A Strong Order 1/2 Method for Multidimensional SDEs with Discontinuous Drift

TL;DR

This paper introduces a novel numerical method with strong order 1/2 convergence for multidimensional SDEs with discontinuous drift, proving existence, uniqueness, and applying a transformation technique to enable Euler-Maruyama approximation.

Contribution

It provides the first strong convergence result for a broad class of SDEs with discontinuous drift using a transformation approach.

Findings

Proves existence and uniqueness for multidimensional SDEs with discontinuous drift.

Develops a transformation technique to handle discontinuities in the drift.

Demonstrates strong order 1/2 convergence of the numerical method.

Abstract

In this paper we consider multidimensional stochastic differential equations (SDEs) with discontinuous drift and possibly degenerate diffusion coefficient. We prove an existence and uniqueness result for this class of SDEs and we present a numerical method that converges with strong order 1/2. Our result is the first one that shows strong convergence for such a general class of SDEs. The proof is based on a transformation technique that removes the discontinuity from the drift such that the coefficients of the transformed SDE are Lipschitz continuous. Thus the Euler-Maruyama method can be applied to this transformed SDE. The approximation can be transformed back, giving an approximation to the solution of the original SDE. As an illustration, we apply our result to an SDE the drift of which has a discontinuity along the unit circle.