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Gerber–Shiu functionals for classical risk processes perturbed by an α-stable motion

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  • Kolkovska, Ekaterina T.
  • Martín-González, Ehyter M.

Abstract

We study the Gerber–Shiu functional of the classical risk process perturbed by a spectrally negative α-stable motion. We provide representations of the scale functions of the process as an infinite series of convolutions of given functions. This, together with a result from Biffis and Kyprianou (2010), allows us to obtain a representation of the Gerber–Shiu functional as an infinite series of convolutions. Moreover, we calculate the Laplace transform and derive a defective renewal equation for the Gerber–Shiu functional, thus extending previous work of Furrer (1998) and of Tsai and Willmot (2002). We also obtain asymptotic expressions for the joint tail distribution of the severity of ruin and the surplus before ruin.

Suggested Citation

  • Kolkovska, Ekaterina T. & Martín-González, Ehyter M., 2016. "Gerber–Shiu functionals for classical risk processes perturbed by an α-stable motion," Insurance: Mathematics and Economics, Elsevier, vol. 66(C), pages 22-28.
    • Handle: RePEc:eee:insuma:v:66:y:2016:i:c:p:22-28
    DOI: 10.1016/j.insmatheco.2015.10.009
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    References listed on IDEAS

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    1. Dickson, David C. M. & Hipp, Christian, 2001. "On the time to ruin for Erlang(2) risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 29(3), pages 333-344, December.
    2. Furrer, Hansjorg & Michna, Zbigniew & Weron, Aleksander, 1997. "Stable Lévy motion approximation in collective risk theory," Insurance: Mathematics and Economics, Elsevier, vol. 20(2), pages 97-114, September.
    3. Tsai, Cary Chi-Liang & Willmot, Gordon E., 2002. "A generalized defective renewal equation for the surplus process perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 30(1), pages 51-66, February.
    4. Biffis, Enrico & Morales, Manuel, 2010. "On a generalization of the Gerber-Shiu function to path-dependent penalties," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 92-97, February.
    5. Biffis, Enrico & Kyprianou, Andreas E., 2010. "A note on scale functions and the time value of ruin for Lévy insurance risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 85-91, February.
    6. Hans Gerber & Elias Shiu, 1998. "On the Time Value of Ruin," North American Actuarial Journal, Taylor & Francis Journals, vol. 2(1), pages 48-72.
    7. Dickson,David C. M., 2010. "Insurance Risk and Ruin," Cambridge Books, Cambridge University Press, number 9780521176750.
    8. Dufresne, Francois & Gerber, Hans U., 1991. "Risk theory for the compound Poisson process that is perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 10(1), pages 51-59, March.
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    Cited by:

    1. He, Yue & Kawai, Reiichiro & Shimizu, Yasutaka & Yamazaki, Kazutoshi, 2023. "The Gerber-Shiu discounted penalty function: A review from practical perspectives," Insurance: Mathematics and Economics, Elsevier, vol. 109(C), pages 1-28.
    2. Yue He & Reiichiro Kawai & Yasutaka Shimizu & Kazutoshi Yamazaki, 2022. "The Gerber-Shiu discounted penalty function: A review from practical perspectives," Papers 2203.10680, arXiv.org, revised Dec 2022.

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