Geometric Ergodicity and Strong Error Estimates for Tamed Schemes of Super-linear SODEs

TL;DR

This paper introduces explicit tamed Euler--Maruyama schemes for super-linear SODEs that preserve Lyapunov structures, inherit geometric ergodicity, and achieve optimal strong convergence, validated by numerical experiments.

Contribution

The paper develops a family of explicit tamed schemes that maintain the Lyapunov structure and ergodic properties of super-linear SODEs, with proven convergence rates.

Findings

Schemes preserve the Lyapunov structure.

Schemes inherit geometric ergodicity.

Numerical experiments confirm theoretical results.

Abstract

We construct a family of explicit tamed Euler--Maruyama (TEM) schemes, which can preserve the same Lyapunov structure for super-linear stochastic ordinary differential equations (SODEs) driven by multiplicative noise.These TEM schemes are shown to inherit the geometric ergodicity of the considered SODEs and converge with optimal strong convergence orders. Numerical experiments verify our theoretical results.