Numerical Methods in the Weak Sense for Stochastic Differential Equations with Small Noise

Abstract

We propose a new approach to constructing weak numerical methods for finding solutions to stochastic systems with small noise. For these methods we prove an error estimate in terms of products $h^i\varepsilon ^j$ (h is a time increment, $\varepsilon $ is a small parameter). We derive various efficient weak schemes for systems with small noise and study the Talay--Tubaro expansion of their global error. An efficient approach to reducing the Monte-Carlo error is presented. Some of the proposed methods are tested by calculating the Lyapunov exponent of a linear system with small noise.