保费随机的相依风险模型的破产问题研究

发布时间：2018-07-08 10:04

本文选题：随机保费收入 + FGM连接函数　；参考：《重庆大学》2013年硕士论文

【摘要】：经典复合泊松风险模型自提出以来已被广泛研究。众所周知，该模型中有两个重要假定：保费率为常数以及理赔时间间隔与理赔额大小相互独立。在这两个假定之下，所研究的风险模型虽得到很好的简化，但与现实情况相差较远。针对于此，随机保费收入风险模型以及相依风险模型随之提出。此外，Gerber-Shiu函数作为一个重要的风险度量工具，自提出以来在破产理论的研究中有着广泛应用，关于破产问题的研究很多都可以归结为基于Gerber-Shiu函数的计算问题。本文将利用Gerber-Shiu函数，针对保费收入随机的相依风险模型的破产问题进行研究。首先，本文对破产理论的研究背景、经典风险模型及其主要研究结果进行了相关介绍。随后对当今破产理论研究的几个推广方向，，如相依风险模型、随机保费收入风险模型、考虑投资、分红风险模型等发展现状进行了相应介绍。接下来，针对本文所研究的特殊相依关系FMG copula函数、主要研究工具Gerber-Shiu函数，以及本文主要研究内容所涉及到的相关理论知识进行介绍。本文主要内容为第三章，该章节中将考虑保费收入随机的风险模型，且进一步假设索赔时间间隔与索赔额的大小不再相互独立，而是具有某种相依关系，该相依关系本文利用FGM连接函数来建立。在基于FMG copula的随机保费相依风险模型下，研究得到了期望折现罚金函数（Gerber-Shiu函数）所满足的积分微分方程以及其拉普拉斯变换。期望折现罚金函数所满足的瑕疵更新方程继而可以得到。特别地，当索赔额服从指数分布时，破产概率等重要风险测度的显式表达式可以得出。最后，针对上文的理论研究结果，进行数值模拟。通过数值模拟对结论进行说明，同时验证文中结论的正确性。
[Abstract]:The classical compound Poisson risk model has been widely studied since it was proposed. It is well known that there are two important assumptions in this model: the constant premium rate and the independence of claim interval and claim amount. Under these two assumptions, the risk model is well simplified, but it is far from the actual situation. In view of this, the stochastic premium income risk model and the dependent risk model are put forward. In addition, Gerber-Shiu function, as an important risk measurement tool, has been widely used in the study of ruin theory since it was put forward. Many researches on the problem of bankruptcy can be attributed to the computing problem based on Gerber-Shiu function. In this paper, we use Gerber-Shiu function to study the ruin of random dependent risk model of premium income. Firstly, this paper introduces the background of bankruptcy theory, classical risk model and its main results. Then it introduces the current development of bankruptcy theory, such as dependent risk model, stochastic premium income risk model, investment, dividend risk model and so on. Next, this paper introduces the special dependent relation copula function, the main research tool Gerber-Shiu function, and the related theoretical knowledge involved in the main research content of this paper. In this chapter, the stochastic risk model of premium income is considered, and it is further assumed that the claim interval and the amount of claim are no longer independent, but are dependent on each other. This dependency is established by FGM connection function. Under the stochastic premium dependent risk model based on FMG copula, the integral differential equation and Laplace transform of the expected discounted penalty function (Gerber-Shiu function) are obtained. The imperfections renewal equation satisfied by the expected discounted penalty function can then be obtained. In particular, when the claim amount is distributed exponentially, the explicit expression of important risk measures such as ruin probability can be obtained. Finally, according to the theoretical results above, numerical simulation is carried out. The conclusion is explained by numerical simulation, and the correctness of the conclusion is verified at the same time.
【学位授予单位】：重庆大学
【学位级别】：硕士
【学位授予年份】：2013
【分类号】：F840.3;F224;O211.67

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