Reflected backward doubly stochastic differential equations with time delayed generators

TL;DR

This paper studies a class of reflected backward doubly stochastic differential equations with generators depending on past solution values, establishing existence and uniqueness under Lipschitz conditions.

Contribution

It introduces and analyzes RBDSDEs with time-delayed generators, extending the theory to include dependence on past solutions.

Findings

Existence and uniqueness of solutions under Lipschitz conditions.

Extension of RBDSDE theory to time-delayed generators.

Framework for future applications in stochastic control and finance.

Abstract

We consider a class of reflected backward doubly stochastic differential equations with time delayed generator (in short RBDSDE with time delayed generator), in this case generator at time $t$ can depend on the values of a solution in the past. Under a Lipschitz condition, we ensure the existence and uniqueness of the solution.

Equations

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

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

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

$${\mathcal{F}}_t:={\mathcal{F}}_t^W\vee {\mathcal{F}}_{t,T}^B,and{\mathcal{G}}_t:={\mathcal{F}}_t^W\vee {\mathcal{F}}_T^B$$

$$\Vert \phi {\Vert }_{M_T^2}^2=E{\int }_0^T{e}^{\beta t}|\phi (t){|}^{2}dt\text{and}\Vert \phi {\Vert }_{S_T^2}^2=E(su{p}_{t\in [0,T]}{e}^{\beta t}|\phi (t){|}^2).$$

$${\left|f(t,y_t^1,z_t^1)-f(t,y_t^2,z_t^2)\right|}^2$$

$$\le L({\int }_{-T}^0{\left|y^1(t+u)-y^2(t+u)\right|}^2\gamma (du)+{\int }_{-T}^0{\left|z^1(t+u)-z^2(t+u)\right|}^2\gamma (du))$$

$${\left|g(t,y_1,z_1)-g(t,y_2,z_2)\right|}^2\le$$

$$L{\int }_{-T}^0{\left|y^1(t+u)-y^2(t+u)\right|}^2\gamma (du)+\alpha {\int }_{-T}^0{\left|z^1(t+u)-z^2(t+u)\right|}^2\gamma (du)$$

$${\int }_0^T|f(t,Y_t,Z_t){|}^{2}dt$$

$$+2L{\int }_0^T{\int }_{-T}^0|Z(t+u){|}^2\gamma (du)dt$$

$$=2{\int }_0^T|f(t,0,0){|}^{2}dt+2L{\int }_{-T}^0{\int }_u^{T+u}|Y(v){|}^{2}dv\gamma (du)$$

$$+2L{\int }_{-T}^0{\int }_u^{T+u}|Z(v){|}^{2}dv\gamma (du)dt$$

$$\le 2{\int }_0^T|f(t,0,0){|}^{2}dt+2L(T\underset{0\le t\le T}{\sup }|Y(t){|}^2+{\int }_0^T|Z(t){|}^{2}dt)$$

$$<\infty .$$

$${\int }_0^T|g(t,Y_t,Z_t){|}^{2}dt<\infty .$$

$$D_1:=\beta -\theta >0,D_2:=(1-2\tilde{\gamma }(\frac{L}{\theta }+\alpha ))>0,$$

$$E(\underset{0\le t\le T}{\sup }{e}^{\beta t}|Y_t{|}^2+{\int }_0^T{e}^{\beta t}|Z_t{|}^{2}dt+e^{\beta T}|K_T{|}^2)$$

$$\displaystyle\leq CE\bigg{(}e^{\beta T}|\xi|^{2}+\int_{0}^{T}e^{\beta t}|f(t,0,0)|^{2}dt+\int_{0}^{T}e^{\beta t}|g(t,0,0)|^{2}dt+\sup_{0\leq t\leq T}e^{\beta t}(S_{t}^{+})^{2}\bigg{)}.$$

$$e^{\beta t}|Y_t{|}^2+{\int }_t^T\beta e^{\beta s}|Y_s{|}^{2}ds$$

$$\le e^{\beta T}|\xi {|}^2+\theta {\int }_t^T{e}^{\beta s}|Y_s{|}^{2}ds+\frac{1}{\theta }{\int }_t^T{e}^{\beta s}|f(s,Y_s,Z_s){|}^{2}ds+2{\int }_t^T{e}^{\beta s}{Y}_{s}d{K}_s$$

