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Fractional martingales and characterization of the fractional Brownian motion
Hu, Yaozhong Nualart, David Song, Jian
Hu, Yaozhong
Nualart, David
Song, Jian
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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.
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This is the published version, also available here: http://dx.doi.org/10.1214/09-AOP464.
Date
2009-11-19
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Institute of Mathematical Statistics
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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. http://dx.doi.org/10.1214/09-AOP464.