Reflected solutions of Anticipated Backward Doubly SDEs driven by Teugels Martingales

TL;DR

This paper investigates reflected solutions and existence results for anticipated backward doubly stochastic differential equations driven by Teugels martingales, with coefficients depending on current and future solution values.

Contribution

It introduces new existence results for anticipated BDSDEs driven by Teugels martingales with coefficients depending on future and present solutions.

Findings

Established existence of solutions for anticipated BDSDEs

Analyzed reflected solutions under Lipschitz conditions

Extended theory to equations driven by Teugels martingales

Abstract

We deal with reflected solutions of anticipated backward doubly stochastic differential equations (RABDSDEs) driven by Teugels martingales associated with L\'evy process under a Lipschitz generator where the coefficients of these BDSDEs depend on the future and present value of the solution $\left( Y,Z\right) $. Also we study the existence of a solution for anticipated BDSDEs.

Equations

$$\begin{array}[]{l}Y_{t}=\xi+\int_{t}^{T}f(s,\Lambda_{s},\Lambda_{s}^{\phi,\psi})ds+\int_{t}^{T}g(s,\Lambda_{s},\Lambda_{s}^{\phi,\psi})d\overleftarrow{B}_{s}-\int_{t}^{T}Z_{s}dW_{s},\text{ }t\in\left[0,T\right],\\ \\ \left(Y_{t},Z_{t}\right)=\left(\eta_{t},\vartheta_{t}\right),\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad t\in\left[T,T+K\right],\end{array}$$

$${\mathcal{G}}_t:={\mathcal{F}}_t^L\vee {\mathcal{F}}_{T+K}^B,$$

$${\left|\left|\left(Y,Z\right)\right|\right|}_{{\mathcal{B}}_{\mathcal{H}}^2\left(\left[0,T\right],\mathbb{R}\right)}={\left|\left|Y\right|\right|}_{{\mathcal{S}}_{\mathcal{H}}^2\left(\left[0,T\right];\mathbb{R}\right)}+{\left|\left|Z\right|\right|}_{{\mathcal{M}}_{\mathcal{H}}^2\left(\left[0,T\right];l^2\right)}.$$

$$⟨H^{\left(i\right)},H^{\left(j\right)}{⟩}_t=t{\delta }_{i,j},$$

$${\alpha }_t={\alpha }_0+{\int }_0^t{\beta }_{s}ds+{\int }_0^t{\gamma }_{s}d{B}_s+\sum _{i=1}^{\infty }{\int }_0^t{\sigma }_s^{\left(i\right)}d{H}_s^{\left(i\right)},$$

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

$$\mathbb{E}{\left|{\alpha }_t\right|}^2\le \mathbb{E}{\left|{\alpha }_0\right|}^2+2\mathbb{E}{\int }_0^t{\alpha }_s{\beta }_{s}ds-\mathbb{E}{\int }_0^t{\left|{\gamma }_s\right|}^{2}ds+\mathbb{E}\sum _{i=1}^{\infty }{\int }_0^t{\left({\sigma }_s^{\left(i\right)}\right)}^{2}ds.$$

$$\left\{\begin{array}[]{l}Y_{t}=\xi+\int_{t}^{T}f(s,\Lambda_{s},\Lambda_{s}^{\phi,\psi})ds+\int_{t}^{T}g(s,\Lambda_{s},\Lambda_{s}^{\phi,\psi})d\overleftarrow{B}_{s}\\ \\ +\int_{t}^{T}dK_{s}-\sum_{i=1}^{\infty}\int_{t}^{T}Z_{s}^{\left(i\right)}dH_{s}^{\left(i\right)},\qquad\qquad\qquad\quad t\in\left[0,T\right],\\ \\ \left(Y_{t},Z_{t}\right)=\left(\eta_{t},\vartheta_{t}\right),\qquad\qquad\qquad\qquad\qquad\qquad\text{\ \ }t\in\left[T,T+K\right],\end{array}\right.$$

$$t+\varphi \left(t\right)\le T+K,t+\psi \left(t\right)\le T+K.$$

$$\left\{\begin{array}[]{c}\int_{t}^{T}h(s+\phi\left(s\right))ds\leq M\int_{t}^{T+K}h(s)ds,\\ \\ \int_{t}^{T}h(s+\psi\left(s\right))ds\leq M\int_{t}^{T+K}h(s)ds.\end{array}\right.$$

$$\left\{\begin{array}[]{c}S_{t}\leq Y_{t},\text{ }0\leq t\leq T+K,\\ \int_{0}^{T}\left(Y_{s-}-S_{s-}\right)dK_{s}=0.\end{array}\right.$$

$${\left|f(t,\omega ,y,z,\pi \left(r\right),\zeta \left(\acute{r}\right))-f(t,\omega ,\acute{y},\acute{z},\acute{\pi }\left(r\right),\acute{\zeta }\left(\acute{r}\right))\right|}^2$$

$${\left|g(t,\omega ,y,z,\pi \left(r\right),\zeta \left(r\right))-g(t,\omega ,\acute{y},\acute{z},\acute{\pi }\left(r\right),\acute{\zeta }\left(\acute{r}\right))\right|}^2$$

