Reflected Discontinuous Backward Doubly Stochastic Differential Equation With Poisson Jumps

TL;DR

This paper establishes the existence and comparison theorems for reflected backward doubly stochastic differential equations with Poisson jumps, including solutions with continuous barriers and linear growth conditions.

Contribution

It introduces new existence results and a comparison theorem for RBDSDEPs with Poisson jumps, expanding the theoretical understanding of these stochastic equations.

Findings

Existence of solutions for RBDSDEPs with continuous barriers.

Comparison theorem for minimal and maximal solutions.

Solutions under linear growth and left continuity conditions.

Abstract

In this paper{\}we prove the existence of a solution for reflected backward doubly stochastic differential equations with poisson jumps (RBDSDEPs) with one continuous barrier where the generator is continuous and also we study the RBDSDEPs with a linear growth condition and left continuity in $y$ on the generator. By a comparison theorem established here for this type of equation we provide a minimal or a maximal solution to RBDSDEPs.

Equations

$$Y_t=\xi +{\int }_t^{T}f(s,Y_s,Z_s)ds+{\int }_t^{T}g(s,Y_s,Z_s)d{\overleftarrow{B}}_s-{\int }_t^T{Z}_{s}d{W}_s,0\le t\le T,$$

$$Y_t=\xi +{\int }_t^{T}f(s,{\Lambda }_s)ds+{\int }_t^{T}g(s,{\Lambda }_s)d{\overleftarrow{B}}_s+{\int }_t^{T}d{K}_s-{\int }_t^T{Z}_{s}d{W}_s-{\int }_t^T{\int }_E{U}_s\left(e\right)\tilde{\mu }\left(ds,de\right),0\le t\le T,$$

$${\left|\left|U_t\right|\right|}_{{\mathcal{L}}^2\left(0,T,\tilde{\mu },{\mathbb{R}}^d\right)}^2=\mathbb{E}{\int }_0^T{\left|\left|U_t\right|\right|}_{L^2\left(E,\mathcal{E},\lambda ,{\mathbb{R}}^d\right)}^{2}dt<\infty ,$$

$${\left|\left|U_t\right|\right|}_{L^2\left(E,\mathcal{E},\lambda ,{\mathbb{R}}^d\right)}^2={\int }_E{\left|U_t\left(e\right)\right|}^2\lambda \left(de\right).$$

$${\left|\left|\left(Y,Z,U,K\right)\right|\right|}_{{\mathcal{D}}^2}={\left|\left|Y\right|\right|}_{{\mathcal{S}}^2}+{\left|\left|Z\right|\right|}_{{\mathcal{M}}^2}+{\left|\left|U\right|\right|}_{{\mathcal{L}}^2}+{\left|\left|K\right|\right|}_{{\mathcal{A}}^2}.$$

$$\left\{\begin{array}[]{l}i)Y\in\mathcal{S}^{2}\left(0,T,\mathbb{R}\right),\text{ }Z\in\mathcal{M}^{2}\left(0,T,\mathbb{R}^{d}\right)\text{ },K\in\mathcal{A}^{2}\text{ },U\in\mathcal{L}^{2}\left(0,T,\tilde{\mu},\mathbb{R}\right),\\ \\ ii)\text{ }Y_{t}=\xi+\int_{t}^{T}f(s,Y_{s},Z_{s},U_{s})ds+\int_{t}^{T}g(s,Y_{s},Z_{s},U_{s})d\overleftarrow{B}_{s}\\ \\ +\int_{t}^{T}dK_{s}-\int_{t}^{T}Z_{s}dW_{s}-\int_{t}^{T}\int_{E}U_{s}\left(e\right)\tilde{\mu}\left(ds,de\right),\ 0\leq t\leq T,\\ \\ iii)\text{ }S_{t}\leq Y_{t},\text{\quad}0\leq t\leq T\text{\quad and\quad}\int_{0}^{T}\left(Y_{t}-S_{t}\right)dK_{t}=0.\end{array}\right.$$

$$f(\cdot ,\omega ,y,z,u)$$

$$g(\cdot ,\omega ,y,z,u)$$

$$\left\{\begin{array}[]{l}\left(i\right)\text{ }\left|f(t,\omega,y,z,u)-f(t,\omega,y^{{}^{\prime}},z^{{}^{\prime}},u^{{}^{\prime}})\right|^{2}\leq C\left[\left|y-y^{{}^{\prime}}\right|^{2}+\left|z-z^{{}^{\prime}}\right|^{2}+\left|u-u^{{}^{\prime}}\right|^{2}\right],\\ \left(ii\right)\text{ }\left|g(t,\omega,y,z,u)-g(t,\omega,y^{{}^{\prime}},z^{{}^{\prime}},u^{{}^{\prime}})\right|^{2}\leq C\left|y-y^{{}^{\prime}}\right|^{2}+\alpha\left(\left|z-z^{{}^{\prime}}\right|^{2}+\left|u-u^{{}^{\prime}}\right|^{2}\right).\end{array}\right.$$

$$\mathbb{E}\left(\underset{0\le t\le T}{\sup }{\left(S_t^{+}\right)}^2\right)<+\infty ,$$

$${\alpha }_t={\alpha }_0+{\int }_0^t{\beta }_{s}ds+{\int }_0^t{\gamma }_{s}d{B}_s+{\int }_0^t{\eta }_{s}d{W}_s+{\int }_0^{t}d{K}_s+{\int }_0^t{\int }_E{\sigma }_s\left(e\right)\tilde{\mu }\left(ds,de\right),$$

$${\left|{\alpha }_t\right|}^2$$

$$\mathbb{E}{\left|{\alpha }_t\right|}^2+\mathbb{E}{\int }_t^T{\left|{\eta }_s\right|}^{2}ds+\mathbb{E}{\int }_t^T{\int }_E{\left|{\sigma }_s\left(e\right)\right|}^2\lambda \left(de\right)ds\le \mathbb{E}{\left|{\alpha }_T\right|}^2+2\mathbb{E}{\int }_t^T⟨{\alpha }_s,{\beta }_s⟩ds+2\mathbb{E}{\int }_t^T⟨{\alpha }_s,d{K}_s⟩+\mathbb{E}{\int }_t^T{\left|{\gamma }_s\right|}^{2}ds.$$

