Lévy's martingale characterization and reflection principle of $G$-Brownian motion
Mingshang Hu, Xiaojun Ji, Guomin Liu
TL;DR
This work addresses Lévy's martingale characterization for $G$-Brownian motion without the nondegenerate condition and derives a reflection principle for $G$-Brownian motion. It introduces a discrete product-space method to handle degeneracy and leverages Krylov's estimate to extend the reflection principle to $\tilde{G}$-Brownian motion. The main contributions are (i) a Lévy-type characterization for $G$-Brownian motion under degeneracy, (ii) a reflection principle for $G$-Brownian motion, and (iii) a Krylov-based reflection principle for $\tilde{G}$-Brownian motion. These results advance stochastic calculus under sublinear expectations and provide distributional descriptions of maximal processes under model uncertainty, with implications for robust financial modeling and uncertain volatility settings.
Abstract
In this paper, we obtain Lévy'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.