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

Abstract

We study the strong convergence order of the Euler–Maruyama (EM) scheme for scalar stochastic differential equations with additive noise and irregular drift. We provide a general 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 nonsmooth part of the drift, our analysis of the quadrature problem yields the convergence order min{3/4,(1+κ)/2}−ϵ for the equidistant EM scheme (for arbitrarily small ϵ>0). The cut-off of the convergence order at 3/4 can be overcome by using a suitable nonequidistant discretization, which yields the strong convergence order of (1+κ)/2−ϵ for the corresponding EM scheme.