Asymptotic properties of integro-differential systems with wideband noise perturbations

Because the class of wideband noise is quite realistic in applications and is a good approximation to white noise, this paper focuses on the asymptotic properties of integro-differential systems with wideband noise perturbations. Using perturbed test function methods combined with the martingale averaging techniques, and weak convergence methods, it is shown that when the small parameter tends to zero, the underlying process converges weakly to a stochastic process satisfying a stochastic integro-differential equation. Inspired by the developments from [3] and [9], the novelty of our paper lies in the treatment of functional differential equations in which differential-integral equations are dealt with. Because of the features of the systems involved, different methods for estimates and proofs from the aforementioned references are developed in this paper. Not only are the results interesting from a mathematical point of view, but also they will be of great utility to a wide range of problems in control and optimization problems.