\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{epic} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 178, pp. 1--16.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2014/178\hfil Existence and uniqueness] {Existence and uniqueness of M-solutions for backward stochastic Volterra integral equations} \author[W. Li, R. Wu, K. Wang \hfil EJDE-2014/178\hfilneg] {Wenxue Li, Ruihua Wu, Ke Wang} % in alphabetical order \address{Wenxue Li \newline Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai 264209, China} \email{wenxuetg@hitwh.edu.cn, Phone +86 0631 5687035, fax +86 0631 5687572} \address{Ruihua Wu \newline Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai 264209, China} \email{wu\_ruihua@hotmail.com} \address{Ke Wang \newline Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai 264209, China} \email{wangke@hitwh.edu.cn} \thanks{Submitted August 2, 2013. Published August 21, 2014.} \subjclass[2000]{45D05, 60H17, 34A12, 60H20} \keywords{Backward stochastic Volterra integral equations; existence;\hfill\break\indent uniqueness; dynamic risk measure} \begin{abstract} In this article, we study general backward stochastic Volterra integral equations (BSVIEs). Combining the contractive-mapping principle, step-by-step iteration method and mathematical induction, we establish the existence and uniqueness theorem of M-solution for the BSVIEs. This theorem could be applied directly to many models, for example, using the result to a kind of financial models provides a new and easy method to discuss the existence of dynamic risk measure. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section {Introduction} Backward stochastic differential equations (BSDEs) and backward stochastic Volterra integral equation (BSVIE) are applied widely in finance and stochastic control etc.\cite{yi,yong,121}. The theoretical foundation of BSDEs have been established by Pardoux and Peng\cite{peng}. The development of BSDEs greatly promoted the evolution of economics and finance. For example, economists Diffie and Epstein introduced BSDEs into economics in 1992, and stochastic analyst El Karoai et al \cite{N.E} discovered the important role of BSDEs in finance. In 2006, Yong \cite{yy} introduce the BSVIEs, in which play a major role in considering the properties of forward stochastic Volterra integral equations, which describe the stochastic optimal control problem with memories, and in the proof of stochastic Pontryagin maximum principle \cite{121} etc.. Besides, in \cite{122}, a class of dynamic convex and coherent risk measures are identified as a component of the adapted M-solutions to certain BSVIEs. In this article, we consider general backward stochastic Volterra integral equation (BSVIE) \begin{equation}\label{53} \begin{split} Y(t)&=F(t,Y(t))+\int_t^Tg(t,s,Y(s),Y(t),Z(t,s),Z(s,t))\,\mathrm{d}s \\ &\quad -\int_t^Th(t,s,Y(s),Y(t),Z(t,s))\,\mathrm{d}W(s),\quad t\in [0,T]. \end{split} \end{equation} A class of important BSVIEs \eqref{53} is the the following \begin{equation}\label{3} Y(t)=\xi+\int_t^Tg(s,Y(s),Z(s))\,\mathrm{d}s+\int_t^TZ(s)\,\mathrm{d}W(s) ,\quad t\in [0,T]. \end{equation} Yong \cite{yy} introduced another form of BSVIE \eqref{53}: \begin{equation}\label{7} Y(t)=\phi(t)-\int_t^Tg\Big(t,s,Y(s),Z(t,s),Z(s,t)\Big)\,\mathrm{d}s -\int_t^TZ(t,s)\,\mathrm{d}W(s),\quad t\in [0,T], \end{equation} and gave the following definition of adapted solution. \begin{definition}\label{de1} \rm Any pair of stochastic processes $(Y(\cdot),Z(\cdot,\cdot))\in \mathcal {H}^{1}[S,T]$ (defined as \eqref{124}) satisfying \eqref{7} is called an adapted solution of \eqref{7}. \end{definition} Also, the conditions of the existence and uniqueness of the adapted solution to \eqref{7} are given. However, it is difficult to consider the uniqueness of the adapted solution to \eqref{7} under Definition \ref{de1}. For example, for the BSVIE \begin{equation}\label{88} Y(t)=\int_t^T g\Big(t,s,Y(s),Z(t,s)\Big)\,\mathrm{d}s- \int_t^TZ(t,s)\,\mathrm{d}W(s),\quad t\in [0,T], \end{equation} in which $g$ satisfies the conditions of existence and uniqueness theorem, suppose $(Y(\cdot),Z(\cdot,\cdot))$ is the uniqueness adapted solution of \eqref{88}. But it is easy to check that $(\hat{Y}(\cdot),\hat{Z}(\cdot,\cdot))$ satisfies \begin{gather*} \hat{Y}(t)=Y(t), \quad t\in [0,T] ,\\ \hat{Z}(t,s)=Z(t,s),\quad (t,s)\in [0,T]\times[t,T] ,\\ \hat{Z}(t,s)=\varsigma(t,s),\quad (t,s)\in [0,T]\times[0,t], \end{gather*} is also an adapted solution of \eqref{88} for any $\varsigma(\cdot)\in L_{\mathbb{F}}^2(0,T;\mathbb{R})$. It is contradictive. The purpose of this article is to discuss the existence and uniqueness of adapted M-solutions (defined later), rather than the adapted solution to \eqref{53} in Definition \ref{de1}. We give some sufficient conditions for the existence and uniqueness of M-solution to \eqref{53}, by combining contractive-mapping principle, step-by-step iteration method and mathematical