Anticipated Generalized Backward Doubly Stochastic Differential Equations

: 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.


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: 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: in which the equations not only involve a standard (forward) stochastic Itô integral dW t but also a symmetric backward stochastic Itô integral d ← − 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: where δ(·) and ζ(·) are given R + -valued continuous functions and for φ(·) = δ(·), ζ(·) such that: (A1) there exists a constant K ≥ 0 such that, for all t ∈ [0, T], t + φ(t) T + K; (A2) there exists a constant M 0 such that, for all t ∈ [0, T] and for all non-negative and integrable g(·), where dK s is a measure generated by K on [0, T + K].
In this paper, we are concerned with the following anticipated GBDSDEs: where δ i (·), i = 1, 2, 3, ζ(·) and ζ(·) are given 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.

Preliminaries
Throughout the paper, we use |x| and A = Tr(AA * ) to denote the norm of a vector x ∈ R k and a matrix A ∈ k × d, respectively, where A * is the transpose of A. Let {W t ; 0 ≤ t ≤ T} and {B t ; 0 ≤ t ≤ T} be two mutually independent standard Browning motions, with values respectively in R d and R l on a complete probability space (Ω, F , P). Let T > 0, K ≥ 0 be fixed constants. Let N denote the class of P-null sets of F . For any t ∈ [0, T + K], we define: Let K t be a continuous, increasing and F t -adapted process on [0, T + K] with K 0 = 0. We will use the following notations: for any n ∈ N, We make the following assumptions about (ξ, f , g, h):

Hypothesis 3 (H3). For any s
where ψ i , i = 1, 2, 3 are three adapted processes with values in [1, +∞) 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.

Existence and Uniqueness Theorem
In order to obtain the existence and uniqueness result, we need the following two priori estimates. where 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 Itô's formula, we have: According to the assumptions (H2), (H3) and Yong's inequality, for any θ > 0, we get: Consequently, we have: T t e µs+λK s |Y s | 2 ds + ( Using the Burkholder-Davis-Gundy's and above inequality, the desired result follows.
where A t := K t + K t , K t is the total variation for process K on the interval [0, t].
With the help of Propositions 1 and 2, we can establish the following existence and uniqueness theorem in this part.
Proof. The uniqueness is easily given by Proposition 2. We now turn to prove its existence. For µ ≥ 0, let M 2 µ (K) represent the set of progressively measurable processes {X(t), 0 ≤ t ≤ T + K}, which satisfy: and M 2 µ represents the space of progressively measurable processes {X(t), 0 ≤ t ≤ T + K} which are such that: µ , by Theorem 2.1 in [9], we can define a map Φ from B 2 µ to B 2 µ through the equation: In the following, we will use the Banach contraction principle to prove the existence. Let g(s) = g(s, Y s , Z s , U s+δ 3 (s) , V s+ζ(s) ) − g(s, Y s , Z s , U s+δ 3 (s) , V s+ζ(s) ) h(s) = h(s, Y s , U s+δ 2 (s) ) − h(s, Y s , U s+δ 2 (s) ).
Consider the following equations: For any λ, θ > 0, in view of Itô's formula, we have: Thus, Φ is a strict contraction on B 2 µ equipped with the norm The proof of existence is complete.