Ergodic Estimates of One-Step Numerical Approximations for Superlinear SODEs

2025/10/24 by Liu, Xin, Liu, Zhihui
#37M25 #60H35 #65C30 #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2510.21279

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 \mathscrO(τ), which is sharp. Our framework applies to several recently studied schemes, including the tamed Euler, projected Euler, and backward Euler methods.

Citations

• Inversions of stochastic processes from ergodic measures of Nonlinear SDEs
• Weak Error Estimates of Ergodic Approximations for Monotone Jump-diffusion SODEs
• Non-asymptotic Error Analysis of Explicit Modified Euler Methods for Superlinear and Non-contractive SODEs
• Geometric Ergodicity and Strong Error Estimates for Tamed Schemes of Super-linear SODEs
• The backward Euler-Maruyama method for invariant measures of stochastic differential equations with super-linear coefficients
• Unadjusted Langevin algorithm with multiplicative noise: Total variation\n and Wasserstein bounds
• The Numerical Invariant Measure of Stochastic Differential Equations With Markovian Switching
• Multivariate approximations in Wasserstein distance by Stein's method and Bismut's formula
• Invariant measures and Euler-Maruyama's approximations of state-dependent regime-switching diffusions
• Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise

Related

• Geometric Ergodicity and Strong Error Estimates for Tamed Schemes of Super-linear SODEs
• Weak Error Estimates of Ergodic Approximations for Monotone Jump-diffusion SODEs
• Non-asymptotic Error Analysis of Explicit Modified Euler Methods for Superlinear and Non-contractive SODEs
• Weak error analysis for strong approximation schemes of SDEs with super-linear coefficients
• Weak error analysis for strong approximation schemes of SDEs with super-linear coefficients