Exponential stability of a kind of stochastic delay difference equations

We present a Razumilchin-type theorem for stochastic delay difference equation, and use it to investigate the mean square exponential stability of a kind of nonautonomous stochastic difference equation which may also be viewed as an approximation of a nonautonomous stochastic delay integrodifferential equations (SDIDEs), and of a difference equation arises from some of the earliest mathematical models of the macroeconomic "trade cycle" with the environmental noise.