A Simple Stochastic Differential Equation with Discontinuous Drift

TL;DR

This paper investigates solutions to stochastic differential equations with discontinuous drift using Euler-Maruyama and Fokker-Planck approaches, introducing a smooth approximation to handle discontinuities.

Contribution

It presents a novel comparison between Euler-Maruyama and Fokker-Planck methods for SDEs with discontinuous drift, including a smooth approximation technique.

Findings

The candidate density functions from both methods closely match.

The smooth approximation effectively handles discontinuities.

The approaches provide consistent results for the specific SDE studied.

Abstract

In this paper we study solutions to stochastic differential equations (SDEs) with discontinuous drift. We apply two approaches: The Euler-Maruyama method and the Fokker-Planck equation and show that a candidate density function based on the Euler-Maruyama method approximates a candidate density function based on the stationary Fokker-Planck equation. Furthermore, we introduce a smooth function which approximates the discontinuous drift and apply the Euler-Maruyama method and the Fokker-Planck equation with this input. The point of departure for this work is a particular SDE with discontinuous drift.