$L^p$ solutions of doubly reflected BSDEs under general assumptions

TL;DR

This paper establishes existence, uniqueness, and approximation results for $L^p$ solutions of doubly reflected backward stochastic differential equations under a generalized Mokobodzki condition, broadening the understanding of their solvability.

Contribution

It introduces a generalized Mokobodzki condition for doubly reflected BSDEs and proves its necessity and sufficiency for $L^p$ solutions, along with approximation methods.

Findings

Existence and uniqueness of $L^p$ solutions under generalized Mokobodzki condition.

Necessity of the Mokobodzki condition for solution existence.

Solutions can be approximated via penalization and solution sequences.

Abstract

Under a generalized Mokobodzki condition for reflected BSDEs with two continuous barriers which relates the growth of the generator $g$ and that of the barriers, we establish several existence and uniqueness results on $L^p\ (p>1)$ solutions of doubly reflected BSDEs with generators satisfying a one-sided Osgood condition together with a general growth in the state variable $y$, and a uniform continuity condition or a linear growth condition in the state variable $z$. This Mokobodzki condition is also proved to be necessary for existence of the $L^p$ solutions. And, we prove that the $L^p$ solutions can be approximated by the penalization method and by some sequences of the $L^p$ solutions of doubly reflected BSDEs.

Equations

$$\mathbb{P}(\{\omega :X_t(\omega )=Y_t(\omega ),t\in [0,T]\})=1,$$

$$\forall t\in [0,T],\mathbb{P}(\{\omega :X_t(\omega )=Y_t(\omega )\})=1.$$

$$\Vert \xi {\Vert }_{{\mathbb{L}}^p}:={\left(\mathbb{E}[|\xi {|}^p]\right)}^{1\wedge 1/p}<+\infty ;\text{vspace}-0.2cm$$

$$\mathbb{P}\left({\int }_0^T|X_t|\mathrm{d}t<+\infty \right)=1;\text{vspace}-0.3cm$$

$$\mathbb{P}\left({\int }_0^T|Z_t{|}^2\mathrm{d}t<+\infty \right)=1;\text{vspace}-0.3cm$$

$$\mathbb{E}\left[{\int }_0^T\text{mathbbm}{1}_D(t,\omega )\mathrm{d}{K}_t^1(\omega )\right]=\mathbb{E}\left[{\int }_0^T\text{mathbbm}{1}_{D^c}(t,\omega )\mathrm{d}{K}_t^2(\omega )\right]=0.$$

$$\mathbb{E}\left[{\int }_0^T\text{mathbbm}{1}_D(t,\omega )\mathrm{d}{K}_t^1(\omega )\right]\le \mathbb{E}\left[{\int }_0^T\text{mathbbm}{1}_D(t,\omega )\mathrm{d}{K}_t^2(\omega )\right],i.e.,K_t^1-K_s^1\le K_t^2-K_s^2,0\le s\le t\le T.$$

$$g(\omega ,t,y,z):\Omega \times [0,T]\times \mathbb{R}\times {\mathbb{R}}^d⟼\mathbb{R}$$

$$Y_t=\xi +{\int }_t^{T}g(s,Y_s,Z_s)\mathrm{d}s+{\int }_t^T\mathrm{d}{V}_s-{\int }_t^T{Z}_s\cdot \mathrm{d}{B}_s,t\in [0,T].$$

$$\left\{\begin{array}[]{l}\displaystyle Y_{t}=\xi+\int_{t}^{T}g(s,Y_{s},Z_{s}){\rm d}s+\int_{t}^{T}{\rm d}V_{s}+\int_{t}^{T}{\rm d}K_{s}-\int_{t}^{T}Z_{s}\cdot{\rm d}B_{s},\ \ t\in[0,T],\vspace{0.2cm}\\ \displaystyle L_{t}\leq Y_{t},\ \ t\in[0,T]\ \ {\rm and}\ \ \int_{0}^{T}(Y_{t}-L_{t}){\rm d}K_{t}=0.\end{array}\right.$$

$$\left\{\begin{array}[]{l}\displaystyle Y_{t}=\xi+\int_{t}^{T}g(s,Y_{s},Z_{s}){\rm d}s+\int_{t}^{T}{\rm d}V_{s}-\int_{t}^{T}{\rm d}A_{s}-\int_{t}^{T}Z_{s}\cdot{\rm d}B_{s},\ \ t\in[0,T],\vspace{0.2cm}\\ \displaystyle Y_{t}\leq U_{t},\ \ t\in[0,T]\ \ {\rm and}\ \ \int_{0}^{T}(U_{t}-Y_{t}){\rm d}A_{t}=0.\end{array}\right.$$

$$\left\{\begin{array}[]{l}\displaystyle Y_{t}=\xi+\int_{t}^{T}g(s,Y_{s},Z_{s}){\rm d}s+\int_{t}^{T}{\rm d}V_{s}+\int_{t}^{T}{\rm d}K_{s}-\int_{t}^{T}{\rm d}A_{s}-\int_{t}^{T}Z_{s}\cdot{\rm d}B_{s},\ \ t\in[0,T],\vspace{0.2cm}\\ \displaystyle L_{t}\leq Y_{t}\leq U_{t},\ \ t\in[0,T],\ \int_{0}^{T}(Y_{t}-L_{t}){\rm d}K_{t}=\int_{0}^{T}(U_{t}-Y_{t}){\rm d}A_{t}=0\ \ {\rm and}\ \ {\rm d}K\bot{\rm d}A.\end{array}\right.\vspace{0.3cm}$$

$$\bar{\xi }:=-\xi ,{\bar{V}}_{\cdot }:=-V_{\cdot },{\bar{U}}_{\cdot }:=-L_{\cdot },\bar{g}(\omega ,t,y,z):=-g(\omega ,t,-y,-z).$$

$$Y_t\le Y_t^{\prime },t\in [0,T](resp.Y_t\ge Y_t^{\prime },t\in [0,T]).$$

$$(g(\omega ,t,y_1,z)-g(\omega ,t,y_2,z))\mathrm{sgn}(y_1-y_2)\le \rho (|y_1-y_2|).$$

$$|g(\omega ,t,y,z_1)-g(\omega ,t,y,z_2)|\le \varphi (|z_1-z_2|).$$

$$|g(\omega ,t,y,z)-g(\omega ,t,y,0)|\le f_t(\omega )+\mu |y|+\lambda |z|.$$

$$\forall x\in {\mathbb{R}}_{+},\rho (x)\le A(x+1)\mathrm{and}\varphi (x)\le A(x+1).$$

$$(g(\omega ,t,y_1,z)-g(\omega ,t,y_2,z))\mathrm{sgn}(y_1-y_2)\le \mu |y_1-y_2|.$$

$$|g(\omega ,t,y,z_1)-g(\omega ,t,y,z_2)|\le \lambda |z_1-z_2|.$$

$$g(\omega ,t,y,z)\mathrm{sgn}(y)\le {\bar{f}}_t(\omega )+\bar{\mu }|y|+\bar{\lambda }|z|,$$

$$|g(\omega ,t,y,z)|\le f_t(\omega )+{\psi }_t(\omega ,|y|)+\lambda |z|,$$

$${\bar{Y}}_t={\bar{Y}}_T+{\int }_t^T\mathrm{d}{\bar{V}}_s-{\int }_t^T{\bar{Z}}_s\cdot \mathrm{d}{B}_s,t\in [0,T].$$

$$\mathbb{E}\left[{\left({\int }_{t\wedge \tau }^{T\wedge \tau }|{\bar{Z}}_s{|}^2\mathrm{d}s\right)}^{\frac{p}{2}}|{\mathcal{F}}_t\right]\le C_1\mathbb{E}\left[\underset{s\in [t,T]}{\sup }|{\bar{Y}}_{s\wedge \tau }{|}^p+\underset{s\in [t,T]}{\sup }{\left[{\left({\int }_{s\wedge \tau }^{T\wedge \tau }{\bar{Y}}_r\mathrm{d}{\bar{V}}_r\right)}^{+}\right]}^{\frac{p}{2}}|{\mathcal{F}}_t\right].$$

$$\begin{array}[]{ll}&\displaystyle\mathbb{E}\left[\left.\sup\limits_{s\in[t,T]}|\bar{Y}_{s\wedge\tau}|^{p}+\int_{t\wedge\tau}^{T\wedge\tau}|\bar{Y}_{s}|^{p-2}\mathbbm{1}_{\{|\bar{Y}_{s}|\neq 0\}}|\bar{Z}_{s}|^{2}{\rm d}s\right|\mathcal{F}_{t}\right]\vspace{0.1cm}\\ \leq&\displaystyle C_{2}\mathbb{E}\left[\left.|\bar{Y}_{\tau}|^{p}+\sup\limits_{s\in[t,T]}\left(\int_{s\wedge\tau}^{T\wedge\tau}|\bar{Y}_{r}|^{p-1}{\rm sgn}(\bar{Y}_{r}){\rm d}\bar{V}_{r}\right)^{+}\right|\mathcal{F}_{t}\right].\end{array}\vspace{0.2cm}$$

$$\begin{array}[]{l}\displaystyle\mathbb{E}\left[\left.\sup\limits_{s\in[t,T]}|Y_{s}|^{p}+\left(\int_{t}^{T}|Z_{s}|^{2}{\rm d}s\right)^{p\over 2}+\left(\int_{t}^{T}|g(s,Y_{s},Z_{s})|{\rm d}s\right)^{p}\right|\mathcal{F}_{t}\right]\\ \ \ \ \ \leq\displaystyle C\mathbb{E}\left[\left.|\xi|^{p}+|V|^{p}_{t,T}+\left(\int_{t}^{T}\bar{f}_{s}\ {\rm d}s\right)^{p}\right|\mathcal{F}_{t}\right].\vspace{0.3cm}\end{array}$$

$$\begin{array}[]{ll}&\displaystyle\mathbb{E}\left[\left.\left(\int_{t}^{T}|Z_{s}|^{2}{\rm d}s\right)^{p\over 2}+|K_{T}-K_{t}|^{p}+\left(\int_{t}^{T}|g(s,Y_{s},Z_{s})|{\rm d}s\right)^{p}\right|\mathcal{F}_{t}\right]\vspace{0.1cm}\\ \leq&\displaystyle C\mathbb{E}\left[\sup\limits_{s\in[t,T]}|Y_{s}|^{p}+|V|^{p}_{t,T}+\sup\limits_{s\in[t,T]}|\bar{X}_{s}|^{p}+\left(\int_{t}^{T}f_{s}{\rm d}s\right)^{p}+1\right.\vspace{0.1cm}\\ &\hskip 28.45274pt\displaystyle+\left.\left.\left(\int_{t}^{T}|g(s,\bar{X}_{s},0)|\ {\rm d}s\right)^{p}+\left(\int_{t}^{T}|g(s,0,0)|\ {\rm d}s\right)^{p}\right|\mathcal{F}_{t}\right].\end{array}$$

$$\begin{array}[]{ll}&\displaystyle\mathbb{E}\left[\left.\left(\int_{t}^{T}|Z_{s}|^{2}{\rm d}s\right)^{p\over 2}+|A_{T}-A_{t}|^{p}+\left(\int_{t}^{T}|g(s,Y_{s},Z_{s})|{\rm d}s\right)^{p}\right|\mathcal{F}_{t}\right]\vspace{0.1cm}\\ \leq&\displaystyle C\mathbb{E}\left[\sup\limits_{s\in[t,T]}|Y_{s}|^{p}+|V|^{p}_{t,T}+\sup\limits_{s\in[t,T]}|\underline{X}_{s}|^{p}+\left(\int_{t}^{T}f_{s}{\rm d}s\right)^{p}+1\right.\vspace{0.1cm}\\ &\hskip 28.45274pt\displaystyle+\left.\left.\left(\int_{t}^{T}|g(s,\underline{X}_{s},0)|\ {\rm d}s\right)^{p}+\left(\int_{t}^{T}|g(s,0,0)|\ {\rm d}s\right)^{p}\right|\mathcal{F}_{t}\right].\end{array}\vspace{0.2cm}$$

$$\begin{array}[]{lll}\displaystyle g(t,y,z){\rm sgn}(y)&\leq&\displaystyle|(g(t,y,z)-g(t,y,0)){\rm sgn}(y)|+(g(t,y,0)-g(t,0,0)){\rm sgn}(y)+|g(t,0,0))|\\ &\leq&\displaystyle f_{t}+\mu|y|+\lambda|z|+\rho(|y|)+|g(t,0,0))|\\ &\leq&\displaystyle f_{t}+|g(t,0,0))|+A+(\mu+A)|y|+\lambda|z|,\ \ \ \forall\ y\in{\mathbb{R}}$, $z\in{\mathbb{R}}^{d}.\end{array}$$

$$\begin{array}[]{lll}\displaystyle-g(t,Y_{t},Z_{t})&\leq&\displaystyle\left(g(t,\bar{X}_{t},Z_{t})-g(t,Y_{t},Z_{t})\right)+\left|g(t,\bar{X}_{t},0)-g(t,\bar{X}_{t},Z_{t})\right|+|g(t,\bar{X}_{t},0)|\\ &\leq&\displaystyle\rho(|\bar{X}_{t}-Y_{t}|)+f_{t}+\mu|\bar{X}_{t}|+\lambda|Z_{t}|+|g(t,\bar{X}_{t},0)|\\ &\leq&\displaystyle|g(t,\bar{X}_{t},0)|+(\mu+A)|\bar{X}_{t}|+f_{t}+A+A|Y_{t}|+\lambda|Z_{t}|.\end{array}$$

Full text

                 Comparison theorem, Penalization method, $L^{p}$ solution

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1 Introduction

Backward stochastic differential equations (BSDEs for short) were first introduced in linear case by Bismut [4] in 1973, and extended to a fully nonlinear version at the first time by Pardoux and Peng [48] in 1990. Later on, as a variation of the notion of nonlinear BSDEs, nonlinear reflected BSDEs (RBSDEs for short) with one and two continuous barriers were introduced by El Karoui et al. [10] and Cvitanić and Karatzas [8] respectively. At present it has been widely recognized that these equations have natural connections with many problems in different mathematical fields, such as partial differential equations, mathematical finance, stochastic control and game theory, optimal switching problem and other optimality problems and others (see, e.g. [2, 3, 10, 11, 12, 21, 25, 27, 29, 30, 35, 43, 47, 49, 50, 51, 52, 53, 54], etc.), and they provide a very useful and efficient tool for studying these problems.

In Pardoux and Peng [48], El Karoui et al. [10] and Cvitanić and Karatzas [8], the existence and uniqueness result of $L^{2}$ solutions of non-reflected BSDEs and RBSDEs with one and two continuous barriers are established under the standard assumption that the generator $g$ satisfies the linear growth condition and is Lipschitz continuous with respect to the state variables $y$ and $z$. Many attempts have been made to relax these assumptions, which are too strong for many interesting applications mentioned above. For example, some works were devoted to solving RBSDEs with less regular barriers, see [3, 21, 22, 23, 38, 42, 52], etc; many scholars were interested in the existence and uniqueness of solutions for non-reflected BSDEs and RBSDEs with data that are in the spaces of $L^{p}\ (p>1)$ and $L^{1}$, see [6, 7, 9, 12, 15, 33, 44] for non-reflected BSDEs, and [1, 2, 28, 37, 38, 55] for RBSDEs; and more papers focused their attention on weakening the linear growth condition and Lipschitz-continuity condition of the generator $g$ with respect to the state variables $y$ and $z$, see [5, 6, 7, 12, 15, 17, 18, 19, 20, 31, 33, 34, 35, 44, 45, 47] for non-reflected BSDEs, and [1, 2, 3, 13, 14, 16, 24, 26, 29, 32, 36, 37, 38, 39, 40, 41, 46, 55, 56, 57] for RBSDEs.

