Fractional martingales and characterization of the fractional Brownian motion

Details Fractional martingales and characterization of the fractional Brownian motion Fractional martingales and characterization of the fractional Brownian motion

2007-11-08

arxiv.org/abs/0711.1313v2

Annals of Probability 2009, Vol. 37, No. 6, 2404-2430

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/09-AOP464 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/09-AOP464

In this paper we introduce the notion of fractional martingale as the fractional derivative of order $\alpha$ of a continuous local martingale, where $\alpha\in(-{1/2},{1/2})$, and we show that it has a nonzero finite variation of order $\frac{2}{1+2\alpha}$, under some integrability assumptions on the quadratic variation of the local martingale. As an application we establish an extension of L\'evy's characterization theorem for the fractional Brownian motion.

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