Anticipated Generalized Backward Doubly Stochastic Differential Equations

Abstract

In this paper, we explore a new class of stochastic differential equations called anticipated generalized backward doubly stochastic differential equations (AGBDSDEs), which not only involve two symmetric integrals related to two independent Brownian motions and an integral driven by a continuous increasing process but also include generators depending on the anticipated terms of the solution (Y, Z). Firstly, we prove the existence and uniqueness theorem for AGBDSDEs. Further, two comparison theorems are obtained after finding a new comparison theorem for GBDSDEs.

1. Introduction

Nonlinear backward stochastic differential equations (BSDEs in short) were introduced by Pardoux and Peng [1] in 1990. Since then, BSDEs have been received considerable research attention due to their application in a lot of different research areas, for example, mathematical finance (see El Karoui et al. [2]), stochastic control, differential games and partial differential equations. Ref. [3] proposed a newly optimized symmetric explicit ten-step method with phase-lag of order infinity to numerically solve the Schrodinger equation. Pardoux and Zhang [4] introduced the following equation:

$$\begin{array}{c}\hfill Y_t=\xi +{\int }_t^{T}f(s,Y_s,Z_s)ds+{\int }_t^{T}h(s,Y_s)d{K}_s-{\int }_t^T{Z}_{s}d{W}_s,t\in [0,T],\end{array}$$

where $K_s$ is an increasing process, to obtain a probabilistic formula for solutions of semilinear partial differential equations (SPDEs) with a Neumann boundary condition. Ren and Xia [5] further investigated the above topic with reflection, then Ren and Otmani [6] extended this problem to Levy setting.

Pardoux and Peng [7] first presented a class of backward doubly stochastic differential equations (BDSDEs in short) to give a probabilistic representation for a class of quasilinear stochastic partial differential equations. Then Shi et al. [8] gave a comparison theorem for BDSDEs with Lipschitz condition on the coefficients. In this way, Boufoussi et al. [9] gave the following generalized backward doubly stochastic differential equation:

$$\begin{array}{cc}\hfill Y_t=& \xi +{\int }_t^{T}f(s,Y_s,Z_s)ds+{\int }_t^{T}h(s,Y_s)d{K}_s+{\int }_t^{T}g(s,Y_s,Z_s)d{\overleftarrow{B}}_s\hfill \\ \hfill & -{\int }_t^T{Z}_{s}d{W}_s,t\in [0,T],\hfill \end{array}$$

(1)

in which the equations not only involve a standard (forward) stochastic It$\widehat{\mathrm{o}}$ integral $d{W}_t$ but also a symmetric backward stochastic It$\widehat{\mathrm{o}}$ integral $d{\overleftarrow{B}}_t$. They first obtained the existence and uniqueness for the above equation, then gave the viscosity solution to one kind of semilinear SPDE, a probabilistic representation. Hu and Ren [10] explored this problem with an integral driven by the Levy process. Aman and Mrhardy [11] investigated the Equation (1) with reflection.

Peng and Yang [12] introduced a new type of BSDE called anticipated BSDEs. The generator of these equations includes not only the values of solutions of the present but also the future. The authors found that these anticipated BSDEs have unique solutions under Lipschitz assumptions, a comparison theorem for their solutions, and a duality between them and stochastic differential delay equations. After the work of Peng and Yang [12], Zhang [13] dealt with the comparison theorem of one dimensional anticipated BSDEs under one kind of non-Lipschitz assumption. Xu [14] and zhang [15] introduced the so-called anticipated BDSDES (ABDSDEs). They proved the existence and uniqueness of the solution to these equations, obtained some comparison theorems in the one dimensional case, and studied the duality between ABDSDEs and delayed SDDEs. Reference [16] investigated a coupled system which is composed by a delayed forward doubly stochastic differential equation and an anticipated backward doubly SDE. Recently, Wu et al. [17] proposed the so-called anticipated GBSDEs (AGBSDEs) of the following form:

$$\left\{\begin{array}{cc}\hfill Y_t=& \xi +{\int }_t^{T}f(s,Y_s,Z_s,Y_{s+\delta \left(s\right)},Z_{s+\zeta \left(s\right)})\mathrm{d}s\hfill \\ \hfill & +{\int }_t^{T}h(s,Y_s,Y_{s+\delta \left(s\right)})d{K}_s-{\int }_t^T{Z}_s\mathrm{d}{W}_s,t\in [0,T],\hfill \\ \hfill Y_t=& {\xi }_t,Z_t={\eta }_t,t\in [T,T+K],\hfill \end{array}\right.$$

where $\delta (·)$ and $\zeta (·)$ are given ${\mathbb{R}}^{+}$-valued continuous functions and for $\varphi (·)=\delta (·),\zeta (·)$ such that:

(A1) there exists a constant $K\ge 0$ such that, for all $t\in [0,T]$, $t+\varphi \left(t\right)⩽T+K$;

(A2) there exists a constant $M⩾0$ such that, for all $t\in [0,T]$ and for all non-negative and integrable $g(·)$,

$${\int }_t^{T}g(s+\varphi \left(s\right))\mathrm{d}s⩽M{\int }_t^{T+K}g\left(s\right)\mathrm{d}s,{\int }_t^{T}g(s+\varphi \left(s\right))d{K}_s⩽M{\int }_t^{T+K}g\left(s\right)d{K}_s,$$

and for any interval $[\alpha ,\beta ],[\alpha +u,\beta +u]\in [0,T+K]$, $u>0$, we have d$K_s\left([\alpha ,\beta ]\right)\le$ d$K_s\left([\alpha +u,\beta +u]\right)$, where d$K_s$ is a measure generated by K on $[0,T+K]$.

In this paper, we are concerned with the following anticipated GBDSDEs:

$$\left\{\begin{array}{c}{Y}_t=Y_T+{\int }_t^{T}f(s,Y_s,Z_s,Y_{s+{\delta }_1\left(s\right)},Z_{s+\zeta \left(s\right)})\mathrm{d}s+{\int }_t^{T}h(s,Y_s,Y_{s+{\delta }_2\left(s\right)})d{K}_s\hfill \\ +{\int }_t^{T}g(s,Y_s,Z_s,Y_{s+{\delta }_3\left(s\right)},Z_{s+\overline{\zeta }\left(s\right)})\mathrm{d}{\overleftarrow{B}}_s-{\int }_t^T{Z}_s\mathrm{d}{W}_s,t\in [0,T],\hfill \\ Y_t={\xi }_t,Z_t={\eta }_t,t\in [T,T+K],\hfill \end{array}\right.$$

(2)

where ${\delta }_i(·)$, $i=1,2,3$, $\zeta (·)$ and $\overline{\zeta }(·)$ are given ${\mathbb{R}}^{+}$-valued continuous functions such that (A1) and (A2). We will prove that the solution of the above AGBDSDE exists uniquely under proper assumptions, and two versions of one dimensional comparison theorems are given. These results are the cornerstones of AGBDSDEs applied to the obstacle problem for some SPDEs with the nonlinear Neumann boundary condition and some stochastic control problems with delay.

The organization of this paper is as follows. In Section 2, some preliminaries, assumptions and definitions are given. In Section 3, we focus on the existence and uniqueness of the solutions of anticipated GBDSDEs. In Section 4, two comparison theorems are given, and in the last section, the conclusion and future work are presented.

2. Preliminaries

Throughout the paper, we use $|x|$ and $\Vert A\Vert =\sqrt{Tr\left(A{A}^{\ast }\right)}$ to denote the norm of a vector $x\in {\mathbb{R}}^k$ and a matrix $A\in k\times d$, respectively, where $A^{\ast }$ is the transpose of A. Let $\{W_t;0\le t\le T\}$ and $\{B_t;0\le t\le T\}$ be two mutually independent standard Browning motions, with values respectively in ${\mathbb{R}}^d$ and ${\mathbb{R}}^l$ on a complete probability space $(\mathsf{\Omega },\mathcal{F},\mathbb{P})$. Let $T>0,K\ge 0$ be fixed constants. Let $\mathcal{N}$ denote the class of P-null sets of $\mathcal{F}$. For any $t\in [0,T+K]$, we define:

$${\mathcal{F}}_t\triangleq {\mathcal{F}}_{0,t}^W\vee {\mathcal{F}}_{t,T+K}^B,$$

where for any processes $\left\{{\eta }_t\right\},{\mathcal{F}}_{s,t}^{\eta }=\sigma \{{\eta }_r-{\eta }_s;s\le r\le t\}\vee \mathcal{N}$. It is worth noting that $\{{\mathcal{F}}_t,t\in [0,T+K]\}$ is not a filtration because $\{{\mathcal{F}}_{0,t}^W,t\in [0,T+K]\}$ is increasing and $\{{\mathcal{F}}_{t,T}^B,t\in [0,T+K]\}$ is decreasing. Let $K_t$ be a continuous, increasing and ${\mathcal{F}}_t$-adapted process on $[0,T+K]$ with $K_0=0$. We will use the following notations: for any $n\in \mathbb{N}$,

(i)

${\mathcal{M}}^2(0,T;{\mathbb{R}}^n)\triangleq \{\phi :\mathsf{\Omega }\times [0,T]\to {\mathbb{R}}^n\mid \phi$ is a ${\mathcal{F}}_t$-progressively measurable processes such that ${\Vert \phi \Vert }_{{\mathcal{M}}^2}^2=\mathbb{E}({\int }_0^T|{\phi }_t{|}^{2}dt)<\infty \}$;

(ii)

${\mathcal{S}}^2([0,T];{\mathbb{R}}^n)\triangleq \{\phi :\mathsf{\Omega }\times [0,T]\to {\mathbb{R}}^n\mid \phi$ is a continuous and ${\mathcal{F}}_t$- progressively measurable processes such that ${\Vert \phi \Vert }_{{\mathcal{S}}^2}^2=\mathbb{E}\left(\underset{0\le t\le T}{sup}{|{\phi }_t|}^2\right)<\infty$;

(iii)

$L^2({\mathcal{F}}_T;{\mathbb{R}}^n)\triangleq \{\xi :\xi \in {\mathbb{R}}^n\mid \xi$ is a ${\mathcal{F}}_T$-measurable random variable with ${\mathbb{E}|\xi |}^2<\infty \}$.

Let $f(t,·,·,·,·,·):\mathsf{\Omega }\times {\mathbb{R}}^k\times {\mathbb{R}}^{k\times d}\times {\mathcal{M}}^2(t,T+K;{\mathbb{R}}^k)\times {\mathcal{M}}^2(t,T+K;{\mathbb{R}}^{k\times d})\to L^2({\mathcal{F}}_t;$${\mathbb{R}}^k)$, $g(t,·,·,·,·,·):\mathsf{\Omega }\times {\mathbb{R}}^k\times {\mathbb{R}}^{k\times d}\times {\mathcal{M}}^2(t,T+K;{\mathbb{R}}^k)\times {\mathcal{M}}^2(t,T+K;{\mathbb{R}}^{k\times d})\to L^2({\mathcal{F}}_t;$${\mathbb{R}}^{k\times l})$, $h(t,·,·,·):\mathsf{\Omega }\times {\mathbb{R}}^k\times {\mathcal{M}}^2(t,T+K;{\mathbb{R}}^k)\to L^2({\mathcal{F}}_t;{\mathbb{R}}^k)$. We make the following assumptions about $(\xi ,f,g,h)$:

Hypothesis 1 (H1). 

