,x))\,ds
% -\int_t^T\psi_k'(u(s,x))v^r(s,x)\,dW^r_s
%\end{align*}
 %in the weak sense. In a similar way to (\ref{checkstochasticregularmeasure1}) and (\ref{checkstochasticregularmeasure2}), we check that {\color{blue}$\psi'_k(u)\mu$ is a stochastic regular measure??}. Denote by $\check{u}_k$ the corresponding stochastic regular potential.
  Thus, by Proposition 3.9(iv) and Remark 3.7 in \cite{qiuwei}, there exist $f^n\in \mathcal{L}^2([0,T];(H^{-1})^+(\mathcal{O}))$, $\check{v}^n\in \mathcal{L}^2([0,T];(L^2(\mathcal{O}))^m)$, $\check{u}^n\in \mathcal{U}(-\infty,f^{n}_1,g^{n}_1,G^n_1)$ and $\phi^n\in\mathcal{U}(-\infty,f^{n}_2,g^{n}_2,G^n_2)$, for some $f^{n}_i\in\mathcal{L}^2([0,T];L^2(\mathcal{O}))$, $g^{n}_i\in\mathcal{L}^2([0,T];(L^2(\mathcal{O}))^n)$, $G^n_i\in\mathcal{L}^2(\mathcal{O})$, $i=1,2$, such that $\phi^n\downarrow 0$ as $n\rightarrow\infty$, $dt\times dx\times d\mathbb{P}$ a.e., $\lim_{n\rightarrow\infty}\sum_{i=1}^m E\int_0^T\|\check{v}^{n,i}(t)\|^2\,dt=0$, $\lim_{n\rightarrow\infty}\|\check{u}^n-\check{u}\|_{\mathcal{L}^2(\mathcal{K})}=0$, $\lim_{n\rightarrow\infty}(\|f^n_2+\nabla\cdot g^n_2\|_{\mathcal{L}^2([0,T];H^{-1}(\mathcal{O}))}+\|G_2^n\|_{\mathcal{L}^2(\mathcal{O})})=0$, $|\check{u}^n-\check{u}|\leq \phi^n$ $dt\times dx\times d\mathbb{P}$ a.e., with $\check{u}^n$ satisfying the SPDE
    \begin{equation*}
    \begin{split}
    \left\{\begin{array}{ll}
      d\check{u}^n(t,x) = & ~ [\Delta \check{u}^n(t,x)+f^n(t,x)]\,dt+\check{v}^n(t,x)\,dW_t, ~~(t,x)\in Q\\
      \check{u}^n(0,x)  =& ~ 0,~~x\in\mathcal{O},\\
      \check{u}^n|_{\partial\mathcal{O}}=&~0.
      \end{array}
      \right.
    \end{split}
    \end{equation*}
Define $u^n:=\hat{u}-\check{u}^n$. Then
    \begin{align*}
      du^n(t,x)=-(-\Delta u^n(t,x)+\bar{f}(t,x)+f^n(t,x)+\nabla\cdot\hat{g}(t,x))\,dt+(v^r(t,x)-\check{v}^{n,r}(t,x))\,dW_t^r.
    \end{align*}