$$\mathbb{E}{\int }_0^T\left|f(s,\omega ,0,0,0,0)\right|ds$$

$$\mathbb{E}{\int }_0^T\left|g(s,\omega ,y,z,\pi ,\zeta )\right|ds$$

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

$$\left\{\begin{array}[]{c}\Delta f\left(s\right)=f(s,\Lambda_{s}^{1},\Lambda_{s}^{1,\phi,\psi})-f(s,\Lambda_{s}^{2},\Lambda_{s}^{2,\phi,\psi}),\\ \\ \Delta g\left(s\right)=g(s,\Lambda_{s}^{1},\Lambda_{s}^{1,\phi,\psi})-g(s,\Lambda_{s}^{2},\Lambda_{s}^{2,\phi,\psi}).\end{array}\right.$$

$$\left\{\begin{array}[]{l}\Delta Y_{t}=\xi_{T}+\int_{t}^{T}\Delta f(s)ds+\int_{t}^{T}\Delta g(s)d\overleftarrow{B}_{s}+\int_{t}^{T}d\left(\Delta K_{s}\right)-\sum_{i=1}^{\infty}\int_{t}^{T}\Delta Z_{s}^{\left(i\right)}dH_{s}^{\left(i\right)},\ t\in\left[0,T\right],\\ \\ \left(\Delta Y_{t},\Delta Z_{t}\right)=\left(0,0\right),\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad t\in\left[T,T+K\right],\end{array}\right.$$

$$\mathbb{E}\left(e^{\beta t}|\Delta Y_t{|}^2\right)$$

$$2\mathbb{E}{\int }_t^T{e}^{\beta s}\Delta Y_s\Delta f(s)ds\le {ϵ}_1\mathbb{E}{\int }_t^{T+K}{e}^{\beta s}{\left|\Delta Y_s\right|}^{2}ds+\left(\frac{c}{{ϵ}_1}+cM\right)\mathbb{E}{\int }_t^{T+K}{e}^{\beta s}\left({\left|\Delta Y_{s-}\right|}^2+{\left|\left|\Delta Z_s\right|\right|}_{l^2}^2\right)ds,$$

$$\mathbb{E}{\int }_t^T{e}^{\beta s}{\left|\Delta g(s)\right|}^{2}ds\le \left(c+cM\right)\mathbb{E}{\int }_t^{T+K}{e}^{\beta s}{\left|\Delta Y_{s-}\right|}^{2}ds+\left({\alpha }_1+{\alpha }_{2}M\right)\mathbb{E}{\int }_t^{T+K}{e}^{\beta s}{\left|\left|\Delta Z_s\right|\right|}_{l^2}^{2}ds.$$

$$\mathbb{E}\left(e^{\beta t}|\Delta Y_t{|}^2\right)$$

$$\mathbb{E}\left(e^{\beta t}|\Delta Y_t{|}^2\right)\le C\mathbb{E}{\int }_t^{T+K}{e}^{\beta s}{\left|\Delta Y_{s-}\right|}^{2}ds,$$

$$f\left(t,\omega ,y,z,\pi \left(r\right),\zeta \left(\acute{r}\right)\right):\left[0,T\right]\times \Omega \times \mathbb{R}\times l^2\times {\mathcal{B}}_{\mathcal{H}}^2([0,T+K],\mathbb{R})\to {\mathcal{M}}_{\mathcal{H}}^2([0,T+K],\mathbb{R}).$$

$${\left|\left|\left(Y,Z\right)\right|\right|}_{\beta }={\left(\mathbb{E}\left[{\int }_0^{T+K}{e}^{\beta s}\left({\left|Y_{s-}\right|}^{2}ds+\sum _{i=1}^{i=\infty }{\left|Z_s^{\left(i\right)}\right|}^2\right)ds\right]\right)}^{\frac{1}{2}}.$$

$$\left\{\begin{array}[]{l}Y_{t}=\xi_{T}+\int_{t}^{T}f(s,\theta_{s},\theta_{s}^{\phi,\psi})ds+\int_{t}^{T}g(s,\theta_{s},\theta_{s}^{\phi,\psi})d\overleftarrow{B}_{s}+\int_{t}^{T}dK_{s}-\sum_{i=1}^{\infty}\int_{t}^{T}Z_{s}^{\left(i\right)}dH_{s}^{\left(i\right)},\ t\in\left[0,T\right],\\ \\ \left(Y_{t},Z_{t}\right)=\left(\eta_{t},\vartheta_{t}\right),\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\qquad\qquad t\in\left[T,T+K\right],\end{array}\right.$$

$$\left\{\begin{array}[]{l}\Delta Y_{t}=\int_{t}^{T}\Delta f(s)ds+\int_{t}^{T}\Delta g(s)d\overleftarrow{B}_{s}+\int_{t}^{T}d\left(\Delta K_{s}\right)-\sum_{i=1}^{i=\infty}\int_{t}^{T}\Delta Z_{s}^{\left(i\right)}dH_{s}^{\left(i\right)},\text{ }t\in\left[0,T\right],\\ \\ \left(\Delta Y_{t},\Delta Z_{t}\right)=\left(0,0\right),\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad t\in\left[T,T+K\right].\end{array}\right.$$