$$+2{\int }_t^T{e}^{\beta s}{Y}_{s}g(s,Y_s,Z_s)d{B}_s-2{\int }_t^T{e}^{\beta s}{Y}_s{Z}_{s}d{W}_s$$

$$+{\int }_t^T{e}^{\beta s}|g(s,Y_s,Z_s){|}^{2}ds-{\int }_t^T{e}^{\beta s}|Z_s{|}^{2}ds$$

$$e^{\beta t}|Y_t{|}^2+(\beta -\theta ){\int }_t^T{e}^{\beta s}|Y_s{|}^{2}ds+{\int }_t^T{e}^{\beta s}|Z_s{|}^{2}ds$$

$$\le e^{\beta T}|\xi {|}^2+(\frac{2LT\tilde{\alpha }+2LT\tilde{\alpha }}{\theta })su{p}_{0\le t\le T}{e}^{\beta t}|Y_t{|}^2$$

$$+2{\int }_t^T|g(s,0,0){|}^{2}ds$$

$$+\frac{2}{\theta }{\int }_t^T|f(s,0,0){|}^{2}ds+2{\int }_t^T{e}^{\beta s}{Y}_{s}d{K}_s$$

$$+2{\int }_t^T{e}^{\beta s}{Y}_{s}g(s,Y_s,Z_s)d{B}_s-2{\int }_t^T{e}^{\beta s}{Y}_s{Z}_{s}d{W}_s$$

$${\int }_t^T({\int }_{-T}^0{e}^{\beta s}|Z_{s+u}{|}^2\gamma (du))ds$$

$$={\int }_{-T}^0{\int }_{(t+u)\vee 0}^{T+u}{e}^{\beta r}{e}^{-\beta u}{1}_{r\ge 0}|Z_r{|}^{2}dr\gamma (du)$$

$$={\int }_0^T{\int }_{r-t}^{(r-t)\wedge 0}{e}^{\beta r}{e}^{-\beta u}{1}_{r\ge 0}|Z_r{|}^2\gamma (du)dr$$

$$={\int }_0^T{e}^{\beta r}|Z_r{|}^2({\int }_{-T}^0{e}^{-\beta u}\gamma (du))dr$$

$$\le {\int }_0^T\tilde{\gamma }{e}^{\beta r}|Z_r{|}^{2}dr.$$

$$e^{\beta t}|Y_t{|}^2+(\beta -\theta ){\int }_t^T{e}^{\beta s}|Y_s{|}^{2}ds+{\int }_t^T{e}^{\beta s}|Z_s{|}^{2}ds$$

$$\le e^{\beta T}|\xi {|}^2+\tilde{\gamma }(\frac{2L}{\theta }+2L)T\underset{0\le t\le T}{\sup }{e}^{\beta t}|Y_t{|}^2+\tilde{\gamma }(\frac{2L}{\theta }+2\alpha ){\int }_0^T{e}^{\beta s}|Z_s{|}^{2}ds$$

$$+\frac{2}{\theta }{\int }_t^T{e}^{\beta s}|f(s,0,0){|}^{2}ds+2{\int }_t^T{e}^{\beta s}{Y}_{s}d{K}_s$$

$$+2{\int }_t^T{e}^{\beta s}{Y}_{s}g(s,Y_s,Z_s)d{B}_s-2{\int }_t^T{e}^{\beta s}{Y}_s{Z}_{s}d{W}_s$$

$$+2{\int }_t^T{e}^{\beta s}|g(s,0,0){|}^{2}ds.$$

$$E{\int }_0^T{e}^{\beta s}|Z_s{|}^{2}ds\le CE{\int }_t^T(e^{\beta s}|f(s,0,0){|}^{2}ds+e^{\beta s}|g(s,0,0){|}^{2}ds+\underset{0\le t\le T}{\sup }{e}^{\beta t}|Y_t{|}^2)$$

$$|Y_0{|}^2+(1-\tilde{\gamma }(\frac{2L}{\theta }+2\alpha )){\int }_0^T{e}^{\beta s}|Z_s{|}^{2}ds$$

$$\le e^{\beta T}|\xi {|}^2+(\tilde{\gamma }(\frac{2L}{\theta }+2L)T+\frac{1}{ϵ})\underset{0\le t\le T}{\sup }{e}^{\beta t}|Y_t{|}^2$$

$$+\frac{2}{\theta }{\int }_0^T{e}^{\beta s}|f(s,0,0){|}^{2}ds+ϵe^{\beta T}|K_T{|}^2$$

$$+2{\int }_t^T{e}^{\beta s}{Y}_{s}g(s,Y_s,Z_s)d{B}_s-2{\int }_t^T{e}^{\beta s}{Y}_s{Z}_{s}d{W}_s$$

$$+2{\int }_0^T{e}^{\beta s}|g(s,0,0){|}^{2}ds$$

Full text

footnotetext: E-mail addresses: 1mansouri.badreddine@gmail.com, 2imensalhi@hotmail.fr, 3tamerlazhar@yahoo.fr

