Almost Sure Exponential Stability of Neutral Stochastic Differential Difference Equations*

A neutral stochastic differential difference equation dw x t . yG x tyt ..x sf t , x t ., x tyt .. dtqs t , x t ., x tyt .. dw t . was introduced by V. B. Kolmanovskii and Y. R. Nosov ‘‘Stability and Periodic Modes of Control Systems with Aftereffect,’’ Nauka, Moscow, 1981. several years ago. However, so far little is known about the almost sure exponential stability for such equations and the aim of this paper is to close this gap. The convergence of nonnegative special semimartingales established by R. Sh. Lipster and A. N. Shiryayev ‘‘Theory of Martingales,’’ Kluwer Academic, Dordrecht, 1989. and the Itˆo formula will play a key role in this paper.