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September  2010, 14(2): 389-407. doi: 10.3934/dcdsb.2010.14.389

Two-sided error estimates for the stochastic theta method

Wolf-Jürgen Beyn 1, and  Raphael Kruse 1,

1. 

Department of Mathematics, Bielefeld University, P.O. Box 100131, 33501 Bielefeld, Germany

Received  April 2009 Revised  August 2009 Published  June 2010

Two-sided error estimates are derived for the strong error of convergence of the stochastic theta method. The main result is based on two ingredients. The first one shows how the theory of convergence can be embedded into standard concepts of consistency, stability and convergence by an appropriate choice of norms and function spaces. The second one is a suitable stochastic generalization of Spijker's norm (1968) that is known to lead to two-sided error estimates for deterministic one-step methods. We show that the stochastic theta method is bistable with respect to this norm and that well-known results on the optimal $\mathcal{O}(\sqrt{h})$ order of convergence follow from this property in a natural way.
Keywords: stochastic Spijker norm., two-sided error estimate, stochastic theta method, SODE, consistency, bistability.
Mathematics Subject Classification: Primary 65C20, 65C30, 65L20 ; Secondary 65L70, 65J1.
Citation: Wolf-Jürgen Beyn, Raphael Kruse. Two-sided error estimates for the stochastic theta method. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 389-407. doi: 10.3934/dcdsb.2010.14.389
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