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First Passage Times for Markov Additive Processes with Positive Jumps of Phase Type

Part of: 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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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

Lévy processfirst passage timesupremumMarkov additive processreflected processstationary distribution

MSC classification

Secondary: 60J25: Continuous-time Markov processes on general state spaces 60G51: Processes with independent increments; L'evy processes 60J55: Local time and additive functionals
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 3 , September 2008 , pp. 779 - 799
Copyright
Copyright © Applied Probability Trust 2008 

References

[1] Asmussen, S. (1995). {Stationary distributions for fluid flow models with or without Brownian motion}. {Stoch. Models} 11, 1–20.Google Scholar
[2] Asmussen, S. (1995). {Stationary distributions via first passage times}. In {Advances in Queueing: Theory, Methods, and Open Problems}, ed. Dshalalow, J., CRC Press, Boca Raton, FL, pp. 79–102.Google Scholar
[3] Asmussen, S. (2003). {Applied Probability and Queues}. Springer, New York.Google Scholar
[4] Asmussen, S. and Kella, O. (2000). {A multi-dimensional martingale for Markov additive processes and its applications}. {Adv. Appl. Prob.} 32, 376–393.Google Scholar
[5] Asmussen, S., Avram, F. and Pistorius, M. (2004). {Russian and American put options under exponential phase-type Lévy models}. {Stoch. Process. Appl.} 109, 79–111.CrossRefGoogle Scholar
[6] Badescu, A. and Breuer, L. (2008). {The use of vector-valued martingales in risk theory}. {Blätter DGVFM} 29, 1–12.Google Scholar
[7] Badescu, A. et al. (2005). {Risk processes analyzed as fluid queues}. {Scand. Actuarial J.} 2005, 127–141.CrossRefGoogle Scholar
[8] Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
[9] Bingham, N. (1975). {Fluctuation theory in continuous time}. {Adv. Appl. Prob.} 7, 705–766.Google Scholar
[10] Çinlar, E. (1972). {Markov additive processes. I, II.} Z. Wahrscheinlichkeitsth. 24, 85–121.CrossRefGoogle Scholar
[11] Dieker, A. (2006). {Applications of factorization embeddings for Lévy processes}. Adv. Appl. Prob. 38, 768–791.Google Scholar
[12] Dieker, A. (2006). Extremes and fluid queues. {. Universiteit van Amsterdam.Google Scholar
[13] Ezhov, I. and Skorokhod, A. (1969). Markov processes with homogeneous second component. I. II. Theory. Prob. Appl. 14, 1–13.Google Scholar
[14] Ezhov, I. and Skorokhod, A. (1969). Markov processes with homogeneous second component. II. Theory. Prob. Appl. 14, 652–667.CrossRefGoogle Scholar
[15] Gerber, H. and Shiu, E. (1998). {On the time value of ruin}. {N. Amer. Actuarial J.} 2, 48–78.CrossRefGoogle Scholar
[16] Jacobsen, M. (2005). {The time to ruin for a class of Markov additive risk processes with two-sided Jumps}. {Adv. Appl. Prob.} 37, 963–992.Google Scholar
[17] Kyprianou, A. and Palmowski, Z. (2008). {Fluctuations of spectrally negative Markov additive processes}. (Lecture Notes Math. 1934), {Séminaire de Probabilités XLI}, Springer, Berlin.Google Scholar
[18] Miyazawa, M. and Takada, H. (2002). {A matrix-exponential form for hitting probabilities and its application to a Markov-modulated fluid queue with downward Jumps}. {J. Appl. Prob.} 39, 604–618.Google Scholar
[19] Mordecki, E. (2002). {The distribution of the maximum of a Lévy process with positive Jumps of phase-type}. {Theory Stoch. Process.} 8, 309–316.Google Scholar
[20] Pistorius, M. (2006). {On maxima and ladder processes for a dense class of Lévy process}. {J. Appl. Prob.} 43, 208–220.Google Scholar