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

Fractional Brownian motion is a non-Markovian Gaussian process X-t, indexed by the Hurst exponent H. It generalizes standard Brownian motion (corresponding to H = 1/2). We study the probability distribution of the maximum m of the process and the time t(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 + epsilon. 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 epsilon.