 Moreover, $|(u^n)^+-u^+|\leq \phi^n$ $dt\times dx\times d\mathbb{P}$ a.e.. The zero Dirichlet conditions of $\check{u}^n$ and $\hat{u}^+$ imply $(u^n)^+|_{\partial\mathcal{O}}=0$. By \cite[Lemma 3.5]{qiutang}, we have almost surely
\begin{align}\label{equation6.13}
&\int_{\mathcal{O}}\Phi(t,x,(u^n)^+(t,x))\,dx+\frac{1}{2}\int_t^T\langle \Phi''(s,(u^n)^+(s)),|(v(s)-\check{v}^n(s))1_{\{u^n>0\}}|^2\rangle \,ds\nonumber\\
=& \int_{\mathcal{O}}\Phi(T,x,(u^n)^+(T,x))\,dx-\int_t^T\int_{\mathcal{O}}\partial_s\Phi(s,x,(u^n)^+(s,x))\,dxds\nonumber\\
&+\int_t^T\langle \Phi'(s,(u^n)^+(s)),\bar{f}(s)1_{\{u^n>0\}}\rangle\,ds+\int_t^T\langle\Phi'(s,(u^n)^+(s)),f^n(s)1_{\{u^n>0\}}\rangle _{1,-1}\,ds\nonumber\\
&-\int_t^T\langle \Phi''(s,(u^n)^+(s))\partial_j (u^n)^+(s)+\partial_j \Phi'(s,(u^n)^+(s)),-\partial_{j}(u^n)^+(s)+\hat{g}^j(s)1_{\{u^n>0\}}\rangle \,ds\nonumber\\
&-\int_t^T\langle\Phi'(s,(u^n)^+(s)),(v^r(s)-\check{v}^{n,r}(s))1_{\{u^n>0\}}\rangle \,dW^r_s,\quad \forall t\in[0,T].
\end{align}
By \cite[Corollary 3.5]{qiuwei}, there exists $\bar{u}\in\mathcal{L}^2(\mathcal{P})$ such that
    \begin{align}\label{U-controlled-by-potential}
      |\hat{u}|+\phi^1 \leq \bar{u},~~dt\times dx\times d\mathbb{P}~ a.e..
    \end{align}
By (\ref{U-controlled-by-potential}) and the properties (2) and (4) of $\Phi$, there holds $dt\times dx\times d\mathbb{P}$ a.e. that
    \begin{align}\label{boundedness-Phi-prime}
      |\Phi'(t,x,(u^n)^+(t,x))| = & ~|\Phi'(t,x,(u^n)^+(t,x))-\Phi'(t,x,0)|  \nonumber\\
       \leq & C|u^n(t,x)|\nonumber\\
       = &~ C|\hat{u}(t,x)-\check{u}^n(t,x)|\nonumber\\
       \leq &~ C|\hat{u}(t,x)|+C|\check{u}(t,x)|+C|\check{u}(t,x)-\check{u}^n(t,x)|\nonumber\\
        \leq &~ C|\hat{u}(t,x)|+C|\check{u}(t,x)|+C\phi^n(t,x)\nonumber\\
        \leq &~ C(|\check{u}(t,x)|+\bar{u}(t,x)).
    \end{align}
By property (4) of $\Phi$, there holds $dt\times dx\times d\mathbb{P}$ a.e. that
    \begin{align}\label{continuity-Phi-prime}
      |\Phi'(t,x,(u^n)^+(t,x))-\Phi'(t,x,u^+(t,x))| \leq & ~C|(u^n)^+(t,x)-u^+(t,x)| \nonumber\\
      \leq & C|\check{u}^n(t,x)-\check{u}(t,x)|\nonumber\\
      \leq & C\phi^n(t,x).
    \end{align}
(\ref{boundedness-Phi-prime}), (\ref{continuity-Phi-prime}) and \cite[Proposition 3.9(ii)]{qiuwei} yield that
    \begin{align*}
        &\lim_{n\rightarrow\infty} \int_t^T\langle\Phi'(s,(u^n)^+(s)),f^n(s)1_{\{u^n>0\}}\rangle _{1,-1}\,ds  \\
     = & ~ \int_{\mathcal{O}_t}\Phi'(s,x,u^+(s,x))\,\mu(dsdx)~a.s..
    \end{align*}
Moreover,
    \begin{align*}
     & E\sup_{0\leq t \leq T}\left|\int_t^T\langle\Phi'(s,(u^n)^+(s)),(v^r(s)-\check{v}^{n,r}(s))1_{\{u^n>0\}}\rangle \,dW^r_s-\int_t^T\langle\Phi'(s,u^+(s)),v^r(s)1_{\{u>0\}}\rangle \,dW^r_s\right|  \\
      \leq & ~CE\left(\int_0^T \left|\langle\Phi'(s,(u^n)^+(s)),(v^r(s)-\check{v}^{n,r}(s))1_{\{u^n>0\}}\rangle -\langle\Phi'(s,u^+(s)),v^r(s)1_{\{u>0\}}\rangle\right|^2\,ds\right)^{\frac{1}{2}}\\
      \leq &~ CE\left(\int_0^T \left|\langle\Phi'(s,(u^n)^+(s)),v(s)1_{\{u^n>0\}}\rangle -\langle\Phi'(s,u^+(s)),v(s)1_{\{u>0\}}\rangle\right|^2\,ds\right)^{\frac{1}{2}}\\
      &+CE\left(\int_0^T \left|\langle\Phi'(s,(u^n)^+(s))-\Phi'(s,0),\check{v}^{n}(s)1_{\{u^n>0\}}\rangle\right|^2\,ds\right)^{\frac{1}{2}}\\
      \leq & C\left(E\textrm{esssup}_{0\leq t\leq T}\|(u^n)^+(t)-u^+(t)\|^2\right)^{\frac{1}{2}}\left(E\int_0^T\|v(t)\|^2\,dt\right)^{\frac{1}{2}}\\
      &+C\left(E\textrm{esssup}_{0\leq t\leq T}\|(u^n)^+(t)\|^2\right)^{\frac{1}{2}}\left(E\int_0^T\|v(t)(1_{\{u^n>0\}}-1_{\{u>0\}})\|^2\,dt\right)^{\frac{1}{2}}\\
      &+C\left(E\textrm{esssup}_{0\leq t\leq T}\|u^n(t)\|^2\right)^{\frac{1}{2}}\left(E\int_0^T\|\check{v}^n(t)\|^2\,dt\right)^{\frac{1}{2}}\\
      \leq & C\left(E\|u^n-u\|^2_{\mathcal{K}}\right)^{\frac{1}{2}}\left(E\int_0^T\|v(t)\|^2\,dt\right)^{\frac{1}{2}}\\
      &+C\left(E\|u^n\|^2_{\mathcal{K}}\right)^{\frac{1}{2}}\left(E\int_0^T\|v(t)(1_{\{u^n>0\}}-1_{\{u>0\}})\|^2\,dt\right)^{\frac{1}{2}}\\
      &+C\left(E\textrm{esssup}_{0\leq t\leq T}\|u^n(t)\|^2\right)^{\frac{1}{2}}\left(E\int_0^T\|\check{v}^n(t)\|^2\,dt\right)^{\frac{1}{2}}\\
      \rightarrow & ~0.
    \end{align*}
By the properties of $\Phi$ and the fact that $|(u^n)^+-u^+|\leq \phi^n$ $dt\times dx\times d\mathbb{P}$ a.e., the convergence of other terms can be treated analogously. Finally by letting $n\rightarrow\infty$, we obtain almost surely that
    \begin{align*}
        &\int_{\mathcal{O}}\Phi(t,x,u^+(t,x))\,dx+\frac{1}{2}\int_t^T\langle \Phi''(s,u^+(s)),|(v(s)1_{\{u>0\}}|^2\rangle \,ds\nonumber\\
        =& \int_{\mathcal{O}}\Phi(T,x,u^+(T,x))\,dx-\int_t^T\int_{\mathcal{O}}\partial_s\Phi(s,x,u^+(s,x))\,dxds\nonumber\\
        &+\int_t^T\langle \Phi'(s,u^+(s)),\bar{f}(s)1_{\{u>0\}}\rangle\,ds+\int_t^T\int_{\mathcal{O}}\Phi'(s,x,u^+(s,x))\,\mu(dsdx)\nonumber\\
        &-\int_t^T\langle \Phi''(s,u^+(s))\partial_j u^+(s)+\partial_j \Phi'(s,u^+(s)),-\partial_{j}u^+(s)+\hat{g}^j(s)1_{\{u^n>0\}}\rangle \,ds\nonumber\\
        &-\int_t^T\langle\Phi'(s,u^+(s)),v^r(s)1_{\{u>0\}}\rangle \,dW^r_s,\quad \forall t\in[0,T].
    \end{align*}
%By mean value theorem, there exists $\eta(k,t,x)$ between $\psi_k(u(t,x))$ and $u^+(t,x)$ such that
%\begin{align*}
%&\left|\int_{\mathcal{O}}\Phi(t,\psi_k(u(t,x)))\,dx-\int_{\mathcal{O}}\Phi(t,u^+(t,x))\,dx\right|\\
%=& \left|\int\Phi'(t,\eta(k,t,x))(\psi_k(u(t,x))-u^+(t,x))\,dx\right|\\
%\leq &\int_{\mathcal{O}}|\Phi'(t,\eta(k,t,x))||\psi_k(u(t,x))-u^+(t,x)|\,dx\\
%= & \int_{\mathcal{O}}|\Phi'(t,\eta(k,t,x))-\Phi'(t,0)||\psi_k(u(t,x))-u^+|\,dx\qquad(by ~the ~property ~(2)~of~ \Phi)\\
%= & \int_{\mathcal{O}}|\Phi''(t,\tilde{\eta}(k,t,x))||\eta(k,t,x)||\psi_k(u)-u^+|\,dx\qquad(by~mean~value~theorem ~and~the ~property ~(3)~of~ \Phi )\\
%\leq & C\int_{\mathcal{O}}|\psi_k(u)+u^+||\psi_k(u)-u^+|\,dx\qquad(by~the ~property ~(3)~of~ \Phi )\\
%\leq & C\int_{\mathcal{O}}|u^+(t,x)|^2\,dx .
%\end{align*}
%Thus by (P-1) and dominated convergence theorem we have
%\begin{align*}
%\int_{\mathcal{O}}\Phi(t,\psi_k(u(t,x)))dx-\int_{\mathcal{O}}\Phi(t,u^+(t,x))\,dx \rightarrow 0, ~~dt\times d\mathbb{P}-a.e.
%\end{align*}
%and similarly,
%\begin{align*}
%\int_{\mathcal{O}}\Phi(t,\psi_k(G(x)))dx-\int_{\mathcal{O}}\Phi(t,G^+(x))\,dx \rightarrow 0, ~~dt\times d\mathbb{P}-a.e..
%\end{align*}

