A Study on a New Class of Backward Stochastic Differential Equation

Pardoux and Peng in [1] first provided the famous backward stochastic differential equations (BSDEs). +e existence and uniqueness for the BSDEs are proved by them. Since then, BSDEs have been discussed and applied to many fields, e.g., Chen and Epstein [2] and Karoui et al. [3–7]. A lot of research has focused on the assumptions on the generator, such as [8–12]. Recently, Delong and Imkeller in [13, 14] obtained many interesting results about the time-delayed equation in which the generator at time t only depends on the past solution. Peng and Yang in [15] discussed anticipated BSDEs, in which the generator includes present and future solutions. +erefore, the natural questions are as follows: can we discuss the backward stochastic differential equations when the generator includes not only the past and the present but also the future solutions? +e comparison theorem for it is still true? Indeed, these questions are answered in the affirmative in this paper. +e equation is called delay and anticipated BSDEs (DABSDEs for short). We obtain existence and uniqueness for delay and anticipated BSDEs, which can be seen as a general version of Delong and Imkeller in [13] or Peng and Yang in [15].


Introduction
Pardoux and Peng in [1] first provided the famous backward stochastic differential equations (BSDEs). e existence and uniqueness for the BSDEs are proved by them. Since then, BSDEs have been discussed and applied to many fields, e.g., Chen and Epstein [2] and Karoui et al. [3][4][5][6][7]. A lot of research has focused on the assumptions on the generator, such as [8][9][10][11][12]. Recently, Delong and Imkeller in [13,14] obtained many interesting results about the time-delayed equation in which the generator at time t only depends on the past solution. Peng and Yang in [15] discussed anticipated BSDEs, in which the generator includes present and future solutions. erefore, the natural questions are as follows: can we discuss the backward stochastic differential equations when the generator includes not only the past and the present but also the future solutions? e comparison theorem for it is still true? Indeed, these questions are answered in the affirmative in this paper. e equation is called delay and anticipated BSDEs (DABSDEs for short). We obtain existence and uniqueness for delay and anticipated BSDEs, which can be seen as a general version of Delong and Imkeller in [13] or Peng and Yang in [15].

Main Notations
Let (Ω, F, P) be a probability space equipped with a natural filtration (F v ) v≥0 and (B v ) v≥0 be a standard Brownian motion. We denote the norm in R m by | · |. Given T > 0, denote the following: In the case n � 1, they are abbreviated to L 2 (F T ), L 2 F (0, T), and S 2 F (0, T), respectively.
(D2) ∃L ≥ 0 s.t., for all nonnegative and integrable f(·), Note that the simple examples for d i (s) satisfying (D1) and (D2) are constant delay and d i (s) � s.
We call system (1) delay and anticipated backward stochastic differential equations (delay and anticipated BSDEs).
Our aim is to search out a pair of processes (Y., Z.) ∈ S 2 F (0, D + T; R m ) × L 2 F (0, D + T; R m×d ) which satisfies the delay and anticipated BSDEs (1).
□ Example 1. Consider a typical delay and anticipated backward stochastic differential equation

Mathematical Problems in Engineering
with Y T � T 0 sin tdB t . We can get the unique solution of DABSDE (15) which is

Comparison Theorem for Delay and Anticipated BSDEs
Next, we deduce the comparison theorem for one-dimensional DABSDEs.