## Martingale property and capacity under $$G$$-framework.(English)Zbl 1226.60085

Summary: The main purpose of this article is to study the symmetric martingale property and capacity defined by $$G$$-expectation introduced by S. Peng [in: Stochastic analysis and applications. The Abel symposium 2005. Proceedings of the second Abel symposium, Oslo, 2005, held in honor of Kiyosi Itô. Berlin: Springer. Abel Symposia 2, 541–567 (2007; Zbl 1131.60057)]. 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 motions without Markovian assumption. This theorem covers Levy’s martingale characterization theorem for Brownian motion, and it also gives a different method to prove Levy’s theorem.

### MSC:

 60H05 Stochastic integrals 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60G44 Martingales with continuous parameter

Zbl 1131.60057
Full Text: