Multiple Stratonovich integral and Hu--Meyer formula for L\'{e}vy processes

TL;DR

This paper develops a unified framework for multiple stochastic integrals with respect to Lévy processes, deriving a general Hu--Meyer formula that relates Itô and Stratonovich integrals, with applications to Brownian motion, Poisson processes, and subordinators.

Contribution

It introduces a combinatorial approach to multiple integrals for Lévy processes and establishes a general Hu--Meyer formula linking Itô and Stratonovich integrals.

Findings

Derived a general Hu--Meyer formula for Lévy processes.

Unified treatment of Itô and Stratonovich integrals in this framework.

Provided pathwise interpretation for integrals with respect to subordinators.

Abstract

In the framework of vector measures and the combinatorial approach to stochastic multiple integral introduced by Rota and Wallstrom [Ann. Probab. 25 (1997) 1257--1283], we present an It\^{o} multiple integral and a Stratonovich multiple integral with respect to a L\'{e}vy process with finite moments up to a convenient order. In such a framework, the Stratonovich multiple integral is an integral with respect to a product random measure whereas the It\^{o} multiple integral corresponds to integrate with respect to a random measure that gives zero mass to the diagonal sets. A general Hu--Meyer formula that gives the relationship between both integrals is proved. As particular cases, the classical Hu--Meyer formulas for the Brownian motion and for the Poisson process are deduced. Furthermore, a pathwise interpretation for the multiple integrals with respect to a subordinator is given.