$$e^{\beta t}{\left|\Delta Y_t\right|}^2+\beta {\int }_t^T{e}^{\beta s}{\left|\Delta Y_{s-}\right|}^{2}ds$$

$$\mathbb{E}{e}^{\beta t}{\left|\Delta Y_t\right|}^2+\beta \mathbb{E}{\int }_t^T{e}^{\beta s}{\left|\Delta Y_{s-}\right|}^{2}ds+\mathbb{E}{\int }_t^T\sum _{i=1}^{i=\infty }{e}^{\beta s}{\left|\Delta Z_s^{\left(i\right)}\right|}^{2}ds$$

$$2\mathbb{E}{\int }_t^T{e}^{\beta s}\Delta Y_s\Delta f(s)ds\le {ϵ}_1\mathbb{E}{\int }_t^{T+K}{e}^{\beta s}{\left|\Delta Y_{s-}\right|}^{2}ds+\left(\frac{c}{{ϵ}_1}+cM\right)\mathbb{E}{\int }_t^{T+K}{e}^{\beta s}\left({\left|\Delta U_{s-}\right|}^2+{\left|\left|\Delta V_s\right|\right|}_{l^2}^2\right)ds,$$

Full text

Reflected solutions of Anticipated Backward Doubly SDEs driven by

Teugels Martingales.

Badreddine Mansouri, Mostapha abd el ouahab Saouli

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

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

Abstract. We deal with reflected solutions of anticipated backward doubly stochastic differential equations (RABDSDEs) driven by Teugels martingales associated with Lévy process under a Lipschitz generator where the coefficients of these BDSDEs depend on the future and present value of the solution $\left(Y,Z\right)$. Also we study the existence of a solution for anticipated BDSDEs.

Keyword Anticipated backward doubly stochastic differential equations, random Lévy measure, comparison theorem, predictable representation, Teugels martingales, Gronwall lemma, the principle of contraction.

1 Introduction

Backward stochastic differential equations (BSDEs in short) were introduced by Bismut for the linear case [2] and by Pardoux and Peng in the general case [7]. Precisely, according to [7], given a data $(\xi,f)$ consisting of a square integrable random variable $\xi$ and a progressively measurable process $f$, a so-called generator, they proved the existence and uniqueness of a solution to these equations. Recently, a new type of BSDE, called anticipated BSDE (ABSDE in short), which can be regarded as a new duality type of stochastic differential delay equations, was introduced by Peng and Yang [9], see also [13, 14]. After they introduced the theory of BSDEs, Pardoux and Peng in [8] considered a new kind of BSDEs, that is a class of backward doubly stochastic dfferential equations (BDSDEs in short) with two different directions of stochastic integrals. They proved existence and uniqueness of solutions for BDSDEs under Lipschitz conditions on the coefficients.

Recently, a new type of BDSDE called anticipated BDSDE (ABDSDE in short), which can be regarded as a new duality type of stochastic differential delay equations, was introduced by Xu [16], The ABDSDE is of the form

[TABLE]

where, $\Lambda_{s}=\left(Y_{s},Z_{s}\right)$ and $\Lambda_{s}^{\phi,\psi}=\left(Y_{s+\phi\left(s\right)},Z_{s+\psi\left(s\right)}\right)$, $\phi\left(\cdot\right):[0,T]\rightarrow\mathbb{R}_{+}\setminus\left\{0\right\}$ and $\psi\left(\cdot\right):[0,T]\rightarrow\mathbb{R}_{+}\setminus\left\{0\right\}$ are continuous functions and $f$ is called a generator.

In the paper of Nulart et al. [5], a martingale representation theorem associated to Lévy processes was proved. Then it is natural to extend BDSDEs driven by Brownian motion to BDSDEs driven by a Lévy process. In the work of Ren et al. [10], the authors proved the existence and uniqueness of solutions of BDSDEs driven by Teugels martingales associated with a Lévy process, under Lipschitz conditions on the generator $f$. These results were important from a pure mathematical point of view as well as from an application point of view in the world of finance.

The first work of Reflected BDSDEsis itroduced by Bahlali et al. [1], after Y. Ren [10] introduced a special class of reflected BDSDEs (RBDSDEs, in short), which is a BDSDE but the solution is forced to stay above a lower barrier.

Motivated by the above results and by the result introduced by Xiaoming Xu [16], we establish firstly the existence and uniqueness of the solution of the Anticipated reflected BDSDE driven by Teugles Martingales (RABDSDEs, in short) in the proof we using the result of Y. Ren [11]. Let us point out that our papier extends the results of Y. Ren [11], Xiaoming Xu [16] and Gaofeng Zong [17]. The main idea of the proof is to the point fixe theorem. And we establish others results for ABDSDEs, we prove the existence and uniqueness of the solution.

