Asymptotic Stability of a Jump-Diffusion Equation and Its Numerical Approximation

Asymptotic linear stability is studied for stochastic differential equations (SDEs) that incorporate Poisson-driven jumps and their numerical simulation using theta-method 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 parameters for which an explicit, forward Euler method has better stability properties than its trapezoidal and backward Euler counterparts. Other computational experiments indicate that all theta methods reproduce the correct asymptotic linear stability for sufficiently small step sizes. By using a recent result of Appleby, Berkolaiko and Rodkina, we give a rigorous verification that both linear stability and instability are reproduced for small step sizes. This property is known not to hold for general, nonlinear problems.