Asymptotic stability of a jump-diffusion equation and its numerical approximation

Graeme, D Chalmers, Desmond J. Higham

Research output: Contribution to journalArticlepeer-review

23 Citations (Scopus)


Asymptotic linear stability is studied for stochastic dierential 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 suffciently small step sizes. By using a recent result of Appleby, Berkolaiko and Rodkina, we give a rigorous verication that both linear stability and instability are reproduced for small step sizes. This property is known not to hold for general, nonlinear problems.
Original languageEnglish
Pages (from-to)1141-1155
Number of pages14
JournalSIAM Journal on Scientific Computing
Issue number2
Publication statusPublished - 17 Dec 2008


  • asymptotic stability
  • backward Euler
  • Euler-Maruyama
  • jump-diusion
  • Poisson process
  • stochastic dierential equation
  • theta method
  • trapezoidal rule


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