An adaptive Euler-Maruyama scheme for stochastic differential equations with discontinuous drift and its convergence analysis

TL;DR

This paper introduces an adaptive Euler-Maruyama scheme tailored for stochastic differential equations with discontinuous drifts, achieving near-optimal strong convergence rates and validated through numerical experiments.

Contribution

The paper develops a novel adaptive step size strategy for Euler-Maruyama to handle discontinuous drifts, improving convergence analysis and computational efficiency.

Findings

Achieves strong convergence order close to 1/2 with logarithmic factors.

Demonstrates effectiveness through multiple numerical examples.

Provides theoretical convergence analysis for the adaptive scheme.

Abstract

We study the strong approximation of stochastic differential equations with discontinuous drift coefficients and (possibly) degenerate diffusion coefficients. To account for the discontinuity of the drift coefficient we construct an adaptive step sizing strategy for the explicit Euler-Maruyama scheme. As a result, we obtain a numerical method which has -- up to logarithmic terms -- strong convergence order $1/2$ with respect to the average computational cost. We support our theoretical findings with several numerical examples.