BIT Numerical Mathematics

, Volume 52, Issue 1, pp 109–140 | Cite as

Characterization of bistability for stochastic multistep methods

  • Raphael Kruse


The focus of this article lies on the bistability of multistep methods applied to stochastic ordinary differential equations. Here bistability is understood in the sense of F. Stummel and leads to two-sided estimates of the strong error of convergence. It is shown that bistability can be characterized by Dahlquist’s strong root condition. The main ingredient of the stability analysis is a stochastic version of Spijker’s norm.

We use our results to discuss the maximum order of convergence for higher order schemes. In particular, we are concerned with the stochastic theta method, BDF2-Maruyama and higher order Itô-Taylor schemes.


Bistability SODE Itô-Taylor schemes BDF2-Maruyama Stochastic multistep method Stochastic theta method Two-sided error estimate Stochastic Spijker norm 

Mathematics Subject Classification (2000)

65C20 65C30 65L06 65L20 65L70 


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© Springer Science + Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of MathematicsBielefeld UniversityBielefeldGermany

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