两类风险过程的混合分红问题

发布时间：2018-11-12 19:06

【摘要】：本文分为两章,主要研究了两类风险模型的混合分红问题. 分红问题首先在1957年被De Finetti提出,自此以后越来越多的学者对分红策略下的风险模型进行了研究,尤其是作为保险精算学中的一个重要课题,分红问题已成为当下研究的热门课题.而Ornstein-Uhlenbeck模型作为一个重要的模型,更是引起众多学者的广泛关注.其中关于Ornstein-Uhlenbeck模型的barrier分红策略和threshold分红策略的相关问题已经有很多研究.本文第一章是对Ornstein-Uhlenbeck模型考虑一种新的分红策略,即混合分红策略.所谓混合分红策略就是假设不同的分红界0b1b2,α0,当保险余额小于b1时,保险公司不支付分红；当保险余额大于b1且小于b2时,保险公司以常数比率α0连续地支付分红；当保险余额大于b2时,保险公司将超出b2的部分用以全部分红.在本章第二节中,我们首先介绍了模型并推导出在混合分红策略下分红值函数的表达式.第三节给出了分红界的极限情况,并与已知结果进行比较.在第四节中,研究了破产时间的拉普拉斯变换,求出了在混合分红策略下拉普拉斯变换的表达式.最后,讨论了累积分红函数的各阶矩和矩母函数,推导出了在混合分红策略下矩母函数所满足的偏微分方程和各阶矩函数满足的微分方程. 在第二章中,我们考虑了更为一般的一维扩散过程的分红问题.在本章第一节中,我们介绍了一般的一维扩散过程.在第二节中应用伊藤公式推导出带一个反射壁的一维扩散过程的首次通过时的拉普拉斯变换的表达式及边界条件,并根据定理对若干过程进行计算求解.第三节是在前一章研究的基础上考虑了一般的扩散过程的混合分红问题,并求得了具体的表达式,最后作为定理的应用,我们分析了相关的例子.
[Abstract]:This paper is divided into two chapters, mainly studying the mixed dividend problem of two kinds of risk models. The dividend issue was first raised by De Finetti in 1957. Since then, more and more scholars have studied the risk model under the dividend strategy, especially as an important subject in the actuarial science of insurance. The issue of dividends has become a hot topic in current research. As an important model, Ornstein-Uhlenbeck model has attracted much attention from many scholars. There have been many studies on the barrier dividend strategy and the threshold dividend strategy of Ornstein-Uhlenbeck model. In the first chapter of this paper, we consider a new dividend strategy for Ornstein-Uhlenbeck model, that is, hybrid dividend strategy. The so-called mixed dividend strategy is to assume that different dividend boundaries 0b1b2, 伪 0, when the insurance balance is less than b1, the insurance company does not pay dividends, when the insurance balance is greater than b1 and less than b2, the insurance company pays dividends continuously with constant ratio 伪 0. When the balance of insurance is greater than B 2, the insurance company will use the excess of B 2 for full dividends. In the second section of this chapter, we first introduce the model and derive the expression of dividend value function under mixed dividend strategy. In the third section, the limit of the dividend boundary is given and compared with the known results. In the fourth section, the Laplace transformation of ruin time is studied, and the expression of Laplace transformation under mixed dividend strategy is obtained. Finally, we discuss the moment and moment generating function of the cumulative dividend function, and derive the partial differential equation of the moment generating function and the differential equation of each order moment function under the mixed dividend strategy. In the second chapter, we consider the more general dividend problem of one-dimensional diffusion process. In the first section of this chapter, we introduce the general one-dimensional diffusion process. In the second section, the expression and boundary conditions of Laplace transformation of one-dimensional diffusion process with a reflection wall are derived by using Ito formula, and some processes are calculated and solved according to the theorems. In the third section, based on the previous chapter, we consider the mixed dividend problem of the general diffusion process, and obtain the concrete expression. Finally, as the application of the theorem, we analyze the relevant examples.
【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】：F224;F840.31

【共引文献】