First Passage Times for Markov Additive Processes with Positive Jumps of Phase Type
- 1 June 2008
- journal article
- research article
- Published by Cambridge University Press (CUP) in Journal of Applied Probability
- Vol. 45 (03), 779-799
- https://doi.org/10.1017/s0021900200004708
Abstract
The present paper generalises some results for spectrally negative Lévy processes to the setting of Markov additive processes (MAPs). A prominent role is assumed by the first passage times, which will be determined in terms of their Laplace transforms. These have the form of a phase-type distribution, with a rate matrix that can be regarded as an inverse function of the cumulant matrix. A numerically stable iteration to compute this matrix is given. The theory is first developed for MAPs without positive jumps and then extended to include positive jumps having phase-type distributions. Numerical and analytical examples show agreement with existing results in special cases.Keywords
This publication has 16 references indexed in Scilit:
- Applications of factorization embeddings for Lévy processesAdvances in Applied Probability, 2006
- On Maxima and Ladder Processes for a Dense Class of Lévy ProcessJournal of Applied Probability, 2006
- The time to ruin for a class of Markov additive risk process with two-sided jumpsAdvances in Applied Probability, 2005
- Russian and American put options under exponential phase-type Lévy modelsStochastic Processes and their Applications, 2004
- A matrix exponential form for hitting probabilities and its application to a Markov-modulated fluid queue with downward jumpsJournal of Applied Probability, 2002
- A multi-dimensional martingale for Markov additive processes and its applicationsAdvances in Applied Probability, 2000
- Fluctuation theory in continuous timeAdvances in Applied Probability, 1975
- Markov additive processes. IProbability Theory and Related Fields, 1972
- Markov Processes with Homogeneous Second Component, IITheory of Probability and Its Applications, 1969
- Markov Processes with Homogeneous Second Component. ITheory of Probability and Its Applications, 1969