Moments of Poisson stochastic integrals with random integrands

• Autorzy: Privault N.
• Języki publikacji: EN

Abstrakty

EN

We show that the moment of order n of the Poisson stochastic integral of a random process (u_x)_x∈X over a metric space X is given by the non-linear Mecke identity [formula]. This formula recovers known results in case (u(x))_x∈X is a deterministic function on X.

Słowa kluczowe

EN

Poisson stochastic integrals; moment identities; Bell polynomials; Poisson–Skorohod integral

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2012
• Tom: Vol. 32, Fasc. 2
• Strony: 227--239
• Opis fizyczny: Bibliogr. 14 poz.

Twórcy

autor

Privault N.

• Nanyang Technological University, School of Physical and Mathematical Sciences, Division of Mathematical Sciences, 21 Nanyang Link, Singapore 637371

Bibliografia

• [1] Ph. Barbe and W. P. McCormick, Ruin probabilities in tough times. Part 1. Heavy-traffic approximation for fractionally integrated random walks in the domain of attraction of a non- Gaussian stable distribution. Preprint arXiv:1101.4437, 2011.
• [2] B. Bassan and E. Bona, Moments of stochastic processes governed by Poisson random measures, Comment. Math. Univ. Carolin. 31 (2) (1990), pp. 337-343.
• [3] M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Algebra Appl. 226/228 (1995), pp. 57-72.
• [4] H. Biermé, Y. Demichel, and A. Estrade, Fractional Poisson field on a finite set. Preprint hal-00597722, 2011.
• [5] Y. Ito, Generalized Poisson functionals, Probab. Theory Related Fields 77 (1998), pp. 1-28.
• [6] E. Lukacs, Characteristic Functions, second edition, revised and enlarged, Hafner Publishing Co., New York 1970.
• [7] J. Mecke, Stationäre zufällige Masse auf lokalkompakten Abelschen Gruppen, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 9 (1967), pp. 36-58.
• [8] G. Peccati and M. Taqqu, Wiener Chaos: Moments, Cumulants and Diagrams: A Survey with Computer Implementation, Bocconi & Springer Series, Springer, 2011.
• [9] J. Picard, Formules de dualité sur l’espace de Poisson, Ann. Inst. H. Poincaré Probab. Statist. 32 (4) (1996), pp. 509-548.
• [10] N. Privault, Moment identities for Poisson-Skorohod integrals and application to measure invariance, C. R. Math. Acad. Sci. Paris 347 (2009), pp. 1071-1074.
• [11] N. Privault, Stochastic Analysis in Discrete and Continuous Settings, Lecture Notes in Math., Vol. 1982, Springer, Berlin 2009.
• [12] N. Privault, Generalized Bell polynomials and the combinatorics of Poisson central moments, Electron. J. Combin. 18 (1): Research Paper 54, 10, 2011.
• [13] N. Privault, Cumulant operators for Lie-Wiener-Itô-Poisson stochastic integrals. Preprint, 2012.
• [14] N. Privault, Invariance of Poisson measures under random transformations, Ann. Inst. H. Poincaré Probab. Statist. 48 (2012), pp. 947-972.