Martingale Property and Capacity under G-Framework

Abstract

The main purpose of this article is to study the symmetric martingale property and capacity defined by G-expectation introduced by Peng (cf. <a href="http://arxiv.org/PS_cache/math/pdf/0601/0601035v2.pdf">http://arxiv.org/PS_cache/math/pdf/0601/0601035v2.pdf</a>) in 2006. We show that the G-capacity can not be dynamic, and also demonstrate the relationship between symmetric G-martingale and the martingale under linear expectation. Based on these results and path-wise analysis, we obtain the martingale characterization theorem for G Brownian motion without Markovian assumption. This theorem covers the Levy's martingale characterization theorem for Brownian motion, and it also gives a different method to prove Levy's theorem.