For $\xi \in S^2([T,T+K];{\mathbb{R}}^k),\eta \in {\mathcal{M}}^2(T,T+K;{\mathbb{R}}^{k\times d})$, we assume for each $\mu \in R^{+}$,

$$\mathbb{E}[\underset{t\in [T,T+K]}{sup}{e}^{\mu K_t}|{\xi }_t{|}^2]<\infty ,\mathbb{E}[{\int }_T^{T+K}{e}^{\mu K_t}|{\xi }_t{|}^{2}d{K}_t]<\infty ,\mathbb{E}[{\int }_T^{T+K}{e}^{\mu K_t}\parallel {\eta }_t{\parallel }^{2}dt]<\infty .$$

Hypothesis 2 (H2). 

For all $s\in [0,T]$, $y,y^{\prime }\in {\mathbb{R}}^k$, $z,z^{\prime }\in {\mathbb{R}}^{k\times d}$, $\theta (·),{\theta }^{\prime }(·)\in {\mathcal{M}}^2(s,T+K;{\mathbb{R}}^k)$, $\vartheta (·),{\vartheta }^{\prime }(·)\in {\mathcal{M}}^2(s,T+K;{\mathbb{R}}^{k\times d})$, $r,\overline{r}\in [s,T+K]$, we have

$$\left\{\begin{array}{c}|f(s,y,z,\theta \left(r\right),\vartheta \left(\overline{r}\right))-f(s,y^{\prime },z^{\prime },{\theta }^{\prime }\left(r\right),{\vartheta }^{\prime }\left(\overline{r}\right)){|}^2\hfill \\ ⩽C(|y-y^{\prime }{|}^2+\parallel z-z^{\prime }{\parallel }^2+{\mathbb{E}}^{{\mathcal{F}}_s}[|\theta \left(r\right)-{\theta }^{\prime }{\left(r\right)|}^2]+{\mathbb{E}}^{{\mathcal{F}}_s}[\parallel \vartheta \left(\overline{r}\right)-{\vartheta }^{\prime }\left(\overline{r}\right){\parallel }^2\left]\right),\hfill \\ \parallel g(s,y,z,\theta \left(r\right),\vartheta \left(\overline{r}\right))-g(s,y^{\prime },z^{\prime },{\theta }^{\prime }\left(r\right),{\vartheta }^{\prime }\left(\overline{r}\right)){\parallel }^2\hfill \\ ⩽C(|y-y^{\prime }{|}^2+{\mathbb{E}}^{{\mathcal{F}}_s}[|\theta \left(r\right)-{\theta }^{\prime }{\left(r\right)|}^2\left]\right)+{\alpha }_1\parallel z-z^{\prime }{\parallel }^2+{\alpha }_2{\mathbb{E}}^{{\mathcal{F}}_s}[\parallel \vartheta \left(\overline{r}\right)-{\vartheta }^{\prime }\left(\overline{r}\right){\parallel }^2],\hfill \\ \langle y-y^{\prime },h(s,y,\theta \left(r\right))-f(s,y^{\prime },{\theta }^{\prime }\left(r\right))\rangle \hfill \\ ⩽\overline{C}|y-y^{\prime }{|}^2+\beta |y-y^{\prime }|{\mathbb{E}}^{{\mathcal{F}}_s}[|\theta \left(r\right)-{\theta }^{\prime }\left(r\right)|],\hfill \end{array}\right.$$

where $C>0,\overline{C}<0,\beta >0$, $0<{\alpha }_1<1$, $0<{\alpha }_1+{\alpha }_{2}M<1$ are five constants.

Hypothesis 3 (H3). 

For any $s\in [0,T],y\in {\mathbb{R}}^k,z\in {\mathbb{R}}^{k\times d},\theta (·)\in {\mathcal{M}}^2(s,T+K;{\mathbb{R}}^k),\vartheta (·)\in {\mathcal{M}}^2(s,T+K;{\mathbb{R}}^{k\times d}),r,\overline{r}\in [s,T+K]$, $\mu \in R^{+}$, we have

$$\left\{\begin{array}{c}|f(s,y,z,\theta \left(r\right),\vartheta \left(\overline{r}\right))|⩽{\psi }_1\left(s\right)+K(|y|+\parallel z\parallel +{\mathbb{E}}^{{\mathcal{F}}_s}[|\theta \left(r\right)|]+{\mathbb{E}}^{{\mathcal{F}}_s}[\parallel \vartheta \left(\overline{r}\right)\parallel \left]\right),\hfill \\ \parallel g(s,y,z,\theta \left(r\right),\vartheta \left(\overline{r}\right))\parallel ⩽{\psi }_2\left(s\right)+K(|y|+\parallel z\parallel +{\mathbb{E}}^{{\mathcal{F}}_s}[|\theta \left(r\right)|]+{\mathbb{E}}^{{\mathcal{F}}_s}[\parallel \vartheta \left(\overline{r}\right)\parallel \left]\right),\hfill \\ |h(s,y,\theta \left(r\right))|⩽{\psi }_3\left(s\right)+K(|y|+{\mathbb{E}}^{{\mathcal{F}}_s}[|\theta \left(r\right)|]),\hfill \\ \mathbb{E}({\int }_0^T{e}^{\mu K_s}|{\psi }_1\left(s\right){|}^{2}ds+{\int }_0^T{e}^{\mu K_s}|{\psi }_2\left(s\right){|}^{2}ds+{\int }_0^T{e}^{\mu K_t}|{\psi }_3\left(s\right){|}^2{K}_s)<\infty ,\hfill \end{array}\right.$$

where ${\psi }_i,i=1,2,3$ are three adapted processes with values in $[1,+\infty )$ and $K>0$ is a constant. The first and second inequalities on the above are the Lipschitze conditions for f and g with anticipated terms, respectively.

Definition 1.

A solution for AGDSDE is a pair $(Y_t,Z_t)\in S^2([0,T+K];{\mathbb{R}}^n)\times {\mathcal{M}}^2(0,T+K;{\mathbb{R}}^{n\times d})$ such that for any $0\le t\le T$,

$$\left\{\begin{array}{c}{Y}_t={\xi }_T+{\int }_t^{T}f(s,Y_s,Z_s,Y_{s+{\delta }_1\left(s\right)},Z_{s+\zeta \left(s\right)})\mathrm{d}s+{\int }_t^{T}h(s,Y_s,Y_{s+{\delta }_2\left(s\right)})d{K}_s\hfill \\ +{\int }_t^{T}g(s,Y_s,Z_s,Y_{s+{\delta }_3\left(s\right)},Z_{s+\overline{\zeta }\left(s\right)})\mathrm{d}{\overleftarrow{B}}_s-{\int }_t^T{Z}_s\mathrm{d}{W}_s,t\in [0,T],\hfill \\ Y_t={\xi }_t,Z_t={\eta }_t,t\in [T,T+K].\hfill \end{array}\right.$$

3. Existence and Uniqueness Theorem

In order to obtain the existence and uniqueness result, we need the following two priori estimates.

Proposition 1.

Assume that (A1), (A2), (H1), (H2) and (H3) hold. Ff $\{(Y_t,Z_t);0\le t\le T+K\}$ is a solution of AGBDSDE (2) and $\mu >0,\lambda >max\{\beta (1+M)-|\overline{C}|,0\}$, we get:

$$\begin{array}{cc}\hfill & \mathbb{E}[\underset{t\in [0,T]}{sup}{e}^{\mu t+\lambda K_t}|Y_t{|}^2+{\int }_0^T{e}^{\mu s+\lambda K_s}\parallel Z_s{\parallel }^{2}ds+{\int }_0^T{e}^{\mu s+\lambda K_s}|Y_s{|}^{2}d{K}_s]\le \tilde{C}\mathbb{E}(\underset{T\le t\le T+K}{sup}{e}^{\mu t+\lambda K_t}{|{\xi }_t|}^2\hfill \\ \hfill & +{\int }_T^{T+K}{e}^{\mu s+\lambda K_s}|{\xi }_s{|}^{2}ds+{\int }_T^{T+K}{e}^{\mu s+\lambda K_s}\parallel {\eta }_s{\parallel }^{2}ds+{\int }_T^{T+K}{e}^{\mu s+\lambda K_s}{|{\xi }_s|}^{2}d{K}_s\hfill \\ \hfill & +{\int }_0^T{e}^{\mu s+\lambda K_s}|{\psi }_1{\left(s\right)|}^{2}ds+{\int }_0^T{e}^{\mu s+\lambda K_s}|{\psi }_2{\left(s\right)|}^{2}ds+{\int }_0^T{e}^{\mu s+\lambda K_s}|{\psi }_3\left(s\right){|}^{2}d{K}_s],\hfill \end{array}$$

where $\tilde{C}$ is a constant.

Proof. 

In the following, we assume $K_{T+K}$ is a bounded random variable, and then apply Fatou’s lemma to obtain the general result. From It$\widehat{\mathrm{o}}$’s formula, we have:

$$\begin{array}{cc}\hfill & e^{\mu t+\lambda K_t}|Y_t{|}^2+{\int }_t^T{e}^{\mu s+\lambda K_s}\parallel Z_s{\parallel }^{2}ds+\lambda {\int }_t^T{e}^{\mu s+\lambda K_s}|Y_s{|}^{2}d{K}_s=e^{\mu T+\lambda K_T}{|{\xi }_T|}^2\hfill \\ \hfill & +2{\int }_t^T{e}^{\mu s+\lambda K_s}\langle Y_s,f(s,Y_s,Z_s,Y_{s+{\delta }_1\left(s\right)},Z_{s+\zeta \left(s\right)})\rangle ds+2{\int }_t^T{e}^{\mu s+\lambda K_s}\langle Y_s,h(s,Y_s,Y_{s+{\delta }_2\left(s\right)})\rangle d{K}_s\hfill \\ \hfill & +2{\int }_t^T{e}^{\mu s+\lambda K_s}\langle Y_s,g(s,Y_s,Z_s,Y_{s+{\delta }_3\left(s\right)},Z_{s+\overline{\zeta }\left(s\right)})\mathrm{d}{\overleftarrow{B}}_s\rangle -2{\int }_t^T{e}^{\mu s+\lambda K_s}\langle Y_s,Z_s\mathrm{d}{W}_s\rangle \hfill \\ \hfill & -\mu {\int }_t^T{e}^{\mu s+\lambda K_s}|Y_s{|}^{2}ds+{\int }_t^T{e}^{\mu s+\lambda K_s}{\parallel g(s,Y_s,Z_s,Y_{s+{\delta }_3\left(s\right)},Z_{s+\overline{\zeta }\left(s\right)})\parallel }^{2}ds.\hfill \end{array}$$

