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arXiv · 0711.1313

Fractional martingales and characterization of the fractional Brownian motion

Yaozhong Hu·David Nualart·Jian Song
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Abstract

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

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Yaozhong Hu, David Nualart, Jian Song. 2009-12-09. Fractional martingales and characterization of the fractional Brownian motion. https://doi.org/10.1214/09-aop464

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