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Regime Switching

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The Cramér–Lundberg Model and Its Variants

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Abstract

This chapter focuses on the evaluation of the transform of the ruin probability in a regime-switching (or Markov modulated) version of the standard Cramér-Lundberg model. In the derivation various elements from Chap. 1 are relied upon, in particular the idea of conditioning on the first event and the Wiener-Hopf decomposition. The results established in this chapter also facilitate the analysis of the conventional (non-modulated, that is) Cramér-Lundberg model over a phase-type horizon (rather than an exponentially distributed horizon). Finally, we comment on a setup in which the model parameters are periodically resampled, and argue that it fits in the modelling framework discussed in the chapter.

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References

  1. S. Asmussen, F. Avram, M. Usabel, Erlangian approximations for finite-horizon ruin probabilities. ASTIN Bull. 32, 267–281 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. H. Albrecher, C. Constantinescu, S. Loisel, Explicit ruin formulas for models with dependence among risks. Insur. Math. Econ. 48, 265–270 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. D. Anick, D. Mitra, M. Sondhi, Stochastic theory of a data–handling system with multiple sources. Bell Syst. Tech. J. 61, 1871–1894 (1982)

    Article  MathSciNet  Google Scholar 

  4. S. Asmussen, Applied Probability and Queues, 2nd edn. (Springer, New York, 2003)

    MATH  Google Scholar 

  5. S. Asmussen, J. Ivanovs, A factorization of a Lévy process over a phase-type horizon. Stoch. Models 34, 397–408 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  6. S. Asmussen, O. Kella, A multi-dimensional martingale for Markov additive processes and its applications. Adv. Appl. Probab. 32, 376–393 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. O. Boxma, M. Mandjes, Affine storage and insurance risk models. Math. Oper. Res. 46, 1282–1302 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  8. O. Boxma, M. Heemskerk, M. Mandjes, Single-server queues under overdispersion in the heavy-traffic regime. Stoch. Models 37, 197–230 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  9. E. Çinlar, Markov additive processes I. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 24, 85–93 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  10. C. Constantinescu, G. Delsing, M. Mandjes, L. Rojas-Nandayapa, A ruin model with a resampled environment. Scand. Actuar. J. 2020, 323–341 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  11. B. D’Auria, J. Ivanovs, O. Kella, M. Mandjes, First passage of a Markov additive process and generalized Jordan chains. J. Appl. Probab. 47, 1048–1057 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. K. Dȩbicki, M. Mandjes, Queues and Lévy Fluctuation Theory (Springer, New York, 2015)

    Google Scholar 

  13. G. Delsing, M. Mandjes, A transient Cramér-Lundberg model with applications to credit risk. J. Appl. Probab. 58, 721–745 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  14. A. Dieker, M. Mandjes, Extremes of Markov-additive processes with one-sided jumps, with queueing applications. Methodol. Comput. Appl. Probab. 13, 221–267 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. J. Ivanovs, O. Boxma, M. Mandjes, Singularities of the matrix exponent of a Markov additive process with one-sided jumps. Stoch. Process. Their Appl. 120, 1776–1794 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. P. Jordanova, Y. Nefedova, M. Stehlík, Risk process approximation with mixing. Appl. Math. Model. 41, 285–298 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. L. Kosten, Stochastic theory of a multi-entry buffer, part 1. Delft Progress Rep. Series F 1, 10–18 (1974)

    MathSciNet  MATH  Google Scholar 

  18. L. Kosten, Stochastic theory of data-handling systems with groups of multiple sources, in Performance of Computer-Communication Systems, ed. by H. Rudin, W. Bux (Elsevier, Amsterdam, 1984), pp. 321–331

    Google Scholar 

  19. L. van Kreveld, M. Mandjes, J.-P. Dorsman, Extreme value analysis for a Markov Additive Process driven by a nonirreducible background chain. Stoch. Syst. 12, 293–317 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  20. Y. Raaijmakers, H. Albrecher, O. Boxma, The single server queue with mixing dependencies. Methodol. Comput. Appl. Probab. 21, 1023–1044 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  21. N. Starreveld, R. Bekker, M. Mandjes, Transient analysis of one-sided Lévy-driven queues. Stoch. Models 32, 481–512 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Mathematical Institute, Leiden University, Leiden, The Netherlands

    Michel Mandjes

  2. Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands

    Onno Boxma

Authors
  1. Michel Mandjes
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  2. Onno Boxma
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Mandjes, M., Boxma, O. (2023). Regime Switching. In: The Cramér–Lundberg Model and Its Variants. Springer Actuarial(). Springer, Cham. https://doi.org/10.1007/978-3-031-39105-7_3

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