Discrete Razumikhin-type technique and stability of the Euler-Maruyama method to stochastic functional differential equations

Fuke Wu, Xuerong Mao, Peter E. Kloeden

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Abstract

A discrete stochastic Razumikhin-type theorem is established to investigate whether the Euler--Maruyama (EM) scheme can reproduce the moment exponential stability of exact solutions of stochastic functional differential equations (SFDEs). In addition, the Chebyshev inequality and the Borel-Cantelli lemma are applied to show the almost sure stability of the EM approximate solutions of SFDEs. To show our idea clearly, these results are used to discuss stability of numerical solutions of two classes of special SFDEs, including stochastic delay differential equations (SDDEs) with variable delay and stochastically perturbed equations.
Original languageEnglish
Pages (from-to)885-903
Number of pages19
JournalDiscrete and Continuous Dynamical Systems - Series A
Volume33
Issue number2
DOIs
Publication statusPublished - Feb 2013

Keywords

  • moment exponential stability
  • razumikhin-type thoerem
  • euler--maruyama method
  • stochastic functional differential equation
  • stochastically perturbed equations
  • differential equations

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