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Abelian-type expansions and non-linear death processes(II)

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Claude Lefèvre  and
Philippe Picard
Claude Lefèvre*
Affiliation:
Université Libre de Bruxelles
Philippe Picard*
Affiliation:
Université De Lyon 1
*
∗ Postal address: Institut de Statistique, C.P.210, Université Libre de Bruxelles, Boulevard du Triomphe, B-1050 Bruxelles, Belgique.
∗∗ Postal address: Mathématiques Appliquées, Université de Lyon 1, 43 Boulevard du 11 Novembre 1918, F-69622 Villeurbanne Cedex, France.
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Abstract

This paper is concerned with the study of death processes with time-homogeneous non-linear death rates. An explicit formula is obtained for the joint distribution of the state XT and the variable ∫T0g(Xt), where g is any given real function and T corresponds to some appropriate stopping time. This is achieved by constructing a family of martingales and then by using a particular family of Abel–Gontcharoff pseudopolynomials (the theory of which has been introduced in a companion paper) and related Abelian-type expansions. Moreover, the distribution of the first crossing level of such a death process through a general upper boundary is also evaluated in terms of pseudopolynomials of that kind. The flexibility of the methods developed makes easy the extension to multidimensional death processes.

Keywords

MARTINGALESABEL–GONTCHAROFF PSEUDOPOLYNOMIALSEXACT DISTRIBUTIONFIRST-CROSSING LEVEL

MSC classification

Primary: 60J75: Jump processes
Type
General Applied Probablity
Information
Advances in Applied Probability , Volume 28 , Issue 3 , September 1996 , pp. 877 - 894
Copyright
Copyright © Applied Probability Trust 1996 

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References

Billard, L., Lacayo, H. and Langberg, N. A. (1980) Generalizations of the simple epidemic process. J. Appl. Prob. 17, 1072–1078.Google Scholar
Godfrey, K. (1983) Compartmental Models and Their Applications. Academic Press, London.Google Scholar
Lefevre, Cl. and Picard, Ph. (1990) A non-standard family of polynomials and the final size distribution of Reed–Frost epidemic processes. Adv. Appl. Prob. 22, 25–48.Google Scholar
Picard, Ph. (1980) Applications of martingale theory to some epidemic models. J. Appl. Prob. 17, 583–599.Google Scholar
Picard, Ph. and Lefevre, Cl. (1996) First crossing of basic counting processes with lower non-linear boundaries: a unified approach with pseudopolynomials (I). Adv. Appl. Prob. 28, 853–876.Google Scholar
Rényi, A. (1970) Probability Theory. North-Holland, New York.Google Scholar