$$\begin{array}[]{l}Y_{t}^{i}=\xi^{i}+\int_{t}^{T}f^{i}(s,Y_{s}^{i},Z_{s}^{i},U_{s}^{i})ds+\int_{t}^{T}g(s,Y_{s}^{i},Z_{s}^{i},U_{s}^{i})d\overleftarrow{B}_{s}\\ \\ +\int_{t}^{T}dK_{s}^{i}-\int_{t}^{T}Z_{s}^{i}dW_{s}-\int_{t}^{T}\int_{E}U_{s}^{i}\left(e\right)\tilde{\mu}\left(ds,de\right),\ 0\leq t\leq T.\end{array}$$

$$\left\{\begin{array}[]{l}\xi^{1}\leq\xi^{2},\\ \forall t\leq T,\text{ }S_{t}^{1}\leq S_{t}^{2},\\ f^{1}\left(t,Y_{t},Z_{t},U_{t}\right)\leq f^{{}^{2}}\left(t,Y_{t},Z_{t},U_{t}\right).\end{array}\right.$$

$$Y_t^1\le Y_t^{{}^2}.$$

$${\bar{Y}}_t$$

$$\mathbb{E}{\left|{\left({\bar{Y}}_t\right)}^{+}\right|}^2+\mathbb{E}{\int }_t^T{1}_{\left\{{\bar{Y}}_s>0\right\}}{\left|\left|{\bar{Z}}_s\right|\right|}^{2}ds+\mathbb{E}{\int }_t^T{\int }_E{1}_{\left\{{\bar{Y}}_s>0\right\}}{\left|{\bar{U}}_s\left(e\right)\right|}^2\lambda \left(de\right)ds$$

$$\left\{\begin{array}[]{l}\left(\xi^{1}-\xi^{2}\right)^{+}=0,\\ \\ \int_{t}^{T}\left(\bar{Y}_{s}\right)^{+}\left(dK_{s}^{1}-dK_{s}^{2}\right)=-\int_{t}^{T}\left(Y_{s}^{1}-Y_{s}^{2}\right)^{+}dK_{s}^{2}\leq 0,\end{array}\right.$$

$$\mathbb{E}\left\{{\left|{\left({\bar{Y}}_t\right)}^{+}\right|}^2+{\int }_t^T{1}_{\left\{{\bar{Y}}_s>0\right\}}{\left|\left|{\bar{Z}}_s\right|\right|}^{2}ds+{\int }_t^T{\int }_E{1}_{\left\{{\bar{Y}}_s>0\right\}}{\left|{\bar{U}}_s\left(e\right)\right|}^2\tilde{\mu }\left(de\right)ds\right\}$$

$$2\mathbb{E}{\int }_t^T{\left({\bar{Y}}_s\right)}^{+}\left(f^1(s,Y_s^1,Z_s^1,U_s^1)-f^2(s,Y_s^2,Z_s^2,U_s^2)\right)ds$$

$${\left|\left|g(s,Y_s^1,Z_s^1,U_s^1)-g(s,Y_s^2,Z_s^2,U_s^2)\right|\right|}^2\le C{\left|{\bar{Y}}_s\right|}^{2}ds+\alpha \left\{{\left|{\bar{Z}}_s\right|}^2+{\left|{\bar{U}}_s\right|}^2\right\}.$$

$$\mathbb{E}\left\{{\left|{\left({\bar{Y}}_t\right)}^{+}\right|}^2+{\int }_t^T{1}_{\left\{{\bar{Y}}_s>0\right\}}{\left|\left|{\bar{Z}}_t\right|\right|}^{2}ds+{\int }_t^T{\int }_E{1}_{\left\{{\bar{Y}}_s>0\right\}}{\left|{\bar{U}}_s\left(e\right)\right|}^2\tilde{\mu }\left(de\right)ds\right\}$$

$$\mathbb{E}{\left|{\left({\bar{Y}}_t\right)}^{+}\right|}^2\le \left(3C+2C^2ϵ\right)\mathbb{E}{\int }_t^T{\left|{\bar{Y}}_s^{+}\right|}^{2}ds,$$

$$\mathbb{E}\left[{\left|{\left({\bar{Y}}_t\right)}^{+}\right|}^2\right]=0,$$

$$Y_t^1\le Y_t^{{}^2}.$$

$$\left\{\begin{array}[]{l}\left|f(t,\omega,y,z,u)\right|\leq C\left(1+\left|y\right|+\left|z\right|+\left|u\right|\right),\\ \left|g(t,\omega,y,z,u)-g(t,\omega,y^{{}^{\prime}},z^{{}^{\prime}},u^{{}^{\prime}})\right|^{2}\leq C\left|y-y^{{}^{\prime}}\right|^{2}+\alpha\left\{\left|z-z^{{}^{\prime}}\right|^{2}+\left|u-u^{{}^{\prime}}\right|^{2}\right\}.\end{array}\right.$$

$$f_n\left(t,\omega ,y,z,u\right)=\underset{\left(y^{{}^{\prime }},z^{\prime },u^{{}^{\prime }}\right)\in {\mathcal{B}}^2\left(\mathbb{R}\right)}{\inf }\left[f\left(t,\omega ,y^{{}^{\prime }},z^{\prime },u^{{}^{\prime }}\right)+n\left(\left|y-y^{{}^{\prime }}\right|+\left|z-z^{\prime }\right|+\left|u-u^{{}^{\prime }}\right|\right)\right],$$

$$\left|f_n\left(t,\omega ,y,z,u\right)-f_n\left(t,\omega ,y^{{}^{\prime }},z^{\prime },u^{{}^{\prime }}\right)\right|\le n\left|y-y^{{}^{\prime }}\right|+\left|z-z^{\prime }\right|+\left|u-u^{{}^{\prime }}\right|.$$

$$\left\{\begin{array}[]{l}Y_{t}^{n}=\xi+\int_{t}^{T}f_{n}(s,Y_{s}^{n},Z_{s}^{n},U_{s}^{n})ds+\int_{t}^{T}g(s,Y_{s}^{n},Z_{s}^{n},U_{s}^{n})d\overleftarrow{B}_{s}\\ \\ +\int_{t}^{T}dK_{s}^{n}-\int_{t}^{T}Z_{s}^{n}dW_{s}-\int_{t}^{T}\int_{E}U_{s}^{n}\left(e\right)\tilde{\mu}\left(ds,de\right),\ 0\leq t\leq T,\\ \\ S_{t}\leq Y_{t}^{n},\text{ }0\leq t\leq T,\text{\quad and\quad}\int_{0}^{T}\left(Y_{t}^{n}-S_{t}\right)dK_{t}^{n}=0.\end{array}\right.$$

$$\left\{\begin{array}[]{l}V_{t}=\xi+\int_{t}^{T}H(s,V_{s},N_{s},M_{s})ds+\int_{t}^{T}g(s,V_{s},N_{s},M_{s})d\overleftarrow{B}_{s}\\ \\ +\int_{t}^{T}dK_{s}-\int_{t}^{T}N_{s}dW_{s}-\int_{t}^{T}\int_{E}M_{s}\left(e\right)\tilde{\mu}\left(ds,de\right),\ 0\leq t\leq T,\\ \\ S_{t}\leq V_{t},\text{ }0\leq t\leq T,\text{\quad and\quad}\int_{0}^{T}\left(V_{t}-S_{t}\right)dK_{t}=0,\end{array}\right.$$

Full text

Reflected Discontinuous Backward Doubly Stochastic Differential

Equation With Poisson Jumps.

