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May 2004 Hitting probabilities in a Markov additive process with linear movements and upward jumps: Applications to risk and queueing processes
Masakiyo Miyazawa
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Ann. Appl. Probab. 14(2): 1029-1054 (May 2004). DOI: 10.1214/105051604000000206

Abstract

Motivated by a risk process with positive and negative premium rates, we consider a real-valued Markov additive process with finitely many background states. This additive process linearly increases or decreases while the background state is unchanged, and may have upward jumps at the transition instants of the background state. It is known that the hitting probabilities of this additive process at lower levels have a matrix exponential form. We here study the hitting probabilities at upper levels, which do not have a matrix exponential form in general. These probabilities give the ruin probabilities in the terminology of the risk process. Our major interests are in their analytic expressions and their asymptotic behavior when the hitting level goes to infinity under light tail conditions on the jump sizes. To derive those results, we use a certain duality on the hitting probabilities, which may have an independent interest because it does not need any Markovian assumption.

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Masakiyo Miyazawa. "Hitting probabilities in a Markov additive process with linear movements and upward jumps: Applications to risk and queueing processes." Ann. Appl. Probab. 14 (2) 1029 - 1054, May 2004. https://doi.org/10.1214/105051604000000206

Information

Published: May 2004
First available in Project Euclid: 23 April 2004

zbMATH: 1057.60073
MathSciNet: MR2052912
Digital Object Identifier: 10.1214/105051604000000206

Subjects:
Primary: 60K25 , 90B22
Secondary: 60G55 , 60K20

Keywords: decay rate , Duality , hitting probability , Markov additive process , Markov renewal theorem , Risk process , stationary marked point process

Rights: Copyright © 2004 Institute of Mathematical Statistics

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Vol.14 • No. 2 • May 2004
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