Euler-type approximation for the invariant measure: An abstract framework

2025/09/04 by Alfonsi, Aurélien, Bally, Vlad, Kohatsu-Higa, Arturo
#37M25 #60G99 #FOS: Mathematics #G5C99 #Probability (math.PR)

paper · doi:10.48550/arxiv.2509.03971

Abstract

We establish a general framework to study the rate of convergence of a Euler type approximation scheme with decreasing time steps to the invariant measure, for a general class of stochastic systems. The error is measured in general Wasserstein distances, which enables to encompass cases with non global contractivity conditions. Our main assumption is a coupling property which is expressed in terms of the one-step approximation. We show that the proposed set-up can be applied to a wide range of equations that may be law dependent, such as Langevin equations, reflected equations, Boltzmann type equations and for a recent McKean Vlasov type model for neuronal activity.

Citations

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Related

• Approximation for the invariant measure with applications for jump processes (convergence in total variation distance)
• Empirical approximation to invariant measures of non-degenerate McKean-Vlasov dynamics
• Computation of the invariant measure for a Lévy driven SDE: Rate of convergence
• Discretization, Uniform-in-Time Estimations and Approximation of Invariant Measures for Nonlinear Stochastic Differential Equations with Non-Uniform Dissipativity
• Approximation of the invariant measure of stable SDEs by an Euler--Maruyama scheme