On the approximation of L\'evy driven Volterra processes and their integrals

TL;DR

This paper introduces a method to approximate non-semimartingale Volterra processes driven by Lévy noise with semimartingales through kernel perturbation, enabling fractional integration analysis and practical simulations.

Contribution

It proposes a novel kernel perturbation approach to approximate Lévy-driven Volterra processes with semimartingales for fractional integration.

Findings

Successful approximation of Volterra processes by semimartingales.

Effective fractional integral approximation demonstrated.

Simulation results validate the approach.

Abstract

Volterra processes appear in several applications ranging from turbulence to energy finance where they are used in the modelling of e.g. temperatures and wind and the related financial derivatives. Volterra processes are in general non-semimartingales and a theory of integration with respect to such processes is in fact not standard. In this work we suggest to construct an approximating sequence of L\'evy driven Volterra processes, by perturbation of the kernel function. In this way, one can obtain an approximating sequence of semimartingales. Then we consider fractional integration with respect to Volterra processes as integrators and we study the corresponding approximations of the fractional integrals. We illustrate the approach presenting the specific study of the Gamma-Volterra processes. Examples and illustrations via simulation are given.