The Euler-Maruyama scheme for SDEs with irregular drift: Convergence rates via reduction to a quadrature problem

Neuenkirch, Andreas ; Szölgyenyi, Michaela

URL:	https://arxiv.org/abs/1904.07784
Dokumenttyp:	Arbeitspapier
Erscheinungsjahr:	2019
Ort der Veröffentlichung:	Ithaca, NY
Verlag:	Cornell University
Sprache der Veröffentlichung:	Englisch
Einrichtung:	Fakultät für Wirtschaftsinformatik und Wirtschaftsmathematik > Wirtschaftsmathematik II: Stochastische Numerik (Neuenkirch 2013-)
Fachgebiet:	510 Mathematik
Abstract:	We study the strong convergence order of the Euler-Maruyama scheme for scalar stochastic differential equations with additive noise and irregular drift. We provide a novel framework for the error analysis by reducing it to a weighted quadrature problem for irregular functions of Brownian motion. Assuming Sobolev-Slobodeckij-type regularity of order κ∈(0,1) for the drift, our analysis of the quadrature problem yields the convergence order min{3/4,(1+κ)/2}−ϵ for the equidistant Euler-Maruyama scheme (for arbitrarily small ϵ>0). The cut-off of the convergence order at κ=3/4 can be overcome by using a suitable non-equidistant discretization, which yields the strong convergence order of (1+κ)/2−ϵ for the corresponding Euler-Maruyama scheme.