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Singularities of the matrix exponent of a Markov additive process with one-sided jumps

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  • Ivanovs, Jevgenijs
  • Boxma, Onno
  • Mandjes, Michel

Abstract

We analyze the number of zeros of det(F([alpha])), where F([alpha]) is the matrix exponent of a Markov Additive Process (MAP) with one-sided jumps. The focus is on the number of zeros in the right half of the complex plane, where F([alpha]) is analytic. In addition, we also consider the case of a MAP killed at an independent exponential time. The corresponding zeros can be seen as the roots of a generalized Cramér-Lundberg equation. We argue that our results are particularly useful in fluctuation theory for MAPs, which leads to numerous applications in queueing theory and finance.

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  • Ivanovs, Jevgenijs & Boxma, Onno & Mandjes, Michel, 2010. "Singularities of the matrix exponent of a Markov additive process with one-sided jumps," Stochastic Processes and their Applications, Elsevier, vol. 120(9), pages 1776-1794, August.
    • Handle: RePEc:eee:spapps:v:120:y:2010:i:9:p:1776-1794
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    References listed on IDEAS

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    1. Lu, Yi & Li, Shuanming, 2005. "On the probability of ruin in a Markov-modulated risk model," Insurance: Mathematics and Economics, Elsevier, vol. 37(3), pages 522-532, December.
    2. Asmussen, Søren & Avram, Florin & Pistorius, Martijn R., 2004. "Russian and American put options under exponential phase-type Lévy models," Stochastic Processes and their Applications, Elsevier, vol. 109(1), pages 79-111, January.
    3. Rajeeva L. Karandikar & Vidyadhar G. Kulkarni, 1995. "Second-Order Fluid Flow Models: Reflected Brownian Motion in a Random Environment," Operations Research, INFORMS, vol. 43(1), pages 77-88, February.
    4. H. R. Gail & S. L. Hantler & B. A. Taylor, 1992. "On a Preemptive Markovian Queue with Multiple Servers and Two Priority Classes," Mathematics of Operations Research, INFORMS, vol. 17(2), pages 365-391, May.
    5. G. J. K. Regterschot & J. H. A. de Smit, 1986. "The Queue M|G|1 with Markov Modulated Arrivals and Services," Mathematics of Operations Research, INFORMS, vol. 11(3), pages 465-483, August.
    6. Dickson,David C. M., 2005. "Insurance Risk and Ruin," Cambridge Books, Cambridge University Press, number 9780521846400.
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    Cited by:

    1. Minghua Wan & Mingxiu Cai & Guowei Yang, 2023. "Robust Exponential Graph Regularization Non-Negative Matrix Factorization Technology for Feature Extraction," Mathematics, MDPI, vol. 11(7), pages 1-14, April.
    2. Feng, Runhuan & Shimizu, Yasutaka, 2014. "Potential measures for spectrally negative Markov additive processes with applications in ruin theory," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 11-26.

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