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

Fuke Wu, Xuerong Mao, Peter E. Kloeden

Research output: Contribution to journalArticle

9 Citations (Scopus)

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.
LanguageEnglish
Pages885-903
Number of pages19
JournalDiscrete and Continuous Dynamical Systems - Series A
Volume33
Issue number2
DOIs
Publication statusPublished - Feb 2013

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Euler-Maruyama Method
Stochastic Functional Differential Equations
Differential equations
Euler
Borel-Cantelli Lemma
Chebyshev's inequality
Almost Sure Stability
Stochastic Delay Differential Equations
Variable Delay
Exponential Stability
Approximate Solution
Exact Solution
Asymptotic stability
Numerical Solution
Moment
Theorem

Keywords

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

Cite this

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title = "Discrete Razumikhin-type technique and stability of the Euler-Maruyama method to stochastic functional differential equations",
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.",
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AU - Wu, Fuke

AU - Mao, Xuerong

AU - Kloeden, Peter E.

PY - 2013/2

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N2 - 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.

AB - 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.

KW - moment exponential stability

KW - razumikhin-type thoerem

KW - euler--maruyama method

KW - stochastic functional differential equation

KW - stochastically perturbed equations

KW - differential equations

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JO - Discrete and Continuous Dynamical Systems - Series A

T2 - Discrete and Continuous Dynamical Systems - Series A

JF - Discrete and Continuous Dynamical Systems - Series A

SN - 1078-0947

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