The present paper is the continuation along the last two research directions, which is devoted to solving RBSDEs with two continuous barriers in the space of $L^{p}\ (p>1)$ under general assumptions. The generator $g$ of doubly reflected BSDEs only needs to satisfy a one-sided Osgood condition together with a general growth in the state variable $y$, and a uniform continuity condition or a linear growth condition in the state variable $z$ (see assumptions (H1), (H3), (H2) and (H2’) in Section 2 respectively), which are weaker than those in many existing works. A generalized Mokobodzki condition (see assumption (H4) in Section 2) for doubly reflected BSDEs which relates the growth of the generator $g$ and that of the barriers is put forward and proved to be necessary and sufficient for existence of the $L^{p}$ solutions. Based on a combination between existing methods, their refinement and perfection, and some new ideas, we prove that the $L^{p}$ solutions of the doubly reflected BSDEs can be approximated by the penalization method and by some sequences of the $L^{p}$ solutions of doubly reflected BSDEs. In particular, we also consider the case that the generator $g$ may be discontinuous and have a general growth in the state variable $y$.

The rest of this paper is organized as follows. Section 2 contains some notations, definitions, assumptions together with some lemmas and propositions which will be frequently used later. In Section 3, by Theorem 3.15 we verify necessity of the generalized Mokobodzki condition to ensure existence of the $L^{p}$ solutions. Then, we prove a general comparison theorem of the $L^{p}$ solutions for RBSDEs with two continuous barriers, which naturally yields uniqueness of the $L^{p}$ solutions under assumptions (H1) and (H2) (ii), see Proposition 3.16 and Theorem 3.19. In Section 4, by Proposition 4.20 we show the convergence of $L^{p}$ solutions of the penalization equations for RBSDEs with one continuous barrier under some elementary conditions. And, by Proposition 4.21 we establish an a priori uniform estimate on the $L^{p}$ solutions of the penalization equations for RBSDEs with one continuous barrier and non-reflected BSDEs under assumptions (H1)-(H4). Then, based on Propositions 3.16, 4.20 and 4.21 together with Remark 4.22, by Theorem 4.23 (resp. Theorem 4.25) we prove existence and uniqueness of the $L^{p}$ solutions (resp. existence of the maximal and minimal $L^{p}$ solutions) for the doubly reflected BSDEs under assumptions (H1), (H2) (resp. (H2’)), (H3) and (H4). In Section 5, by Proposition 5.30 we establish a general approximation result for the $L^{p}$ solutions of the doubly reflected BSDEs under some elementary assumptions, and based on it we prove an existence result of the minimal (maximal) $L^{p}$ solutions for the doubly reflected BSDEs under several weaker assumptions, see Theorem 5.33, where the generator $g$ has a general growth in $y$ and a linear growth in $z$, but it is interesting that $g$ may be discontinuous in $y$ as considered in Fan and Jiang [19] and Zheng and Zhou [57], see assumptions (A1a) and (A1b) in Section 5. Finally, at the end of Sections 4 and 5, we provide several examples which the results of this paper can (but any existing works can not) be applied to, and give some remarks to illustrate further our theoretical results, see Remarks 4.29 and 5.37 for more details. And, some details for the proof of Propositions 4.20 and 5.30 are provided in Appendix.

At the end of the introduction, we would like to mention that the results obtained in this paper improve considerably some known works. In particular, these results extend the corresponding results in Fan [16] for RBSDEs with one continuous barrier to the case of RBSDEs with two continuous barriers, and strengthen the corresponding results in Klimsiak [38], where the generator $g$ needs to satisfy some stronger assumptions (see assumptions (H1s) and (H2s) in Section 2) than ours.

2 Preliminaries

2.1 Notations

In the whole paper we fix a real number $T>0$ and a positive integer $d$, and let $(B_{t})_{t\in[0,T]}$ be a standard $d$-dimensional Brownian motion defined on some complete filtered probability space $(\Omega,\mathcal{F},\mathbb{P})$, where $(\mathcal{F}_{t})_{t\in[0,T]}$ is the completed $\sigma$-algebra filtration generated by $(B_{t})_{t\in[0,T]}$ and $\mathcal{F}=\mathcal{F}_{T}$. We assume that if there is not a special illustration, all processes of this paper are defined on $\Omega\times[0,T]$, all notions whose definitions are related to some filtration are understood with respect to the filtration $(\mathcal{F}_{t})_{t\in[0,T]}$, and all equalities and inequalities between random elements are understood to hold $\mathbb{P}-a.s.$ To avoid ambiguity we stress that writing $X_{t}=Y_{t},\ t\in[0,T]$ we mean that

[TABLE]

while writing $X_{t}=Y_{t}$ for each $t\in[0,T]$ we mean that

[TABLE]

It is clear that they are equivalent if $(X_{t})_{t\in[0,T]}$ and $(Y_{t})_{t\in[0,T]}$ are both continuous processes. In what follows, the variable $\omega$ in random elements is often omitted as usually done.

Denote ${\mathbb{R}}_{+}:=[0,+\infty)$, $a^{+}:=\max\{a,0\}$ and $a^{-}:=(-a)^{+}$ for any real number $a$. For a set $A$, we denote by $A^{c}$ the complement of $A$ and by $\mathbbm{1}_{A}$ the indicator function of $A$. Let ${\rm sgn}(x)$ represent the sign of a real number $x$. For $n\geq 1$, the Euclidean norm of an element $y\in{\mathbb{R}}^{n}$ will be denoted by $|y|$.

For $p>0$, we define the following spaces:

• •

${\mathbb{L}}^{p}(\mathcal{F}_{T})$ the set of all $\mathcal{F}_{T}$-measurable real-valued random variables $\xi$ satisfying

[TABLE]

• •

$\mathcal{H}$ the set of all progressively measurable real-valued processes $X_{\cdot}$ satisfying

[TABLE]

• •

$\mathcal{H}^{p}$ the set of all processes $X_{\cdot}\in\mathcal{H}$ satisfying $\|X\|_{\mathcal{H}^{p}}:=\left\{\mathbb{E}\left[\left(\int_{0}^{T}|X_{t}|{\rm d}t\right)^{p}\right]\right\}^{1\wedge 1/p}<+\infty;$

• •

$\mathcal{S}$ the set of all progressively measurable and continuous real-valued processes;

• •

$\mathcal{S}^{p}$ the set of all processes $Y_{\cdot}\in\mathcal{S}$ satisfying $\|Y\|_{{\mathcal{S}}^{p}}:=\left(\mathbb{E}[\sup_{t\in[0,T]}|Y_{t}|^{p}]\right)^{1\wedge 1/p}<+\infty;$

• •

${\rm M}$ the set of all progressively measurable ${\mathbb{R}}^{d}$-valued processes $Z_{\cdot}$ satisfying

[TABLE]

• •

${\rm M}^{p}$ the set of all processes $Z_{\cdot}\in{\rm M}$ satisfying $\|Z\|_{{\rm M}^{p}}:=\left\{\mathbb{E}\left[\left(\int_{0}^{T}|Z_{t}|^{2}{\rm d}t\right)^{p/2}\right]\right\}^{1\wedge 1/p}<+\infty;$

• •

$\mathcal{M}$ the set of all continuous real-valued local martingales;

• •

$\mathcal{M}^{p}$ the set of all martingales $M_{\cdot}\in\mathcal{M}$ satisfying $\mathbb{E}\left[\left(\langle M\rangle_{T}\right)^{p/2}\right]<+\infty$;

• •

$\mathcal{V}$ the set of all progressively measurable and continuous real-valued processes of finite variation;

• •

$\mathcal{V}^{+}$ the set of all increasing processes $V_{\cdot}\in\mathcal{V}$ valued [math] at [math];

• •

$\mathcal{V}^{p}$ the set of all processes $V_{\cdot}\in\mathcal{V}$ satisfying $\mathbb{E}\left[|V|^{p}_{T}\right]<+\infty$;

• •

$\mathcal{V}^{+,p}$ the set of all processes $V_{\cdot}\in\mathcal{V}^{+}$ satisfying $\mathbb{E}\left[|V|^{p}_{T}\right]<+\infty$.

Here and hereafter, for each $V_{\cdot}\in\mathcal{V}$ and $t\in[0,T]$, $|V|_{t,T}$ denotes the random finite variation of $V_{\cdot}$ on the interval $[t,T]$, and $|V|_{0,T}$ is denoted simply by $|V|_{T}$. Clearly, if $V_{\cdot}\in\mathcal{V}^{+}$, then $|V|_{t,T}=V_{T}-V_{t}$.

2.2 Definitions

We now recall a definition used in Essaky and Hassani [14].

Definition 2.1**.**

For any two processes $K^{1}_{\cdot}$ and $K^{2}_{\cdot}$ in $\mathcal{V}^{1}$, we say that

• •

${\rm d}K^{1}\bot{\rm d}K^{2}$* if and only if there exists a progressively measurable set $D\subset\Omega\times[0,T]$ such that*

[TABLE]

• •

${\rm d}K^{1}\leq{\rm d}K^{2}$* if and only if for each progressively measurable set $D\subset\Omega\times[0,T]$,*

[TABLE]

In the rest of this paper, we always assume that $\xi$ is an $\mathcal{F}_{T}$-measurable random variable, $V_{\cdot}\in\mathcal{V}$, $L_{\cdot}\in\mathcal{S}$ (or $L_{\cdot}\equiv-\infty$) and $U_{\cdot}\in\mathcal{S}$ (or $U_{\cdot}\equiv+\infty$) with $L_{\cdot}\leq U_{\cdot}$, and that a random function

[TABLE]

is progressively measurable for each $(y,z)$, which is usually called a generator.

Definition 2.2**.**

By a solution to BSDE $(\xi,g+{\rm d}V)$ we understand a pair of progressively measurable processes $(Y_{t},Z_{t})_{t\in[0,T]}$ belonging to $\mathcal{S}\times{\rm M}$ such that $\mathbb{P}-a.s.$,

[TABLE]

By a solution to RBSDE $(\xi,g+{\rm d}V,L)$ we understand a triple of progressively measurable processes $(Y_{t},Z_{t},K_{t})_{t\in[0,T]}$ belonging to $\mathcal{S}\times{\rm M}\times\mathcal{V}^{+,1}$ such that $\mathbb{P}-a.s.$,

[TABLE]

By a solution to $\bar{R}$BSDE $(\xi,g+{\rm d}V,U)$ we understand a triple of progressively measurable processes $(Y_{t},Z_{t},A_{t})_{t\in[0,T]}$ belonging to $\mathcal{S}\times{\rm M}\times\mathcal{V}^{+,1}$ such that $\mathbb{P}-a.s.$,

[TABLE]

By a solution to DRBSDE $(\xi,g+{\rm d}V,L,U)$ we understand a quadruple of progressively measurable processes $(Y_{t},Z_{t},K_{t},A_{t})_{t\in[0,T]}$ belonging to $\mathcal{S}\times{\rm M}\times\mathcal{V}^{+,1}\times\mathcal{V}^{+,1}$ such that $\mathbb{P}-a.s.$,

[TABLE]

Remark 2.3**.**

It is not hard to verify the following assertions.

• (i)

The claim that $(Y_{t},Z_{t},K_{t})_{t\in[0,T]}$ is a solution of RBSDE $(\xi,g+{\rm d}V,L)$ is equivalent to the claim that $(Y_{t},Z_{t},K_{t},0)_{t\in[0,T]}$ is a solution of DRBSDE $(\xi,g+{\rm d}V,L,+\infty)$.

• (ii)

*The claim that $(Y_{t},Z_{t},A_{t})_{t\in[0,T]}$ is a solution of *$\bar{R}$BSDE $(\xi,g+{\rm d}V,U)$ is equivalent to the claim that $(Y_{t},Z_{t},0,A_{t})_{t\in[0,T]}$ is a solution of DRBSDE $(\xi,g+{\rm d}V,-\infty,U)$.

• (iii)

The claim that $(Y_{t},Z_{t})_{t\in[0,T]}$ is a solution of BSDE $(\xi,g+{\rm d}V)$ is equivalent to anyone of the following three claims:

• –

$(Y_{t},Z_{t},0)_{t\in[0,T]}$* is a solution of RBSDE $(\xi,g+{\rm d}V,-\infty)$;*

• –

$(Y_{t},Z_{t},0)_{t\in[0,T]}$* is a solution of *$\bar{R}$BSDE $(\xi,g+{\rm d}V,+\infty)$;

• –

$(Y_{t},Z_{t},0,0)_{t\in[0,T]}$* is a solution of DRBSDE $(\xi,g+{\rm d}V,-\infty,+\infty)$.*

• (iv)

*The claim that $(Y_{t},Z_{t},K_{t})_{t\in[0,T]}$ is a solution of RBSDE $(\xi,g+{\rm d}V,L)$ is equivalent to the claim that $(\bar{Y}_{t},\bar{Z}_{t},\bar{A}_{t})_{t\in[0,T]}:=(-Y_{t},-Z_{t},K_{t})_{t\in[0,T]}$ is a solution of *$\bar{R}$BSDE $(\bar{\xi},\bar{g}+{\rm d}\bar{V},\bar{U})$, where

[TABLE]

Definition 2.4**.**

A solution $(Y_{t},Z_{t})_{t\in[0,T]}$ of BSDE $(\xi,g+{\rm d}V)$ is called the minimal (resp. maximal) one in some space if $(Y_{\cdot},Z_{\cdot})$ belongs to this space, and for any solution $(Y^{\prime}_{t},Z^{\prime}_{t})_{t\in[0,T]}$ of BSDE $(\xi,g+{\rm d}V)$ in this space, we have

[TABLE]

Similarly, we can define that

• •

A solution $(Y_{t},Z_{t},K_{t})_{t\in[0,T]}$ of RBSDE $(\xi,g+{\rm d}V,L)$ is called the minimal (resp. maximal) one in some space if $(Y_{\cdot},Z_{\cdot},K_{\cdot})$ belongs to this space, and (2.5) holds for any solution $(Y^{\prime}_{t},Z^{\prime}_{t},K^{\prime}_{t})_{t\in[0,T]}$ of RBSDE $(\xi,g+{\rm d}V,L)$ in this space.

• •

*A solution $(Y_{t},Z_{t},A_{t})_{t\in[0,T]}$ of *$\bar{R}$*BSDE $(\xi,g+{\rm d}V,U)$ is called the minimal (resp. maximal) one in some space if $(Y_{\cdot},Z_{\cdot},A_{\cdot})$ belongs to this space, and (2.5) holds for any solution $(Y^{\prime}_{t},Z^{\prime}_{t},A^{\prime}_{t})_{t\in[0,T]}$ of *$\bar{R}$BSDE $(\xi,g+{\rm d}V,U)$ in this space.

• •

A solution $(Y_{t},Z_{t},K_{t},A_{t})_{t\in[0,T]}$ of DRBSDE $(\xi,g+{\rm d}V,L,U)$ is called the minimal (resp. maximal) one in some space if $(Y_{\cdot},Z_{\cdot},K_{\cdot},A_{\cdot})$ belongs to this space, and (2.5) holds for any solution $(Y^{\prime}_{t},Z^{\prime}_{t},K^{\prime}_{t},A^{\prime}_{t})_{t\in[0,T]}$ of DRBSDE $(\xi,g+{\rm d}V,L,U)$ in this space.

2.3 Assumptions

In this paper, we will mainly use the following assumptions on the generator, the terminal condition and the barriers, where $p>1$.

