The Annals of Probability

Fractional martingales and characterization of the fractional Brownian motion

Yaozhong Hu, David Nualart, and 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∈(−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.

Article information

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

Dates
First available in Project Euclid: 16 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.aop/1258380793

Digital Object Identifier
doi:10.1214/09-AOP464

Mathematical Reviews number (MathSciNet)
MR2573562

Zentralblatt MATH identifier
1196.60075

Subjects
Primary: 60G44: Martingales with continuous parameter 60J65: Brownian motion [See also 58J65] 60G15: Gaussian processes 26A45: Functions of bounded variation, generalizations

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

Citation

Hu, Yaozhong; Nualart, David; Song, Jian. Fractional martingales and characterization of the fractional Brownian motion. Ann. Probab. 37 (2009), no. 6, 2404--2430. doi:10.1214/09-AOP464. https://projecteuclid.org/euclid.aop/1258380793


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