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Conditioned stable Lévy processes and the Lamperti representation

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

M. E. Caballero  and
L. Chaumont
M. E. Caballero*
Affiliation:
Universidad Nacional Autónoma de México
L. Chaumont*
Affiliation:
Université Pierre et Marie Curie
*
∗Postal address: Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico 04510 DF. Email address: emilia@servidor.unam.mx
∗∗Postal address: Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris Cedex 05, France. Email address: chaumont@ccr.jussieu.fr
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Abstract

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By variously killing a stable Lévy process when it leaves the positive half-line, conditioning it to stay positive, and conditioning it to hit 0 continuously, we obtain three different, positive, self-similar Markov processes which illustrate the three classes described by Lamperti (1972). For each of these processes, we explicitly compute the infinitesimal generator and from this deduce the characteristics of the underlying Lévy process in the Lamperti representation. The proof of this result bears on the behaviour at time 0 of stable Lévy processes before their first passage time across level 0, which we describe here. As an application, for a certain class of Lévy processes we give the law of the minimum before an independent exponential time. This provides the explicit form of the spatial Wiener-Hopf factor at a particular point and the value of the ruin probability for this class of Lévy processes.

Keywords

Positive, self-similar Markov processLamperti representationinfinitesimal generatorstable Lévy process conditioned to stay positivestable Lévy processconditioning to hit 0 continuously

MSC classification

Primary: 60G18: Self-similar processes 60G51: Processes with independent increments; L'evy processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 4 , December 2006 , pp. 967 - 983
Copyright
© Applied Probability Trust 2006 

References

Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
Bertoin, J. (2002). Self-similar fragmentations. Ann. Inst. H. Poincaré Prob. Statist. 38, 319–340.Google Scholar
Bingham, N. H. (1973). Maxima of sums of random variables and suprema of stable processes. Z. Wahrscheinlichkeitsth. 26, 273–296.Google Scholar
Bryn-Jones, A. and Doney, R. A. (2006). A functional limit theorem for random walk conditioned to stay non-negative. J. London Math. Soc. 74, 244–258.Google Scholar
Caravenna, F. and Chaumont, L. (2006). Invariances principles for conditioned random walks. Preprint 1050, LPMA.Google Scholar
Carmona, P., Petit, F. and Yor, M. (2001). Exponential functionals of Lévy processes. In Lévy Processes, Birkhäuser, Boston, MA, pp. 41–55.Google Scholar
Chaumont, L. (1996). Conditionings and path decompositions for Lévy processes. Stoch. Process. Appl. 64, 39–54.Google Scholar
Doney, R. A. (2005). Fluctuation theory for Lévy processes. To appear in École d'été de Probabilités de Saint-Flour.Google Scholar
Duquesne, T. and Le Gall, J. F. (2002). Random Trees, Lévy Processes and Spatial Branching Processes (Astérisque 281). Société Mathématique de France, Paris.Google Scholar
Lamperti, J. W. (1972). Semi-stable Markov processes. Z. Wahrscheinlichkeitsth. 22, 205–225.Google Scholar
Lewis, A. and Mordecki, E. (2005). Wiener–Hopf factorization for Lévy processes having negative Jumps with rational transforms. Tech. Rep.Google Scholar
Rivero, V. (2005). Recurrent extensions of self-similar Markov processes and Cramér's condition. Bernoulli 11, 471–509.Google Scholar
Sato, K. I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
Zolotarev, V. M. (1986). One-Dimensional Stable Distributions (Transl. Math. Monogr. 65). American Mathematical Society, Providence, RI.Google Scholar