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 $t_{\rm max}$ 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.