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Birth and death processes as projections of higher-dimensional Poisson processes

Part of:Geometric probability and stochastic geometryStochastic processes

Published online by Cambridge University Press:  01 July 2016

Nancy Lopes Garcia
Nancy Lopes Garcia*
Affiliation:
Universidade Estadual de Campinas
*
* Postal address: Departamento de Estatística, IMECC, Cidade Universitária ‘Zeferino Vaz', Caixa Postal 6065, 13.081-970, Campinas, SP, Brazil.
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Abstract

Birth and death processes can be constructed as projections of higher-dimensional Poisson processes. The existence and uniqueness in the strong sense of the solutions of the time change problem are obtained. It is shown that the solution of the time change problem is equivalent to the solution of the corresponding martingale problem. Moreover, the processes obtained by the projection method are ergodic under translations.

Keywords

POISSON POINT PROCESSSPATIAL BIRTH AND DEATH PROCESSESRANDOM TIME CHANGEMARTINGALE PROBLEMSPATIAL ERGODICITY

MSC classification

Primary:60D05: Geometric probability and stochastic geometry
Secondary:60G55: Point processes

Information

Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 27 , Issue 4 , December 1995, pp. 911 - 930
Copyright
Copyright © Applied Probability Trust 1995 

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References

Daley, D. J. and Vere-Jones, D. (1988) An Introduction to the Theory of Point Processes. Springer-Verlag, New York.Google Scholar
Ethier, S. N. and Kurtz, T. G. (1986) Markov Processes: Characterization and Convergence. Wiley, New York.CrossRefGoogle Scholar
Gaver, D. P., Jacobs, P. A. and Latouche, G. (1984) Finite birth-and-death models in randomly changing environments. Adv. Appl. Prob. 16, 715–731.CrossRefGoogle Scholar
Karr, A. F. (1986) Point Processes and Their Statistical Inference. Marcel Dekker, New York.Google Scholar
Kurtz, T. G. (1980a) The optional sampling theorem for martingales indexed by direct sets. Ann. Prob. 8, 675–681.CrossRefGoogle Scholar
Kurtz, T. G. (1980b) Representation of Markov processes as multiparameter time changes. Ann. Prob. 8, 682–715.CrossRefGoogle Scholar
Kurtz, T. G. (1989) Stochastic processes as projections of Poisson random measures. Special invited paper at IMS meeting, Washington, D.C. Unpublished.Google Scholar
Liggett, T.M. (1972) Existence theorems for infinite particle systems. Trans. Amer. Math. Soc. 165, 471–481.CrossRefGoogle Scholar
Lotwick, H. W. and Silverman, B. W. (1981) Convergence of spatial birth-and-death processes. Math. Proc. Camb. Phil. Soc. 90, 155–165.CrossRefGoogle Scholar
Meyer, P. A. (1971) Demonstration simplifiée d'un théorème de Knith. In Seminaire de Probabilités V, Univ. Strasbourg, Lecture Notes in Mathematics 191, pp. 191–195. Springer-Verlag, Berlin.Google Scholar
Walsh, J. B. (1984) Martingale measures. In école d'été de Probabilités de Saint Flour XIV-1984, Lecture Notes in Mathematics 1180, 286–307. Springer-Verlag, Berlin.Google Scholar