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We study the strong convergence order of the Euler-Maruyama scheme for scalarstochastic differential equations with additive noise and irregular drift. Weprovide a general framework for the error analysis by reducing it to a weightedquadrature problem for irregular functions of Brownian motion. AssumingSobolev-Slobodeckij-type regularity of order $\kappa \in (0,1)$ for thenon-smooth part of the drift, our analysis of the quadrature problem yields theconvergence order $\min\{3/4,(1+\kappa)/2\}-\epsilon$ for the equidistantEuler-Maruyama scheme (for arbitrarily small $\epsilon>0$). The cut-off of theconvergence order at $3/4$ can be overcome by using a suitable non-equidistantdiscretization, which yields the strong convergence order of$(1+\kappa)/2-\epsilon$ for the corresponding Euler-Maruyama scheme.

The Euler-Maruyama Scheme for SDEs with Irregular Drift: Convergence Rates via Reduction to a Quadrature Problem