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First Passage of a Markov Additive Process and Generalized Jordan Chains

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Bernardo D‘Auria ,
Jevgenijs Ivanovs ,
Offer Kella  and
Michel Mandjes
Bernardo D‘Auria*
Affiliation:
Universidad Carlos III de Madrid
Jevgenijs Ivanovs*
Affiliation:
Eurandom and University of Amsterdam
Offer Kella*
Affiliation:
The Hebrew University of Jerusalem
Michel Mandjes*
Affiliation:
University of Amsterdam, Eurandom and CWI
*
∗Postal address: Universidad Carlos III de Madrid, Avda Universidad 30, 28911 Leganes (Madrid), Spain.
∗∗Postal address: Eurandom, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address: ivanovs@eurandom.tue.nl
∗∗∗Postal address: Department of Statistics, The Hebrew University of Jerusalem, Jerusalem 91905, Israel.
∗∗∗∗Postal address: Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands.
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Abstract

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In this paper we consider the first passage process of a spectrally negative Markov additive process (MAP). The law of this process is uniquely characterized by a certain matrix function, which plays a crucial role in fluctuation theory. We show how to identify this matrix using the theory of Jordan chains associated with analytic matrix functions. This result provides us with a technique that can be used to derive various further identities.

Keywords

Lévy processfluctuation theoryMarkov additive process

MSC classification

Secondary: 60K25: Queueing theory 60K37: Processes in random environments
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 4 , December 2010 , pp. 1048 - 1057
Copyright
Copyright © Applied Probability Trust 2010 

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