Badreddine Mansouri, Mostapha abd elouahab Saouli

University Mohamed Khider, PO BOX 145, 07000 Biskra, Algeria

footnotetext: E-mail: mansouri.badreddine@gmail.com (Badreddine Mansouri), saoulimoustapha@yahoo.fr (Mostapha abd el ouahab Saouli)

Abstract. In this paper we prove the existence of a solution for reflected backward doubly stochastic differential equations with poisson jumps (RBDSDEPs) with one continuous barrier where the generator is continuous and also we study the RBDSDEPs with a linear growth condition and left continuity in $y$ on the generator. By a comparison theorem established here for this type of equation we provide a minimal or a maximal solution to RBDSDEPs.

**Keyword **Reflected Backward doubly stochastic differential equations, random Poisson measure, minimal solution, comparison theorem, discontinuous generator.

1 Introduction.

A new kind of backward stochastic differential equations was introduced by Pardoux and Peng [11] in 1994, which is a class of backward doubly stochastic differential equations (BDSDEs for short)

[TABLE]

where $\xi$ is a random variable termed the terminal condition, $f:\Omega\times[0,T]\times\mathbb{R}\times\mathbb{R}^{d}\rightarrow\mathbb{R}$, $g:\Omega\times[0,T]\times\mathbb{R}\times\mathbb{R}^{d}\rightarrow\mathbb{R}$ are two jointly measurable processes, $W$ and $B$ are two mutually independent standard Brownian motion, with values, respectively in $\mathbb{R}^{d}$ and $\mathbb{R}$. Several authors interested in weakening this assumption see Bahlali et al [3], Boufoussi et al. [5], Lin. Q [8] and [9], N’zi el al. [10], Shi et al. [12], Wu et al. [14], Zhu et al. [16]. A class of backward doubly stochastic differential equations with jumps was study by Sun el al. [13], Zhu et al. [15] They have proved the existence and uniqueness of solutions for this type of BDSDEs under uniformly Lipschitz conditions.

In addition, Bahlali et al [2] prove the existence and uniqueness of solutions to reflected backward doubly stochastic differential equations (RBDSDEs) with one continuous barrier and uniformly Lipschitz coefficients. The existence of a maximal and a minimal solution for RBDSDEs with continuous generator is also established.

In this paper, we study the now well-know reflected backward doubly stochastic differential equations with jumps (RBDSDEPs for short):

[TABLE]

where $\Lambda_{s}=\left(Y_{s},Z_{s},U_{s}\right)$.

Motivated by the above results and by the result introduced by Fan. X, Ren. Y [6] and Zhu, Q., Shi, Y [15, 16], we establish firstly the existence of the solution of the reflected BDSDE with Poisson jumps (RBDSDEP in short) under the continuous coefficient, also we prove the existence solution of a RBDSDEP where the coefficient $f$ satisfy a linear growth and left continuity in $y$ conditions on the generator of this type of equation.

The organization of the paper is as follows. In Section 2, we give some preliminaires and we consider the spaces of processus also we define the Itô’s formula. In Section 3, we proof a comparison theorem, section 4 under a continuous conditions on $f$ we obtain the existence of a minimal solution of RBDSDEP, and finally in section 5, we study RBDSDEP where the generator $f$ satisfied a left continuity in $y$ and linear growth conditions.

2 Notation, assumptation and definition.

Let $\left(\Omega,\mathcal{F},P\right)$ be a complete probability space. For $T>0$, We suppose that $\left(\mathcal{F}_{t}\right)_{t\geq 0}$ is generated by the following three mutually independent processes:

(i) Let $\left\{W_{t},0\leq t\leq T\right\}$ and $\left\{B_{t},0\leq t\leq T\right\}$ be two standard Brownian motion defined on $\left(\Omega,\mathcal{F},P\right)$ with values in $\mathbb{R}^{d}$ and $\mathbb{R}$, respectively.

(ii) Let random Poisson measure $\mu$ on $E\times\mathbb{R}_{+}$ with compensator $\nu\left(dt,de\right)=\lambda\left(de\right)dt$, where the space $E=\mathbb{R}-\left\{0\right\}$ is equipped with its Borel field $\mathcal{E}$ such that $\left\{\tilde{\mu}\left(\left[0,t\right]\times A\right)=\left(\mu-\nu\right)\left(\left[0,t\right]\times A\right)\right\}$ is a martingale for any $A\in\mathcal{E}$ satisfying $\lambda\left(A\right)<\infty$. $\lambda$ is a $\sigma$ finite measure on $\mathcal{E}$ and satisfies $\int_{E}\left(1\wedge\left|e\right|^{2}\right)\lambda\left(de\right)<\infty.$

Let  $\mathcal{F}_{t}^{W}:=\sigma(W_{s};0\leq s\leq t)$, $\mathcal{F}_{t}^{\mu}:=\sigma(\mu_{s};0\leq s\leq t)$ and  $\mathcal{F}_{t,T}^{B}:=\sigma(B_{s}-B_{t};t\leq s\leq T),$ completed with $P$-null sets. We put,  $\mathcal{F}_{t}:=\mathcal{F}_{t}^{W}\vee\mathcal{F}_{t,T}^{B}\vee\mathcal{F}_{t}^{\mu}$. It should be noted that $\left(\mathcal{F}_{t}\right)$ is not an increasing family of sub $\sigma-$fields, and hence it is not a filtration.

For $d\in\mathbb{N}^{\ast},$ $\left|\cdot\right|$ stands for the Euclidian norm in $\mathbb{R}^{d}\times\left[0,T\right].$

We consider the following spaces of processus:

• •

We denote by $\mathcal{S}^{2}\left(0,T,\mathbb{R}^{d}\right)$, the set of continuous $\mathcal{F}_{t}-$measurable processes $\left\{\varphi_{t};t\in\left[0,T\right]\right\}$, which satisfy $\mathbb{E}(\sup_{0\leq t\leq T}\left|\varphi_{t}\right|^{2})<\infty$.

