IDEAS home Printed from https://ideas.repec.org/a/gam/jrisks/v7y2019i4p104-d276247.html
   My bibliography  Save this article

Ruin Probability Approximations in Sparre Andersen Models with Completely Monotone Claims

Author

Listed:
  • Hansjörg Albrecher

    (Department of Actuarial Science, Faculty of Business and Economics, University of Lausanne, Quartier UNIL-Chamberonne Bâtiment Extranef, 1015 Lausanne, Switzerland)

  • Eleni Vatamidou

    (Department of Actuarial Science, Faculty of Business and Economics, University of Lausanne, Quartier UNIL-Chamberonne Bâtiment Extranef, 1015 Lausanne, Switzerland)

Abstract

We consider the Sparre Andersen risk process with interclaim times that belong to the class of distributions with rational Laplace transform. We construct error bounds for the ruin probability based on the Pollaczek–Khintchine formula, and develop an efficient algorithm to approximate the ruin probability for completely monotone claim size distributions. Our algorithm improves earlier results and can be tailored towards achieving a predetermined accuracy of the approximation.

Suggested Citation

  • Hansjörg Albrecher & Eleni Vatamidou, 2019. "Ruin Probability Approximations in Sparre Andersen Models with Completely Monotone Claims," Risks, MDPI, vol. 7(4), pages 1-14, October.
  • Handle: RePEc:gam:jrisks:v:7:y:2019:i:4:p:104-:d:276247
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-9091/7/4/104/pdf
    Download Restriction: no

    File URL: https://www.mdpi.com/2227-9091/7/4/104/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Willmot, Gordon E., 2007. "On the discounted penalty function in the renewal risk model with general interclaim times," Insurance: Mathematics and Economics, Elsevier, vol. 41(1), pages 17-31, July.
    2. Asmussen, Soren & Rolski, Tomasz, 1992. "Computational methods in risk theory: A matrix-algorithmic approach," Insurance: Mathematics and Economics, Elsevier, vol. 10(4), pages 259-274, January.
    3. Albrecher, Hansjoerg & Constantinescu, Corina & Thomann, Enrique, 2012. "Asymptotic results for renewal risk models with risky investments," Stochastic Processes and their Applications, Elsevier, vol. 122(11), pages 3767-3789.
    4. 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.
    5. Hansjörg Albrecher & Hans Gerber & Hailiang Yang, 2010. "A Direct Approach to the Discounted Penalty Function," North American Actuarial Journal, Taylor & Francis Journals, vol. 14(4), pages 420-434.
    6. Peralta, Oscar & Rojas-Nandayapa, Leonardo & Xie, Wangyue & Yao, Hui, 2018. "Approximation of ruin probabilities via Erlangized scale mixtures," Insurance: Mathematics and Economics, Elsevier, vol. 78(C), pages 136-156.
    7. Landriault, David & Willmot, Gordon, 2008. "On the Gerber-Shiu discounted penalty function in the Sparre Andersen model with an arbitrary interclaim time distribution," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 600-608, April.
    8. Dickson, D. C. M., 2001. "Lundberg Approximations for Compound Distributions with Insurance Applications. By G. E. Willmot and X. S. Lin. (Springer, 2000)," British Actuarial Journal, Cambridge University Press, vol. 7(4), pages 690-691, October.
    9. Embrechts, P. & Veraverbeke, N., 1982. "Estimates for the probability of ruin with special emphasis on the possibility of large claims," Insurance: Mathematics and Economics, Elsevier, vol. 1(1), pages 55-72, January.
    10. Albrecher, Hansjorg & Boxma, Onno J., 2005. "On the discounted penalty function in a Markov-dependent risk model," Insurance: Mathematics and Economics, Elsevier, vol. 37(3), pages 650-672, December.
    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. Tautvydas Kuras & Jonas Sprindys & Jonas Šiaulys, 2020. "Martingale Approach to Derive Lundberg-Type Inequalities," Mathematics, MDPI, vol. 8(10), pages 1-18, October.
    2. Josef Anton Strini & Stefan Thonhauser, 2020. "On Computations in Renewal Risk Models—Analytical and Statistical Aspects," Risks, MDPI, vol. 8(1), pages 1-20, March.

