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

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Fractional Brownian motion is a non-Markovian Gaussian process Xₜ, 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 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 + ε. 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 ε.