• •

Let $\mathcal{M}^{2}\left(0,T,\mathbb{R}^{d}\right)$ denote the set of  $d-$ dimensional, $\mathcal{F}_{t}-$measurable processes $\left\{\varphi_{t};t\in\left[0,T\right]\right\}$, such that  $\mathbb{E}\int_{0}^{T}\left|\varphi_{t}\right|^{2}dt<\infty$.

• •

$\mathcal{A}^{2}$ set of continuous, increasing, $\mathcal{F}_{t}-$measurable process $K:\left[0,T\right]\times\Omega\rightarrow[0,+\infty($ with $K_{0}=0,$ $\mathbb{E}\left(K_{T}\right)^{2}<+\infty.$

• •

$\mathbb{L}^{2}$ set of $\mathcal{F}_{T}$- measurable random variables $\xi:\Omega\rightarrow\mathbb{R}$ with $\mathbb{E}\left|\xi\right|^{2}<+\infty.$

• •

We denote by $\mathcal{L}^{2}\left(0,T,\tilde{\mu},\mathbb{R}^{d}\right)$, the space of mappings $U:\Omega\times\left[0,T\right]\times E\rightarrow\mathbb{R}^{d}$ which are $\mathcal{P}\otimes\mathcal{E}$ measurable such that

[TABLE]

where $\mathcal{P}\otimes\mathcal{E}$ denoted the $\sigma-$algebra of $\mathcal{F}_{t}-$predectable sets of $\Omega\times\left[0,T\right]$ and

[TABLE]

• •

Notice also the space $\mathcal{D}^{2}\left(\mathbb{R}\right)=\mathcal{S}^{2}\left(0,T,\mathbb{R}\right)\times\mathcal{M}^{2}\left(0,T,\mathbb{R}^{d}\right)\times\mathcal{L}^{2}\left(0,T,\tilde{\mu},\mathbb{R}\right)\times\mathcal{A}^{2}$ endowed with the norm

[TABLE]

is a Banach space.

Definition 2.1**.**

A solution of a reflected BDSDEPs is a quadruple of processes $\left(Y,Z,K,U\right)$ wich satisfies

[TABLE]

We give the following assumptions (H) on the data $\left(\xi,f,g,S\right)$:

(H.1) $f:[0,T]\times\Omega\times\mathbb{R}\times\mathbb{R}^{d}\times\mathcal{L}^{2}\left(0,T,\tilde{\mu},\mathbb{R}\right)\rightarrow\mathbb{R};$ $g:[0,T]\times\Omega\times\mathbb{R}\times\mathbb{R}^{d}\times\mathcal{L}^{2}\left(0,T,\tilde{\mu},\mathbb{R}\right)\rightarrow\mathbb{R}$ be jointly measurable such that for any $\left(y,z,u\right)\in\mathbb{R}\times\mathbb{R}^{d}\times\mathcal{L}^{2}\left(0,T,\tilde{\mu},\mathbb{R}\right)$

[TABLE]

**(H.2) **There exist constant $C>0$ and a constant $0<\alpha<1$ such that for every $\left(\omega,t\right)\in\Omega\times[0,T]$ and $\left(y,y^{{}^{\prime}}\right)\in\mathbb{R}^{2},$ $\left(z,z^{{}^{\prime}}\right)\in\left(\mathbb{R}^{d}\right)^{2},$ $\left(u,u^{{}^{\prime}}\right)\in\left(\mathcal{L}^{2}\left(0,T,\tilde{\mu},\mathbb{R}\right)\right)^{2}$

[TABLE]

(H.3) The terminal value $\xi$ be a given random variable in $L^{2}$.

(H.4) $\left(S_{t}\right)_{t\geq 0},$ is a continuous progressively measurable real valued process satisfying

[TABLE]

(H.5) $S_{T}\leq\xi$, $\mathbb{P}$-almost surely.

Theorem 2.1**.**

[6] Assume that $\left(H.1\right)-\left(H.5\right)$ holds. Then Eq $\left(1.1\right)$ admits a unique solution $\left(Y,Z,U,K\right)\in\mathcal{D}^{2}\left(\mathbb{R}\right)$.

The result depends on the following extension of the well-krown Itô’s formula. Its proof follows the same way as lemma 1.3 of [11]

Lemma 2.1**.**

Let $\alpha\in\mathcal{S}^{2}\left(0,T,\mathbb{R}^{k}\right),$ $\left(\beta,\gamma\right)\in\left(\mathcal{M}^{2}\left(\mathbb{R}^{k}\right)\right)^{2},$ $\eta\in\mathcal{M}^{2}\left(\mathbb{R}^{k\times d}\right)$ and $\sigma\in\mathcal{L}^{2}\left(0,T,\tilde{\mu},\mathbb{R}^{k}\right)$ such that:

[TABLE]

then $\left(i\right)$

[TABLE]

$\left(ii\right)$

[TABLE]

3 Comparison theorem.

Given two parameters $\left(\xi^{1},f^{1},g,T\right)$ and $\left(\xi^{2},f^{2},g,T\right),$ we considere the reflected BDSDEPs, $i=1,2$

[TABLE]

Theorem 3.1**.**

Assume that the reflected BDSDEP associated with dates $(\xi^{1},f^{1},g,T),$ $\left(resp\text{ }\left(\xi^{{}^{2}},f^{{}^{2}},g,T\right)\right)$ has a solution $(Y_{t}^{1},Z_{t}^{1},K_{t}^{1},U_{t}^{1})_{t\in[0,T)]},$ $($ $resp$ $\left(Y_{t}^{2},Z_{t}^{2},K_{t}^{2},U_{t}^{2}\right)_{t\in[0,T]}).$ Each one satisfying the assumption (H), assume moreover that:

[TABLE]

Then we have $\mathbb{P}-a.s.,$

[TABLE]

**Proof: **Let us show that $\left(Y_{t}^{1}-Y_{t}^{{}^{2}}\right)^{+}=0,$ using the equation (3.1), we get

[TABLE]

where $\bar{\xi}=\xi^{1}-\xi^{2},\ \bar{Z}=Z^{1}-Z^{2}$ and $\bar{U}=U^{1}-U^{2}.$