1. (H1)

$g$ satisfies the one-sided Osgood condition in $y$, i.e., there exists a nondecreasing concave function $\rho(\cdot):{\mathbb{R}}_{+}\mapsto{\mathbb{R}}_{+}$ with $\rho(0)=0$, $\rho(u)>0$ for $u>0$ and $\int_{0^{+}}{{\rm d}u\over\rho(u)}=+\infty$ such that ${\rm d}\mathbb{P}\times{\rm d}t-a.e.$, $\forall\ y_{1},y_{2}\in{\mathbb{R}},z\in{\mathbb{R}}^{d}$,

[TABLE] 2. (H2)

• (i)

$g$ is continuous in $y$, i.e, ${\rm d}\mathbb{P}\times{\rm d}t-a.e.$, $\forall\ z\in{{\mathbb{R}}^{d}},\ \ g(\omega,t,\cdot,z)$ is continuous;

• (ii)

$g$ is uniformly continuous in $z$, i.e., there exists a nondecreasing continuous function $\phi(\cdot):{\mathbb{R}}_{+}\mapsto{\mathbb{R}}_{+}$ with $\phi(0)=0$ such that ${\rm d}\mathbb{P}\times{\rm d}t-a.e.$, $\forall\ y\in{\mathbb{R}},z_{1},z_{2}\in{\mathbb{R}}^{d}$,

[TABLE] 3. (H2’)

• (i)

$g$ is stronger continuous in $(y,z)$, i.e., ${\rm d}\mathbb{P}\times{\rm d}t-a.e.$, $\forall\ y\in{\mathbb{R}},\ g(\omega,t,y,\cdot)$ is continuous, and $g(\omega,t,\cdot,z)$ is continuous uniformly with respect to $z$;

• (ii)

$g$ has a stronger linear growth in $z$, i.e., there exist two constants $\mu,\lambda\geq 0$ and a nonnegative process $f_{\cdot}\in\mathcal{H}^{p}$ such that ${\rm d}\mathbb{P}\times{\rm d}t-a.e.$, $\forall\ y\in{\mathbb{R}},z\in{\mathbb{R}}^{d}$,

[TABLE] 4. (H3)

• (i)

$g$ has a general growth in $y$, i.e, $\forall r>0,\ \psi_{\cdot}(r):=\sup\limits_{|y|\leq r}|g(\cdot,y,0)-g(\cdot,0,0)|\ \in\mathcal{H}$;

• (ii)

$g(\cdot,0,0)\in\mathcal{H}^{p}$. 5. (H4)

• (i)

$L_{\cdot}\in\mathcal{S}\ ({\rm or}\ L_{\cdot}\equiv-\infty)$, $U_{\cdot}\in\mathcal{S}\ ({\rm or}\ U_{\cdot}\equiv+\infty)$, $\xi\in{\mathbb{L}}^{p}(\mathcal{F}_{T})$ and $L_{T}\leq\xi\leq U_{T}$;

• (ii)

There exists an $X_{\cdot}\in\mathcal{M}^{p}+\mathcal{V}^{p}$ such that $g(\cdot,X_{\cdot},0)\in\mathcal{H}^{p}$ and $L_{t}\leq X_{t}\leq U_{t}$ for each $t\in[0,T]$.

Remark 2.5**.**

Without loss of generality, we will always assume that the functions $\rho(\cdot)$ and $\phi(\cdot)$ defined respectively in (H1) and (H2) are of linear growth, i.e., there exists a constant $A>0$ such that

[TABLE]

And, we note that assumption (H4) is usually called the generalized Mokobodzki condition for doubly reflected BSDEs, which relate the growth of $g$ and that of $L_{\cdot}$ and $U_{\cdot}$

In order to illustrate our results more clearly, the following several assumptions will also be used.

1. (H1s)

$g$ satisfies the monotonicity condition in $y$, i.e., there exists a constant $\mu\in{\mathbb{R}}$ such that ${\rm d}\mathbb{P}\times{\rm d}t-a.e.$, $\forall\ y_{1},y_{2}\in{\mathbb{R}},z\in{\mathbb{R}}^{d}$,

[TABLE] 2. (H2s)

• (i)

$g$ is continuous in $y$, i.e, ${\rm d}\mathbb{P}\times{\rm d}t-a.e.$, $\forall\ z\in{{\mathbb{R}}^{d}},\ \ g(\omega,t,\cdot,z)$ is continuous;

• (ii)

$g$ satisfies the uniform Lipschitz condition in $z$, i.e., there exists a constant $\lambda\geq 0$ such that ${\rm d}\mathbb{P}\times{\rm d}t-a.e.$, $\forall\ y\in{\mathbb{R}},z_{1},z_{2}\in{\mathbb{R}}^{d}$,

[TABLE] 3. (H3s)

$g$ has a linear growth in $y$, i.e., there exists a constant $\mu\geq 0$ and a nonnegative process $f_{\cdot}\in\mathcal{H}^{p}$ such that ${\rm d}\mathbb{P}\times{\rm d}t-a.e.$, $\forall\ y\in{\mathbb{R}}$, $|g(\omega,t,y,0)|\leq f_{t}(\omega)+\mu|y|$.

Remark 2.6**.**

It is clear that assumptions (H1s), (H2s) and (H3s) are respectively (strictly) stronger than (H1), (H2) and (H3). And, (ii) of (H2) implies (ii) of (H2’).

Moreover, the following several assumptions will be used in some technical results of this paper.

1. (AA)

There exist two nonnegative constants $\bar{\mu}$ and $\bar{\lambda}$ such that ${\rm d}\mathbb{P}\times{\rm d}t-a.e.$, $\forall\ y\in{\mathbb{R}},\ z\in{\mathbb{R}}^{d}$,

[TABLE]

where $\bar{f}_{\cdot}$ is a nonnegative process belonging to $\mathcal{H}^{p}$. 2. (HH)

• (i)

$g$ is continuous in $(y,z)$, i.e, ${\rm d}\mathbb{P}\times{\rm d}t-a.e.$,  $g(\omega,t,\cdot,\cdot)$ is continuous;

• (ii)

$g$ has a general growth in $(y,z)$, i.e., there exists a constant $\lambda\geq 0$, a nonnegative process $f_{\cdot}\in\mathcal{H}^{p}$ and a nonnegative function $\psi_{\cdot}(r)\in{\bf S}$ such that ${\rm d}\mathbb{P}\times{\rm d}t-a.e.$, $\forall\ y\in{\mathbb{R}},\ z\in{\mathbb{R}}^{d}$,

[TABLE]

here and hereafter, ${\bf S}$ denotes the set of nonnegative functions $\psi_{t}(\omega,r):\Omega\times[0,T]\times{\mathbb{R}}_{+}\mapsto{\mathbb{R}}_{+}$ satisfying the following two conditions:

• –

${\rm d}\mathbb{P}\times{\rm d}t-a.e.$, the function $r\mapsto\psi_{t}(\omega,r)$ is increasing and $\psi_{t}(\omega,0)=0$;

• –

for each $r\geq 0$, $\psi_{\cdot}(r)\in\mathcal{H}$. 3. (H4L)

• (i)

$L_{\cdot}\in\mathcal{S}\ ({\rm or}\ L_{\cdot}\equiv-\infty)$, $\xi\in{\mathbb{L}}^{p}(\mathcal{F}_{T})$ and $L_{T}\leq\xi$;

• (ii)

There exists an $X_{\cdot}\in\mathcal{M}^{p}+\mathcal{V}^{p}$ such that $g(\cdot,X_{\cdot},0)\in\mathcal{H}^{p}$ and $L_{t}\leq X_{t}$ for each $t\in[0,T]$. 4. (H4U)

• (i)

$U_{\cdot}\in\mathcal{S}\ ({\rm or}\ U_{\cdot}\equiv+\infty)$, $\xi\in{\mathbb{L}}^{p}(\mathcal{F}_{T})$ and $\xi\leq U_{T}$;

• (ii)

There exists an $X_{\cdot}\in\mathcal{M}^{p}+\mathcal{V}^{p}$ such that $g(\cdot,X_{\cdot},0)\in\mathcal{H}^{p}$ and $X_{t}\leq U_{t}$ for each $t\in[0,T]$.

Remark 2.7**.**

It is not hard to verify that, see also Remark 2.2 in Fan [16] for details,

• (i)

(H2)+(H3)* $\Rightarrow$ (HH);  (H2’)+(H3) $\Rightarrow$ (HH);  (H1)+(HH)(ii) $\Rightarrow$ (AA);  (HH)(ii) $\Rightarrow$ (H3);*

• (ii)

(H4)* $\Rightarrow$ (H4L) + (H4U);   (H4L)(ii) $\Rightarrow$ $L_{\cdot}^{+}\in\mathcal{S}^{p}$;  (H4U)(ii) $\Rightarrow$ $U_{\cdot}^{-}\in\mathcal{S}^{p}$;*

• (iii)

$L_{\cdot}^{+}\in\mathcal{S}^{p}$* together with $\left(g(t,\sup\limits_{s\in[0,t]}L_{s}^{+},0)\right)_{t\in[0,T]}\in\mathcal{H}^{p}$ implies (H4L)(ii);*

• (iv)

$U_{\cdot}^{-}\in\mathcal{S}^{p}$* together with $\left(g(t,-\inf\limits_{s\in[0,t]}U_{s}^{-},0)\right)_{t\in[0,T]}\in\mathcal{H}^{p}$ implies (H4U)(ii);*

• (v)

If (H3s) holds, then (H4L)(ii) $\Leftrightarrow$ $L_{\cdot}^{+}\in\mathcal{S}^{p}$ and (H4U)(ii) $\Leftrightarrow$ $U_{\cdot}^{-}\in\mathcal{S}^{p}$.

2.4 Lemma and propositions

Let us first introduce the following lemma, which comes from Lemma 3.1 in Fan [16].

Lemma 2.8**.**

Let $(\bar{Y}_{\cdot},\bar{Z}_{\cdot},\bar{V}_{\cdot})\in\mathcal{S}\times{\rm M}\times\mathcal{V}$ satisfy the following equation:

[TABLE]

Then, the following two assertions hold.

• (i)

There exists a constant $C_{1}>0$ depending only on $p>0$ such that for each $t\in[0,T]$ and each stopping time $\tau$ valued in $[0,T]$,

[TABLE]

• (ii)

If $\bar{Y}_{\cdot}\in\mathcal{S}^{p}$ for some $p>1$, then there exists a constant $C_{2}>0$ depending only on $p$ such that for each $t\in[0,T]$ and each stopping time $\tau$ valued in $[0,T]$,

[TABLE]

The following proposition is a direct corollary of Lemma 3.2 in Fan [16].

Proposition 2.9**.**

Assume that $p>1$, $\xi\in{\mathbb{L}}^{p}(\mathcal{F}_{T})$, $V_{\cdot}\in\mathcal{V}^{p}$ and the generator $g$ satisfies assumption (AA). Let $(Y_{\cdot},Z_{\cdot})\in\mathcal{S}^{p}\times{\rm M}^{p}$ be a solution of BSDE $(\xi,g+{\rm d}V)$. Then, there exists a constant $C>0$ depending only on $p,\bar{\mu},\bar{\lambda},T$ such that for each $t\in[0,T]$,

[TABLE]

By Lemma 3.4 in Fan [16], we can verify the following a priori estimate.

Proposition 2.10**.**

Assume that $p>1$, $\xi\in{\mathbb{L}}^{p}(\mathcal{F}_{T})$, $V_{\cdot}\in\mathcal{V}^{p}$, $K_{\cdot},A_{\cdot}\in\mathcal{V}^{+,p}$, and the generator $g$ satisfies assumptions (H1) with $\rho(\cdot)$, (ii) of (H2’) with $f_{\cdot}$, $\mu$ and $\lambda$, and (ii) of (H3).

• (i)

Let $(Y_{\cdot},Z_{\cdot})\in\mathcal{S}^{p}\times{\rm M}^{p}$ be a solution of BSDE $(\xi,g+{\rm d}\bar{V})$ with $\bar{V}_{\cdot}=V_{\cdot}+K_{\cdot}$, and the following assumption (B1) hold:

1. (B1)

There exists an $\bar{X}_{\cdot}\in\mathcal{S}^{p}$ such that $g(\cdot,\bar{X}_{\cdot},0)\in\mathcal{H}^{p}$ and $Y_{t}\leq\bar{X}_{t}$ for each $t\in[0,T]$.

Then, there exists a constant $C>0$ depending only on $p,\mu,\lambda,A,T$ such that for each $t\in[0,T]$,

[TABLE]

• (ii)

Let $(Y_{\cdot},Z_{\cdot})\in\mathcal{S}^{p}\times{\rm M}^{p}$ be a solution of BSDE $(\xi,g+{\rm d}\underline{V})$ with $\underline{V}_{\cdot}=V_{\cdot}-A_{\cdot}$, and the following assumption (B2) hold:

1. (B2)

There exists an $\underline{X}_{\cdot}\in\mathcal{S}^{p}$ such that $g(\cdot,\underline{X}_{\cdot},0)\in\mathcal{H}^{p}$ and $\underline{X}_{t}\leq Y_{t}$ for each $t\in[0,T]$.

Then, there exists a constant $C>0$ depending only on $p,\mu,\lambda,A,T$ such that for each $t\in[0,T]$,

[TABLE]

Proof.

Since $g$ satisfies (H1) with $\rho(\cdot)$ and (H2’)(ii) with $f_{\cdot}$, $\mu$ and $\lambda$, it follows that ${\rm d}\mathbb{P}\times{\rm d}t-a.e.$,

[TABLE]

Furthermore, if $(Y_{\cdot},Z_{\cdot})\in\mathcal{S}^{p}\times{\rm M}^{p}$ is a solution of BSDE $(\xi,g+{\rm d}\bar{V})$ with $\bar{V}_{\cdot}=V_{\cdot}+K_{\cdot}$, and assumption (B1) holds, then it follows from (H1) and (H2’)(ii) that ${\rm d}\mathbb{P}\times{\rm d}t-a.e.$,

[TABLE]

With the above two inequalities in hand and in view of the fact of $f_{\cdot},g(\cdot,0,0),g(\cdot,\bar{X}_{\cdot},0)\in\mathcal{H}^{p}$ and $\bar{X}_{\cdot}\in\mathcal{S}^{p}$, the desired estimate (2.7) follows immediately from Lemma 3.4 in Fan [16]. Finally, in view of (iv) in Remark 2.3, we know that (ii) of Proposition 2.10 is also true. ∎

Now, let us recall several important results on reflected BSDEs with one continuous barrier and non-reflected BSDEs obtained in Fan [16], by using (iv) in Remark 2.3 for the case of $\bar{R}$BSDEs, see Theorem 4.4, Corollary 4.5, Remark 4.6, Theorem 5.2, Corollary 5.4, Theorem 5.8, Corollary 5.9, Remark 5.10, Theorem 5.11, Theorem 5.13, Remark 5.14 and Proposition 5.15 in Fan [16] for more details.

Proposition 2.11**.**

Assume that $p>1$, $\xi\in{\mathbb{L}}^{p}(\mathcal{F}_{T})$, $V_{\cdot}\in\mathcal{V}^{p}$ and the generator $g$ satisfies assumptions (H1), (H2) and (H3). We have the following assertions.

• (i)

BSDE $(\xi,g+{\rm d}V)$ admits a unique solution $(Y_{t},Z_{t})_{t\in[0,T]}$ in $\mathcal{S}^{p}\times{\rm M}^{p}$.

• (ii)

Assume further that (H4L) holds for $\xi$, $L_{\cdot}$ and $X_{\cdot}$ Then, RBSDE $(\xi,g+{\rm d}V,L)$ admits a unique solution $(Y_{t},Z_{t},K_{t})_{t\in[0,T]}$ in $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}$.

• (iii)

*Assume further that (H4U) holds for $\xi$, $U_{\cdot}$ and $X_{\cdot}$ Then, *$\bar{R}$BSDE $(\xi,g+{\rm d}V,U)$ admits a unique solution $(Y_{t},Z_{t},A_{t})_{t\in[0,T]}$ in $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}$.

Proposition 2.12**.**

Let $p>1$ and assume that for $i=1,2$, $\xi^{i}\in{\mathbb{L}}^{p}(\mathcal{F}_{T})$ with $\xi^{1}\leq\xi^{2}$, $V_{\cdot}^{i}\in\mathcal{V}^{p}$ with ${\rm d}V^{1}\leq{\rm d}V^{2}$, and the generator $g^{i}$ satisfies assumptions (H1)-(H3). We have the following assertions.

• (i)

For $i=1,2$, let $(Y_{\cdot}^{i},Z_{\cdot}^{i})$ be the unique solution of BSDE $(\xi^{i},g^{i}+{\rm d}V^{i})$ in $\mathcal{S}^{p}\times{\rm M}^{p}$. If

[TABLE]

then $Y_{t}^{1}\leq Y_{t}^{2}$ for each $t\in[0,T]$.