1 Introduction

After the earlier work of Pardoux & Peng (1990)[8], the theory of backward stochastic differential equations (BSDEs in short) has a significant headway thanks to the many application areas. Several authors contributed in weakening the Lipschitz assumption required on the drift of the equation (see Lepaltier & San martin (1996)[5], Kobylanski (1997)[4], Mao (1995)[7], Bahlali (2000)[1]).

A new kind of backward stochastic differential equations was introduced by Pardoux & Peng [10] (1994),

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with two different directions of stochastic integrals, i.e., the equation involves both a standard (forward) stochastic integral $dW_{t}$ and a backward stochastic integral $dB_{t}.$ They have proved the existence and uniqueness of solutions for backward doubly stochastic differential equations under uniformly Lipschitz conditions. Shi et al [11](2005) provided a comparison theorem which is very important in studying viscosity solution of SPDEs with stochastic tools. Bahlali et al [3] provided the existence and uniqueness in the case with a superlinear growth generator and a square integrable terminal datum.

Bahlali et al [2] (2009) proved the existence and uniqueness of the solution to the following reflected backward doubly stochastic differential equations (RBDSDEs in short) with one continuous barrier and uniformly Lipschitz coefficients:

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Recently, LuO, Zhang and Li [6] introduced the following BDSDE

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where the generator $f$ at time $s$ depends arbitrary on the past values of a solution $(Y_{s},Z_{s})=(Y(s+u),Z(s+u)),T\leq u\leq 0$. This class of BDSDEs is called as BDSDEs with time delayed generators. In Luo et al[6], the authors proved the existence and uniqueness of a solution for the above equation.

In this paper, we establish the existence and uniqueness of the solutions for RBDSDEs with time delayed generators. The paper is organized as follows. In Section 2, we propose some preliminaries and notations. The section 3 we gives a priori estimates of the solution. Finally the section 4 our main result is stated and proved.

2 Notations, assumptions and Definitions

Let $\left(\Omega,\mathcal{F},P\right)$ be a complete probability space, and $T>0$. Let $\left\{W_{t},0\leq t\leq T\right\}$ and $\left\{B_{t},0\leq t\leq T\right\}$ be two independent standard Brownian motions defined on $\left(\Omega,\mathcal{F},P\right)$ with values in $\mathbb{R}$ and $\mathbb{R}$, respectively. For $t\in\left[0,T\right]$, we put,

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where $\mathcal{F}_{t}^{W}=\sigma(W_{s};0\leq s\leq t)$ and $\mathcal{F}_{t,T}^{B}=\sigma(B_{s}-B_{t};t\leq s\leq T),$ completed with $P$-null sets. 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. However $(\mathcal{G}_{t})$ is a filtration.

Let $M_{T}^{2}\left(0,T,\mathbb{R}\right)$ denote the set of jointly measurable stochastic processes $\left\{\varphi_{t};t\in\left[0,T\right]\right\}$, which satisfy :

(a)  $E\int_{0}^{T}\left|\varphi_{t}\right|^{2}dt<\infty.$

**(b) ** $\varphi_{t}$ is $\mathcal{F}_{t}-$measurable, for any $t\in\left[0,T\right].$

We denote by $S_{T}^{2}\left(\left[0,T\right],\mathbb{R}\right),$ the set of continuous stochastic processes $\varphi_{t}$, which satisfy :

(a’) $E\left(\sup_{0\leq t\leq T}\left|\varphi_{t}\right|^{2}\right)<\infty.$

(b’) For every $t\in\left[0,T\right],\ \varphi_{t}$ is $\mathcal{F}_{t}-$measurable.