Since $\int_{t}^{T}\left(\bar{Y}_{s}\right)^{+}\left(g(s,Y_{s}^{1},Z_{s}^{1},U_{s}^{1})-g(s,Y_{s}^{2},Z_{s}^{2},U_{s}^{2})\right)d\overleftarrow{B}_{s}$ and $\int_{t}^{T}\left(\bar{Y}_{s}\right)^{+}\bar{Z}_{s}dW_{s}$ are a uniformly integrable martingale then taking expectation, we get by applying Lemma 2.1

[TABLE]

Since

[TABLE]

we get

[TABLE]

we obtain, by hypothesis $\left(H.2\right),$ and Young’s inequality the following inequality

[TABLE]

also we applying the assumption $\left(H.2\right)$ for $g,$ we get

[TABLE]

Then, we have the following inequality

[TABLE]

choosing $\epsilon$ such that $0<\epsilon^{-1}+\alpha\leq 1$, we have

[TABLE]

using Gronwall’s lemma implies that

[TABLE]

finally, we have

[TABLE]

4 Reflected BDSDEPs with continuous coefficient.

In this section we are interested in weakening the conditions on $f$. We assume that $f$ and $g$ satisfy the following assumptions:

**(H.6) **There exists $0<\alpha<1$ and $C>0$ $s.t$. for all $\left(t,\omega,y,z,u\right)$ $\in\left[0,T\right]\times\Omega\times\mathbb{R}\times\mathbb{R}^{d}\times L^{2}\left(E,\mathcal{E},\lambda,\mathbb{R}\right),$

[TABLE]

**(H.7) **For fixed $\omega$ and $t$, $f(t,\omega,\cdot,\cdot,\cdot)$ is continuous.

The theree next Lemmas will be useful in the sequel.

we recall the following classical lemma. It can be proved by adapting the proof given in J. J. Alibert and K. Bahlali [1].

Lemma 4.1**.**

Let $f:\Omega\times[0,T]\times\mathbb{R}\times\mathbb{R}^{d}\times\mathcal{L}^{2}\left(E,\mathcal{E},\lambda,\mathbb{R}\right)\rightarrow\mathbb{R}$ be a mesurable function such that:

For a.s. every $\left(t,\omega\right)\in[0,T]\times\Omega,$ $f\left(t,\omega,y,z,u\right)$ is a continuous. 2. 2.

There exists a constant $C>0$ such that for every $\left(t,\omega,y,z,u\right)\in[0,T]\times\Omega\times\mathbb{R}\times\mathbb{R}^{d}\times L^{2}\left(E,\mathcal{E},\lambda,\mathbb{R}\right),$ $\left|f\left(t,\omega,y,z,u\right)\right|\leq C\left(1+\left|y\right|+\left|z\right|+\left|u\right|\right).$

Then exists the sequence of fonction $f_{n}$

[TABLE]

is well defined for each $n\geq C$, and it satisfies, $d\mathbb{P}\times dt-a.s.$

$\left(i\right)$* Linear growth: $\forall n\geq 1,$ $\left(y,z,u\right)\in\mathbb{R}\times\mathbb{R}^{d}\times\mathcal{L}^{2}$, $\left|f_{n}\left(t,\omega,y,z,u\right)\right|\leq C\left(1+\left|y\right|+\left|z\right|+\left|u\right|\right).$*

$\left(ii\right)$* Monotonicity in $n:\forall$ $y,z,u,$ $f_{n}\left(t,\omega,y,z,u\right)$ is increases in $n.$*

$\left(iii\right)$* Convergence: $\forall\left(t,\omega,y,z,u\right)\in\left[0,T\right]\times\Omega\times\mathcal{B}^{2}\left(\mathbb{R}\right)$, if $\left(t,\omega,y_{n},z_{n},u_{n}\right)\rightarrow\left(t,\omega,y,z,u\right),$ then $f_{n}\left(t,\omega,y_{n},z_{n},u_{n}\right)\rightarrow f\left(t,\omega,y,z,u\right).$*

$\left(iv\right)$* Lipschitz condition: $\forall n\geq 1,$ $\left(t,\omega\right)\in[0,T]\times\Omega,$ $\forall\left(y,z,u\right)\in\mathcal{B}^{2}\left(\mathbb{R}\right)$ and $\left(y^{{}^{\prime}},z^{\prime},u^{{}^{\prime}}\right)\in\mathcal{B}^{2}\left(\mathbb{R}\right)$, we have*

[TABLE]

Now given $\xi\in$ $\mathbb{L}^{2}$, $n\in N$, we consider $\left(Y^{n},Z^{n},K^{n},U^{n}\right)$ and $\left(\text{resp }\left(V,N,K,M\right)\right)$ be solutions of the following reflected BDSDEPs:

[TABLE]

respectively

[TABLE]

where $H(s,\omega,V,N,M)=C\left(1+\left|V\right|+\left|N\right|+\left|M\right|\right).$

Lemma 4.2**.**

$\left(i\right)$* $a.s.$ for all, $t$ and $\forall n\leq m,$ $Y_{t}^{n}\leq Y_{t}^{m}\leq V_{t}.$*

$\left(ii\right)$* Assume that $\left(H.1\right),$ $\left(H.3\right)-\left(H.7\right)$ is in force. Then there exists a constant $A>0$ depending only on $C,$ $\alpha,$ $\xi$ and $T$ such that:*

[TABLE]

Proof: The prove of the $\left(i\right)$ follow from comparison theorem. It remains to prove $\left(ii\right)$, by lemma 2.1, we have

[TABLE]

By $\left(i\right)$ in lemma 4.1, we have

[TABLE]

also by the hypothesis associated with $g$, we get

[TABLE]

Chossing $\gamma_{1}=\gamma_{2}=\frac{\epsilon^{2}}{2}$. Then, we obtain the following inequality

[TABLE]

Consequently, we have

[TABLE]

where

$\Lambda=\mathbb{E}\left|\xi\right|^{2}+TC^{2}+\frac{1+\epsilon}{\epsilon}\mathbb{E}\int_{t}^{T}\left|\left|g(s,0,0,0)\right|\right|^{2}ds+\frac{1}{\theta}\mathbb{E}\left(\sup_{0\leq s\leq T}\left(S_{s}\right)^{2}\right)+T\left(1+2C+\frac{4C^{2}}{\epsilon^{2}}+\left(1+\epsilon\right)C\right)\mathbb{E}\left(\sup_{t}\left|Y_{t}^{n}\right|^{2}\right).$ Now chossing $\epsilon$ and $\alpha$ such that $0\leq\frac{\epsilon^{2}}{2}+\left(1+\epsilon\right)\alpha<1$, we obtain

[TABLE]

On the other hand, we have from Eq.$\left(4.1\right)$

[TABLE]

Using the Hölder’s inequality and assupmtion $\left(H.6\right)$, we have

[TABLE]

From inequality $\left(4.2\right)$, we get

[TABLE]

Finally chossing $\theta$ such that $0\leq\theta C_{2}\leq 1$, we obtain

[TABLE]

The prove of lemma 4.2 is complet.

