Mean-Field and Anticipated BSDEs with Time-Delayed Generator

Abstract

In this paper, we discuss a new type of mean-field anticipated backward stochastic differential equation with a time-delayed generator (MF-DABSDEs) which extends the results of the anticipated backward stochastic differential equation to the case of mean-field limits, and in which the generator considers not only the present and future times but also the past time. By using the fixed point theorem, we shall demonstrate the existence and uniqueness of the solutions to these equations. Finally, we shall establish a comparison theorem for the solutions.

General Mean-Field BDSDEs with Stochastic Linear Growth and Discontinuous Generator

“…By Lemma 3, there exists at least one solution of (11). Here, we only consider the minimal solution, denoted as (Y n , Z n ).…”

“…Remark 1. Proposition 2 implies that the minimal solution sequence of general mean-field BDSDEs (11) is increasing and has an upper bound, i.e.,…”

“…Therefore, {Z n } ∞ n=1 converges in H 2,β (0, T; R), we denote it by Z. Now, we pass to the limit, as n → ∞ on both sides of (11), it follows that…”

“…Obviously, (Y, Z) solves the general mean-field BDSDEs (5). Let, (Y, Z) ∈ M 2,β (0, T) be any solution of (5) and consider (11) with its minimal solution (Y n , Z n ) ∈ M 2,β (0, T) for every n ≥ 0.…”

“…Inspired by this idea, Buckdahn, Djehiche, Li, Peng [8] and Buckdahn, Li, Peng [9] used purely random methods to investigate a special class of mean-field problems, and proposed a new type of BSDEs, called mean-field BSDEs. Since then, more and more scholars have devoted their energies to the study of mean-field problems (see [10,11], etc.). Li, Liang and Zhang [12] studied mean-field BSDEs under continuous conditions and proposed a technical lemma by which the existence of solutions was obtained.…”