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 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 $\kappa \in (0,1)$ for the non-smooth part of the drift, our analysis of the quadrature problem yields the convergence order $\min{3/4,(1+\kappa)/2}-\epsilon$ for the equidistant Euler-Maruyama scheme (for arbitrarily small $\epsilon>0$). The cut-off of the convergence order at $3/4$ can be overcome by using a suitable non-equidistant discretization, which yields the strong convergence order of $(1+\kappa)/2-\epsilon$ for the corresponding Euler-Maruyama scheme.