Perturbative Expansion for the Maximum of Fractional Brownian Motion

TL;DR

This paper introduces a perturbative method to analyze the maximum and its timing in fractional Brownian motion, providing analytical results validated by numerical simulations across various Hurst exponents.

Contribution

We develop a perturbative expansion around H=1/2 to analytically study extreme value statistics of fractional Brownian motion, extending beyond traditional scaling exponents.

Findings

Analytical expressions for maximum distribution and timing derived.

Excellent agreement between theory and simulations for different H values.

Method applicable even for H far from 1/2.

Abstract

Brownian motion is the only random process which is Gaussian, stationary and Markovian. Dropping the Markovian property, i.e. allowing for memory, one obtains a class of processes called fractional Brownian motion, indexed by the Hurst exponent $H$. For $H=1/2$, Brownian motion is recovered. We develop a perturbative approach to treat the non-locality in time in an expansion in $\varepsilon = H-1/2$. This allows us to derive analytic results beyond scaling exponents for various observables related to extreme value statistics: The maximum $m$ of the process and the time $t_{\text{max}}$ at which this maximum is reached, as well as their joint distribution. We test our analytical predictions with extensive numerical simulations for different values of $H$. They show excellent agreement, even for $H$ far from $1/2$.