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A Risk Model with an Observer in a Markov Environment

Author

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  • Hansjörg Albrecher

    (Department of Actuarial Science, University of Lausanne, Lausanne CH-1015, Switzerland
    Swiss Finance Institute, University of Lausanne, Lausanne CH-1015, Switzerland)

  • Jevgenijs Ivanovs

    (Department of Actuarial Science, University of Lausanne, Lausanne CH-1015, Switzerland)

Abstract

We consider a spectrally-negative Markov additive process as a model of a risk process in a random environment. Following recent interest in alternative ruin concepts, we assume that ruin occurs when an independent Poissonian observer sees the process as negative, where the observation rate may depend on the state of the environment. Using an approximation argument and spectral theory, we establish an explicit formula for the resulting survival probabilities in this general setting. We also discuss an efficient evaluation of the involved quantities and provide a numerical illustration.

Suggested Citation

  • Hansjörg Albrecher & Jevgenijs Ivanovs, 2013. "A Risk Model with an Observer in a Markov Environment," Risks, MDPI, vol. 1(3), pages 1-14, November.
    • Handle: RePEc:gam:jrisks:v:1:y:2013:i:3:p:148-161:d:30342
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    References listed on IDEAS

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    Cited by:

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    2. Ivanovs, Jevgenijs, 2017. "Splitting and time reversal for Markov additive processes," Stochastic Processes and their Applications, Elsevier, vol. 127(8), pages 2699-2724.
    3. Landriault, David & Shi, Tianxiang, 2015. "Occupation times in the MAP risk model," Insurance: Mathematics and Economics, Elsevier, vol. 60(C), pages 75-82.
    4. Bladt, Mogens & Ivanovs, Jevgenijs, 2021. "Fluctuation theory for one-sided Lévy processes with a matrix-exponential time horizon," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 105-123.
    5. Eric C. K. Cheung & Jeff T. Y. Wong, 2023. "A Note on a Modified Parisian Ruin Concept," Risks, MDPI, vol. 11(3), pages 1-15, March.
    6. Lesław Gajek & Marcin Rudź, 2018. "Risk-switching insolvency models," Collegium of Economic Analysis Annals, Warsaw School of Economics, Collegium of Economic Analysis, issue 51, pages 129-146.
    7. Avram, Florin & Pérez, José-Luis & Yamazaki, Kazutoshi, 2018. "Spectrally negative Lévy processes with Parisian reflection below and classical reflection above," Stochastic Processes and their Applications, Elsevier, vol. 128(1), pages 255-290.
    8. Choi, Michael C.H. & Cheung, Eric C.K., 2014. "On the expected discounted dividends in the Cramér–Lundberg risk model with more frequent ruin monitoring than dividend decisions," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 121-132.
    9. 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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    11. Landriault, David & Li, Bin & Wong, Jeff T.Y. & Xu, Di, 2018. "Poissonian potential measures for Lévy risk models," Insurance: Mathematics and Economics, Elsevier, vol. 82(C), pages 152-166.

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