Convergence of the Euler-Maruyama method for multidimensional SDEs with discontinuous drift and degenerate diffusion coefficient

TL;DR

This paper establishes the strong convergence rate of the Euler-Maruyama method for multidimensional SDEs with discontinuous drift and degenerate diffusion, using a novel transformation-based proof technique.

Contribution

It proves a convergence rate of order 1/4 - epsilon for such SDEs, extending the understanding of numerical methods under irregular coefficients.

Findings

Strong convergence of order 1/4 - epsilon proven

Applicable to multidimensional SDEs with discontinuous drift

Introduces a transformation-based analysis approach

Abstract

We prove strong convergence of order $1/4-\epsilon$ for arbitrarily small $\epsilon>0$ of the Euler-Maruyama method for multidimensional stochastic differential equations (SDEs) with discontinuous drift and degenerate diffusion coefficient. The proof is based on estimating the difference between the Euler-Maruyama scheme and another numerical method, which is constructed by applying the Euler-Maruyama scheme to a transformation of the SDE we aim to solve.