The invariance of asymptotic laws of stochastic systems under discretization

The stochastic trapezoidal rule provides the only discretization scheme from the family of implicit Euler methods (see [11]) which possesses the same asymptotic (stationary) law as underlying linear continuous time stochastic systems with white or coloured noise. This identity is shown for systems with multiplicative (para- metric) and additive noise using fixed point principles and the theory of positive operators. The key result is useful for adequate implementation of stochastic algorithms applied to numerical solution of autonomous stochastic differential equations. In particular it has practical importance when accurate long time integration is required such as in the process of estimation of Lyapunov exponents or stationary measures for oscillators in Mechanical Engineering