Weak backward error analysis for SDEs

Arnaud Debussche (1) , Erwan Faou (2, 1)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 IPSO - Invariant Preserving SOlvers

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.

Mots clés: backward error analysis; stochastic differential equations; exponential mixing; numerical scheme; Kolmogorov equation; weak error

Domaines: Analyse numérique [math.NA]; Probabilités [math.PR]

Arnaud Debussche, Erwan Faou. Weak backward error analysis for SDEs. SIAM Journal on Numerical Analysis, 2012, 50 (3), pp.1735-1752. ⟨10.1137/110831544⟩. ⟨hal-00589881⟩