Stochastic Volterra equations under perturbations

Anna Karczewska,Carlos Lizama

doi:10.1214/ecp.v19-3365

Abstract

We study stochastic perturbed Volterra equations of convolution type in an infinite dimensional case. Our interest is directed towards the existence and regularity of stochastic convolutions connected to the equations considered under some kind of perturbations. We use an operator theoretical method for the representation of solutions.

Highlights

Since abstract linear Volterra equations are being used in many applications, it is becoming increasingly desirable to study stochastic perturbations
Recently it has been demonstrated that the use of one-parameter systems of bounded and linear operators can help to analyze the existence of strong solutions to stochastic Volterra equations in the infinite dimensional framework [10], [11]
In the current paper we study stochastic Volterra equations (SVEs) of the convolution type with some perturbations

Summary

Introduction

Since abstract linear Volterra equations are being used in many applications, it is becoming increasingly desirable to study stochastic perturbations. Liu has generalized classical results in a different direction than we have done He has studied SRDEs with some kind of retarding part, some times referred to as the Hale type operator. This result will be used in the remaining part of the paper where we study the stochastic equation (1.2) in Hilbert spaces.

Preliminaries

Representation of the solution and qualitative properties

Stochastic convolution

Itô-type formula

Continuity of trajectories of stochastic convolution