Lévy Systems and Moment Formulas for Mixed Poisson Integrals
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.
KeywordsLévy system Poisson-Skorochod integral
2010Mathematics Subject Classification60G51 60G57
The inspiration to study Lévy systems came from the joint work of the first named author with Rodrigo Bañuelos , and our interest in moment formulas is due to the work of Nicolas Privault . The present paper is based in part on the PhD dissertation of the fourth named author . 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.
- 8.C. Dellacherie, P.-A. Meyer, Probabilities and Potential. B (North-Holland, Amsterdam, 1982)Google Scholar
- 12.O. Kallenberg, Foundations of Modern Probability. Probability and Its Applications, 2nd edn. (Springer, New York, 2002)Google Scholar
- 17.W. Linde, Probability in Banach Spaces—Stable and Infinitely Divisible Distributions. A Wiley-Interscience Publication, 2nd edn. (Wiley, Chichester, 1986)Google Scholar
- 19.G. Peccati, M.S. Taqqu, Wiener Chaos: Moments, Cumulants and Diagrams. Bocconi & Springer Series, vol. 1 (Springer, Milan/Bocconi University Press, Milan, 2011). A survey with computer implementation, Supplementary material available onlineGoogle Scholar
- 20.N. Privault, Stochastic Analysis in Discrete and Continuous Settings with Normal Martingales. Lecture Notes in Mathematics, vol. 1982 (Springer, Berlin, 2009)Google Scholar
- 23.R.L. Schilling, R. Song, Z. Vondraček, Bernstein Functions: Theory and Applications. de Gruyter Studies in Mathematics, vol. 37 (Walter de Gruyter, Berlin, 2010)Google Scholar
- 24.Ł. Wojciechowski, Stochastic analysis of Lévy processes. PhD dissertation, University of Wrocław, 2014Google Scholar