基于延迟索赔风险模型的破产理论研究

发布时间：2018-05-30 07:28

本文选题：延迟索赔 + 拉普拉斯变换　；参考：《华北电力大学》2014年硕士论文

【摘要】：保险在当今社会的发展中发挥着不容忽视的作用,破产理论是对破产概率及相关问题的研究,而破产概率则是衡量保险公司稳定性的一个重要指标,越是能够刻画保险公司所面临的真实情况的风险模型,对保险公司的价值就越大。在现实中,索赔过程往往并不是平稳独立增量过程,索赔有可能延迟发生,延迟索赔可以看作是IBNR(Incurred But Not Reported)索赔,保险公司有必要为这类索赔建立储备以防范风险。大多数关于延迟索赔风险模型研究均假设主索赔引起的副索赔的种类只有一种,然而在现实中,受随机因素的影响,主索赔X可能引起副索赔Y,也有可能引起副索赔Z,即主索赔引起的副索赔的种类可能不止一种。基于以上考虑,本文研究了几类具有延迟索赔的风险模型。本文首先研究了一类复合泊松风险模型,假设副索赔的种类为两种,主索赔的发生可能不引起副索赔,也可能引起两种副索赔中的一种。通过对全概率公式的应用,首先得到一组生存概率满足的微积分方程,再利用拉普拉斯变换、拉普拉斯终值定理和儒歇定理,最终可以求得生存概率的表达式。假设主索赔和副索赔满足相同的指数分布,得到了生存概率的具体表达式。最后通过数值算例分析了不同参数的变化对生存概率的影响。随后本文研究了具有随机保费的Gerber-Shiu罚金折现函数,同样假设副索赔的种类为两种。主索赔的发生分别以一定的概率引起两种副索赔中的一种,副索赔延迟发生与否取决于对应的主索赔与设定的门限值大小的对比。给出了Gerber-Shiu罚金折现函数的求解过程并讨论了Gerber-Shiu罚金折现函数所满足的瑕疵更新方程,最后给出了当Gerber-Shiu罚金折现函数退化为破产概率时的数值算例及分析。本文最后研究了两类副索赔的种类为n种的风险模型,每个主索赔的发生均会引起一个副索赔。第一类风险模型假设主索赔的计数过程满足泊松分布且副索赔延迟发生与否取决于对应的主索赔与设定的门限值大小的对比；第二类风险模型则假设主索赔到达时刻满足Erlang(2)分布,副索赔以一定的概率延迟发生。分析得到生存概率的表达式,最后给出了数值算例及分析。
[Abstract]:Insurance plays an important role in the development of today's society. Bankruptcy theory is the study of bankruptcy probability and related problems, and bankruptcy probability is an important index to measure the stability of insurance companies. The more the risk model can describe the real situation faced by the insurance company, the greater the value to the insurance company. In reality, the claim process is often not a stable independent increment process, claims may be delayed, delay claims can be regarded as IBNR(Incurred But Not Reported) claims, it is necessary for insurance companies to establish reserves for such claims to guard against risks. Most studies on the risk model of delayed claims assume that there is only one category of sub-claims arising from the main claim, but in reality, it is affected by random factors. Main claim X may give rise to sub-claim Yor sub-claim Z. that is, there may be more than one type of sub-claim arising from the main claim. Based on the above considerations, this paper studies several risk models with delay claims. In this paper, we first study a kind of compound Poisson risk model. Assuming that there are two kinds of sub-claims, the occurrence of the main claim may not cause the sub-claim or one of the two kinds of sub-claims. Through the application of the total probability formula, a set of calculus equations of survival probability satisfied is obtained first, and then the expression of survival probability can be obtained by using Laplace transformation, Laplace final value theorem and Ruch theorem. Assuming that the main claim and the sub-claim satisfy the same exponential distribution, the concrete expression of survival probability is obtained. Finally, the influence of different parameters on survival probability is analyzed by numerical examples. Then we study the discounted Gerber-Shiu penalty function with random premium, and assume that there are two kinds of sub-claims. The occurrence of the main claim causes one of the two sub-claims with a certain probability. The delay in the occurrence of the sub-claim depends on the comparison between the corresponding main claim and the threshold value set. In this paper, the process of solving the Gerber-Shiu penalty discounting function is given, and the defect renewal equation satisfied by the Gerber-Shiu penalty discount function is discussed. Finally, the numerical example and analysis of the Gerber-Shiu penalty discounting function when it degenerates to the ruin probability is given. At the end of this paper, we study the risk model of two kinds of sub-claims, each main claim will cause a sub-claim. The first kind of risk model assumes that the counting process of the main claim satisfies the Poisson distribution and the delay of the sub-claim depends on the comparison between the corresponding main claim and the set threshold value. The second type of risk model assumes that the arrival time of the main claim satisfies Erlang2) distribution, and the sub-claim occurs with a certain probability delay. The expression of survival probability is obtained. Finally, numerical examples and analysis are given.
【学位授予单位】：华北电力大学
【学位级别】：硕士
【学位授予年份】：2014
【分类号】：F840;F224

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