Exact Pathwise and Mean--Square Asymptotic Behaviour of Stochastic Affine Volterra and Functional Differential Equations

The almost sure rate of exponential-polynomial growth or decay of affine stochastic Volterra and affine stochastic finite-delay equations is investigated. These results are achieved under suitable smallness conditions on the intensities of the deterministic and stochastic perturbations diffusion, given that the asymptotic behaviour of the underlying deterministic resolvent is determined by the zeros of its characteristic equation. The results rely heavily upon a stochastic variant of the admissibility theory for linear Volterra operators.