On the exit times of SDEs driven by $G$-Brownian motion

TL;DR

This paper investigates the exit times of stochastic differential equations driven by $G$-Brownian motion, establishing their quasi-continuity and providing a probabilistic representation for certain nonlinear elliptic equations.

Contribution

It introduces the quasi-continuity property of exit times for $G$-SDEs and links this to solutions of nonlinear elliptic equations with Dirichlet boundary conditions.

Findings

Exit times of $G$-SDEs are quasi-continuous.

Probabilistic representation for nonlinear elliptic equations.

Application to fully nonlinear PDEs with boundary conditions.

Abstract

This paper is devoted to studying the properties of the exit times of stochastic differential equations driven by $G$-Brownian motion ($G$-SDEs). In particular, we prove that the exit times of $G$-SDEs has the quasi-continuity property. As an application, we give a probabilistic representation for a large class of fully nonlinear elliptic equations with Dirichlet boundary.