Lemma 4.3**.**

Assume that $\left(H.1\right),$ $\left(H.3\right)-\left(H.7\right)$ is in force. Then the sequence $\left(Z^{n},U^{n}\right)$ converges a.s. in $\mathcal{M}^{2}\left(0,T,\mathbb{R}^{d}\right)\times\mathcal{L}^{2}\left(0,T,\tilde{\mu},\mathbb{R}\right).$

**Proof: **Let $n_{0}\geq C$. From Eq.$\left(4.1\right)$, we deduce that there exists a process $Y\in\mathcal{S}^{2}\left(0,T,\mathbb{R}\right)$ such that $Y^{n}\rightarrow Y$ a.s., as $n\rightarrow\infty$. Applying Lemma 2.1 to $\left|Y_{t}^{n}-Y_{t}^{m}\right|^{2}$, for $n,m\geq n_{0}$

[TABLE]

Since $\int_{t}^{T}\left(Y_{s}^{n}-Y_{s}^{m}\right)\left(dK_{s}^{n}-dK_{s}^{m}\right)\leq 0$, we deduce that

[TABLE]

Using Hölder’s inequality and assumption $\left(H.6\right)$ for $g$, we deduce that

[TABLE]

Applying assumption $\left(H.6\right)$ for $f$ and the boundedness of the sequence $\left(Y^{n},Z^{n},U^{n}\right)$, we deduce that

[TABLE]

where the constant $C^{te}>0$ depend only $\xi,\ C,$ $\alpha$ and $T.$

Which yields that $\left(Z^{n}\right)_{n\geq 0}$ respectively $\left(U^{n}\right)_{n\geq 0}$ is a Cauchy sequence in $\mathcal{M}^{2}\left(0,T,\mathbb{R}^{d}\right),$ respectively in $\mathcal{L}^{2}\left(0,T,\tilde{\mu},\mathbb{R}\right).$ Then there exists $\left(Z,U\right)\in\mathcal{M}^{2}\left(0,T,\mathbb{R}^{d}\right)\times\mathcal{L}^{2}\left(0,T,\tilde{\mu},\mathbb{R}\right)$ such that

[TABLE]

Theorem 4.1**.**

Assume that $\left(H.1\right),$ $\left(H.3\right)-\left(H.7\right)$ holds. Then Eq $\left(1.1\right)$ admits a solution $\left(Y,Z,U,K\right)\in\mathcal{D}^{2}\left(\mathbb{R}\right)$. Moreover there is a minimal solution $\left(Y^{\ast},Z^{\ast},U^{\ast}\right)$ of RBDSDEP $\left(1.1\right)$ in the sense that for any other solution $\left(Y,Z,U\right)$ of Eq. $\left(1.1\right)$, we have $Y^{\ast}\leq Y$.

Proof :

From Eq.$\left(4.1\right)$, it’s readily seen that $\left(Y^{n}\right)$ converges in $\mathcal{S}^{2}\left(0,T,\mathbb{R}\right)$, $dt\otimes d\mathbb{P}-a.s.$ to $Y\in\mathcal{S}^{2}\left(0,T,\mathbb{R}\right)$. Otherwise thanks to Lemma 4.3 there exists two subsequences still noted as the whole sequence $\left(Z^{n}\right)_{n\geq 0}$ respectively $\left(U^{n}\right)_{n\geq 0}$ such that

[TABLE]

Applying Lemma 4.1, we have $f_{n}\left(t,Y^{n},Z^{n},U^{n}\right)\rightarrow f\left(t,Y,Z,U\right)$ and the linear growth of $f_{n}$ implies

[TABLE]

Thus by Lebesgue’s dominated convergence theorem, we deduce that for almost all $\omega$ and uniformly in $t,$ we have

[TABLE]

We have by $\left(H.6\right)$ the following estimation

[TABLE]

using Burkholder-Davis-Gundy inequality, we have

[TABLE]

Let the following reflected BDSDEPs with data $\left(\xi,f,g,S\right)$

[TABLE]

By Itô’s formula, we derive that

[TABLE]

Using the fact that $\mathbb{E}\int_{t}^{T}\left(Y_{s}^{n}-\hat{Y}_{s}\right)\left(dK_{s}^{n}-dK_{s}\right)\leq 0,$ we get

[TABLE]

letting $n\rightarrow\infty$, we have $Y_{t}=\hat{Y}_{t},$ $U_{t}=\hat{U}_{t}$ and $Z_{t}=\hat{Z}_{t}$ $d\mathbb{P}\times dt-a.e$.

Let $\left(Y^{\ast},Z^{\ast},U^{\ast},K^{\ast}\right)$ be a solution of $\left(1.1\right)$. Then by Theorem 3.1, we have for any $n\in\mathbb{N}^{\ast}$, $Y^{n}\leq Y^{\ast}.$ Therefore, $Y$ is a minimal solution of $\left(1.1\right).$

5 RBDSDEPs with discontinuous coefficient.

In this section we are interested in weakening the conditions on $f$. We assume that $f$ satisfy the following assumptions:

**(H.8) **There exists a nonnegative process $f_{t}\in\mathcal{M}^{2}\left(0,T,\mathbb{R}\right)$ and constant $C>0$, such that

[TABLE]

(H.9) $f\left(t,\cdot,z,u\right):\mathbb{R}\rightarrow\mathbb{R}$ is a left continuous and ** $f\left(t,y,\cdot,\cdot\right)$** is a continuous.

(H.10) There exists a continuous fonction $\pi:\left[0,T\right]\times\mathcal{B}^{2}\left(\mathbb{R}\right)$ satisfying for $y\geq y^{{}^{\prime}}$, $\left(z,u\right)\in\mathbb{R}^{d}\times L^{2}\left(E,\mathcal{E},\lambda,\mathbb{R}\right)$

[TABLE]

(H.11) $g$ satisfies $\left(H.2,(ii)\right).$

5.1 Existence result.

The two next Lemmas will be useful in the sequel.