According to the assumptions (H2), (H3) and Yong’s inequality, for any $\theta >0$, we get:

$$\begin{array}{cc}\hfill 2\langle Y_s,f(s,Y_s,Z_s,Y_{s+{\delta }_1\left(s\right)},Z_{s+\zeta \left(s\right)})\rangle \le & \theta |Y_s{|}^2+\frac{2C}{\theta }(|Y_s{|}^2+|Z_s{|}^2+{\mathbb{E}}^{{\mathcal{F}}_s}|Y_{s+{\delta }_1\left(s\right)}{|}^2+{\mathbb{E}}^{{\mathcal{F}}_s}|Z_{s+\zeta \left(s\right)}{|}^2)\hfill \\ \hfill & +\frac{2}{\theta }{|{\psi }_1\left(s\right)|}^2,\hfill \\ \hfill 2\langle Y_s,h(s,Y_s,Y_{s+{\delta }_2\left(s\right)})\rangle \le & 2\overline{C}|Y_s{|}^2+2\beta |Y_s|{\mathbb{E}}^{{\mathcal{F}}_s}|Y_{s+{\delta }_2\left(s\right)}|+2|Y_s||{\psi }_3\left(t\right)|\hfill \\ \hfill \le & (2\overline{C}+|\overline{C}|+\beta )|Y_s{|}^2+\beta {\mathbb{E}}^{{\mathcal{F}}_s}|Y_{s+{\delta }_2\left(s\right)}{|}^2+\frac{1}{|\overline{C}|}{|{\psi }_3\left(s\right)|}^2,\hfill \\ \hfill \parallel g(s,Y_s,Z_s,Y_{s+{\delta }_3\left(s\right)},Z_{s+\overline{\zeta }\left(s\right)}){\parallel }^2\le & (1+\frac{1}{\theta })\left(C\right(|Y_s{|}^2+{\mathbb{E}}^{{\mathcal{F}}_s}|Y_{s+{\delta }_3\left(s\right)}{|}^2)+{\alpha }_1\parallel Z_s{\parallel }^2+{\alpha }_2{\mathbb{E}}^{{\mathcal{F}}_s}|Z_{s+\overline{\zeta }\left(s\right)}{|}^2)\hfill \\ \hfill & +(1+\theta )|{\psi }_2{\left(s\right)|}^2.\hfill \end{array}$$

Consequently, we have:

$$\begin{array}{cc}\hfill & E[e^{\mu t+\lambda K_t}|Y_t{|}^2+(1-{\alpha }_1-{\alpha }_{2}M-\frac{2C(1+M)+{\alpha }_1+{\alpha }_{2}M}{\theta }){\int }_t^T{e}^{\mu s+\lambda K_s}\parallel Z_s{\parallel }^{2}ds\hfill \\ \hfill & +(\lambda +|\overline{C}|-\beta (1+M)){\int }_t^T{e}^{\mu s+\lambda K_s}|Y_s{|}^{2}d{K}_s]\le E[e^{\mu T+\lambda K_T}{|{\xi }_T|}^2\hfill \\ \hfill & +(\theta +\frac{3C(1+M)}{\theta }+C(1+M)-\mu ){\int }_t^T{e}^{\mu s+\lambda K_s}|Y_s{|}^{2}ds+(\frac{3CM}{\theta }+CM){\int }_T^{T+K}{e}^{\mu s+\lambda K_s}{|{\xi }_s|}^{2}ds\hfill \\ \hfill & +(\frac{2CM}{\theta }+(1+\frac{1}{\theta }){\alpha }_{2}M){\int }_T^{T+K}{e}^{\mu s+\lambda K_s}\parallel {\eta }_s{\parallel }^{2}ds+\frac{2}{\theta }{\int }_t^T{e}^{\mu s+\lambda K_s}{|{\psi }_1(s)|}^{2}ds\hfill \\ \hfill & +\beta M{\int }_T^{T+K}{e}^{\mu s+\lambda K_s}|{\xi }_s{|}^{2}d{K}_s+\frac{1}{|\overline{C}|}{\int }_t^T{e}^{\mu s+\lambda K_s}|{\psi }_3{(s)|}^{2}d{K}_s+(1+\theta ){\int }_t^T{e}^{\mu s+\lambda K_s}|{\psi }_2(s){|}^{2}ds].\hfill \end{array}$$

Thus, choosing $\theta =\frac{2C(1+M)+{\alpha }_1+{\alpha }_{2}M}{1-{\alpha }_1-{\alpha }_{2}M}+|\mu |+1$, and from Gronwall’s inequality, we can obtain:

$$\begin{array}{cc}\hfill & \underset{t\in [0,T]}{sup}\mathbb{E}[e^{\mu t+\lambda K_t}|Y_t{|}^2+{\int }_0^T{e}^{\mu s+\lambda K_s}\parallel Z_s{\parallel }^{2}ds+{\int }_0^T{e}^{\mu s+\lambda K_s}|Y_s{|}^{2}d{K}_s]\le \widehat{C}\mathbb{E}(\underset{T\le t\le T+K}{sup}{e}^{\mu t+\lambda K_t}{|{\xi }_t|}^2\hfill \\ \hfill & +{\int }_T^{T+K}{e}^{\mu s+\lambda K_s}|{\xi }_s{|}^{2}ds+{\int }_T^{T+K}{e}^{\mu s+\lambda K_s}\parallel {\eta }_s{\parallel }^{2}ds+{\int }_T^{T+K}{e}^{\mu s+\lambda K_s}{|{\xi }_s|}^{2}d{K}_s\hfill \\ \hfill & +{\int }_0^T{e}^{\mu s+\lambda K_s}|{\psi }_1{\left(s\right)|}^{2}ds+{\int }_0^T{e}^{\mu s+\lambda K_s}|{\psi }_2{\left(s\right)|}^{2}ds+{\int }_0^T{e}^{\mu s+\lambda K_s}|{\psi }_3\left(s\right){|}^{2}d{K}_s].\hfill \end{array}$$

Using the Burkholder–Davis–Gundy’s and above inequality, the desired result follows. □

Proposition 2.

Denote $(\overline{Y},\overline{Z},\overline{\xi },\overline{\eta },\overline{f},\overline{g},\overline{h},\overline{K})=(Y-Y^{\prime },Z-Z^{\prime },\xi -{\xi }^{\prime },\eta -{\eta }^{\prime },f-f^{\prime },g-g^{\prime },K-K^{\prime })$, then, for any $\mu >|2\overline{C}+\beta (1+M)|$, there exists a constant $\tilde{C}>0$ such that:

$$\begin{array}{cc}\hfill & \mathbb{E}[\underset{t\in [0,T]}{sup}{e}^{\mu A_t}|{\overline{Y}}_t{|}^2+{\int }_0^T{e}^{\mu A_s}\parallel {\overline{Z}}_s{\parallel }^{2}ds]\le \tilde{C}\mathbb{E}[e^{\mu K_T}{|{\overline{\xi }}_T|}^2\hfill \\ \hfill & +{\int }_T^{T+K}{e}^{\mu A_s}\parallel {\overline{\eta }}_s{\parallel }^{2}ds+{\int }_T^{T+K}{e}^{\mu A_s}|{\overline{\xi }}_s{|}^{2}d{K}_s^{\prime }+{\int }_0^T{e}^{\mu A_s}|h(s,Y_s,Y_{s+{\delta }_2\left(s\right)}){|}^2{\parallel \overline{K}\parallel }_s\hfill \\ \hfill & +{\int }_0^T{e}^{\mu A_s}{|f(s,Y_s,Z_s,Y_{s+{\delta }_1\left(s\right)},Z_{s+\zeta \left(s\right)})-f^{\prime }(s,Y_s,Z_s,Y_{s+{\delta }_1\left(s\right)},Z_{s+\zeta \left(s\right)})|}^{2}ds\hfill \\ \hfill & +{\int }_0^T{e}^{\mu A_s}{\parallel g(s,Y_s,Z_s,Y_{s+{\delta }_3\left(s\right)},Z_{s+\overline{\zeta }\left(s\right)})-g^{\prime }(s,Y_s,Z_s,Y_{s+{\delta }_3\left(s\right)},Z_{s+\overline{\zeta }\left(s\right)})\parallel }^{2}ds\hfill \\ \hfill & +{\int }_0^T{e}^{\mu A_s}|h(s,Y_s,Y_{s+{\delta }_2\left(s\right)})-h^{\prime }(s,Y_s,Y_{s+{\delta }_2\left(s\right)}){|}^{2}d{K}_s^{\prime }],\hfill \end{array}$$

where $A_t:=\parallel \overline{K}{\parallel }_t+K_t^{\prime },{\parallel \overline{K}\parallel }_t$ is the total variation for process $\overline{K}$ on the interval $[0,t]$.

Proof. 

Similar to Proposition 1, we assume $K_{T+K}$ is bounded random variable. From It$\widehat{\mathrm{o}}$’s formula, we have:

$$\begin{array}{cc}\hfill & e^{\mu A_t}|{\overline{Y}}_t{|}^2+{\int }_t^T{e}^{\mu A_s}\parallel {\overline{Z}}_s{\parallel }^{2}ds+\mu {\int }_t^T{e}^{\mu A_s}|{\overline{Y}}_s{|}^{2}d{A}_s=e^{\mu K_T}{|{\overline{\xi }}_T|}^2-2{\int }_t^T{e}^{\mu A_s}\langle {\overline{Y}}_s,{\overline{Z}}_s\mathrm{d}{W}_s\rangle \hfill \\ \hfill & +2{\int }_t^T{e}^{\mu A_s}\langle {\overline{Y}}_s,f(s,Y_s,Z_s,Y_{s+{\delta }_1(s)},Z_{s+\zeta (s)})-f^{\prime }(s,Y_s^{\prime },Z_s^{\prime },Y_{s+{\delta }_1(s)}^{\prime },Z_{s+\zeta (s)}^{\prime })\rangle ds\hfill \\ \hfill & +2{\int }_t^T{e}^{\mu A_s}\langle {\overline{Y}}_s,h(s,Y_s,Y_{s+{\delta }_2(s)})\rangle d{\overline{K}}_s+2{\int }_t^T{e}^{\mu A_s}\langle {\overline{Y}}_s,h(s,Y_s,Y_{s+{\delta }_2(s)})-h^{\prime }(s,Y_s^{\prime },Y_{s+{\delta }_2(s)}^{\prime })\rangle d{K}_s^{\prime }\hfill \\ \hfill & +2{\int }_t^T{e}^{\mu A_s}\langle {\overline{Y}}_s,(g(s,Y_s,Z_s,Y_{s+{\delta }_3(s)},Z_{s+\overline{\zeta }(s)})-g^{\prime }(s,Y_s^{\prime },Z_s^{\prime },Y_{s+{\delta }_3(s)}^{\prime },Z_{s+\overline{\zeta }(s)}^{\prime }))\mathrm{d}{\overleftarrow{B}}_s\rangle \hfill \\ \hfill & +{\int }_t^T{e}^{\mu A_s}{\parallel g(s,Y_s,Z_s,Y_{s+{\delta }_3(s)},Z_{s+\overline{\zeta }(s)})-g^{\prime }(s,Y_s^{\prime },Z_s^{\prime },Y_{s+{\delta }_3(s)}^{\prime },Z_{s+\overline{\zeta }(s)}^{\prime })\parallel }^{2}ds\hfill \\ \hfill & =e^{\mu K_T}{|{\overline{\xi }}_T|}^2-2{\int }_t^T{e}^{\mu A_s}\langle {\overline{Y}}_s,{\overline{Z}}_s\mathrm{d}{W}_s\rangle +2{\int }_t^T{e}^{\mu A_s}\langle {\overline{Y}}_s,h(s,Y_s,Y_{s+{\delta }_2(s)})\rangle d{\overline{K}}_s\hfill \\ \hfill & +2{\int }_t^T{e}^{\mu A_s}\langle {\overline{Y}}_s,f(s,Y_s,Z_s,Y_{s+{\delta }_1(s)},Z_{s+\zeta (s)})-f^{\prime }(s,Y_s,Z_s,Y_{s+{\delta }_1(s)},Z_{s+\zeta (s)})\rangle ds\hfill \\ \hfill & +2{\int }_t^T{e}^{\mu A_s}\langle {\overline{Y}}_s,f^{\prime }(s,Y_s,Z_s,Y_{s+{\delta }_1(s)},Z_{s+\zeta (s)})-f^{\prime }(s,Y_s^{\prime },Z_s^{\prime },Y_{s+{\delta }_1(s)}^{\prime },Z_{s+\zeta (s)}^{\prime })\rangle ds\hfill \\ \hfill & +2{\int }_t^T{e}^{\mu A_s}\langle {\overline{Y}}_s,h(s,Y_s,Y_{s+{\delta }_2(s)})-h^{\prime }(s,Y_s,Y_{s+{\delta }_2(s)})\rangle d{K}_s^{\prime }\hfill \\ \hfill & +2{\int }_t^T{e}^{\mu A_s}\langle {\overline{Y}}_s,h^{\prime }(s,Y_s,Y_{s+{\delta }_2(s)})-h^{\prime }(s,Y_s^{\prime },Y_{s+{\delta }_2(s)}^{\prime })\rangle d{K}_s^{\prime }\hfill \\ \hfill & +2{\int }_t^T{e}^{\mu A_s}\langle {\overline{Y}}_s,(g(s,Y_s,Z_s,Y_{s+{\delta }_3(s)},Z_{s+\overline{\zeta }(s)})-g^{\prime }(s,Y_s^{\prime },Z_s^{\prime },Y_{s+{\delta }_3(s)}^{\prime },Z_{s+\overline{\zeta }(s)}^{\prime }))\mathrm{d}{\overleftarrow{B}}_s\rangle \hfill \\ \hfill & +{\int }_t^T{e}^{\mu A_s}{\parallel g(s,Y_s,Z_s,Y_{s+{\delta }_3(s)},Z_{s+\overline{\zeta }(s)})-g^{\prime }(s,Y_s^{\prime },Z_s^{\prime },Y_{s+{\delta }_3(s)}^{\prime },Z_{s+\overline{\zeta }(s)}^{\prime })\parallel }^{2}ds.\hfill \end{array}$$