We denote by $L_{-T}^{2}(\mathbb{R})$ the set of measurable function $\varphi:[-T,0]\rightarrow\mathbb{R}$, which satisfy : $\int_{-T}^{0}|\varphi(t)|^{2}dt<\infty.$

We denote by $S_{-T}^{\infty}(\mathbb{R})$ the set of measurable function $\varphi(t):[-T,0]\rightarrow\mathbb{R}$, which satisfy : $sup_{t\in[-T,0]}|\varphi(t)|^{2}dt<\infty.$

We denote by $L_{T}^{2}(\mathbb{R})$ the set of $\mathfrak{F}_{T}$-measurable random $\eta$ $\mathbb{R}$-valued, which satisfy : $E|\eta|^{2}<\infty$.

The spaces $M_{T}^{2}(\mathbb{R})$ and $S_{T}^{2}(\mathbb{R})$ are respectively endowed with the norms

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where $\beta>0$.

We denote by $\lambda$ the Lebesgue measure on $([-T,0],\mathbb{B}([-T,0]))$, where $\mathbb{B}([-T,0])$ is the Borel sets of $[-T,0]$.

We consider the following assumptions,

H1) Let $\ f:\Omega\times\left[0,T\right]\times S_{-T}^{\infty}(\mathbb{R})\times L_{-T}^{2}(\mathbb{R})\mathbb{\rightarrow R}$ and $g:\Omega\times\left[0,T\right]\times S_{-T}^{\infty}(\mathbb{R})\times L_{-T}^{2}\mathbb{\rightarrow R}$ be two measurable functions and such that for every $\left(y,z\right)\in\mathbb{R}\times\mathbb{R}$, $f(.,y,z)$ and, $g(.,y,z)$ belongts $M^{2}(0,T,\mathbb{R})$

H2) There exist constants $L>0$ and $0<\alpha<1,$ such that  for every $\left(t,\omega\right)\in\Omega\times\left[0,T\right]$ and $\left(y,z\right)\in S_{-T}^{\infty}(\mathbb{R})\times L_{-T}^{2},$ and for a probability measure $\gamma$ on $([-T,0],\mathbb{B}([-T,0]))$

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H3) $E\int_{0}^{T}|f(t,0,0)|^{2}dt<\infty,$ and $E\int_{0}^{T}|g(t,0,0)|^{2}dt<\infty,$

H4) $f(t,.,.)=0$ and $g(t,.,.)=0$ for $t<0.$

H5) Let $\xi$ be a square integrable random variable which is $\mathcal{F}_{T}-$mesurable.

H6) The obstacle $\left\{S_{t},0\leq t\leq T\right\}$, is a continuous $\mathcal{F}_{t}-$progressively measurable real-valued process satisfying $E\left(\sup_{0\leq t\leq T}e^{\beta t}\left(S_{t}^{+}\right)^{2}\right)<\infty$, for $\beta>0$.

We assume also that $S_{T}\leq\xi\ a.s.$

We define now RBDSDE with time delayed generators.

Definition 1

*A solution of RBDSDE with time delayed generators is a $\left(\mathbb{R}\times\mathbb{R}\times\mathbb{R}_{+}\right)$ -valued $\mathcal{F}_{t}-$progressively measurable process $\left(Y_{t},Z_{t},K_{t}\right)_{0\leq t\leq T}$ which satisfies :

i) $(Y,Z,K_{T})\in S^{2}_{T}\times M^{2}_{T}\times L^{2}(\mathbb{R})$.

ii) $Y_{t}=\xi+\int_{t}^{T}f(s,Y_{s},Z_{s})ds+K_{T}-K_{t}+\int_{t}^{T}g(s,Y_{s},Z_{s})dB_{s}-\int_{t}^{T}Z_{s}dW_{s},\ \ \ 0\leq t\leq T.$

iii)$Y_{t}\geq S_{t},\ \ 0\leq t\leq T.$

iv) $K_{t}$ is adapted, continuous and nondecreasing, $K_{0}=0$ and $\int_{0}^{T}e^{\beta t}\left(Y_{t}-S_{t}\right)dK_{t}=0,$ for $\beta>0$.*

Note that for $(Y,Z)\in S^{2}(\mathbb{R})\times M^{2}(\mathbb{R})$ the generator $f,\ g$ are well-defined and $P-a.s$. integrable as

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with the same argument we get

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3 Priori estimates of the solution

Using standard arguments of RBDSDEs one can prove the following estimates

Proposition 2

Let $(Y_{t},Z_{t},K_{t},0\leq t\leq T)$ be a solution of the RBDSDE with time delayed generator. If the Lipschitz constant $L$ of the generator $f$ and $g$ is small enough, then there exist two positive constants $\beta$ and $\theta$ satisfying that

