IDEAS home Printed from https://ideas.repec.org/a/eee/insuma/v43y2008i1p116-120.html
   My bibliography  Save this article

A uniform asymptotic estimate for discounted aggregate claims with subexponential tails

Author

Listed:
  • Hao, Xuemiao
  • Tang, Qihe

Abstract

In this paper we study the tail probability of discounted aggregate claims in a continuous-time renewal model. For the case that the common claim-size distribution is subexponential, we obtain an asymptotic formula, which holds uniformly for all time horizons within a finite interval. Then, with some additional mild assumptions on the distributions of the claim sizes and inter-arrival times, we further prove that this formula holds uniformly for all time horizons. In this way, we significantly extend a recent result of Tang [Tang, Q., 2007. Heavy tails of discounted aggregate claims in the continuous-time renewal model. J. Appl. Probab. 44 (2), 285-294].

Suggested Citation

  • Hao, Xuemiao & Tang, Qihe, 2008. "A uniform asymptotic estimate for discounted aggregate claims with subexponential tails," Insurance: Mathematics and Economics, Elsevier, vol. 43(1), pages 116-120, August.
    • Handle: RePEc:eee:insuma:v:43:y:2008:i:1:p:116-120
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167-6687(08)00047-4
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. Nilsen, Trygve & Paulsen, Jostein, 1996. "On the distribution of a randomly discounted compound Poisson process," Stochastic Processes and their Applications, Elsevier, vol. 61(2), pages 305-310, February.
    2. 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.
    3. Gjessing, Håkon K. & Paulsen, Jostein, 1997. "Present value distributions with applications to ruin theory and stochastic equations," Stochastic Processes and their Applications, Elsevier, vol. 71(1), pages 123-144, October.
    4. Paulsen, Jostein, 1993. "Risk theory in a stochastic economic environment," Stochastic Processes and their Applications, Elsevier, vol. 46(2), pages 327-361, 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. Bai, Xiaodong & Song, Lixin, 2011. "The asymptotic estimate for the sum of two correlated classes of discounted aggregate claims with heavy tails," Statistics & Probability Letters, Elsevier, vol. 81(12), pages 1891-1898.
    2. Dawei Lu & Ting Li & Meng Yuan & Xinmei Shen, 2024. "Asymptotic Finite-Time Ruin Probabilities for a Multidimensional Risk Model with Subexponential Claims," Methodology and Computing in Applied Probability, Springer, vol. 26(3), pages 1-28, September.
    3. Jiang, Tao & Wang, Yuebao & Chen, Yang & Xu, Hui, 2015. "Uniform asymptotic estimate for finite-time ruin probabilities of a time-dependent bidimensional renewal model," Insurance: Mathematics and Economics, Elsevier, vol. 64(C), pages 45-53.
    4. Li, Jinzhu, 2017. "A note on the finite-time ruin probability of a renewal risk model with Brownian perturbation," Statistics & Probability Letters, Elsevier, vol. 127(C), pages 49-55.
    5. Gao, Qingwu & Liu, Xijun, 2013. "Uniform asymptotics for the finite-time ruin probability with upper tail asymptotically independent claims and constant force of interest," Statistics & Probability Letters, Elsevier, vol. 83(6), pages 1527-1538.
    6. Gao Qingwu & Gu Peng & Jin Na, 2012. "Asymptotic Behavior of the Finite-Time Ruin Probability with Constant Interest Force and WUOD Heavy-Tailed Claims," Asia-Pacific Journal of Risk and Insurance, De Gruyter, vol. 6(1), pages 1-16, February.
    7. Bingzhen Genga & Shijie Wanga & Yang Yang, 2025. "Second order asymptotics for discounted aggregate claims of continuous-time renewal risk models with constant interest force," Papers 2501.02545, arXiv.org.
    8. Fu, Ke-Ang & Ng, Cheuk Yin Andrew, 2017. "Uniform asymptotics for the ruin probabilities of a two-dimensional renewal risk model with dependent claims and risky investments," Statistics & Probability Letters, Elsevier, vol. 125(C), pages 227-235.
    9. Kaiyong Wang & Yuebao Wang & Qingwu Gao, 2013. "Uniform Asymptotics for the Finite-Time Ruin Probability of a Dependent Risk Model with a Constant Interest Rate," Methodology and Computing in Applied Probability, Springer, vol. 15(1), pages 109-124, March.
    10. Peng, Jiangyan & Huang, Jin, 2010. "Ruin probability in a one-sided linear model with constant interest rate," Statistics & Probability Letters, Elsevier, vol. 80(7-8), pages 662-669, April.
    11. Dan Zhu & Ming Zhou & Chuancun Yin, 2023. "Finite-Time Ruin Probabilities of Bidimensional Risk Models with Correlated Brownian Motions," Mathematics, MDPI, vol. 11(12), pages 1-18, June.
    12. Li, Jinzhu, 2022. "Asymptotic analysis of a dynamic systemic risk measure in a renewal risk model," Insurance: Mathematics and Economics, Elsevier, vol. 107(C), pages 38-56.
    13. Hongmin Xiao & Lin Xie, 2018. "Asymptotic Ruin Probability of a Bidimensional Risk Model Based on Entrance Processes with Constant Interest Rate," Risks, MDPI, vol. 6(4), pages 1-12, November.
    14. Wei, Li, 2009. "Ruin probability in the presence of interest earnings and tax payments," Insurance: Mathematics and Economics, Elsevier, vol. 45(1), pages 133-138, August.
    15. Yang, Haizhong & Li, Jinzhu, 2019. "On asymptotic finite-time ruin probability of a renewal risk model with subexponential main claims and delayed claims," Statistics & Probability Letters, Elsevier, vol. 149(C), pages 153-159.
    16. Liu, Xijun & Gao, Qingwu & Wang, Yuebao, 2012. "A note on a dependent risk model with constant interest rate," Statistics & Probability Letters, Elsevier, vol. 82(4), pages 707-712.
    17. Yang, Haizhong & Li, Jinzhu, 2014. "Asymptotic finite-time ruin probability for a bidimensional renewal risk model with constant interest force and dependent subexponential claims," Insurance: Mathematics and Economics, Elsevier, vol. 58(C), pages 185-192.
    18. Dawei Lu & Meng Yuan, 2022. "Asymptotic Finite-Time Ruin Probabilities for a Bidimensional Delay-Claim Risk Model with Subexponential Claims," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 2265-2286, December.

