Certain aspects of application of numerical methods for solving SDE systems

The problems of determining the structure of numerical method, the choice of its parameters, analysis of meansquare or weak convergence of the numerical solution to the true one are much more complicated for systems of SDE, than for those of ODE. Nevertheless, many theoretical and practical ideas of the numerical methods of ODE solution can be transferred or extended onto the numerical methods of SDE solution. In particular, the notion of absolute stability has induced searching for the methods conserving the one-dimensional distribution of the model SDE solution. Elementary transfer of the absolute stability notion onto the numerical methods of SDE solution is not constructive, since it is possible that the numerical method be absolutely stable for the given integration stepsize, while the critical situations occur in the process of trajectory simulation. A cause of this phenomena is an unstable growth of the variance of the numerical solution. Thus it is necessary to define for a numerical method of SDE solution not only the stability region in the sense of ODE, but also the conservation conditions of the variance of the stationary solution of the model equations.