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Article

A Note on a Generalized Gerber–Shiu Discounted Penalty Function for a Compound Poisson Risk Model

1
Department of Humanities and Social Sciences, Yibin Vocational & Technical College, Yibin 644003, China
2
School of Insurance, Shandong University of Finance and Economics, Jinan 250014, China
3
School of Science, Shandong Jiaotong University, Jinan 250357, China
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(10), 891; https://doi.org/10.3390/math7100891
Received: 13 August 2019 / Revised: 19 September 2019 / Accepted: 20 September 2019 / Published: 24 September 2019
(This article belongs to the Special Issue Stochastic Processes: Theory and Applications)
In this paper, we propose a new generalized Gerber–Shiu discounted penalty function for a compound Poisson risk model, which can be used to study the moments of the ruin time. First, by taking derivatives with respect to the original Gerber–Shiu discounted penalty function, we construct a relation between the original Gerber–Shiu discounted penalty function and our new generalized Gerber–Shiu discounted penalty function. Next, we use Laplace transform to derive a defective renewal equation for the generalized Gerber–Shiu discounted penalty function, and give a recursive method for solving the equation. Finally, when the claim amounts obey the exponential distribution, we give some explicit expressions for the generalized Gerber–Shiu discounted penalty function. Numerical illustrations are also given to study the effect of the parameters on the generalized Gerber–Shiu discounted penalty function. View Full-Text
Keywords: compound Poisson risk model; generalized Gerber–Shiu discounted penalty function; Laplace transform; Dickson–Hipp operator; recursive formula compound Poisson risk model; generalized Gerber–Shiu discounted penalty function; Laplace transform; Dickson–Hipp operator; recursive formula
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MDPI and ACS Style

Ruan, J.; Yu, W.; Song, K.; Sun, Y.; Huang, Y.; Yu, X. A Note on a Generalized Gerber–Shiu Discounted Penalty Function for a Compound Poisson Risk Model. Mathematics 2019, 7, 891. https://doi.org/10.3390/math7100891

AMA Style

Ruan J, Yu W, Song K, Sun Y, Huang Y, Yu X. A Note on a Generalized Gerber–Shiu Discounted Penalty Function for a Compound Poisson Risk Model. Mathematics. 2019; 7(10):891. https://doi.org/10.3390/math7100891

Chicago/Turabian Style

Ruan, Jiechang, Wenguang Yu, Ke Song, Yihan Sun, Yujuan Huang, and Xinliang Yu. 2019. "A Note on a Generalized Gerber–Shiu Discounted Penalty Function for a Compound Poisson Risk Model" Mathematics 7, no. 10: 891. https://doi.org/10.3390/math7100891

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