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and a positive constant $C=C(\beta,\theta,\widetilde{\gamma},L,T,\alpha)$, ($\gamma$ as in H2) and $\widetilde{\gamma}=\int_{-T}^{0}e^{-\beta u}\gamma(du)),$ such that

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Proof. Applying the Ito formula to $e^{\beta t}|Y_{t}|^{2}$ yields that

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by equation (2) and (3) we get

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By a change of integration order, we obtain

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return back to (4)

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Firstly, we say that

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where $C>0$ depends on $\beta,\theta,\widetilde{\gamma},\alpha,L$ and $T$. The estimate of (7) can be obtain as follows : Since $D_{1}=\beta-\theta>0$ in (3) putting $t=0$ we get

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where $\epsilon>0$, by (1)

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there exist a constant $\widetilde{C}$ depending on $L,\alpha$ and $T$ such that

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we replace the last inequality in previous

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choosing $\epsilon$ small and $\theta$ such that $D_{2}=(1-\widetilde{\gamma}(\frac{2L}{\theta}+2\alpha))>0$, we obtain

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where $C_{1}$ depends on $\beta,\widetilde{\gamma},\theta,\alpha,L$ and $T$. By the Burkholder Davis Gundy inequality, we have

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and

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where $\lambda>0$. Finally, we return back to the last inequality, choosing $\lambda$ small and using the Fatou lemma, we obtain that there exist a constant $C>0$ depending on $\beta,\theta,\alpha,\widetilde{\gamma},L$ and $T$ such that

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In the second part of proof, we claim that

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holds for a positive constant $\widehat{C}$ depending on $\beta,\theta,\alpha,\widetilde{\gamma},L$ and $T$. To prove (8), going back to (6), using the fact $\int_{0}^{T}e^{\beta t}(Y_{t}-S_{t})dK_{t}=0$, we get

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where $\delta>0$ is a constant. Using [2] and (7), we have

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a constant $K$ depending on $\beta,\theta,\alpha,\widetilde{\gamma},L$ and $T$. Now taking expectation in (9) and taking into account of (10), we obtain

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thus

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Next going back to (9) and (10), we obtain

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Using the Burkholder Davis Gundy inequality , there exist a constant $C_{4}>0$ such that

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and

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where $\rho>0$. Plugging this inequality in (12), we get

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Finally it is enough to choose $\rho=\frac{1}{2C_{4}}$ and $\delta$ small to obtain (8). From the above inequalities (7), (8) and (10), we obtain that there exists a positive constant $\widehat{C}$ depending on $\beta,\theta,\alpha,\widetilde{\gamma},L$ and $T$ such that

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Proposition 3

Let $(\xi,f,g,S)$ and $(\xi^{\prime},f^{\prime},g,S^{\prime})$ be two triplets satisfying the above assumptions (H1)-(H5). Suppose that $(Y,Z,K)$ is a solution of the RBDSDE with time delayed generator $(\xi,f,g,S)$ and $(Y^{\prime},Z^{\prime},K^{\prime})$ is a solution of the RBDSDE with time delayed generator $(\xi^{\prime},f^{\prime},g,S^{\prime})$. Define

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If the Lipschitz constant $L$ of the generator $f$ and $g$ is small enough, then there exist two positive constants $\beta$ and $\epsilon$ satisfying that

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and a positive constant $D$ depending on $\beta,\epsilon,\widetilde{\gamma},L,\alpha$ and $T$ such that

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where

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Proof. Applying the It formula to $e^{\beta t}|Y_{t}|^{2}$ yields that

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where $\epsilon>0$, now we use the relation $\int_{t}^{T}e^{\beta s}(\Delta Y_{s}-\Delta S_{s})d\Delta K_{s}\leq 0$ and (5) and taking expectation, we have

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with the Holder inequality

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and since E|\Delta K_{T}|^{2}\leq C\big{(}E|K_{T}|^{2}+E|K^{\prime}_{T}|^{2}\big{)}, using inequalities (10) and proposition 2, we deduce that

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where $C_{5}>0$ depends on $\beta,\epsilon,\widetilde{\gamma},\alpha,L$ and $T$. Therefore, we obtain

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where $C_{6}>0$ depends on $\beta,\epsilon,\widetilde{\gamma},\alpha,L$ and $T$. But

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where $\varepsilon>0$. With (14), the Burkholder Davis Gundy inequality and the two previous inequalities, we get

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where $c,\ \rho$ are constants.

