Regularity of stochastic Volterra equations by functional calculus methods

TL;DR

This paper investigates the regularity of solutions to stochastic Volterra equations with additive noise, using functional calculus and dilation theorems to establish pathwise continuity properties.

Contribution

It introduces a novel approach combining $H^ abla$-calculus and dilation theorems to analyze the regularity of stochastic Volterra equations.

Findings

Established pathwise continuity of solutions.

Applied $H^ abla$-calculus to stochastic equations.

Used dilation theorem for positive definite operator families.

Abstract

We establish pathwise continuity properties of solutions to a stochastic Volterra equation with an additive noise term given by a local martingale. The deterministic part is governed by an operator with an $H^\infty$-calculus and a scalar kernel. The proof relies on the dilation theorem for positive definite operator families on a Hilbert space.