Lemma 5.1**.**

Assume that $\pi$ satisfies $\left(H.10\right),$ $g$ satisfies $\left(H.11\right)$ and $h$ belongs in $\mathcal{M}^{2}\left(0,T,\mathbb{R}\right)$. For a continuous processes of finite variation $A$ belong in $\mathcal{A}^{2}$ we consider the processes $\left(\bar{Y},\bar{Z},\bar{U}\right)\in\mathcal{S}^{2}\left(0,T,\mathbb{R}\right)\times\mathcal{M}^{2}\left(0,T,\mathbb{R}^{d}\right)\times\mathcal{L}^{2}\left(0,T,\tilde{\mu},\mathbb{R}\right)$ such that:

[TABLE]

Then we have,

$\left(1\right)$* The RBDSDEPs $\left(5.1\right)$ admits a minimal solution $\left(\overline{Y}_{t},\overline{Z}_{t},A_{t},\overline{U}_{t}\right)\in\mathcal{D}^{2}\left(\mathbb{R}\right).$*

$\left(2\right)$* if $h\left(t\right)\geq 0$ and $\xi\geq 0,$ we have $\bar{Y}_{t}\geq 0,$ $d\mathbb{P}\times dt-a.s.$*

Proof: $(1)$ Obtained from a previous part.

$(2)$ Applying lemma 2.1 to $\left|\bar{Y}_{t}^{-}\right|^{2}$, we have

[TABLE]

Since $h\left(t\right)\geq 0$, $\xi\geq 0$ and using the fact that $\int_{0}^{T}\bar{Y}_{t}^{-}dA_{s}\geq 0$, we obtain

[TABLE]

According to assumptions $\left(H.11\right)$, we get

[TABLE]

applying assumption $\left(H.10\right)$ and using Young’s inequality, we have

[TABLE]

Then

[TABLE]

Therefore, choosing $\epsilon$, $\alpha$ and $C$ such that $0<\alpha+2\epsilon C^{2}<1$ and using Gronwall’s inequality, we have

[TABLE]

$\mathbf{P}-a.s.$ for all $t\in\left[0,T\right].$ Finally implies that $\bar{Y}_{t}\geq 0,$ $\mathbf{P}-a.s.$ for all $t\in\left[0,T\right].$

Now by theorem (4.1), we consider the processes $\left(\tilde{Y}_{t}^{0},\tilde{Z}_{t}^{0},\tilde{K}_{t}^{0},\tilde{U}_{t}^{0}\right),$ $\left(Y_{t}^{0},Z_{t}^{0},K_{t}^{0},U_{t}^{0}\right)$ and sequence of processes $\left(\tilde{Y}_{t}^{n},\tilde{Z}_{t}^{n},\tilde{K}_{t}^{n},\tilde{U}_{t}^{n}\right)_{n\geq 0}$ respectively minimal solution of the following RBDSDEPs for all $t\in\left[0,T\right]$

[TABLE]

[TABLE]

and

[TABLE]

Lemma 5.2**.**

Under the assumptions $(H.1),\ (H.3),\ (H.5)$ and $\left(H.8\right)-\left(H.11\right)$, we have for any $n\geq 1$ and $t\in\left[0,T\right]$

[TABLE]

**Proof: **For any $n\geq 0$, we set

[TABLE]

Using Eq.$\left(5.4\right)$, we have

[TABLE]

where

[TABLE]

According to it definition, on can show that $\theta_{s}^{0}$ and $\Delta g^{n}$, $\forall n\geq 0$ satisfy all assumption of lemma 5.1. Moreover, since $\tilde{K}_{t}^{n}$ is a continuous and increasing process, for all $n\geq 0$, $\delta\tilde{K}_{s}^{n+1}$ is a continuous process of finite variation. Using the same argument as in first part, On can show that

[TABLE]

Applying lemma 5.1 we deduce that $\delta\tilde{Y}_{t}^{n+1}\geq 0,$ i.e. $\tilde{Y}_{t}^{n+1}\geq\tilde{Y}_{t}^{n}$ , for all $t\in\left[0,T\right],$ we have $\tilde{Y}_{t}^{n+1}\geq\tilde{Y}_{t}^{n}\geq\tilde{Y}_{t}^{0}.$

Now we show prove that $\tilde{Y}_{t}^{n+1}\leq Y_{t}^{0},$ by definition, we obtain

[TABLE]

where

$\Lambda_{s}^{n+1}=C\left(\left|Y_{s}^{0}-\tilde{Y}_{s}^{n+1}\right|+\left|Z_{s}^{0}-\tilde{Z}_{s}^{n+1}\right|+\left|U_{s}^{0}-\tilde{U}_{s}^{n+1}\right|+\left|Y_{s}^{0}\right|+\left|Z_{s}^{0}\right|+\left|U_{s}^{0}\right|\right)+f_{s}-f(s,\tilde{Y}_{s}^{n},\tilde{Z}_{s}^{n},\tilde{U}_{s}^{n})-\pi\left(s,\delta\tilde{Y}_{s}^{n+1},\delta\tilde{Z}_{s}^{n+1},\delta\tilde{U}_{s}^{n+1}\right).$

Also using lemma 5.1 we deduce that $Y_{t}^{0}-\tilde{Y}_{t}^{n+1}\geq 0$, i.e. $Y_{t}^{0}\geq\tilde{Y}_{t}^{n+1},$ for all $t\in\left[0,T\right]$. Thus, we have for all $n\geq 0$

[TABLE]

Now, our main result

Theorem 5.1**.**

Under assumption $\left(H.1\right),\ \left(H.3\right),\ \left(H.5\right)$ and $\left(H.8\right)-\left(H.11\right)$, the RBDSDEPs $\left(1.1\right)$ has a minimal solution $(Y_{t},Z_{t},K_{t},U_{t})_{0\leq t\leq T}\in\mathcal{D}^{2}\left(\mathbb{R}\right).$

**Proof: **Since $\left|\tilde{Y}_{t}^{n}\right|\leq\max\left(\tilde{Y}_{t}^{0},Y_{t}^{0}\right)\leq\left|\tilde{Y}_{t}^{0}\right|+\left|Y_{t}^{0}\right|$ for all $t\in\left[0,T\right]$, we have

[TABLE]

Therefore, we deduce from the Lebesgue’s dominated convergence theorem that $\left(\tilde{Y}_{t}^{n}\right)_{n\geq 0}$ converges in $\mathcal{S}^{2}\left(0,T,\mathbb{R}\right)$ to a limit $Y.$

On the other hand from $(5.4)$, we deduce that

[TABLE]

applying lemma 2.1, we obtain

[TABLE]

From $\left(H.8\right)$ and $\left(H.10\right)$, we get

[TABLE]

