A note on the Euler–Maruyama scheme for stochastic differential equations with a discontinuous monotone drift coefficient

Abstract

It is shown that the Euler–Maruyama scheme applied to a stochastic differential equation with a discontinuous monotone drift coefficient, such as a Heaviside function, and additive noise converges strongly to a solution of the stochastic differential equation with the same initial condition. The proof uses upper and lower solutions of the stochastic differential equations and the Euler–Maruyama scheme.