Statistical modeling of random processes with invariants

The main goal of this paper is to study and to test the numerical methods for stochastic differential equations with solutions on a given smooth manifold. Second-order cylindrical surfaces such as elliptic, hyperbolic, and parabolic cylinders are chosen as manifolds in three-dimensional space provided that the phase space is two-dimensional space. The classes of stochastic differential equations are formed for the considered manifolds and linear equations with multiplicative noise that are the particular case of these classes are concerned. The numerical methods accuracy as the mean distance between solutions and the given smooth manifold is estimated. The comparative analysis of the results obtained by using the different numerical methods is given.