%Moreover,
%\begin{align*}
%&E\left|\int_t^T\langle \Phi''(s,\psi_k(u(s))),|\psi_k'(u(s))v(s)|^2\rangle ds-\int_t^T\langle \Phi''(s,u^+(s)),|v^u(s)|^2\rangle\, ds\right|\nonumber\\
%\leq & E\int_t^T\int_{\mathcal{O}}|\Phi''(s,\psi_k(u(s,x)))-\Phi''(s,u^+(s,x))||v^u(s)|^2\, dxds\nonumber\\
%&+ E\int_t^T\int_{\mathcal{O}}|\Phi''(x,u^+(s,x))|\left|\psi_k'(u(s,x))|^2-1_{\{u>0\}}\right||v(s,x)|^2\,dxds\nonumber\\
%\leq & CE\int_t^T\int_{\mathcal{O}}|v(s,x)|^2\, dxds\qquad (by ~the ~property ~(3) ~of ~\Phi ~and ~(\textrm{P-5}))\nonumber\\
%< & \infty.
%\end{align*}
%Then, by (P-1), (P-2) and dominated convergence,
%\begin{align*}
%&E\left|\int_t^T\langle \Phi''(s,\psi_k(u(s))),|\psi_k'(u(s))v(s)|^2\rangle \,ds-\int_t^T\langle \Phi''(s,u^+(s)),|v^u(s)|^2\rangle\, ds\right|\rightarrow 0.
%\end{align*}
%Furthermore, by the BDG inequality, one has
%\begin{align*}
%&E\sup_{0\leq t\leq T}\left|\int_t^T\langle \Phi'(s,\psi_k(u(s))),\psi_k'(u(s))v^r(s)\rangle dW_s^r-\int_t^T\langle\Phi'(s,u^+(s)),1_{\{u>0\}}v^r(s)\rangle \, dW_s^r\right|^2\\
%\leq & CE\left(\int_0^T|\langle\Phi'(s,\psi_k(u(s)))v(s),\psi'_k(u(s))-1\rangle|^2\,ds\right)\\
%&+CE\left(\int_0^T|\langle\Phi'(s,\psi_k(u(s)))-\Phi'(s,u^+(s)),v(s)\rangle|^2\, ds\right).
%\end{align*}
%By (P-4), (P-5), the property (2) of $\Phi$ and the mean value theorem,
%\begin{align*}
%&E\left(\int_t^T|\langle\Phi'(s,\psi_k(u(s)))v(s),\psi'_k(u(s))-1\rangle|^2ds\right)\nonumber\\
%\leq & E\int_0^T\left(\int_{\mathcal{O}}|\Phi'(s,u^+(s,x))|^2\,dx\int_{\mathcal{O}}|\psi'_k(u(s,x))-1|^2|v(s,x)|^2\,dx\right)\,ds\nonumber\\
%\leq & C\textrm{esssup}_{\Omega\times[0,T]}\|u^+\|^2E\int_0^T\int_{\mathcal{O}}|v|^2\,dxds<\infty
%\end{align*}
%and
%\begin{align*}
%&E\left(\int_0^T|\langle\Phi'(s,\psi_k(u(s)))-\Phi'(s,u^+(s)),v(s)\rangle|^2 \,ds\right)\\
%\leq & E\left(\int_0^T\|\Phi'(s,\psi_k(u(s)))-\Phi'(s,u^+(s))\|^2\|v(s)\|^2\,ds\right)\\
%\leq & CE\int_0^T\|u(s)\|^2\|v(s)\|^2\,ds\\
%\leq & C\textrm{esssup}_{\Omega\times[0,T]}\|u\|^2E\int_0^T\int_{\mathcal{O}}|v|^2\,dxds<\infty.
%\end{align*}
%Thus by (P-1), (P-2) and dominated convergence we get that
%\begin{align*}
%&E\sup_{0\leq t\leq T}\left|\int_t^T\langle \Phi'(s,\psi_k(u(s))),\psi_k'(u(s))v^r(s)\rangle dW_s^r-\int_t^T\langle\Phi'(s,u^+(s)),1_{\{u>0\}}v^r(s)\rangle \, dW_s^r\right|^2\rightarrow 0.
%\end{align*}
%By properties (2) (3) of $\Phi$, (P-4), (P-5) and the mean value theorem, we have for $i=1,2$ that
%\begin{align*}
%&\left|\int_t^T\int_{\mathcal{O}}\Phi'(s,\psi_k(u(s,x)))\psi'_k(u(s,x))\,\mu_i(ds,dx)-\int_t^T\int_{\mathcal{O}}\Phi'(s,u^+(s,x))1_{\{u>0\}}\,\mu_i(ds,dx)\right|\nonumber\\
%\leq & \int_t^T\int_{\mathcal{O}}\left|\Phi'(s,\psi_k(u(s,x)))-\Phi'(s,u^+(s,x))\right|\left|\psi'_k(u(s,x))\right|\,\mu_i(ds,dx)\nonumber\\
%&+\int_t^T\int_{\mathcal{O}}\left|\Phi'(s,u^+(s,x))\right|\left|\psi'_k(u(s,x))-1_{\{u>0\}}\right|\,\mu_i(ds,dx)\nonumber\\
%\leq & C\int_0^T\int_{\mathcal{O}}u^+(s,x)\,\mu_i(ds,dx).
%\end{align*}
%Therefore by (P-1), (P-2) and dominated convergence,
%\begin{align}
%&\int_t^T\int_{\mathcal{O}}\Phi'(s,\psi_k(u(s,x)))\psi'_k(u(s,x))\,\mu_i(ds,dx)\rightarrow\int_t^T\int_{\mathcal{O}}\Phi'(s,u^+(s,x))\,\mu_i(ds,dx),~~~i=1,2.\nonumber
%\end{align}
%By the above arguments we can also get
%\begin{align*}
%\int_t^T\int_{\mathcal{O}}\partial_s\Phi(s,\psi_k(u(s,x)))\,dxds\rightarrow\int_t^T\int_{\mathcal{O}}\partial_s\Phi(s,u^+(s,x))\,dxds
%\end{align*}
%and
%\begin{align}
%&\int_t^T\langle \Phi'(s,\psi_k(u(s))),\psi'_k(u(s))\bar{f}(s)+\psi'_k(u(s))\partial_{i}(a^{ij}(s)\partial_{j}u(s)+\sigma^{ir}(s)v^r(s)+\bar{g}^i(s))\rangle \,ds\nonumber\\
%\rightarrow & \int_t^T\langle \Phi'(s,u^+(s,x)),\bar{f}^u(s)\rangle \,ds\nonumber\\
%&-\int_t^T\langle \Phi''(s,u^+(s))\partial_{j}u^+(s),a^{ij}(s)\partial_{i}u^+(s)+\sigma^{jr}(s)v^{r,u}(s)+\bar{g}^{i,u}(s)\rangle ds\nonumber
%\end{align}


%Finally we get (\ref{equation6.5}) by taking limit $k\rightarrow\infty$ in (\ref{equation6.13}).
\end{proof}
%The following theorem is It\^o formula in local case.


%\begin{theorem}\label{theorem6.4}
%Assume $\mathcal{A}_1$-$\mathcal{A}_4$ hold. $\zeta$ is a cut-off function on $Q_{\rho,\tau}:=[t_0-\tau,t_0]\times B_{\rho}(x_0)\subset Q$ and %${B_{\rho}(x_0)}$ is a ball centering at $x_0$ with radius $\rho$, where $\rho$, $\tau\in(0,1)$. The triple $(u,v,\mu)$ is the weak solution to RBSPDE %(1.1) as defined in Definition \ref{definition3.1}. Then we have
%\begin{align}\label{equation5.1}
%&\|(u\zeta)^+(s)\|^2_{L^2(B_{\rho}(x_0))}+\int_t^{t_0}\|\zeta(s)v^u(s)\|^2_{L^2(B_{\rho}(x_0))}ds\nonumber\\
%&=-2\int_t^{t_0}\langle\zeta(s)\partial _s\zeta(s),|u^+(s)|^2\rangle_{B_{\rho}(x_0)}ds
%+2\int_t^{t_0}\langle\zeta^2(s)u^+(s),f^u(s,u(s),\nabla u(s),v(s))\rangle_{B_{\rho}(x_0)}ds\nonumber\\
%&-2\int_t^{t_0}\langle\partial _j(\zeta^2(s)u^+(s)),a^{ij}(s)\partial_iu(s)+\sigma ^{jr}(s)v^{u,r}(s)
%+g^{u,j}(s,u(s),\nabla u(s),v(s))\rangle_{B_{\rho}(x_0)}ds\nonumber\\
%&-2\int_t^{t_0}\langle\zeta^2(s)u^+(s),v^u(s)\rangle_{B_{\rho}(x_0)}dW_s
%+2\int_t^{t_0}\int_{B_{\rho}(x_0)}u^+(s,x)\zeta^2\mu(ds,dx),
%\end{align}
%where $v^u:=v1_{\{u>0\}}$, $f^u:=f1_{\{u>0\}}$ and $g^u:=f1_{\{u>0\}}$.
%\end{theorem}

