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

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