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

Xu, Jing; Zhang, Bo

Electron. J. Probab. 15, Paper No. 67, 2041-2068 (2010).

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

Keywords:

\(G\)-Brownian motion; \(G\)-expectation; martingale characterization; capacity

Citations:

Zbl 1131.60057