Properties of convolutions arising in stochastic Volterra equations

The aim of the paper is to provide some regularity results for stochastic convolutions corresponding to stochastic Volterra equations in separable Hilbert space. We study convolutions of the form WΨ(t) := ∫ t 0 S(t − τ)Ψ(τ)dW (τ), t ≥ 0, where S(t), t ≥ 0, is so-called resolvent for Volterra equation considered,Ψ is an appropriate process and W is a cylindrical Wiener process. In the paper we extend the semigroup approach to stochastic convolutions with resolvent operators. 1 Definitions and notation In the paper we consider the following stochastic Volterra equation in a separable Hilbert space H : X(t) = X0 + ∫ t 0 a(t− τ)AX(τ)dτ +Ψ(t)W (t), (1) where t ∈ R+, a ∈ L 1 loc(R+), A is a closed unbounded linear operator in H , Ψ is an adapted integrable stochastic process specified below, W is a cylindrical Wiener process with respect to t and X0 belongs to H .