Mean Field Backward Stochastic Differential Equations with Double Mean Reflections

TL;DR

This paper studies complex mean field backward stochastic differential equations with double mean reflections, establishing existence and uniqueness results under various growth conditions using advanced mathematical techniques.

Contribution

It introduces new methods for solving doubly mean reflected MFBSDEs, including contraction mapping, penalization, and fixed-point approaches for different growth scenarios.

Findings

Existence and uniqueness for Lipschitz continuous generator cases.

Solutions constructed via penalization for linear constraints.

Results extended to quadratic growth cases using BMO martingale theory.

Abstract

In this paper, we analyze the mean field backward stochastic differential equations (MFBSDEs) with double mean reflections, whose generator and constraints both depend on the distribution of the solution. When the generator is Lipschitz continuous, based on the backward Skorokhod problem with nonlinear constraints, we investigate the solvability of the doubly mean reflected MFBSDEs by constructing a contraction mapping. Furthermore, if the constraints are linear, the solution can also be constructed by a penalization method. For the case of quadratic growth, we obtain the existence and uniqueness results by using a fixed-point argument, the BMO martingale theory and the {\theta}-method.