• (ii)

For $i=1,2$, suppose that (H4L)(i) holds for $\xi^{i}$ and $L^{i}$, and that $(Y_{\cdot}^{i},Z_{\cdot}^{i},K_{\cdot}^{i})$ is the unique solution of RBSDE $(\xi^{i},g^{i}+{\rm d}V^{i},L^{i})$ in the space $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}$. If $L^{1}_{\cdot}\leq L^{2}_{\cdot}$ and (2.9) is satisfied, then

[TABLE]

Moreover, if $L^{1}_{\cdot}=L^{2}_{\cdot}$ and for each $(y,z)\in{\mathbb{R}}\times{\mathbb{R}}^{d}$,

[TABLE]

then ${\rm d}K^{1}\geq{\rm d}K^{2}$.

• (iii)

*For $i=1,2$, suppose that (H4U)(i) holds for $\xi^{i}$ and $U^{i}$, and that $(Y_{\cdot}^{i},Z_{\cdot}^{i},A_{\cdot}^{i})$ is the unique solution of *$\bar{R}$BSDE $(\xi^{i},g^{i}+{\rm d}V^{i},U^{i})$ in the space $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}$. If $U^{1}_{\cdot}\leq U^{2}_{\cdot}$ and (2.9) is satisfied, then (2.10) holds. Moreover, if $U^{1}_{\cdot}=U^{2}_{\cdot}$ and (2.11) holds, then ${\rm d}A^{1}\leq{\rm d}A^{2}$.

Remark 2.13**.**

From the related proof in Fan [16], it can be observed that in order to obtain (2.10) in (ii) of Proposition 2.12 we do not need the condition that

[TABLE]

and in order to obtain (2.10) in (iii) of Proposition 2.12 we do not need the condition that

[TABLE]

This important observation will be used several times later.

Remark 2.14**.**

Proposition 2.11*, Proposition 2.12 and Remark 2.13 still hold if we replace (H2) with (H2’), (2.9) with (2.11), and the word “unique” with “maximal (minimal)” in their statements.*

3 Necessity of assumption (H4) (ii) and uniqueness of the $L^{p}$ solution

3.1 Necessity of the assumption

By the following theorem we show that under conditions of (H1), (H2) (ii) (resp. (H2’)(ii)), (H3)(ii) and (H4)(i), (H4)(ii) is necessary to ensure the existence of $L^{p}$ solutions for DRBSDEs.

Theorem 3.15** (Necessity of (H4)(ii)).**

Assume that $p>1$, $V_{\cdot}\in\mathcal{V}^{p}$, the generator $g$ satisfies assumptions (H1), (H2)(ii) (resp. (H2’)(ii)) and (H3)(ii), and that assumption (H4)(i) holds for $L_{\cdot}$, $U_{\cdot}$ and $\xi$. If DRBSDE $(\xi,g+{\rm d}V,L,U)$ admits a solution $(Y_{\cdot},Z_{\cdot},K_{\cdot},A_{\cdot})$ in the space $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}\times\mathcal{V}^{+,p}$, then $g(\cdot,Y_{\cdot},0)\in\mathcal{H}^{p}$. That is to say, (H4)(ii) holds true.

Proof.

We only prove the case of (H2’)(ii). The case of (H2)(ii) can be proved in a same way. In fact, by assumptions of Theorem 3.15, it is easy to verify that $g$ satisfies (AA) with $\bar{f}_{\cdot}:=|g(\cdot,0,0)|+f_{\cdot}+A$, $\bar{\mu}:=\mu+A$ and $\bar{\lambda}:=\lambda$ (see also the proof of Proposition 2.10), and $(Y_{\cdot},Z_{\cdot})\in\mathcal{S}^{p}\times{\rm M}^{p}$ is a solution of BSDE$(\xi,g+{\rm d}\bar{V})$ with $\bar{V}_{\cdot}=V_{\cdot}+K_{\cdot}-A_{\cdot}\in\mathcal{V}^{p}$. Then, it follows from Proposition 2.9 that

[TABLE]

Furthermore, by (H2’)(ii) together with Hölder’s inequality we deduce that

[TABLE]

Then, (H4)(ii) holds for $X_{\cdot}=Y_{\cdot}$. Theorem 3.15 is then proved. ∎

3.2 Comparison theorem

In this subsection, we first establish a general comparison theorem for the $L^{p}\ (p>1)$ solutions of doubly RBSDEs, and then verify the uniqueness of the $L^{p}$ solution under assumptions (H1) and (H2)(ii).

Proposition 3.16** (Comparison Theorem for $L^{p}$ solutions of DRBSDEs).**

Let $p>1$, $V_{\cdot}^{i}\in\mathcal{V}^{p}$, (H4)(i) hold for $\xi^{i}$, $L_{\cdot}^{i}$ and $U_{\cdot}^{i}$, and $(Y_{\cdot}^{i},Z_{\cdot}^{i},K_{\cdot}^{i},A_{\cdot}^{i})$ be a solution of DRBSDE $(\xi^{i},g^{i}+{\rm d}V^{i},L^{i},U^{i})$ in $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}\times\mathcal{V}^{+,p}$ for $i=1,2$. If $\xi^{1}\leq\xi^{2}$, ${\rm d}V^{1}\leq{\rm d}V^{2}$, $L^{1}_{\cdot}\leq L^{2}_{\cdot}$, $U^{1}_{\cdot}\leq U^{2}_{\cdot}$, and either

[TABLE]

or

[TABLE]

is satisfied, then $Y_{t}^{1}\leq Y_{t}^{2}$ for each $t\in[0,T]$.

Proof.

It follows from Itô-Tanaka’s formula that

[TABLE]

Since $L_{t}^{1}\leq L_{t}^{2}\leq Y_{t}^{2}$, $L_{t}^{1}\leq Y_{t}^{1}$, $t\in[0,T]$ and $\int_{0}^{T}(Y^{1}_{s}-L_{s}^{1}){\rm d}K_{s}^{1}=0$, we know that for $t\in[0,T]$,

[TABLE]

Similarly, since $Y^{1}_{t}\leq U_{t}^{1}\leq U_{t}^{2}$, $Y^{2}_{t}\leq U_{t}^{2}$, $t\in[0,T]$ and $\int_{0}^{T}(U^{2}_{s}-Y_{s}^{2}){\rm d}A_{s}^{2}=0$, we know that for $t\in[0,T]$,

[TABLE]

Thus, in view of $\xi^{1}\leq\xi^{2}$ and ${\rm d}V^{1}\leq{\rm d}V^{2}$, by the previous three inequalities we obtain

[TABLE]

Now, in view of the assumptions of $g^{1}$ and $g^{2}$, the rest proof runs as that in the proof of Theorem 4.4 in Fan [16]. The proof is complete. ∎

By Proposition 3.16, the following corollary follows immediately.

Corollary 3.17**.**

Let $p>1$, $V_{\cdot}^{i}\in\mathcal{V}^{p}$, (H4)(i) hold for $\xi^{i}$, $L_{\cdot}^{i}$ and $U_{\cdot}^{i}$, and $(Y_{\cdot}^{i},Z_{\cdot}^{i},K_{\cdot}^{i},A_{\cdot}^{i})$ be a solution of DRBSDE $(\xi^{i},g^{i}+{\rm d}V^{i},L^{i},U^{i})$ in $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}\times\mathcal{V}^{+,p}$ for $i=1,2$. If $\xi^{1}\leq\xi^{2}$, ${\rm d}V^{1}\leq{\rm d}V^{2}$, $L^{1}_{\cdot}\leq L^{2}_{\cdot}$, $U^{1}_{\cdot}\leq U^{2}_{\cdot}$, $g^{1}$ or $g^{2}$ satisfies (H1) and (H2)(ii), and for each $(y,z)\in{\mathbb{R}}\times{\mathbb{R}}^{d}$,

[TABLE]

then $Y_{t}^{1}\leq Y_{t}^{2}$ for each $t\in[0,T]$.

Remark 3.18**.**

We note that in the proof of Proposition 3.16 the following two assumptions are not utilized:

[TABLE]

In addition, it follows from Remarks 2.3 and 2.7 that if the comparison of ${\rm d}K^{i}$ and ${\rm d}A^{i}$ is not taken into account, then Proposition 3.16 and Corollary 3.17 improve Proposition 2.12.

Theorem 3.19** (Uniqueness).**

Let $p>1$, $V_{\cdot}\in\mathcal{V}^{p}$, (H4)(i) hold for $\xi$, $L_{\cdot}$ and $U_{\cdot}$, and the generator $g$ satisfy assumptions (H1) and (H2)(ii). Then DRBSDE $(\xi,g+{\rm d}V,L,U)$ admits at most one solution in $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}\times\mathcal{V}^{+,p}$, i.e, if both $(Y_{\cdot},Z_{\cdot},K_{\cdot},A_{\cdot})$ and $(Y^{\prime}_{\cdot},Z^{\prime}_{\cdot},K^{\prime}_{\cdot},A^{\prime}_{\cdot})$ are solutions of DRBSDE $(\xi,g+{\rm d}V,L,U)$ in $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}\times\mathcal{V}^{+,p}$, then ${\rm d}\mathbb{P}\times{\rm d}t-a.e.$, $Y_{\cdot}=Y^{\prime}_{\cdot},\ Z_{\cdot}=Z^{\prime}_{\cdot}\ K_{\cdot}=K^{\prime}_{\cdot}$ and $A_{\cdot}=A^{\prime}_{\cdot}$.

Proof.

Firstly, it follows from Corollary 3.17 that $Y_{t}=Y^{\prime}_{t}$ for each $t\in[0,T]$. Furthermore, by Itô’s formula we know that ${\rm d}\mathbb{P}\times{\rm d}t-a.e.$, $Z_{\cdot}=Z^{\prime}_{\cdot}$, and then $K_{t}-A_{t}=K^{\prime}_{t}-A^{\prime}_{t}$ for each $t\in[0,T]$. Finally, the conclusion of $K_{\cdot}=K^{\prime}_{\cdot}$ and $A_{\cdot}=A^{\prime}_{\cdot}$ follows from the Ham-Bananch Composition of Sign Measure. ∎

4 Existence of the $L^{p}$ solution: penalization method

In this section, we will prove the existence of $L^{p}$ solutions for DRBSDEs under assumptions (H1), (H2) (resp. (H2’)), (H3) and (H4) by showing the convergence of the sequence of $L^{p}$ solutions for the penalized RBSDEs with one continuous barrier and the penalized BSDEs.

4.1 Convergence of the sequence of $L^{p}$ solutions for the penalized RBSDEs

The following proposition is a general convergence result on the sequence of $L^{p}\ (p>1)$ solutions of penalized RBSDEs with one continuous barrier under some elementary conditions.

Proposition 4.20** (Penalization method).**

Assume that $p>1$, $V_{\cdot}\in\mathcal{V}^{p}$, (H4)(i) holds for $L_{\cdot},U_{\cdot}$ and $\xi$, and the generator $g$ satisfies (HH) with $f_{\cdot},\psi_{\cdot}(r)$ and $\lambda$.

• (i)

*For each $n\geq 1$, let $(Y^{n}_{\cdot},Z^{n}_{\cdot},A^{n}_{\cdot})$ be a solution of *$\bar{R}$BSDE $(\xi,\bar{g}_{n}+{\rm d}V,U)$ in $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}$ with $\bar{g}_{n}(t,y,z):=g(t,y,z)+n(y-L_{t})^{-}$, i.e.,

[TABLE]

If for each $n\geq 1$, $Y^{n}_{\cdot}\leq Y^{n+1}_{\cdot}\leq\bar{X}_{\cdot}$ for some $\bar{X}_{\cdot}\in\mathcal{S}^{p}$ and ${\rm d}A^{n}\leq{\rm d}A^{n+1}$, and

[TABLE]

then there exists a quadruple $(Y_{\cdot},Z_{\cdot},K_{\cdot},A_{\cdot})\in\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}\times\mathcal{V}^{+,p}$ which solves DRBSDE $(\xi,g+{\rm d}V,L,U)$,

[TABLE]

and there exists a subsequence $\{K_{\cdot}^{n_{j}}\}$ of $\{K_{\cdot}^{n}\}$ such that

[TABLE]

• (ii)

*For each $n\geq 1$, let $(Y^{n}_{\cdot},Z^{n}_{\cdot},K^{n}_{\cdot})$ be a solution of *$\underline{R}$BSDE $(\xi,\underline{g}_{n}+{\rm d}V,L)$ in $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}$ with $\underline{g}_{n}(t,y,z):=g(t,y,z)-n(y-U_{t})^{+}$, i.e.,

[TABLE]

If for each $n\geq 1$, $\underline{X}_{\cdot}\leq Y^{n+1}_{\cdot}\leq Y^{n}_{\cdot}$ for some $\underline{X}_{\cdot}\in\mathcal{S}^{p}$ and ${\rm d}K^{n}\leq{\rm d}K^{n+1}$, and (4.2) holds, then there exists a quadruple $(Y_{\cdot},Z_{\cdot},K_{\cdot},A_{\cdot})\in\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}\times\mathcal{V}^{+,p}$ which solves DRBSDE $(\xi,g+{\rm d}V,L,U)$,

[TABLE]

and there exists a subsequence $\{A_{\cdot}^{n_{j}}\}$ of $\{A_{\cdot}^{n}\}$ such that

[TABLE]

Proof.

We only prove the claim (i). The claim (ii) can be proved in the same way. Now, we assume that $p>1$, $V_{\cdot}\in\mathcal{V}^{p}$, (i) of (H4) holds for $L_{\cdot},U_{\cdot}$ and $\xi$, the generator $g$ satisfies (HH) with $f_{\cdot},\psi_{\cdot}(r)$ and $\lambda$, and all the assumptions in (i) are satisfied.

Since $Y_{\cdot}^{n}$ increases in $n$ and is bounded above by $\bar{X}_{\cdot}$, we know the existence of a progressively measurable real-valued process $Y_{\cdot}$ such that for each $t\in[0,T]$,

[TABLE]

Since ${\rm d}A^{n}\leq{\rm d}A^{n+1}$ for each $n\geq 1$, we get the existence of a progressively measurable and increasing real-valued process $(A_{t})_{t\in[0,T]}$ with $A_{0}=0$ such that $A_{t}^{n}\uparrow A_{t}$ for each $t\in[0,T]$, and for each $j\geq n\geq 1$,

[TABLE]

In the above inequality, first letting $j\rightarrow\infty$, and then taking the superume with respect to $t$ in $[0,T]$, finally letting $n\rightarrow\infty$, we obtain that

[TABLE]

which means that $A_{\cdot}\in\mathcal{V}^{+}$. On the other hand, by Fatou’s lemma and (4.2) we deduce that

[TABLE]

So, $A_{\cdot}\in\mathcal{V}^{+,p}$. Then, in view of (4.4) and the fact that $|A^{n}_{\cdot}|\leq|A_{\cdot}|$ for each $n\geq 1$, Lebesgue’s dominated convergence theorem yields that

[TABLE]

For each integer $l,q\geq 1$, we introduce the following two stopping times:

[TABLE]

with the convention $\inf\emptyset=+\infty$. Then we have, $\tau_{l}\rightarrow T$ as $l\rightarrow\infty$, $\sigma_{l,q}\rightarrow\tau_{l}$ as $q\rightarrow\infty$ for each $l\geq 1$,

[TABLE]

and

[TABLE]

Furthermore, since $g$ satisfies assumption (HH) with $f_{\cdot},\psi_{\cdot}(r)$ and $\lambda$, and (4.3) holds, it follows from the definitions of $\tau_{l}$ and $\sigma_{l,q}$ that ${\rm d}\mathbb{P}\times{\rm d}t-a.e.$, for each $n\geq 1$,

[TABLE]

with $h^{n;l,q}_{t}:=\mathbbm{1}_{t\leq\sigma_{l,q}}g(t,Y_{t}^{n},Z_{t}^{n})$,

[TABLE]

The rest proof is divided into 6 steps, which will be detailed in Appendix.

