Mean Reflected Backward Stochastic Differential Equations Driven by G-Brownian Motion with Double Constraints

TL;DR

This paper investigates mean reflected backward stochastic differential equations driven by G-Brownian motion with dual constraints on the solution's law, establishing existence, uniqueness, and key properties using advanced stochastic analysis techniques.

Contribution

It introduces a novel framework for mean reflected BSDEs driven by G-Brownian motion with double constraints, including new comparison theorems and connections to optimization.

Findings

Proved existence and uniqueness of solutions.

Developed a new comparison theorem.

Connected solutions to deterministic optimization problems.

Abstract

In this paper, we study the backward stochastic differential equations driven by G-Brownian motion with double mean reflections, which means that the constraints are made on the law of the solution. Making full use of the backward Skorokhod problem with two nonlinear reflecting boundaries and the fixed-point theory, the existence and uniqueness of solutions are established. We also consider the case where the coefficients satisfy a non-Lipschitz condition using the Picard iteration argument only for the Y component. Moreover, some basic properties including a new version of comparison theorem and connection with a deterministic optimization problem are also obtained.