Mean-square convergence of stochastic multi-step methods with variable step-size

Authors:
Thorsten Sickenberger
Affiliations:
Humboldt-Universität zu Berlin, Institut für Mathematik, Unter den Linden 6, 10099 Berlin, Germany
Venue:
Journal of Computational and Applied Mathematics
Year:
2008

Citing 6
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Abstract

We study mean-square consistency, stability in the mean-square sense and mean-square convergence of drift-implicit linear multi-step methods with variable step-size for the approximation of the solution of Ito stochastic differential equations. We obtain conditions that depend on the step-size ratios and that ensure mean-square convergence for the special case of adaptive two-step-Maruyama schemes. Further, in the case of small noise we develop a local error analysis with respect to the h-@e approach and we construct some stochastic linear multi-step methods with variable step-size that have order 2 behaviour if the noise is small enough.