Step 1. Based on (4.1)-(4.8), by using a weak convergence argument together with Lemma 4.4 of Klimsiak [37] and Lemma A.3 in Bayraktar and Yao [3], we show that $Y_{\cdot}$ is a càdlàg process.

Step 2. Making use of the conclusion of step 1 together with the definition of $K^{n}_{\cdot}$, (4.2) and Dini’s theorem, we show that $Y_{t}\geq L_{t}$ for each $t\in[0,T]$ and $\lim\limits_{n\rightarrow\infty}\sup\limits_{t\in[0,T]}(Y^{n}_{t}-L_{t})^{-}=0$.

Step 3. By virtue of (ii) of Lemma 2.8, the definition of $K_{\cdot}^{n}$ and $A_{\cdot}^{n}$ with (4.6)-(4.8), Hölder’s inequality, the conclusion of step 2, (4.2), (4.3) and Lebesgue’s dominated convergence theorem, we show that the sequence $\{Y_{\cdot}^{n}\}$ converges to the process $Y_{\cdot}$ in $\mathcal{S}^{p}$ as $n\to\infty$.

Step 4. By virtue of (i) of Lemma 2.8, Hölder’s inequality, (4.2) and the conclusion of step 3, we show that the sequence $\{Z_{\cdot}^{n}\}$ converges to a process $Z_{\cdot}$ in ${\rm M}^{p}$ as $n\to\infty$.

Step 5. Making use of the continuity of $g$, (4.2), (4.5)-(4.8), and the conclusions of steps 3 and 4, we show that there exists a subsequence $\{K_{\cdot}^{n_{j}}\}$ of the sequence $\{K_{\cdot}^{n}\}$ which converges almost surely to a process $K_{\cdot}\in\mathcal{V}^{+,p}$ uniformly in $t$ as $j\to\infty$.

Step 6. Based on all the conclusions of steps 1-6, we finally show that $(Y_{\cdot},Z_{\cdot},K_{\cdot},A_{\cdot})$ is a solution of DRBSDE $(\xi,g+{\rm d}V,L,U)$ in the space $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}\times\mathcal{V}^{+,p}$. The proof is then completed. ∎

4.2 Uniform estimate on the $L^{p}$ solutions of the penalized equations

In this subsection, we establish the following uniform estimate on the $L^{p}\ (p>1)$ solutions of the penalization equations for reflected BSDEs with one continuous barrier and non-reflected BSDEs.

Proposition 4.21**.**

Assume that $p>1$, $V_{\cdot}\in\mathcal{V}^{p}$, the generator $g$ satisfies assumptions (H1), (H2) and (H3), and assumption (H4) holds for $L_{\cdot},U_{\cdot},\xi$ and $X_{\cdot}$.

• (i)

*For each $n\geq 1$, let $(\underline{Y}^{n}_{\cdot},\underline{Z}^{n}_{\cdot},\underline{A}^{n}_{\cdot})$ be the unique solution of *$\bar{R}$BSDE $(\xi,\bar{g}_{n}+{\rm d}V,U)$ in $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}$ with $\bar{g}_{n}(t,y,z):=g(t,y,z)+n(y-L_{t})^{-}$, i.e.,

[TABLE]

(Recall (ii) of Remark 2.7 and (iii) of Proposition 2.11). Then, $\underline{Y}^{n}_{\cdot}$ increases in $n$ and is bounded above by a process $\bar{X}_{\cdot}\in\mathcal{S}^{p}$, ${\rm d}\underline{A}^{n}\leq{\rm d}\underline{A}^{n+1}$, and there exists a random variable $\underline{\eta}\in L^{1}(\mathcal{F}_{T})$ such that for each $n\geq 1$ and $t\in[0,T]$,

[TABLE]

• (ii)

*For each $n\geq 1$, let $(\bar{Y}^{n}_{\cdot},\bar{Z}^{n}_{\cdot},\bar{K}^{n}_{\cdot})$ be the unique solution of *$\underline{R}$BSDE $(\xi,\underline{g}_{n}+{\rm d}V,L)$ in $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}$ with $\underline{g}_{n}(t,y,z):=g(t,y,z)-n(y-U_{t})^{+}$, i.e.,

[TABLE]

(Recall (ii) of Remark 2.7 and (ii) of Proposition 2.11). Then, $\bar{Y}^{n}_{\cdot}$ decreases in $n$ and is bounded below by a process $\underline{X}_{\cdot}\in\mathcal{S}^{p}$, ${\rm d}\bar{K}^{n}\leq{\rm d}\bar{K}^{n+1}$, and there exists a random variable $\bar{\eta}\in L^{1}(\mathcal{F}_{T})$ such that for each $n\geq 1$ and $t\in[0,T]$,

[TABLE]

• (iii)

For each $n\geq 1$, let $(Y^{n}_{\cdot},Z^{n}_{\cdot})$ be the unique solution of BSDE $(\xi,g_{n}+{\rm d}V)$ in the space $\mathcal{S}^{p}\times{\rm M}^{p}$ with $g_{n}(t,y,z):=g(t,y,z)+n(y-L_{t})^{-}-n(y-U_{t})^{+}$ (Recall (i) of Proposition 2.11), i.e.,

[TABLE]

Then, for each $n\geq 1$ and $t\in[0,T]$, $\underline{Y}^{n}_{t}\leq Y^{n}_{t}\leq\bar{Y}^{n}_{t}$, ${\rm d}K^{n}\leq{\rm d}\underline{K}^{n}$ and ${\rm d}A^{n}\leq{\rm d}\bar{A}^{n}$. And, there exists a random variable $\eta\in L^{1}(\mathcal{F}_{T})$ such that for each $n\geq 1$ and $t\in[0,T]$,

[TABLE]

Proof.

Let $p>1$, $V_{\cdot}\in\mathcal{V}^{p}$, the generator $g$ satisfy assumptions (H1) with $\rho(\cdot)$, (H2) with $\phi(\cdot)$, and (H3) with $\varphi_{\cdot}(r)$, and assumption (H4) hold for $L_{\cdot},U_{\cdot},\xi$ and $X_{\cdot}$.

By representation property of Brownian filtration, we can let $(C_{\cdot},H_{\cdot})$ be the unique pair of processes in the space $\mathcal{V}^{p}\times{\rm M}^{p}$ such that

[TABLE]

It follows from (ii) of (H2) that ${\rm d}\mathbb{P}\times{\rm d}t-a.e.$,

[TABLE]

from which together with (H4) we know that $g(\cdot,X_{\cdot},H_{\cdot})\in\mathcal{H}^{p}$, and then

[TABLE]

and

[TABLE]

where $C_{\cdot}-C_{0}=C^{0,+}_{\cdot}-C^{0,-}_{\cdot}$ and $V_{\cdot}-V_{0}=V^{0,+}_{\cdot}-V^{0,-}_{\cdot}$ with $V_{\cdot}^{0,+},V_{\cdot}^{0,-},C_{\cdot}^{0,+},C_{\cdot}^{0,-}\in\mathcal{V}^{+,p}$. Thus, the equation (4.15) can be rewritten as

[TABLE]

On the other hand, by (i) of Proposition 2.11, let $(\bar{X}_{\cdot},\bar{Z}_{\cdot})$ be the unique solution in $\mathcal{S}^{p}\times{\rm M}^{p}$ of the BSDE

[TABLE]

and $(\underline{X}_{\cdot},\underline{Z}_{\cdot})$ be the unique solution in $\mathcal{S}^{p}\times{\rm M}^{p}$ of the following BSDE

[TABLE]

It follows from (i) of Remark 2.7 that $g$ satisfies assumption (AA). Then, Proposition 2.9 yields that $g(\cdot,\bar{X}_{\cdot},\bar{Z}_{\cdot})\in\mathcal{H}^{p}$ and $g(\cdot,\underline{X}_{\cdot},\underline{Z}_{\cdot})\in\mathcal{H}^{p}$, which together with (H2) leads to

[TABLE]

and

[TABLE]

In what follows, for each $n\geq 1$, by (i) of Proposition 2.11, let $(\dot{Y}^{n}_{\cdot},\dot{Z}^{n}_{\cdot})$ and $(\ddot{Y}^{n}_{\cdot},\ddot{Z}^{n}_{\cdot})$ be respectively the unique solution in the space $\mathcal{S}^{p}\times{\rm M}^{p}$ of the following BSDEs:

[TABLE]

and

[TABLE]

with

[TABLE]

Note that $L_{\cdot}\leq X_{\cdot}\leq U_{\cdot}$. We have, with $t\in[0,T]$ and $n\geq 1$,

[TABLE]

and

[TABLE]

It then follows from (ii) of (H4) and (i) of Proposition 2.12 that for each $n\geq 1$,

[TABLE]

and

[TABLE]

which means that for each $n\geq 1$, (B1) in Proposition 2.10 holds for $X_{\cdot}$, the generator $g$ and $\dot{Y}^{n}_{\cdot}$, and (B2) in Proposition 2.10 holds for $X_{\cdot}$, the generator $g$ and $\ddot{Y}^{n}_{\cdot}$. Thus, in view of the fact that $g$ satisfies (H1), (H2’) with $f_{\cdot}=A$, $\mu=0$ and $\lambda=A$, and (H3), by Proposition 2.10 together with (4.18) and (4.19) we obtain that there exists a constant $C>0$ depending only on $p,A,T$ such that for each $t\in[0,T]$,

[TABLE]

and

[TABLE]

In the sequel, we will prove (i)-(iii) respectively.

(i) For each $n\geq 1$, let $(\underline{Y}^{n}_{\cdot},\underline{Z}^{n}_{\cdot},\underline{A}^{n}_{\cdot})$ be the unique solution of $\bar{R}$BSDE $(\xi,\bar{g}_{n}+{\rm d}V,U)$ in the space $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}$ with $\bar{g}_{n}(t,y,z):=g(t,y,z)+n(y-L_{t})^{-}$, i.e., (4.9).

Firstly, since $\bar{X}_{\cdot}\geq L_{\cdot}$ by (4.19), it follows from (iii) and (i) of Proposition 2.12 that for each $n\geq 1$,

[TABLE]

Then, in view of (4.18), by (iii) of Proposition 2.12 with Remark 2.13 we deduce that for each $n\geq 1$,

[TABLE]

which means that for each $t\in[0,T]$,

[TABLE]

Furthermore, it follows from (4.17) and (4.23) that for each $n\geq 1$, (B2) in Proposition 2.10 holds for $\underline{X}_{\cdot}$, the generator $g$ and $\underline{Y}^{n}_{\cdot}$. Thus, in view of the fact that $g$ satisfies (H1), (H2’) with $f_{\cdot}=A$, $\mu=0$ and $\lambda=A$, and (H3), by (ii) of Proposition 2.10 we know that for each $t\in[0,T]$ and $n\geq 1$,

[TABLE]

It then follows from (4.22)– (4.25) and (4.20) that (4.10) holds with

[TABLE]

for some constant $\bar{C}>0$ depending only on $p,A,T$.

(ii) For each $n\geq 1$, let $(\bar{Y}^{n}_{\cdot},\bar{Z}^{n}_{\cdot},\bar{K}^{n}_{\cdot})$ be the unique solution of $\underline{R}$BSDE $(\xi,\underline{g}_{n}+{\rm d}V,L)$ in the space $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}$ with $\underline{g}_{n}(t,y,z):=g(t,y,z)-n(y-U_{t})^{+}$, i.e., (4.11).

Firstly, since $\underline{X}_{\cdot}\leq U_{\cdot}$ by (4.18), it follows from (i) and (ii) of Proposition 2.12 that for each $n\geq 1$,

[TABLE]

Then, in view of (4.19), by (ii) of Proposition 2.12 with Remark 2.13 we deduce that for each $n\geq 1$,

[TABLE]

which means that for each $t\in[0,T]$,

[TABLE]

Furthermore, it follows from (4.16) and (4.27) that for each $n\geq 1$, (B1) in Proposition 2.10 holds for $\bar{X}_{\cdot}$, the generator $g$ and $\bar{Y}^{n}_{\cdot}$. Thus, in view of the fact that $g$ satisfies (H1), (H2’) with $f_{\cdot}=A$, $\mu=0$ and $\lambda=A$, and (H3), by (i) of Proposition 2.10 we know that for each $t\in[0,T]$ and $n\geq 1$,

[TABLE]

It then follows from (4.26)– (4.29) and (4.21) that (4.12) holds with

[TABLE]

for some constant $\bar{C}>0$ depending only on $p,A,T$.

(iii) For each $n\geq 1$, let $(Y^{n}_{\cdot},Z^{n}_{\cdot})$ be the unique solution of BSDE $(\xi,g_{n}+{\rm d}V)$ in the space $\mathcal{S}^{p}\times{\rm M}^{p}$ with $g_{n}(t,y,z):=g(t,y,z)+n(y-L_{t})^{-}-n(y-U_{t})^{+}$, i.e., (4.13).

It follows from (4.11) and (4.9) that, with $t\in[0,T]$ and $n\geq 1$,

[TABLE]

and

[TABLE]

Then, (i) of Proposition 2.12 together with (4.23) and (4.27) yields that for each $n\geq 1$,

[TABLE]

which means that for each $t\in[0,T]$, in view of (4.24) and (4.28),

[TABLE]

and

[TABLE]

Furthermore, note that $g$ satisfies assumption (AA) with $\bar{f}_{\cdot}=|g(\cdot,0,0)|+2A$, $\bar{\mu}=A$ and $\bar{\lambda}=A$ by (i) of Remark 2.7. It follows from Proposition 2.9 that there exists a constant $C^{\prime}>0$ depending only on $p,A,T$ such that for each $t\in[0,T]$ and $n\geq 1$,

[TABLE]

Finally, in view of (4.30)– (4.33) and (4.20)–(4.21), we can deduce that (4.14) holds with

[TABLE]

for some constant $\bar{C}>0$ depending only on $p,A,T$. The proof of Proposition 4.21 is then complete. ∎

Remark 4.22**.**

In view of Remark 2.14, all conclusions of Proposition 4.21 still hold if we replace (H2) with (H2’), and the expression “the unique solution” with “the maximal (minimal) solution” in its statement. Furthermore, (4.10), (4.12) and (4.14) still hold if we replace (H2) with (H2’), and the expression “the unique solution” with “any solution” in its statement. In fact, in this case, it is enough to let $(\underline{X}_{\cdot},\underline{Z}_{\cdot})$ and $(\dot{Y}^{n}_{\cdot},\dot{Z}^{n}_{\cdot})$ be respectively the minimal solution of the corresponding equation instead of the unique solution, and let $(\ddot{Y}^{n}_{\cdot},\ddot{Z}^{n}_{\cdot})$ and $(\bar{X}_{\cdot},\bar{Z}_{\cdot})$ be respectively the maximal solution of the corresponding equation instead of the unique solution in the procedure of proof of Proposition 4.21, with omitting the comparisons between the processes indexed with $n$ and $n+1$.

4.3 Existence and uniqueness

We are now at a position to state and prove a general existence and unique result on the $L^{p}$ solutions of doubly RBSDEs, which, in view of Remarks 2.3 and 2.7, strengthens Proposition 2.11.

Theorem 4.23**.**

Assume that $p>1$, $V_{\cdot}\in\mathcal{V}^{p}$, the generator $g$ satisfies assumptions (H1), (H2) and (H3), and assumption (H4) holds for $L_{\cdot},U_{\cdot},\xi$ and $X_{\cdot}$ Then, DRBSDE $(\xi,g+{\rm d}V,L,U)$ admits a unique solution $(Y_{\cdot},Z_{\cdot},K_{\cdot},A_{\cdot})$ in $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}\times\mathcal{V}^{+,p}$. Moreover, we have the following assertions.

