Geometric ergodicity of modified Euler schemes for SDEs with super-linearity

TL;DR

This paper establishes the geometric ergodicity of modified Euler schemes, including tamed and truncated versions, for SDEs with super-linear coefficients, and provides quantitative error bounds for invariant measures.

Contribution

It introduces a unified framework for modified Euler schemes applicable to super-linear SDEs and proves their geometric ergodicity using coupling methods, also deriving error bounds for invariant measures.

Findings

All proposed Euler schemes are geometrically ergodic under mixed probability distances.

The tamed Euler scheme is geometrically ergodic under the $L^1$-Wasserstein distance.

Quantitative $L^1$-Wasserstein error bounds between invariant measures are established.

Abstract

As a well-known fact, the classical Euler scheme works merely for SDEs with coefficients of linear growth. In this paper, we study a general framework of modified Euler schemes, which is applicable to SDEs with super-linear drifts and encompasses numerical methods such as the tamed Euler scheme and the truncated Euler scheme. On the one hand, by exploiting an approach based on the refined basic coupling, we show that all Euler recursions within our proposed framework are geometrically ergodic under a mixed probability distance (i.e., the total variation distance plus the $L^1$-Wasserstein distance) and the weighted total variation distance. On the other hand, by utilizing the coupling by reflection, we demonstrate that the tamed Euler scheme is geometrically ergodic under the $L^1$-Wasserstein distance. In addition, as an important application, we provide a quantitative $L^1$-Wasserstein error bound between the exact invariant probability measure of an SDE with super-linearity, and the invariant probability measure of the tamed Euler scheme which is its numerical counterpart.