Multiple Stratonovich integral and Hu--Meyer formula for Lévy processes

Details Multiple Stratonovich integral and Hu--Meyer formula for Lévy processes Multiple Stratonovich integral and Hu--Meyer formula for Lévy processes

2008-02-21

arxiv.org/abs/0802.3112v2

Annals of Probability 2010, Vol. 38, No. 6, 2136-2169

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/10-AOP528 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/10-AOP528

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.

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