Home | Current | Past volumes | About | Login | Notify | Contact | Search
 Probability Surveys > Vol. 6 (2009) open journal systems 


Proof(s) of the Lamperti representation of continuous-state branching processes

Ma. Emilia Caballero, Instituto de Matemáticas, Universidad Nacional Autónoma de México
Amaury Lambert, Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Ma
Gerónimo Uribe Bravo, Departamento de Probabilidad y Estadística, Instituto de Investigaciones en Mat


Abstract
This paper uses two new ingredients, namely stochastic differential equations satisfied by continuous-state branching processes (CSBPs), and a topology under which the Lamperti transformation is continuous, in order to provide self-contained proofs of Lamperti's 1967 representation of CSBPs in terms of spectrally positive Lévy processes. The first proof is a direct probabilistic proof, and the second one uses approximations by discrete processes, for which the Lamperti representation is evident.

AMS 2000 subject classifications: Primary 60J80; secondary 60B10, 60G44, 60G51, 60H20.

Keywords: Continuous-state branching processes; spectrally positive Lévy processes; random time change; stochastic integral equations; Skorohod topology.

Creative Common LOGO

Full Text: PDF


Caballero, Ma. Emilia, Lambert, Amaury, Bravo, Gerónimo Uribe, Proof(s) of the Lamperti representation of continuous-state branching processes, Probability Surveys, 6, (2009), 62-89 (electronic). DOI: 10.1214/09-PS154.

References

[1]     Bertoin, J. (1996). Lévy processes. Cambridge Tracts in Mathematics, Vol. 121. Cambridge University Press, Cambridge. MR1406564 (98e:60117)

[2]     Bertoin, J. (1997). Cauchy’s principal value of local times of Lévy processes with no negative jumps via continuous branching processes. Electron. J. Probab. 2, no. 6, 12 pp. (electronic). MR1475864 (99b:60120)

[3]     Bertoin, J. (2000). Subordinators, Lévy processes with no negative jumps and branching processes. MaPhySto Lecture Notes Series No. 8.

[4]     Bertoin, J. and Le Gall, J.-F. (2005). Stochastic flows associated to coalescent processes. II. Stochastic differential equations. Ann. Inst. H. Poincaré Probab. Statist. 41, 3, 307–333. MR2139022 (2005m:60067)

[5]     Bertoin, J. and Le Gall, J.-F. (2006). Stochastic flows associated to coalescent processes. III. Limit theorems. Illinois J. Math. 50, 1-4, 147–181 (electronic). MR2247827 (2008c:60032)

[6]     Bertoin, J. and Yor, M. (2005). Exponential functionals of Lévy processes. Probab. Surv. 2, 191–212 (electronic). MR2178044

[7]     Billingsley, P. (1999). Convergence of probability measures, Second ed. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons Inc., New York. A Wiley-Interscience Publication. MR1700749 (2000e:60008)

[8]     Bingham, N. H. (1976). Continuous branching processes and spectral positivity. Stochastic Processes Appl. 4, 3, 217–242. MR0410961 (53 #14701)

[9]     Dawson, D. A. and Li, Z. (2006). Skew convolution semigroups and affine Markov processes. Ann. Probab. 34, 3, 1103–1142. MR2243880

[10]     Dynkin, E. B. (1965). Markov processes. Vols. I, II. Translated with the authorization and assistance of the author by J. Fabius, V. Greenberg, A. Maitra, G. Majone. Die Grundlehren der Mathematischen Wi ssenschaften, Bände 121, Vol. 122. Academic Press Inc., Publishers, New York. MR0193671 (33 #1887)

[11]     Ethier, S. N. and Kurtz, T. G. (1986). Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons Inc., New York. MR838085 (88a:60130)

[12]     Gˉı   hman, Ĭ. ˉI   . and Skorohod, A. V. (1980). The theory of stochastic processes. I, English ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 210. Springer-Verlag, Berlin. Translated from the Russian by Samuel Kotz. MR636254 (82k:60005)

[13]     Grimvall, A. (1974). On the convergence of sequences of branching processes. Ann. Probability 2, 1027–1045. MR0362529 (50 #14969)

[14]     Helland, I. S. (1978). Continuity of a class of random time transformations. Stochastic Processes Appl. 7, 1, 79–99. MR0488203 (58 #7765)

[15]     Ikeda, N. and Watanabe, S. (1989). Stochastic differential equations and diffusion processes, Second ed. North-Holland Mathematical Library, Vol. 24. North-Holland Publishing Co., Amsterdam. MR1011252 (90m:60069)

[16]     Jacod, J. and Shiryaev, A. N. (2003). Limit theorems for stochastic processes, Second ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 288. Springer-Verlag, Berlin. MR1943877 (2003j:60001)

[17]     Kallenberg, O. (2002). Foundations of modern probability, Second ed. Probability and its Applications (New York). Springer-Verlag, New York. MR1876169 (2002m:60002)

[18]     Kawazu, K. and Watanabe, S. (1971). Branching processes with immigration and related limit theorems. Teor. Verojatnost. i Primenen. 16, 34–51. MR0290475 (44 #7656)

[19]     Kyprianou, A. E. (2006). Introductory lectures on fluctuations of Lévy processes with applications. Universitext. Springer-Verlag, Berlin. MR2250061

[20]     Lamperti, J. (1967a). Continuous state branching processes. Bull. Amer. Math. Soc. 73, 382–386. MR0208685 (34 #8494)

[21]     Lamperti, J. (1967b). The limit of a sequence of branching processes. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 7, 271–288. MR0217893 (36 #982)

[22]     Lamperti, J. (1967c). Limiting distributions for branching processes. In Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), Vol. II: Contributions to Probability Theory, Part 2. Univ. California Press, Berkeley, Calif., 225–241. MR0219148 (36 #2231)

[23]     Lamperti, J. (1972). Semi-stable Markov processes. I. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 22, 205–225. MR0307358 (46 #6478)

[24]     Le Gall, J.-F. (1999). Spatial branching processes, random snakes and partial differential equations. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel. MR1714707 (2001g:60211)

[25]     Pagès, G. (1986). Un théorème de convergence fonctionnelle pour les intégrales stochastiques. In Séminaire de Probabilités, XX, 1984/85. Lecture Notes in Math., Vol. 1204. Springer, Berlin, 572–611. MR942045 (89k:60046)

[26]     Silverstein, M. L. (1967/1968). A new approach to local times. J. Math. Mech. 17, 1023–1054. MR0226734 (37 #2321)

[27]     Whitt, W. (1980). Some useful functions for functional limit theorems. Math. Oper. Res. 5, 1, 67–85. MR561155 (81e:60035)




Home | Current | Past volumes | About | Login | Notify | Contact | Search

Probability Surveys. ISSN: 1549-5787