High order numerical approximation of the invariant measure of ergodic SDEs
A. Abdulle, G. Vilmart, and K.C. Zygalakis
Abstract.
We introduce new sufficient conditions for a numerical method to approximate
with high order of accuracy the invariant measure of an ergodic system of stochastic
differential equations, independently of the weak order of accuracy of the method. We
then present a systematic procedure based on the framework of modified differential
equations for the construction of stochastic integrators that capture the invariant
measure of a wide class of ergodic SDEs (Brownian and Langevin dynamics) with
an accuracy independent of the weak order of the underlying method. Numerical
experiments confirm our theoretical findings.
Key Words. stochastic differential equations, weak convergence, modified differential
equations, backward error analysis, invariant measure, ergodicity.