Through the assumptions (A1), (A2), (H2) and Yong’s inequality, for any $\theta >0$, we get

$$\begin{array}{cc}\hfill & \mathbb{E}[e^{\mu A_t}|{\overline{Y}}_t{|}^2+{\int }_t^T{e}^{\mu A_s}\parallel {\overline{Z}}_s{\parallel }^{2}ds+\mu {\int }_t^T{e}^{\mu A_s}|{\overline{Y}}_s{|}^{2}d{A}_s]\hfill \\ \hfill & \le \mathbb{E}[e^{\mu K_T}|{\overline{\xi }}_T{|}^2+\mu {\int }_t^T{e}^{\mu A_s}|{\overline{Y}}_s{|}^{2}d\parallel \overline{K}{\parallel }_s+\frac{1}{\mu }{\int }_t^T{e}^{\mu A_s}|h(s,Y_s,Y_{s+{\delta }_2(s)}){|}^{2}d\parallel \overline{K}{\parallel }_s\hfill \\ \hfill & +{\int }_t^T{e}^{\mu A_s}|{\overline{Y}}_s{|}^{2}ds+{\int }_t^T{e}^{\mu A_s}{|f(s,Y_s,Z_s,Y_{s+{\delta }_1(s)},Z_{s+\zeta (s)})-f^{\prime }(s,Y_s,Z_s,Y_{s+{\delta }_1(s)},Z_{s+\zeta (s)})|}^{2}ds\hfill \\ \hfill & +\frac{4(M+1)C}{1-{\alpha }_1-{\alpha }_{2}M}{\int }_t^T{e}^{\mu A_s}|{\overline{Y}}_s{|}^{2}ds+\frac{1-{\alpha }_1-{\alpha }_{2}M}{4(M+1)}({\int }_t^T{e}^{\mu A_s}|{\overline{Y}}_s{|}^{2}ds+{\int }_t^T{e}^{\mu A_s}{\parallel {\overline{Z}}_s\parallel }^{2}ds\hfill \\ \hfill & +{\int }_t^T{e}^{\mu A_s}{\mathbb{E}}^{{\mathcal{F}}_s}|{\overline{Y}}_{s+{\delta }_1(s)}{|}^{2}ds+{\int }_t^T{e}^{\mu A_s}{\mathbb{E}}^{{\mathcal{F}}_s}\parallel {\overline{Z}}_{s+\zeta (s)}{\parallel }^{2}ds)+\theta {\int }_t^T{e}^{\mu A_s}{|{\overline{Y}}_s|}^{2}d{K}_s^{\prime }\hfill \\ \hfill & +\frac{1}{\theta }{\int }_t^T{e}^{\mu A_s}{|h(s,Y_s,Y_{s+{\delta }_2(s)})-h^{\prime }(s,Y_s,Y_{s+{\delta }_2(s)})|}^{2}d{K}_s^{\prime }\hfill \\ \hfill & +(2\overline{C}+\beta ){\int }_t^T{e}^{\mu A_s}|{\overline{Y}}_s{|}^{2}d{K}_s^{\prime }+\beta {\int }_t^T{e}^{\mu A_s}{\mathbb{E}}^{{\mathcal{F}}_s}{|{\overline{Y}}_{s+{\delta }_2(s)}|}^{2}d{K}_s^{\prime }\hfill \\ \hfill & +(1+\frac{2({\alpha }_1+{\alpha }_{2}M)}{1-{\alpha }_1-{\alpha }_{2}M}){\int }_t^T{e}^{\mu A_s}{\parallel g(s,Y_s,Z_s,Y_{s+{\delta }_3(s)},Z_{s+\overline{\zeta }(s)})-g^{\prime }(s,Y_s,Z_s,Y_{s+{\delta }_3(s)},Z_{s+\overline{\zeta }(s)})\parallel }^{2}ds\hfill \\ \hfill & +(1+\frac{1-{\alpha }_1-{\alpha }_{2}M}{2({\alpha }_1+{\alpha }_{2}M)})(C({\int }_t^T{e}^{\mu A_s}|{\overline{Y}}_s{|}^{2}ds+{\int }_t^T{e}^{\mu A_s}{\mathbb{E}}^{{\mathcal{F}}_s}|{\overline{Y}}_{s+{\delta }_3(s)}{|}^{2}ds)\hfill \\ \hfill & +{\alpha }_1{\int }_t^T{e}^{\mu A_s}\parallel {\overline{Z}}_s{\parallel }^{2}ds+{\alpha }_2{\int }_t^T{e}^{\mu A_s}{\mathbb{E}}^{{\mathcal{F}}_s}\parallel {\overline{Z}}_{s+\overline{\zeta }(s)}{\parallel }^{2}ds)].\hfill \end{array}$$

Choosing $\theta =\mu -|2\overline{C}+\beta (1+M)|$, we have:

$$\begin{array}{cc}\hfill & \mathbb{E}[e^{\mu A_t}|{\overline{Y}}_t{|}^2+\frac{1-{\alpha }_1-{\alpha }_{2}M}{4}{\int }_t^T{e}^{\mu A_s}\parallel {\overline{Z}}_s{\parallel }^{2}ds]\le \mathbb{E}[e^{\mu K_T}|{\overline{\xi }}_T{|}^2+\frac{1}{\mu }{\int }_t^T{e}^{\mu A_s}|h(s,Y_s,Y_{s+{\delta }_2(s)}){|}^{2}d{\parallel \overline{K}\parallel }_s\hfill \\ \hfill & +{\int }_t^T{e}^{\mu A_s}{|f(s,Y_s,Z_s,Y_{s+{\delta }_1(s)},Z_{s+\zeta (s)})-f^{\prime }(s,Y_s,Z_s,Y_{s+{\delta }_1(s)},Z_{s+\zeta (s)})|}^{2}ds\hfill \\ \hfill & +(\frac{5-{\alpha }_1-{\alpha }_{2}M}{4}+\frac{4(M+1)C}{1-{\alpha }_1-{\alpha }_{2}M}+\frac{1+{\alpha }_1+{\alpha }_{2}M}{2({\alpha }_1+{\alpha }_{2}M)}C(1+M)){\int }_t^T{e}^{\mu A_s}{|{\overline{Y}}_s|}^{2}ds\hfill \\ \hfill & +(\frac{1+{\alpha }_1+{\alpha }_{2}M}{2({\alpha }_1+{\alpha }_{2}M)}C+\frac{1-{\alpha }_1-{\alpha }_{2}M}{4(M+1)})M{\int }_T^{T+K}{e}^{\mu A_s}{|{\overline{\xi }}_s|}^{2}ds\hfill \\ \hfill & +(\frac{1+{\alpha }_1+{\alpha }_{2}M}{2({\alpha }_1+{\alpha }_{2}M)}{\alpha }_2+\frac{1-{\alpha }_1-{\alpha }_{2}M}{4(M+1)})M{\int }_T^{T+K}{e}^{\mu A_s}\parallel {\overline{\eta }}_s{\parallel }^{2}ds+\beta M{\int }_T^{T+K}{e}^{\mu A_s}{|{\overline{\xi }}_s|}^{2}d{K}_s^{\prime }\hfill \\ \hfill & +\frac{1}{\mu -|2\overline{C}+\beta (1+M)|}{\int }_t^T{e}^{\mu A_s}{|h(s,Y_s,Y_{s+{\delta }_2(s)})-h^{\prime }(s,Y_s,Y_{s+{\delta }_2(s)})|}^{2}d{K}_s^{\prime }\hfill \\ \hfill & +\frac{1+{\alpha }_1+{\alpha }_{2}M}{1-{\alpha }_1-{\alpha }_{2}M}{\int }_t^T{e}^{\mu A_s}\parallel g(s,Y_s,Z_s,Y_{s+{\delta }_3(s)},Z_{s+\overline{\zeta }(s)})-g^{\prime }(s,Y_s,Z_s,Y_{s+{\delta }_3(s)},Z_{s+\overline{\zeta }(s)}){\parallel }^{2}ds].\hfill \end{array}$$

From Gronwall’s lemma, we get:

$$\begin{array}{cc}\hfill & \underset{t\in [0,T]}{sup}\mathbb{E}[e^{\mu A_t}|{\overline{Y}}_t{|}^2]+\mathbb{E}[{\int }_0^T{e}^{\mu A_s}\parallel {\overline{Z}}_s{\parallel }^{2}ds]\le \widehat{C}\mathbb{E}[e^{\mu K_T}|{\overline{\xi }}_T{|}^2+{\int }_0^T{e}^{\mu A_s}|h(s,Y_s,Y_{s+{\delta }_2(s)}){|}^{2}d{\parallel \overline{K}\parallel }_s\hfill \\ \hfill & +{\int }_0^T{e}^{\mu A_s}{|f(s,Y_s,Z_s,Y_{s+{\delta }_1(s)},Z_{s+\zeta (s)})-f^{\prime }(s,Y_s,Z_s,Y_{s+{\delta }_1(s)},Z_{s+\zeta (s)})|}^{2}ds\hfill \\ \hfill & +{\int }_T^{T+K}{e}^{\mu A_s}|{\overline{\xi }}_s{|}^{2}ds+{\int }_T^{T+K}{e}^{\mu A_s}\parallel {\overline{\eta }}_s{\parallel }^{2}ds+{\int }_T^{T+K}{e}^{\mu A_s}{|{\overline{\xi }}_s|}^{2}d{K}_s^{\prime }\hfill \\ \hfill & +{\int }_0^T{e}^{\mu A_s}{|h(s,Y_s,Y_{s+{\delta }_2(s)})-h^{\prime }(s,Y_s,Y_{s+{\delta }_2(s)})|}^{2}d{K}_s^{\prime }\hfill \\ \hfill & +{\int }_0^T{e}^{\mu A_s}\parallel g(s,Y_s,Z_s,Y_{s+{\delta }_3(s)},Z_{s+\overline{\zeta }(s)})-g^{\prime }(s,Y_s,Z_s,Y_{s+{\delta }_3(s)},Z_{s+\overline{\zeta }(s)}){\parallel }^{2}ds].\hfill \end{array}$$

Using the Burkholder–Davis–Gundy’s and above inequality, the desired result follows. □

With the help of Propositions 1 and 2, we can establish the following existence and uniqueness theorem in this part.

