Fractional martingales and characterization of the fractional Brownian motion

Fractional martingales and characterization of the fractional Brownian motion

Yaozhong Hu, David Nualart, and Jian Song

Source: Ann. Probab. Volume 37, Number 6 (2009), 2404-2430.

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∈(−½, ½), and we show that it has a nonzero finite variation of order 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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Primary Subjects: 60G44, 60J65, 60G15, 26A45

Keywords: Fractional Brownian motion; fractional martingale; Lévy’s characterization theorem; β-variation

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aop/1258380793
Digital Object Identifier: doi:10.1214/09-AOP464
Mathematical Reviews number (MathSciNet): MR2573562
Zentralblatt MATH identifier: 05708805

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