Reflected Stochastic Differential Equations Driven by G-Brownian Motion with Nonlinear Constraints

TL;DR

This paper investigates reflected stochastic differential equations driven by G-Brownian motion with nonlinear constraints, establishing existence, uniqueness, and comparison results within the G-expectation framework.

Contribution

It introduces a pathwise construction of doubly reflected G-Brownian motion and extends the theory to reflected G-SDEs with nonlinear constraints.

Findings

Constructed doubly reflected G-Brownian motion pathwise.

Proved existence and uniqueness of solutions to reflected G-SDEs.

Established a comparison theorem for solutions and constraining processes.

Abstract

In this paper, we study the reflected stochastic differential equations driven by G-Brownian motion (reflected G-SDEs) with two nonlinear constraints. With the help of the Skorokhod problem with nonlinear constraints, we first study the doubly reflected G-Brownian motion, which is constructed pathwise and lies in the same G-expectation space as the G-Brownian motion. For the reflected G-SDE, the uniqueness is derived from some a priori estimate and the existence is obtained by a Picard iteration method. The comparison theorem of the solution and the individual constraining processes are provided.