Theorem 1.

Assume that (A1), (A2), (H1), (H2) and (H3) hold. Then, AGDDSDE (2) admits a unique solution $(Y,Z)\in {\mathcal{S}}^2([0,T+K],{\mathbb{R}}^k)\times {\mathcal{M}}^2(0,T+K,{\mathbb{R}}^{k\times d})$.

Proof. 

The uniqueness is easily given by Proposition 2. We now turn to prove its existence. For $\mu \ge 0$, let $M_{\mu }^2(K)$ represent the set of progressively measurable processes $\{X(t),0\le t\le T+K\}$, which satisfy:

$$\mathbb{E}{\int }_0^{T+K}{e}^{\mu K_t}{|X_t|}^{2}dt+\mathbb{E}{\int }_0^{T+K}{e}^{\mu K_t}d{K}_t<\infty ,$$

and $M_{\mu }^2$ represents the space of progressively measurable processes $\{X(t),0\le t\le T+K\}$ which are such that:

$$\mathbb{E}{\int }_0^{T+K}{e}^{\mu K_t}{|X_t|}^{2}dt<\infty .$$

We define

$${\mathcal{B}}_{\mu }^2={(M_{\mu }^2(K))}^n\times {(M_{\mu }^2)}^{n\times d}.$$

Giving $(U,V)\in {\mathcal{B}}_{\mu }^2$, by Theorem 2.1 in [9], we can define a map $\mathsf{\Phi }$ from ${\mathcal{B}}_{\mu }^2$ to ${\mathcal{B}}_{\mu }^2$ through the equation:

$$\left\{\begin{array}{c}{Y}_t={\xi }_T+{\int }_t^{T}f(s,Y_s,Z_s,U_{s+{\delta }_1(s)},V_{s+\zeta (s)})\mathrm{d}s+{\int }_t^{T}h(s,Y_s,U_{s+{\delta }_2(s)})d{K}_s\hfill \\ +{\int }_t^{T}g(s,Y_s,Z_s,U_{s+{\delta }_3(s)},V_{s+\overline{\zeta }(s)})\mathrm{d}{\overleftarrow{B}}_s-{\int }_t^T{Z}_s\mathrm{d}{W}_s,t\in [0,T],\hfill \\ Y_t={\xi }_t,Z_t={\eta }_t,t\in [T,T+K].\hfill \end{array}\right.$$

In the following, we will use the Banach contraction principle to prove the existence. Let $(U,V),(U^{\prime },V^{\prime })\in {\mathcal{B}}_{\mu }^2$, $(Y,Z)=\mathsf{\Phi }(U,V),(Y^{\prime },Z^{\prime })=\mathsf{\Phi }(U^{\prime },V^{\prime })$,

$$\begin{array}{cc}\hfill & \overline{U}=U-U^{\prime },\overline{V}=V-V^{\prime },\overline{Y}=Y-Y^{\prime },\overline{Z}=Z-Z^{\prime },\hfill \\ \hfill & \overline{f}(s)=f(s,Y_s,Z_s,U_{s+{\delta }_1(s)},V_{s+\zeta (s)})-f(s,Y_s^{\prime },Z_s^{\prime },U_{s+{\delta }_1(s)}^{\prime },V_{s+\zeta (s)}^{\prime }),\hfill \\ \hfill & \overline{g}(s)=g(s,Y_s,Z_s,U_{s+{\delta }_3(s)},V_{s+\overline{\zeta }(s)})-g(s,Y_s^{\prime },Z_s^{\prime },U_{s+{\delta }_3(s)}^{\prime },V_{s+\overline{\zeta }(s)}^{\prime })\hfill \\ \hfill & \overline{h}(s)=h(s,Y_s,U_{s+{\delta }_2(s)})-h(s,Y_s^{\prime },U_{s+{\delta }_2(s)}^{\prime }).\hfill \end{array}$$

Consider the following equations:

$$\left\{\begin{array}{c}{\overline{Y}}_t={\int }_t^T\overline{f}(s)\mathrm{d}s+{\int }_t^T\overline{h}(s)d{K}_s+{\int }_t^T\overline{g}(s)\mathrm{d}{\overleftarrow{B}}_s-{\int }_t^T{\overline{Z}}_s\mathrm{d}{W}_s,t\in [0,T],\hfill \\ Y_t=0,Z_t=0,t\in [T,T+K].\hfill \end{array}\right.$$

For any $\lambda ,\theta >0$, in view of It$\widehat{\mathrm{o}}$’s formula, we have:

$$\begin{array}{cc}\hfill & \mathbb{E}[e^{\lambda t+\mu K_t}|{\overline{Y}}_t{|}^2+\lambda {\int }_t^T{e}^{\lambda s+\mu K_s}|{\overline{Y}}_s{|}^{2}ds+\mu {\int }_t^T{e}^{\lambda s+\mu K_s}|{\overline{Y}}_s{|}^{2}d{K}_s+{\int }_t^T{e}^{\lambda s+\mu K_s}\parallel {\overline{Z}}_s{\parallel }^{2}ds]\hfill \\ \hfill & =2\mathbb{E}{\int }_t^T{e}^{\lambda s+\mu K_s}\langle {\overline{Y}}_s,\overline{f}(s)\rangle ds+2\mathbb{E}{\int }_t^T{e}^{\lambda s+\mu K_s}\langle {\overline{Y}}_s,\overline{h}(s)\rangle d{K}_s+\mathbb{E}{\int }_t^T{e}^{\lambda s+\mu K_s}{\parallel \overline{g}(s)\parallel }^{2}ds\hfill \\ \hfill & \le (\theta +\frac{C}{\theta }+C)\mathbb{E}{\int }_t^T{e}^{\lambda s+\mu K_s}|{\overline{Y}}_s{|}^{2}ds+(\frac{C}{\theta }+{\alpha }_1)\mathbb{E}{\int }_t^T{e}^{\lambda s+\mu K_s}{\parallel {\overline{Z}}_s\parallel }^{2}ds\hfill \\ \hfill & +(\frac{CM}{\theta }+CM)\mathbb{E}{\int }_t^T{e}^{\lambda s+\mu K_s}|{\overline{U}}_s{|}^{2}ds+(\frac{CM}{\theta }+{\alpha }_{2}M)\mathbb{E}{\int }_t^T{e}^{\lambda s+\mu K_s}{\parallel {\overline{V}}_s\parallel }^{2}ds\hfill \\ \hfill & +(2\overline{C}+\beta \theta )\mathbb{E}{\int }_t^T{e}^{\lambda s+\mu K_s}|{\overline{Y}}_s{|}^{2}d{K}_s+\frac{\beta M}{\theta }\mathbb{E}{\int }_t^T{e}^{\lambda s+\mu K_s}{|{\overline{U}}_s|}^{2}d{K}_s\hfill \end{array}$$

Choosing $\theta =16\frac{C(M+1)}{1-{\alpha }_1-{\alpha }_{2}M}+2\beta M,\lambda =\theta +\frac{C}{\theta }+C+2(\frac{CM}{\theta }+CM)+2,\mu =|2\overline{C}+\beta \theta |+2(\frac{CM}{\theta }+CM)+4$, we have

$$\begin{array}{cc}\hfill & \mathbb{E}{\int }_t^T{e}^{\lambda s+\mu K_s}|{\overline{Y}}_s{|}^{2}ds+\mathbb{E}{\int }_t^T{e}^{\lambda s+\mu K_s}|{\overline{Y}}_s{|}^{2}d{K}_s+\mathbb{E}{\int }_t^T(\frac{C}{\theta }+{\alpha }_1)e^{\lambda s+\mu K_s}{\parallel {\overline{Z}}_s\parallel }^{2}ds\hfill \\ \hfill & \le \frac{1}{2}\mathbb{E}{\int }_t^T{e}^{\lambda s+\mu K_s}|{\overline{Y}}_s{|}^{2}ds+\frac{1}{2}\mathbb{E}{\int }_t^T{e}^{\lambda s+\mu K_s}|{\overline{Y}}_s{|}^{2}d{K}_s+\frac{1}{2}\mathbb{E}{\int }_t^T(\frac{C}{\theta }+{\alpha }_1)e^{\lambda s+\mu K_s}{\parallel {\overline{Z}}_s\parallel }^{2}ds.\hfill \end{array}$$

Thus, $\mathsf{\Phi }$ is a strict contraction on ${\mathcal{B}}_{\mu }^2$ equipped with the norm

$$\begin{array}{c}\hfill {\parallel (Y,Z)\parallel }_{\lambda ,\mu }^2=\mathbb{E}[{\int }_0^T{e}^{\lambda s+\mu K_s}|Y_s{|}^{2}ds+{\int }_0^T{e}^{\lambda s+\mu K_s}|Y_s{|}^{2}d{K}_s+{\int }_0^T(\frac{C}{\theta }+{\alpha }_1)e^{\lambda s+\mu K_s}\parallel Z_s{\parallel }^{2}ds].\end{array}$$

The proof of existence is complete. □

4. Comparison Theorems

In this section, we consider one dimensional AGBDSDEs, that is, $k=1$. Let us first give a comparison theorem of GBDSDEs, which will play a key role in what follows. Assume that, for $i=1,2$, ${\xi }^i\in L^2({\mathcal{F}}_T;\mathbb{R})$ and $f^i(t,y,z):[0,T]\times \mathsf{\Omega }\times \mathbb{R}\times {\mathbb{R}}^d\to \mathbb{R}$ satisfies (H1) and (H3). Then, according to Theorem 2.1 in [9], the following GBDSDE,

$$Y_t^i={\xi }^i+{\int }_t^T{f}^i(s,Y_s^i,Z_s^i)\mathrm{d}s+{\int }_t^{T}h(s,Y_s)d{K}_s+{\int }_t^T{g}^i(s,Y_s^i,Z_s^i)\mathrm{d}{\overleftarrow{B}}_s-{\int }_t^T{Z}_s^i\mathrm{d}{W}_s,t\in [0,T],$$

(3)

admits a unique solution $(Y_{·}^i,Z_{·}^i)\in {\mathcal{S}}^2([0,T];{\mathbb{R}}^n)\times {\mathcal{M}}^2(0,T;{\mathbb{R}}^{n\times d})$ for $i=1,2$. We can assert the following comparison theorem, which generalizes the Theorem 2.2 in [11].