    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. Zhehao Zhang, 2025. "The Aggregate Discounted Claims Process Under Multiple and Terminable Arrivals Renewal Processes," Methodology and Computing in Applied Probability, Springer, vol. 27(1), pages 1-23, March.
    2. Constantinos Kardaras & Scott Robertson, 2017. "Continuous-time perpetuities and time reversal of diffusions," Finance and Stochastics, Springer, vol. 21(1), pages 65-110, January.
    3. Kardaras, Constantinos & Robertson, Scott, 2017. "Continuous-time perpetuities and time reversal of diffusions," LSE Research Online Documents on Economics 67495, London School of Economics and Political Science, LSE Library.
    4. Zhang, Zhehao, 2019. "On the stochastic equation L(Z)=L[V(X+Z)] and properties of Mittag–Leffler distributions," Applied Mathematics and Computation, Elsevier, vol. 361(C), pages 365-376.
    5. Gjessing, Håkon K. & Paulsen, Jostein, 1997. "Present value distributions with applications to ruin theory and stochastic equations," Stochastic Processes and their Applications, Elsevier, vol. 71(1), pages 123-144, October.
    6. Nyrhinen, Harri, 2001. "Finite and infinite time ruin probabilities in a stochastic economic environment," Stochastic Processes and their Applications, Elsevier, vol. 92(2), pages 265-285, April.
    7. Bae, Taehan & Kim, Changki & Kulperger, Reginald J., 2009. "Securitization of motor insurance loss rate risks," Insurance: Mathematics and Economics, Elsevier, vol. 44(1), pages 48-58, February.
    8. Nyrhinen, Harri, 1999. "On the ruin probabilities in a general economic environment," Stochastic Processes and their Applications, Elsevier, vol. 83(2), pages 319-330, October.
    9. Paulsen, Jostein, 1998. "Sharp conditions for certain ruin in a risk process with stochastic return on investments," Stochastic Processes and their Applications, Elsevier, vol. 75(1), pages 135-148, June.
    10. Jostein Paulsen, 2008. "Ruin models with investment income," Papers 0806.4125, arXiv.org, revised Dec 2008.
    11. Wang, Guojing & Wu, Rong, 2001. "Distributions for the risk process with a stochastic return on investments," Stochastic Processes and their Applications, Elsevier, vol. 95(2), pages 329-341, October.
    12. Charupat, Narat & Milevsky, Moshe A., 2002. "Optimal asset allocation in life annuities: a note," Insurance: Mathematics and Economics, Elsevier, vol. 30(2), pages 199-209, April.
    13. Klüppelberg, Claudia & Kostadinova, Radostina, 2008. "Integrated insurance risk models with exponential Lévy investment," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 560-577, April.
    14. Milevsky, Moshe Arye, 1999. "Martingales, scale functions and stochastic life annuities: a note," Insurance: Mathematics and Economics, Elsevier, vol. 24(1-2), pages 149-154, March.
    15. Paulsen, Jostein, 1998. "Ruin theory with compounding assets -- a survey," Insurance: Mathematics and Economics, Elsevier, vol. 22(1), pages 3-16, May.
    16. Paulsen, Jostein & Gjessing, Hakon K., 1997. "Optimal choice of dividend barriers for a risk process with stochastic return on investments," Insurance: Mathematics and Economics, Elsevier, vol. 20(3), pages 215-223, October.
    17. Yang, Haizhong & Sun, Suting, 2013. "Subexponentiality of the product of dependent random variables," Statistics & Probability Letters, Elsevier, vol. 83(9), pages 2039-2044.
    18. Yuri Kabanov & Platon Promyslov, 2023. "Ruin probabilities for a Sparre Andersen model with investments: the case of annuity payments," Finance and Stochastics, Springer, vol. 27(4), pages 887-902, October.
    19. Yang, Yang & Wang, Yuebao, 2010. "Asymptotics for ruin probability of some negatively dependent risk models with a constant interest rate and dominatedly-varying-tailed claims," Statistics & Probability Letters, Elsevier, vol. 80(3-4), pages 143-154, February.
    20. Yuchao Dong & Jérôme Spielmann, 2020. "Weak Limits of Random Coefficient Autoregressive Processes and their Application in Ruin Theory," Post-Print hal-02170829, HAL.

    More about this item

    Statistics

    Access and download statistics

    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:eee:insuma:v:43:y:2008:i:1:p:116-120. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/inca/505554 .

    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.