BSDEs driven by $G$-Brownian motion with uniformly continuous generators

TL;DR

This paper establishes existence and uniqueness results for a class of $G$-BSDEs with non-Lipschitz generators, extending the theory to uniformly continuous cases and providing comparison theorems and Feynman-Kac formulas.

Contribution

It introduces a method to prove existence and uniqueness of $G$-BSDE solutions with uniformly continuous generators, broadening previous Lipschitz-based results.

Findings

Proved existence and uniqueness of solutions for non-Lipschitz $G$-BSDEs.

Established a comparison theorem for these equations.

Derived a Feynman-Kac formula connecting $G$-BSDEs to PDEs.

Abstract

The present paper is devoted to investigating the existence and uniqueness of solutions to a class of non-Lipschitz scalar valued backward stochastic differential equations driven by $G$-Brownian motion ($G$-BSDEs). In fact, when the generators are Lipschitz continuous in $y$ and uniformly continuous in $z$, we construct the unique solution to such equations by monotone convergence argument. The comparison theorem and related Feynman-Kac formula are stated as well.