Asymptotic linear stability is studied for stochastic differential equations (SDEs) that incorporate Poisson-driven jumps and their numerical simulation using Eulertype discretisations. The property is shown to have a simple explicit characterisation for the SDE, whereas for the discretisation a condition is found that is amenable to numerical evaluation. This allows us to evaluate the asymptotic stability behaviour of the methods. One surprising observation is that there exist problem

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... s for which an explicit, forward Euler-based method has better stability than its trapezoidal and backward Euler counterparts. Other computational experiments indicate that all Euler-type methods reproduce the correct asymptotic stability for sufficiently small step sizes. By using a recent result of Appleby, Berkolaiko and Rodkina, we give a rigorous verification that both stability and instability are reproduced for small step sizes. This property is known not to hold for general, nonlinear problems.