Almost sure stability of linear Itô-Volterra equations with damped stochastic perturbations (Q1860599)

The author considers the following stochastic Volterra differential equation \[ dX(t)= \Biggl(AX(t)+\int_0^t K(t-s)X(s) ds\Biggr) dt +\Sigma(t) dW(t), \] where \(A\) is a matrix, \(K\) is a continuous and integrable matrix-valued function and \(W\) is a Brownian motion. Under the assumptions that the underlying deterministic Volterra differential equation is uniformly asymptotically stable and that its resolvent decreases polynomially, the almost sure stability of the zero solution of the stochastic equation is derived if the volatility \(\Sigma\) obeys a certain rate of decay. Under a monotonicity restriction on \(\Sigma\) a converse result is established that the rate of decay for \(\Sigma\) is necessary. Some extensions of the results to linear equations with bounded delay and to linear neutral difference equations are discussed.