Abstract: Asymptotic characteristic of solution of the stochastic functional differential equation was discussed and sufficient condition was established by multiple Lyapunov functions for locating the limit set of the solution. Moreover, from them many effective criteria on stochastic asymptotic stability, which enable us to construct the Lyapunov functions much more easily in application, were obtained. The results show that the well-known classical theorem on stochastic asymptotic stability is a special case of our more general results. In the end, application in stochastic Hopfield neural networks is given to verify our results.

Key words: stochastic functional differential equation, stochastic neural network, asymptotic stability, semi-martingale convergence theorem, Itô, formula