基于跳跃扩散风险模型的Gerber-Shiu函数等问题的讨论

【作者】 张莉；

【导师】 吕玉华；

【作者基本信息】 曲阜师范大学 ， 概率论与数理统计， 2013， 硕士

【摘要】 破产论中最经典的风险模型是Cramer-Lundberg风险模型.随着破产论研究的深入Cramer-Lundberg风险模型也被进行了各种各样的改造,使其更接近实际情况Levy过程就是Cramer-Lundberg风险模型的一种推广,更具有实际意义,因而Levy过程成为一个研究热点.另外,最初Cramer-Lundberg风险模型中的跳跃部分只有索赔额即跳跃部分仅为向下跳跃,但随着时代的发展,保险公司也会从事投资活动,向上跳跃也被引入到模型中.文章[1]、[5]分别研究Levy过程下跳和上跳部分分别服从比例变换型分布时的Wiener-Hopf分解.近年来,双边跳跃扩散风险模型得到了更深入的研究.更多细节可参见[3]、[4],这两篇文章主要使用了拉普拉斯变换和Wiener-Hopf分解等方法.受上述论文的启发,本文旨在研究上跳部分服从任意分布函数,下跳部分服从具有n个参数的指数混合分布的双边跳Levy过程.为了达到这个目的,文章首先分析了单参数的情况.然后利用拉普拉斯变换的相关性质得到了双参数指数分布情形的一些结论.最后讨论了下跳部分服从n个参数的指数混合分布情形,并且将结果简洁地以矩阵的形式给出.第一章简单的介绍了相关的背景,符号,预备知识,并且给出了盈余过程：或者,其中｛Yt｝是本文主要研究的Leuy过程.假设进一步求出了｛Yt｝的Lundberg基本方程并对该方程的解进行了分析.第二章利用拉普拉斯变换的知识与相关的引理对第一章提出的模型进行讨论,研究了一段随机时间[0,e(υ)]内最小值的概率密度函数,首次击中阈值与该时刻赤字的二维拉普拉斯变换,赤字的折现概率密度,最后对Gerber-Shiu函数进行讨论.第三章对下跳服从双参数指数分布函数的情形做了与第一章类似的讨论.第四章是本文最重要的部分,旨在研究一般情形,即下跳部分服从具有n个参数的指数混合分布.本章得到以下结论：在[0,e(υ)]内,最小值的概率密度函数为：首次击中时与赤字的二维拉普拉斯变换为：其中,为任意实数.赤字的折现概率密度为：当U0=u时,Gerber—Shiu函数更多还原

【Abstract】 One of the most classical models in ruin theory is the Cramer-Lundberg risk mod-el. As ruin theory is studied extensively, the Cramer-Lundberg risk model is improved a lot so that it is more actual. The Levy risk models are a natural generalization of the Cramer-Lundberg risk model and more practical, so Levy processes are researched deep. In addition, jumps in the models are explained as the claim amounts ar first, that is, only downward jumps are allowed in these models. However, with the development of the society, upward jumps are also allowed result from investments of insurance companies. In the papers of [1][5], Wiener-Hopf factorization for Levy processes, which respec-tively have negative and positive jumps with rational transforms, is researched. Recently, problems of two-side jumps are discussed in depth. See [3][4] for more details. The main methods used in them is the Laplace transform, Wiener-Hopf factorization and so on.Being enlightened by the above paper, this paper is aimed at doing research on Levy processes with two-side jumps, whose positive jumps have an arbitrary distribution and negative ones have exponential distribution with n exponential parameters. In order to hit the target, the paper analyses the situation of one exponential parameter firstly. Then it makes use of the Laplace transform to give several conclusions. At last, the situation in which the negative jumps have exponential distribution with n exponential parameters is discussed, and the results are compactly given in term of matrix.Chapter1, we simply introduce relevant backgrounds, symbols, preliminaries and a surplus process: and, where{Yt} is the Levy processes we mainly discuss in this paper. It is assumed that furthermore, the Lundberg fundamental equation of{Yt} is given and its roots are anal-ysed. Chapter2, making use of the knowledge of the Laplace transform and related lemmas, this chapter researchs some problems during a random time interval [0, e(v)] base on the model of Chapter1, such as the probability density function of minimal value, the two-dimensional Laplace transform of the lower barrier hitting time and the deficit, the discounted density of the deficit. Besides, Gerber-Shiu function is discussed at last.Chapter3, in the situation of exponential distribution with two exponential param-eters, we do similar research to Chapter2.Chapter4is the most important part in this paper, It is aimed at researching general situation, that is, the negative jumps have exponential distribution with n exponential pa-rameters. Some results are given:during a random time interval [0, e(v)], the probability density function of the minimal value is: the two-dimensional Laplace transform of the lower barrier hitting time and the deficit is given by where, here s can be any real number. the discounted density of the deficit in this situation isWhen U0=u, get Gerber-Shiu function更多还原