L\'{e}vy's martingale characterization and reflection principle of $G$-Brownian motion

TL;DR

This paper extends Lévy's martingale characterization to G-Brownian motion without the nondegeneracy condition and establishes the reflection principle for G-Brownian motion and its variant using Krylov's estimate.

Contribution

It provides a novel Lévy's martingale characterization for G-Brownian motion without the nondegeneracy assumption and proves the reflection principle for G-Brownian motion and G-Brownian motion.

Findings

Lévy's martingale characterization of G-Brownian motion obtained

Reflection principle for G-Brownian motion proved

Reflection principle for G-Brownian motion established using Krylov's estimate

Abstract

In this paper, we obtain L\'{e}vy's martingale characterization of $G$-Brownian motion without the nondegenerate condition. Base on this characterization, we prove the reflection principle of $G$-Brownian motion. Furthermore, we use Krylov's estimate to get the reflection principle of $\tilde{G}$-Brownian motion.