The organization of the paper is as follows. In Section 2, we give some preliminaires on the martingales $\left\{H^{\left(i\right)},t\geq 0\right\}$ and we consider the spaces of processus also we define the Itô’s lamma. In Section 3, under certain assumptions, we obtain the existence and uniqueness solution for the associated Anticipated reflected BDSDEs (RABDSDEs) and ABDSDEs.

2 Preliminaries

Let $\left(\Omega,\mathcal{F},P,B_{t},L_{t};0\leq t\leq T\right)$ be a complete Brownien-Lévy space in $\mathbb{R}\times\mathbb{R}^{\ast}$ with Lévy mesure. For $T>0$, let $\left\{B_{t},0\leq t\leq T\right\}$ is a standard Brownian motion defined on $\left(\Omega,\mathcal{F},P\right)$ with values in $\mathbb{R}$ and $\left\{L_{t};0\leq t\leq T\right\}$ is a $\mathbb{R-}$valued pure jump-Lévy process of the form $L_{t}=bt+l_{t}$ independent of $\left\{B_{t},0\leq t\leq T\right\}$.

Let  $\mathcal{F}_{t}^{L}:=\sigma(L_{s};0\leq s\leq t)$  and  $\mathcal{F}_{t,T}^{B}:=\sigma(B_{s}-B_{t};t\leq s\leq T),$ completed with $P$-null sets. We put, $\mathcal{F}_{t}:=\mathcal{F}_{t}^{L}\vee\mathcal{F}_{t,T}^{B}$, for each $t\in\left[0,T\right]$, and $\mathcal{H}_{t}:=\mathcal{F}_{0,t}^{L}\vee\mathcal{F}_{t,T+K}^{B}$ for each $t\in\left[0,T+K\right]$. It should be noted that $\left(\mathcal{H}_{t}\right)$ is not an increasing family of sub $\sigma-$fields, and hence it is not a filtration.

For each $t\in[0,T+K]$, we define

[TABLE]

the collection $\left(\mathcal{G}_{t}\right)_{t\in\left[0,T+K\right]}$ is a filtration.

For any $d,$ $k\geq 1$, we consider the following spaces of processus:

• •

Let $\mathcal{M}_{\mathcal{H}}^{2}\left(\left[0,T\right];\mathbb{R}\right)$ denote the set of  $1-$dimensional, $\mathcal{H}_{t}-$progressively measurable stochastic processes $\left\{\varphi_{t};t\in\left[0,T\right]\right\}$, such that  $\mathbb{E}\int_{0}^{T}\left|\varphi_{t}\right|^{2}dt<\infty$.

• •

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

• •

$l^{2}$ be the space of real valued sequences $\left(x_{n}\right)_{n\geq 0}$ such that $\sum_{i=1}^{i=\infty}x_{i}^{2}<\infty,$ and $\left|\left|x\right|\right|_{l^{2}}^{2}=\sum_{i=1}^{i=\infty}x_{i}^{2}.$

• •

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

• •

$\mathcal{M}_{\mathcal{H}}^{2}\left(\left[0,T\right];l^{2}\right)$ and $\mathcal{S}_{\mathcal{H}}^{2}\left(\left[0,T\right];l^{2}\right)$; are the corresponding spaces of $l^{2}$-valued processes equipped with the norm $\left|\left|\varphi\right|\right|_{l^{2}}^{2}=\mathbb{E}\int_{0}^{T}\sum_{i=1}^{i=\infty}\left|\varphi_{t}^{\left(i\right)}\right|^{2}dt<\infty.$

• •

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

• •

Notice that the space $\mathcal{B}_{\mathcal{H}}^{2}\left(\left[0,T\right],\mathbb{R}\right)=\mathcal{S}_{\mathcal{H}}^{2}\left(\left[0,T\right];\mathbb{R}\right)\times\mathcal{M}_{\mathcal{H}}^{2}\left(\left[0,T\right];l^{2}\right)$ endowd with the norm

[TABLE]

We denote by $\left(H^{i}\right)_{i\geq 1}$ the Teugels martingale associated with the lévy process $\left\{L_{t},t\in\left[0,T\right]\right\}$ with is given by $H^{\left(i\right)}=c_{i,i}Y_{t}^{\left(i\right)}+c_{i,i-1}Y_{t}^{\left(i-1\right)}+...+c_{i,1}Y_{t}^{\left(1\right)},$ where $Y_{t}^{\left(i\right)}=L_{t}^{\left(i\right)}-\mathbb{E}\left[L_{t}^{\left(i\right)}\right]=L_{t}^{\left(i\right)}-t\mathbb{E}\left[L_{1}^{\left(i\right)}\right]$ for all $i\geq 1$ and $L_{t}^{\left(i\right)}$ are power-jump processes. That is, $L_{t}^{\left(1\right)}=L_{t}$ and $L_{t}^{\left(i\right)}=\sum_{0\leq s\leq t}\left(\Delta L_{s}\right)^{i}$ for $i\geq 2,$ and $\left[H^{\left(i\right)},H^{\left(j\right)}\right],i\neq j$ and $\left\{\left[H^{\left(i\right)},H^{\left(i\right)}\right]_{t}-t,\text{ }t\geq 0\right\}$ are uniformly integrable martingale with initial value $0,$ i.e.,

[TABLE]

where it was shown in [6] that the coefficients $c_{i,k}$ correspond to the orthonormalization of the polynomials $1,x,x^{2},...$with respect to the measure $\mu\left(dx\right)=x^{2}v\left(dx\right)+\sigma^{2}\delta_{0}(dx).$ the resulting processes $H^{\left(i\right)}=\left\{H^{\left(i\right)},t\geq 0\right\}$ are called the orthonormalized ith-power-jump processes.

