Extreme events for fractional Brownian motion with drift: Theory and numerical validation.

Abstract

We study the first-passage time, the distribution of the maximum, and the absorption probability of fractional Brownian motion of Hurst parameter H with both a linear and a nonlinear drift. The latter appears naturally when applying nonlinear variable transformations. Via a perturbative expansion in ɛ=H-1/2, we give the first-order corrections to the classical result for Brownian motion analytically. Using a recently introduced adaptive-bisection algorithm, which is much more efficient than the standard Davies-Harte algorithm, we test our predictions for the first-passage time on grids of effective sizes up to N_{eff}=2^{28}≈2.7×10^{8} points. The agreement between theory and simulations is excellent, and by far exceeds in precision what can be obtained by scaling alone.