几类风险模型下Gerber-Shiu函数的应用

发布时间：2018-05-08 07:25

本文选题：分红策略 + Gerber-Shiu函数　；参考：《重庆理工大学》2015年硕士论文

【摘要】：著名精算大师Hans Gerber与Elias Shiu于1998年在“On the time value of ruin”这篇文章中,提出用期望折现罚函数来研究破产时间的Laplace变换、破产前瞬时盈余和破产后赤字的联合分布,因而该函数也被称作Gerber-Shiu函数.随着风险理论的日益发展,许多学者尝试利用该函数去探讨一些风险模型中的分红问题,并取得了相当大的理论成果.因此Gerber-Shiu函数成为研究分红和破产问题的一个风险度量工具,具有重要的理论研究价值.本文将对风险模型中的三种分红策略进行研究,并给出Gerber-Shiu函数在这几类风险模型中的实际应用.主要内容如下:(1)考虑带干扰经典风险模型中的Barrier分红策略,分别推导出期望折现分红函数以及Gerber-Shiu函数所满足的积分表达式,并证明其关于变量u二次连续可微,然后利用?Ito公式得出它们所满足的积分-微分方程,最后给出??b,d?u关于其积分-微分方程特殊形式下解的表达式.(2)研究Threshold分红策略下带干扰的广义Erlang(n)分布的更新风险模型,得到破产前折现分红总额的矩母函数、Gerber-Shiu函数以及m阶矩的积分-微分方程,并讨论它们的边界条件,最后给出Erlang(2)风险模型的具体实例.(3)在复合二项风险模型中,将服从线性函数分布的保费收入推广到服从二项分布的模型中,研究此模型下的随机分红问题.利用全概率公式和控制收敛定理,推导出0?u?a和u?a时,Gerber-Shiu函数所满足的递推公式,并进一步给出这两种情况下Gerber-Shiu函数的瑕疵更新方程,最后分别计算出更新方程以及最终破产概率解的表达式,这也是本文的主要创新之处.
[Abstract]:The famous actuarial master Hans Gerber and Elias Shiu, in the article "On the time value of ruin" in 1998, proposed the use of the expected discounted penalty function to study the Laplace transformation of the bankruptcy time, the joint distribution of the instantaneous surplus before bankruptcy and the deficit after bankruptcy, thus the function is also called the Gerber-Shiu function. With the increasing risk theory, the function is also called. Development, many scholars try to use this function to discuss the problem of dividend in some risk models, and have obtained considerable theoretical results. Therefore, Gerber-Shiu function becomes a risk measurement tool to study the problem of dividend and bankruptcy. It has important theoretical research value. This paper will study three kinds of dividend strategies in the risk model. The practical application of Gerber-Shiu functions in these types of risk models is given. The main contents are as follows: (1) considering the Barrier dividend strategy in the classical risk model with interference, the integral expressions of the expected discounted dividend function and the Gerber-Shiu function are derived respectively, and it is proved that the variable U is continuously differentiable with respect to the variable U. Use the Ito formula to derive the integral differential equation they satisfy, and finally give the expression of the solution in the special form of the integral differential equation of? B, D? U. (2) the renewal risk model of the generalized Erlang (n) distribution with interference under the Threshold dividend strategy is studied, and the moment mother function, Gerber-Shiu function and m order of the total amount of the discounted bonus before the production are obtained. The boundary condition of the moment integral differential equation and their boundary conditions are discussed. Finally, a concrete example of the Erlang (2) risk model is given. (3) in the compound two term risk model, the premium income that obeys the linear function distribution is extended to the model that obeys the two distribution, and the random dividend problem under this model is studied. The full probability formula and control convergence are used. The theorem, derives the recurrence formula which the Gerber-Shiu function satisfies when 0? U? A and u? A, and further gives the defect renewal equation of the Gerber-Shiu function under these two cases. Finally, the expression of the renewal equation and the final ruin probability solution are calculated respectively. This is the main innovation of this paper.

【学位授予单位】：重庆理工大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：F224

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