Lemma 1.

Let $(Y^1,Z^1)$ and $(Y^2,Z^2)$ be solutions of GBDSDEs (3) respectively. We suppose that (1) ${\xi }_t^1⩾{\xi }_t^2,\mathit{a.s.}$; (2) $f^1(t,Y_t^1,Z_t^1)⩾f^2(t,Y_t^1,Y_t^1)\mathit{or}{f}^1(t,Y_t^2,Z_t^2)⩾f^2(t,Y_t^2,Y_t^2),\mathit{a.s.},\mathit{a.e.}t\in [0,T]$; (3) $h^1(t,Y_t^1)\ge h^2(t,Y_t^1)\mathit{or}{h}^1(t,Y_t^2)\ge h^2(t,Y_t^2),\mathit{a.s.},\mathit{a.e.}t\in [0,T]$; (4) $g^1$$(t,Y_t^1,Z_t^1)=g^2(t,Y_t^1,Y_t^1)\mathit{or}{g}^1(t,Y_t^2,Z_t^2)=g^2(t,Y_t^2,Y_t^2),\mathit{a.s.},\mathit{a.e.}t\in [0,T]$. Then, $Y_t^1⩾Y_t^2,\mathit{a.s.},\mathit{for}\mathit{all}t\in [0,T]$.

Proof. 

Without loss of generality, we assume that ${\xi }^1⩾{\xi }^2$, a.s., and $f^1(t,Y_t^1,Z_t^1)⩾f^2(t,Y_t^1,Z_t^1),h^1(t,Y_t^1)\ge h^2(t,Y_t^1)$ a.s., for all $t\in [0,T]$. Denote

$$\begin{array}{cc}\hfill & {\widehat{Y}}_t=Y_t^2-Y_t^1,{\widehat{Z}}_t=Z_t^2-Z_t^1,\widehat{\xi }={\xi }^2-{\xi }^1.\hfill \\ \hfill & {\widehat{f}}_t=f^2(t,Y_t^2,Z_t^2)-f^1(t,Y_t^1,Z_t^1),{\widehat{g}}_t=g^2(t,Y_t^2,Z_t^2)-g^1(t,Y_t^1,Z_t^1),{\widehat{h}}_t=h^2(t,Y_t^2)-h^1(t,Y_t^1).\hfill \end{array}$$

Using It$\widehat{\mathrm{o}}$-Meyer’s formula and ${\xi }^1⩾{\xi }^2$, a.s., we have:

$$\mathbb{E}|{\widehat{Y}}_t^{+}{|}^2+\mathbb{E}{\int }_t^T{1}_{\{{\widehat{Y}}_s\ge 0\}}|{\widehat{Z}}_s{|}^{2}ds=2\mathbb{E}{\int }_t^T{\widehat{Y}}_s^{+}{\widehat{f}}_{s}ds+2\mathbb{E}{\int }_t^T{\widehat{Y}}_s^{+}{\widehat{h}}_{s}d{K}_s+\mathbb{E}{\int }_t^T{1}_{\{{\widehat{Y}}_s\ge 0\}}{|{\widehat{g}}_s|}^{2}ds.$$

In view of (H3), Young’s inequality $2ab\le \frac{1}{\theta }{a}^2+\theta b^2$ and Jensen’s inequality, for any $\theta >0$, we have:

$$\begin{array}{cc}\hfill 2\mathbb{E}{\int }_t^T{\widehat{Y}}_s^{+}{\widehat{f}}_{s}ds\le & 2\mathbb{E}{\int }_t^T{\widehat{Y}}_s^{+}(f^2(s,Y_s^2,Z_s^2)-f^2(t,Y_s^1,Z_s^1))ds\hfill \\ \hfill \le & \frac{2C}{1-{\alpha }_1}\mathbb{E}{\int }_t^T|{\widehat{Y}}_s^{+}{|}^{2}ds+\frac{1-{\alpha }_1}{2C}{\int }_t^T{1}_{\{{\widehat{Y}}_s\ge 0\}}{|f^2(s,Y_s^2,Z_s^2)-f^2(t,Y_s^1,Z_s^1)|}^{2}ds\hfill \\ \hfill \le & (\frac{2C}{1-{\alpha }_1}+\frac{1-{\alpha }_1}{2})\mathbb{E}{\int }_t^T|{\widehat{Y}}_s^{+}{|}^{2}ds+\frac{1-{\alpha }_1}{2}\mathbb{E}{\int }_t^T{1}_{\{{\widehat{Y}}_s\ge 0\}}{|{\widehat{Z}}_s|}^{2}ds,\hfill \end{array}$$

$$\begin{array}{c}\hfill 2\mathbb{E}{\int }_t^T{\widehat{Y}}_s^{+}{\widehat{h}}_{s}d{K}_s\le 2\mathbb{E}{\int }_t^T{\widehat{Y}}_s^{+}(h^2(s,Y_s^2)-h^2(t,Y_s^1))ds\le 2\overline{C}\mathbb{E}{\int }_t^T{1}_{\{{\widehat{Y}}_s\ge 0\}}{|{\widehat{Y}}_s|}^{2}ds,\end{array}$$

and

$$\begin{array}{cc}\hfill \mathbb{E}{\int }_t^T{1}_{\{{\widehat{Y}}_s\ge 0\}}{|{\widehat{g}}_s|}^{2}ds=& \mathbb{E}{\int }_t^T{1}_{\{{\widehat{Y}}_s\ge 0\}}{|g^1(s,Y_s^2,Z_s^2)-g^1(t,Y_s^1,Z_s^1)|}^{2}ds\hfill \\ \hfill \le & C\mathbb{E}{\int }_t^T|{\widehat{Y}}_s^{+}{|}^{2}ds+{\alpha }_1\mathbb{E}{\int }_t^T{1}_{\{{\widehat{Y}}_s\ge 0\}}{|{\widehat{Z}}_s|}^{2}ds.\hfill \end{array}$$

Then, thanks to the above inequalities, we obtain:

$$\mathbb{E}|{\widehat{Y}}_t^{+}{|}^2\le (C+\frac{2C}{1-{\alpha }_1}+\frac{1-{\alpha }_1}{2})\mathbb{E}{\int }_t^T{|{\widehat{Y}}_s^{+}|}^{2}ds.$$

From the Gronwall’s inequality, we can obtain:

$$\mathbb{E}|{\widehat{Y}}_t^{+}{|}^2=0,\mathrm{for}\mathrm{all}t\in [0,T].$$

Hence

$$Y_t^1⩾Y_t^2,\mathit{a.s.},\mathrm{for}\mathrm{all}t\in [0,T].$$

□

Now let us turn to the study of the comparison theorem for anticipated GBDSDEs. For $i=1,2$, we first consider the following anticipated BDSDE:

$$\left\{\begin{array}{c}{Y}_t^i={\xi }_T^i+{\int }_t^T{f}^i(s,Y_s^i,Z_s^i,Y_{s+{\delta }_1^i(s)}^i,Z_{s+{\zeta }^i(s)}^i)\mathrm{d}s+{\int }_t^T{h}^i(s,Y_s^i,Y_{s+{\delta }_2^i(s)}^i)d{K}_s\hfill \\ +{\int }_t^T{g}^i(s,Y_s^i,Z_s^i,Y_{s+{\delta }_3^i(s)}^i,Z_{s+{\overline{\zeta }}^i(s)}^i)\mathrm{d}{\overleftarrow{B}}_s-{\int }_t^T{Z}_s^i\mathrm{d}{W}_s,t\in [0,T],\hfill \\ Y_t^i={\xi }_t^i,Z_t^i={\eta }_t^i,t\in [T,T+K].\hfill \end{array}\right.$$

(4)

Let us assume that ${\delta }^i(·),{\overline{\delta }}^i(·),{\zeta }^i(·),{\overline{\zeta }}^i(·)$ satisfy (A1) and (A2), ${\xi }_{·}^i\in {\mathcal{S}}^2([T,T+K];\mathbb{R})$, ${\eta }_{·}^i\in {\mathcal{M}}^2(T,T+K;{\mathbb{R}}^d)$, and $(f^i,g^i)$ satisfies (H1) and (H3). Then, by Theorem 1, anticipated GBDSDE (4) admits a unique solution $(Y_{·}^i,Z_{·}^i)\in {\mathcal{S}}^2([0,T+K];\mathbb{R})\times {\mathcal{M}}^2(0,T+K;{\mathbb{R}}^d)$ for $i=1,2$.

Theorem 2.

Let $(Y^1,Z^1)$ and $(Y^2,Z^2)$ be solutions of AGBDSDEs (4) respectively. We suppose that (1) ${\xi }_t^1⩾{\xi }_t^2,\mathit{a.s.},\mathit{for}\mathit{all}t\in [T,T+K]$; (2) $f^1(t,Y_t^1,Z_t^1,Y_{t+{\delta }_1^1(t)}^1,Z_{t+{\zeta }^1(t)}^1)⩾f^2(t,Y_t^1,Z_t^1,Y_{t+{\delta }_1^2(t)}^2,Z_{t+{\zeta }^2(t)}^2)\mathit{or}{f}^1(t,Y_t^2,Z_t^2,Y_{t+{\delta }_1^1(t)}^1,Z_{t+{\zeta }^1(t)}^1)⩾f^2(t,Y_t^2,Y_t^2,Y_{t+{\delta }_1^2(t)}^2,$$Z_{t+{\zeta }^2(t)}^2),a.s.,a.e.t\in [0,T]$; (3) $h^1(t,Y_t^1,Y_{t+{\delta }_2^1(t)}^1)⩾h^2(t,Y_t^1,Y_{t+{\delta }_2^2(t)}^2)\mathit{or}{f}^1(t,Y_t^2,Y_{t+{\delta }_2^1(t)}^1)⩾f^2(t,Y_t^2,Y_{t+{\delta }_2^2(t)}^2),\mathit{a.s.},\mathit{a.e.}t\in [0,T]$; (4) $g^1(t,Y_t^1,Z_t^1,Y_{t+{\overline{\delta }}^1(t)}^1,Z_{t+{\overline{\zeta }}^1(t)}^1)=g^2(t,Y_t^1,Y_t^1,Y_{t+{\overline{\delta }}^2(t)}^2,$ $Z_{t+{\overline{\zeta }}^2(t)}^2)$ or $g^1(t,Y_t^2,Z_t^2,Y_{t+{\overline{\delta }}^1(t)}^1,Z_{t+{\overline{\zeta }}^1(t)}^1)=g^2(t,Y_t^2,Y_t^2,Y_{t+{\overline{\delta }}^1(t)}^1,Z_{t+{\overline{\zeta }}^1(t)}^1),\mathit{a.s.},\mathit{a.e.}t\in [0,T]$. Then $Y_t^1⩾Y_t^2,\mathit{a.s.},\mathit{for}\mathit{all}t\in [0,T+K]$.

