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A scale function based approach for solving integral-differential equations in insurance risk models

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  • Zhang, Aili
  • Li, Shuanming
  • Wang, Wenyuan

Abstract

In risk theory, the resolutions of many interesting problems are reduced to solving some integro-differential equations (IDE), see [4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 21, 22, 23, 24, 25, 26, 27, 29, 31], and references therein. Meanwhile, due to the recent advances made on Lévy processes, explicit analytical expressions of the scale functions associated with Lévy processes become on offer (see [1, 2, 11, 16, 17], Chapter 8 of [19], [20], etc). This paper aims at bridging together the scale functions and the IDEs by presenting a unified scale function based approach for solving IDEs that arise in risk theory. In particular, to demonstrate the effectiveness of this approach, a dividend and capital injection problem is considered under a jump-diffusion risk model. We first derive the IDEs satisfied by the expected accumulated discounted difference between the net dividends and the costs of capital injections, and then solve the IDEs with its solution being expressed in compact and transparent forms.

Suggested Citation

  • Zhang, Aili & Li, Shuanming & Wang, Wenyuan, 2023. "A scale function based approach for solving integral-differential equations in insurance risk models," Applied Mathematics and Computation, Elsevier, vol. 450(C).
    • Handle: RePEc:eee:apmaco:v:450:y:2023:i:c:s0096300323001340
    DOI: 10.1016/j.amc.2023.127965
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    References listed on IDEAS

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    1. Wan, Ning, 2007. "Dividend payments with a threshold strategy in the compound Poisson risk model perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 40(3), pages 509-523, May.
    2. Lin, X.Sheldon & Pavlova, Kristina P., 2006. "The compound Poisson risk model with a threshold dividend strategy," Insurance: Mathematics and Economics, Elsevier, vol. 38(1), pages 57-80, February.
    3. Shuanming Li & Yi Lu & Kristina P. Sendova, 2019. "The expected discounted penalty function: from infinite time to finite time," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2019(4), pages 336-354, April.
    4. Wang, Wenyuan & Ming, Ruixing & Hu, Yijun, 2011. "On the expected discounted penalty function for risk process with tax," Statistics & Probability Letters, Elsevier, vol. 81(4), pages 489-501, April.
    5. Ng, Andrew C.Y., 2009. "On a dual model with a dividend threshold," Insurance: Mathematics and Economics, Elsevier, vol. 44(2), pages 315-324, April.
    6. 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.
    7. Dickson, David C.M. & Waters, Howard R., 2004. "Some Optimal Dividends Problems," ASTIN Bulletin, Cambridge University Press, vol. 34(1), pages 49-74, May.
    8. Jevgenijs Ivanovs, 2021. "On scale functions for Lévy processes with negative phase-type jumps," Queueing Systems: Theory and Applications, Springer, vol. 98(1), pages 3-19, June.
    9. Peng, Dan & Liu, Donghai & Liu, Zaiming, 2013. "Dividend problems in the dual risk model with exponentially distributed observation time," Statistics & Probability Letters, Elsevier, vol. 83(3), pages 841-849.
    10. Hans Gerber & Elias Shiu, 1998. "On the Time Value of Ruin," North American Actuarial Journal, Taylor & Francis Journals, vol. 2(1), pages 48-72.
    11. Zhang, Zhimin & Han, Xiao, 2017. "The compound Poisson risk model under a mixed dividend strategy," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 1-12.
    12. David Landriault & Gordon E. Willmot, 2020. "On series expansions for scale functions and other ruin-related quantities," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2020(4), pages 292-306, April.
    13. 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.
    14. Corina Constantinescu & Gennady Samorodnitsky & Wei Zhu, 2018. "Ruin probabilities in classical risk models with gamma claims," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2018(7), pages 555-575, August.
    15. Gerber, Hans U. & Lin, X. Sheldon & Yang, Hailiang, 2006. "A Note on the Dividends-Penalty Identity and the Optimal Dividend Barrier," ASTIN Bulletin, Cambridge University Press, vol. 36(2), pages 489-503, November.
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