    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. Albrecher, Hansjörg & Constantinescu, Corina & Pirsic, Gottlieb & Regensburger, Georg & Rosenkranz, Markus, 2010. "An algebraic operator approach to the analysis of Gerber-Shiu functions," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 42-51, February.
    2. Ehyter Matías Martín-González & Antonio Murillo-Salas & Henry Pantí, 2022. "Gerber-Shiu Function for a Class of Markov-Modulated Lévy Risk Processes with Two-Sided Jumps," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 2779-2800, December.
    3. 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.
    4. 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.
    5. Franck Adékambi & Essodina Takouda, 2020. "Gerber–Shiu Function in a Class of Delayed and Perturbed Risk Model with Dependence," Risks, MDPI, vol. 8(1), pages 1-25, March.
    6. Georgios Psarrakos, 2016. "An Operator Property of the Distribution of a Nonhomogeneous Poisson Process with Applications," Methodology and Computing in Applied Probability, Springer, vol. 18(4), pages 1197-1215, December.
    7. Cheung, Eric C.K., 2011. "A generalized penalty function in Sparre Andersen risk models with surplus-dependent premium," Insurance: Mathematics and Economics, Elsevier, vol. 48(3), pages 384-397, May.
    8. Tang, Qihe & Wei, Li, 2010. "Asymptotic aspects of the Gerber-Shiu function in the renewal risk model using Wiener-Hopf factorization and convolution equivalence," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 19-31, February.
    9. Ren, Jiandong, 2009. "A connection between the discounted and non-discounted expected penalty functions in the Sparre Andersen risk model," Statistics & Probability Letters, Elsevier, vol. 79(3), pages 324-330, February.
    10. Cheung, Eric C.K. & Landriault, David & Willmot, Gordon E. & Woo, Jae-Kyung, 2010. "Structural properties of Gerber-Shiu functions in dependent Sparre Andersen models," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 117-126, February.
    11. Kim, So-Yeun & Willmot, Gordon E., 2016. "On the analysis of ruin-related quantities in the delayed renewal risk model," Insurance: Mathematics and Economics, Elsevier, vol. 66(C), pages 77-85.
    12. Cheung, Eric C.K., 2013. "Moments of discounted aggregate claim costs until ruin in a Sparre Andersen risk model with general interclaim times," Insurance: Mathematics and Economics, Elsevier, vol. 53(2), pages 343-354.
    13. Cheung, Eric C.K. & Liu, Haibo & Willmot, Gordon E., 2018. "Joint moments of the total discounted gains and losses in the renewal risk model with two-sided jumps," Applied Mathematics and Computation, Elsevier, vol. 331(C), pages 358-377.
    14. Zhang, Zhimin & Yang, Hu, 2010. "A generalized penalty function in the Sparre-Andersen risk model with two-sided jumps," Statistics & Probability Letters, Elsevier, vol. 80(7-8), pages 597-607, April.
    15. Zan Yu & Lianzeng Zhang, 2024. "Computing the Gerber-Shiu function with interest and a constant dividend barrier by physics-informed neural networks," Papers 2401.04378, arXiv.org.
    16. Anna Castañer & M.Mercè Claramunt & Maite Mármol, 2014. "Some optimization and decision problems in proportional reinsurance," UB School of Economics Working Papers 2014/310, University of Barcelona School of Economics.
    17. Lee, Wing Yan & Willmot, Gordon E., 2014. "On the moments of the time to ruin in dependent Sparre Andersen models with emphasis on Coxian interclaim times," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 1-10.
    18. Xin Zhang, 2008. "On the Ruin Problem in a Markov-Modulated Risk Model," Methodology and Computing in Applied Probability, Springer, vol. 10(2), pages 225-238, June.
    19. Landriault, David & Willmot, Gordon, 2008. "On the Gerber-Shiu discounted penalty function in the Sparre Andersen model with an arbitrary interclaim time distribution," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 600-608, April.
    20. Jae-Kyung Woo & Haibo Liu, 2018. "Discounted Aggregate Claim Costs Until Ruin in the Discrete-Time Renewal Risk Model," Methodology and Computing in Applied Probability, Springer, vol. 20(4), pages 1285-1318, 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:gam:jrisks:v:7:y:2019:i:4:p:104-:d:276247. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    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.