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Ergodic Estimates of One-Step Numerical Approximations for Superlinear SODEs

Xin Liu, Zhihui Liu

TL;DR

We study ergodic error for stochastic ODEs with superlinear growth and multiplicative noise by developing a Stein-method framework based on a Stein equation $\varphi-\pi(\varphi)=\mathcal{A}f_\varphi$ and its explicit solution $f_\varphi$. A one-step error representation using a discretized generator $\mathcal{A}_\tau$ and a continuous interpolation yields a sharp first-order bound $|\pi_\tau(\varphi)-\pi(\varphi)|=O(\tau)$ for a class of schemes. The framework applies to tamed Euler, projected Euler, and backward Euler methods under dissipativity and smoothness conditions, with detailed estimates for the Stein equation and variational processes. The results provide rigorous, sharp ergodic error bounds and extend ergodic analysis of numerical schemes to superlinear SODEs, with implications for long-time sampling tasks in diffusion models and related areas.

Abstract

This paper establishes the first-order convergence rate for the ergodic error of numerical approximations to a class of stochastic ODEs (SODEs) with superlinear coefficients and multiplicative noise. By leveraging the generator approach to the Stein method, we derive a general error representation formula for one-step numerical schemes. Under suitable dissipativity and smoothness conditions, we prove that the error between the accurate invariant measure $π$ and the numerical invariant measure $π_τ$ is of order $\mathscr{O}(τ)$, which is sharp. Our framework applies to several recently studied schemes, including the tamed Euler, projected Euler, and backward Euler methods.

Ergodic Estimates of One-Step Numerical Approximations for Superlinear SODEs

TL;DR

We study ergodic error for stochastic ODEs with superlinear growth and multiplicative noise by developing a Stein-method framework based on a Stein equation and its explicit solution . A one-step error representation using a discretized generator and a continuous interpolation yields a sharp first-order bound for a class of schemes. The framework applies to tamed Euler, projected Euler, and backward Euler methods under dissipativity and smoothness conditions, with detailed estimates for the Stein equation and variational processes. The results provide rigorous, sharp ergodic error bounds and extend ergodic analysis of numerical schemes to superlinear SODEs, with implications for long-time sampling tasks in diffusion models and related areas.

Abstract

This paper establishes the first-order convergence rate for the ergodic error of numerical approximations to a class of stochastic ODEs (SODEs) with superlinear coefficients and multiplicative noise. By leveraging the generator approach to the Stein method, we derive a general error representation formula for one-step numerical schemes. Under suitable dissipativity and smoothness conditions, we prove that the error between the accurate invariant measure and the numerical invariant measure is of order , which is sharp. Our framework applies to several recently studied schemes, including the tamed Euler, projected Euler, and backward Euler methods.
Paper Structure (9 sections, 4 theorems, 71 equations)

This paper contains 9 sections, 4 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Main Idea and Results
  3. Notations
  4. Stein Equation
  5. Ergodic Estimate of One-step Approximations
  6. Applications to Superlinear SODEs
  7. Assumptions and Preliminary Estimates
  8. Estimates of Stein Equation
  9. Ergodic Estimates of Numerical Schemes

Key Result

Theorem 2.1

For any $\varphi \in \mathcal{B}_b(\mathbb R^d)$ or ${\rm Lip}(\mathbb R^d)$ such that $P_\cdot \varphi \in L^1(0, \infty; \mathcal{C}_b^2(\mathbb R^d))$, Eq. ste admits a solution $f_\varphi \in \mathcal{C}_b^2(\mathbb R^d)$ given by

Theorems & Definitions (14)

  • Theorem 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Remark 3.1
  • Proposition 3.2
  • proof
  • Example 3.3
  • Remark 3.4
  • ...and 4 more