Electronic Journal of Probability

Martingale Property and Capacity under G-Framework

Jing Xu and Bo Zhang

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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.

Article information

Electron. J. Probab., Volume 15 (2010), paper no. 67, 2041-2068.

Accepted: 3 December 2010
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H05: Stochastic integrals
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05]

G-Brownian motion G-expectation Martingale characterization Capacity

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Xu, Jing; Zhang, Bo. Martingale Property and Capacity under G-Framework. Electron. J. Probab. 15 (2010), paper no. 67, 2041--2068. doi:10.1214/EJP.v15-832. https://projecteuclid.org/euclid.ejp/1464819852

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