Annals of Probability

Multiple Stratonovich integral and Hu–Meyer formula for Lévy processes

Mercè Farré, Maria Jolis, and Frederic Utzet

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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.

Article information

Ann. Probab., Volume 38, Number 6 (2010), 2136-2169.

First available in Project Euclid: 24 September 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G51: Processes with independent increments; Lévy processes 60H99: None of the above, but in this section
Secondary: 05A18: Partitions of sets 60G57: Random measures

Lévy processes Stratonovich integral Hu–Meyer formula random measures Teugels martingales


Farré, Mercè; Jolis, Maria; Utzet, Frederic. Multiple Stratonovich integral and Hu–Meyer formula for Lévy processes. Ann. Probab. 38 (2010), no. 6, 2136--2169. doi:10.1214/10-AOP528.

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