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Communications in Mathematical Physics

, Volume 205, Issue 1, pp 97–111 | Cite as

Maximum of a Fractional Brownian Motion: Probabilities of Small Values

  • G. M. Molchan
Article

Abstract:

Let bγ (t), bγ(0)= 0 be a fractional Brownian motion, i.e., a Gaussian process with the structure function E|b γ (t) - b γ (s)|2 = |t - s| γ , 0 < γ < 2. We study the logarithmic asymptotics of P T = P{bγ (t) < 1,□tTΔ} as T→∞, where Δ is either the interval (0,1) or a bounded region that contains a vicinity of 0 for the case of multidimensional time. It is shown that ln P T = - D ln T(1 + o(1)), where D is the dimension of zeroes of bγ (t) in the former case and the dimension of time in the latter.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • G. M. Molchan
    • 1
  1. 1.International Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences, Warshavskoye sh. 79, kor. 2, Moscow, 113556, Russia. E-mail: molchan@mitp.ruRussia

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