Backward Stochastic Differential Equations with Double Mean Reflections

TL;DR

This paper investigates backward stochastic differential equations with two nonlinear mean reflection constraints on the distribution of solutions, establishing existence, uniqueness, and approximation methods.

Contribution

It introduces a novel framework for BSDEs with double mean reflections and provides solutions via contraction mapping and penalized mean-field BSDEs.

Findings

Existence and uniqueness of solutions are proven.

Solutions can be approximated by penalized mean-field BSDEs.

Framework extends classical BSDE theory to mean reflection constraints.

Abstract

In this paper, we study the backward stochastic differential equation (BSDE) with two nonlinear mean reflections, which means that the constraints are imposed on the distribution of the solution but not on its paths. Based on the backward Skorokhod problem with nonlinear constraints, we obtain the existence and uniqueness result by constructing a contraction mapping. When the constraints are linear, the solution can be approximated by a family of penalized mean-field BSDEs.