IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v8y2020i10p1742-d426123.html
   My bibliography  Save this article

Martingale Approach to Derive Lundberg-Type Inequalities

Author

Listed:
  • Tautvydas Kuras

    (Institute of Mathematics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania
    All authors contributed equally to this work.)

  • Jonas Sprindys

    (Institute of Mathematics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania
    All authors contributed equally to this work.)

  • Jonas Šiaulys

    (Institute of Mathematics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania
    All authors contributed equally to this work.)

Abstract

In this paper, we find the upper bound for the tail probability P sup n ⩾ 0 ∑ I = 1 n ξ I > x with random summands ξ 1 , ξ 2 , … having light-tailed distributions. We find conditions under which the tail probability of supremum of sums can be estimated by quantity ϱ 1 exp { − ϱ 2 x } with some positive constants ϱ 1 and ϱ 2 . For the proof we use the martingale approach together with the fundamental Wald’s identity. As the application we derive a few Lundberg-type inequalities for the ultimate ruin probability of the inhomogeneous renewal risk model.

Suggested Citation

  • Tautvydas Kuras & Jonas Sprindys & Jonas Šiaulys, 2020. "Martingale Approach to Derive Lundberg-Type Inequalities," Mathematics, MDPI, vol. 8(10), pages 1-18, October.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:10:p:1742-:d:426123
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/8/10/1742/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/8/10/1742/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Raducan, Anisoara Maria & Vernic, Raluca & Zbaganu, Gheorghita, 2015. "Recursive Calculation Of Ruin Probabilities At Or Before Claim Instants For Non-Identically Distributed Claims," ASTIN Bulletin, Cambridge University Press, vol. 45(2), pages 421-443, May.
    2. 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.
    3. Zhongyang Sun, 2019. "Upper bounds for ruin probabilities under model uncertainty," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 48(18), pages 4511-4527, September.
    4. Sgibnev, M. S., 1997. "Submultiplicative moments of the supremum of a random walk with negative drift," Statistics & Probability Letters, Elsevier, vol. 32(4), pages 377-383, April.
    5. Claude Lefèvre & Stéphane Loisel & Muhsin Tamturk & Sergey Utev, 2018. "A Quantum-Type Approach to Non-Life Insurance Risk Modelling," Risks, MDPI, vol. 6(3), pages 1-17, September.
    6. Ambagaspitiya, Rohana S., 2009. "Ultimate ruin probability in the Sparre Andersen model with dependent claim sizes and claim occurrence times," Insurance: Mathematics and Economics, Elsevier, vol. 44(3), pages 464-472, June.
    7. Edita Kizinevič & Jonas Šiaulys, 2018. "The Exponential Estimate of the Ultimate Ruin Probability for the Non-Homogeneous Renewal Risk Model," Risks, MDPI, vol. 6(1), pages 1-17, March.
    8. 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.
    9. Christian Hipp, 2018. "Company Value with Ruin Constraint in Lundberg Models," Risks, MDPI, vol. 6(3), pages 1-15, July.
    Full references (including those not matched with items on IDEAS)

    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. Edita Kizinevič & Jonas Šiaulys, 2018. "The Exponential Estimate of the Ultimate Ruin Probability for the Non-Homogeneous Renewal Risk Model," Risks, MDPI, vol. 6(1), pages 1-17, March.
    2. Andrius Grigutis & Jonas Šiaulys, 2020. "Ultimate Time Survival Probability in Three-Risk Discrete Time Risk Model," Mathematics, MDPI, vol. 8(2), pages 1-30, January.
    3. Leipus, Remigijus & Siaulys, Jonas, 2007. "Asymptotic behaviour of the finite-time ruin probability under subexponential claim sizes," Insurance: Mathematics and Economics, Elsevier, vol. 40(3), pages 498-508, May.
    4. Remigijus Leipus & Jonas Šiaulys, 2009. "Asymptotic behaviour of the finite‐time ruin probability in renewal risk models," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 25(3), pages 309-321, May.
    5. 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.
    6. Grandell, Jan, 2000. "Simple approximations of ruin probabilities," Insurance: Mathematics and Economics, Elsevier, vol. 26(2-3), pages 157-173, May.
    7. S. Pitts, 1994. "Nonparametric estimation of compound distributions with applications in insurance," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 46(3), pages 537-555, September.
    8. Dutang, C. & Lefèvre, C. & Loisel, S., 2013. "On an asymptotic rule A+B/u for ultimate ruin probabilities under dependence by mixing," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 774-785.
    9. Julien Trufin & Stéphane Loisel, 2013. "Ultimate ruin probability in discrete time with Bühlmann credibility premium adjustments," Post-Print hal-00426790, HAL.
    10. repec:hal:wpaper:hal-00746251 is not listed on IDEAS
    11. Yuen, Kam C. & Wang, Guojing & Ng, Kai W., 2004. "Ruin probabilities for a risk process with stochastic return on investments," Stochastic Processes and their Applications, Elsevier, vol. 110(2), pages 259-274, April.
    12. Florin Avram & Romain Biard & Christophe Dutang & Stéphane Loisel & Landy Rabehasaina, 2014. "A survey of some recent results on Risk Theory," Post-Print hal-01616178, HAL.
    13. Zhimin Zhang & Hailiang Yang & Hu Yang, 2012. "On a Sparre Andersen Risk Model with Time-Dependent Claim Sizes and Jump-Diffusion Perturbation," Methodology and Computing in Applied Probability, Springer, vol. 14(4), pages 973-995, December.
    14. 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.
    15. Zhu, Lingjiong, 2013. "Ruin probabilities for risk processes with non-stationary arrivals and subexponential claims," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 544-550.
    16. Konstantinides, Dimitrios & Tang, Qihe & Tsitsiashvili, Gurami, 2002. "Estimates for the ruin probability in the classical risk model with constant interest force in the presence of heavy tails," Insurance: Mathematics and Economics, Elsevier, vol. 31(3), pages 447-460, December.
    17. Vaios Dermitzakis & Konstadinos Politis, 2011. "Asymptotics for the Moments of the Time to Ruin for the Compound Poisson Model Perturbed by Diffusion," Methodology and Computing in Applied Probability, Springer, vol. 13(4), pages 749-761, December.
    18. J. Cerda-Hernandez & A. Sikov & A. Ramos, 2022. "An optimal investment strategy aimed at maximizing the expected utility across all intermediate capital levels," Papers 2207.02947, arXiv.org, revised Jun 2024.
    19. M. S. Sgibnev, 1998. "On the Asymptotic Behavior of the Harmonic Renewal Measure," Journal of Theoretical Probability, Springer, vol. 11(2), pages 371-382, April.
    20. Griffin, Philip S. & Maller, Ross A. & Schaik, Kees van, 2012. "Asymptotic distributions of the overshoot and undershoots for the Lévy insurance risk process in the Cramér and convolution equivalent cases," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 382-392.
    21. Muhsin Tamturk & Dominic Cortis & Mark Farrell, 2020. "Examining the Effects of Gradual Catastrophes on Capital Modelling and the Solvency of Insurers: The Case of COVID-19," Risks, MDPI, vol. 8(4), pages 1-13, 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:jmathe:v:8:y:2020:i:10:p:1742-:d:426123. 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.