induction. These results could be applied directly to many models, such as those described as BSVIEs, in which a component of the M-solution of BSVIEs has a close relation with the dynamic risk measure. By applying the main result to the financial models, it provides a new and easy method to discuss the existence of dynamic risk measure. \section{Motivation and main results} \subsection{Experimental motivation} In this article, we use \eqref{53} to describe a class of economic problems as certain portfolio, such as European option, some current cash flows, mutual funds etc. Here, $Y(t)$ denotes the price of merchandize, $F(t,Y(t))$ stands for the total wealth of certain portfolio and $g$ is referred to as the generator of \eqref{53}. Since the component $Y(t)$ of the M-solution to \eqref{53} has a close relationship with the dynamic risk measure, it is significant to consider the existence and uniqueness of the M-solution. Here the dynamic risk measure is defined as follows: \begin{definition}[\cite{122}] \rm A map $\rho:L^2_{\mathcal {F}_{T}}(0,T)\to L^2_{\mathbb{F}}(0,T)$ is called a dynamic risk measure if the following conditions hold: \begin{itemize} \item[1.] For any $\psi(\cdot), \bar{\psi}(\cdot) \in L^2_{\mathcal {F}_{T}}(0,T)$, if $\psi(\cdot)=\bar{\psi}(\cdot)$, a.s. $s\in [t,T]$, for some $t\in [0,T)$, then $$ \rho(t,\psi(\cdot))=\rho(t,\bar{\psi}(\cdot)),\quad\text{a.s.}. $$ \item[2.] For any $\psi(\cdot), \bar{\psi}(\cdot) \in L^2_{\mathcal {F}_{T}}(0,T)$, if $\psi(\cdot)\leq\bar{\psi}(\cdot)$, a.s. $s\in [t,T]$, for some $t\in [0,T)$, then $$ \rho(t,\psi(\cdot))\geq\rho(t,\bar{\psi}(\cdot)),\quad\text{a.s. }s\in [t,T]. $$ \end{itemize} \end{definition} In fact, the model is on the basis of some classical financial models. El Karoui et al \cite{N.E} described the problem of European option pricing by applying BSDE \eqref{3}. In this case, $\xi$ represents square-integrable contingent claim and $Y(t)$ represents the price of European option. For their model, if there exists unique solution, then the map $\rho:\xi\to Y(t)$ defined by \eqref{3} is a dynamic risk measure. Yong \cite{122}, extended \eqref{3} into \eqref{7}. There $\phi(t)$ represents the total wealth of certain portfolio. The author gave dynamic risk measure $\rho(t,\phi(t))=Y(t)$ for \eqref{7}. \subsection {Model assumptions and novelty} Now we give some assumptions for \eqref{53} such that it has unique solution. \begin{itemize} \item [(H1)] Let $g:\Delta ^{c}\times \mathbb{R}^{m}\times \mathbb{R}^{m}\times \mathbb{R}^{m\times d}\times \mathbb{R}^{m\times d}\times \Omega\to \mathbb{R}^{m} $ be $\mathfrak{B}(\Delta ^{c}\times \mathbb{R}^{m}\times \mathbb{R}^{m}\times \mathbb{R}^{m\times d}\times \mathbb{R}^{m\times d}) \otimes \mathcal {F}_{T}$-measurable, and for all $(t,\zeta,\eta,\xi,\varsigma)\in[0,T]\times \mathbb{R}^{m}\times \mathbb{R}^{m}\times \mathbb{R}^{m\times d}\times \mathbb{R}^{m\times d}$, $g(t,\cdot,\zeta,\eta,\xi,\varsigma)$ is $\mathbb{F}$-adapted and satisfies \begin{equation}\label{22} E\int_{0}^T\Big(\int_t^T| g(t,s,0,0,0,0)| \,\mathrm{d}s\Big)^2\,\mathrm{d}t<\infty, \end{equation} and for all $(t,s)\in \Delta ^{c},\zeta,\bar{\zeta},\eta,\bar{\eta}\in \mathbb{R}^{m},\xi,\bar{\xi},\varsigma,\bar{\varsigma}\in \mathbb{R}^{m\times d}$ \begin{equation}\label{23} \begin{split} &|g(t,s,\zeta,\eta,\xi,\varsigma) -g(t,s,\bar{\zeta},\bar{\eta},\bar{\xi},\bar{\varsigma}))|\\ &\leq L_{\zeta}(t,s)| \zeta-\bar{\zeta}|+L_{\eta}(t,s)| \eta-\bar{\eta}| +L_{\xi}(t,s)|\xi-\bar{\xi}| +L_{\varsigma}(t,s)|\varsigma-\bar{\varsigma}|~~\text{a.s.}, \end{split} \end{equation} where for some $\varepsilon>0$, $L_{\zeta}(t,s),L_{\eta}(t,s),L_{\xi}(t,s),L_{\varsigma}(t,s):\Delta ^{c}\to \mathbb{R}$ satisfy $$\sup_{t\in [0,T]}\int_t^T\Big[L_{\zeta}(t,s)^{2+\varepsilon}+L_{\eta}(t,s)^{2+\varepsilon} +L_{\xi}(t,s)^{2+\varepsilon}+ L_{\varsigma}(t,s)^{2+\varepsilon}\Big]\,\mathrm{d}s=A<\infty,$$ and \begin{equation}\label{24} \sup_{t\in [0,T]}\Big(\int_{0}^TL_{\eta}(t,s) \,\mathrm{d}s\Big)^2=K<\frac{1}{8C^2}, \end{equation} in which $C$ is the same as the one in \eqref{15}. \item [(H2)] Let $F:\mathbb{R}^{1}\times \mathbb{R}^{m}\times \Omega\to \mathbb{R}^{m} $ be $\mathfrak{B}(\mathbb{R}^{1}\times \mathbb{R}^{m})\otimes\mathcal {F}_{T}$-measurable, and \begin{equation}\label{45} E\int_{0}^T| F(t,0)|\,\mathrm{d}t<\infty. \end{equation} Moreover, \begin{equation}\label{46} \begin{gathered} | F(t,\varsigma) -F(t,\bar{\varsigma}))|\leq L_{\varsigma}(t)| \varsigma-\bar{\varsigma}|,\quad t\in \mathbb{R},\varsigma,\bar{\varsigma}\in \mathbb{R}^{m} \text{ a.s.