The invariance of asymptotic laws of linear stochastic systems under discretization

The stochastic trapezoidal rule provides the only equidistant discretizatio n scheme from the family of implicit Euler methods (see [12]) which possess es the same asymptotic (stationary) law as underlying continuous time, line ar and autonomous stochastic systems with white or coloured noise. This ide ntity holds even when integration time goes to infinity, independent of use d integration step size! Especially, the asymptotic behaviour of first two moments of corresponding probability distributions is rigorously examined a nd compared in this paper. The coincidence of asymptotic moments is shown f or autonomous systems with multiplicative (parametric) and additive noise u sing fixed point principles and the theory of positive operators. The I;key result turns out to be useful for adequate implementation of stochastic al gorithms applied to numerical solution of autonomous stochastic differentia l equations. In particular, it has practical importance when accurate long time integration is required such as in the process of estimation of Lyapun ov exponents or stationary measures for oscillators in mechanical engineeri ng.