The compound binomial model with randomly paying dividends to shareholders and policyholders☆
Introduction
The compound binomial model which is the classic risk model has been numerously studied, for examples Bara et al. (2008), Cai and Dickson (2002), Cheng et al. (2000), DeVylder (1996), Gong and Yang (2001), Isckson (1994), Shiu (1989). Based on the compound binomial risk, Tan and Yang (2006) considered that insurance company randomly paid dividends to policyholder: given a threshold, when the surplus is no less than the threshold, the company randomly decides whether to pay dividends to policyholders or not. Tan and Yang (2006) had studied the ruin problem for this model.
In this paper, we extend the model in Tan and Yang (2006), and consider a joint stock insurance company which randomly pays dividends to shareholders and policyholders on the basis of the compound binomial model: given two thresholds, the company randomly does decide to pay dividends to shareholders and policyholders according to the thresholds. We have built this sort of model, and derived the recursive formulas of both the expected discount penalty function and the ruin probability, and the distribution function of the deficit at ruin.
Section snippets
The model and preliminaries
Let the following essential factors be given:
1. Three non-negative integers ; without loss of generality, we assume that and moreover that .
2. Four stochastic processes in some probability space .
(a) is independent and identically distributed; the common distribution is the binomial distribution , .
(b) is independent and identically distributed; the common distribution is the binomial distribution ,
Recursive formulas
Let , , In this article, we always assume , which leads to a positive security loading. We denote the security loading by :
Remark (iii). The following fact will be used in the proof of Theorem 1: For the compound binomial model with randomly paying dividends to shareholders and policyholders. We have the following result.
Theorem 1 Let
Application of formulas
Example 1 Letting , the , when , by Theorem 1, can be obtained by the following linear equations where , When , can be obtained by the following linear equations
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Cited by (0)
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Supported by National Natural Science Foundation of China (NSF10871064) and Key Laboratory of Comput. and Stoch. Math. and its Appl., Universities of Hunan Province, Hunan Normal University.