Ergodic Estimates of One-Step Numerical Approximations for Superlinear SODEs

TL;DR

This paper proves the first-order convergence rate of ergodic errors for numerical schemes approximating superlinear stochastic ODEs with multiplicative noise, using a novel generator-based Stein method approach.

Contribution

It introduces a general error representation formula for one-step schemes and establishes sharp $ au$ order convergence under dissipativity and smoothness conditions.

Findings

Error between invariant measures is of order $ au$

Applicable to tamed Euler, projected Euler, and backward Euler methods

Framework provides sharp convergence rate results

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 $\pi$ and the numerical invariant measure $\pi_\tau$ is of order $\mathscr{O}(\tau)$, which is sharp. Our framework applies to several recently studied schemes, including the tamed Euler, projected Euler, and backward Euler methods.