几类风险模型的分红问题

发布时间：2018-06-02 03:10

本文选题：一维扩散过程 + 逸出时　；参考：《曲阜师范大学》2014年硕士论文

【摘要】：近年来,分红问题在精算数学中受到了广泛的关注.本文考虑了几类风险模型的分红问题.文中讨论的风险模型有一维扩散风险模型,经典风险模型和带扰动的经典风险模型;主要涉及的分红策略有障碍分红策略,阈值分红策略和混合分红策略;研究的主要问题有对应风险模型的逸出时,破产前期望折现分红,破产前折现分红的矩母函数和各阶矩,Gerber-Shiu函数和破产时的拉普拉斯变换等;用到的工具主要包括逸出时,特殊函数,马氏过程,泰勒公式和Dynkin公式等.文中的有些问题得到了具体的结果.根据文章的具体内容本文可分为以下三章: 1)一维扩散过程的逸出时和分红值函数. 在这一章我们考虑一维时齐的扩散过程在区间上的逸出时问题,以及它们在风险理论中分红问题的应用.首先,我们利用Dynkin公式推导出了逸出时的拉普拉斯变换满足的常微分方程.然后,我们列举了几个在精算学和金融市场模型中经常用到的扩散过程,得到了它们的逸出时的拉普拉斯变换的明确表达式.最后,我们将逸出时的拉普拉斯变换与分红值函数联系起来,利用逸出时的拉普拉斯变换分别表示出了障碍分红值函数和阈值分红值函数. 2)混合分红策略下的古典风险模型. 第二章研究了在混合分红策略下的古典风险模型.我们先详细地介绍了古典风险模型、混合分红策略以及分红策略问题的研究背景.然后定义了在混合分红策略下的古典风险模型中的破产前折现分红函数,破产前期望折现分红函数,折现分红函数的矩母函数和各阶矩, Gerber-Shiu函数和破产时的拉普拉斯变换.进一步推导出了破产前的期望折现分红函数满足的积分微分方程和边界条件.我们也得到了破产前折现分红函数的矩母函数及各阶矩分别满足的积分微分方程和边界条件.最后我们讨论了Gerber-Shiu函数和破产时的拉普拉斯变换.其中关于个体索赔额服从指数的情况,我们得到了一些具体的结果. 3)混合分红策略下的带扰动的古典风险模型. 在第二章的基础上,本章考虑了在混合分红策略下的带扰动的古典风险模型.为方便,我们沿用了第二章的部分记号,由于模型的变化破产时的定义不同,我们首先介绍了带扰动的古典风险模型,指出了由于模型的变化导致的与第二章的记号的变化.我们利用Dynkin公式推导出了破产前期望折现分红函数满足的积分微分方程和边界条件.我们在例子中得到了当索赔服从指数分布时破产前预期折现分红的具体表达式.然后分别推导出了破产前折现分红的矩母函数和k阶矩满足的积分微分方程和边界条件.最后,我们讨论了著名的Gerber-Shiu函数,具体计算了当个体索赔额为指数分布时破产时的拉普拉斯变换.另外,本章的一些边界条件的证明利用了第二章的结果.
[Abstract]:In recent years, the issue of dividend has received extensive attention in actuarial mathematics. In this paper, the dividend problem of several risk models is considered. The risk models discussed in this paper include one-dimensional diffusion risk model, classical risk model and perturbed classical risk model, including barrier dividend strategy, threshold dividend strategy and mixed dividend strategy. The main problems studied include the expected discounted dividend before bankruptcy, the moment generating function of discounted dividend before bankruptcy, the Gerber-Shiu function of each order and the Laplace transformation of ruin, etc. Special function, Markov process, Taylor formula and Dynkin formula. Some of the problems in this paper have obtained concrete results. According to the specific content of the article, this article can be divided into the following three chapters: 1) the escape time and dividend value function of one dimensional diffusion process. In this chapter we consider the escape time problems of one-dimensional homogeneous diffusion processes on the interval and their application to the dividend problem in risk theory. First, we derive the ordinary differential equation of Laplace transform when escaping by using Dynkin formula. Then we enumerate several diffusion processes often used in actuarial and financial market models and obtain the explicit expressions of Laplace transformation when they escape. Finally, we associate Laplace transform of escape time with dividend value function, and express obstacle dividend value function and threshold dividend value function by using Laplace transformation when escaping. 2) the classical risk model under the mixed dividend strategy. The second chapter studies the classical risk model under the mixed dividend strategy. First, we introduce the classical risk model, the mixed dividend strategy and the research background of dividend strategy in detail. Then we define the discounted dividend function in the classical risk model under the mixed dividend strategy, the expected discounted dividend function before ruin, the moment mother function of the discounted dividend function and each moment, the Gerber-Shiu function and the Laplace transformation at the time of ruin. The integro-differential equations and boundary conditions of the expected discounted dividend function before ruin are derived. We also obtain the moment generating function of the discounted dividend function before ruin and the integro-differential equations and boundary conditions satisfied by each order moment respectively. Finally, we discuss the Gerber-Shiu function and the Laplace transformation in ruin. Among them, we get some concrete results about the individual claim amount from the index. 3) the classical risk model with disturbance under mixed dividend strategy. Based on the second chapter, we consider the classical risk model with disturbance under mixed dividend strategy. For convenience, we use the partial notation of the second chapter. Because the definition of the model is different, we first introduce the classical risk model with disturbance, and point out the change of the symbol caused by the change of the model and the symbol of the second chapter. By using the Dynkin formula, we derive the integro-differential equations and boundary conditions of the expected discounted dividend function before ruin. In this example, we obtain the expression of the expected discount dividend before bankruptcy when the claim service is distributed exponentially. Then the moment generating function of discounted dividend before ruin and the integral differential equation and boundary condition of k-order moment are derived respectively. Finally, we discuss the famous Gerber-Shiu function and calculate the Laplace transformation when the individual claim amount is exponential distribution. In addition, the proof of some boundary conditions in this chapter is based on the results of the second chapter.
【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;F271;F275

【共引文献】