Lévy Systems and Moment Formulas for Mixed Poisson Integrals

  • Krzysztof Bogdan
  • Jan Rosiński
  • Grzegorz Serafin
  • Łukasz Wojciechowski
Conference paper
Part of the Progress in Probability book series (PRPR, volume 72)


We propose Mecke-Palm formula for multiple integrals with respect to the Poisson random measure and its intensity measure performed, or mixed, in an arbitrary order. We apply the formulas to mixed Lévy systems of Lévy processes and obtain moment formulas for mixed Poisson integrals.


Lévy system Poisson-Skorochod integral 

2010Mathematics Subject Classification

60G51 60G57 



The inspiration to study Lévy systems came from the joint work of the first named author with Rodrigo Bañuelos [1], and our interest in moment formulas is due to the work of Nicolas Privault [21]. The present paper is based in part on the PhD dissertation of the fourth named author [24]. We thank Mateusz Kwaśnicki for many discussions and suggestions, Jean Jacod and Alexandre Popier for references to literature, Anita Behme for discussions on the Lévy integral in Sect. 4.2, Nicolas Privault for a discussion of our results and Aleksander Janicki and Rodrigo Bañuelos for their inspirations. K. Bogdan and Ł. Wojciechowski were supported in part by NCN grant 2012/07/B/ST1/03356. J. Rosiński’s research was partially supported by a grant # 281440 from the Simons Foundation.


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Krzysztof Bogdan
    • 1
  • Jan Rosiński
    • 2
  • Grzegorz Serafin
    • 1
  • Łukasz Wojciechowski
    • 3
  1. 1.Faculty of Pure and Applied MathematicsWrocław University of Science and TechnologyWrocławPoland
  2. 2.Department of MathematicsUniversity of TennesseeKnoxvilleUSA
  3. 3.Mathematical InstituteUniversity of WrocławWrocławPoland

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