IDEAS home Printed from https://ideas.repec.org/a/eee/insuma/v49y2011i1p126-131.html
   My bibliography  Save this article

A numerical method for the expected penalty-reward function in a Markov-modulated jump-diffusion process

Author

Listed:
  • Diko, Peter
  • Usábel, Miguel

Abstract

A generalization of the Cramér-Lundberg risk model perturbed by a diffusion is proposed. Aggregate claims of an insurer follow a compound Poisson process and premiums are collected at a constant rate with additional random fluctuation. The insurer is allowed to invest the surplus into a risky asset with volatility dependent on the level of the investment, which permits the incorporation of rational investment strategies as proposed by Berk and Green (2004). The return on investment is modulated by a Markov process which generalizes previously studied settings for the evolution of the interest rate in time. The Gerber-Shiu expected penalty-reward function is studied in this context, including ruin probabilities (a first-passage problem) as a special case. The second order integro-differential system of equations that characterizes the function of interest is obtained. As a closed-form solution does not exist, a numerical procedure based on the Chebyshev polynomial approximation through a collocation method is proposed. Finally, some examples illustrating the procedure are presented.

Suggested Citation

  • Diko, Peter & Usábel, Miguel, 2011. "A numerical method for the expected penalty-reward function in a Markov-modulated jump-diffusion process," Insurance: Mathematics and Economics, Elsevier, vol. 49(1), pages 126-131, July.
    • Handle: RePEc:eee:insuma:v:49:y:2011:i:1:p:126-131
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167668711000382
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    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. Ren, Jiandong, 2005. "The expected value of the time of ruin and the moments of the discounted deficit at ruin in the perturbed classical risk process," Insurance: Mathematics and Economics, Elsevier, vol. 37(3), pages 505-521, December.
    2. Wang, Guojing, 2001. "A decomposition of the ruin probability for the risk process perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 28(1), pages 49-59, February.
    3. Jonathan B. Berk & Richard C. Green, 2004. "Mutual Fund Flows and Performance in Rational Markets," Journal of Political Economy, University of Chicago Press, vol. 112(6), pages 1269-1295, December.
    4. Hans Gerber & Elias Shiu, 1998. "On the Time Value of Ruin," North American Actuarial Journal, Taylor & Francis Journals, vol. 2(1), pages 48-72.
    5. 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.
    6. Wang, Guojing & Wu, Rong, 2008. "The expected discounted penalty function for the perturbed compound Poisson risk process with constant interest," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 59-64, February.
    7. 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.
    8. Li, Shuanming & Garrido, José, 2005. "Ruin Probabilities for Two Classes of Risk Processes," ASTIN Bulletin, Cambridge University Press, vol. 35(1), pages 61-77, May.
    9. 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.
    10. 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.
    11. Dickson,David C. M., 2005. "Insurance Risk and Ruin," Cambridge Books, Cambridge University Press, number 9780521846400.
    12. 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.
    13. Usabel, Miguel, 1999. "Calculating multivariate ruin probabilities via Gaver-Stehfest inversion technique," Insurance: Mathematics and Economics, Elsevier, vol. 25(2), pages 133-142, November.
    14. Gerber, Hans U. & Shiu, Elias S. W., 1997. "The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin," Insurance: Mathematics and Economics, Elsevier, vol. 21(2), pages 129-137, November.
    15. 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. 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.
    2. J. Cerda-Hernandez & A. Sikov, 2022. "Optimal investment strategy to maximize the expected utility of an insurance company under Cramer Lundberg dynamic," Papers 2207.02947, arXiv.org.
    3. Simon Pojer & Stefan Thonhauser, 2023. "The Markovian Shot-noise Risk Model: A Numerical Method for Gerber-Shiu Functions," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-26, March.
    4. Chen, Mi & Yuen, Kam Chuen & Guo, Junyi, 2014. "Survival probabilities in a discrete semi-Markov risk model," Applied Mathematics and Computation, Elsevier, vol. 232(C), pages 205-215.
    5. 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.
    6. Chen, Xu & Xiao, Ting & Yang, Xiang-qun, 2014. "A Markov-modulated jump-diffusion risk model with randomized observation periods and threshold dividend strategy," Insurance: Mathematics and Economics, Elsevier, vol. 54(C), pages 76-83.
    7. Zhou, Zhongbao & Xiao, Helu & Deng, Yingchun, 2015. "Markov-dependent risk model with multi-layer dividend strategy," Applied Mathematics and Computation, Elsevier, vol. 252(C), pages 273-286.

    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. 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.
    2. 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.
    3. 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.
    4. 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.
    5. Yin, Chuancun & Wen, Yuzhen, 2013. "An extension of Paulsen–Gjessing’s risk model with stochastic return on investments," Insurance: Mathematics and Economics, Elsevier, vol. 52(3), pages 469-476.
    6. 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.
    7. Chuancun Yin & Yuzhen Wen, 2013. "An extension of Paulsen-Gjessing's risk model with stochastic return on investments," Papers 1302.6757, arXiv.org.
    8. 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.
    9. 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.
    10. 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.
    11. 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.
    12. Zhang, Chunsheng & Wang, Guojing, 2003. "The joint density function of three characteristics on jump-diffusion risk process," Insurance: Mathematics and Economics, Elsevier, vol. 32(3), pages 445-455, July.
    13. Jostein Paulsen, 2008. "Ruin models with investment income," Papers 0806.4125, arXiv.org, revised Dec 2008.
    14. Claude Lefèvre & Philippe Picard, 2013. "Ruin Time and Severity for a Lévy Subordinator Claim Process: A Simple Approach," Risks, MDPI, vol. 1(3), pages 1-21, December.
    15. 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.
    16. Biffis, Enrico & Kyprianou, Andreas E., 2010. "A note on scale functions and the time value of ruin for Lévy insurance risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 85-91, February.
    17. Li, Shuanming & Ren, Jiandong, 2013. "The maximum severity of ruin in a perturbed risk process with Markovian arrivals," Statistics & Probability Letters, Elsevier, vol. 83(4), pages 993-998.
    18. Ben Salah, Zied & Garrido, José, 2018. "On fair reinsurance premiums; Capital injections in a perturbed risk model," Insurance: Mathematics and Economics, Elsevier, vol. 82(C), pages 11-20.
    19. Lu, Zhaoyang & Xu, Wei & Zhang, Yan & Sun, Yingling, 2009. "On the ruin probability for the Cox correlated risk model perturbed by diffusion," Statistics & Probability Letters, Elsevier, vol. 79(3), pages 381-389, February.
    20. Tsai, Cary Chi-Liang, 2001. "On the discounted distribution functions of the surplus process perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 28(3), pages 401-419, June.

    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:49:y:2011:i:1:p:126-131. 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.