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A connection between the discounted and non-discounted expected penalty functions in the Sparre Andersen risk model

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  • Ren, Jiandong

Abstract

In this paper, we study the Gerber-Shiu expected discounted penalty function in a Sparre Andersen risk model probabilistically. This is implemented through the modified random walk defined herein. As a result, we derive explicit formulas for the discounted probability of ruin and the discounted distribution of the deficit at ruin when the distribution of the interclaim times is arbitrary and the distribution of claim sizes is phase-type. In addition, we provide iterative schemes for evaluating the formulas.

Suggested Citation

  • Ren, Jiandong, 2009. "A connection between the discounted and non-discounted expected penalty functions in the Sparre Andersen risk model," Statistics & Probability Letters, Elsevier, vol. 79(3), pages 324-330, February.
    • Handle: RePEc:eee:stapro:v:79:y:2009:i:3:p:324-330
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    References listed on IDEAS

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    1. Willmot, Gordon E., 2007. "On the discounted penalty function in the renewal risk model with general interclaim times," Insurance: Mathematics and Economics, Elsevier, vol. 41(1), pages 17-31, July.
    2. Shuanming Li, 2008. "“The Discounted Joint Distribution of the Surplus Prior to Ruin and the Deficit at Ruin in a Sparre Andersen Model,” Jiandong Ren, July 2007," North American Actuarial Journal, Taylor & Francis Journals, vol. 12(2), pages 208-210.
    3. Hans Gerber & Elias Shiu, 1998. "On the Time Value of Ruin," North American Actuarial Journal, Taylor & Francis Journals, vol. 2(1), pages 48-72.
    4. Dickson, David C. M. & Hipp, Christian, 2001. "On the time to ruin for Erlang(2) risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 29(3), pages 333-344, December.
    5. Hans Gerber & Elias Shiu, 2005. "The Time Value of Ruin in a Sparre Andersen Model," North American Actuarial Journal, Taylor & Francis Journals, vol. 9(2), pages 49-69.
    6. Landriault, David & Willmot, Gordon, 2008. "On the Gerber-Shiu discounted penalty function in the Sparre Andersen model with an arbitrary interclaim time distribution," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 600-608, April.
    7. Ng, Andrew C.Y. & Yang, Hailiang, 2005. "Lundberg-type Bounds for the Joint Distribution of Surplus Immediately Before and at Ruin under a Markov-modulated Risk Model," ASTIN Bulletin, Cambridge University Press, vol. 35(2), pages 351-361, November.
    8. Bangwon Ko, 2007. "“The Discounted Joint Distribution of the Surplus Prior to Ruin and the Deficit at Ruin in a Sparre Andersen Model”, Jiandong Ren, July 2007," North American Actuarial Journal, Taylor & Francis Journals, vol. 11(3), pages 136-137.
    9. Andrew Ng & Hailiang Yang, 2005. "Lundberg-Type Bounds for the Joint Distribution of Surplus Immediately Before and at Ruin Under the Sparre Andersen Model," North American Actuarial Journal, Taylor & Francis Journals, vol. 9(2), pages 85-100.
    10. Avram, F. & Usábel, M., 2004. "Ruin Probabilities and Deficit for the Renewal Risk Model with Phase-type Interarrival Times," ASTIN Bulletin, Cambridge University Press, vol. 34(2), pages 315-332, November.
    11. Jiandong Ren, 2007. "The Discounted Joint Distribution of the Surplus Prior to Ruin and the Deficit at Ruin in a Sparre Andersen Model," North American Actuarial Journal, Taylor & Francis Journals, vol. 11(3), pages 128-136.
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