Paper

Extreme-Value Statistics of Fractional Brownian Motion Bridges

Wiese, Kay Jörg, Delorme, Mathieu

2016

Overview

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_ max\\) when this maximum is taken. Using a perturbative expansion around Brownian motion (\\(H=12\\)), we give the first-order result for the probability distribution of these three variables, and the joint distribution of \\(m\\) and \\(t_ max\\). Our analytical results are tested, and found in excellent agreement, with extensive numerical simulations, both for \\(H>12\\) and \\(H<12\\). 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.

• Publisher: Cornell University Library, arXiv.org
• Subject: Brownian motion / Computer simulation / Extreme values / Markov processes / Random walk theory