Also applying $\left(H.11\right)$, we obtain the following inequality

[TABLE]

Using Young’s inequality, we get

[TABLE]

Therefore, there exists a constant $C$ independent of $n$ such that for any $\epsilon_{i},$ where $i=1:4$, we derive

[TABLE]

Moreover, since

[TABLE]

Using Hölder’s inequality and assumption $\left(H.8\right),$ $\left(H.10\right)$, there exists two constants $C_{1}$ and $C_{2}$ depend of $\xi,C,\alpha,\epsilon_{i},i=1,...,4$, and we have that

[TABLE]

we come back to inequality $\left(5.5\right)$, we obtain

[TABLE]

we taking $\epsilon_{1}=\epsilon_{2}=\epsilon_{0}$ and $\epsilon_{3}=\epsilon_{4}=\bar{\epsilon},$ we have

[TABLE]

we chossing $\bar{\epsilon},$ $\theta$ and $\alpha$ such that $0\leq\left(\bar{\epsilon}+\theta C_{2}+\alpha\right)<1,$ we get

[TABLE]

Now chossing $\epsilon_{0},$ $\theta$ and $C_{2}$ such that $\epsilon_{0}+\theta C_{2}<1$ and notting $\mathbb{E}\int_{0}^{T}\left(\left|\tilde{Z}_{s}^{0}\right|^{2}+\int_{E}\left|\tilde{U}_{s}^{0}\right|^{2}\lambda\left(de\right)\right)ds<\infty.$ We obtain

[TABLE]

consequently, we deduce that

[TABLE]

Now we shall prove that $\left(\tilde{Z}^{n},\tilde{K}^{n},\tilde{U}^{n}\right)$ is a Cauchy sequence in $\mathcal{M}^{2}\left(0,T,\mathbb{R}^{d}\right)\times\mathcal{A}^{2}\times\mathcal{L}^{2}\left(0,T,\tilde{\mu},\mathbb{R}\right),$ set $\Gamma_{s}^{n}=f(s,\tilde{Y}_{s}^{n-1},\tilde{Z}_{s}^{n-1},\tilde{U}_{s}^{n-1})+\pi\left(s,\delta\tilde{Y}_{s}^{n},\delta\tilde{Z}_{s}^{n},\delta\tilde{U}_{s}^{n}\right),$ we have

[TABLE]

applying Lemma 2.1 to $\left|\delta\tilde{Y}_{s}^{n,m}\right|^{2}=\left|\tilde{Y}_{s}^{n}-\tilde{Y}_{s}^{m}\right|^{2}$, we have

[TABLE]

since $\int_{t}^{T}\left(\tilde{Y}_{s}^{n+1}-\tilde{Y}_{s}^{n}\right)\left(d\tilde{K}_{s}^{n}-d\tilde{K}_{s}^{m}\right)\leq 0,$ we obtain

[TABLE]

Applying Hölder’s inequality and assumption $\left(H.11\right)$, we deduce that

[TABLE]

The boundedness of the sequence $\left(\tilde{Y}^{n},\tilde{Z}^{n},\tilde{K}^{n},\tilde{U}^{n}\right)$, we deduce that $\Lambda=\sup_{n\in\mathbb{N}}\left(\mathbb{E}\int_{0}^{T}\left|\Gamma_{s}^{n}\right|^{2}ds\right)<\infty,$ this yields that

[TABLE]

Which yields that $\left(\tilde{Z}^{n}\right)_{n\geq 0}$ respectively $\left(\tilde{U}^{n}\right)_{n\geq 0}$ is a Cauchy sequence in $\mathcal{M}^{2}\left(0,T,\mathbb{R}^{d}\right)$ respectively in $\mathcal{L}^{2}\left(0,T,\tilde{\mu},\mathbb{R}\right).$ Then there exists $\left(Z,U\right)\in\mathcal{M}^{2}\left(0,T,\mathbb{R}^{d}\right)\times\mathcal{L}^{2}\left(0,T,\tilde{\mu},\mathbb{R}\right)$ such that,

[TABLE]

On the other hand, applying Burkholder-Davis-Gundy inequality and $\left(5.6\right)$, we obtain

[TABLE]

Therefore, from the properieties of $\left(f,\pi\right)$, we have

[TABLE]

$P-a.s.,$ for all $t\in\left[0,T\right]$ as $n\rightarrow\infty$. Then follows by dominated convergence theorem that

[TABLE]

Since $\left(\tilde{Y}_{s}^{n},\tilde{Z}_{s}^{n},\tilde{U}_{s}^{n},\Gamma_{s}^{n}\right)$ converges in $\mathcal{B}^{2}\left(\mathbb{R}\right)\times\mathcal{M}^{2}\left(0,T,\mathbb{R}\right)$ and

[TABLE]

for any $n,m\geq 0$, we deduce that

[TABLE]

as $n,m\rightarrow\infty.$ Consequently, there exists a $\mathcal{F}_{t}-$measurable process $K$ wich value in $\mathbb{R}$ such that

[TABLE]

Finally, we have

[TABLE]

Obviously, $K_{0}=0$ and $\{K_{t};0\leq t\leq T\}$ is a increasing and continuous process. From $(5.4)$, we have for all $n\geq 0$, $\tilde{Y}_{t}^{n}\geq S_{t}$, $\forall t\in[0,T]$, then $Y_{t}\geq S_{t}$, $\forall t\in[0,T]$.

On the other hand, from the result of Saisho, we have

[TABLE]

Using the identity $\int_{0}^{T}\left(\tilde{Y}_{s}^{n}-S_{s}\right)d\tilde{K}_{s}^{n}=0$ for all $n\geq 0,$ we obtain $\int_{0}^{T}\left(Y_{s}-S_{s}\right)dK_{s}=0.$ letting $n\rightarrow+\infty$ in equation $\left(1.1\right)$, we prove that $\left(Y_{t},Z_{t},K_{t},U_{t}\right)_{t\in[0,T]}$ is the solution to $\left(1.1\right).$

Let $\left(Y_{\ast},Z_{\ast},U_{\ast},K_{\ast}\right)$ be a solution of $\left(1.1\right)$. Then by Theorem 3.1, we have for any $n\in\mathbb{N}^{\ast}$, $Y^{n}\leq Y_{\ast}.$ Therefore, $Y$ is a minimal solution of $\left(1.1\right).$

Remark 5.1**.**

Using the same arguments and the following approximating sequence

[TABLE]

one can prove that the RBDSDE (1.1) has a maximal solution.