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Splitting and time reversal for Markov additive processes

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  • Ivanovs, Jevgenijs

Abstract

We consider a Markov additive process with a finite phase space and study its path decompositions at the times of extrema, first passage and last exit. For these three families of times we establish splitting conditional on the phase, and provide various relations between the laws of post- and pre-splitting processes using time reversal. These results offer valuable insight into the behaviour of the process, and while being structurally similar to the Lévy process case, they demonstrate various new features. As an application we formulate the Wiener–Hopf factorization, where time is counted in each phase separately and killing of the process is phase dependent. Restricting to the case of no positive jumps, we find concise formulas for these factors, and also characterize the time of last exit from the negative half-line. The latter result is obtained using three quite different approaches based on the established path decomposition theory, which further demonstrates its applicability.

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  • Ivanovs, Jevgenijs, 2017. "Splitting and time reversal for Markov additive processes," Stochastic Processes and their Applications, Elsevier, vol. 127(8), pages 2699-2724.
    • Handle: RePEc:eee:spapps:v:127:y:2017:i:8:p:2699-2724
    DOI: 10.1016/j.spa.2016.12.007
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    References listed on IDEAS

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    1. A. B. Dieker & M. Mandjes, 2011. "Extremes of Markov-additive Processes with One-sided Jumps, with Queueing Applications," Methodology and Computing in Applied Probability, Springer, vol. 13(2), pages 221-267, June.
    2. D'Auria, Bernardo & Kella, Offer & Ivanovs, Jevgenijs & Mandjes, Michel, 2010. "First passage of a Markov additive process and generalized Jordan chains," DES - Working Papers. Statistics and Econometrics. WS ws103923, Universidad Carlos III de Madrid. Departamento de Estadística.
    3. Hansjörg Albrecher & Jevgenijs Ivanovs, 2013. "A Risk Model with an Observer in a Markov Environment," Risks, MDPI, vol. 1(3), pages 1-14, November.
    4. Bertoin, Jean, 1993. "Splitting at the infimum and excursions in half-lines for random walks and Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 47(1), pages 17-35, August.
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    Cited by:

    1. Berkelmans, Wouter & Cichocka, Agata & Mandjes, Michel, 2020. "The correlation function of a queue with Lévy and Markov additive input," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1713-1734.
    2. Lyu, Hanbaek & Sivakoff, David, 2019. "Persistence of sums of correlated increments and clustering in cellular automata," Stochastic Processes and their Applications, Elsevier, vol. 129(4), pages 1132-1152.

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