%\begin{proof}

%\end{proof}

%\subsection{Maximum Principle for Linear BSPDE within General Domain}

%This subsection deals with the following linear BSPDE,
%\begin{equation}\label{linearBSPDE}
%\left\{\begin{array}{ll}
%\begin{split}
%-du(t,x)&=[\partial _j(a^{ij}\partial _iu(t,x)+\sigma ^{jr}v^r(t,x))
%+f(t,x)+\nabla \cdot g(t,x)]dt
%-v^r(t,x)dW^r_t,\\
%&~~~\quad (t,x)\in Q,\\
%u(T,x)&=G(x),~~~x\in \mathcal{O},
%\end{split}
%\end{array}
%\right.
%\end{equation}
%with the random functions $f$, $g$ and $G$ satisfy $G\in L^{\infty}(\Omega , \mathcal{F}_T, L^2(\mathcal{O}))\cap L^{\infty}(\Omega,\mathcal{O})$, $f\in \mathcal{M}^{0,\frac{p(n+2)}{p+n+2}}(Q)\cap\mathcal{M}^{0,2}(Q)$ and $g\in \mathcal{M}^{0,p}(Q)\cap\mathcal{M}^{0,2}(Q)$ and $a$ and $\sigma$ satisfying $(\mathcal{A}_2)$.

