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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
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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 

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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