Proof. 

For $i=1,2$, denote

$$\begin{array}{cc}\hfill & F^i(t,y,z)=f^i(t,y,z,Y_{s+{\delta }_1^i(s)}^i,Z_{s+{\zeta }^i(s)}^i),G^i(t,y,z)=g^i(t,y,z,Y_{s+{\delta }_3^i(s)}^i,Z_{s+{\overline{\zeta }}^i(s)}^i),\hfill \\ \hfill & H^i(t,y)=h^i(s,y,Y_{s+{\delta }_2^i(s)}^i),\hfill \end{array}$$

then $(Y^i,Z^i)$ is the unique solution of the following GBDSDE,

$$Y_t^i={\xi }_T^i+{\int }_t^T{F}^i(s,Y_s^i,Z_s^i)\mathrm{d}s+{\int }_t^T{H}^i(s,Y_s^i)d{K}_s+{\int }_t^T{G}^i(s,Y_s^i,Z_s^i)\mathrm{d}{\overleftarrow{B}}_s-{\int }_t^T{Z}_s^i\mathrm{d}{W}_s,t\in [0,T].$$

According to Lemma 1, we can get

$$Y_t^1⩾Y_t^2,\mathit{a.s.},\mathit{for}\mathit{all}t\in [0,T],$$

which implies

$$Y_t^1⩾Y_t^2,\mathit{a.s.},\mathit{for}\mathit{all}t\in [0,T+K].$$

□

Let us give an example.

Example 1.

Let $f^1(t,y,z,\theta (r),\vartheta (\overline{r}))=|y|+|z|+{\mathbb{E}}^{{\mathcal{F}}_t}[|\theta (r)|+1]+{\mathbb{E}}^{{\mathcal{F}}_t}[|cos\vartheta (\overline{r})|]$, $f^2(t,y,z,\theta (r),\vartheta (\overline{r}))=y-|z|+{\mathbb{E}}^{{\mathcal{F}}_t}[sin\theta (r)]$, $g^1(t,y,z,\theta (r),\vartheta (\overline{r}))=g^2(t,y,z,\theta (r),\vartheta (\overline{r}))=y+|z|$, $h^1(t,y,\theta (r))=|y|+{\mathbb{E}}^{{\mathcal{F}}_t}[|arctan\theta (r)|]$, $h^2(t,y,\theta (r))=y-1$. Then by Theorem 2, we can obtain $Y_t^1⩾Y_t^2,a.s.,\mathrm{for}\mathrm{all}t\in [0,T+K]$ as long as the assumption (1) of Theorem 2 holds.

Next, we turn to the study of another comparison theorem for anticipated GBDSDEs. For $i=1,2$, we consider the following anticipated GBDSDE:

$$\left\{\begin{array}{c}{Y}_t^i={\xi }_T^i+{\int }_t^T{f}^i(s,Y_s^i,Z_s^i,Y_{s+{\delta }_1(s)}^i,Z_{s+{\zeta }^i(s)}^i)\mathrm{d}s+{\int }_t^T{h}^i(s,Y_s^i,Y_{s+{\delta }_2^i(s)}^i)d{K}_s\hfill \\ +{\int }_t^T{g}^i(s,Y_s^i,Z_s^i,Y_{s+{\overline{\delta }}_3^i(s)}^i,Z_{s+{\overline{\zeta }}^i(s)}^i)\mathrm{d}{\overleftarrow{B}}_s-{\int }_t^T{Z}_s^i\mathrm{d}{W}_s,t\in [0,T],\hfill \\ Y_t^i={\xi }_t^i,Z_t^i={\eta }_t^i,t\in [T,T+K].\hfill \end{array}\right.$$

(5)

We always assume that $(\delta (·),{\zeta }^i(·),{\overline{\delta }}^i(·),{\overline{\zeta }}^i(·))$ satisfy (A1) and (A2), ${\xi }_{·}^i\in {\mathcal{S}}^2([T,T+K];\mathbb{R})$, ${\eta }_{·}^i\in {\mathcal{M}}^2(T,T+K;{\mathbb{R}}^d)$ and $(f^i,g^i)$ satisfy (H1) and (H3). Then, by Theorem 1, anticipated GBDSDE (5) admits a unique solution $(Y_{·}^i,Z_{·}^i)\in {\mathcal{S}}^2([0,T+K];\mathbb{R})\times {\mathcal{M}}^2(0,T+K;{\mathbb{R}}^d)$ for $i=1,2$.

Theorem 3.

Let $(Y^1,Z^1)$ and $(Y^2,Z^2)$ be solutions of AGBDSDEs (5) respectively. We suppose that (1) ${\xi }_t^1⩾{\xi }_t^2,\mathit{a.s.},\mathit{for}\mathit{all}t\in [T,T+K]$; (2) $f^1(t,Y_t^1,Z_t^1,Y_{t+{\delta }_1(t)}^1,Z_{t+{\zeta }^1(t)}^1)⩾f^2(t,Y_t^1,Z_t^1,Y_{t+{\delta }_1(t)}^1,Z_{t+{\zeta }^2(t)}^2),\mathit{a.s.},\mathit{a.e.}t\in [0,T]$; (3) $h^1(t,Y_t^1,Y_{t+{\delta }_2^1(t)}^1)⩾h^2(t,Y_t^1,Y_{t+{\delta }_2^2(t)}^2),\mathit{a.s.},\mathit{a.e.}t\in [0,T]$; (4) for any $(t,y,z)\in [0,T]\times R\times R^d,\gamma \in L^2({\mathcal{F}}_s;{\mathbb{R}}^d)$ and $s\in [t,T+K],f^2(t,y,z,·,\gamma )$ is increasing, that is, $f^2(t,y,z,y_2,\gamma )\ge f^2(t,y,z,y_1,\gamma )$, if $y_2\ge y_1$ with $y_1,y_2\in R$; (5) $g^1(t,Y_t^1,Z_t^1,Y_{t+{\overline{\delta }}^1(t)}^1,Z_{t+{\overline{\zeta }}^1(t)}^1)=g^2(t,Y_t^1,Y_t^1,Y_{t+{\overline{\delta }}^2(t)}^2,Z_{t+{\overline{\zeta }}^2(t)}^2),\mathit{a.s.},\mathit{a.e.}t\in [0,T]$. Then, $Y_t^1⩾Y_t^2,\mathit{a.s.},\mathit{for}\mathit{all}t\in [0,T+K]$.

Proof. 

For $i=1,2$, denote:

$$\begin{array}{cc}\hfill & F^i(t,y,z)=f^i(t,y,z,Y_{t+{\delta }_1(t)}^i,Z_{t+{\zeta }^i(t)}^i),G^i(t,y,z)=g^i(t,y,z,Y_{t+{\overline{\delta }}_3^i(t)}^i,Z_{t+{\overline{\zeta }}^i(t)}^i),\hfill \\ \hfill & H^i(t,y)=h^i(s,y,Y_{s+{\delta }_2^i(s)}^i),\hfill \end{array}$$

then $(Y^i,Z^i)$ is the unique solution of the following GBDSDEs,

$$\left\{\begin{array}{cc}\hfill Y_t^i=& {\xi }_T^i+{\int }_t^T{F}^i(s,Y_s^i,Z_s^i)\mathrm{d}s+{\int }_t^T{H}^i(s,Y_s^i)d{K}_s+{\int }_t^T{G}^i(s,Y_s^i,Z_s^i)\mathrm{d}{\overleftarrow{B}}_s\hfill \\ \hfill & -{\int }_t^T{Z}_s^i\mathrm{d}{W}_s,t\in [0,T],\hfill \\ \hfill Y_t^i=& {\xi }_t^i,Z_t^i={\eta }_t^i,t\in [T,T+K].\hfill \end{array}\right.$$

Let ${\overline{f}}^2(t,y,z)=f^2(t,y,z,Y_{t+{\delta }_1(t)}^1,Z_{t+{\zeta }^2(t)}^2)$, then the following GBDSDE admits a unique solution $(Y^3,Z^3)$,

$$\left\{\begin{array}{cc}\hfill Y_t^3=& {\xi }_T^2+{\int }_t^T{\overline{f}}^2(s,Y_s^3,Z_s^3)\mathrm{d}s+{\int }_t^T{H}^2(s,Y_s^3)d{K}_s+{\int }_t^T{G}^2(s,Y_s^3,Z_s^3)\mathrm{d}{\overleftarrow{B}}_s\hfill \\ \hfill & -{\int }_t^T{Z}_s^3\mathrm{d}{W}_s,t\in [0,T],\hfill \\ \hfill Y_t^3=& {\xi }_t^2,Z_t^3={\eta }_t^2,t\in [T,T+K].\hfill \end{array}\right.$$

According to the assumptions (1), (2), (4) and Lemma 1, we can get $Y_t^1⩾Y_t^3,\mathit{a.s.},\mathrm{for}\mathrm{all}t\in [0,T],$ which implies $Y_t^1⩾Y_t^3,\mathit{a.s.},\mathrm{for}\mathrm{all}t\in [0,T+K].$ Let $(Y^4,Z^4)$ be the unique solution for the following GBDSDE:

$$\left\{\begin{array}{cc}\hfill Y_t^4=& {\xi }_T^2+{\int }_t^T{f}^2(s,Y_s^4,Z_s^4,Y_{t+{\delta }_1(t)}^3,Z_{t+{\zeta }^2(t)}^2)\mathrm{d}s+{\int }_t^T{H}^2(s,Y_s^4)d{K}_s+{\int }_t^T{G}^2(s,Y_s^4,Z_s^4)\mathrm{d}{\overleftarrow{B}}_s\hfill \\ \hfill & -{\int }_t^T{Z}_s^4\mathrm{d}{W}_s,t\in [0,T],\hfill \\ \hfill Y_t^4=& {\xi }_t^2,Z_t^4={\eta }_t^2,t\in [T,T+K].\hfill \end{array}\right.$$

From Lemma 1 and assumptions (3), we can get $Y_t^3⩾Y_t^4,\mathit{a.s.},\mathrm{for}\mathrm{all}t\in [0,T],$ which implies $Y_t^3⩾Y_t^4,\mathit{a.s.},\mathrm{for}\mathrm{all}t\in [0,T+K].$ For $j\ge 5$, define:

$$\left\{\begin{array}{cc}\hfill Y_t^j=& {\xi }_T^2+{\int }_t^T{f}^2(s,Y_s^j,Z_s^j,Y_{t+{\delta }_1(t)}^{j-1},Z_{t+{\zeta }^2(t)}^2)\mathrm{d}s+{\int }_t^T{H}^2(s,Y_s^j)d{K}_s+{\int }_t^T{G}^2(s,Y_s^j,Z_s^j)\mathrm{d}{\overleftarrow{B}}_s\hfill \\ \hfill & -{\int }_t^T{Z}_s^j\mathrm{d}{W}_s,t\in [0,T],\hfill \\ \hfill Y_t^j=& {\xi }_t^2,Z_t^j={\eta }_t^2,t\in [T,T+K].\hfill \end{array}\right.$$

