Extreme-Value Statistics of Fractional Brownian Motion Bridges
Mathieu Delorme, Kay Jörg Wiese
CNRS-Laboratoire de Physique Théorique de
l'Ecole Normale Supérieure, PSL Research University, Sorbonne Universités, UPMC,
24 rue Lhomond, 75005 Paris, France.
Abstract
Fractional Brownian motion is a self-affine, non-Markovian and translationally invariant generalization of Brownian motion, depending on the Hurst exponent $H$.
Here we investigate fractional Brownian motion where both the starting and the end point are zero, commonly referred to as bridge processes. Observables are the time $t_+$ the process is positive, the maximum $m$ it achieves, and the time $t_{\rm max}$ when this maximum is taken.
Using a perturbative expansion around Brownian motion ($H=\frac12$), we give the first-order result for the probability distribution of these three variables, and the joint distribution of $m$ and $t_{\rm max}$. Our analytical results are tested, and found in excellent agreement, with extensive numerical simulations, both for $H>\frac12$ and $H<\frac12$. This precision is achieved by sampling processes with a free endpoint, and then converting each realization to a bridge process, in generalization to what is usually done for Brownian motion.
arXiv:1605.04132 [pdf]
Phys. Rev. E 94 (2016) 052105 [pdf]
Copyright (C)
by Kay Wiese. Last edited May 16, 2016.