Tamed Euler-Maruyama method for SDEs with non-globally Lipschitz drift and multiplicative noise

TL;DR

This paper analyzes a tamed Euler-Maruyama numerical scheme for SDEs with superlinear drift and multiplicative noise, proving uniform convergence under ergodicity conditions in Wasserstein and total variation distances.

Contribution

It introduces and analyzes a tamed Euler-Maruyama method for SDEs with non-globally Lipschitz coefficients, establishing convergence rates under ergodic conditions.

Findings

Uniform-in-time convergence rates in Wasserstein distance

Uniform-in-time convergence rates in total variation distance

Applicable to SDEs with superlinear drift and multiplicative noise

Abstract

Consider the following stochastic differential equation driven by multiplicative noise on $\mathbb{R}^d$ with a superlinearly growing drift coefficient, \begin{align*} \mathrm{d} X_t = b (X_t) \, \mathrm{d} t + \sigma (X_t) \, \mathrm{d} B_t. \end{align*} It is known that the corresponding explicit Euler schemes may not converge. In this article, we analyze an explicit and easily implementable numerical method for approximating such a stochastic differential equation, i.e. its tamed Euler-Maruyama approximation. Under partial dissipation conditions ensuring the ergodicity, we obtain the uniform-in-time convergence rates of the tamed Euler-Maruyama process under $L^{1}$-Wasserstein distance and total variation distance.