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 [8]

Lemma 2.1**.**

Let $\alpha\in\mathcal{S}_{\mathcal{H}}^{2}\left(\left[0,T\right];\mathbb{R}\right),$ $\beta,$ $\gamma$ and $\sigma\in\mathcal{M}_{\mathcal{H}}^{2}\left(\left[0,T\right];l^{2}\right)$ such that

[TABLE]

then

[TABLE]

note that $\langle H^{\left(i\right)},H^{\left(j\right)}\rangle_{t}=\delta_{ij}t,$ we have

[TABLE]

3 Main result

3.1 Anticipated BDSDE with Lower barrier.

In this subsection, we consider the following $1-$dimensional anticipated reflected backward doubly stochastic differential equation

[TABLE]

where $f$ called the generator, $\Lambda_{s}=\left(Y_{s-},Z_{s}\right)$ and $\Lambda_{s}^{\phi,\psi}=\left(Y_{s+\phi\left(s\right)-},Z_{s+\psi\left(s\right)}\right).$

Let $\phi:\left[0,T\right]\rightarrow\mathbb{R}_{+}^{\ast}$, and $\psi:\left[0,T\right]\rightarrow\mathbb{R}_{+}^{\ast}$ are continuous functions satisfying:

(A) There exists a constant $K\geq 0$ such that for all $t\in\left[0,T\right],$

[TABLE]

(B) There exists a constant $M\geq 0$ such that for each $t\in\left[0,T\right]$ and for all nonnegative integrable functions $h\left(\cdot\right)$,

[TABLE]

Definition 3.1**.**

A solution of equation $(3.1)$ is a triple $(Y,Z,K)$ which belongs to the space $\mathcal{B}_{\mathcal{H}}^{2}([0,\,T+K],\mathbb{R})\times\mathcal{A}^{2}$ and satisfies $(3.1)$ sach that:

[TABLE]

In this subsection we study the ABDSDEs with reflection under Lipschitz continuous generator. We consider the following assumptions (H1):

**(H1.1) **(i) There exist a constant $c>0$ such that for any $\left(r,\acute{r}\right)\in\left[0,T+K\right]^{2}$, $\left(t,\omega,y,z,\pi,\zeta\right),$ $\left(t,\omega,y,z,\acute{\pi},\acute{\zeta}\right)\in\left[0,T\right]\times\Omega\times\mathbb{R}\times l^{2}\times\mathcal{B}_{\mathcal{H}}^{2}\left(\left[0,T+K\right],\mathbb{R}\right),$

[TABLE]

(ii) There exists a constant $c>0$, $0<\alpha_{1}<\frac{1}{2}$ and $0<\alpha_{2}<\frac{1}{M}$ satisfying $0<\alpha_{1}+\alpha_{2}M<\frac{1}{2}$, such that

[TABLE]

(H1.2) For any $\left(t,\omega,y,z,\pi,\zeta\right),$

[TABLE]

(H1.3) The terminal value $\xi$ be a given random variable in $\mathbb{L}^{2}$.

Also we consider the following assumptions (H2):

(H2.1) $\left(S_{t}\right)_{t\geq 0},$ is a continuous progressively measurable real valued process satisfying

[TABLE]

(H2.2) For any $t\in\left[T,T+K\right],$ $S_{t}\leq\eta_{t}$, $\mathbb{P}$-almost surely.

(H2.3) $\left(\eta_{t},\vartheta_{t}\right)\in\mathcal{B}_{\mathcal{H}}^{2}\left(\left[T,T+K\right],\mathbb{R}\right).$

(H2.4) $(K_{t})_{t\in\left[0,T\right]}$ is a continuous, increasing process with $K_{0}=0$ and $\mathbb{E}\left(K_{T}\right)^{2}<+\infty.$

3.1.1 Existence and uniqueness of solutions.

Theorem 3.1**.**

Let $f,$ $g$ satisfies the hypothesis $\left(H1\right)$, $\left(H2\right)$ and (A), (B) are hold. Then the Anticipated RBDSDEs $\left(3.1\right)$ has a unique solution $\left(Y_{t},Z_{t},K_{t}\right)_{t\in\left[0,T+K\right]}.$

