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First crossing of basic counting processes with lower non-linear boundaries: A unified approach through pseudopolynomials (I)

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  01 July 2016

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

The paper is concerned with the distribution of the level N of the first crossing of a counting process trajectory with a lower boundary. Compound and simple Poisson or binomial processes, gamma renewal processes, and finally birth processes are considered. In the simple Poisson case, expressing the exact distribution of N requires the use of a classical family of Abel–Gontcharoff polynomials. For other cases convenient extensions of these polynomials into pseudopolynomials with a similar structure are necessary. Such extensions being applicable to other fields of applied probability, the central part of the present paper has been devoted to the building of these pseudopolynomials in a rather general framework.

Keywords

LOWER AND UPPER BOUNDARIESABEL–GONTCHAROFF POLYNOMIALSAPPELL POLYNOMIALS

MSC classification

Primary: 60G17: Sample path properties
Secondary: 60J75: Jump processes 60G40: Stopping times; optimal stopping problems; gambling theory
Type
General Applied Probablity
Information
Advances in Applied Probability , Volume 28 , Issue 3 , September 1996 , pp. 853 - 876
Copyright
Copyright © Applied Probability Trust 1996 

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Footnotes

Research partially supported by the Institut National de la Santé et de la Recherche Médicale under contract No. 921011.

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