According to Lemma 1 and by induction, we can get $Y_t^4⩾Y_t^5\ge \cdots \ge Y_t^{j-1}\ge Y_t^j,\mathit{a.s.},\mathrm{for}\mathrm{all}$ $t\in [0,T+K];$ hence, for all $j\ge 3$

$$Y_t^1⩾Y_t^j,\mathit{a.s.},\mathrm{for}\mathrm{all}t\in [0,T+K].$$

(6)

Set

$$\begin{array}{cc}\hfill & {\widehat{Y}}_t^j=Y_t^j-Y_t^{j-1},{\widehat{Z}}_t^j=Z_t^j-Z_t^{j-1},\hfill \\ \hfill & {\widehat{F}}_t^j=f^2(t,Y_t^j,Z_t^j,Y_{t+{\delta }_1(t)}^{j-1},Z_{t+{\zeta }^2(t)}^2)-f^2(t,Y_t^{j-1},Z_t^{j-1},Y_{t+{\delta }_1(t)}^{j-2},Z_{t+{\zeta }^2(t)}^2),\hfill \\ \hfill & {\widehat{G}}_t^j=G^2(t,Y_t^j,Z_t^j)-G^2(t,Y_t^{j-1},Z_t^{j-1}),{\widehat{H}}_t^j=H^2(t,Y_t^j)-H^2(t,Y_t^{j-1}).\hfill \end{array}$$

Then for $j\ge 5$, $({\widehat{Y}}^j,{\widehat{Z}}^j)$ satisfies

$$\left\{\begin{array}{c}{\widehat{Y}}_t^j={\int }_t^T{\widehat{F}}_s^j\mathrm{d}s+{\int }_t^T{\widehat{H}}_s^{j}d{K}_s+{\int }_t^T{\widehat{G}}_s^j\mathrm{d}{\overleftarrow{B}}_s-{\int }_t^T{\widehat{Z}}_s^j\mathrm{d}{W}_s,t\in [0,T],\hfill \\ {\widehat{Y}}_t^j=0,{\widehat{Z}}_t^j=0,t\in [T,T+K].\hfill \end{array}\right.$$

For any $\lambda ,\mu ,\theta >0$, apply It$\widehat{\mathrm{o}}$’s formula to $e^{\lambda t+\mu K_t}{|{\widehat{Y}}_t^j|}^2$, in view of (H3), Young’s inequality, Jensen’s inequality, we have:

$$\begin{array}{cc}\hfill & \mathbb{E}[e^{\lambda t+\mu K_t}|{\widehat{Y}}_t^j{|}^2+\lambda {\int }_t^T{e}^{\lambda s+\mu K_s}|{\widehat{Y}}_s^j{|}^{2}ds+\mu {\int }_t^T{e}^{\lambda s+\mu K_s}|{\widehat{Y}}_s^j{|}^{2}d{K}_s+{\int }_t^T{e}^{\lambda s+\mu K_s}|{\widehat{Z}}_s^j{|}^{2}ds]\hfill \\ \hfill & =2\mathbb{E}{\int }_t^T{e}^{\lambda s+\mu K_s}{\widehat{Y}}_s^j{\widehat{F}}_s^{j}ds+2\mathbb{E}{\int }_t^T{e}^{\lambda s+\mu K_s}{\widehat{Y}}_s^j{\widehat{H}}_s^{j}d{K}_s+\mathbb{E}{\int }_t^T{e}^{\lambda s+\mu K_s}{|{\widehat{G}}_s^j|}^{2}ds\hfill \\ \hfill & \le (\theta +\frac{C}{\theta }+C)\mathbb{E}{\int }_t^T{e}^{\lambda s+\mu K_s}|{\overline{Y}}_s{|}^{2}ds+(\frac{C}{\theta }+{\alpha }_1)\mathbb{E}{\int }_t^T{e}^{\lambda s+\mu K_s}{|{\overline{Z}}_s|}^{2}ds\hfill \\ \hfill & +\frac{CM}{\theta }\mathbb{E}{\int }_t^T{e}^{\lambda s+\mu K_s}|{\widehat{Y}}_s^{j-1}{|}^{2}ds+2\overline{C}\mathbb{E}{\int }_t^T{e}^{\lambda s+\mu K_s}{|{\widehat{Y}}_s^j|}^{2}d{K}_s.\hfill \end{array}$$

Choosing $\theta =\frac{2C}{1-{\alpha }_1}+2CM$, $\lambda =\theta +\frac{C}{\theta }+C+1$, $\mu =1$, we have:

$$\begin{array}{cc}\hfill & \mathbb{E}[{\int }_t^T{e}^{\lambda s+\mu K_s}|{\widehat{Y}}_s^j{|}^{2}ds+{\int }_t^T{e}^{\lambda s+\mu K_s}|{\widehat{Y}}_s^j{|}^{2}d{K}_s+{\int }_t^T(1-\frac{C}{\theta }-{\alpha }_1)e^{\lambda s+\mu K_s}|{\widehat{Z}}_s^j{|}^{2}ds]\hfill \\ \hfill & \le \frac{1}{2}\mathbb{E}[{\int }_t^T{e}^{\lambda s+\mu K_s}|{\widehat{Y}}_s^{j-1}{|}^{2}ds+{\int }_t^T{e}^{\lambda s+\mu K_s}|{\widehat{Y}}_s^{j-1}{|}^{2}d{K}_s+{\int }_t^T(1-\frac{C}{\theta }-{\alpha }_1)e^{\lambda s+\mu K_s}|{\widehat{Z}}_s^{j-1}{|}^{2}ds].\hfill \end{array}$$

Thus, we have proved that $(Y_{·}^k,Z_{·}^k)$ is a Cauchy sequence in ${\mathcal{B}}_{\mu }^2$ with the norm,

$$\begin{array}{c}\hfill {\parallel (Y,Z)\parallel }_{\lambda ,\mu }^2=\mathbb{E}[{\int }_0^T{e}^{\lambda s+\mu K_s}|Y_s{|}^{2}ds+{\int }_0^T{e}^{\lambda s+\mu K_s}|Y_s{|}^{2}d{K}_s+{\int }_0^T(1-\frac{C}{\theta }-{\alpha }_1)e^{\lambda s+\mu K_s}|Z_s{|}^{2}ds],\end{array}$$

so it is also a Cauchy sequence in $M^2(0,T+K;\mathbb{R})\times M^2(0,T+K;{\mathbb{R}}^d)$. Therefore, there exists $(Y_{·},Z_{·})\in M^2(0,T+K;\mathbb{R})\times M^2(0,T+K;{\mathbb{R}}^d)$ such that $Y_t={\xi }_t^2$, $Z_t={\eta }_t^2$ for $T⩽t⩽T+K$ and

$$\mathbb{E}[{\int }_0^T|Y_t^j-Y_t{|}^2\mathrm{d}{K}_t]+\mathbb{E}[{\int }_0^T|Y_t^j-Y_t{|}^2\mathrm{d}t]+\mathbb{E}[{\int }_0^T|Z_t^j-Z_t{|}^2\mathrm{d}t]\to 0\mathrm{as}j\to \infty .$$

Hence, it is easy to check that, as $j\to \infty$,

$$\mathbb{E}{\int }_t^T{|f^2(s,Y_s^j,Z_s^j,Y_{s+{\delta }_1(s)}^{j-1},Z_{s+{\zeta }^2(s)}^2)-f^2(s,Y_s,Z_s,Y_{s+\delta (s)},Z_{s+{\zeta }^2(s)}^2)|}^2\mathrm{d}s\to 0,$$

$$\mathbb{E}{\int }_t^T{|g^2(s,Y_s^j,Z_s^j,Y_{s+{\overline{\delta }}^2(s)}^2,Z_{s+{\overline{\zeta }}^2(s)}^2)-g^2(s,Y_s,Z_s,Y_{s+{\overline{\delta }}^2(s)}^2,Z_{s+{\overline{\xi }}^2(s)}^2)|}^2\mathrm{d}s\to 0,$$

$$\mathbb{E}[|{\int }_t^T{Z}_s^j·\mathrm{d}{W}_s-{\int }_t^T{Z}_s·\mathrm{d}{W}_s{|}^2]\to 0.$$

Furthermore, we can prove $Y_{·}\in S^2(0,T+K;\mathbb{R})$ through Burkholder–Davis–Gundy inequality. So, we conclude that $(Y_{·},Z_{·})$ solves the following AGBDSDE:

$$\left\{\begin{array}{c}{Y}_t={\xi }_T^2+{\int }_t^T{f}^2(s,Y_s,Z_s,Y_{s+{\delta }_1(s)},Z_{s+{\zeta }^2(s)}^2)\mathrm{d}s+{\int }_t^T{h}^2(s,Y_s,Y_{s+{\delta }_2^2(s)}^2)d{K}_s\hfill \\ +{\int }_t^T{g}^2(s,Y_s,Z_s,Y_{s+{\overline{\delta }}_3^2(s)}^2,Z_{s+{\overline{\zeta }}^2(s)}^2)\mathrm{d}{\overleftarrow{B}}_s-{\int }_t^T{Z}_s\mathrm{d}{W}_s,t\in [0,T],\hfill \\ \hfill & Y_t={\xi }_t^2,Z_t={\eta }_t^2,t\in [T,T+K].\hfill \end{array}\right.$$

Then the uniqueness part of Theorem 1 shows that $Y_t=Y_t^2,\mathit{a.s.},\mathrm{for}\mathrm{all}t\in [0,T+K]$. Finally, letting $j\to \infty$ in (6) yields $Y_t^1\ge Y_t^2,\mathit{a.s.},\mathrm{for}\mathrm{all}t\in [0,T+K]$. □

Let us give an example.

Example 2.

Let $f^1(t,y,z,\theta (r),\vartheta (\overline{r}))=|y|+|z|+{\mathbb{E}}^{{\mathcal{F}}_t}[|\theta (r)|]+{\mathbb{E}}^{{\mathcal{F}}_t}[|cos\vartheta (\overline{r})|]$, $f^2(t,y,z,\theta (r),\vartheta (\overline{r}))=y+|z|+{\mathbb{E}}^{{\mathcal{F}}_t}\left[\theta (r)\right]$, $g^1(t,y,z,\theta (r),\vartheta (\overline{r}))=g^2(t,y,z,\theta (r),\vartheta (\overline{r}))=\arctan y+|z|$, $h^1(t,y,\theta (r))=|y|+{\mathbb{E}}^{{\mathcal{F}}_t}\left[\arctan \theta (r)\right]$, $h^2(t,y,\theta (r))=y-\frac{\pi }{2}$. Then by Theorem 3, we can derive $Y_t^1⩾Y_t^2,a.s.,\mathrm{for}\mathrm{all}t\in [0,T+K]$ as long as the assumption (1) of Theorem 3 holds.

5. Conclusions

In this paper, we explore a class of anticipated AGBDSDEs. We proved the existence and uniqueness of the solutions of this kind of AGBDSDE. Moreover, two comparison theorems are also proved. In the coming future papers, we will focus on studying this topic and pay more attention to the applications of such equations.