%The existence and uniqueness of solutions to BSPDE (\ref{linearBSPDE}) is obvious due to Theorem \ref{theorem3.2} if we let the obstacle $\xi\rightarrow-\infty$. Now we state the maximum principle as follows, whose proof is similar to that of Theorem \ref{Theorem_MP_GD}, but we still prove it due to the convenience of the readers.
%\begin{lemma}\label{MP_GD_L}
%Under the above conditions for the coefficients in the linear BSPDE (\ref{linearBSPDE}), there is a positive constant $C$ such that
%\begin{align}
%\textrm{esssup}_{\Omega\times Q}u^{\pm}&\leq C\left(\textrm{esssup}_{\Omega\times\partial_{p}Q}u^{\pm}+\|f^{\pm}\|_{0,2;Q}+\|g\|_{0,2;Q}+\|f^{\pm}\|_{0,\frac{p(n+2)}{p+n+2};Q}+\|g\|_{0,p;Q}\right),
%\end{align}
%where $C$ only depends on $\lambda$, $\varrho$, $n$ and $p$.
%\end{lemma}
%\begin{proof}
%Letting $k_m=k(2-2^{-m})$ with $m\in \mathbb{N}$ and $k\geq \textrm{esssup}_{\Omega\times\partial_{p}Q}u^+$, for each $t\in[0,T]$ there holds almost surely by the It\^o formula (\ref{equation6.5}),
%\begin{align}
%&\|(u-k_m)^+(t)\|^2+\int_t^T\|v^{k_m}(s)\|^2ds
%\nonumber
%\\
%&=-2\int_t^T\langle\partial_j(u-k_m)^+(s),a^{ij}\partial_i(u-k_m)^+(s)+\sigma ^{jr}(s)v^{k_m,r}{s}\rangle\, ds\nonumber\\
%&-2\int_t^T\langle\partial_j(u-k_m)^+(s),g^{j}(s)\rangle\, ds\nonumber+2\int_t^T\langle (u-k_m)^+(s),f(s)\rangle \,ds\\
%&-2\int_t^T\langle (u-k_m)^+(s),v^{k_m,r}(s)dW^r_s\rangle,
%\end{align}
%where $v^{k_m}:=v1_{\{u>k_m\}}$. Taking conditional expectations on both sides w.r.t. $\mathcal{F}_t$, with similar estimate for $I_1$ in the proof of Theorem \ref{theorem3.2} we get
%\begin{align}
%B_1:=&-2E\left[\int_t^T\langle\partial_j(u-k_m)^+(s),a^{ij}\partial_i(u-k_m)^+(s)+\sigma ^{jr}(s)v^{k_m,r}{s}\rangle ds|\mathcal{F}_t\right]\nonumber\\
%\leq &-\lambda E\left[\int_t^T\|\nabla(u-k_m)^+(s)\|^2ds|\mathcal{F}_t\right]+\frac{1}{\varrho}E\left[\int_t^T\|v^{k_m}(s)\|^2ds|\mathcal{F}_t\right].
%\end{align}
%From the fact that
%\begin{align*}
%\left[(u-k_{m-1})^+-(u-k_m)^+\right]1_{\{u>k_m\}}=(k_m-k_{m-1})1_{\{u>k_m\}}=2^{-m}k1_{\{u>k_m\}}
%\end{align*}
%we have
%\begin{align}\label{estimate_indicator}
%1_{\{u>k_m\}}=\frac{2^m(u-k_{m-1})^+}{k}1_{\{u>k_m\}}\leq\frac{2^m(u-k_{m-1})^+}{k}.
%\end{align}
%By (\ref{estimate_indicator}) we have
%With similar approach to $J_2$, we have
%\begin{align}
%B_2:=&-2E\left(\int_t^T\langle\partial_j(u-k_m)^+(s),g^{j}(s)\rangle\, ds|\mathcal{F}_t\right)\nonumber\\
%\leq & \left(E\left[\int_t^T\|\nabla(u-k_m)^+\|^2ds|\mathcal{F}_t\right]\right)^{\frac{1}{2}}\left(E\left[\int_t^T\int_{\mathcal{O}}|g(s,x)1_{\{u>k_m\}}|^2dxds|\mathcal{F}_t\right]\right)^{\frac{1}{2}}\nonumber\\
%\leq & \|\nabla (u-k_m)^+\|_{0,2;\mathcal{O}_t}\|g\|_{0,p;\mathcal{O}_t}\left(E\left[\int_t^T\int_{\mathcal{O}}1_{\{u>k_m\}}dxds|\mathcal{F}_t\right]\right)^{\frac{1}{2}-\frac{1}{p}}\nonumber\\
%\leq & \|\nabla (u-k_m)^+\|_{0,2;\mathcal{O}_t}\|g\|_{0,p;\mathcal{O}_t}\left(E\left[\int_t^T\int_{\mathcal{O}}\left(\frac{2^m(u-k_{m-1})^+}{k}\right)^{\frac{2(n+2)}{n}}dxds|\mathcal{F}_t\right]\right)^{\frac{1}{2}-\frac{1}{p}}\nonumber\\
%\leq & \left(\frac{2^m}{k}\right)^{1+\frac{2(p-n-2)}{np}}\|\nabla (u-k_m)^+\|_{0,2;\mathcal{O}_t}\|g\|_{0,p;\mathcal{O}_t}\|(u-k_{m-1})^+\|_{0,\frac{2(n+2)}{n};\mathcal{O}_t}^{1+\frac{2(p-n-2)}{np}}\nonumber\\
%\leq & \left(\frac{2^m}{k}\right)^{1+\frac{2(p-n-2)}{np}}\|\nabla (u-k_{m-1})^+\|_{0,2;\mathcal{O}_t}\|g\|_{0,p;\mathcal{O}_t}\|(u-k_{m-1})^+\|_{0,\frac{2(n+2)}{n};\mathcal{O}_t}^{1+\frac{2(p-n-2)}{np}}.\nonumber\\
%\end{align}
%With similar approach to $J_3$, we have
%\begin{align}
%B_3:=&2E\left(\int_t^T\langle (u-k_m)^+(s),f^+(s)\rangle \,ds|\mathcal{F}_t\right)\nonumber\\
%\leq & \left(E\left[\int_t^T\int_{\mathcal{O}}|(u-k_m)^+|^{\frac{2(n+2)}{n}}dxds|\mathcal{F}_t\right]\right)^{\frac{n}{2(n+2)}}\left(E\left[\int_t^T\int_{\mathcal{O}}|f^+|^{\frac{p(n+2)}{p+n+2}}dxds|\mathcal{F}_t\right]\right)^{\frac{p+n+2}{p(n+2)}}\nonumber\\
%&\times\left(E\left[\int_t^T\int_{\mathcal{O}}1_{\{u>k_m\}}dxds|\mathcal{F}_t\right]\right)^{\frac{1}{2}-\frac{1}{p}}\nonumber\\
%\leq & \|(u-k_m)^+\|_{0,\frac{2(n+2)}{n};\mathcal{O}_t}\|f^+\|_{0,\frac{p(n+2)}{p+n+2};\mathcal{O}_t}\left(E\left[\int_t^T\int_{\mathcal{O}}\left(\frac{2^m(u-k_{m-1})^+}{k}\right)^{\frac{2(n+2)}{n}}dxds|\mathcal{F}_t\right]\right)^{\frac{1}{2}-\frac{1}{p}}\nonumber\\
%\leq & \left(\frac{2^m}{k}\right)^{1+\frac{2(p-n-2)}{np}}\|(u-k_m)^+\|_{0,\frac{2(n+2)}{n};\mathcal{O}_t}\|(u-k_{m-1})^+\|_{0,\frac{2(n+2)}{n};\mathcal{O}_t}^{1+\frac{2(p-n-2)}{np}}\|f^+\|_{0,\frac{p(n+2)}{p+n+2};\mathcal{O}_t}\nonumber\\
%\leq & \left(\frac{2^m}{k}\right)^{1+\frac{2(p-n-2)}{np}}\|(u-k_{m-1})^+\|_{0,\frac{2(n+2)}{n};\mathcal{O}_t}^{2+\frac{2(p-n-2)}{np}}\|f^+\|_{0,\frac{p(n+2)}{p+n+2};\mathcal{O}_t}.
%\end{align}
%Combining the estimates above, we get
%\begin{align}
%&\|(u-k_m)^+\|^2_{\mathcal{V}_2(\mathcal{O}_t)}+\|v^{k_m}\|^2_{0,2;\mathcal{O}_t}\nonumber\\
%\leq& C(\lambda,\varrho)\left(\frac{2^m}{k}\right)^{1+\frac{2(p-n-2)}{np}}\left(\|\nabla(u-k_{m-1})^+\|_{0,2;\mathcal{O}_t}\|(u-k_{m-1})^+\|_{0,\frac{2(n+2)}{n};\mathcal{O}_t}^{1+\frac{2(p-n-2)}{np}}\|g\|_{0,p;\mathcal{O}_t}\right.\nonumber\\
%&\left.+\|(u-k_{m-1})^+\|_{0,\frac{2(n+2)}{n};\mathcal{O}_t}^{2+\frac{2(p-n-2)}{np}}\|f^+\|_{0,\frac{p(n+2)}{p+n+2};\mathcal{O}_t}\right).
%\end{align}
%By the embedding lemma \ref{lemma2.6} and the definition of $\mathcal{V}_2(\mathcal{O}_t)$, we have
%\begin{equation}
%\|(u-k_m)^+\|^2_{\mathcal{V}_2(\mathcal{O}_t)}+\|v^{k_m}\|^2_{0,2;\mathcal{O}_t}\leq C(\lambda,\varrho)\left(\frac{2^m}{k}\right)^{1+\frac{2(p-n-2)}{np}}\left(\|g\|_{0,p;\mathcal{O}_t}+\|f^+\|_{0,\frac{p(n+2)}{p+n+2};\mathcal{O}_t}\right)\|(u-k_{m-1})^+\|_{\mathcal{V}_2(\mathcal{O}_t)}^{2+\frac{2(p-n-2)}{np}}.
%\end{equation}
%Let $\phi(m)=\|(u-k_m)^+\|^2_{\mathcal{V}_2(\mathcal{O}_t)}$ and $k\geq \|g\|_{0,p;\mathcal{O}_t}+\|f^+\|_{0,\frac{p(n+2)}{p+n+2};\mathcal{O}_t}$, then we have
%\begin{equation}\label{De Giorgi iteration}
%\phi(m)\leq C(\lambda,\varrho)\frac{2^{\left(1+\frac{2(p-n-2)}{np}\right)m}}{k^{\frac{2(p-n-2)}{np}}}\phi(m-1)^{1+\frac{p-n-2}{np}}.
%\end{equation}
%In Lemma \ref{lemma4.1}, by taking $\delta=\frac{p-n-2}{np}>0$, $b=2^{1+\frac{p-n-2}{np}}>1$ and $C_0=C(\lambda,\varrho)\frac{2^{1+\frac{2(p-n-2)}{np}}}{k^{\frac{2(p-n-2)}{np}}}$ we can get
%$$\lim\limits_{m\rightarrow\infty}\phi(m)=0$$
%if we can verify $\phi(0) \leq C_0^{-\frac{1}{\delta}}b^{-\frac{1}{\delta^2}}$, which is implied by letting $k\geq C(\lambda,\varrho,p,n)\|u^+\|_{\mathcal{V}_2(\mathcal{O}_t)}$. From $\lim\limits_{m\rightarrow\infty}\phi(m)=0$ we have $\textrm{esssup}_{\Omega\times\mathcal{O}_t}u^+\leq 2k$, together with the above estimates for $k$ and (\ref{estimate_u+_2}) we get
%\begin{align}
%\textrm{esssup}_{\Omega\times\mathcal{O}_t}u^+\leq & C\left(\|u^+\|_{\mathcal{V}_2(\mathcal{O}_t)}+\textrm{esssup}_{\Omega\times\partial_p\mathcal{O}_t}u^++\|g\|_{0,p;\mathcal{O}_t}+\|f^+\|_{0,\frac{p(n+2)}{p+n+2};\mathcal{O}_t}\right)\nonumber\\
%\leq & C\left(\textrm{esssup}_{\Omega\times\partial_p\mathcal{O}_t}u^++\|g\|_{0,2;\mathcal{O}_t}+\|f^+\|_{0,2;\mathcal{O}_t}+\|g\|_{0,p;\mathcal{O}_t}+\|f^+\|_{0,\frac{p(n+2)}{p+n+2};\mathcal{O}_t}\right).
%\end{align}
%The result for $u^-$ can be obtained in a similar way.
%\end{proof}
\section{Some definitions associated with stochastic regular measures}\label{appendix-measure}

In general the random measure $\mu$ in (\ref{equation1.1}) can be a local time, which is not absolutely continuous w.r.t. Lebesgue measure. Hence, the Skorokhod condition $\int_Q(u-\xi)\,\mu(dt,dx)=0$ might not make sense. To give a precise meaning to the Skorohod condition, the theory of parabolic potential and capacity introduced by \cite{pierre,pierre1980} was generalized by \cite{qiuwei} to a backward stochastic framework. This subsection recalls the notion of quasi continuity  and stochastic regular measure, which are repeatedly used in the main text and in the proof of Theorem \ref{Ito-u+}. Moreover, spaces used in the proof of Theorem \ref{Ito-u+} are also presented.

