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

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Abstract

This paper is mainly concerned with whether the almost sure exponential stability of stochastic differential equations (SDEs) is shared with that of a numerical method. Under the global Lipschtiz condition, we first show that the SDE is pth moment exponentially stable (for p 2 (0; 1)) if and only if the stochastic theta method is pth moment exponentially stable for a sufficiently small step size. We then show that the pth moment exponential stability of the SDE or the stochastic theta method implies the almost sure exponential stability of the SDE or the stochastic theta method, respectively. Hence, our new theory enables us to study the almost sure exponential stability of the SDEs using the stochastic theta method, instead of the method of the Lyapunov functions. That is, we can now carry out careful numerical simulations using the stochastic theta method with a sufficiently small step size t. If the stochastic theta method is pth moment exponentially stable for a sufficiently small p 2 (0; 1), we can then infer that the underlying SDE is almost surely exponentially stable. Our new theory also enables us to show the ability of the stochastic theta method to reproduce the almost sure exponential stability of the SDEs. In particular, we give positive answers to two open problems (P1) and (P2) listed on page 2.
Original languageEnglish
Pages (from-to)370-389
Number of pages20
JournalSIAM Journal on Numerical Analysis
Volume53
Issue number1
DOIs
Publication statusPublished - 3 Feb 2015

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Almost Sure Exponential Stability
θ-method
Asymptotic stability
Stochastic Equations
Differential equations
Differential equation
Numerical Simulation
Computer simulation
Moment
Lyapunov functions
Exponential Stability
Lyapunov Function
Open Problems
Numerical methods
Numerical Methods
If and only if
Imply

Keywords

  • almost sure exponential stability
  • linear growth condition
  • Lipschitz condition
  • stochastic theta method
  • moment exponential stability

Cite this

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Almost sure exponential stability in the numerical simulation of stochastic differential equations. / Mao, Xuerong.

In: SIAM Journal on Numerical Analysis, Vol. 53, No. 1, 03.02.2015, p. 370-389.

Research output: Contribution to journalArticle

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