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. http://arxiv.org/PS_cache/math/pdf/0601/ 0601035v2.pdf) 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 Lèvy’s martingale characterization theorem for Brownian motion, and it also gives a different method to prove Lèvy’s theorem. © 2010 Applied Probability Trust.