First some spaces are introduced. Denote by $H_0^1(\mathcal{O})$ the first order Sobolev space vanishing on the boundary $\partial\mathcal{O}$ equipped with the norm $\|\upsilon\|^2_1:=\|\upsilon\|^2+\|\nabla\upsilon\|^2$ and by $H^{-1}(\mathcal{O})$ the dual space of $H_0^1(\mathcal{O})$. The dual pair between $H_0^1(\mathcal{O})$ and $H^{-1}(\mathcal{O})$ is denoted by $\langle\cdot,\cdot\rangle_{1,-1}$. Define $(H^{-1})^+(\mathcal{O})=\{v\in H^{-1}(\mathcal{O}):\langle \varphi,v\rangle_{1,-1}\geq 0,~\textrm{for~each}~\varphi\in H^1_0(\mathcal{O})~\textrm{and~}\varphi\geq 0\}$.

For a Hilbert space $V$, denote by $\mathcal{L}^2([0,T];V)$ the set of all $L^2([0,T];V)$ valued $(\mathcal{F}_t)$ adapted process $u$ with the norm defined as $\|u\|_{\mathcal{L}^2([0,T];V)}:=\left(E\|u\|_{L^2([0,T];V)}^2\right)^{\frac{1}{2}}<\infty$. Denote by $\mathcal{L}^2(\mathcal{O})$ the set of all $L^2(\mathcal{O})$ valued $(\mathcal{F}_t)$ adapted process $u$ with the norm $\|u\|_{\mathcal{L}^2(\mathcal{O})}:=\left(E\|u\|^2\right)^{\frac{1}{2}}<\infty$

Denote  $\mathcal{K}:=L^{\infty}([0,T];L^2(\mathcal{O}))\cap L^2([0,T];H_0^1(\mathcal{O}))$, equipped with the norm
$$\|\upsilon\|_{\mathcal{K}}:=\left(\|\upsilon\|^2_{L^{\infty}([0,T];L^2(\mathcal{O}))}+\|\upsilon\|^2_{L^2([0,T];H_0^1(\mathcal{O}))}\right)^{\frac{1}{2}}.$$
Set $\mathcal{W}=\{\upsilon\in L^2(0,T;H_0^1):\partial_t\upsilon \in L^2(0,T;H^{-1})\}$ endowed with the norm
$$\|\upsilon\|_{\mathcal{W}}=\left(\|\upsilon\|^2_{L^2(0,T;H_0^1)}+\|\partial_t\upsilon\|^2_{L^2(0,T;H^{-1})}\right)^{\frac{1}{2}},$$
where $H^{-1}$ is the dual space of $H_0^1$. Furthermore, we set
$$\mathcal{W}_T=\{\upsilon\in\mathcal{W}:\upsilon(T)=0\},~\mathcal{W}^+=\{\upsilon\in\mathcal{W}:\upsilon\geq 0\},~\mathcal{W}_T^+=\mathcal{W}_T\cap\mathcal{W}^+.$$


\begin{definition} %definition of potential
We denote by $\mathcal{P}$ the set of parabolic potentials, which is the class of $\upsilon\in\mathcal{K}$ such that
$$\int_0^T-\langle\partial_t\varphi(t),\upsilon(t)\rangle \,dt+\int_0^T\langle\partial_i\varphi(t),\partial_i\upsilon(t)\rangle \,dt\geq 0,~\forall\varphi\in\mathcal{W}^+_T.$$
\end{definition}


Denote by $\mathcal{C}(Q)$ the class of continuously differentiable functions in $Q$ with compact support.
By the Hahn-Banach theorem and because $\mathcal{C}(Q)\cap\mathcal{W}_T$ is dense in $\mathcal{C}(Q)$, parabolic potentials can be represented by associated Radon measures. This leads to the following proposition, due to Pierre \cite{pierre1980}.


\begin{proposition}\label{radon-measure}
Let $\upsilon\in\mathcal{P}$. Then there exists a unique Radon measure on $[0,T)\times\mathcal{O}$, denoted by $\mu ^{\upsilon}$, such that
\begin{align*}
\forall~\varphi\in\mathcal{W}_T\cap\mathcal{C}(Q),~\int_0^T-\langle \partial_t\varphi(t),\upsilon(t)\rangle+\int_0^T\langle\partial_i\varphi(t),\partial_i\upsilon(t)\rangle \,dt=\int_0^T\int_{\mathcal{O}}\varphi(t,x)\mu^{\upsilon}(dt,dx)
\end{align*}
\end{proposition}


\begin{definition}
For any open set $A\subset[0,T)\times\mathcal{O}$, the parabolic capacity of $A$ is defined as
$$\textrm{cap}(A)=\inf\{\|\varphi\|^2_{\mathcal{W}}:\varphi\in\mathcal{W}^+,~ \varphi\geq 1 ~a.e. \textrm{~on}~ A\}.$$
For any Borel set $B\subset[0,T)\times\mathcal{O}$, its parabolic capacity is defined as
$$\textrm{cap}(B)=\inf\{\textrm{cap}(A):A\supset B, ~ A~\textrm{is~open}\}.$$
\end{definition}


\begin{definition}
A real valued function $\phi$ on $[0,T)\times\mathcal{O}$ is said to be quasi-continuous, if there exists a sequence of non-increasing open sets $A_n\subset[0,T)\times\mathcal{O}$ such that
\begin{itemize}
	\item[(1)] $\phi$ is continuous on the complement of each $A_n$;
	\item[(2)] $\lim\limits_{n\rightarrow \infty}\textrm{cap}(A_n)=0$.
\end{itemize}
\end{definition}

Denote by $\mathcal{P}_0$ the class of $\upsilon\in\mathcal{P}$ such that $\upsilon$ is quasi-continuous and $\upsilon(0)=0$ in $L^2$. Each element $\upsilon\in\mathcal{P}_0$ is called a regular potential and the associated Radon measure in Definition \ref{radon-measure} is called a regular measure.
Furthermore, let $\mathcal{L}^0(\mathcal{K})$ be the class of the measurable maps from $(\Omega, \mathcal{F}_T)$ to $\mathcal{K}$, such that each element $\upsilon\in\mathcal{L}^0(\mathcal{K})$ is an $L^2$ valued adapted process. $\mathcal{L}^0(\mathcal{P})$ and $\mathcal{L}^0(\mathcal{P}_0)$ are similarly defined as $\mathcal{L}^0(\mathcal{K})$. Moreover, set
$$\mathcal{L}^2(\mathcal{K}):=L^2(\Omega,\mathcal{F}_T;\mathcal{K})\cap \mathcal{L}^0(\mathcal{K})$$
endowed with the norm $$\|\upsilon\|_{\mathcal{L}^2(\mathcal{K})}=\left( E\|\upsilon\|^2_{\mathcal{K}}\right)^{1/2}.$$
%Similarly, we define $\mathcal{L}^2(\mathcal{K})=L^2(\Omega,\mathcal{F}_T;\mathcal{W})\cap \mathcal{L}^0(\mathcal{K})$ endowed with the norm
%$$\|u\|_{\mathcal{L}^2(\mathcal{K})}=\left( E\|u\|^2_{\mathcal{K}}\right)^{1/2}.$$
%where as before $\mathcal{L}^0(\mathcal{K})$ is the totality of the measurable maps from $(\Omega, \mathcal{F}_T)$ to $\mathcal{K}$ such that each element $u\in\mathcal{L}^0(\mathcal{K})$ is an $L^2$ valued adapted process.

The stochastic parabolic potential is defined as
$$\mathcal{L}^2(\mathcal{P}):=\mathcal{L}^2(\mathcal{K})\cap\mathcal{L}^0(\mathcal{P}),$$
endowed with the norm $$\|u\|_{\mathcal{L}^2(\mathcal{P})}=\|u\|_{\mathcal{L}^2(\mathcal{K})}.$$
In addition, we define the stochastic regular parabolic potential as
$$\mathcal{L}^2(\mathcal{P}_0):=\mathcal{L}^2(\mathcal{P})\cap\mathcal{L}^0(\mathcal{P}_0),$$
and the associated random Radon measure is called a stochastic regular measure.
\end{appendix}











\begin{thebibliography}{10}

\bibitem{AronsonSerrin1967}
{\sc D.~G. Aronson and J.~Serrin}, { Local behavior of solutions of
  quasilinear parabolic equations}, Arch. Rational Mech. Anal., 25 (1967),
  pp.~81--122.

