Strong solutions to stochastic Volterra equations

Abstract

In this paper stochastic Volterra equations admitting exponentially bounded resolvents are studied. After obtaining convergence of resolvents, some properties for stochastic convolutions are studied. Our main result provide sufficient conditions for strong solutions to stochastic Volterra equations.Comment: 16 pages. The existence of strong solutions under some general assumptions is proved. Some proofs changed and precise

Global Solutions to Stochastic Volterra Equations Driven by Lévy Noise

“…. , N. Therefore, using Assumption 4.1, estimate (28) together with Bernoulli's inequality, we get for n large enough that…”

“…Existence and uniqueness of solutions for Gaussian noise driven Volterra equations in various frameworks have been treated by several authors, see, for example, [4,5,6,7,8,9,14,19,26,27,28,29] for an incomplete list of papers and the references therein. For an additive fractional Brownian motion driven linear parabolic stochastic Volterra equation we refer to [50] while for linear additive square-integrable local martingale driven stochastic Volterra equations we refer [49].…”

Regularity of solutions to stochastic Volterra equations of convolution type

“…In the case a(0) = 0, from the formula (19), passing to the limit we would have only W S (t) = t 0 a(t − τ )AW S (τ )dτ + W (t), t ∈ [0, T ] , that is, the formula like (11).…”

Stochastic Volterra equations driven by cylindrical Wiener process

“…When a(t) is a completely positive function, sufficient conditions for existence of strong solutions for (1) were obtained in [9]. This was done using a method which involves the use of a resolvent family associated to the deterministic version of equation (1):…”

“…the results in [9] cannot be used directly for α > 1. On the other hand, for α ∈ (0, 1), we have a singularity of the kernel in t = 0.…”