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Une preuve simple d’un résultat de Dufresne

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Séminaire de Probabilités XLI

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 1934))

Abstract

We give a simple proof of the following result by Dufresne [Duf90]: if {w s { s ≥0 is a linear Brownian motion and c a positive constant,

$$\mathbb{P}(\int_0^\infty {e^{ - w_s - cs} ds \in [a,b[} ) = \frac{{2^{2c} }}{{\Gamma (2c)}}\int_a^b {\frac{{e^{ - 2/u} }}{{u^{1 + 2c} }}du.}$$

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References

  • [Bai07] Ismaël Bailleul. Poisson boundary of a relativistic diffusion. A paraitre dans P.T.R.F., 2007.

    Google Scholar 

  • [Bas98] Richard F. Bass. Diffusions and elliptic operators. Probability and its Applications (New York). Springer-Verlag, New York, 1998.

    MATH  Google Scholar 

  • [Bel87] Denis R. Bell. The Malliavin calculus, volume 34 of Pitman Monographs and Surfeys in Pure and Applied Mathematics. Longman Scientific & Technical, Harlow, 1987.

    Google Scholar 

  • [CPY01] Philippe Carmona, Frédérique Petit, and Marc Yor. Exponential functionals of Lévy processes. In Lévy processes, pages 41–55. Birkhäuser Boston, Boston, MA, 2001.

    Chapter  Google Scholar 

  • [Duf90] Daniel Dufresne. The distribution of a perpetuity, with applications to risk theory and pension funding. Scand. Actuar. J., (1–2): 39–79, 1990.

    Google Scholar 

  • [Mal97] Paul Malliavin. Stochastic analysis, volume 313 of Grundlehren der Mathemotischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1997.

    Google Scholar 

  • [MY05a] Hirojuki Matsumoto and Marc Yor. Exponential functionals of Brownian motion. I. Probability laws at fixed time. Probab. Surv., 2: 312–347 (electronic), 2005.

    Article  MATH  MathSciNet  Google Scholar 

  • [MY05b] Hirojuki Matsumoto and Marc Yor. Exponential functionals of Brownian motion. II. Some related diffusion processes. Probab. Surv., 2: 348–384 (electronic), 2005.

    Article  MATH  MathSciNet  Google Scholar 

  • [Nua06] David Nualart. The Malliavin calculus and related topics. Probability and its Applications (New York). Springer-Verlag, Berlin, second edition, 2006.

    Google Scholar 

  • [Yor92] Marc Yor. Sur certaines fonctionnelles du mouvement brownien réel. J. Appl. Probab., 29(1): 202–208, 1992.

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Université Paris Sud, Paris, France

    Ismael Bailleul

Authors
  1. Ismael Bailleul
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Editors and Affiliations

  1. Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, Boîte courrier I88 4, place Jussieu, 75252, Paris cedex 05, France

    Catherine Donati-Martin

  2. Institut de Recherche Mathématique Avancée, Université Louis Pasteur, 7, rue René Descartes, 67084, Strasbourg cedex, France

    Michel Émery

  3. Laboratoire de Mathématiques Bâtiment Fermat, Université Versailles-Saint-Quentin, 45, avenue des Etats-Unis, 78035, Versailles cedex, France

    Alain Rouault

  4. UFR Sciences et techniques, Université de Besançon, 16, route de Gray, 25030, Besançon cedex, France

    Christophe Stricker

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Bailleul, I. (2008). Une preuve simple d’un résultat de Dufresne. In: Donati-Martin, C., Émery, M., Rouault, A., Stricker, C. (eds) Séminaire de Probabilités XLI. Lecture Notes in Mathematics, vol 1934. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77913-1_10

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