It\^o formula for integral processes related to space-time L\'evy white noise

TL;DR

This paper presents a new proof of Itô's formula for integral processes driven by space-time Lévy white noise, facilitating analysis of SPDEs with such noise through maximal inequalities and chaos expansions.

Contribution

It provides an alternative proof of Itô's formula for space-time Lévy white noise processes, expanding tools for SPDE analysis with non-Gaussian noise.

Findings

New proof of Itô formula for Lévy white noise processes

Maximal inequality for stochastic integrals with Lévy noise

Itô chaos expansion similar to Gaussian case

Abstract

In this article, we give a new proof of the It\^o formula for some integral processes related to the space-time L\'evy white noise introduced in Balan (2015) as an alternative for the Gaussian white noise perturbing an SPDE. We discuss two applications of this result, which are useful in the study of SPDEs driven by a space-time L\'evy noise with finite variance: a maximal inequality for the $p$-th moment of the stochastic integral, and the It\^o representation theorem leading to a chaos expansion similar to the Gaussian case.