Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations

D.J. Higham, X. Mao, C. Yuan

Research output: Contribution to journalArticlepeer-review

113 Citations (Scopus)
157 Downloads (Pure)


Relatively little is known about the ability of numerical methods for stochastic differential equations (SDEs) to reproduce almost sure and small-moment stability. Here, we focus on these stability properties in the limit as the timestep tends to zero. Our analysis is motivated by an example of an exponentially almost surely stable nonlinear SDE for which the Euler-Maruyama (EM)method fails to reproduce this behavior for any nonzero timestep. We begin by showing that EM correctly reproduces almost sure and small-moment exponential stability for sufficiently small timesteps on scalar linear SDEs. We then generalize our results to multidimensional nonlinear SDEs. We show that when the SDE obeys a linear growth condition, EM recovers almost surely exponential stability very well. Under the less restrictive condition that the drift coefficient of the SDE obeys a one-sided Lipschitz condition, where EM may break down, we show that the backward Euler method maintains almost surely exponential stability.
Original languageEnglish
Pages (from-to)592-609
Number of pages18
JournalSIAM Journal on Numerical Analysis
Issue number2
Publication statusPublished - 3 Apr 2007


  • backward Euler
  • Euler-Maruyama
  • implicit
  • one-sided Lipschitz condition
  • linear growth condition
  • Lyapunov exponent
  • stochastic theta method
  • numerical mathematics

Cite this