Mean square characterisation of a stochastic Volterra integrodifferential equation with delay

Abstract

In this paper the asymptotic behaviour of the mean square of the solution of a linear stochastic Volterra integro-differential equation with delay is entirely characterised. In the case when the solution is mean-square asymptotically stable or unstable the exact rate of growth or decay can be determined by the real solution of a transcendental equation which is constructed as a by-product of the proof. The proof of the mean square stability of an equation with fading memory is also sketched.