Martingale Characterization Of Multi-dimensional G-Brownian Motion

We mainly discuss the question of martingale characterization of multi-dimensional G-Brownian motion in this article. As we known, Levy gave a martingale character-ization for classical Brownian motion:As a continuous martingale, M is a Brownian motion if and only if its quadratic variation process (M)t= t,(?)t. Researches about G-framework have become more and more popular since Peng introduced sublinear expectation, the so-called G-expectation. So we also want to know if there is a similar martingale characterization for G-Brownian motion. Xu and Zhang [21,22] stated the martingale characterization of one-dimensional G-Brownian motion. We try to explore martingale characterization for multi-dimensional G-Brownian motion based their re-sults.At the same time,we state the research of Bayraktar and Munk[1], who explained the differences of multi-dimensional G-normal distribution and classical normal distri-bution, and we also make a further discussion about two questions derived from [1].