In this paper the numerical approximation of solutions of Itô stochastic differential
equations is considered, in particular for equations with a small parameter ? in the noise coex-
cient. We construct stochastic linear multi-step methods and develop the fundamental numerical
analysis concerning their mean-square consistency, numerical stability in the mean-square sense and
mean-square convergence. For the special case of two-step Maruyama schemes we derive conditions
guaranteeing their mean-square consistency. Further, for the small noise case we obtain expansions
of the local error in terms of the stepsize and the small parameter ?. Simulation results using several
explicit and implicit stochastic linear k-step schemes, k = 1; 2, illustrate the theoretical findings.
A strategy for controlling the stepsize in the numerical integration of stochastic
differential equations (SDEs) is presented. It is based on estimating the p-th mean of
local errors. The strategy leads to deterministic stepsize sequences that are identical
for all paths. For the family of Euler schemes for SDEs with small noise we derive
computable estimates for the dominating term of the p-th mean of local errors
and show that the strategy becomes efficient for reasonable stepsizes. Numerical
experience is reported for test examples including scalar SDEs and a stochastic
circuit model.