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

TL;DR

This paper develops a perturbative analytical approach to study the distribution of the maximum and the time of maximum of fractional Brownian motion, validated by extensive numerical simulations across different Hurst exponents.

Contribution

It introduces a perturbation theory method around standard Brownian motion to analytically compute distributions for fractional Brownian motion's maximum and its timing.

Findings

Analytic expressions derived for maximum distribution and timing.

Excellent agreement between theory and simulations for various H values.

Method extends understanding beyond scaling exponents.

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 $\varepsilon$.