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 journalArticle

99 Citations (Scopus)

Abstract

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.
LanguageEnglish
Pages592-609
Number of pages17
JournalSIAM Journal on Numerical Analysis
Volume45
Issue number2
DOIs
Publication statusPublished - 2007

Fingerprint

Exponential Stability
Asymptotic stability
Stochastic Equations
Differential equations
Differential equation
Moment
Numerical Simulation
Computer simulation
Euler
Euler-Maruyama Method
Backward Euler Method
Lipschitz condition
Growth Conditions
Breakdown
Numerical methods
Numerical Methods
Scalar
Tend
Generalise
Zero

Keywords

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

Cite this

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Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations. / Higham, D.J.; Mao, X.; Yuan, C.

In: SIAM Journal on Numerical Analysis, Vol. 45, No. 2, 2007, p. 592-609.

Research output: Contribution to journalArticle

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AU - Mao, X.

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