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 Electronic Communications in Probability > Vol. 13 (2008) > Paper 59 open journal systems 

Renewal series and square-root boundaries for Bessel processes

Nathanael Enriquez, Universite Paris 10
Christophe Sabot, Université de Lyon 1
Marc Yor, Universite Paris 6

We show how a description of Brownian exponential functionals as a renewal series gives access to the law of the hitting time of a square-root boundary by a Bessel process. This extends classical results by Breiman and Shepp, concerning Brownian motion, and recovers by different means, extensions for Bessel processes, obtained independently by Delong and Yor.

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Pages: 649-652

Published on: December 17, 2008

  1. Breiman, Leo. First exit times from a square root boundary. 1967 Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), Vol. II: Contributions to Probability Theory, Part 2 pp. 9--16 Univ. California Press, Berkeley, Calif. MR0212865 (35 #3730)
  2. DeLong, David M. Crossing probabilities for a square root boundary by a Bessel process. Comm. Statist. A---Theory Methods 10 (1981), no. 21, 2197--2213. MR0629897 (82i:62119)
  3. DeLong, David M. Erratum: ``Crossing probabilities for a square root boundary by a Bessel process'' [Comm. Statist. A---Theory Methods 10 (1981), no. 21, 2197--2213; MR 82i:62119]. Comm. Statist. A---Theory Methods 12 (1983), no. 14, 1699. MR0711257 (84i:62106)
  4. Dufresne, Daniel. The distribution of a perpetuity, with applications to risk theory and pension funding. Scand. Actuar. J. 1990, no. 1-2, 39--79. MR1129194 (92i:62195)
  5. Lamperti, John. Semi-stable Markov processes. I. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 22 (1972), 205--225. MR0307358 (46 #6478)
  6. Lebedev, N. N. Special functions and their applications.Revised edition, translated from the Russian and edited by Richard A. Silverman.Unabridged and corrected republication.Dover Publications, Inc., New York, 1972. xii+308 pp. MR0350075 (50 #2568)
  7. Shepp, L. A. A first passage problem for the Wiener process. Ann. Math. Statist. 38 1967 1912--1914. MR0217879 (36 #968)
  8. Vervaat, Wim. On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv. in Appl. Probab. 11 (1979), no. 4, 750--783. MR0544194 (81b:60064)
  9. Yor, Marc. On square-root boundaries for Bessel processes, and pole-seeking Brownian motion. Stochastic analysis and applications (Swansea, 1983), 100--107, Lecture Notes in Math., 1095, Springer, Berlin, 1984. MR0777516 (86h:60155)
  10. Yor, Marc. On some exponential functionals of Brownian motion. Adv. in Appl. Probab. 24 (1992), no. 3, 509--531. MR1174378 (94b:60095)
  11. Yor, Marc. Sur certaines fonctionnelles exponentielles du mouvement brownien réel.(French) [On some exponential functionals of real Brownian motion] J. Appl. Probab. 29 (1992), no. 1, 202--208. MR1147781 (93g:60179)

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Electronic Communications in Probability. ISSN: 1083-589X