Exponential stability of Itô-type linear functional difference equations

We study the stability properties of rather general linear stochastic functional difference equations and offer a partial justification of an important result in the stability analysis, which is known as “the Bohl–Perron principle” and which helps us to deduce exponential Lyapunov stability from the input-to-state stability with respect to non-weighted functional spaces. We use a special technique based on integral regularization, which proved to be powerful in the general theory of linear functional differential and difference equations. In addition to the general framework, we provide a number of examples demonstrating the efficiency of our results.