3.1.2 Proof of the existence and uniqueness result.

Proof : Uniqueness. Let $\left(Y^{j},Z^{j},K^{j}\right)\in\mathcal{B}_{\mathcal{H}}^{2}([0,T+K],\mathbb{R})\times\mathcal{A}^{2}$ for $j=1,2$ be any two solutions. Define $\Delta Y_{t}=Y_{t}^{1}-Y_{t}^{2},$ $\Delta Z_{t}^{\left(i\right)}=Z_{t}^{1,\left(i\right)}-Z_{t}^{2,\left(i\right)}$ and $\Delta K_{s}=K_{t}^{1}-K_{t}^{2}$ and for a function:

[TABLE]

We consider the following equation

[TABLE]

where $\Lambda_{s}^{j}=\left(Y_{s-}^{j},Z_{s}^{j}\right)$ and $\Lambda_{s}^{j,\phi,\psi}=\left(Y_{s+\phi\left(s\right)-}^{j},Z_{s+\psi\left(s\right)}^{j}\right)$ for $j=1,2.$

It follows from Itô’s formula that

[TABLE]

Since$\int_{t}^{T}e^{\beta s}\Delta Y_{s-}d\left(\Delta K_{s}\right)\leq 0,$ we have

[TABLE]

Using Young’s inequality $2ab\leq\epsilon_{1}a^{2}+\frac{b^{2}}{\epsilon_{1}}$ and hypothesis $\left(H.1\right),$ we have

[TABLE]

and also

[TABLE]

Then, we have the following inequality

[TABLE]

chossing $\epsilon_{1}>0$ such that, $\left(\alpha_{1}+\alpha_{2}M+\frac{c}{\epsilon_{1}}+cM\right)<1$ and $\beta-\epsilon_{1}>0,$ we get

[TABLE]

where $C=\frac{c}{\epsilon_{1}}+2cM+c.$ The uniqueness of solution follows from Gronwall’s lemma.

Existence. Before we start proving equation $(3.1)$ has a unique solution with $f,$ $g$ independent on the value and the futur value of $\left(Y,Z\right)$, i.e., P-a.s.,$f\left(t,\omega,y,z,\pi,\zeta\right)=f\left(t,\omega\right)$ and $g\left(t,\omega,y,z,\pi,\zeta\right)=g\left(t,\omega\right),$ for any $\left(t,y,z,\pi,\zeta\right).$ Then by Y. Ren [11] and the provious proof, we deduce that the l’equation $(3.1)$ where $f$ and $g$ independent on the value and the futur value of $\left(Y,Z\right)$ has a unique solution.

Now, we shall prove the existence in the general case. For all $\left(r,\acute{r}\right)\in\left[t,T+K\right]^{2},$ $\forall t\in\left[0,T+K\right]$

[TABLE]

Let $\mathcal{S}_{\mathcal{H}}^{2}\left(\left[0,T+K\right];\mathbb{R}^{d}\right)\times\mathcal{M}_{\mathcal{H}}^{2}\left(\left[0,T+K\right];l^{2}\right)$ endowed with the norm

[TABLE]

Therefore, for given $U\in\mathcal{S}_{\mathcal{H}}^{2}([0,T+K],\mathbb{R})$, $V\in\mathcal{M}_{\mathcal{H}}^{2}\left(\left[0,T+K\right];l^{2}\right)$, there exists a unique solution $\left(Y,Z,K\right)$ for the following ABDSDEs with reflection

[TABLE]

We can construct the mapping $\Phi$ is well defined, let $\left(Y_{t},Z_{t}\right)$ and $\left(\tilde{Y}_{t},\tilde{Z}_{t}\right)$ be two solution of system $(3.2)$ such that $\left(Y_{t},Z_{t}\right)=\Phi\left(U_{t-},V_{t}\right)$ and $\left(\tilde{Y}_{t},\tilde{Z}_{t}\right)=\Phi\left(\tilde{U}_{t-},\tilde{V}_{t}\right)$.

For $\beta\in\mathbb{R}$. The couple $\left(\Delta Y_{t},\Delta Z_{t}\right)$ solve the ABDSDEs with reflection

[TABLE]

where for a function $h\in\left\{f,g\right\}$, $\Delta h(s)=h(s,\theta_{s},\theta_{s}^{\phi,\psi})-h(s,\tilde{\theta}_{s},\tilde{\theta}_{s}^{\phi,\psi}),$ $\theta_{s}=\left(U_{s-},V_{s}\right),$ $\theta_{s}^{\phi,\psi}=\left(U_{s+\phi\left(s\right)-},V_{s+\psi\left(s\right)}\right),$ $\tilde{\theta}_{s}=\left(\tilde{U}_{s-},\tilde{V}_{s}\right)$, $\tilde{\theta}_{s}^{\phi,\psi}=\left(\tilde{U}_{s+\phi\left(s\right)-},\tilde{V}_{s+\psi\left(s\right)}\right)$ and $\Delta\Psi_{s}=\Psi_{s}-\tilde{\Psi}_{s}$.

