On a discrete risk model with delayed claims and a randomized dividend strategy
- Chaolin Liu^{1}Email author and
- Zhimin Zhang^{1}
https://doi.org/10.1186/s13662-015-0614-4
© Liu and Zhang 2015
Received: 29 April 2015
Accepted: 20 August 2015
Published: 15 September 2015
Abstract
In this paper, we consider a discrete risk model with delayed claims and randomized dividend strategy. The expected discounted dividends before ruin are studied. Difference equations for the expected discounted dividends are derived and solved.
Keywords
1 Introduction
The compound binomial risk model has been studied by many authors, for example, Gerber [1], Shiu [2], Willmot [3] and Dickson [4]. Recently, some extensions have been made on this model. Yang et al. [5] study the ruin probabilities in a discrete Markov risk model. Yang and Zhang [6] consider a discrete renewal risk model with two-sided jumps. Gerber et al. [7] modify the compound binomial risk model by dividend payments. Chen et al. [8] study the survival probabilities in a discrete semi-Markov risk model.
2 Difference equations
- (1)
no claim occurs in \((0, 1]\) and no dividend is paid in \((0, 1]\);
- (2)
no claim occurs in \((0, 1]\) and a dividend of 1 is paid in \((0, 1]\) (if \(u < b\), this case does not exist);
- (3)
a main claim and its by-claim occur simultaneously in \((0, 1]\), and no dividend is paid in \((0, 1]\);
- (4)
a main claim and its by-claim occur simultaneously in \((0, 1]\), and a dividend of 1 is paid in \((0, 1]\) (if \(u < b\), this case does not exist);
- (5)
a main claim occurs in \((0, 1]\) and its by-claim is delayed to the next period, and no dividend is paid in \((0, 1]\);
- (6)
a main claim occurs in \((0, 1]\) and its by-claim is delayed to the next period, and a dividend of 1 is paid in \((0, 1]\) (if \(u < b\), this case does not exist).
3 The case \(0\leq u< b\)
4 The case \(u\geq b\)
Theorem 1
5 Conclusion
Dividend problems are hot topics in insurance risk theory. In this paper, we consider a compound binomial model with delayed claims. Suppose that the insurance company will possibly pay dividends when the surplus level is larger than a given barrier b. The expected present values of dividends paid before ruin are studied. We derive systems of difference equations for \(V(u;b)\) and \(\bar{V}(u;b)\), and get the solutions by generating function method. The main results given in Theorem 1 show that the analytic expressions for \(V(u;b)\) and \(\bar {V}(u;b)\) can be obtained.
Declarations
Acknowledgements
The authors would like to thank two anonymous referees for their helpful comments and suggestions, which improved an earlier version of the paper. This work is supported by the National Natural Science Foundation of China (11101451, 11471058, 11426051), the Natural Science Foundation Project of CQ CSTC of China (cstc2014jcyjA00007), Scientific and Technological Research Program of Chongqing Municipal Education Commission (KJ1400521) and the Fundamental Research Funds for the Central Universities (106112015CDJXY100006).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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