  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.
