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

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.