Stochastic linear multi-step methods Adaptive methods Mean-square convergence Mean-square numerical stability Mean-square consistency Small noise Two-step Maruyama methods. Mean-square convergence of stochastic multi-step methods with variable step-size Thorsten Sickenberger Sickenberger Thorsten Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 2005-20, 18

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

Thorsten Sickenberger

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 2005-20, 18

MSC 2000

60H35 Computational methods for stochastic equations
65C30 Stochastic differential and integral equations

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 $h-\varepsilon$ approach and we construct some stochastic linear multi-step methods with variable step-size that have order 2 behavior if the noise is small enough.

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