Almost sure exponential stability of the Euler–Maruyama approximations for stochastic functional differential equations

Fuke Wu, Xuerong Mao, Peter E. Kloeden

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By the continuous and discrete nonnegative semimartingale convergence theorems,
this paper investigates conditions under which the Euler–Maruyama (EM) approximations of stochastic functional differential equations (SFDEs) can share the almost sure exponential stability of the exact solution. Moreover, for sufficiently small stepsize, the decay rate as measured by the Lyapunov exponent can be reproduced arbitrarily accurately.
Original languageEnglish
Pages (from-to)105-216
Number of pages22
JournalRandom Operator and Stochastic Equations
Issue number2
Publication statusPublished - 2011


  • stochastic functional differential equations (SFDEs)
  • nonnegative semimartingale convergence theorem
  • almost sure stability
  • EM method

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