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 $1/2$.