It\^o integral for a two-sided L\'evy process

TL;DR

This paper develops an Itô integral for two-sided Lévy processes without Gaussian parts, providing moment estimates and linking it to Poisson-Malliavin calculus.

Contribution

It introduces a new Itô integral for two-sided Lévy processes and connects it to existing stochastic calculus frameworks.

Findings

Provides $p$-th moment estimates for the integral.

Shows the integral extends the Hitsuda-Skorohod integral.

Establishes a connection with Poisson-Malliavin calculus.

Abstract

In this article, we construct an It\^o integral with respect to a two-sided finite-variance L\'evy process $\{L(x)\}_{x\in \mathbb{R}}$, without a Gaussian component. Using Rosenthal inequality for discrete-time martingales, we give an estimate for the $p$-th moment of this integral, for any even integer $p\geq 2$. Then, using Poisson-Malliavin calculus, we show that the It\^o integral is an extension of the Hitsuda-Skorohod integral with respect to the compensated Poisson random measure associated to the L\'evy process.