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A quintuple law for Markov additive processes with phase-type jumps

Part of: Mathematical economics Markov processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Lothar Breuer
Lothar Breuer*
Affiliation:
University of Kent
*
∗Postal address: Institute of Mathematics and Statistics, University of Kent, Canterbury CT2 7NF, UK. Email address: l.breuer@kent.ac.uk
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Abstract

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We consider a Markov additive process (MAP) with phase-type jumps, starting at 0. Given a positive level u, we determine the joint distribution of the undershoot and overshoot of the first jump over the level u, the maximal level before this jump, the time of attaining this maximum, and the time between the maximum and the jump. The analysis is based on first passage times and time reversion of MAPs. A marginal of the derived distribution is the Gerber-Shiu function, which is of interest to insurance risk. Several examples serve to compare the present result with the literature.

Keywords

Markov additive processundershootsurplusruinGerber-Shiu function

MSC classification

Secondary: 60J25: Continuous-time Markov processes on general state spaces 60G51: Processes with independent increments; L'evy processes 91B30: Risk theory, insurance
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 2 , June 2010 , pp. 441 - 458
Copyright
Copyright © Applied Probability Trust 2010 

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