Now applying Itô’s formula for $e^{\beta t}\left|\Delta Y_{t}\right|^{2},$ we get

[TABLE]

Note that $\int_{0}^{t}e^{\beta s}\Delta Y_{s-}\Delta g(s)d\overleftarrow{B}_{s},$ $\int_{0}^{t}\sum_{i=1}^{i=\infty}e^{\beta s}\Delta Y_{s-}\Delta Z_{s}^{\left(i\right)}dH_{s}^{\left(i\right)}$ $\forall$ $i\geq 1$ and

$\int_{0}^{t}\sum_{i,j=1}^{\infty}e^{\beta s}\Delta Z_{s}^{\left(i\right)}\Delta Z_{s}^{\left(j\right)}d\left(\left[H_{s}^{\left(i\right)},H_{s}^{\left(j\right)}\right]-\langle H_{s}^{\left(i\right)},H_{s}^{\left(j\right)}\rangle\right)$ for $i\neq j$ are uniformly integrable martingales. Since $\int_{t}^{T}e^{\beta s}\Delta Y_{s-}d\left(\Delta K_{s}\right)\leq 0$, taking the mathematical expectation on bath sides, we obtain

[TABLE]

Firstly using $2ab\leq\epsilon_{1}a^{2}+\frac{b^{2}}{\epsilon_{1}}$ and hypothesis $\left(H.1\right),$ we have

[TABLE]

and also

[TABLE]

Then, we have

[TABLE]

we noting that $\Delta Y_{t}=\Delta Z_{t}=0,$ for all $t\in\left[T,T+K\right]$

[TABLE]

where $\epsilon_{2}=\frac{\frac{c}{\epsilon_{1}}+c+2cM}{\left(\alpha_{1}+\alpha_{2}M\right)+\left(\frac{c}{\epsilon_{1}}+cM\right)}$. Hence, if we choose $\epsilon_{0}=\epsilon_{1}$ satisfying $\hat{c}=\left(\alpha_{1}+\alpha_{2}M\right)+\left(\frac{c}{\epsilon_{0}}+cM\right)<1$, chosse $\beta=\epsilon_{0}+\epsilon_{2}$. Then, we deduce

[TABLE]

Thus, the mapping $\Phi$ is a strict contraction on $\mathcal{S}_{\mathcal{H}}^{2}\left(\left[0,T+K\right];\mathbb{R}^{d}\right)\times\mathcal{M}_{\mathcal{H}}^{2}\left(\left[0,T+K\right];l^{2}\right)$ and it has a unique fixed point $\left(Y_{t},Z_{t}\right)\in\mathcal{S}_{\mathcal{H}}^{2}\left(\left[0,T+K\right];\mathbb{R}^{d}\right)\times\mathcal{M}_{\mathcal{H}}^{2}\left(\left[0,T+K\right];l^{2}\right).$

3.2 Anticipated BDSDE.

In this subsection we considere the anticipated BDSDE as follows

[TABLE]

where $f$ the generator, $\Lambda_{s}=\left(Y_{s-},Z_{s}\right)$ and $\Lambda_{s}^{\phi,\psi}=\left(Y_{s+\phi\left(s\right)-},Z_{s+\psi\left(s\right)}\right).$

Definition 3.2**.**

A solution of equation $(3.3)$ is a couple $(Y,Z)$ which belongs to the space $S_{\mathcal{H}}^{2}([0,T+K],\mathbb{R})\times\mathcal{M}_{\mathcal{H}}^{2}([0,T+K];l^{2})$ and satisfies $(3.3)$.

3.2.1 Existence and uniqueness of solutions.

Theorem 3.2**.**

Assume that (A), (B) and (H1) are satisfied. Then for given $\left(\eta_{t},\vartheta_{t}\right)\in\mathcal{B}_{\mathcal{H}}^{2}\left(\left[T,T+K\right],\mathbb{R}\right)$ l’equation $(3.3)$ has a unique solution $\left(Y_{t},Z_{t}\right)\in\mathcal{B}_{\mathcal{H}}^{2}\left(\left[0,T+K\right],\mathbb{R}\right)$.

3.2.2 Proof of the existence and uniqueness result.

Before we start proving equation $(3.3)$ has a unique solution with $f,$ $g$ independent on the value and the futur value of $\left(Y,Z\right)$, i.e., P-a.s.,$f\left(t,\omega,y,z,\pi,\zeta\right)=f\left(t,\omega\right)$ and $g\left(t,\omega,y,z,\pi,\zeta\right)=g\left(t,\omega\right),$ for any $\left(t,y,z,\pi,\zeta\right).$ More precisely, given $f,$ $g$ such that

[TABLE]

Under the above assumptation on $f,$ $g,$ $\xi.$

Proposition 3.1**.**

Given $\xi\in\mathbb{L}^{2}\left(\mathcal{H}_{T}\right),$ there exists a unique couple of processes $\left(Y_{t},Z_{t}\right)\in\mathcal{B}_{\mathcal{H}}^{2}\left(\left[0,T+K\right],\mathbb{R}\right),$ to solve the following BDSDEs,

[TABLE]

Proof: We consider the following filtration

[TABLE]

and the $\mathcal{G}_{t}$ square integrable martingale

[TABLE]

Thank’s to the prédictable representation property in Nualart et al. [5] yields that there existe $Z\in\mathcal{M}_{\mathcal{G}}^{2}\left(\left[0,T\right];l^{2}\right)$ such that

[TABLE]

hence

[TABLE]

Let

[TABLE]

from which, we deduce that

[TABLE]

Now by the same procedure of Xu [16], we can obtain the uniqueness and $\mathcal{H}_{t}-$ measurable of $Y_{t}$ and $Z_{t}$.

