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

Zhihui Liu, Xiaoming Wu

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.

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

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.

Paper Structure

This paper contains 9 sections, 9 theorems, 81 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Geometric Ergodicity of Tamed Euler--Maruyama Schemes
  3. Lyapunov Structure
  4. Geometric Ergodicity of TEM
  5. Error Estimates
  6. Moment's Estimate and Hölder Continuity
  7. Strong Error Estimates of Multiplicative Noise Case
  8. Strong Error Estimates of Additive Noise Case
  9. Numerical Experiments

Key Result

Lemma 2.1

Under Assumption ap, there exists $\tau_{\max} \le 1$ such that for any $\tau \in (0, \tau_{\max}]$, or are both of linear growth and coe-tau+ holds.

Figures (4)

  • Figure 1: Empirical density of \ref{['tem']} for \ref{['sde']}
  • Figure 2: Strong error estimates of \ref{['tem']} for \ref{['sde']}
  • Figure 3: Empirical density of \ref{['tem*']} for \ref{['sde']}
  • Figure 4: Strong error estimates of \ref{['tem*']} for \ref{['sde']}

Theorems & Definitions (23)

  • Remark 2.1
  • Lemma 2.1
  • proof
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Corollary 2.1
  • proof
  • Theorem 2.2
  • Remark 2.3
  • ...and 13 more