Strong convergence of the tamed Euler scheme for scalar SDEs with superlinearly growing and discontinuous drift coefficient

TL;DR

This paper proves strong convergence of the tamed Euler scheme for scalar SDEs with superlinear, discontinuous drift, establishing existence, uniqueness, and an $L_p$ error rate of 1/2, supported by numerical evidence.

Contribution

It extends the analysis of the tamed Euler scheme to SDEs with superlinear, discontinuous drift, providing convergence results and error rates.

Findings

Existence and uniqueness of solutions for the considered SDEs.

The tamed Euler scheme achieves an $L_p$ error rate of 1/2.

Numerical example confirms theoretical results.

Abstract

In this paper, we consider scalar stochastic differential equations (SDEs) with a superlinearly growing and piecewise continuous drift coefficient. Existence and uniqueness of strong solutions of such SDEs are obtained. Furthermore, the classical $L_p$-error rate 1/2 for all $p\in [1, +\infty)$ is recovered for the tamed Euler scheme. A numerical example is provided to support our conclusion.