The Maximum of a Fractional Brownian Motion: Analytic Results from Perturbation Theory

Mathieu Delorme, Kay Jörg Wiese
CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, 24 rue Lhomond, 75005 Paris, France.

Abstract

Fractional Brownian motion is a non-Markovian Gaussian process $X_t$, indexed by the Hurst exponent $H$. It generalises standard Brownian motion (corresponding to $H=1/2$). We study the probability distribution of the maximum $m$ of the process and the time $t_{\rm max}$ at which the maximum is reached. They are encoded in a path integral, which we evaluate perturbatively around a Brownian, setting $H=1/2 + \varepsilon$. This allows us to derive analytic results beyond the scaling exponents. Extensive numerical simulations for different values of $H$ test these analytical predictions and show excellent agreement, even for large $\varepsilon$.


arXiv:1507.06238 [pdf]
Phys. Rev. Lett. 115 (2015) 210601 [pdf]


Copyright (C) by Kay Wiese. Last edited December 15, 2015.