带广义F-G-M Copula函数风险模型的分红策略

发布时间：2018-08-01 11:10

【摘要】：在研究经典复合泊松风险模型时,一般我们都假定索赔额和索赔时间间隔二者是相互独立的.然而,事实上,索赔额和索赔时问间隔之间可能具有某种相依性.这种打破索赔额和索赔时间间隔之间独立性的风险模型就是一种相依风险模型.相依风险模型自提出以来得到了广泛研究.近年来,学者们提出了索赔额和索赔时间间隔的联合分布满足Copula函数的相依复合泊松风险模型.后来,有学者又研究了带广义F-G-M Copula函数的相依复合泊松风险模型的Gerber-Shiu函数.这类风险模型是由经典复合泊松风险模型延伸而来,它的相依结构是基于一种广义的Farlie-Gumbel-Morgenstern Copula函数建立的.针对带有广义F-G-M Copula函数的相依复合泊松风险模型,学者们又进一步研究了其障碍分红策略下的折现罚金Gerber-Shiu函数. 本文继续对带广义F-G-M Copula函数的相依复合泊松风险模型的有关分红策略问题进行探讨.这篇论文主要研究了这种相依风险模型的三种分红策略：障碍分红策略、阈值分红策略和混合分红策略,得出了期望折现分红函数分别满足的积分-微分方程及边界条件.除此之外,在混合分红策略下,我们还求出了期望折现罚金Gerber-Shiu函数所满足的积分-微分方程及边界条件.本文最重要的结果就是,针对索赔额服从指数分布这一特殊情况,我们得到了期望折现分红函数所满足的微分方程.但是,当索赔额服从其它分布时,本文尚未得到较好的结果. 这篇文章的结构如下. 第一章主要阐述了本篇论文所研究问题的背景知识. 第二章详细介绍了带广义F-G-M Copula函数的相依复合泊松风险模型. 第三章详细阐述了三种分红策略即障碍分红策略、阈值分红策略和混合分红策略.在这三种分红策略下,我们分别推导出了期望折现分红函数满足的积分-微分方程及边界条件.更进一步,当索赔额服从指数分布时,我们将期望折现分红函数所满足的积分-微分方程化成了微分方程,并举例说明了如何求出期望折现分红函数的具体表达式. 第四章推导出具有混合分红策略的Gerber-Shiu函数满足的积分-微分方程和边界条件.
[Abstract]:When we study the classical compound Poisson risk model, we generally assume that the claim amount and the claim interval are independent of each other. In fact, however, there may be some dependency between the amount claimed and the time interval between the claim and the claim. This risk model, which breaks the independence between claim amount and claim interval, is a dependent risk model. The dependent risk model has been widely studied since it was proposed. In recent years, scholars have proposed a dependent compound Poisson risk model, in which the joint distribution of claim amount and claim interval satisfies the Copula function. Later, some scholars have studied the Gerber-Shiu function of the dependent compound Poisson risk model with generalized F-G-M Copula function. This kind of risk model is extended from the classical compound Poisson risk model and its dependent structure is based on a generalized Farlie-Gumbel-Morgenstern Copula function. For the dependent compound Poisson risk model with generalized F-G-M Copula function, the discounted penalty Gerber-Shiu function under the barrier dividend strategy is further studied. In this paper, we continue to discuss the dividend strategy of the dependent compound Poisson risk model with generalized F-G-M Copula function. In this paper, we mainly study three dividend strategies of this dependent risk model: obstacle dividend strategy, threshold dividend strategy and hybrid dividend strategy, and obtain the integro-differential equations and boundary conditions which are satisfied by the expected discount dividend function respectively. In addition, under the mixed dividend strategy, we also obtain the integro-differential equations and boundary conditions satisfied by the expected discounted penalty Gerber-Shiu function. The most important result of this paper is that we obtain the differential equation of the expected discounted dividend function for the special case of the exponential distribution of the claim amount. However, when the amount claimed from other distribution, this paper has not obtained a better result. The structure of the article is as follows. The first chapter mainly elaborates the background knowledge of the problems studied in this paper. In the second chapter, the dependent compound Poisson risk model with generalized F-G-M Copula function is introduced in detail. In the third chapter, three kinds of dividend strategies, i.e. obstacle dividend strategy, threshold dividend strategy and hybrid dividend strategy, are described in detail. Under these three dividend strategies, we derive the integro-differential equations and boundary conditions of the expected discounted dividend function. Furthermore, when the claim amount is distributed exponentially, we transform the integro-differential equation satisfied by the expected discounted dividend function into a differential equation, and illustrate how to obtain the concrete expression of the expected discounted dividend function. In chapter 4, the integro-differential equations and boundary conditions of Gerber-Shiu function with mixed dividend strategy are derived.
【学位授予单位】：曲阜师范大学
【学位级别】：硕士
【学位授予年份】：2014
【分类号】：O211.6;F830.91

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