Multiple stratonovich integral and Hu-Meyer formula for Lévy processes

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 Itô multiple integral and a Stratonovich multiple integral with respect to a Lévy 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 Itô 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. © Institute of Mathematical Statistics, 2010.