\bibitem{Bensousan_83}
{\sc A.~Bensoussan}, { Maximum principle and dynamic programming approaches
  of the optimal control of partially observed diffusions}, Stoch., 9 (1983),
  pp.~169--222.

%\bibitem{bismut}J. M. Bismut \textit{Conjugate convex functions in optimal stochastic control}, J. Math. Anal. Appl. \textbf{44} (1973), 384-404.

\bibitem{changpangyong}M. Chang, Y. Pang and J. Yong { Optimal stopping problem for stochastic differential equations with random coefficients}, SIAM J. Control Optim. \textbf{48} (2009), 941-971.

\bibitem{chen}Y. Chen, { Parabolic partial differential equations of second order}, Peking University Press, 2005 (in Chinese).

%\bibitem{cvitanickaratzas}J. Cvitanic, I. Karatzas \textit{Backward stochastic differential equations with reflection and Dynkin games}, Ann. Probab. \textbf{24} (1996), 2024-2056.

%\bibitem{dlebaenqiutang}F. Delbaen, J. Qiu and S. Tang, \textit{Forward-backward stochastic differential systems associated to Navier-Stokes equations in the whole space}, arXiv:1303.5329v2 [math-ph].



\bibitem{DenisMatoussiStoica2005}
{\sc L.~Denis, A.~Matoussi, and L.~Stoica}, { $\textrm{L}^p$ estimates for
  the uniform norm of solutions of quasilinear $\textrm{SPDE}$'s}, Probab.
  Theory Relat. Fields, 133 (2005), pp.~437--463.


\bibitem{denis-matoussi-stoica-2009}L. Denis, A. Matoussi, and L. Stoica { Maximum principle and comparison theorem for quasi-linear stochastic PDEs}, Electron. J. Probab. \textbf{14} (2009), 500-530.



\bibitem{denismzhang}L. Denis, A. Matoussi and J. Zhang { The obstacle problem for quasi-linear stochastic PDEs: analytical approach}, Ann. Probab. \textbf{42} (2014), 865-905.

\bibitem{denismatoussizhang}L. Denis, A. Matoussi and J. Zhang { Maximum principle for quasi-linear stochastic PDEs with obstacle}, Electon. J. Probab. \textbf{19} (2014), 1-32.

%\bibitem{duqiutang}K. Du, J. Qiu and J. Tang \textit{$L^p$ theory for super-parabolic backward stochastic partial differential equations in the whole space}, Appl. Math. Optim. \textbf{65} (2011), 175-219.

%\bibitem{karouikapapeng}N. El Karoui, C. Kapoudjian,  E. Pardoux, S. Peng and M. C. Quenez \textit{Reflected solutions of backward SDE's, and related obstacle problems for PDE's}, Ann. Probab. \textbf{25} (1997), 702-737.

%\bibitem{englezoskaratzas}N. Englezos, I. Karatzas \textit{Utility maximization with habit formation: dynamic programming and stochastic PDEs}, SIAM. J. Control Optim. \textbf{48} (2009), 481-520.

\bibitem{paulwinqiuhorst} P. Graewe, U. Horst, and J. Qiu { A Non-Markovian liquidation problem and backward SPDEs with singular terminal conditions}, SIAM J. Control Optim. \textbf{53} (2014), 690-711.

%\bibitem{hsuwangwang} E.P. Hsu, Y. Wang, and Z. Wang \textit{Stochastic De Girogi Iteration and Regularity of Stochastic Partial Differential Equation}, arXiv:1312.3311, 2013.

%\bibitem{Hu_Ma_Yong02}
%{\sc Y.~Hu, J.~Ma, and J.~Yong}, {\em On semi-linear degenerate backward
%  stochastic partial differential equations}, Probab. Theory Relat. Fields, \textbf{123}
%  (2002), pp.~381--411.

%\bibitem{klimsiak} T. Klimsiak \textit{Reflected BSDEs and the obstacle problem for semilinear PDEs in divergence form}, Stochastic Process. Appl. \textbf{122} (2012), 134-169.

%\bibitem{Lieberman}
%{\sc G.~M. Lieberman}, {\em Second Order Parabolic Differential Equations},
%  World Scientific, 1996.

\bibitem{oksendalsulemzhang}B. {\O}ksendal, A. Sulem and T. Zhang { Singular Control and Optimal Stopping of SPDEs, and Backward SPDEs with Reflection}, Math. Oper. Res. \textbf{39} (2014), 464-486.

%\bibitem{pardouxpeng}E. Pardoux, S. Peng \textit{Adapted solution of a backward stochastic differential equation}, Systems Control Lett. \textbf{14} (1990), 55-61.

\bibitem{peng}S. Peng { Stochastic Hamilton-Jacobi-Bellman equations}, SIAM. J. Control Optim. \textbf{30} (1992), 284-304.

\bibitem{pierre}M. Pierre { Prob\'{e}mes d'evolution avec contraintes unilaterales et potentiels parabolique}, Comm. Partial Differential Equations. \textbf{4} (1979), 1149-1197.

%\bibitem{pierre1980}M. Pierre \textit{Repr\'{e}sentant pr\'{e}cis d'un potentiels parabolique}, in: S\'{e}minaire de th\'{e}orie du potentiel, in: Lect. Notes Math. \textbf{814} (1980), 186-228.
%

\bibitem{pierre1980}M. Pierre { Repr\'esentant pr\'ecis d'un potentiel parabolique, in:S\'eminaire de th\'eorie du potentiel}, Lect. Notes Math., \textbf{814} (1980), 186-228.

%\bibitem{denismzhang}L. Denis, A. Matoussi and J. Zhang \textit{The obstacle problem for quasi-linear stochastic PDEs: analytical approach}, Ann. Probab. \textbf{42} (2014), 865-905.
%
%\bibitem{pratozabczyk}G.D. Prato, J. Zabczyk, \textit{Stochastic equations in infinite dimensions}, Encyclopedia Math. Appl., Cambridge University Press, 1992.

%\bibitem{qiu2014}J. Qiu, \textit{Weak solution for fully nonlinear stochastic Hamilton-Jacobi-Bellman equations}, arXiv:1410.6967 [math.OC].

\bibitem{qiu-2015-DSPDE}J. Qiu, { {L}$^{2} $-Theory of Linear Degenerate {SPDEs} and {L}$^p $ Estimates for the Uniform Norm of Weak Solutions}, arXiv:1503.06162, 2015.

\bibitem{qiutang}J. Qiu, S. Tang, { Maximum principle for quasi-linear backward stochastic partial differential equations}, J. Funct. Anal. \textbf{262} (2012), 2436-2480.

\bibitem{qiuwei}J. Qiu, W. Wei, { On the quasi-linear reflected backward stochastic partial differential equations}, J. Funct. Anal. \textbf{267} (2014), 3598-3656.

%\bibitem{tangwei}S. Tang, W. Wei, \textit{On the Cauchy problem for backward stochastic partial differential equations in H\"older spaces}, to appear in Ann. Probab.
%

%\bibitem{Tang-Yang-2011}
%{\sc Z.~Yang and S.~Tang}, {\em Dynkin game of stochastic differential
%  equations with random coefficients, and associated backward stochastic
%  partial differential variational inequality}, \textrm{SIAM} J. Control
%  Optim., 51 (2013), pp.~64--95.