• (i)

*For each $n\geq 1$, let $(\underline{Y}^{n}_{\cdot},\underline{Z}^{n}_{\cdot},\underline{A}^{n}_{\cdot})$ be the unique solution of *$\bar{R}$BSDE $(\xi,\bar{g}_{n}+{\rm d}V,U)$ in $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}$ with $\bar{g}_{n}(t,y,z):=g(t,y,z)+n(y-L_{t})^{-}$, i.e., (4.9). Then,

[TABLE]

• (ii)

*For each $n\geq 1$, let $(\bar{Y}^{n}_{\cdot},\bar{Z}^{n}_{\cdot},\bar{K}^{n}_{\cdot})$ be the unique solution of *$\underline{R}$BSDE $(\xi,\underline{g}_{n}+{\rm d}V,L)$ in $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}$ with $\underline{g}_{n}(t,y,z):=g(t,y,z)-n(y-U_{t})^{+}$, i.e., (4.11). Then,

[TABLE]

• (iii)

For each $n\geq 1$, let $(Y^{n}_{\cdot},Z^{n}_{\cdot})$ be the unique solution of BSDE $(\xi,g_{n}+{\rm d}V)$ in the space $\mathcal{S}^{p}\times{\rm M}^{p}$ with $g_{n}(t,y,z):=g(t,y,z)+n(y-L_{t})^{-}-n(y-U_{t})^{+}$, i.e., (4.13). Then,

[TABLE]

Proof.

The uniqueness part follows from Theorem 3.19. With regard to (i), combining Propositions 4.21 and 4.20, in view of (i) in Remark 2.7, we can deduce that there exists a quadruple $(Y_{\cdot},Z_{\cdot},K_{\cdot},A_{\cdot})\in\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}\times\mathcal{V}^{+,p}$ which solves DRBSDE $(\xi,g+{\rm d}V,L,U)$,

[TABLE]

and there exists a subsequence $\{\underline{K}_{\cdot}^{n_{j}}\}$ of $\{\underline{K}_{\cdot}^{n}\}$ such that

[TABLE]

Furthermore, using a similar argument to that in the proof of Theorem 5.8 of Fan [16], we obtain that

[TABLE]

Then, (4.34) follows from (4.37) and (4.38). And, (4.35) can be proved in a same way.

In the sequel, we prove (iii). Firstly, it follows from Proposition 4.21 that for each $n\geq 1$ and $t\in[0,T]$, $\underline{Y}^{n}_{t}\leq Y^{n}_{t}\leq\bar{Y}^{n}_{t}$, ${\rm d}K^{n}\leq{\rm d}\underline{K}^{n}$ and ${\rm d}A^{n}\leq{\rm d}\bar{A}^{n}$. Then, (4.34) and (4.35) yield that

[TABLE]

Now, we show the convergence of the sequence $\{Z_{\cdot}^{n}\}$ in ${\rm M}^{p}$. Indeed, for each $n\geq 1$, observe that

[TABLE]

solves equation (2.6). It follows from (i) of Lemma 2.8 with $t=0$ and $\tau=T$ that there exists a constant $C^{\prime}>0$ such that for each $n\geq 1$,

[TABLE]

It then follows from the definitions of $K^{n}_{\cdot}$ and $A^{n}_{\cdot}$ as well as the fact of $L_{\cdot}\leq Y_{\cdot}\leq U_{\cdot}$ together with Hölder’s inequality that

[TABLE]

Thus, in view of (4.40), (4.39), (4.14) together with Proposition 2.9, it follows that

[TABLE]

Furthermore, in view of (4.39) and (4.41), by a similar argument to (4.38) we deduce that

[TABLE]

Finally, (4.36) follows from (4.39), (4.41) and (4.42). Theorem 4.23 is then proved. ∎

By Theorem 4.23 we can prove the following corollary, which together with Proposition 3.16, in view of Remarks 2.3 and 2.7, strengthens Proposition 2.12.

Corollary 4.24**.**

Let $p>1$, $V_{\cdot}^{1},V_{\cdot}^{2},\in\mathcal{V}^{p}$ and both $g^{1}$ and $g^{2}$ satisfy assumptions (H1), (H2) and (H3). For $i=1,2$, assume that (H4) holds for $\xi^{i}$, $L_{\cdot}^{i}$, $U_{\cdot}^{i}$ and $X_{\cdot}^{i}$ associated with $g^{i}$, and that $(Y_{\cdot}^{i},Z_{\cdot}^{i},K_{\cdot}^{i},A_{\cdot}^{i})$ is the unique solution of DRBSDE $(\xi^{i},g^{i}+{\rm d}V^{i},L^{i},U^{i})$ in $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}\times\mathcal{V}^{+,p}$. If $\xi^{1}\leq\xi^{2}$, ${\rm d}V^{1}\leq{\rm d}V^{2}$, $L^{1}_{\cdot}=L^{2}_{\cdot}$, $U^{1}_{\cdot}=U^{2}_{\cdot}$ and

[TABLE]

for each $(y,z)\in{\mathbb{R}}\times{\mathbb{R}}^{d}$, then ${\rm d}K^{1}\geq{\rm d}K^{2}$ and ${\rm d}A^{1}\leq{\rm d}A^{2}$.

Proof.

For each $n\geq 1$ and $i=1,2$, let $(Y^{i,n}_{\cdot},Z^{i,n}_{\cdot},A^{i,n}_{\cdot})\in\mathcal{S}^{p}\times{\rm M}^{p}$ be the unique solution of $\bar{R}$BSDE $(\xi^{i},\bar{g}_{n}^{i}+{\rm d}V^{i},U^{i})$ with $\bar{g}_{n}^{i}(t,y,z):=g^{i}(t,y,z)+n(y-L_{t}^{i})^{-}$. In view of assumptions of Corollary 4.24, it follows from (iii) of Proposition 2.12 that for each $n\geq 1$, $Y^{1,n}_{\cdot}\leq Y^{2,n}_{\cdot}$ and ${\rm d}A^{1,n}\leq{\rm d}A^{2,n}$, and then for each progressively measurable set $D\subset\Omega\times[0,T]$ and each $n\geq 1$, we have

[TABLE]

and

[TABLE]

Since

[TABLE]

and

[TABLE]

as $n\rightarrow\infty$ by (i) of Theorem 4.23, it follows that

[TABLE]

which is the desired result. ∎

At the end of this subsection, we put forward and prove a general existence result of the $L^{p}$ solutions for DRBSDEs under assumptions (H1), (H2’), (H3) and (H4).

Theorem 4.25**.**

Assume that $p>1$, $V_{\cdot}\in\mathcal{V}^{p}$, the generator $g$ satisfies assumptions (H1), (H2’) and (H3), and assumption (H4) holds for $L_{\cdot},U_{\cdot},\xi$ and $X_{\cdot}$

• (i)

*For each $n\geq 1$, let $(\underline{Y}^{n}_{\cdot},\underline{Z}^{n}_{\cdot},\underline{A}^{n}_{\cdot})$ be the minimal (resp.maximal) solution of *$\bar{R}$BSDE $(\xi,\bar{g}_{n}+{\rm d}V,U)$ in $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}$ with $\bar{g}_{n}(t,y,z):=g(t,y,z)+n(y-L_{t})^{-}$, i.e., (4.9), recalling Proposition 4.21 and Remark 4.22. Then, DRBSDE $(\xi,g+{\rm d}V,L,U)$ admits a minimal solution (resp. a solution) $(\underline{Y}_{\cdot},\underline{Z}_{\cdot},\underline{K}_{\cdot},\underline{A}_{\cdot})$ in the space $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}\times\mathcal{V}^{+,p}$ such that

[TABLE]

and there exists a subsequence $\{\underline{K}_{\cdot}^{n_{j}}\}$ of $\{\underline{K}_{\cdot}^{n}\}$ such that

[TABLE]

• (ii)

*For each $n\geq 1$, let $(\bar{Y}^{n}_{\cdot},\bar{Z}^{n}_{\cdot},\bar{K}^{n}_{\cdot})$ be the maximal (resp. minimal) solution of *$\underline{R}$BSDE $(\xi,\underline{g}_{n}+{\rm d}V,L)$ in $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}$ with $\underline{g}_{n}(t,y,z):=g(t,y,z)-n(y-U_{t})^{+}$, i.e., (4.11), recalling Proposition 4.21 and Remark 4.22. Then, DRBSDE $(\xi,g+{\rm d}V,L,U)$ admits a maximal solution (resp. a solution) $(\bar{Y}_{\cdot},\bar{Z}_{\cdot},\bar{K}_{\cdot},\bar{A}_{\cdot})$ in the space $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}\times\mathcal{V}^{+,p}$ such that

[TABLE]

and there exists a subsequence $\{\bar{A}_{\cdot}^{n_{j}}\}$ of $\{\bar{A}_{\cdot}^{n}\}$ such that

[TABLE]

Proof.

We only prove (i), and (ii) can be proved in the same way.

In view of Remarks 2.14 and 2.7, using Propositions 4.20-4.21 and Remark 4.22, we can prove that all the conclusions in (i) of Theorem 4.25 hold expect the minimal property of the solution $(\underline{Y}_{\cdot},\underline{Z}_{\cdot},\underline{K}_{\cdot},\underline{A}_{\cdot})$ of DRBSDE $(\xi,g+{\rm d}V,L,U)$ in $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}\times\mathcal{V}^{+,p}$ when $(\underline{Y}^{n}_{\cdot},\underline{Z}^{n}_{\cdot},\underline{A}^{n}_{\cdot})$ is the minimal solution of $\bar{R}$BSDE $(\xi,\bar{g}_{n}+{\rm d}V,U)$ in $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}$ for each $n\geq 1$. Now, we show this property.

Indeed, for any solution $(Y_{\cdot},Z_{\cdot},K_{\cdot},A_{\cdot})$ of DRBSDE $(\xi,g+{\rm d}V,L,U)$ in $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}\times\mathcal{V}^{+,p}$, it is not difficult to check that $(Y_{\cdot},Z_{\cdot},A_{\cdot})$ is a solution of $\bar{R}$BSDE $(\xi,\bar{g}_{n}+{\rm d}\bar{V},U)$ in $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}$ with $\bar{V}_{\cdot}:=V_{\cdot}+K_{\cdot}$ for each $n\geq 1$. Thus, in view of the assumption that $(\underline{Y}^{n}_{\cdot},\underline{Z}^{n}_{\cdot},\underline{A}^{n}_{\cdot})$ is the minimal solution of $\bar{R}$BSDE $(\xi,\bar{g}_{n}+{\rm d}V,U)$ in $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}$ for each $n\geq 1$, using (iii) of Proposition 2.12 together with Remark 2.14 yields that for each $n\geq 1$,

[TABLE]

Furthermore, since $\lim\limits_{n\rightarrow\infty}\|\underline{Y}_{\cdot}^{n}-\underline{Y}_{\cdot}\|_{\mathcal{S}^{p}}=0$, we have

[TABLE]

which is the desired result. ∎

4.4 Examples and remarks

We first introduce several examples which Theorems 4.23 and 4.25 can be applied to. However, to the best of our knowledge, all of their conclusions can not be obtained by any existing results. In these examples, we always assume that $p>1$, $V_{\cdot}\in\mathcal{V}^{p}$, and (H4) holds for $L_{\cdot},U_{\cdot},\xi$ and $X_{\cdot}$

Example 4.26**.**

Let the generator

[TABLE]

Clearly, this $g$ satisfies assumptions (H1s) with $\mu=1$, (H2) with $\phi(x)=x+\sqrt{x}$, and (H3), but does not satisfy assumption (H2s). Then, in view of Remark 2.6, it then follows from Theorem 4.23 that DRBSDE $(\xi,g+{\rm d}V,L,U)$ admits a unique solution in $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}\times\mathcal{V}^{+,p}$.

Example 4.27**.**

Let the generator

[TABLE]

where, with $\delta>0$ small enough,

[TABLE]

It is not very hard to verify that this $g$ satisfies assumptions (H1) with $\rho(\cdot)=h(\cdot)$, (H2) with $\phi(x)=x+\sqrt[3]{x}$, and (H3), but does not satisfy assumptions (H1s) and (H2s). It then follows from Theorem 4.23 that DRBSDE $(\xi,g+{\rm d}V,L,U)$ admits a unique solution in $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}\times\mathcal{V}^{+,p}$.

Example 4.28**.**

Let the generator

[TABLE]

It is not hard to check that this $g$ satisfies assumptions (H1s) with $\mu=0$, (H2’) with $f_{\cdot}\equiv 0$, $\mu=0$ and $\lambda=1$, and (H3), but does not satisfy (H2) (ii). Then, in view of Remark 2.6, it follows from Theorem 4.25 that DRBSDE $(\xi,g+{\rm d}V,L,U)$ admits a minimal and a maximal solutions in $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}\times\mathcal{V}^{+,p}$.

Remark 4.29**.**

With respect to this section, we would like to mention the following things.

Compared with that in Proposition 4.1 of Fan [16], the assumption (4.2) in our Proposition 4.20 is weaker and more natural, although some ideas of the proof are borrowed from there. And, in view of Remark 2.3, Proposition 4.20 includes Proposition 4.1 in Fan [16] as its particular case.

• 2)

Theorem 4.23* strengthens (v) of Theorem 6.5 in Klimsiak [38], where the generator $g$ needs to satisfy stronger assumptions (H1s) and (H2s) than assumptions (H1) and (H2) in Theorem 4.23. And, in view of Remark 2.3, Theorems 4.23 and 4.25 include respectively Theorems 5.8 and 5.11 in Fan [16] as its particular case.*

• 3)

Proposition 4.21* is a key and difficult step to verify Theorems 4.23 and 4.25. In order to prove the conclusions of Proposition 4.21 under general assumptions (H1)-(H4), several auxiliary BSDEs need to be introduced, Propositions 2.9-2.12 need to be comprehensively and repeatedly applied, and Remarks 2.13 and 2.7 need to be always kept in mind.*

• 4)

In (iii) of Theorem 4.23, it is not clear whether the sequences of processes $\{K^{n}_{\cdot}\}$ and $\{A^{n}_{\cdot}\}$ converge respectively to $K_{\cdot}$ and $A_{\cdot}$ as $n$ tends to infinity.

• 5)

*Under the assumptions of Theorem 4.25, we do not know whether the minimal (resp. maximal) solution of DRBSDE $(\xi,g+{\rm d}V,L,U)$ in $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}$ can be approximated by a sequence of solutions of *$\underline{R}$*BSDEs with lower barrier $L$ (resp. *$\bar{R}$BSDEs with upper barrier $U$). In particular, under the same assumptions we also do not know whether a solution of DRBSDE $(\xi,g+{\rm d}V,L,U)$ in $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}$ can be approximated by a sequence of solutions of BSDEs.

5 Existence of the minimal (maximal) $L^{p}$ solution: approximation method

In this section, we will establish a general approximation result for the $L^{p}\ (p>1)$ solutions of DRBSDEs under some elementary conditions, and study existence of the $L^{p}$ solution of DRBSDEs where the generator $g$ may be discontinuous and have a general growth in $y$.

5.1 Convergence of the sequence of $L^{p}$ solutions for the approximation DRBSDEs

The following proposition is a general approximation result for the $L^{p}$ solutions of DRBSDEs under some elementary conditions.

Proposition 5.30** (Approximation method).**

Let $p>1$, $V_{\cdot}\in\mathcal{V}^{p}$ and (H4)(i) hold for $L_{\cdot}$, $U_{\cdot}$ and $\xi$. Assume that for each $n\geq 1$, the generator $g_{n}$ satisfies (ii) of (HH) with the same $f_{\cdot},\psi_{\cdot}(r)$ and $\lambda$, and $(Y_{\cdot}^{n},Z_{\cdot}^{n},K_{\cdot}^{n},A_{\cdot}^{n})$ is a solution of DRBSDE $(\xi,g_{n}+{\rm d}V,L,U)$ in $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}\times\mathcal{V}^{+,p}$. If $Y_{\cdot}^{n}\leq Y_{\cdot}^{n+1}\leq\bar{Y}_{\cdot}$, ${\rm d}A^{n}\leq{\rm d}A^{n+1}$ and ${\rm d}K^{n}\geq{\rm d}K^{n+1}$ for each $n\geq 1$ and a process $\bar{Y}_{\cdot}\in\mathcal{S}^{p}$, $g_{n}$ tends locally uniformly in $(y,z)$ to the generator $g$ as $n\rightarrow\infty$ in the following sense:

[TABLE]

and

[TABLE]

then there exists a quadruple $(Y_{\cdot},Z_{\cdot},K_{\cdot},A_{\cdot})\in\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}\times\mathcal{V}^{+,p}$ which verifies DRBSDE $(\xi,g+{\rm d}V,L,U)$ such that

[TABLE]

Proof.

