Asymptotic behaviours of stochastic differential delay equations

Yi Shen, Xuerong Mao

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Most of the existing results on stochastic stability use a single Lyapunov function, but we shall instead use multiple Lyapunov functions in this paper. We shall establish the sufficient condition, in terms of multiple Lyapunov functions, for the asymptotic behaviours of solutions of stochastic differential delay equations. Moreover, from them follow many effective criteria on stochastic asymptotic stability, which enable us to construct the Lyapunov functions much more easily in applications. In particular, the well-known classical theorem on stochastic asymptotic stability is a special case of our more general results. These show clearly the power of our new results. Two examples are also given for illustration.
Original languageEnglish
Pages (from-to)21–27
Number of pages7
JournalAsian Journal of Control
Issue number1
Publication statusPublished - Mar 2006


  • Lyapunov function
  • asymptotic stability
  • semi-martingale convergence theorem
  • Ito formula


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