High weak order methods for stochastic differential equations based on modified equations

Assyr Abdulle (1) , David Cohen (2) , Gilles Vilmart (3, 4) , Konstantinos Zygalakis (5)

1 EPFL - Ecole Polytechnique Fédérale de Lausanne
2 Umeå University
3 IPSO - Invariant Preserving SOlvers
4 IRMAR - Institut de Recherche Mathématique de Rennes
5 School of Mathematics [Southampton]

Inspired by recent advances in the theory of modified differential equations, we propose a new methodology for constructing numerical integrators with high weak order for the time integration of stochastic differential equations. This approach is illustrated with the constructions of new methods of weak order two, in particular, semi-implicit integrators well suited for stiff (mean-square stable) stochastic problems, and implicit integrators that exactly conserve all quadratic first integrals of a stochastic dynamical system. Numerical examples confirm the theoretical results and show the versatility of our methodology.

Mots clés: weak convergence; modified equations; backward error analysis; stiff integrator; invariant preserving integrator

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

Assyr Abdulle, David Cohen, Gilles Vilmart, Konstantinos Zygalakis. High weak order methods for stochastic differential equations based on modified equations. SIAM Journal on Scientific Computing, 2012, 34 (3), pp.1800-1823. ⟨10.1137/110846609⟩. ⟨hal-00746563⟩