We are now in a position to give the proof of Theorem 3.2.

Proof: Let $\mathcal{S}_{\mathcal{H}}^{2}\left(\left[0,T+K\right];\mathbb{R}^{d}\right)\times\mathcal{M}_{\mathcal{H}}^{2}\left(\left[0,T+K\right];l^{2}\right)$ endowed with the norm

[TABLE]

Let we consider the following mapping:

[TABLE]

where the couple $\left(Y_{t},Z_{t}\right)_{T\leq t\leq T+K}\in\mathcal{S}_{\mathcal{H}}^{2}\left(\left[0,T+K\right];\mathbb{R}\right)\times\mathcal{M}_{\mathcal{H}}^{2}\left(\left[0,T+K\right];l^{2}\right)$ such that

$\left(Y_{t},Z_{t}\right)_{T\leq t\leq T+K}=\left(\eta_{t},\vartheta_{t}\right)$ and satisfies the equation $(E^{\xi,f^{\phi,\psi},g^{\phi,\psi}})$. Thanks to Proposition $\left(3.1\right)$, the mapping $\Phi$ is well defined. Let $\left(Y_{t},Z_{t}\right)$ and $\left(\tilde{Y}_{t},\tilde{Z}_{t}\right)$ be two solution of $(3.3)$ such that $\left(Y_{t},Z_{t}\right)=\Phi\left(y_{t-},z_{t}\right)$ and $\left(\tilde{Y}_{t},\tilde{Z}_{t}\right)=\Phi\left(\tilde{y}_{t-},\tilde{z}_{t}\right)$.

For $\beta\in\mathbb{R}$. The couple $\left(\Delta Y_{t},\Delta Z_{t}\right)$ solve the ABDSDEs with teugles martingale

[TABLE]

where for a function $h\in\left\{f,g\right\}$, $\Delta h(s)=h(s,\theta_{s},\theta_{s}^{\phi,\psi})-h(s,\tilde{\theta}_{s},\tilde{\theta}_{s}^{\phi,\psi}),$ $\theta_{s}=\left(y_{s-},z_{s}\right),$ $\theta_{s}^{\phi,\psi}=\left(y_{s+\phi\left(s\right)-},z_{s+\psi\left(s\right)}\right),$ $\tilde{\theta}_{s}=\left(\tilde{y}_{s-},\tilde{z}_{s}\right)$, $\tilde{\theta}_{s}^{\phi,\psi}=\left(\tilde{y}_{s+\phi\left(s\right)-},\tilde{z}_{s+\psi\left(s\right)}\right)$ and $\Delta\Psi_{s}=\Psi_{s}-\tilde{\Psi}_{s}$. Applying Itô’s formula to $e^{\beta t}\left|\Delta Y_{t}\right|^{2}$, we obtain

[TABLE]

note that $\int_{0}^{t}e^{\beta s}\Delta Y_{s-}\Delta g(s)d\overleftarrow{B}_{s},$ $\int_{0}^{t}\sum_{i=1}^{i=\infty}e^{\beta s}\Delta Y_{s-}\Delta Z_{s}^{\left(i\right)}dH_{s}^{\left(i\right)}$ $\forall$ $i\geq 1$ and

$\int_{0}^{t}\sum_{i,j=1}^{\infty}e^{\beta s}\Delta Z_{s}^{\left(i\right)}\Delta Z_{s}^{\left(j\right)}d\left(\left[H_{s}^{\left(i\right)},H_{s}^{\left(j\right)}\right]-\langle H_{s}^{\left(i\right)},H_{s}^{\left(j\right)}\rangle\right)$ for $i\neq j$ are uniformly integrable martingales. Now taking the mathematical expectation on bath sides, we obtain

[TABLE]

Now by the same computation of Lipschitz coefficient for Anticipated reflected BDSDEs in general case, we deduce that

[TABLE]

where $0<\varepsilon<1.$ Thus, the mapping $\Phi$ is a strict contraction on $\mathcal{S}_{\mathcal{H}}^{2}\left(\left[0,T+K\right];\mathbb{R}\right)\times\mathcal{M}_{\mathcal{H}}^{2}\left(\left[0,T+K\right];l^{2}\right)$ and it has a unique fixed point $\left(Y_{t},Z_{t}\right)\in\mathcal{S}_{\mathcal{H}}^{2}\left(\left[0,T+K\right];\mathbb{R}\right)\times\mathcal{M}_{\mathcal{H}}^{2}\left(\left[0,T+K\right];l^{2}\right).$

Finally we complete the proof of theorem 3.2.