IDEAS home Printed from https://ideas.repec.org/p/hal/journl/hal-00412977.html
   My bibliography  Save this paper

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
    as

    Download full text from publisher

    To our knowledge, this item is not available for download. To find whether it is available, there are three options:
    1. Check below whether another version of this item is available online.
    2. Check on the provider's web page whether it is in fact available.
    3. Perform a search for a similarly titled item that would be available.

    Other versions of this item:

    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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. Nikita Ratanov, 2008. "Option Pricing Model Based on a Markov-modulated Diffusion with Jumps," Papers 0812.0761, arXiv.org.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    9. Pierre-Olivier Goffard, 2019. "Fraud risk assessment within blockchain transactions," Working Papers hal-01716687, HAL.
    10. Benjamin Avanzi & Debbie Kusch Falden & Mogens Steffensen, 2022. "Stable Dividends under Linear-Quadratic Optimization," Papers 2210.03494, arXiv.org.
    11. 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.
    12. Alessandro De Gregorio & Stefano Iacus, 2007. "Change point estimation for the telegraph process observed at discrete times," UNIMI - Research Papers in Economics, Business, and Statistics unimi-1053, Universitá degli Studi di Milano.
    13. 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.
    14. 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.
    15. 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.
    16. 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.
    17. 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.
    18. 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.
    19. Pierre-Olivier Goffard & Claude Lefèvre, 2018. "Duality in ruin problems for ordered risk models," Post-Print hal-01398910, HAL.
    20. 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.
    21. 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.
    22. Pierre-O. Goffard, 2019. "Fraud risk assessment within blockchain transactions," Post-Print hal-01716687, HAL.
    23. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Jiang Hui & Xu Lihu & Yang Qingshan, 2024. "Functional Large Deviations for Kac–Stroock Approximation to a Class of Gaussian Processes with Application to Small Noise Diffusions," Journal of Theoretical Probability, Springer, vol. 37(4), pages 3015-3054, November.
    2. De Gregorio, Alessandro & Iafrate, Francesco, 2021. "Telegraph random evolutions on a circle," Stochastic Processes and their Applications, Elsevier, vol. 141(C), pages 79-108.
    3. Iacus, Stefano Maria, 2001. "Statistical analysis of the inhomogeneous telegrapher's process," Statistics & Probability Letters, Elsevier, vol. 55(1), pages 83-88, November.
    4. 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.
    5. Cinque, Fabrizio & Orsingher, Enzo, 2021. "On the exact distributions of the maximum of the asymmetric telegraph process," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 601-633.
    6. Zhang, Huiming & Liu, Yunxiao & Li, Bo, 2014. "Notes on discrete compound Poisson model with applications to risk theory," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 325-336.
    7. Nikita Ratanov, 2020. "First Crossing Times of Telegraph Processes with Jumps," Methodology and Computing in Applied Probability, Springer, vol. 22(1), pages 349-370, March.
    8. De Gregorio, Alessandro & Macci, Claudio, 2012. "Large deviation principles for telegraph processes," Statistics & Probability Letters, Elsevier, vol. 82(11), pages 1874-1882.
    9. Kolesnik, Alexander D. & Turbin, Anatoly F., 1998. "The equation of symmetric Markovian random evolution in a plane," Stochastic Processes and their Applications, Elsevier, vol. 75(1), pages 67-87, June.
    10. Li Qin & Susan M. Pitts, 2012. "Nonparametric Estimation of the Finite-Time Survival Probability with Zero Initial Capital in the Classical Risk Model," Methodology and Computing in Applied Probability, Springer, vol. 14(4), pages 919-936, December.
    11. Ratanov, Nikita, 2021. "On telegraph processes, their first passage times and running extrema," Statistics & Probability Letters, Elsevier, vol. 174(C).
    12. Cinque, Fabrizio, 2022. "A note on the conditional probabilities of the telegraph process," Statistics & Probability Letters, Elsevier, vol. 185(C).
    13. Bogachev, Leonid & Ratanov, Nikita, 2011. "Occupation time distributions for the telegraph process," Stochastic Processes and their Applications, Elsevier, vol. 121(8), pages 1816-1844, August.
    14. Venegas-Martínez, Francisco & Franco-Arbeláez, Luis Ceferino & Franco-Ceballos, Luis Eduardo & Murillo-Gómez, Juan Guillermo, 2015. "Riesgo operativo en el sector salud en Colombia: 2013," eseconomía, Escuela Superior de Economía, Instituto Politécnico Nacional, vol. 0(43), pages 7-36, segundo s.
    15. Dhaene, Jan & Vandebroek, Martina, 1995. "Recursions for the individual model," Insurance: Mathematics and Economics, Elsevier, vol. 16(1), pages 31-38, April.
    16. Vernic, Raluca, 2018. "On the evaluation of some multivariate compound distributions with Sarmanov’s counting distribution," Insurance: Mathematics and Economics, Elsevier, vol. 79(C), pages 184-193.
    17. Julien Trufin & Stéphane Loisel, 2013. "Ultimate ruin probability in discrete time with Bühlmann credibility premium adjustments," Post-Print hal-00426790, HAL.
    18. Eric Ghysels & Christian Gouriéroux & Joann Jasiak, 1995. "Market Time and Asset Price Movements Theory and Estimation," CIRANO Working Papers 95s-32, CIRANO.
    19. Nabil Kazi-Tani, 2020. "Indifference Pricing of Reinsurance with Reinstatements Using Coherent Monetary Criteria," Working Papers hal-01742638, HAL.
    20. 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.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:hal:journl:hal-00412977. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: CCSD (email available below). General contact details of provider: https://hal.archives-ouvertes.fr/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.