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

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 hiεj (h is a time increment, ε 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.