The Annals of Probability

Girsanov identities for Poisson measures under quasi-nilpotent transformations

Nicolas Privault

Full-text: Open access

Abstract

We prove a Girsanov identity on the Poisson space for anticipating transformations that satisfy a strong quasi-nilpotence condition. Applications are given to the Girsanov theorem and to the invariance of Poisson measures under random transformations. The proofs use combinatorial identities for the central moments of Poisson stochastic integrals.

Article information

Source
Ann. Probab., Volume 40, Number 3 (2012), 1009-1040.

Dates
First available in Project Euclid: 4 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.aop/1336136057

Digital Object Identifier
doi:10.1214/10-AOP640

Mathematical Reviews number (MathSciNet)
MR2962085

Zentralblatt MATH identifier
1248.60056

Subjects
Primary: 60G57: Random measures 60G30: Continuity and singularity of induced measures 60H07: Stochastic calculus of variations and the Malliavin calculus 28D05: Measure-preserving transformations 28C20: Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11] 11B73: Bell and Stirling numbers

Keywords
Poisson measures random transformations Girsanov identities quasi-invariance invariance Skorohod integral moment identities Stirling numbers

Citation

Privault, Nicolas. Girsanov identities for Poisson measures under quasi-nilpotent transformations. Ann. Probab. 40 (2012), no. 3, 1009--1040. doi:10.1214/10-AOP640. https://projecteuclid.org/euclid.aop/1336136057


Export citation

References

  • [1] Abramowitz, M. and Stegun, I. A. (1972). Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, 9th ed. 55 Dover, New York.
  • [2] Albeverio, S. and Smorodina, N. V. (2006). A distributional approach to multiple stochastic integrals and transformations of the Poisson measure. Acta Appl. Math. 94 1–19.
  • [3] Bernstein, M. and Sloane, N. J. A. (1995). Some canonical sequences of integers. Linear Algebra Appl. 226/228 57–72.
  • [4] Bichteler, K. (2002). Stochastic Integration with Jumps. Encyclopedia of Mathematics and Its Applications 89. Cambridge Univ. Press, Cambridge.
  • [5] Boyadzhiev, K. N. (2009). Exponential polynomials, Stirling numbers, and evaluation of some gamma integrals. Abstr. Appl. Anal. Art. ID 168672, 18.
  • [6] Davydov, Y. and Nagaev, S. (2000). On the convex hulls of point processes. Unpublished manuscript, Université des Sciences et Technologies de Lille.
  • [7] Dermoune, A., Krée, P. and Wu, L. (1988). Calcul stochastique non adapté par rapport à la mesure aléatoire de Poisson. In Séminaire de Probabilités, XXII. Lecture Notes in Math. 1321 477–484. Springer, Berlin.
  • [8] Di Nardo, E. and Senato, D. (2001). Umbral nature of the Poisson random variables. In Algebraic Combinatorics and Computer Science 245–266. Springer Italia, Milan.
  • [9] Ito, Y. (1988). Generalized Poisson functionals. Probab. Theory Related Fields 77 1–28.
  • [10] Kusuoka, S. (1982). The nonlinear transformation of Gaussian measure on Banach space and absolute continuity. I. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 567–597.
  • [11] Lépingle, D. and Mémin, J. (1978). Sur l’intégrabilité uniforme des martingales exponentielles. Z. Wahrsch. Verw. Gebiete 42 175–203.
  • [12] Mecke, J. (1967). Stationäre zufällige Masse auf lokalkompakten Abelschen Gruppen. Z. Wahrsch. Verw. Gebiete 9 36–58.
  • [13] Meyer, P. A. (1976). Un cours sur les intégrales stochastiques. In Séminaire de Probabilités, X (Seconde Partie: Théorie des Intégrales Stochastiques, Univ. Strasbourg, Strasbourg, Année Universitaire 1974/1975). Lecture Notes in Math. 511 245–400. Springer, Berlin.
  • [14] Nualart, D. and Vives, J. (1995). A duality formula on the Poisson space and some applications. In Seminar on Stochastic Analysis, Random Fields and Applications (Ascona, 1993). Progress in Probability 36 205–213. Birkhäuser, Basel.
  • [15] Picard, J. (1996). Formules de dualité sur l’espace de Poisson. Ann. Inst. Henri Poincaré Probab. Stat. 32 509–548.
  • [16] Privault, N. (1996). Girsanov theorem for anticipative shifts on Poisson space. Probab. Theory Related Fields 104 61–76.
  • [17] Privault, N. (1998). Absolute continuity in infinite dimensions and anticipating stochastic calculus. Potential Anal. 8 325–343.
  • [18] Privault, N. (2003). Quasi-invariance for Lévy processes under anticipating shifts. In Stochastic Analysis and Related Topics VIII (U. Çapar and A. S. Üstünel, eds.). Progress in Probability 53 181–202. Birkhäuser, Basel.
  • [19] Privault, N. (2009). Moment identities for Poisson–Skorohod integrals and application to measure invariance. C. R. Math. Acad. Sci. Paris 347 1071–1074.
  • [20] Privault, N. (2009). Moment identities for Skorohod integrals on the Wiener space and applications. Electron. Commun. Probab. 14 116–121.
  • [21] Privault, N. (2009). Stochastic Analysis in Discrete and Continuous Settings with Normal Martingales. Lecture Notes in Math. 1982. Springer, Berlin.
  • [22] Privault, N. (2012). Invariance of Poisson measures under random transformations. Ann. Inst. Henri Poincaré Probab. Stat. To appear. Available at arXiv:1004.2588v2.
  • [23] Ramer, R. (1974). On nonlinear transformations of Gaussian measures. J. Funct. Anal. 15 166–187.
  • [24] Segall, A. and Kailath, T. (1976). Orthogonal functionals of independent-increment processes. IEEE Trans. Inform. Theory IT-22 287–298.
  • [25] Üstünel, A. S. and Zakai, M. (1995). Random rotations of the Wiener path. Probab. Theory Related Fields 103 409–429.
  • [26] Üstünel, A. S. and Zakai, M. (2000). Transformation of Measure on Wiener Space. Springer, Berlin.
  • [27] Zakai, M. and Zeitouni, O. (1992). When does the Ramer formula look like the Girsanov formula? Ann. Probab. 20 1436–1440.