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A link between wave governed random motions and ruin processes

Author

Listed:
  • Christian Mazza

    (Département de Mathématiques - Albert-Ludwigs-Universität Freiburg = University of Freiburg)

  • Didier Rullière

    (SAF - Laboratoire de Sciences Actuarielle et Financière - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon)

Abstract

This article establishes a link between hitting times associated with the risk process (time of ruin) and wave governed random motions, which are widely used in physics. Concerning risk theory, another link holds between processes corresponding to models called positive and negative risk sums. Some classical results appear to be strongly interconnected. An original algorithm is proposed for computing finite-time ruin probabilities in renewal non-Poissonian risk model with exponential claims. Concerning wave-governed random motions, we analyze the distribution of the maxima of the processes. New bounds are directly derived from risk theory and appear to be more accurate than the ones proposed recently in the probabilistic literature. Finally, we propose applications of these notions in finance.

Suggested Citation

  • Christian Mazza & Didier Rullière, 2004. "A link between wave governed random motions and ruin processes," Post-Print hal-00412977, HAL.
    • Handle: RePEc:hal:journl:hal-00412977
    DOI: 10.1016/j.insmatheco.2004.07.014
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    References listed on IDEAS

    as
    1. Foong, S. K. & Kanno, S., 1994. "Properties of the telegrapher's random process with or without a trap," Stochastic Processes and their Applications, Elsevier, vol. 53(1), pages 147-173, September.
    2. Rulliere, Didier & Loisel, Stephane, 2004. "Another look at the Picard-Lefevre formula for finite-time ruin probabilities," Insurance: Mathematics and Economics, Elsevier, vol. 35(2), pages 187-203, October.
    3. Orsingher, Enzo, 1990. "Probability law, flow function, maximum distribution of wave-governed random motions and their connections with Kirchoff's laws," Stochastic Processes and their Applications, Elsevier, vol. 34(1), pages 49-66, February.
    4. Panjer, Harry H., 1981. "Recursive Evaluation of a Family of Compound Distributions," ASTIN Bulletin, Cambridge University Press, vol. 12(1), pages 22-26, June.
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    Citations

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    Cited by:

    1. Goffard, Pierre-Olivier & Lefèvre, Claude, 2018. "Duality in ruin problems for ordered risk models," Insurance: Mathematics and Economics, Elsevier, vol. 78(C), pages 44-52.
    2. Loisel, Stéphane & Mazza, Christian & Rullière, Didier, 2008. "Robustness analysis and convergence of empirical finite-time ruin probabilities and estimation risk solvency margin," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 746-762, April.
    3. Alessandro De Gregorio & Stefano M. Iacus, 2007. "Change point estimation for the telegraph process observed at discrete times," Papers 0705.0503, arXiv.org.
    4. Nikita Ratanov, 2008. "Option Pricing Model Based on a Markov-modulated Diffusion with Jumps," Papers 0812.0761, arXiv.org.
    5. Avanzi, Benjamin & Lau, Hayden & Wong, Bernard, 2020. "Optimal periodic dividend strategies for spectrally positive Lévy risk processes with fixed transaction costs," Insurance: Mathematics and Economics, Elsevier, vol. 93(C), pages 315-332.
    6. Cheung, Eric C.K. & Wong, Jeff T.Y., 2017. "On the dual risk model with Parisian implementation delays in dividend payments," European Journal of Operational Research, Elsevier, vol. 257(1), pages 159-173.
    7. Benjamin Avanzi & Hayden Lau & Bernard Wong, 2020. "Optimal periodic dividend strategies for spectrally positive L\'evy risk processes with fixed transaction costs," Papers 2003.13275, arXiv.org, revised May 2020.
    8. Christophette Blanchet-Scalliet & Diana Dorobantu & Didier Rullière, 2013. "The density of the ruin time for a renewal-reward process perturbed by a diffusion," Post-Print hal-00625099, HAL.
    9. Loisel, Stéphane & Mazza, Christian & Rullière, Didier, 2009. "Convergence and asymptotic variance of bootstrapped finite-time ruin probabilities with partly shifted risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 45(3), pages 374-381, December.
    10. Antonio Di Crescenzo & Barbara Martinucci, 2013. "On the Generalized Telegraph Process with Deterministic Jumps," Methodology and Computing in Applied Probability, Springer, vol. 15(1), pages 215-235, March.
    11. Pierre-Olivier Goffard & Claude Lefèvre, 2018. "Duality in ruin problems for ordered risk models," Post-Print hal-01398910, HAL.
    12. Avanzi, Benjamin & Cheung, Eric C.K. & Wong, Bernard & Woo, Jae-Kyung, 2013. "On a periodic dividend barrier strategy in the dual model with continuous monitoring of solvency," Insurance: Mathematics and Economics, Elsevier, vol. 52(1), pages 98-113.
    13. Macci, Claudio, 2009. "Convergence of large deviation rates based on a link between wave governed random motions and ruin processes," Statistics & Probability Letters, Elsevier, vol. 79(2), pages 255-263, January.
    14. Pierre-Olivier Goffard, 2019. "Fraud risk assessment within blockchain transactions," Working Papers hal-01716687, HAL.
    15. Dimitrova, Dimitrina S. & Kaishev, Vladimir K. & Zhao, Shouqi, 2015. "On finite-time ruin probabilities in a generalized dual risk model with dependence," European Journal of Operational Research, Elsevier, vol. 242(1), pages 134-148.
    16. Benjamin Avanzi & Debbie Kusch Falden & Mogens Steffensen, 2022. "Stable Dividends under Linear-Quadratic Optimization," Papers 2210.03494, arXiv.org.
    17. Dickson, David C.M. & Li, Shuanming, 2010. "Finite time ruin problems for the Erlang(2) risk model," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 12-18, February.
    18. Pierre-O. Goffard, 2019. "Fraud risk assessment within blockchain transactions," Post-Print hal-01716687, HAL.
    19. Borovkov, Konstantin A. & Dickson, David C.M., 2008. "On the ruin time distribution for a Sparre Andersen process with exponential claim sizes," Insurance: Mathematics and Economics, Elsevier, vol. 42(3), pages 1104-1108, June.
    20. Benjamin Avanzi & Jos'e-Luis P'erez & Bernard Wong & Kazutoshi Yamazaki, 2016. "On optimal joint reflective and refractive dividend strategies in spectrally positive L\'evy models," Papers 1607.01902, arXiv.org, revised Nov 2016.
    21. Hansjörg Albrecher & Dina Finger & Pierre-Olivier Goffard, 2022. "Blockchain mining in pools: Analyzing the trade-off between profitability and ruin," Working Papers hal-03336851, HAL.
    22. Wong, Jeff T.Y. & Cheung, Eric C.K., 2015. "On the time value of Parisian ruin in (dual) renewal risk processes with exponential jumps," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 280-290.
    23. Avanzi, Benjamin & Pérez, José-Luis & Wong, Bernard & Yamazaki, Kazutoshi, 2017. "On optimal joint reflective and refractive dividend strategies in spectrally positive Lévy models," Insurance: Mathematics and Economics, Elsevier, vol. 72(C), pages 148-162.

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