Extreme-Value Statistics of Fractional Brownian Motion Bridges

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 $t_{\max }$ when this maximum is taken. Using a perturbative expansion around Brownian motion ($H=\frac{1}{2}$), 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 $H>\frac{1}{2}$ and $H<\frac{1}{2}$. 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.

16 pages, 7 figures

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