Since $Y_{\cdot}^{n}$ increases in $n$ and is bounded above by a process $\bar{Y}_{\cdot}$, there exists a progressively measurable real-valued process $Y_{\cdot}$ such that for each $t\in[0,T]$,

[TABLE]

Furthermore, since ${\rm d}A^{n}\leq{\rm d}A^{n+1}$ and ${\rm d}K^{n}\geq{\rm d}K^{n+1}$ for each $n\geq 1$, a same analysis as that in proving (4.5) yields that there exist two processes $K_{\cdot}$ and $A_{\cdot}$ in $\mathcal{V}^{+,p}$ such that

[TABLE]

For each positive integer $l,q\geq 1$, as in the proof of Proposition 4.20, we introduce the following two stopping times:

[TABLE]

Then we have

[TABLE]

Furthermore, since all $g_{n}$ satisfy (HH) with the same $f_{\cdot},\psi_{\cdot}(r)$ and $\lambda$, and (5.3) holds, in view of the definitions of $\tau_{l}$ and $\sigma_{l,q}$, we know that ${\rm d}\mathbb{P}\times{\rm d}t-a.e.$, for each $l,q,n\geq 1$,

[TABLE]

with

[TABLE]

The rest proof is divided into 3 steps, which will be detailed in Appendix.

Step 1. Based on (5.2)-(5.7), making use of (ii) of Lemma 2.8, Hölder’s inequality and Lebesgue’s dominated convergence theorem, we show that $\{Y_{\cdot}^{n}\}$ converges to the process $Y_{\cdot}$ in $\mathcal{S}^{p}$ as $n\to\infty$.

Step 2. By virtue of (i) of Lemma 2.8, (5.2) and the conclusion of step 1, we show that $\{Z_{\cdot}^{n}\}$ converges to a process $Z_{\cdot}$ in ${\rm M}^{p}$ as $n\to\infty$.

Step 3. Making use of (5.1)- (5.7) and conclusions of steps 1 and 2, we show that $(Y_{\cdot},Z_{\cdot},K_{\cdot},A_{\cdot})$ is a solution of DRBSDE $(\xi,g+{\rm d}V,L,U)$ in $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}\times\mathcal{V}^{+,p}$. Proposition 5.30 is then proved. ∎

Remark 5.31**.**

We remark that the conclusion of Proposition 5.30 still holds if we replace the expression that $Y_{\cdot}^{n}\leq Y_{\cdot}^{n+1}\leq\bar{Y}_{\cdot}$, ${\rm d}A^{n}\leq{\rm d}A^{n+1}$ and ${\rm d}K^{n}\geq{\rm d}K^{n+1}$ for each $n\geq 1$ and a process $\bar{Y}_{\cdot}\in\mathcal{S}^{p}$ with $\underline{Y}_{\cdot}\leq Y_{\cdot}^{n+1}\leq Y_{\cdot}^{n}$, ${\rm d}A^{n}\geq{\rm d}A^{n+1}$ and ${\rm d}K^{n}\leq{\rm d}K^{n+1}$ for each $n\geq 1$ and a process $\underline{Y}_{\cdot}\in\mathcal{S}^{p}$, and replace the expression that $y^{n}\leq y$ in (5.1) with $y^{n}\geq y$.

5.2 Existence of the minimal (maximal) solution

We now consider the DRBSDEs where the generator $g$ may be discontinuous and have a general growth in $y$. Let us first introduce the following assumptions introduced by Fan and Jiang [19]:

1. (A1a)

$g$ is left-continuous and lower semi-continuous in $y$, and continuous in $z$, i.e., ${\rm d}\mathbb{P}\times{\rm d}t-a.e.$, for each $(y_{0},z_{0})\in{\mathbb{R}}^{1+d}$, we have

[TABLE]

and

[TABLE] 2. (A1b)

$g$ is right-continuous and upper semi-continuous in $y$, and continuous in $z$, i.e., ${\rm d}\mathbb{P}\times{\rm d}t-a.e.$, for each $(y_{0},z_{0})\in{\mathbb{R}}^{1+d}$, we have

[TABLE]

and

[TABLE] 3. (A2)

$g$ has a linear growth in $(y,z)$, i.e., there exist two constants $\tilde{\mu},\tilde{\lambda}\geq 0$ and a process $\tilde{f}_{\cdot}\in\mathcal{H}^{p}$ such that ${\rm d}\mathbb{P}\times{\rm d}t-a.e.$, for each $(y,z)\in{\mathbb{R}}^{1+d}$,

[TABLE]

Remark 5.32**.**

It is clear that (HH)(i) $\Leftrightarrow$ (A1a) $+$ (A1b), and that (A2) $\Leftrightarrow$ (H2’)(ii) $+$ (H3s).

Theorem 5.33**.**

Assume that $p>1$, $V_{\cdot}\in\mathcal{V}^{p}$, $g^{1}$ satisfies assumptions (H1), (H2’) and (H3), $g^{2}$ satisfies assumptions (A1a) (resp. (A1b)) and (A2), and that $g=g^{1}+g^{2}$. Assume further that assumption (H4) holds for $L_{\cdot}$, $U_{\cdot}$, $\xi$, $X_{\cdot}$ and $g$ (or $g^{1}$). Then, DRBSDE $(\xi,g+{\rm d}V,L_{\cdot},U_{\cdot})$ admits a minimal (resp. maximal) solution $(Y_{\cdot},Z_{\cdot},K_{\cdot},A_{\cdot})$ in the space $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}\times\mathcal{V}^{+,p}$.

Proof.

We only prove the case of the minimal solution. Another case can be proved in a similar way in view of Remark 5.31. Assume now that $p>1$, $V_{\cdot}\in\mathcal{V}^{p}$, $g^{1}$ satisfies (H1) with $\rho(\cdot)$, (H2’) with $f_{\cdot}$, $\mu$ and $\lambda$, and (H3) with $\psi_{\cdot}(r)$, $g^{2}$ satisfies (A1a) and (A2) with $\tilde{f}_{\cdot}$, $\tilde{\mu}$ and $\tilde{\lambda}$, and that $g=g^{1}+g^{2}$. Assume further that (H4) holds for $L_{\cdot}$, $U_{\cdot}$, $\xi$, $X_{\cdot}$ and $g^{1}$.

In view of the assumptions of $g^{1}$ and $g^{2}$ together with the proof of Theorem 1 in Fan and Jiang [19], it is not very difficult to verify that for each $n\geq 1$ and $(y,z)\in{\mathbb{R}}\times{\mathbb{R}}^{d}$, the following function

[TABLE]

with

[TABLE]

and

[TABLE]

is well defined and progressively measurable, ${\rm d}\mathbb{P}\times{\rm d}t-a.e.$, $g_{n}$ increases in $n$ and converges locally uniformly in $(y,z)$ to the generator $g=g^{1}+g^{2}$ as $n\rightarrow\infty$ in the sense of (5.1), all $g^{1}_{n}$ satisfy (H1) with the same $\rho(\cdot)$, (H2s) with $n+2\lambda$, (H3) with the same $\psi_{\cdot}(r)+\mu r+2f_{\cdot}$, and ${\rm d}\mathbb{P}\times{\rm d}t-a.e.$, $\forall\ n\geq 1$, $\forall\ (y,z)\in{\mathbb{R}}\times{\mathbb{R}}^{d}$,

[TABLE]

and all $g^{2}_{n}$ satisfy (H1) with $(n+2\tilde{\mu})x$, (H2s) with $n+2\tilde{\lambda}$, and ${\rm d}\mathbb{P}\times{\rm d}t-a.e.$, $\forall\ n\geq 1$, $\forall\ (y,z)\in{\mathbb{R}}\times{\mathbb{R}}^{d}$,

[TABLE]

Then, in view of (5.8), (5.9) and (H3) for $g^{1}$, we know that ${\rm d}\mathbb{P}\times{\rm d}t-a.e.$, $\forall\ n\geq 1$, $\forall\ (y,z)\in{\mathbb{R}}\times{\mathbb{R}}^{d}$,

[TABLE]

That is to say, all $g_{n}$ satisfy (HH) with the same parameters.

Note that for each $n\geq 1$, $g_{n}$ satisfies (H1), (H2) and (H3) by Remark 2.6 and Remark 5.32 together with (5.9). Furthermore, by (5.10) we know that for each $n\geq 1$, $g_{n}(\cdot,X_{\cdot},0)\in\mathcal{H}^{p}$ and then (H4) holds for $L_{\cdot}$, $U_{\cdot}$, $\xi$, $X_{\cdot}$ and $g_{n}$. It then follows from Theorem 4.23 that there exists a unique solution $(Y_{\cdot}^{n},Z_{\cdot}^{n},K_{\cdot}^{n},A_{\cdot}^{n})$ of DRBSDE $(\xi,g_{n}+{\rm d}V,L,U)$ in the space $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}\times\mathcal{V}^{+,p}$ for each $n\geq 1$. And, noticing that $g_{n}$ increases in $n$, by Corollary 3.17 and Corollary 4.24 we can deduce that $Y_{\cdot}^{n}\leq Y_{\cdot}^{n+1}$, ${\rm d}A^{n}\leq{\rm d}A^{n+1}$ and ${\rm d}K^{n}\geq{\rm d}K^{n+1}$ for each $n\geq 1$.

In the sequel, we show that inequality (5.2) appearing in Proposition 5.30 holds. In fact, let

[TABLE]

and

[TABLE]

It then follows from (5.8) and (5.9) that $\underline{g}\leq g_{n}\leq\bar{g}$ for each $n\geq 1$, and both $\underline{g}$ and $\bar{g}$ satisfy assumptions (H1), (H2s) and (H3) with

[TABLE]

and

[TABLE]

Thus, it follows from Theorem 4.23 that DRBSDE $(\xi,\underline{g}+{\rm d}V,L,U)$ and DRBSDE $(\xi,\bar{g}+{\rm d}V,L,U)$ admit respectively a unique solution $(\underline{Y}_{\cdot},\underline{Z}_{\cdot},\underline{K}_{\cdot},\underline{A}_{\cdot})$ and $(\bar{Y}_{\cdot},\bar{Z}_{\cdot},\bar{K}_{\cdot},\bar{A}_{\cdot})$ in the space $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}\times\mathcal{V}^{+,p}$, and by Corollary 3.17 and Corollary 4.24, we have that for each $t\in[0,T]$ and $n\geq 1$,

[TABLE]

Furthermore, note that for each $n\geq 1$, $g_{n}$ satisfies assumption (AA) with the same $\bar{f}_{\cdot}=|g^{1}(\cdot,0,0)|+f_{\cdot}+\tilde{f}_{\cdot}+A$, $\bar{\mu}=\tilde{\mu}+A$ and $\bar{\lambda}=\lambda+\tilde{\lambda}$ since $g^{n}_{1}$ satisfies assumption (H1) with the same $\rho(\cdot)$, and inequalities (5.8) and (5.9) hold. It follows from Proposition 2.9 that there exists a positive constant $C>0$ depending only on $p,A,\tilde{\mu},\tilde{\lambda},T$ such that for each $t\in[0,T]$ and $n\geq 1$,

[TABLE]

Finally, combining (5.11) and (5.12) yields that (5.2) in Proposition 5.30 holds.

Up to now, we have checked all conditions in Proposition 5.30. It then follows from Proposition 5.30 that DRBSDE $(\xi,g+{\rm d}V,L,U)$ admits a solution $(Y_{\cdot},Z_{\cdot},K_{\cdot},A_{\cdot})$ in $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}\times\mathcal{V}^{+,p}$ such that

[TABLE]

Finally, let us show that $(Y_{\cdot},Z_{\cdot},K_{\cdot},A_{\cdot})$ is just the minimal solution of RBSDE $(\xi,g+{\rm d}V,L,U)$ in the space $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}\times\mathcal{V}^{+,p}$. In fact, if $(Y^{\prime}_{\cdot},Z^{\prime}_{\cdot},K^{\prime}_{\cdot},A^{\prime}_{\cdot})$ is also a solution of DRBSDE $(\xi,g+{\rm d}V,L,U)$ in the space $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}\times\mathcal{V}^{+,p}$, then noticing that for each $n\geq 1$, $g_{n}\leq g$ and $g_{n}$ satisfies (H1) and (H2), it follows from Corollary 3.17 that for each $n\geq 1$ and $t\in[0,T]$, $Y_{t}^{n}\leq Y^{\prime}_{t}$. Thus, by (5.13) we obtain that $Y_{t}\leq Y^{\prime}_{t}$ for each $t\in[0,T]$, which is the desired result. Theorem 5.33 is then proved. ∎

By Corollary 3.17, Corollary 4.24 and the proof of Theorem 5.33, it is not hard to verify the following comparison theorem for the minimal (resp. maximal) $L^{p}$ solutions of DRBSDEs.

Proposition 5.34**.**

Let $p>1$ and for $i=1,2$, assume that $V_{\cdot}^{i}\in\mathcal{V}^{p}$, $g^{1,i}$ satisfies (H1), (H2’) and (H3), $g^{2,i}$ satisfies (A1a) (resp. (A1b)) and (A2), and that $g^{i}:=g^{1,i}+g^{2,i}$. Assume further that for $i=1,2$, (H4) holds for $\xi^{i}$, $L_{\cdot}^{i}$, $U_{\cdot}^{i}$, and $X_{\cdot}^{i}$ associated with $g^{i}$ (or $g^{1,i}$), and $(Y_{\cdot}^{i},Z_{\cdot}^{i},K_{\cdot}^{i},A_{\cdot}^{i})$ is the minimal (resp. maximal) solution of DRBSDE $(\xi^{i},g^{i}+{\rm d}V^{i},L^{i},U^{i})$ in the space $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}\times\mathcal{V}^{+,p}$. If $\xi^{1}\leq\xi^{2}$, ${\rm d}V^{1}\leq{\rm d}V^{2}$, $L^{1}_{\cdot}\leq L^{2}_{\cdot}$, $U^{1}_{\cdot}\leq U^{2}_{\cdot}$, and for each $(y,z)\in{\mathbb{R}}\times{\mathbb{R}}^{d}$,

[TABLE]

then $Y_{t}^{1}\leq Y_{t}^{2}$, $t\in[0,T]$. Furthermore, if $L^{1}_{\cdot}=L^{2}_{\cdot}$ and $U^{1}_{\cdot}=U^{2}_{\cdot}$, then ${\rm d}K^{1}\geq{\rm d}K^{2}$ and ${\rm d}A^{1}\leq{\rm d}A^{2}$.

5.3 Examples and remarks

We introduce two examples which Theorem 5.33 can be applied to, but any existing results can not be applied to.

Example 5.35**.**

Let the generator $g:=g^{1}+g^{2}$ with

[TABLE]

and

[TABLE]

where, with $\delta>0$ small enough,

[TABLE]

It is not very hard to verify that $g^{1}$ satisfies (H1) with $\rho(\cdot)=\bar{h}(\cdot)$, (H2’) with $f_{\cdot}\equiv 0$, $\mu=0$ and $\lambda=1$, and (H3), and that $g^{2}$ satisfies (A1a) and (A2) with $\tilde{f}_{\cdot}=|B_{\cdot}|+2$, $\tilde{\mu}=2$ and $\tilde{\lambda}=1$ for each $p>1$. Thus, if (H4) holds for $g^{1}$ and some $\xi$, $L_{\cdot}$, $U_{\cdot}$, $X_{\cdot}$ and $p>1$, and $V_{\cdot}\in\mathcal{V}^{p}$, then by Theorem 5.33 we know that DRBSDE $(\xi,g+{\rm d}V,L_{\cdot},U_{\cdot})$ admits a minimal solution in $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}\times\mathcal{V}^{+,p}$.

