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Asymptotic mean‐square stability of linear multi‐step methods for SODEs
Author(s) -
Buckwar Evelyn,
Winkler Renate
Publication year - 2006
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200610310
Subject(s) - mathematics , exponential stability , mean square , stability (learning theory) , stochastic differential equation , square (algebra) , ordinary differential equation , lyapunov function , linear multistep method , invariant (physics) , mathematical analysis , differential equation , nonlinear system , computer science , differential algebraic equation , physics , geometry , quantum mechanics , mathematical physics , machine learning
In this article we present results of a linear stability analysis of stochastic linear multi‐step methods for stochastic ordinary differential equations. As in deterministic numerical analysis we use a linear time‐invariant test equation and study when the numerical approximation shares asymptotic properties in the mean‐square sense of the exact solution of that test equation. Sufficient conditions for asymptotic mean‐square stability of stochastic linear two‐step‐Maruyama methods are obtained with the aide of Lyapunov‐type functionals. In particular we study the asymptotic mean‐square stability of stochastic counterparts of two‐step Adams‐Bashforth‐ and Adams‐Moulton‐methods and the BDF method. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)