On the Admissibility of Linear Stochastic Volterra Operators

TL;DR

This paper investigates the conditions under which linear stochastic Volterra operators converge, establishing necessary and sufficient criteria for mean square and almost sure convergence, and exploring their sharpness through examples.

Contribution

It provides a comprehensive analysis of convergence conditions for stochastic Volterra operators, including new necessary and sufficient criteria and their validation via examples.

Findings

Necessary and sufficient conditions for mean square convergence.

Almost sure convergence implies mean square convergence.

Examples demonstrating the sharpness of the conditions.

Abstract

Conditions guaranteeing convergence of linear stochastic Volterra operators are studied. Necessary and sufficient conditions for mean square convergence are established, while almost sure convergence of the linear operator is shown to imply mean square convergence. Sufficient conditions for almost sure convergence of the stochastic linear operator are established. The sharpness or necessity of these conditions is explored by means of examples.