IDEAS home Printed from https://ideas.repec.org/a/spr/metcap/v18y2016i1d10.1007_s11009-014-9412-9.html
   My bibliography  Save this article

Saddlepoint Approximations to the Probability of Ruin in Finite Time for the Compound Poisson Risk Process Perturbed by Diffusion

Author

Listed:
  • Riccardo Gatto

    (University of Bern)

  • Benjamin Baumgartner

    (University of Bern)

Abstract

A large deviations type approximation to the probability of ruin within a finite time for the compound Poisson risk process perturbed by diffusion is derived. This approximation is based on the saddlepoint method and generalizes the approximation for the non-perturbed risk process by Barndorff-Nielsen and Schmidli (Scand Actuar J 1995(2):169–186, 1995). An importance sampling approximation to this probability of ruin is also provided. Numerical illustrations assess the accuracy of the saddlepoint approximation using importance sampling as a benchmark. The relative deviations between saddlepoint approximation and importance sampling are very small, even for extremely small probabilities of ruin. The saddlepoint approximation is however substantially faster to compute.

Suggested Citation

  • Riccardo Gatto & Benjamin Baumgartner, 2016. "Saddlepoint Approximations to the Probability of Ruin in Finite Time for the Compound Poisson Risk Process Perturbed by Diffusion," Methodology and Computing in Applied Probability, Springer, vol. 18(1), pages 217-235, March.
    • Handle: RePEc:spr:metcap:v:18:y:2016:i:1:d:10.1007_s11009-014-9412-9
    DOI: 10.1007/s11009-014-9412-9
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11009-014-9412-9
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s11009-014-9412-9?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Feng, Runhuan, 2011. "An operator-based approach to the analysis of ruin-related quantities in jump diffusion risk models," Insurance: Mathematics and Economics, Elsevier, vol. 48(2), pages 304-313, March.
    2. Dufresne, François & Gerber, Hans U., 1989. "Three Methods to Calculate the Probability of Ruin," ASTIN Bulletin, Cambridge University Press, vol. 19(1), pages 71-90, April.
    3. Gerber, Hans U. & Landry, Bruno, 1998. "On the discounted penalty at ruin in a jump-diffusion and the perpetual put option," Insurance: Mathematics and Economics, Elsevier, vol. 22(3), pages 263-276, July.
    4. Riccardo Gatto & Benjamin Baumgartner, 2014. "Value at Ruin and Tail Value at Ruin of the Compound Poisson Process with Diffusion and Efficient Computational Methods," Methodology and Computing in Applied Probability, Springer, vol. 16(3), pages 561-582, September.
    5. Dufresne, Francois & Gerber, Hans U., 1991. "Risk theory for the compound Poisson process that is perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 10(1), pages 51-59, March.
    6. Tsai, Cary Chi-Liang & Willmot, Gordon E., 2002. "A generalized defective renewal equation for the surplus process perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 30(1), pages 51-66, February.
    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. Youngjun Choe & Henry Lam & Eunshin Byon, 2018. "Uncertainty Quantification of Stochastic Simulation for Black-box Computer Experiments," Methodology and Computing in Applied Probability, Springer, vol. 20(4), pages 1155-1172, December.
    2. He, Yue & Kawai, Reiichiro, 2022. "Moment and polynomial bounds for ruin-related quantities in risk theory," European Journal of Operational Research, Elsevier, vol. 302(3), pages 1255-1271.

    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. Chi, Yichun & Lin, X. Sheldon, 2011. "On the threshold dividend strategy for a generalized jump-diffusion risk model," Insurance: Mathematics and Economics, Elsevier, vol. 48(3), pages 326-337, May.
    2. Chiu, S. N. & Yin, C. C., 2003. "The time of ruin, the surplus prior to ruin and the deficit at ruin for the classical risk process perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 33(1), pages 59-66, August.
    3. Biffis, Enrico & Morales, Manuel, 2010. "On a generalization of the Gerber-Shiu function to path-dependent penalties," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 92-97, February.
    4. Tsai, Cary Chi-Liang, 2003. "On the expectations of the present values of the time of ruin perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 32(3), pages 413-429, July.
    5. Kam C. Yuen & Yuhua Lu & Rong Wu, 2009. "The compound Poisson process perturbed by a diffusion with a threshold dividend strategy," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 25(1), pages 73-93, January.
    6. Tsai, Cary Chi-Liang & Willmot, Gordon E., 2002. "On the moments of the surplus process perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 31(3), pages 327-350, December.
    7. Schmidli, Hanspeter, 2010. "On the Gerber-Shiu function and change of measure," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 3-11, February.
    8. 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.
    9. Yang, Hu & Zhang, Zhimin, 2009. "The perturbed compound Poisson risk model with multi-layer dividend strategy," Statistics & Probability Letters, Elsevier, vol. 79(1), pages 70-78, January.
    10. Morales, Manuel, 2007. "On the expected discounted penalty function for a perturbed risk process driven by a subordinator," Insurance: Mathematics and Economics, Elsevier, vol. 40(2), pages 293-301, March.
    11. Franck Adékambi & Essodina Takouda, 2022. "On the Discounted Penalty Function in a Perturbed Erlang Renewal Risk Model With Dependence," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 481-513, June.
    12. Sarkar, Joykrishna & Sen, Arusharka, 2005. "Weak convergence approach to compound Poisson risk processes perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 36(3), pages 421-432, June.
    13. Tsai, Cary Chi-Liang, 2006. "On the stop-loss transform and order for the surplus process perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 39(1), pages 151-170, August.
    14. Runhuan Feng & Yasutaka Shimizu, 2013. "On a Generalization from Ruin to Default in a Lévy Insurance Risk Model," Methodology and Computing in Applied Probability, Springer, vol. 15(4), pages 773-802, December.
    15. Chi, Yichun & Jaimungal, Sebastian & Lin, X. Sheldon, 2010. "An insurance risk model with stochastic volatility," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 52-66, February.
    16. Chi, Yichun, 2010. "Analysis of the expected discounted penalty function for a general jump-diffusion risk model and applications in finance," Insurance: Mathematics and Economics, Elsevier, vol. 46(2), pages 385-396, April.
    17. Gatto, Riccardo, 2008. "A saddlepoint approximation to the probability of ruin in the compound Poisson process with diffusion," Statistics & Probability Letters, Elsevier, vol. 78(13), pages 1948-1954, September.
    18. 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.
    19. Yacine Koucha & Alfredo D. Egidio dos Reis, 2021. "Approximations to ultimate ruin probabilities with a Wienner process perturbation," Papers 2107.02537, arXiv.org.
    20. Gerber, Hans U. & Shiu, Elias S.W. & Yang, Hailiang, 2013. "Valuing equity-linked death benefits in jump diffusion models," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 615-623.

    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:spr:metcap:v:18:y:2016:i:1:d:10.1007_s11009-014-9412-9. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.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.