Example 5.36**.**

Let the generator

[TABLE]

It is easy to check that this $g$ satisfies (A1b) and (A2) with $\tilde{f}_{\cdot}=2$, $\tilde{\mu}=0$ and $\tilde{\lambda}=1$ for each $p>1$. Thus, if (H4) holds for $g$ and some $\xi$, $L_{\cdot}$, $U_{\cdot}$, $X_{\cdot}$ and $p>1$, and $V_{\cdot}\in\mathcal{V}^{p}$, it then follows from Theorem 5.33 that DRBSDE $(\xi,g+{\rm d}V,L_{\cdot},U_{\cdot})$ admits a maximal solution in $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}\times\mathcal{V}^{+,p}$.

Remark 5.37**.**

With respect to this section, we would like to mention the following things.

Compared with that in Proposition 4.2 of Fan [16], the assumptions (5.1) and (5.2) in Proposition 5.30 is weaker and more natural, although some ideas of the proof are borrowed from there. And, in view of Remark 2.3, Proposition 5.30 together with Remark 5.31 includes Proposition 4.2 in Fan [16] as its particular case.

• 2)

In view of Remark 2.3, Theorem 5.33 and Proposition 5.34 include respectively Theorem 5.13 and Proposition 5.15 in Fan [16]* as its special case . In particular, Theorem 5.33 and Proposition 5.34 also consider the case that the generator $g$ may be discontinuous in $y$.*

• 3)

It follows from Remarks 2.3, 2.6, 2.7 and 5.32 that since the associated assumptions are more general, Theorem 5.33 and Proposition 5.34 strengthen and unify some existing corresponding results on DRBSDEs, RBSDEs with one continuous barrier, and non-reflected BSDEs obtained, for example, in Briand et al. [6], El Asri et al. [9], Fan [16], Fan and Jiang [19], Hamadène et al. [26], Hamadène and Popier [28], Klimsiak [37], Ma et al. [44] and Rozkosz and Słomiński [55]**.

Appendix A

In this section, we will supply the details omitted in the proof procedures of Proposition 4.20 and Proposition 5.30.

Complementary of the details for the proof of Proposition 4.20. Now, we will detail the proof of steps 1-7 after the inequality (4.8).

Step 1. We show that $Y_{\cdot}$ is a càdlàg process. Let us fix a pair of $l,q\geq 1$ arbitrarily. From (4.7) and (4.8) together with (4.2), it follows that there exists a subsequence $\{h^{n_{j};l,q}_{\cdot}\}_{j=1}^{\infty}$ of the sequence $\{h^{n;l,q}_{\cdot}\}_{n=1}^{\infty}$ which converges weakly to a process $h^{l,q}_{\cdot}$ in the space $\mathcal{H}^{1}$. Now, take any bounded linear functional $\Phi(\cdot)$ defined on $\mathbb{L}^{1}(\mathcal{F}_{T})$. Then there exists a constant $b>0$ such that for each $\overline{h}_{\cdot}\in\mathcal{H}^{1}$ and each stopping time $\bar{\tau}$ taking values in $[0,T]$, we have

[TABLE]

Hence, for each stopping time $\bar{\tau}$ taking values in $[0,T]$, $\Phi(\int_{0}^{\bar{\tau}}\cdot\ {\rm d}s)$ is a bounded linear functional defined on $\mathcal{H}^{1}$, which means that

[TABLE]

As a result, for every stopping time $\tau$ with $0\leq\tau\leq\sigma_{l,q}$, as $j\rightarrow\infty$,

[TABLE]

Furthermore, it follows from (4.2) and Lemma 4.4 of Klimsiak [37] that there exists a process $Z_{\cdot}\in{\rm M}^{p}$ together with a subsequence of the sequence $\{n_{j}\}_{j=1}^{\infty}$, still denoted by itself, such that for any stopping time $\bar{\tau}$ taking values in $[0,T]$, as $j\rightarrow\infty$,

[TABLE]

In the sequel, we define

[TABLE]

Then, in view of (4.1), (4.5), (1.14), (1.15) and the fact that for each stopping time $\bar{\tau}$ taking values in $[0,T]$, $Y^{n}_{\bar{\tau}}\uparrow Y_{\bar{\tau}}$ in $\mathbb{L}^{1}(\mathcal{F}_{T})$, we can deduce that for every stopping time $\tau$ with $0\leq\tau\leq\sigma_{l,q}$, the sequence

[TABLE]

converges weakly to $K^{l,q}_{\tau}$ in $\mathbb{L}^{1}(\mathcal{F}_{T})$ as $j\rightarrow\infty$. Thus, since $K_{\cdot}^{n}\in\mathcal{V}^{+,p}$ for each $n\geq 1$, we have

[TABLE]

for any stopping times $\tau_{1}\leq\tau_{2}$ valued in $[0,T]$. Furthermore, in view of the definition of $K_{\cdot}^{l,q}$ as well as the facts that $V_{\cdot}\in\mathcal{V}^{p}$, $A_{\cdot}\in\mathcal{V}^{+,p}$, $Y^{n}_{\cdot}\uparrow Y_{\cdot}$ and $Y^{n}_{\cdot}\in\mathcal{S}^{p}$ for each $n\geq 1$, it is not very hard to verify that $K^{l,q}_{\cdot}$ is a optional process with $\mathbb{P}-a.s.$ upper semi-continuous paths. Thus, by Lemma A.3 in Bayraktar and Yao [3] we know that $K^{l,q}_{\cdot\wedge\sigma_{l,q}}$ is nondecreasing, and should have $\mathbb{P}-a.s.$ right lower semi-continuous paths. Hence, $K^{l,q}_{\cdot\wedge\sigma_{l,q}}$ is càdlàg and so is $Y_{\cdot\wedge\sigma_{l,q}}$ from the definition of $K^{l,q}_{\cdot}$. Finally, it follows from (4.6) that $Y_{\cdot}$ is also a càdlàg process.

Step 2. We show that $Y_{t}\geq L_{t}$ for each $t\in[0,T]$ and as $n\rightarrow\infty$,

[TABLE]

In fact, it follows from (4.2) and the definition of $K^{n}_{\cdot}$ that for each $n\geq 1$,

[TABLE]

Hence, by Fatou’s lemma and Hölder’s inequality, we have

[TABLE]

which implies that

[TABLE]

Since $Y_{\cdot}-L_{\cdot}$ is a càdlàg process, we get that $(Y_{t}-L_{t})^{-}=0$ and hence $Y_{t}\geq L_{t}$ for each $t\in[0,T)$. Moreover, note that $Y_{T}=Y^{n}_{T}=\xi\geq L_{T}$. Hence

[TABLE]

for each $t\in[0,T]$, and (1.16) follows by Dini’s theorem.

Step 3. We show the convergence of the sequence $\{Y_{\cdot}^{n}\}$ in $\mathcal{S}^{p}$. Let $\tau_{l}$ and $\sigma_{l,q}$ be the sequences of stopping times defined in Step 1. For each $n,m\geq 1$, observe from (4.1) that

[TABLE]

solves equation (2.6). It then follows from (ii) of Lemma 2.8 with $p=2$, $t=0$ and $\tau=\sigma_{l,q}$ that there exists a constant $C>0$ such that for each $n,m,l,q\geq 1$,

[TABLE]

Furthermore, in view of the definition of $K_{\cdot}^{n}$ and $A_{\cdot}^{n}$ with (4.1), we deduce that for each $t\in[0,T]$,

[TABLE]

and

[TABLE]

Combining (4.7), (1.18), (1.19), (1.20) and Hölder’s inequality leads to that for each $n,m,l,q\geq 1$,

[TABLE]

[TABLE]

Thus, in view of the definitions of $\tau_{l}$ and $\sigma_{l,q}$, (4.3), (4.2), (4.8) and (1.16), by (1.21) and Lebesgue’s dominated convergence theorem we can deduce that for each $l,q\geq 1$, as $n,m\rightarrow\infty$,

[TABLE]

which implies that for each $l,q\geq 1$, as $n,m\rightarrow\infty$,

[TABLE]

And, by (4.3) and (4.6) we get that

[TABLE]

So, $Y_{\cdot}$ is a continuous process. Finally, in view of (4.3) and (1.22), Lebesgue’s dominated convergence theorem yields that

[TABLE]

Step 4. We show the convergence of the sequence $\{Z_{\cdot}^{n}\}$ in ${\rm M}^{p}$. Note that (1.17) verifies (2.6). It follows from (i) of Lemma 2.8 with $t=0$ and $\tau=T$ that there exists a positive constant $C^{\prime}>0$ such that for each $m,n\geq 1$,

[TABLE]

Then, it follows from Hölder’s inequality and (1.20) that for each $m,n\geq 1$,

[TABLE]

[TABLE]

from which together with (4.2), (1.23) and (1.15) leads to the existence of a process $Z_{\cdot}\in{\rm M}^{p}$ such that

[TABLE]

Step 5. We show the desired convergence of the sequence $\{K_{\cdot}^{n}\}$. Let $\tau_{l}$ and $\sigma_{l,q}$ be the sequences of stopping times defined in Step 1. Since $g$ satisfies (HH), by (4.7), (4.8), (4.2), (1.22) and (1.24) we can deduce the existence of a subsequence $\{n_{j}\}$ of $\{n\}$ such that for each $l,q\geq 1$,

[TABLE]

Then, in view of (4.6), we have

[TABLE]

Combining (4.5), (1.22), (1.24) and (1.25) leads to that $\mathbb{P}-a.s.$, for each $t\in[0,T]$,

[TABLE]

tends to

[TABLE]

as $j\rightarrow\infty$ and that

[TABLE]

Hence, $K_{\cdot}$ is a continuous process. Furthermore, by Fatou’s lemma with (1.26) and (4.2) we get that

[TABLE]

Thus, $K_{\cdot}\in\mathcal{V}^{+,p}$.

Step 6. We show that $(Y_{\cdot},Z_{\cdot},K_{\cdot},A_{\cdot})$ is a desired solution of DRBSDE $(\xi,g+{\rm d}V,L,U)$ in the space $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}\times\mathcal{V}^{+,p}$. In fact, note that $(Y_{\cdot},Z_{\cdot},K_{\cdot},A_{\cdot})\in\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}\times\mathcal{V}^{+,p}$ solves

[TABLE]

It follows from Step 2 that $Y_{t}\geq L_{t}$ for each $t\in[0,T]$, and then

[TABLE]

And, in view of (1.22) and (1.26), it follows from the definition of $K_{\cdot}^{n}$ that

[TABLE]

Consequently,

[TABLE]

Furthermore, in view of the fact that $Y^{n}_{\cdot}\leq U_{\cdot}$ and $\int_{0}^{T}(U_{t}-Y^{n}_{t})\ {\rm d}A^{n}_{t}=0$ for each $n\geq 1$, from (1.23) and (4.5) we derive that $Y_{t}\leq U_{t}$ for each $n\geq 1$, and

[TABLE]

Finally, let us verify that ${\rm d}K\bot{\rm d}A$. In fact, for each $n\geq 1$, we define the following progressively measurable set

[TABLE]

It then follows from the definition of $K^{n}_{\cdot}$ that for each $n\geq 1$,

[TABLE]

and, in view of $\int_{0}^{T}(U_{t}-Y^{n}_{t}){\rm d}A^{n}_{t}=0$,

[TABLE]

Thus, noticing that $D_{n}\subset D_{n+1}$ for each $n\geq 1$ due to $Y^{n}_{\cdot}\leq Y^{n+1}_{\cdot}$, by (1.26) and (4.5) we have

[TABLE]

and

[TABLE]

Hence, ${\rm d}K\bot{\rm d}A$. The proof of Proposition 4.20 is then complete.

Complementary of the details for the proof of Proposition 5.30. Now, we will detail the proof of steps 1-3 after the inequality (5.7).

Step 1. We show the convergence of the sequence $\{Y_{\cdot}^{n}\}$ in $\mathcal{S}^{p}$. For each $n,m\geq 1$, observe that

[TABLE]

solves equation (2.6). It then follows from (ii) of Lemma 2.8 with $p=2$, $t=0$ and $\tau=\sigma_{l,q}$ that there exists a constant $C>0$ such that for each $n,m,l,q\geq 1$,

[TABLE]

[TABLE]

Furthermore, in view of the facts that $L_{t}\leq Y_{t}^{n}\leq U_{t}$ for each $t\in[0,T]$ and $n\geq 1$, and $\int_{0}^{T}(Y_{t}^{n}-L_{t}){\rm d}K_{t}^{n}=\int_{0}^{T}(U_{t}-Y_{t}^{n}){\rm d}A_{t}^{n}=0$ for each $n\geq 1$, we have that for each $t\in[0,T]$ and $l,q,m,n\geq 1$,

[TABLE]

and

[TABLE]

Combining (5.6), (1.28), (1.29) and (1.30) together with Hölder’s inequality leads to that

[TABLE]

Thus, in view of the definitions of $\tau_{l}$ and $\sigma_{l,q}$, (5.3), (5.2) and (5.7), it follows from (1.31) and Lebesgue’s dominated convergence theorem that for each $l,q\geq 1$, as $n,m\rightarrow\infty$,

[TABLE]

which implies that for each $l,q\geq 1$, as $n,m\rightarrow\infty$,

[TABLE]

And, in view of (5.5) and (5.3), we have

[TABLE]

So, $Y_{\cdot}$ is a continuous process. Finally, in view of (1.32) and (5.3), Lebesgue’s dominated convergence theorem yields that

[TABLE]

Step 2. We show the convergence of the sequence $\{Z_{\cdot}^{n}\}$ in ${\rm M}^{p}$. Note that (1.27) verifies (2.6). It follows from (i) of Lemma 2.8 with $t=0$ and $\tau=T$ that there exists a positive constant $C^{\prime}>0$ such that for each $m,n\geq 1$,

[TABLE]

Then, in view of (1.29) and (1.30), it follows from Hölder’s inequality that for each $m,n\geq 1$,

[TABLE]

from which together with (1.33) and (5.2) yields the existence of a process $Z_{\cdot}\in{\rm M}^{p}$ such that

[TABLE]

Step 3. We show that $(Y_{\cdot},Z_{\cdot},K_{\cdot},A_{\cdot})$ is a solution of DRBSDE $(\xi,g+{\rm d}V,L,U)$ in the space $\mathcal{S}^{p}\times{\rm M}^{p}\times\mathcal{V}^{+,p}\times\mathcal{V}^{+,p}$. By (5.1)-(5.3), (1.34) and (5.6)-(5.7) we derive the existence of a subsequence $\{n_{j}\}$ of $\{n\}$ such that for each $l,q\geq 1$,

[TABLE]

Then, in view of (5.5), we have

[TABLE]

Combining (5.4), (1.32), (1.34) and (1.35) leads to that

[TABLE]

Since $L_{t}\leq Y_{t}^{n}\leq U_{t},\ n\geq 1$ and $Y_{t}^{n}\uparrow Y_{t}$ for each $t\in[0,T]$, we know that $L_{t}\leq Y_{t}\leq U_{t}$ for each $t\in[0,T]$. Furthermore, in view of (1.33) and (5.4), we have

[TABLE]

and

[TABLE]

Finally, let us show that ${\rm d}K\bot{\rm d}A$. In fact, for each $n\geq 1$, since ${\rm d}K^{n}\bot{\rm d}A^{n}$, we know that there exists a progressively measurable set $D_{n}\subset\Omega\times[0,T]$ such that

[TABLE]

Then, in view of (5.4) and the fact that ${\rm d}K\leq{\rm d}K^{n}$ for each $n\geq 1$, we have

[TABLE]

and

[TABLE]

Hence, ${\rm d}K\bot{\rm d}A$. The proof of Proposition 5.30 is then complete.