Electronic Communications in Probability

Fractional Brownian Motion and the Markov Property

Philippe Carmona and Laure Coutin

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Abstract

Fractional Brownian motion belongs to a class of long memory Gaussian processes that can be represented as linear functionals of an infinite dimensional Markov process. This leads naturally to:

  • An efficient algorithm to approximate the process.
  • An ergodic theorem which applies to functionals of the type $$\int_0^t \phi(V_h(s)),ds \quad\text{where}\quad V_h(s)=\int_0^s h(s-u), dB_u,.$$
where $B$ is a real Brownian motion.

Article information

Source
Electron. Commun. Probab., Volume 3 (1998), paper no. 12, 95-107.

Dates
Accepted: 27 October 1998
First available in Project Euclid: 2 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1456935918

Digital Object Identifier
doi:10.1214/ECP.v3-998

Mathematical Reviews number (MathSciNet)
MR1658690

Zentralblatt MATH identifier
0921.60067

Subjects
Primary: 60FXX
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60G15: Gaussian processes 65U05 26A33: Fractional derivatives and integrals 60A10: Probabilistic measure theory {For ergodic theory, see 28Dxx and 60Fxx}

Keywords
Gaussian processes Markov Processes Numerical Approximation Ergodic Theorem

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Carmona, Philippe; Coutin, Laure. Fractional Brownian Motion and the Markov Property. Electron. Commun. Probab. 3 (1998), paper no. 12, 95--107. doi:10.1214/ECP.v3-998. https://projecteuclid.org/euclid.ecp/1456935918


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References

  • J. Audounet and G. Montseny, Modèles non héréditaires pour l'analyse et le contrôle de systèmes à mémoire longue, April 1995, Journées d'etude: les systemes non entiers en automatique, Bordeaux, France.
  • J. Audounet, G. Montseny, and B.Mbodje, Optimal models of fractional integrators and application to systems with fading memory, 1993, IEEE SMC's conference, Le Touquet (France).
  • J. Beran and N. Terrin, Testing for a change of the long-memory parameter, Biometrika 83 (1996), no. 3, 627–638.
  • V.S. Borkar, Probability Theory: an advanced course, Universitext, Springer, 1995.
  • N. Bouleau and D. L'epingle, Numerical methods for stochastic processes, Wiley series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., 1994, ISBN 0-471-54641-0.
  • Ph. Carmona and L. Coutin, Simultaneous approximation of a family of (stochastic) differential equations, Unpublished, June 1998.
  • Ph. Carmona, L. Coutin, and G. Montseny, Applications of a representation of long memory Gaussian processes, Submitted to Stochastic Processes and their Applications, June 1997.
  • Ph. Carmona, L. Coutin, and G. Montseny, A diffusive Markovian representation of fractional Brownian motion with Hurst parameter less than $1/2$, Submitted to Statistic and Probability Letters, November 1998.
  • N. Dunford and J.T. Schwartz, Linear Operators, Part I: General Theory, Wiley Classics Library Edition Published 1988, John Wiley and Sons, New York, 1988.
  • H. Graf, Long range correlations and estimation of the self-similarity parameter, Ph.D. thesis, ETH Zürich, 1983.
  • W.E. Leland, M.S. Taqqu, W. Willinger, and D.V. Wilson, On the self-similar nature of Ethernet traffic, IEEE/ACM Trans. Networking 2 (1994), no. 1, 1–15.
  • B. Mandelbrot, Self-similar error clusters in communication and the concept of conditional stationarity, IEEE Trans. Commun. Technol. COM–13 (1965), 71–90.
  • B.B. Mandelbrot and Van Ness, Fractional brownian motions, fractional noises and applications, SIAM Review 10 (1968), no. 4, 422–437.
  • Yu.A. Rozanov, Stationary random processes, Holden-Day, San Francisco, 1966.