}, \\ \sup_{t\in[0,T]}| L_{\varsigma}(t)|^2\leq D, \\ 2C_{L}D<1. \end{gathered} \end{equation} hold. Here $C_{L}$ is determined by \eqref{25}. \item [(H3)] Let $h:\Delta ^{c}\times \mathbb{R}^{m}\times \mathbb{R}^{m}\times \mathbb{R}^{m\times d}\times \Omega\to \mathbb{R}^{m\times d} $ be $\mathfrak{B}(\Delta ^{c}\times \mathbb{R}^{m}\times \mathbb{R}^{m}\times \mathbb{R}^{m\times d}) \otimes \mathcal {F}_{T}$-measurable, and for all $(t,\zeta,\eta,\xi)\in[0,T]\times \mathbb{R}^{m}\times \mathbb{R}^{m}\times \mathbb{R}^{m\times d}$, $h(t,\cdot,\zeta,\eta,\xi)$ is $\mathbb{F}$-adapted and has the following relations: \begin{equation}\label{54} \begin{gathered} |h(t,s,\zeta,\eta,\xi) -h(t,s,\bar{\zeta},\bar{\eta},\bar{\xi})-(\xi-\bar{\xi})|^2\leq L_{\zeta}|\zeta-\bar{\zeta}|^2+L_{\eta}| \eta-\bar{\eta}|^2+L_{\xi}| \xi-\bar{\xi}|^2, \\ \forall(t,s)\in \Delta ^{c},\zeta,\bar{\zeta},\eta,\bar{\eta}\in \mathbb{R}^{m},\; \xi,\bar{\xi}\in \mathbb{R}^{m\times d},\text{ a.s.}, \\ E\int_{0}^T\int_t^T| h(t,s,0,0,0)|^2\,\mathrm{d}s\,\mathrm{d}t <\infty, \\ \max\Big\{4C_{F}L_{\xi},C_{F}(L_{\zeta}+L_{\eta})T\Big\}<1, \end{gathered} \end{equation} where $C_{F}$ is determined by \eqref{47}. \end{itemize} It is easy to verify that conditions (H1)--(H3) will degenerate into (H) in \cite{zzoo} as \eqref{53} equal to \eqref{7}. In the rest of the subsection, we will show the novelty in this paper from the following points: theory and application. (1) It is noted that \eqref{3} can not show the rule of the total wealth changing with the time; and both \eqref{3} and \eqref{7} cannot build up the relation between the total wealth with the price of merchandise. To get rid of the two defects, we reconstruct the model as BSVIE \eqref{53}. In \cite{N.E} (or \cite{122}), by building the relation between $\xi$ and $Y(t)$, (or $\phi(t)$ and $Y(t)$), the dynamic risk measure for \eqref{3} (or \eqref{7}) is found. However, model \eqref{53} gives directly the explicit relation of $\varphi(t)$ and $F(t,Y(t))$, i. e. $\varphi(t)=F(t,Y(t)).$ Hence, if $F(t,\cdot)$ has well properties, we could arrive a dynamic risk measure $F^{-1}: \varphi(t)\to Y(t)$. Sequently, in order to study the properties of the dynamic risk measure, such as dynamic convex or coherent risk measures, it only needs to restrict the $F(t,\cdot)$ further. Then applying the main results to this model, it is convenient to find the dynamic risk measure. (2) It is well-known that step-by-step iteration method and fixed point theorem are important method to prove the existence and uniqueness of solution to equations \cite{kh,maox,peng,yy,zzoo}. But sometimes, for the equations having complicated forms, it is hard to derive the existence and uniqueness theorem, by using only method of them. However, in this paper we combine step-by-step iteration method, fixed point theorem, mathematic induction and Martingale representation theorem to provide a proof of the existence and uniqueness of M-solution to \eqref{53}. For different domain of definition of $Z(\cdot,\cdot)$, we use different method as in Figure \ref{fig1}. \begin{figure}[ht] \begin{center} \setlength{\unitlength}{1mm} \begin{picture}(75,45)(-5,0) \put(0,35){\line(1,0){64}} \dashline{1}(0,22)(64,22) \put(-5,5){\line(1,0){74}} \put(68.5,4.1){$\to$} \put(0,0){\line(0,1){40}} \put(-.85,40){$\uparrow$} \dashline{1}(32,5)(32,35) \put(64,5){\line(0,1){30}} \put(3,30){Step-by-step} \put(3,26){iteration method} \put(37,30){Fixed point} \put(37,26){theorem} \put(3,17){Mathematical} \put(3,13){induction} \put(37,17){Martingale} \put(37,13){representation} \put(37,8){theorem} \put(31,1){$S$} \put(62.6,1){$T$} \put(70,1){$t$} \put(-4,21){$S$} \put(-4,34){$T$} \end{picture} \end{center} \caption{Diagram of methods used in different domains of definition of $Z(\cdot,\cdot)$ to prove the main result} \label{fig1} \end{figure} \subsection{Main results} Firstly, we introduce the definition of M-solution, and then give the main results of the paper: existence and uniqueness theorem of M-solution to BSVIE \eqref{53}. The proof of it is left in the next section. \begin{definition}\label{M} \rm A pair $(Y(\cdot),Z(\cdot,\cdot))\in \mathcal {H}^{1}[S,T]$ is called an adapted M-solution of BSVIE \eqref{53} on $[S,T]$, if $(Y(\cdot),Z(\cdot,\cdot))$ satisfies \eqref{53} in the usual $It\hat{o}s$ sense for almost all $t\in[S,T]$ and, in addition, the following holds: $$ Y(t)=E(Y(t)| \mathcal{F}_{S})+\int_{S}^{t}Z(t,s)\,\mathrm{d}W(s),\quad \text{a.e. }t\in [S,T]. $$ \end{definition} \begin{theorem}\label{main} Let {\rm(H1)--(H3)} hold, then \eqref{53} admits a unique adapted M-solution $(Y(\cdot),Z(\cdot,\cdot))$, and the following estimate holds: \begin{equation}\label{55} \begin{split} &\| (Y(\cdot),Z(\cdot,\cdot))\| ^2_{\mathcal {H}^2[R,T]}\\ &\leq C_{h}E\Big\{\int_{R}^T| F(t,0)|^2\,\mathrm{d}t+ \int_{R}^T\Big(\int_t^T| g(t,s,0,0,0,0)| \,\mathrm{d}s\Big)^2\,\mathrm{d}t\\ &\quad +\int_{R}^T\int_t^T| h(t,s,0,0,0)|^2\,\mathrm{d}s\,\mathrm{d}t\Big\},\quad R\in[0,T]. \end{split} \end{equation} Furthermore, if $\bar{g}, \bar{F}, \bar{h}$ satisfy {\rm (H1)--(H3)}, respectively, and $(\bar{Y}(\cdot),\bar{Z}(\cdot,\cdot))$ is the M-solution, in which $(g,F,h)$ is replaced by $(\bar{g},\bar{F},\bar{h})$. Then \begin{equation}\label{56} \begin{split} &E\Big[\int_{R}^T| Y(t)-\bar{Y}(t)|^2\,\mathrm{d}t+ \int_{R}^T\int_{R}^T| Z(t,s)-\bar{Z}(t,s)|^2\,\mathrm{d}s\,\mathrm{d}t\Big] \\ &\leq C_{h}E\Big\{\int_{R}^T| F(t,Y(t))-\bar{F}(t,Y(t))|^2\,\mathrm{d}t\\ &\quad +\int_{R}^T\Big(\int_t^T|g(t,s,Y(s),Y(t),Z(t,s),Z(t,s))\\ &\quad -\bar{g}(t,s,Y(s),Y(t),Z(t,s),Z(t,s))| \,\mathrm{d}s\Big)^2\,\mathrm{d}t\\ &\quad +\int_{R}^T\int_t^T| h(t,s,Y(s),Y(t),Z(t,s))-\bar{h}(t,s,Y(s),Y(t),Z(t,s))|^2 \,\mathrm{d}s\,\mathrm{d}t\Big\}, \end{split} \end{equation} for $R\in[0,T]$. \end{theorem} \section{Proof of the main theorem} Before proving Theorem \ref{main}, we show some useful notion from \cite{zzoo}, and some lemmas. Let \begin{gather*} \begin{aligned} L^{p}_{\mathcal {F}_{S}}(\Omega; L^{q}(0,T)) =\Big\{&\phi:(0,T)\times\Omega\to \mathbb{R}^{m} :\phi(\cdot) \text{ is }\mathcal {B}([ 0,T])\otimes \mathcal {F}_{S} \text{-measurable}, \\ & E\Big(\int_{0}^T| \phi(t)|^{q}\,\mathrm{d}t\Big)^{\frac{p}{q}}< \infty\Big\}. \end{aligned} \\ L^{p}_{\mathbb{F}}(\Omega; L^{q}(0,T))=\Big\{\phi(\cdot)\in L^{p}(\Omega; L^{q}(0,T)): \phi(\cdot)\text{ is } \mathbb{F}\text{-adapted} \Big\}. \end{gather*} For any $p,q \geq1$, let $L^{q}(0,T;L^{p}_{\mathbb{F}}(\Omega;L^2(0,T)))$ be the set of all processes $Z:[0,T]^2\times \Omega\to \mathbb{R}^{m\times d} $, such that for almost all $t\in [0,T],Z(t,\cdot)\in L^{p}_{\mathbb{F}}(\Omega;L^2(0,T))$, there is $$ \int_{0}^T\Big\{E\Big(\int_{0}^T |Z(t,s)|^2\,\mathrm{d}s\Big) ^{p/2}\Big\}^{q/p}\,\mathrm{d}t<\infty. $$ For convenience, denote \begin{gather*} %\label{123} \Delta[R,S]=\{(t,s)\in [R,S]^2: R\leq s\leq t\leq S\}, \\ \Delta ^{c}[R,S]=\{(t,s)\in [R,S]^2: R\leq t1$. Moreover, the following holds: $$| k(t,s,z)-k(t,s,\bar{z})|\leq L_{z}(t,s)| z-\bar{z}|,~~(t,s)\in [S,T]\times[R,T],~ z, \bar{z}\in \mathbb{R}^{m\times d},a.s,$$ where $L_{z}:[S,T]\times[R,T]\to [0,\infty)$ is a deterministic function, such that for some $\varepsilon>0$,$$\sup_{t\in [S,T]}\int_{R}^TL_{z}(t,s)^{2+\varepsilon}\,\mathrm{d}s<\infty.$$ \end{itemize} Let $r=S\in[R,T)$ be fixed. Define $$ \psi^{S}(t)=\lambda(t,S),~Z(t,s)=\mu(t,s),\quad t\in [R,S],\;s\in [S,T]. $$ Then \eqref{9} is rewritten as stochastic Fredholm integral equations (SFIEs): \begin{equation}\label{14} \psi^{S}(t)=\psi(t)+\int_{r}^Tk(t,s,Z(t,s))\,\mathrm{d}s -\int_{r}^TZ(t,s)\,\mathrm{d}W(s),\quad t\in [S,T]. \end{equation} We call $(\psi^{S}(\cdot),Z(\cdot,\cdot))\in L^{p}_{\mathcal {F}_{S}}(R,S)\times L^{p}(R,S;L^2_{\mathbb{F}}(S,T))$ as an adapted solution of \eqref{14}, if it satisfies \eqref{14} in the sense of It\^{o}. \begin{lemma}[\cite{zzoo}]\label{le3.1} Let {\rm (H0)} hold. Then for any $\psi(\cdot)\in L_{\mathcal {F}_{T}}^{p}(R,S)$, \eqref{14} admits a unique adapted solution $(\psi^{S}(\cdot),Z(\cdot,\cdot))\in L_{\mathcal {F}_{S}}^{p}(R,S)\times L^{p}(R,S;L^2_{\mathbb{F}}(S,T))$, and the following estimate holds: \begin{equation}\label{15} E\Big\{|\psi^{S}(t)|^{p}+\Big(\int_{S}^T|Z(t,s)|^2 \,\mathrm{d}s\Big)^{p/2}\Big\} \leq CE\Big\{|\psi(t)|^{p}+\Big(\int_{S}^T| k(t,s,0)| \,\mathrm{d}s\Big)^{p}\Big\}, \end{equation} for $t\in[R,S]$. If $\bar{k}:[R,S]\times [S,T]\times \mathbb{R}^{m\times d}\times \Omega\to \mathbb{R}^{m}$ satisfies (H0), $\bar{\psi}(\cdot)\in L_{\mathcal {F}_{T}}^{p}(R,S)$, and $(\bar{\psi}^{S}(\cdot), \bar{Z}(\cdot,\cdot))\in L_{\mathcal {F}_{S}}^{p}(R,S)\times L^{p}(R,S;L^2_{\mathbb{F}}(S,T))$ is the unique adapted solution of \eqref{14} in which $(k,\psi)$ is replaced by $(\bar{k},\bar{\psi})$, then \begin{equation}\label{16} \begin{split} &E\Big\{| \psi^{S}(t)-\bar{\psi^{S}}(t)|^{p}+ \Big(\int_{S}^T| Z(t,s)-\bar{Z}(t,s)|^2\,\mathrm{d}s\Big)^{p/2}\Big\}\\ &\leq CE\Big\{|\psi(t)-\bar{\psi}(t)|^{p}+\Big(\int_{S}^T| k(t,s,Z(t,s))-\bar{k}(t,s,Z(t,s))| \,\mathrm{d}s\Big)^{p}\Big\}, \end{split} \end{equation} for $t\in[R,S]$. \end{lemma} Let $S=R$, and define \begin{gather*} Y(t)=\lambda(t,t), \quad t\in[S,T], \\ Z(t,s)=\mu(t,s), \quad (t,s)\in\Delta ^{c}[S,T]. \end{gather*} Then \eqref{9} can be rewritten as \begin{equation}\label{18} Y(t)=\psi(t)+\int_t^Tk(t,s,Z(t,s))\,\mathrm{d}s -\int_t^TZ(t,s)\,\mathrm{d}W(s),\quad t\in [S,T]. \end{equation} \begin{lemma}[\cite{zzoo}]\label{le3.2} Let {\rm (H0)} hold. Then for any $\psi(\cdot)\in L_{\mathcal{F}_{T}}^{p}(R,S)$, \eqref{18} admits a unique adapted M-solution $(Y(\cdot),Z(\cdot,\cdot))\in \mathcal {H}^{p}[S,T]$, and the following estimate holds: \begin{equation}\label{19} E\Big\{| Y(t)|^{p}+\Big(\int_{S}^T| Z(t,s)|^2\,\mathrm{d}s\Big)^{p/2}\Big\} \leq CE\Big\{|\psi(t)|^{p}+\Big(\int_t^T| k(t,s,0)| \,\mathrm{d}s\Big)^{p}\Big\}, \end{equation} for $t\in[S,T]$. If $\bar{k}:[R,S]\times [S,T]\times \mathbb{R}^{m\times d}\times \Omega\to \mathbb{R}^{m}$ also satisfies {\rm (H0)}, $\bar{\psi}(\cdot)\in L_{\mathcal {F}_{T}}^{p}(S,T)$, and $(\bar{Y}(\cdot), \bar{Z}(\cdot,\cdot))\in \mathcal {H}^{p}[S,T] $ is the unique adapted M-solution of BSVIE \eqref{18} in which $(k,\psi)$ is replaced by $(\bar{k},\bar{\psi})$, then \begin{equation}\label{20} \begin{split} &E\Big\{| Y(t)-\bar{Y}(t)|^{p}+ \Big(\int_{S}^T| Z(t,s)-\bar{Z}(t,s)|^2\,\mathrm{d}s\Big)^{p/2}\Big\} \\ &\leq CE\Big\{|\psi(t)-\bar{\psi}(t)|^{p}+\Big(\int_t^T| k(t,s,Z(t,s))-\bar{k}(t,s,Z(t,s))| \,\mathrm{d}s\Big)^{p}\Big\}, \end{split} \end{equation} for $t\in[S,T]$. \end{lemma} The proof of Theorem \ref{main} is split into three steps, in which we find solutions for the three BSVIE's: \eqref{21}, \eqref{44}, and \eqref{53}. \subsection{Existence and uniqueness of M-solution for the BSVIE} \begin{equation}\label{21} Y(t)=\psi(t)+\int_t^Tg(t,s,Y(s),Y(t),Z(t,s),Z(s,t))\,\mathrm{d}s -\int_t^TZ(t,s)\,\mathrm{d}W(s), \end{equation} for $t\in [0,T]$. \begin{theorem}\label{dingli2} Let {\rm (H1)} hold. Then for any $\psi(\cdot)\in L_{\mathcal {F}_{T}}^2(0,T)$, \eqref{21} admits a unique adapted M-solution $(Y(\cdot),Z(\cdot,\cdot))\in \mathcal {H}^2[0,T]$. Moreover, the following estimate holds: \begin{equation}\label{25} \begin{split} &\| (Y(\cdot),Z(\cdot,\cdot))\| ^2_{\mathcal{H}^2[R,T]}\\ &\equiv E\Big\{\int_{R}^T| Y(t)|^2\,\mathrm{d}t+\int_{R}^T \int_{R}^T| Z(t,s)|^2\,\mathrm{d}s\,\mathrm{d}t\Big\}\\ &\leq C_{L}E\Big\{\int_{R}^T|\psi(t)|^2\,\mathrm{d}t+\int_{R}^T\Big(\int_t^T| g(t,s,0,0,0,0)| \,\mathrm{d}s\Big)^2\,\mathrm{d}t\Big\},\quad R\in[0,T]. \end{split} \end{equation} If $\bar{g}$ also satisfies (H1), $\bar{\psi}(\cdot)\in L_{\mathcal {F}_{T}}^2(0,T)$, and $(\bar{Y}(\cdot),\bar{Z}(\cdot,\cdot))\in \mathcal {H}^2[0,T] $ is the adapted M-solution of \eqref{21} in which $(g,\psi)$ is replaced by $(\bar{g},\bar{\psi})$, then \begin{equation}\label{26} \begin{split} &E\Big\{\int_{R}^T| Y(t)-\bar{Y}(t)|^2\,\mathrm{d}t+ \int_{R}^T\int_{R}^T| Z(t,s)-\bar{Z}(t,s)|^2\,\mathrm{d}s\,\mathrm{d}t\Big\} \\ &\leq C_{L}E\Big\{\int_{R}^T|\psi(t)-\bar{\psi}(t)|^2\,\mathrm{d}t+\int_{R}^T\Big(\int_t^T| g(t,s,Y(s),Y(t),Z(t,s),Z(s,t))\\ &\quad -\bar{g}(t,s,Y(s),Y(t),Z(t,s),Z(s,t))| \,\mathrm{d}s\Big)^2\,\mathrm{d}t\Big\},\quad R\in[0,T]. \end{split} \end{equation} \end{theorem} \begin{proof} We split the proof into three steps. \smallskip \noindent\textbf{Step 1:} Let $\mathcal {M}^2[S,T]$ be a subspace of $\mathcal {H}^2[S,T]$, and for any $(y(\cdot),z(\cdot,\cdot))\in\mathcal {M}^2[S,T]$ satisfies \begin{equation}\label{27} y(t)=E(y(t)| \mathcal {F}_{S})+\int_{S}^{t}z(t,s)\,\mathrm{d}W(s),\quad ~a.e.~ t\in [S,T],\text{a.s.}, \end{equation} and define $$ \| y(\cdot),z(\cdot,\cdot)\|_{\mathcal {M}^2[S,T]} \equiv \Big\{ E\big[\int_{S}^T| y(t)|^2\,\mathrm{d}t+\int_{S}^T \int_t^T| z(t,s)|^2\,\mathrm{d}s\,\mathrm{d}t\big]\Big\}^{1/2}. $$ Clearly, $\mathcal{M}^2[S,T]$ is a nontrivial closed subspace of $\mathcal{H}^2[S,T]$. For any $\psi(\cdot)\in L_{\mathcal {F}_{T}}^2(S,T)$, $(y(\cdot),z(\cdot,\cdot))\in\mathcal {M}^2[S,T]$, consider the BSVIE: $$ Y(t)=\psi(t)+\int_t^Tg(t,s,y(s),y(t),Z(t,s),z(s,t))\,\mathrm{d}s -\int_t^TZ(t,s)\,\mathrm{d}W(s),\quad t\in [S,T]. $$ Applying Lemma \ref{le3.2}, this BSVIE admits a unique adapted M-solution $(Y(\cdot),Z(\cdot,\cdot))$ in $\mathcal {H}^2[S,T]$. On the other hand, $(Y(\cdot),Z(\cdot,\cdot))$ salsifies \eqref{27}, hence $(Y(\cdot),Z(\cdot,\cdot))\in \mathcal {M}^2[S,T]$. Define map $\Lambda:\mathcal {M}^2[S,T]\to\mathcal {M}^2[S,T]$ by \begin{equation}\label{30} \Lambda(y(\cdot),z(\cdot,\cdot))=(Y(\cdot),Z(\cdot,\cdot)). \end{equation} If $(\bar{y}(\cdot),\bar{z}(\cdot,\cdot))\in\mathcal {M}^2[S,T]$, and $\Lambda(\bar{y}(\cdot),\bar{z}(\cdot,\cdot))=(\bar{Y}(\cdot),\bar{Z}(\cdot,\cdot))$, then by \eqref{20} we obtain \begin{align*} %\label{31} &E\Big[| Y(t)-\bar{Y}(t)|^2+ \int_{S}^T| Z(t,s)-\bar{Z}(t,s)|^2\,\mathrm{d}s\Big] \\ &\leq CE\Big[\int_t^T| g(t,s,y(s),y(t),Z(t,s),z(s,t))-\bar{g}(t,s,\bar{y}(s),\bar{y}(t),Z(t,s),\bar{z}(s,t))| \,\mathrm{d}s\Big]^2\\ &\leq CE\Big[\int_t^T\Big( L_{\zeta}(t,s)| y(s)-\bar{y}(s)|+L_{\eta}(t,s)| y(t)-\bar{y}(t)| +L_{\varsigma}(t,s)| z(s,t)\\ &\quad -\bar{z}(s,t)| \Big) \,\mathrm{d}s\Big]^2 \\ &\leq 9CA^{\frac{2}{2+\varepsilon}}(T-t)^{\frac{\varepsilon}{2+\varepsilon}}E\Big[\int_t^T\Big(| y(s)-\bar{y}(s)|^2+| y(t)-\bar{y}(t)|^2+| z(s,t)\\ &\quad -\bar{z}(s,t)|^2\Big)\,\mathrm{d}s\Big]. \end{align*} Consequently, we obtain that \begin{align*} %\label{32} &\|\Lambda(y(\cdot),z(\cdot,\cdot))-\Lambda(\bar{y}(\cdot),\bar{z}(\cdot,\cdot)) \|_{\mathcal {M}^2[S,T]}^2 \\ &\equiv E\Big[\int_{S}^T| Y(t)-\bar{Y}(t)|^2\,\mathrm{d}t+\int_{S}^T \int_t^T| Z(t,s)-\bar{Z}(t,s)|^2\,\mathrm{d}s\,\mathrm{d}t\Big] \\ &\leq 9CA^{\frac{2}{2+\varepsilon}}(T-t)^{\frac{\varepsilon}{2+\varepsilon}} \max\{1,2(T-S)\}E\Big\{\int_{S}^T| y(t)-\bar{y}(t)|^2\,\mathrm{d}t\\ &\quad +\int_{S}^T \int_t^T| z(s,t)-\bar{z}(s,t)|^2\,\mathrm{d}s\,\mathrm{d}t\Big\}\\ &\leq C_{3}(T-S)^{\frac{\varepsilon}{2+\varepsilon}}\| (y(\cdot),z(\cdot,\cdot))-(\bar{y}(\cdot),\bar{z}(\cdot,\cdot)) \|_{\mathcal {M}^2[S,T]}^2, \end{align*} where $C_{3}=9CA^{\frac{2}{2+\varepsilon}}\max\{1,2(T-S)\}$. So $\Lambda:\mathcal {M}^2[S,T]\to\mathcal {M}^2[S,T]$ is contracting, if $T-S$ is sufficiently small. Then there exists a unique fixed point in $\mathcal {M}^2[S,T]$. Hence \eqref{21} admits a unique adapted M-solution $(Y(\cdot),Z(\cdot,\cdot))\in \mathcal {M}^2[S,T]$. Now \eqref{19} yields \begin{equation}\label{99} \begin{split} &E\Big[| Y(t)|^2+\int_t^T| Z(t,s)|^2\,\mathrm{d}s\Big]\\ &\leq CE\Big\{|\psi(t)|^2+\Big(\int_t^T| g(t,s,Y(s),Y(t),0,Z(s,t))| \,\mathrm{d}s\Big)^2\Big\} \\ &\leq 4CE\Big\{|\psi(t)|^2+\Big(\int_t^T| g(t,s,0,0,0,0)|\,\mathrm{d}s\Big)^2\\ &\quad +\Big(\int_t^TL_{\zeta}(t,s)^2\,\mathrm{d}s\Big)\Big( \int_t^T| Y(s)|^2\,\mathrm{d}s\Big) +\Big(\int_t^TL_{\eta}(t,s)^2\,\mathrm{d}s\Big)\Big(\int_t^T| Y(t)|^2\,\mathrm{d}s\Big)\\ &\quad +\Big(\int_t^TL_{\varsigma}(t,s)^2\,\mathrm{d}s\Big)\Big(\int_t^T| Z(s,t)|^2\,\mathrm{d}s\Big)\Big\}. \end{split} \end{equation} Consequently, \begin{align*} %\label{100} &\| (Y(\cdot),Z(\cdot,\cdot))\| ^2_{\mathcal {H}^2[S,T]}\\ &\equiv E\Big[\int_{S}^T| Y(t)|^2\,\mathrm{d}t+\int_{S}^T \int_{S}^T| Z(t,s)|^2\,\mathrm{d}s\,\mathrm{d}t\Big]\\ &\leq 4CE\Big\{\int_{S}^T|\psi(t)|^2\,\mathrm{d}t+\int_{S}^T\Big(\int_t^T| g(t,s,0,0,0,0)| \,\mathrm{d}s\Big)^2\,\mathrm{d}t\\ &\quad +2(T-S)^{^{\frac{\varepsilon}{2+\varepsilon}+1}}A^{\frac{2}{2+\varepsilon}}\int_{S}^T| Y(s)|^2\,\mathrm{d}s\\ &\quad + (T-S)^{^{\frac{\varepsilon}{2+\varepsilon}}}A^{\frac{2}{2+\varepsilon}}\int_{S}^T\int_{S}^T| Z(s,t)|^2\,\mathrm{d}s\,\mathrm{d}t\Big\}. \end{align*} If $T-S$ is so small that $4C(T-S)^{^{\frac{\varepsilon}{2+\varepsilon}}}A\max\{2(T-S),1\}<1/2$, then \eqref{25} holds with $C_{L}=8C$. Next, let us talk about the stability estimate. Let $(Y(\cdot),Z(\cdot,\cdot))$ and $(\bar{Y}(\cdot),\bar{Z}(\cdot,\cdot))$ be adapted M-solutions of \eqref{21} corresponding to $(g,\psi)$ and $(\bar{g},\bar{\psi})$, respectively. Denote \begin{gather*} \hat{Y}(t)=Y(t)-\bar{Y}(t) ,\quad \hat{Z}(t,s)=Z(t,s)-\bar{Z}(t,s),\quad \hat{\psi}(t)=\psi(t)-\bar{\psi}(t)\\ \hat{g}(t,s)=g(t,s,Y(s),Y(t),Z(t,s),Z(s,t))-\bar{g}(t,s,Y(s),Y(t),Z(t,s),Z(s,t)). \end{gather*} Then by Hadamard formula, we obtain \begin{align*} \hat{Y}(t) &=\hat{\psi}(t)+\int_t^T\Big[\alpha_{1}(t,s)\hat{Y}(s)+\alpha_{2}(t,s)\hat{Y}(t)\\ &\quad +\sum_{i=1}^{d} (\beta_{i}(t,s)\hat{Z}_{i}(t,s)+\bar{\beta}_{i}(t,s)\hat{Z}_{i}(s,t))+\hat{g}(t,s)\Big]\,\mathrm{d}s -\int_t^T\hat{Z}(t,s)\,\mathrm{d}W(s). \end{align*} Applying \eqref{25}, we have the stability estimate \eqref{26} holds with $C_{L}=8C$. In this step we determine the unique solution $(Y(t),Z(t,s))$ to \eqref{21} for $t,s\in [S,T]$. \smallskip \noindent\textbf{Step 2:} Since $E[Y(t)| \mathcal {F}_{S}]\in L^2(S,T; L^2_{\mathcal {F}_{S}}(\Omega))$, by the Martingale Representation Theorem we could find a unique $Z(\cdot,\cdot)\in L^2(S,T; L^2_{\mathcal {F}_{S}}(2S-T,S))$ such that \begin{equation}\label{33} E[Y(t)| \mathcal {F}_{S}]=E[Y(t)| \mathcal {F}_{2S-T}]+\int_{2S-T}^{S}Z(t,s)\,\mathrm{d}W(s),\quad t\in [S,T]. \end{equation} By \eqref{33} we conclude that $$ E\int_{2S-T}^{S}| Z(t,s)|^2\,\mathrm{d}s=E| Y(t)|^2-| EY(t)|^2, \quad t\in [S,T]. $$ Furthermore, \begin{align*} &E\int_{S}^T\int_{2S-T}^{S}|Z(t,s)|^2\,\mathrm{d}s\,\mathrm{d}t\\ &\leq E\int_{S}^T| Y(t)|^2\,\mathrm{d}t \\ &\leq 8CE\Big\{\int_{S}^T|\psi(t)|^2\,\mathrm{d}t+\int_{S}^T\Big(\int_t^T| g(t,s,0,0,0,0)| \,\mathrm{d}s\Big)^2\,\mathrm{d}t\Big\}. \end{align*} In this step we determine the unique solution $(Y(t),Z(t,s))$ of \eqref{21} for $t\in [S,T],s\in[2S-T,S]$. \smallskip \noindent\textbf{Step 3:} Denote \begin{gather*} Y_{1}(t)=Y(t),\quad t\in [2S-T,S],\quad Y_{2}(t)=Y(t),\quad t\in [S,T],\\ Z_{11}(t,s)=Z(t,s),t\in [2S-T,S]\times[S,T],\quad Z_{12}(t,s)=Z(t,s),t\in [S,T]\times[S,T], \\ Z_{21}(t,s)=Z(t,s),t\in [2S-T,S]\times[2S-T,S],\\ Z_{22}(t,s)=Z(t,s),t\in [S,T]\times[2S-T,S]. \end{gather*} Set $Y_{1}^{(0)}(t)=0$, and for $n=1,2,\cdots,$ define the Picard iterations: \begin{equation}\label{10000} \begin{gathered} \begin{aligned} \psi^{S}_{n}(t) &=\psi(t)+\int_{S}^T g(t,s,Y_{2}(s),Y_{1}^{(n-1)}(t),Z_{11}(t,s),Z_{22}(s,t))\,\mathrm{d}s\\ &\quad -\int_{S}^TZ_{11}(t,s)\,\mathrm{d}W(s), \end{aligned} \\ \begin{aligned} Y_{1}^{(n)}(t) &=\psi^{S}_{n}(t)+\int_t^{S}g(t,s,Y_{1}^{(n)}(s),Y_{1} ^{(n)}(t),Z_{21}^{(n)}(t,s),Z_{21}^{(n)}(s,t))\,\mathrm{d}s \\ &\quad -\int_t^{S}Z_{21}^{(n)}(t,s)\,\mathrm{d}W(s), \end{aligned} \end{gathered} \end{equation} for $t\in [2S-T,S]$. Obviously, $(Y_{1}^{(n)}(t),Z_{21}^{(n)}(t,s))\in\mathcal {M}^2[2S-T,S]$. Moreover, by stability estimate \eqref{26} we can obtain \begin{align*} %\label{37} &E\Big[\int_{2S-T}^{S}| Y_{1}^{n}(t)-Y_{1}^{n-1}(t)|^2\,\mathrm{d}t+ \int_{2S-T}^{S}\int_{2S-T}^{S}| Z_{21}^{(n)}(t,s)-Z_{21}^{(n-1)}(t,s)|^2 \,\mathrm{d}s\,\mathrm{d}t\Big] \\ &\leq 8CE\Big\{\int_{2S-T}^{S}| \psi^{S}_{n}(t)-\psi^{S}_{n-1}(t)|^2\,\mathrm{d}t\Big\}.\\ \end{align*} By \eqref{24} and \eqref{16}, it is not difficult to see that \begin{align*} %\label{38} &E\Big[| \psi^{S}_{n}(t)-\psi^{S}_{n-1}(t)|^2+ \int_{S}^T| Z_{11}^{n}(t,s)-Z_{11}^{n-1}(t,s)|^2\,\mathrm{d}s\Big]\\ &\leq CKE|Y_{1}^{n-1}(t)-Y_{1}^{n-2}(t)|^2, \end{align*} and \begin{equation*} 8CE\int_{2S-T}^{S}|\psi_{n}^{S}(t)-\bar{\psi}_{n-1}^{S}(t)|^2\,\mathrm{d}t \leq 8C^2KE\int_{2S-T}^{S}| Y_{1}^{n-1}(t)-Y_{1}^{n-2}(t)|^2\,\mathrm{d}t. \end{equation*} Consequently, \begin{align*} &E\Big[\int_{2S-T}^{S}| Y_{1}^{n}(t)-Y_{1}^{n-1}(t)|^2\,\mathrm{d}t+ \int_{2S-T}^{S}\int_{2S-T}^{S}| Z_{21}^{(n)}(t,s)-Z_{21}^{(n-1)}(t,s)|^2\,\mathrm{d}s\,\mathrm{d}t\Big] \\ &\leq 8C^2KE\int_{2S-T}^{S}| Y_{1}^{n-1}(t)-Y_{1}^{n-2}(t)|^2\,\mathrm{d}t\\ &\leq \cdots \leq (8C^2K)^{n-1}E\int_{2S-T}^{S}| Y_{1}^{1}(t)-Y_{1}^{0}(t)|^2\,\mathrm{d}t. \end{align*} By \eqref{24} it is easy to see that $8C^2K<1$, then we obtain $(Y^{(n)}_{1}(\cdot),Z^{(n)}_{21}(\cdot,\cdot))$ and $(\psi^{S}_{n}(\cdot),Z_{11}^{(n)}(\cdot,\cdot))$ are Cauchy sequences on $ \mathcal {M}^2[2S-T,S]$ and $L_{\mathcal {F}_{T}}^{p}(S,T)\times L^{p}(2S-T,S;L^2_{\mathbb{F}}(S,T))$, respectively. If $n\to \infty$ in \eqref{10000}, we could obtain the unique adapted M-solution $(Y(t),Z(t,s))$ of \eqref{21} for $t\in[2S-T,S]$, $s\in[2S-T,T]$. Now, we give the estimate of solution for $(t,s)\in[2S-T,S]\times[2S-T,T]$. Since \begin{equation*}\label{250} \begin{split} &E\Big[\int_{2S-T}^{S}| Y_{1}(t)|^2\,\mathrm{d}t+\int_{2S-T}^{S} \int_{2S-T}^{S}| Z_{21}(t,s)|^2\,\mathrm{d}s\,\mathrm{d}t\Big]\\ &\leq 8CE\Big\{\int_{2S-T}^{S}|\psi^{S}(t)|^2\,\mathrm{d}t+\int_{2S-T}^{S}\Big(\int_t^{S}| g(t,s,0,0,0,0)| \,\mathrm{d}s\Big)^2\,\mathrm{d}t\Big\}, \end{split} \end{equation*} and \begin{align*}%\label{115} &E\Big[|\psi^{S}(t)|^2+\int_{S}^T| Z_{11}(t,s)|^2\,\mathrm{d}s\Big]\\ &\leq CE\Big\{| \psi(t)|^2+\Big(\int_{S}^T| g(t,s,Y_{2}(s),Y_{1}(t),0,Z_{22}(t,s))| \,\mathrm{d}s\Big)^2\Big\}\\ &\leq CE\Big\{|\psi(t)|^2+4\Big(\int_{S}^T| g(t,s,0,0,0,0)| \,\mathrm{d}s\Big)^2\Big\}\\ &\quad +4C(T-S)^{\frac{\varepsilon}{2+\varepsilon}}A^{\frac{2}{2+\varepsilon}}E\Big[\int_{S}^T| Y_{2}(s)|^2 \,\mathrm{d}s\\ &\quad +\int_{S}^T| Z_{22}(s,t)|^2 \,\mathrm{d}s\Big]+4C(T-S)^{\frac{\varepsilon}{2+\varepsilon}+1}A^{\frac{2}{2+\varepsilon}}E| Y_{1}(t)|^2. \end{align*} So, if $32(T-S)^{\frac{2\varepsilon}{2+\varepsilon}}A^{\frac{2}{2+\varepsilon}}C^2 \max\{1,T-S\}<\frac{1}{2}$, we have \begin{equation}\label{10001} \begin{split} &E\Big[\int_{2S-T}^{S}| \psi^{S}(t)|^2\,\mathrm{d}t+\int_{2S-T}^{S} \int_{S}^T| Z_{11}(t,s)|^2\,\mathrm{d}s\,\mathrm{d}t\Big] \\ &\leq (8C+3)E\Big\{\int_{2S-T}^T|\psi(t)|^2\,\mathrm{d}t+\int_{2S-T}^T\Big(\int_t^T| g(t,s,0,0,0,0)| \,\mathrm{d}s\Big)^2\,\mathrm{d}t\Big\}, \end{split} \end{equation} and \begin{equation}\label{10002} \begin{split} &E\Big[\int_{2S-T}^{S}| Y_{1}(t)|^2\,\mathrm{d}t+\int_{2S-T}^{S} \int_{2S-T}^{S}| Z_{21}(t,s)|^2\,\mathrm{d}s\,\mathrm{d}t\Big]\\ &\leq (8C+4)E\Big\{\int_{2S-T}^T|\psi(t)|^2\,\mathrm{d}t+\int_{2S-T}^T\Big(\int_t^T| g(t,s,0,0,0,0)| \,\mathrm{d}s\Big)^2\,\mathrm{d}t\Big\}. \end{split} \end{equation} Combining \eqref{10001} and \eqref{10002}, we show that \begin{align*} &E\Big\{\int_{2S-T}^T| Y(t)|^2\,\mathrm{d}t+\int_{2S-T}^T \int_{2S-T}^T| Z(t,s)|^2\,\mathrm{d}s\,\mathrm{d}t\Big\}\\ &\leq (8C+4)E\Big\{\int_{2S-T}^T|\psi(t)|^2\,\mathrm{d}t+ \int_{2S-T}^T\Big(\int_t^T| g(t,s,0,0,0,0)| \,\mathrm{d}s\Big)^2\,\mathrm{d}t\Big\} \end{align*} Similar to Step 1, stability estimate \eqref{26} holds for $t\in [2S-T,T]$. Then we can use induction method to finish the theorem. \end{proof} \subsection{Existence and uniqueness of M-solution for the BSVIE} \begin{equation}\label{44} \begin{aligned} Y(t)&=F(t,Y(t))+\int_t^Tg(t,s,Y(s),Y(t),Z(t,s),Z(s,t))\,\mathrm{d}s\\ &\quad -\int_t^TZ(t,s)\,\mathrm{d}W(s),~t\in [0,T]. \end{aligned}\end{equation} \begin{theorem} \label{thm3} If {\rm (H1)} and {\rm (H2)} hold for \eqref{44}. Then there admits unique adapted M-solution $(Y(\cdot),Z(\cdot,\cdot))\in \mathcal {H}^2[0,T]$, and the following estimate holds: \begin{equation}\label{47} \begin{split} &\| (Y(\cdot),Z(\cdot,\cdot))\| ^2_{\mathcal {H}^2[R,T]}\\ &\leq C_{F}E\Big\{\int_{R}^T| F(t,0)|^2\,\mathrm{d}t+\int_{R}^T\Big(\int_t^T| g(t,s,0,0,0,0)| \,\mathrm{d}s\Big)^2\,\mathrm{d}t\Big\},~~R\in[0,T], \end{split} \end{equation} where $C_{F}=\frac{2C_{L}}{1-2C_{L}D}$. Furthermore, if $\bar{g}$ also satisfies (H1), $\bar{F}$ satisfies (H2), and $(\bar{Y}(\cdot),\bar{Z}(\cdot,\cdot))\in \mathcal {H}^2[0,T] $ is the adapted M-solution of \eqref{44}, in which $(g,F)$ is replaced by $(\bar{g},\bar{F})$. Then \begin{equation}\label{484848} \begin{split} &E\Big[\int_{R}^T|Y(t)-\bar{Y}(t)|^2\,\mathrm{d}t+ \int_{R}^T\int_{R}^T|Z(t,s)-\bar{Z}(t,s)|^2\,\mathrm{d}s\,\mathrm{d}t\Big] \\ &\leq C_{F}E\Big\{\int_{R}^T| F(t,Y(t))-\bar{F}(t,Y(t))|^2\,\mathrm{d}t\\ &\quad +\int_{R}^T\Big(\int_t^T| g(t,s,Y(s),Y(t),Z(t,s),Z(t,s))\\ &\quad -\bar{g}(t,s,Y(s),Y(t),Z(t,s),Z(t,s))| \,\mathrm{d}s\Big)^2\,\mathrm{d}t\Big\},~~R\in[0,T]. \end{split} \end{equation} \end{theorem} \begin{proof} Let $Y^{(0)}(t)=0$. Since $F(t,Y^{(0)}(t)) \in L_{\mathcal {F}_{T}}^2(0,T)$, then for $n=1,2,\cdots$ we can define the Picard iterations: \begin{equation}\label{51} \begin{split} Y^{(n)}(t) &=F(t,Y^{(n-1)}(t))+\int_t^Tg(t,s,Y^{(n)}(s),Y^{(n)}(t),Z^{(n)}(t,s),Z^{(n)}(s,t))\,\mathrm{d}s \\ &\quad -\int_t^TZ^{(n)}(t,s)\,\mathrm{d}W(s),\quad t\in [0,T]. \end{split} \end{equation} In view of \eqref{26}, we have that \begin{align*}%\label{52} &E\Big[\int_{0}^T| Y^{(n)}(t)-Y^{n-1}(t)|^2\,\mathrm{d}t+ \int_{0}^T\int_{0}^T| Z^{n}(t,s)-Z^{n-1}(t,s)|^2\,\mathrm{d}s\,\mathrm{d}t\Big] \\ &\leq C_{L}D E\int_{0}^T| Y_{1}^{n-1}(t)-Y_{1}^{n-2}(t)|^2 \,\mathrm{d}t \leq \cdots \\ &\leq (C_{L}D)^{n-1}E\int_{0}^T| Y_{1}^{1}(t)-Y_{1}^{0}(t)|^2 \,\mathrm{d}t. \end{align*} By \eqref{46} we are sure that $(Y^{(n)}(\cdot),Z^{(n)}(\cdot,\cdot))$ is a Cauchy sequence on $\mathcal {M}^2[0,T]$. Let $n\to \infty$ in \eqref{51} we could obtain the unique solution of \eqref{44}. From \eqref{25}, we have \begin{equation}\label{101} \begin{aligned} &E\Big[\int_{0}^T| Y(t)|^2\,\mathrm{d}t+\int_{0}^T \int_{0}^T| Z(t,s)|^2\,\mathrm{d}s\,\mathrm{d}t\Big]\\ &\leq C_{L}E\Big\{2\int_{0}^T|F(t,0)|^2\,\mathrm{d}t+2D\int_{0}^T| Y(t)|^2\,\mathrm{d}t\\ &\quad +\int_{0}^T\Big(\int_t^T| g(t,s,0,0,0,0)| \,\mathrm{d}s\Big)^2\,\mathrm{d}t\Big\}. \end{aligned} \end{equation} So from \eqref{46} and \eqref{101} we conclude that \begin{align*} &E\Big[\int_{0}^T| Y(t)|^2\,\mathrm{d}t+\int_{0}^T \int_{0}^T| Z(t,s)|^2\,\mathrm{d}s\,\mathrm{d}t\Big]\\ &\leq C_{F}E\Big\{\int_{0}^T| F(t,0)|^2\,\mathrm{d}t +\int_{0}^T\Big(\int_t^T| g(t,s,0,0,0,0)| \,\mathrm{d}s\Big)^2\,\mathrm{d}t\Big\}, \end{align*} where $C_{F}=\frac{2C_{L}}{1-2C_{L}D}$. Similar to Step 1 of Theorem \ref{dingli2}, the stability estimate holds. The proof is complete. \end{proof} \subsection{Existence and uniqueness of M-solution for \eqref{53}} \begin{proof} Let $(Y^{(0)}(t),Z^{(0)}(t,s))=(0,0)$, then for $n=1,2,\cdots$ we can have the Picard iterations: \begin{equation}\label{60} \begin{split} Y^{n}(t) &=F(t,Y^{n-1}(t))-\int_t^T\Big[h(t,s,Y^{(n-1)}(s),Y^{(n-1)}(t),Z^{(n-1)}(t,s))\\ &\quad -Z^{(n-1)}(t,s)\Big] \,\mathrm{d}W(s)\\ &\quad +\int_t^Tg(t,s,Y^{n}(s),Y^{n}(t),Z^{n}(t,s),Z^{n}(s,t))\,\mathrm{d}s\\ &\quad -\int_t^TZ^{n}(t,s)\,\mathrm{d}W(s),\quad t\in [0,T]. \end{split} \end{equation} By \eqref{484848} and It\^{o}'s isometry, we have \begin{equation}\label{61} \begin{split} E&\Big[\int_{0}^T| Y^{(n)}(t)-Y^{(n-1)}(t)|^2\,\mathrm{d}t+ \int_{0}^T\int_{0}^T| Z^{(n)}(t,s)-Z^{(n-1)}(t,s)|^2\,\mathrm{d}s\,\mathrm{d}t\Big] \\ &\leq C_{F}E\Big\{\int_{0}^T\int_t^T| h(t,s,Y^{(n-1)}(s),Y^{(n-1)}(t),Z^{(n-1)}(t,s))-Z^{(n-1)}(t,s)\\ &\quad - [h(t,s,Y^{(n-2)}(s),Y^{(n-2)}(t),Z^{(n-2)}(t,s))-Z^{(n-2)}(t,s)]|^2\,\mathrm{d}s\,\mathrm{d}t\Big\}\\ &\leq C_{F}L_{\xi}E\int_{0}^T\int_{0}^T| Z^{(n-1)}(t,s)-Z^{(n-2)}(t,s)|^2\,\mathrm{d}s\,\mathrm{d}t\\ &\quad +C_{F}(L_{\zeta}+L_{\eta})TE\int_{0}^T(Y^{(n-1)}(s)-Y^{(n-2)}(s)) \,\mathrm{d}s. \end{split} \end{equation} By \eqref{54}, we see that $\max\{C_{F}L_{\zeta},C_{F}(L_{\zeta}+L_{\eta})T\}=M<1$, and then \begin{align*} &E\Big[\int_{0}^T| Y^{(n)}(t)-Y^{(n-1)}(t)|^2\,\mathrm{d}t+ \int_{0}^T\int_{0}^T| Z^{(n)}(t,s)-Z^{(n-1)}(t,s)|^2\,\mathrm{d}s\,\mathrm{d}t\Big] \\ &\leq ME\Big[\int_{0}^T| Y^{(n-1)}(t)-Y^{(n-2)}(t)|^2\,\mathrm{d}t\\ &\quad +\int_{0}^T\int_{0}^T|Z^{(n-1)}(t,s)-Z^{(n-2)}(t,s)|^2\,\mathrm{d}s \,\mathrm{d}t\Big] \leq \cdots \\ &\leq M^{n-1}E\Big[\int_{0}^T| Y^{(1)}(t)-Y^{(0)}(t)|^2\,\mathrm{d}t+ \int_{0}^T\int_{0}^T| Z^{(1)}(t,s)-Z^{(0)}(t,s)|^2\,\mathrm{d}s\,\mathrm{d}t\Big] \end{align*} holds. So we obtain $(Y^{(n)}(\cdot),Z^{(n)}(\cdot,\cdot))$ is a Cauchy sequence on $ \mathcal {M}^2[0,T]$. Let $n\to \infty$ in \eqref{60} we obtain the unique adapted M-solution of \eqref{53}. From \eqref{47}, we have \begin{align*} %\label{470} &E\Big[\int_{0}^T| Y(t)|^2\,\mathrm{d}t+\int_{0}^T \int_{0}^T| Z(t,s)|^2\,\mathrm{d}s\,\mathrm{d}t\Big]\\ &\leq C_{F}E\Big\{\int_{0}^T| F(t,0)-\int_t^T\Big[h(t,s,0,0,Z(t,s))-Z(t,s)\Big]\,\mathrm{d}W(s)|^2\,\mathrm{d}t\\ &\quad +\int_{0}^T\Big(\int_t^T| g(t,s,0,0,0,0)| \,\mathrm{d}s\Big)^2\,\mathrm{d}t\Big\}\\ &\leq C_{F}E\Big\{\int_{0}^T| F(t,0)|^2\,\mathrm{d}t+4L_{\xi}\int_{0}^T\int_{0}^T| Z(t,s)|^2\,\mathrm{d}s\,\mathrm{d}t\\ &\quad +4\int_{0}^T\int_t^T| h(t,s,0,0,0)|^2\,\mathrm{d}s\,\mathrm{d}t+\int_{0}^T\Big(\int_t^T| g(t,s,0,0,0,0)| \,\mathrm{d}s\Big)^2\,\mathrm{d}t\Big\}. \end{align*} So \begin{equation}\label{55b} \begin{split} &E\Big[\int_{0}^T| Y(t)|^2\,\mathrm{d}t+\int_{0}^T \int_{0}^T| Z(t,s)|^2\,\mathrm{d}s\,\mathrm{d}t\Big]\\ &\leq C_{h}E\Big\{\int_{0}^T| F(t,0)|^2\,\mathrm{d}t+ \int_{0}^T\Big(\int_t^T| g(t,s,0,0,0,0)| \,\mathrm{d}s\Big)^2\,\mathrm{d}t\\ &\quad +\int_{0}^T\int_t^T| h(t,s,0,0,0)|^2\,\mathrm{d}s\,\mathrm{d}t\Big\} \end{split} \end{equation} where $C_{h}=\frac{4C_{F}}{1-4L_{\xi}C_{F}}$. 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