\bibitem{xhou}X. Xhou, { A duality analysis on stochastic partial differential equations}, J. Funct. Anal. \textbf{103} (1992), 275-293.

\end{thebibliography}
\end{document}

1604.02425.extracted_bib
{AronsonSerrin1967 {\sc D.~G. Aronson and J.~Serrin, { Local behavior of solutions of quasilinear parabolic equations, Arch. Rational Mech. Anal., 25 (1967), pp.~81--122.
1604.02425.extracted_bib
{Bensousan_83 {\sc A.~Bensoussan, { Maximum principle and dynamic programming approaches of the optimal control of partially observed diffusions, Stoch., 9 (1983), pp.~169--222. %
1604.02425.extracted_bib
{bismutJ. M. Bismut Conjugate convex functions in optimal stochastic control, J. Math. Anal. Appl. 44 (1973), 384-404.
1604.02425.extracted_bib
{changpangyongM. Chang, Y. Pang and J. Yong { Optimal stopping problem for stochastic differential equations with random coefficients, SIAM J. Control Optim. 48 (2009), 941-971.
1604.02425.extracted_bib
{chenY. Chen, { Parabolic partial differential equations of second order, Peking University Press, 2005 (in Chinese). %
1604.02425.extracted_bib
{cvitanickaratzasJ. Cvitanic, I. Karatzas Backward stochastic differential equations with reflection and Dynkin games, Ann. Probab. 24 (1996), 2024-2056. %
1604.02425.extracted_bib
{dlebaenqiutangF. Delbaen, J. Qiu and S. Tang, Forward-backward stochastic differential systems associated to Navier-Stokes equations in the whole space, arXiv:1303.5329v2 [math-ph].
1604.02425.extracted_bib
{DenisMatoussiStoica2005 {\sc L.~Denis, A.~Matoussi, and L.~Stoica, { $L^p$ estimates for the uniform norm of solutions of quasilinear $SPDE$'s, Probab. Theory Relat. Fields, 133 (2005), pp.~437--463.
1604.02425.extracted_bib
{denis-matoussi-stoica-2009L. Denis, A. Matoussi, and L. Stoica { Maximum principle and comparison theorem for quasi-linear stochastic PDEs, Electron. J. Probab. 14 (2009), 500-530.
1604.02425.extracted_bib
{denismzhangL. Denis, A. Matoussi and J. Zhang { The obstacle problem for quasi-linear stochastic PDEs: analytical approach, Ann. Probab. 42 (2014), 865-905.
1604.02425.extracted_bib
{denismatoussizhangL. Denis, A. Matoussi and J. Zhang { Maximum principle for quasi-linear stochastic PDEs with obstacle, Electon. J. Probab. 19 (2014), 1-32. %
1604.02425.extracted_bib
{duqiutangK. Du, J. Qiu and J. Tang $L^p$ theory for super-parabolic backward stochastic partial differential equations in the whole space, Appl. Math. Optim. 65 (2011), 175-219. %
1604.02425.extracted_bib
{karouikapapengN. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng and M. C. Quenez Reflected solutions of backward SDE's, and related obstacle problems for PDE's, Ann. Probab. 25 (1997), 702-737. %
1604.02425.extracted_bib
{englezoskaratzasN. Englezos, I. Karatzas Utility maximization with habit formation: dynamic programming and stochastic PDEs, SIAM. J. Control Optim. 48 (2009), 481-520.
1604.02425.extracted_bib
{paulwinqiuhorst P. Graewe, U. Horst, and J. Qiu { A Non-Markovian liquidation problem and backward SPDEs with singular terminal conditions, SIAM J. Control Optim. 53 (2014), 690-711. %
1604.02425.extracted_bib
{hsuwangwang E.P. Hsu, Y. Wang, and Z. Wang Stochastic De Girogi Iteration and Regularity of Stochastic Partial Differential Equation, arXiv:1312.3311, 2013. %
1604.02425.extracted_bib
{Hu_Ma_Yong02 %{\sc Y.~Hu, J.~Ma, and J.~Yong, {\em On semi-linear degenerate backward % stochastic partial differential equations, Probab. Theory Relat. Fields, 123 % (2002), pp.~381--411. %
1604.02425.extracted_bib
{klimsiak T. Klimsiak Reflected BSDEs and the obstacle problem for semilinear PDEs in divergence form, Stochastic Process. Appl. 122 (2012), 134-169. %
1604.02425.extracted_bib
{Lieberman %{\sc G.~M. Lieberman, {\em Second Order Parabolic Differential Equations, % World Scientific, 1996.
1604.02425.extracted_bib
{oksendalsulemzhangB. {\Oksendal, A. Sulem and T. Zhang { Singular Control and Optimal Stopping of SPDEs, and Backward SPDEs with Reflection, Math. Oper. Res. 39 (2014), 464-486. %
1604.02425.extracted_bib
{pardouxpengE. Pardoux, S. Peng Adapted solution of a backward stochastic differential equation, Systems Control Lett. 14 (1990), 55-61.
1604.02425.extracted_bib
{pengS. Peng { Stochastic Hamilton-Jacobi-Bellman equations, SIAM. J. Control Optim. 30 (1992), 284-304.
1604.02425.extracted_bib
{pierreM. Pierre { Prob\'{emes d'evolution avec contraintes unilaterales et potentiels parabolique, Comm. Partial Differential Equations. 4 (1979), 1149-1197. %
1604.02425.extracted_bib
{pierre1980M. Pierre Repr\'{esentant pr\'{ecis d'un potentiels parabolique, in: S\'{eminaire de th\'{eorie du potentiel, in: Lect. Notes Math. 814 (1980), 186-228. %
1604.02425.extracted_bib
{pierre1980M. Pierre { Repr\'esentant pr\'ecis d'un potentiel parabolique, in:S\'eminaire de th\'eorie du potentiel, Lect. Notes Math., 814 (1980), 186-228. %
1604.02425.extracted_bib
{denismzhangL. Denis, A. Matoussi and J. Zhang The obstacle problem for quasi-linear stochastic PDEs: analytical approach, Ann. Probab. 42 (2014), 865-905. % %
1604.02425.extracted_bib
{pratozabczykG.D. Prato, J. Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia Math. Appl., Cambridge University Press, 1992. %
1604.02425.extracted_bib
{qiu2014J. Qiu, Weak solution for fully nonlinear stochastic Hamilton-Jacobi-Bellman equations, arXiv:1410.6967 [math.OC].
1604.02425.extracted_bib
{qiu-2015-DSPDEJ. Qiu, { {L$^{2 $-Theory of Linear Degenerate {SPDEs and {L$^p $ Estimates for the Uniform Norm of Weak Solutions, arXiv:1503.06162, 2015.
1604.02425.extracted_bib
{qiutangJ. Qiu, S. Tang, { Maximum principle for quasi-linear backward stochastic partial differential equations, J. Funct. Anal. 262 (2012), 2436-2480.
1604.02425.extracted_bib
{qiuweiJ. Qiu, W. Wei, { On the quasi-linear reflected backward stochastic partial differential equations, J. Funct. Anal. 267 (2014), 3598-3656. %
1604.02425.extracted_bib
{tangweiS. Tang, W. Wei, On the Cauchy problem for backward stochastic partial differential equations in H\"older spaces, to appear in Ann. Probab. % %
1604.02425.extracted_bib
{Tang-Yang-2011 %{\sc Z.~Yang and S.~Tang, {\em Dynkin game of stochastic differential % equations with random coefficients, and associated backward stochastic % partial differential variational inequality, SIAM J. Control % Optim., 51 (2013), pp.~64--95.
1604.02425.extracted_bib
{xhouX. Xhou, { A duality analysis on stochastic partial differential equations, J. Funct. Anal. 103 (1992), 275-293.