Characterization of bistability for stochastic multistep methods

Abstract

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.