On the Dynamic Consistency of the Split Step Method for Classifying the Asymptotic Behaviour of Globally Stable Differential Equations perturbed by State--independent Stochastic terms

In this paper we classify the pathwise asymptotic behaviour of the discretisation of a general autonomous scalar differential equation which has a unique and globally stable equilibrium. The underlying continuous equation is subjected to a stochastic perturbation whose intensity is state--independent. In the main result, it is shown that when the split--step--method is applied to the resulting stochastic differential equation, and the stochastic intensity is decreasing, the solutions of the discretised equation inherit the asymptotic behaviour of the continuous equation, regardless of whether the continuous equation has stable, bounded but unstable, or unbounded solutions, provided the step size is chosen sufficiently small.