Doubly Reflected Backward SDEs Driven by $G$-Brownian Motion with Quadratic Generator

TL;DR

This paper investigates doubly reflected backward stochastic differential equations driven by $G$-Brownian motion with quadratic generator growth, establishing existence, uniqueness, and approximation methods linked to nonlinear PDEs.

Contribution

It introduces new existence and uniqueness results for doubly reflected $G$-BSDEs with quadratic growth, using $G$-BMO martingale theory and Girsanov theorem, and connects solutions to nonlinear PDEs.

Findings

Established existence and uniqueness of solutions.

Developed a monotone approximation scheme.

Linked solutions to fully nonlinear PDEs with double obstacles.

Abstract

In this paper, we study the doubly reflected backward stochastic differential equations driven by $G$-Brownian motion ($G$-BSDEs for short) when the generator has quadratic growth in the $z$-component. Based on the theory of $G$-BMO martingale and $G$-Girsanov theorem, we establish the existence and uniqueness result when the upper obstacle is almost a generalized $G$-It\^{o}'s process. Moreover, the solution can be approximated monotonically by the solutions to a family of penalized reflected $G$-BSDEs with a lower obstacle, which plays an important role to establish the relation between doubly reflected $G$-BSDEs and fully nonlinear partial differential equations with double obstacles.