Fractional calculus and path-wise integration for Volterra processes driven by L\'evy and martingale noise

TL;DR

This paper develops a pathwise integration framework for Volterra processes driven by Lévy or martingale noise, combining fractional calculus with stochastic and deterministic techniques to enhance modeling clarity and computational potential.

Contribution

It introduces a general pathwise integration method for Volterra processes driven by various noises, including Lévy and martingale, bridging stochastic and deterministic approaches.

Findings

Framework applicable to diverse noise types

Inclusion of subordinated Wiener processes

Foundation for future computational rules

Abstract

We introduce a pathwise integration for Volterra processes driven by L\'evy noise or martingale noise. These processes are widely used in applications to turbulence, signal processes, biology, and in environmental finance. Indeed they constitute a very flexible class of models, which include fractional Brownian and L\'evy motions and it is part of the so-called ambit fields. A pathwise integration with respect of such Volterra processes aims at producing a framework where modelling is easily understandable from an information perspective. The techniques used are based on fractional calculus and in this there is a bridging of the stochastic and deterministic techniques. The present paper aims at setting the basis for a framework in which further computational rules can be devised. Our results are general in the choice of driving noise. Additionally we propose some further details in the relevant context subordinated Wiener processes.