Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-23T05:44:36.606Z Has data issue: false hasContentIssue false
  • English
  • Français

The distribution of Brownian quantiles

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Marc Yor
Marc Yor*
Affiliation:
Université Paris VI
*
∗Postal address: Laboratoire de Probabilités, Université Paris VI, 4, place Jussieu, Tour 56, 3ème Etage, 75252 Paris CEDEX 05, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

The distribution of Brownian quantiles is determined, simplifying related integral expressions obtained by Lévy [9], [10] and more recently by Miura [11]. Three proofs are given, two of them involving last-passage times of Brownian motion, before time 1, at a given level.

Keywords

ARC-SINE LAWBETA AND GAMMA VARIABLES

MSC classification

Primary: 60J65: Brownian motion
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 2 , June 1995 , pp. 405 - 416
Copyright
Copyright © Applied Probability Trust 1995 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

References

[1] Akahori, J. (1995) Some formulae for a new type of path-dependent option. Ann. Appl. Prob. To appear.Google Scholar
[2] Basbakova, P. E. and Borodin, A. N. (1991) Tables of the distributions of the functionals of Brownian motion. Zap. Nauch. Sem. Pom. 13, 8–20.Google Scholar
[3] Biane, Ph. and Yor, M. (1987) Valeurs principales associées aux temps locaux Browniens. Bull. Sci. Maths. 2ème série 111, 23–101.Google Scholar
[4] Denes, J. (1991) Evaluation of distributions of Wiener functionals. , Carleton University, Ottawa, Canada.Google Scholar
[5] Dynkin, E. B. (1961) Some limit theorems for sums of independent random variables with infinite mathematical expectations. I.M.S.-A.M.S. Selected Trans. Math. Stat. Prob. 1, 171–189.Google Scholar
[6] Geman, H. and Yor, M. (1993) Bessel processes, Asian options and perpetuities. Math. Finance 3, 349–375.Google Scholar
[7] Jeulin, Th. (1980) Semi-martingales et grossissement d'une filtration. Lecture Notes in Mathematics 833. Springer-Verlag, Berlin.Google Scholar
[8] Jeulin, Th. (1985) Application du grossissement de filtrations à l'étude des temps locaux du mouvement brownien. In Grossissements de filtrations et applications, ed. Jeulin, T. and Yor, M. Lecture Notes in Mathematics 1118, Springer-Verlag, Berlin.Google Scholar
[9] Levy, P. (1939) Sur un problème de M. Marcinkiewicz. C.R. Acad. Sci. Paris 208, 318–321.Google Scholar
[10] Levy, P. (1939) Sur certains processus stochastiques homogènes. Compositio Math. 7(2), 283–339.Google Scholar
[11] Miura, R. (1992) A note on look-back options based on order statistics. Hitotsubashi Journal of Commerce and Management 27, 15–28.Google Scholar
[12] Yor, M. (1992) On some exponential functionals of Brownian motion. Adv. Appl. Prob. 24, 509–531.Google Scholar
[13] Embrechts, P., Rogers, L. C. G. and Yor, M. (1995) A proof of Dassios's representation of the a-quantile of Brownian motion with drift. Ann. Appl. Prob. Google Scholar