Uniform error bounds for numerical schemes applied to multiscale SDEs in a Wong-Zakai diffusion approximation regime

TL;DR

This paper develops and analyzes numerical schemes for multiscale stochastic differential equations, providing uniform error bounds in a Wong-Zakai diffusion approximation regime, which improves understanding of their accuracy as the scale separation vanishes.

Contribution

The paper introduces and rigorously analyzes schemes that are asymptotic preserving and provide uniform strong error estimates for multiscale SDEs in a Wong-Zakai regime.

Findings

Schemes satisfy asymptotic preserving property.

Error estimates are uniform with respect to scale separation.

Fills a gap in the theoretical analysis of these numerical methods.

Abstract

We study a family of numerical schemes applied to a class of multiscale systems of stochastic differential equations. When the time scale separation parameter vanishes, a well-known homogenization or Wong--Zakai diffusion approximation result states that the slow component of the considered system converges to the solution of a stochastic differential equation driven by a real-valued Wiener process, with Stratonovich interpretation of the noise. We propose and analyse schemes for effective approximation of the slow component. Such schemes satisfy an asymptotic preserving property and generalize the methods proposed in a recent article. We fill a gap in the analysis of these schemes and prove strong error estimates, which are uniform with respect to the time scale separation parameter.