$G$-BSDEs with mean constraints in time-dependent intervals

TL;DR

This paper investigates mean-reflected $G$-BSDEs with time-dependent constraints, establishing well-posedness and extending results to multi-dimensional cases using advanced stochastic analysis techniques.

Contribution

It introduces a novel framework for mean-reflected $G$-BSDEs with time-dependent constraints and develops methods for their well-posedness and multi-dimensional extensions.

Findings

Established well-posedness for doubly mean-reflected $G$-BSDEs.

Extended results to multi-dimensional cases with diagonal generators.

Developed new techniques involving backward Skorokhod problems and fixed-point methods.

Abstract

In this paper, we study a collection of mean-reflected backward stochastic differential equations driven by $G$-Brownian motions ($G$-BSDEs), where $G$-expectations are constrained in some time-dependent intervals. To establish well-posedness results, we firstly construct a backward Skorokhod problem with sublinear expectation, and then apply that in the study of doubly mean-reflected $G$-BSDEs involving Lipschitz and quadratic generators under bounded and unbounded terminal conditions. Also we utilize fixed-point argumentations and $\theta$-methods while solving these equations. Finally, we extend the results to multi-dimensional doubly mean-reflected $G$-BSDEs with diagonal generators.