The Annals of Probability

Fractional martingales and characterization of the fractional Brownian motion

Yaozhong Hu, David Nualart, and Jian Song

Full-text: Open access


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

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

First available in Project Euclid: 16 November 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

Export citation


  • [1] Berman, S. M. (1973/74). Local nondeterminism and local times of Gaussian processes. Indiana Univ. Math. J. 23 69–94.
  • [2] Durrett, R. (2005). Probability: Theory and Examples, 3nd ed. Duxbury Press, Belmont, CA.
  • [3] Geman, D. and Horowitz, J. (1980). Occupation densities. Ann. Probab. 8 1–67.
  • [4] Hu, Y. (2005). Integral transformations and anticipative calculus for fractional Brownian motions. Mem. Amer. Math. Soc. 175.
  • [5] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Springer, New York.
  • [6] Mandelbrot, B. B. and Van Ness, J. W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 422–437.
  • [7] Mishura, J. and Valkeila, E. (2007). An extension of the Lévy characterization to fractional Brownian motion. Preprint.
  • [8] Norros, I., Valkeila, E. and Virtamo, J. (1999). An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions. Bernoulli 5 571–587.
  • [9] Nualart, D. (2003). Stochastic integration with respect to fractional Brownian motion and applications. Contemp. Math. 336 3–39.
  • [10] Rogers, L. C. G. (1997). Arbitrage with fractional Brownian motion. Math. Finance 7 95–105.
  • [11] Samko, S. G., Kilbas, A. A. and Marichev, O. I. (1993). Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon.