Mean-square approximation for stochastic differential equations with small noises

New approach to construction of mean-square numerical methods for solution of stochastic differential equations with small noises is proposed. The approach is based on expanding of the exact solution of the system with small noises by powers of time increment and regrouping of expansion terms according to powers of time increment and small parameter. The theorem on mean-square estimate of method errors is proved. Various efficient numerical schemes are derived for a general system with small noises and for systems with small additive and small colored noises. The proposed methods are tested by calculation of Lyapunov exponents and simulation of a laser Langevin equation with multiplicative noises