# American Institute of Mathematical Sciences

February  2013, 33(2): 885-903. doi: 10.3934/dcds.2013.33.885

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

 1 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074 2 Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, United Kingdom 3 Institut für Mathematik, Johann Wolfgang Goethe Universität, D-60054 Frankfurt am Main, Germany

Received  August 2011 Revised  December 2011 Published  September 2012

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.
Citation: Fuke Wu, Xuerong Mao, Peter E. Kloeden. Discrete Razumikhin-type technique and stability of the Euler--Maruyama method to stochastic functional differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 885-903. doi: 10.3934/dcds.2013.33.885
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