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

## TL;DR

This paper establishes convergence rates for the Euler-Maruyama scheme applied to SDEs with irregular drift by reducing the problem to a quadrature analysis, showing improved rates with non-uniform discretization.

## Contribution

It introduces a novel framework linking SDE approximation errors to quadrature problems, enabling precise convergence rate analysis for irregular drifts.

## Key findings

- Convergence order is min{3/4, (1+κ)/2} - ε for equidistant schemes.
- Non-equidistant discretization improves convergence to (1+κ)/2 - ε.
- Framework applies Sobolev-Slobodeckij regularity assumptions to analyze irregular drifts.

## 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.