Weak backward error analysis for SDEs

TL;DR

This paper develops a weak backward error analysis for Euler approximations of elliptic and hypoelliptic SDEs, showing the generator matches a modified Kolmogorov equation and the scheme's invariant measure is close to a modified one.

Contribution

It provides a high-order weak backward error analysis for Euler schemes applied to elliptic and hypoelliptic SDEs, linking numerical solutions to modified equations.

Findings

Generator of the numerical scheme matches a modified Kolmogorov equation up to high order.

Invariant measures of the scheme are close to modified invariant measures.

The Euler scheme exhibits exponential mixing up to negligible terms.

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.