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 Levy's martingale characterization theorem for Brownian motion, and it also gives a different method to prove Levy's theorem.
Highlights
Cox and Ross [6] notice that, without loss of generality, to price some derivative security as an option, one would get the correct result by assuming that all of the securities have the same expected rate of return. This is the “risk neutral” pricing method. This notion is first given by Harrison and Kreps [14], who formalize the near equivalence of the absence arbitrage with the existence of some new “risk neutral” probability measure under which all expected rates of return are equal to the current risk free rate
We give the martingale characterization theorem for G-Brownian motion without Markovian condition which improves the related result in [29]
G-expectation is a kind of time consistent nonlinear expectations, which has the properties of linear expectation in Weiner space except linearity
Summary
The most famous pricing formula for the European call option was given by Black and Scholes in 1973, see [2] As these developments unfolded, Cox and Ross [6] notice that, without loss of generality, to price some derivative security as an option, one would get the correct result by assuming that all of the securities have the same expected rate of return. We investigate the relationship between symmetric G-martingales and martingales under linear expectation, when the corresponding G-heat equation is uniformly parabolic Based on these results, we give the martingale characterization theorem for G-Brownian motion without Markovian condition which improves the related result in [29]. The last section is conclusion and discussion about this work and future work
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