Advertisement

A Remark on the 1/H-Variation of the Fractional Brownian Motion

  • Maurizio Pratelli
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2006)

Abstract

We give an elementary proof of the following property of H-fractional Brownian motion: almost all sample paths have infinite 1/H-variation on every interval.

Fractional Brownian motion p-Variation Ergodic theorem 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ciesielski, Z., Kerkyacharian, G., Roynette, B.: Quelques espaces fonctionnelles associés a des processus Gaussiens. Studia Math. 107(2), 171–204 (1993)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Coutin, L.: An introduction to (stochastic) calculus with respect to fractional Brownian motion. In: Séminaire de Probabilités XL, Lecture Notes in Mathematics, vol. 1899, pp. 3–65. Springer (2007)Google Scholar
  3. 3.
    Dudley, N.R.: An introduction to P-variation and Young integrals. Tech. rep. 1., Maphysto, Center for Mathematical Physics and Stochastics, University of Aarhus (1998)Google Scholar
  4. 4.
    Freedman, D.: Brownian Motion and Diffusion. Holden-Day (1971)Google Scholar
  5. 5.
    Guerra, J., Nualart, D.: The 1 ∕ H-variation of the divergence integral with respect to the fractional Brownian motion for H > 1 ∕ 2 and fractional Bessel processes. Stoch. Process. Appl. 115(1), 91–115 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Lévy, P.: Processus stochastiques et Mouvement Brownien. Gauthier-Villars (1965)Google Scholar
  7. 7.
    Nualart, D.: The Malliavin Calculus and Related Topics. Springer, New York (2006)zbMATHGoogle Scholar
  8. 8.
    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin (1999)zbMATHGoogle Scholar
  9. 9.
    Rogers, L.C.G.: Arbitrage with fractional Brownian motion. Math. Finance 7, 95–105 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Shiryaev, A.N.: Probability, 2nd edn. Springer, Berlin (1996)Google Scholar
  11. 11.
    Taylor, S.J.: Exact asymptotic estimates of Brownian path variation. Duke Math. J. 39, 219–241 (1972)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Dipartimento di MatematicaPisaItaly

Personalised recommendations