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The expected discounted penalty function: from infinite time to finite time

Author

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  • Shuanming Li
  • Yi Lu
  • Kristina P. Sendova

Abstract

In this paper we study the finite-time expected discounted penalty function (EDPF) and its decomposition in the classical risk model perturbed by diffusion. We first give the solution to a class of second-order partial integro-differential equations (PIDEs) with certain boundary conditions. We then show that the finite-time EDPFs as well as their decompositions satisfy this specific class of PIDEs so that their explicit expressions are obtained. Furthermore, we demonstrate that the finite-time EDPF may be expressed in terms of its ordinary counterpart (infinite-time) under the same risk model. Especially, the finite-time ruin probability due to oscillations and the finite-time ruin probability caused by a claim may also be expressed in terms of the corresponding quantities under the infinite-time horizon. Numerical examples are given when claims follow an exponential distribution.

Suggested Citation

  • 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.
    • Handle: RePEc:taf:sactxx:v:2019:y:2019:i:4:p:336-354
    DOI: 10.1080/03461238.2018.1560955
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    Cited by:

    1. 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).
    2. Kang Hu & Ya Huang & Yingchun Deng, 2023. "Estimating the Gerber–Shiu Function in the Two-Sided Jumps Risk Model by Laguerre Series Expansion," Mathematics, MDPI, vol. 11(9), pages 1-30, April.
    3. Onno Boxma & Fabian Hinze & Michel Mandjes, 2023. "Gerber-Shiu Metrics for a Bivariate Perturbed Risk Process," Risks, MDPI, vol. 12(1), pages 1-17, December.

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