【作者】 贺丽娟；

【导师】 李萍；

【作者基本信息】 华中科技大学 ， 概率论与数理统计， 2006， 硕士

【摘要】 在保险数学,也称为精算数学的范畴内,破产论是风险理论的核心内容。1903年,Lundberg首次在他的博士论文中提出一类重要的随机过程,即Poisson过程。Cramer在Lundberg工作的基础上,发展了严格的随机过程理论。继Cramer之后,Feller和Gerber引入的更新论证技巧和鞅证明技巧研究破产论,这也成为研究破产论的主要数学工具。Lundberg-Cramer经典模型中,保险公司的单位时间常数速率收到保费,理赔的发生次数为Poisson过程,且与索赔发生过程相互独立。现在大量研究文献所研究的模型较经典模型有不同程度的推广。Gerber考虑到保险公司的不确定性收益,提出了带干扰风险过程。不少文章分别从保单到达理赔的发生,保费的厘定等方面进行推广,也有文章考虑到保险公司险种的多元化及新险种的不断开发,提出了多险种风险模型。本文首先对风险模型中保险公司按单位时间常数速率收到保费的假设做了改进,考虑变保费率的情形,同时用Cox过程代替一般风险模型中的Poisson过程,描述保险公司的理赔次数,并且,考虑到利率因素和不确定因素的影响,得到一个新的Cox风险模型,证明了该模型的最终生存概率满足的积微分方程,并得到了破产概率的估计及上下界。在此基础上,文章又考虑利率随机的情况,建立了一个更符合实际的风险模型。当随机利率为Brownian运动时,得到了该情况下保险公司破产概率满足的积微分方程,并给出了该积微分方程的应用。更多还原

【Abstract】 In Actuarial Mathematics, the study of ruin theory has been the centre of interest of risk theory. In 1903, Lundberg proposed an important stochastic process, namely, Poisson process in his doctor paper. On the base of his, Cramer developed strict stochastic process theory. After Cramer, Feller and Gerber introduced Renew and Martingale methods. All of these are important methods in the study of ruin theory.In Lundberg-Cramer model, insurer company receives the premium with a constant rate, the occurrence of claim is a Poisson process. The risk models are extensively discussed by many authors. Gerber considered uncertainly profit and presented a ruin model perturbed by diffusion. Many papers improved it from the arrival of the guarantee and the occurrence of claims. Other fellows considered all kinds of insurance and proposed multiple risk models.In this thesis, we improve the assumption that the insurance company receives the premium with a differentiable rate. At the same time, we instead Cox process of Poisson process to describe the occurrence of claims and consider the diffusion. It should be noted that the company also receives interest of its reserves with a constant force. Think of all of these, in the considered Cox risk model, we discuss equations for the ultimate survival probability as well as approximations and upper and lower bound.Based on it, the probability of ruin is investigated under the influence of a variable premium rate and the stochastic rates of interest. In the present risk model, the occurrence of claims is described by the Cox process. When the interest process is a Brownian motion with drift, we derive an integro-differential equation for ruin probability. Applications of the integro-differential equation are given.更多还原