Perturbative Expansion for the Maximum of Fractional Brownian Motion

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 $1/2$.