Extreme Events for Fractional Brownian Motion with Drift: Theory and Numerical Validation

Maxence Arutkin¹, Benjamin Walter², Kay Jörg Wiese³
¹ UMR CNRS 7083 Gulliver, ESPCI Paris, 10 rue Vauquelin, 75005 Paris, France.
² Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom.
³ Laboratoire de Physique de l'Ecole normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris-Diderot, Sorbonne Paris Cité, Paris, France.

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 non-linear drift. The latter appears naturally when applying non-linear variable transformations. Via a perturbative expansion in $\epsilon = H-1/2$, 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 $N_{\rm eff}=2^{28}\approx 2.7\times 10^{8}$ points. The agreement between theory and simulations is excellent, and by far exceeds in precision what can be obtained by scaling alone.

arXiv:1908.10801v2 [pdf]
Phys. Rev. E 102 (2020) 022102 [pdf]