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Continuous-State Branching Processes and Self-Similarity

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

A. E. Kyprianou  and
J. C. Pardo
A. E. Kyprianou*
Affiliation:
University of Bath
J. C. Pardo*
Affiliation:
University of Bath
*
∗Postal address: Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK.
∗Postal address: Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK.
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Abstract

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In this paper we study the α-stable continuous-state branching processes (for α ∈ (1, 2]) and the α-stable continuous-state branching processes conditioned never to become extinct in the light of positive self-similarity. Understanding the interaction of the Lamperti transformation for continuous-state branching processes and the Lamperti transformation for positive, self-similar Markov processes gives access to a number of explicit results concerning the paths of α-stable continuous-state branching processes and α-stable continuous-state branching processes conditioned never to become extinct.

Keywords

Positive, self-similar Markov processLamperti representationstable Lévy processconditioning to stay positivecontinuous-state branching process

MSC classification

Secondary: 60G18: Self-similar processes 60G51: Processes with independent increments; L'evy processes
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 4 , December 2008 , pp. 1140 - 1160
Copyright
Copyright © Applied Probability Trust 2008 

References

[1] Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
[2] Bertoin, J. and Yor, M. (2002). The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes. Potential Anal. 17, 389–400.Google Scholar
[3] Bingham, N. (1976). Continuous branching processes and spectral positivity. Stoch. Process. Appl. 4, 217–242.CrossRefGoogle Scholar
[4] Caballero, M. E. and Chaumont, L. (2006). Conditioned stable Lévy processes and the Lamperti representation. J. Appl. Prob. 43, 967–983.CrossRefGoogle Scholar
[5] Caballero, M. E. and Chaumont, L. (2006). Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes. Ann. Prob. 34, 1012–1034.Google Scholar
[6] Caballero, M. E., Pardo, J. C. and Pérez, J. L. (2008). On the Lamperti stable processes. Preprint. Available at http://arxiv.org/abs/0802.0851.Google Scholar
[7] Chaumont, L. and Doney, R. (2005). On Lévy processes conditioned to stay positive. Electron. J. Prob. 10, 948–961.Google Scholar
[8] Chaumont, L. and Pardo, J. C. (2006). The lower envelope of positive self-similar Markov processes. Electron. J. Prob. 11, 1321–1341.Google Scholar
[9] Chaumont, L., Kyprianou, A. E. and Pardo, J. C. (2008). Some explicit identities associated with positive self-similar Markov processes. To appear in Stoch. Process. Appl. Google Scholar
[10] Donati-Martin, C. and Yor, M. (2007). Further examples of explicit Krein representations of certain subordinators. Publ. Res. Inst. Math. Sci. 2, 315–328.Google Scholar
[11] Grey, D. R. (1974). Asymptotic behaviour of continuous time, continuous state-space branching processes. J. Appl. Prob. 11, 669–677.CrossRefGoogle Scholar
[12] Grimvall, A. (1974). On the convergence of sequences of branching processes. Ann. Prob. 2, 1027–1045.Google Scholar
[13] Itô, K. and McKean, H. P. (1965). Diffusion Processes and Their Sample Paths. Springer, Berlin.Google Scholar
[14] Jirina, M. (1958). Stochastic branching processes with continuous state space. Czech. Math. J. 8, 292–313.Google Scholar
[15] Lambert, A. (2001). Arbres, excursions, et processus de Lévy complètement asymétriques. , l'Université Pierre et Marie Curie.Google Scholar
[16] Lambert, A. (2002). The genealogy of continuous-state branching processes with immigration. Prob. Theory Relat. Fields 122, 42–70.Google Scholar
[17] Lambert, A. (2007). Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct. Electron. J. Prob. 12, 420–446.CrossRefGoogle Scholar
[18] Lamperti, J. (1967). Continuous state branching processes. Bull. Amer. Math. Soc. 73, 382–386.CrossRefGoogle Scholar
[19] Lamperti, J. (1972). Semi-stable Markov processes. I. Z. Wahrscheinlichkeitsth. 22, 205–225.Google Scholar
[20] Li, Z.-H. (2000). Asymptotic behaviour of continuous time and continuous-state branching processes. J. Austral. Math. Soc. 68, 68–84.CrossRefGoogle Scholar
[21] Pakes, A. (1999). Conditional limit theorems for continuous time and state branching processes. Res. Rep. 9/99, Department of Mathematics and Statistics, The University of Western Australia.Google Scholar
[22] Pardo, J. C. (2006). On the future infimum of positive self-similar Markov processes. Stochastics 78, 123–155.Google Scholar
[23] Pardo, J. C. (2008). The upper envelope of positive self-similar Markov processes. To appear in J. Theoret. Prob. Google Scholar
[24] Patie, P. (2007). Law of the exponential functional of a new family of one-sided Lévy processes via self-similar continuous-state branching processes with immigration and the Wright hypergeometric functions. Preprint. Available at http://arxiv.org/abs/0712.1115.Google Scholar
[25] Rivero, V. M. (2005). Recurrent extensions of self-similar Markov processes and Cramér's condition. Bernoulli 11, 471–509.Google Scholar
[26] Roelly, S. and Rouault, A. (1989). Processus de Dawnson–Watanabe conditionné par le futur lointain. C. R. Acad. Sci. Paris 309, 867–872.Google Scholar
[27] Sato, K. I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
[28] Silverstein, M. L. (1968). A new approach to local time. J. Math. Mech. 17, 1023–1054.Google Scholar