Weak backward error analysis for SDEs

2011/05/03 by Arnaud Debussche, Erwan Faou, Debussche, Arnaud +1 · 1 citation
Economics, Econometrics and Finance · Engineering · Mathematics · #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Numerical Analysis (math.NA) #Numerical methods for differential equations #Probability (math.PR) #Stochastic processes and financial applications

paper · doi:10.48550/arxiv.1105.0489

openalex publication_date 2011/05/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider numerical approximations of stochastic differential equations by the Euler method. In the case where the SDE is elliptic or hypoelliptic, we show a weak backward error analysis result in the sense that the generator associated with the numerical solution coincides with the solution of a modified Kolmogorov equation up to high order terms with respect to the stepsize. This implies that every invariant measure of the numerical scheme is close to a modified invariant measure obtained by asymptotic expansion. Moreover, we prove that, up to negligible terms, the dynamic associated with the Euler scheme is exponentially mixing.

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