Two-sided error estimates for the  stochastic theta method

Wolf-Jürgen Beyn; Raphael Kruse

doi:10.3934/dcdsb.2010.14.389

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2010, Volume 14, Issue 2: 389-407. Doi: 10.3934/dcdsb.2010.14.389

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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

Received: March 31, 2009

Revised: July 31, 2009

Published: August 2010

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Abstract

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.

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Mathematics Subject Classification: Primary 65C20, 65C30, 65L20 ; Secondary 65L70, 65J15.

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