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

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 $\varepsilon$.