Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-27T07:40:32.533Z Has data issue: false hasContentIssue false
  • English
  • Français

Sparre Andersen identity and the last passage time

Part of: Stochastic processes

Published online by Cambridge University Press:  21 June 2016

Jevgenijs Ivanovs
Jevgenijs Ivanovs*
Affiliation:
University of Lausanne
*
* Postal address: Department of Actuarial Science, Faculty of Business and Economics, University of Lausanne, CH-1015 Lausanne, Switzerland. Email address: jevgenijs.ivanovs@unil.ch
Get access Rights & Permissions [Opens in a new window]

Abstract

It is shown that the celebrated result of Sparre Andersen for random walks and Lévy processes has intriguing consequences when the last time of the process in (-∞, 0], say σ, is added to the picture. In the case of no positive jumps this leads to six random times, all of which have the same distribution—the uniform distribution on [0, σ]. Surprisingly, this result does not appear in the literature, even though it is based on some classical observations concerning exchangeable increments.

Keywords

Last passagetime reversalexchangeable incrementsuniform law

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 60G50: Sums of independent random variables; random walks
Type
Short Communications
Information
Journal of Applied Probability , Volume 53 , Issue 2 , June 2016 , pp. 600 - 605
Copyright
Copyright © Applied Probability Trust 2016 

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

[1]Alili, L., Chaumont, L. and Doney, R. A. (2005).On a fluctuation identity for random walks and Lévy processes.Bull. London Math. Soc. 37, 141–148.Google Scholar
[2]Andersen, E. S. (1953).On sums of symmetrically dependent random variables.Skand. Aktuarietidski. 36, 123–138.Google Scholar
[3]Andersen, E. S. (1953).On the fluctuations of sums of random variables.Math. Scand. 1, 263–285.CrossRefGoogle Scholar
[4]Asmussen, S. (2003).Applied Probability and Queues, 2nd edn.Springer, New York.Google Scholar
[5]Asmussen, S. and Albrecher, H. (2010).Ruin Probabilities, 2nd edn.World Scientific, Hackensack, NJ.Google Scholar
[6]Bertoin, J. (1996).Lévy Processes.Cambridge University Press.Google Scholar
[7]Chaumont, L., Hobson, D. G. and Yor, M. (2001).Some consequences of the cyclic exchangeability property for exponential functionals of Lévy processes. In Séminaire de Probabilités, XXXV (lecture Notes Math.1755), Springer, Berlin, pp.334–347.Google Scholar
[8]Chiu, S. N. and Yin, C. (2005).Passage times for a spectrally negative Lévy process with applications to risk theory.Bernoulli 11, 511–522.Google Scholar
[9]Feller, W. (1966).An Introduction to Probability Theory and Its Applications Vol. II.John Wiley, New York.Google Scholar
[10]Knight, F. B. (1996).The uniform law for exchangeable and Lévy process bridges.Astérisque 236, 171–188.Google Scholar
[11]Marchal, P. (2001).Two consequences of a path transform.Bull. London Math. Soc. 33, 213–220.Google Scholar
[12]Nagasawa, M. (1964).Time reversions of Markov processes.Nagoya Math. J. 24, 177–204.Google Scholar
[13]Whitt, W. (2002).Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues.Springer, New York.Google Scholar