By (15) and (16), we have

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Finally, it is enough to choose $\rho=\frac{1}{2c}$ and $\varepsilon$ small in the above inequalities (15) and (16) to obtain (13). The proposition is proved

We deduce immediately the following uniqueness result from proposition 3 with $\xi^{\prime}=\xi,\ f=f^{\prime}$ and $S=S^{\prime}$.

Corollary 4

Under assumptions $H1)-H5)$ , if the Lipschitz constant L of the generator f is small enough and for two positive constants and the conditions of Proposition 3 are satisfied, then there exists at most one solution of the RBSDE with time delayed generators $i)-iv).$

4 Existence and uniqueness of the solution

To begin with, let us first assume that $f$ does not depend on $(y,z)$, that is, it is a given $\mathfrak{F}_{t}$-progressively measurable process satisfying that

(H2’ ) $E\int^{T}_{0}|f(t)|^{2}dt<\infty$ and $E\int^{T}_{0}|g(t)|^{2}dt<\infty$.

A solution to the backward reflection problem is a triple $(Y,Z,K)$ which satisfies (i), (iii), (iv) and

(ii’) $Y(t)=\xi+\int^{T}_{t}f(s)ds+K(T)-K(t)+\int^{T}_{t}g(s)dB_{s}-\int^{T}_{t}Z(s)dW(s),0\leq t\leq T.$

The following proposition is from Bahlali et al. (2009) .

Proposition 5

Under assumptions (H1), (H2’ ) and (H5), the Backward Reflected Problem (i), (ii’), (iii), (iv) has a unique solution ${(Y_{t},Z_{t},K_{t});0\leq t\leq T}.$

We now deal with the general case of generator i.e. f depends on (y, z).

Theorem 6

Assume assumptions H1)-H5) hold. If the Lipschitz constant $L$ of the generator $f$ and $g$ is small enough and for two positive constants $\beta$ and $\epsilon$ the conditions of Proposition 3 are satisfied, then the RBDSDE with time delayed generator (i)-(iv) has a unique solution $(Y_{t},Z_{t},K_{t});0\leq t\leq T.$

Proof. For any $\left(Y_{t},Z_{t},K_{t}\right);0\leq t\leq T$ which satisfy (i) and (iii). Given $(U,V)\in S^{2}\times M^{2}~{}$, let

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the solution of this equation exists and is unique by proposition 5. Hence, if we define $\Theta(U,V)=(Y,Z)$, then $\Theta$ maps $S^{2}\times M^{2}~{}$ to itself. We show now that $\Theta$ is contractive. To this end, take any $\left(U^{i},V^{i}\right)\in S^{2}\times M^{2}$ and $\left(Y^{i},Z^{i}\right)\in S^{2}\times M^{2},\ \ (i=1,2)$, and let $\Theta(U^{i},V^{i})=(Y^{i},Z^{i}).$ We denote  $\left(\overline{Y},\overline{Z},\overline{K}\right)=\left(Y^{1}-Y^{2},Z^{1}-Z^{2},K^{1}-K^{2}\right)$ and $\left(\overline{U},\overline{V},\right)=\left(U^{1}-U^{2},V^{1}-V^{2}\right)$. Therefore, Itô’s formula applied to $\left|\overline{Y}\right|^{2}e^{\beta t}$ and the inequality $2ab\leq\left(\frac{1}{\epsilon}\right)a^{2}+\epsilon b^{2}$, lead to

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where $\epsilon>0$, and from $2\int_{t}^{T}e^{\beta s}\overline{Y}_{s}d\overline{K}_{s}\leq 0$, we obtain

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taking the expectation we have

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By choosing $\beta-\epsilon>0$, we get

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Taking supremum in (17), and the expectation, we get

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By the Burkholder Davis Gundy inequality, there exist real number $\widehat{C}$, such that

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Plugging now the inequality in (18) and (20), we obtain

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with good choice of $\epsilon$, we deduce that

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Consequently the mapping $\Theta$ is a strict contraction on $S^{2}\times M^{2}$ equipped with the norm

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Moreover, it has a unique fixed point, which is the unique solution of the RBDSDE with delayed time generator $\left(\xi,f,g,S\right).$