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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
- randomized dividend strategy
- expected discounted dividends
- difference equations
- delayed claims
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
References
- Gerber, HU: Mathematical fun with the compound binomial process. ASTIN Bull. 24, 177-184 (1988) Google Scholar
- Shiu, ESW: The probability of eventual ruin in the compound binomial model. ASTIN Bull. 19, 179-190 (1989) View ArticleGoogle Scholar
- Willmot, GE: Ruin probabilities in the compound binomial model. Insur. Math. Econ. 12, 133-142 (1993) MATHMathSciNetView ArticleGoogle Scholar
- Dickson, DCM: Some comments on the compound binomial models. ASTIN Bull. 24, 33-45 (1994) View ArticleGoogle Scholar
- Yang, H, Zhang, Z, Lan, C: Ruin problems in a discrete Markov risk model. Stat. Probab. Lett. 79, 21-28 (2009) MATHMathSciNetView ArticleGoogle Scholar
- Yang, H, Zhang, Z: On a discrete risk model with two-sided jumps. J. Comput. Appl. Math. 234, 835-844 (2010) MATHMathSciNetView ArticleGoogle Scholar
- Gerber, HU, Shiu, ESW, Yang, H: An elementary approach to discrete models of dividend strategies. Insur. Math. Econ. 46, 109-116 (2010) MATHMathSciNetView ArticleGoogle Scholar
- Chen, M, Yuen, KC, Guo, J: Survival probabilities in a discrete semi-Markov risk model. Appl. Math. Comput. 232, 205-215 (2014) MathSciNetView ArticleGoogle Scholar
- Yuen, KC, Guo, J: Ruin probabilities for time-correlated claims in the compound binomial model. Insur. Math. Econ. 29, 47-57 (2001) MATHMathSciNetView ArticleGoogle Scholar
- Yuen, KC, Guo, J, Ng, KW: On ultimate ruin in a delayed-claims risk model. J. Appl. Probab. 42, 163-174 (2005) MATHMathSciNetView ArticleGoogle Scholar
- Xiao, Y, Guo, J: The compound binomial risk model with time-correlated claims. Insur. Math. Econ. 41, 124-133 (2007) MATHMathSciNetView ArticleGoogle Scholar
- Albrecher, H, Cheung, ECK, Thonhauser, S: Randomized observation periods for the compound Poisson risk model: dividends. ASTIN Bull. 41, 645-672 (2011) MATHMathSciNetGoogle Scholar
- Avanzi, B, Cheung, ECK, Wong, B, Woo, JK: On a periodic dividend barrier strategy in the dual model with continuous monitoring of solvency. Insur. Math. Econ. 52, 98-113 (2013) MATHMathSciNetView ArticleGoogle Scholar
- Zhang, Z: On a risk model with randomized dividend-decision times. J. Ind. Manag. Optim. 10, 1041-1058 (2014) MATHMathSciNetView ArticleGoogle Scholar
- Zhang, Z, Cheung, ECK: The Markov additive risk process under an Erlangized dividend barrier strategy. Methodol. Comput. Appl. Probab. (2014). doi:10.1007/s11009-014-9414-7 Google Scholar
- Tan, J, Yang, X: The compound binomial model with randomized decisions on paying dividends. Insur. Math. Econ. 39, 1-18 (2006) MATHMathSciNetView ArticleGoogle Scholar
- He, L, Yang, X: The compound binomial model with randomly paying dividends to shareholders and policyholders. Insur. Math. Econ. 46, 443-449 (2010) MATHMathSciNetView ArticleGoogle Scholar
- Li, S: On a class of discrete time renewal risk models. Scand. Actuar. J. 2005, 241-260 (2005) MATHView ArticleGoogle Scholar