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

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-ε 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.