【作者】 佘志芳；

【导师】 耿显民；

【作者基本信息】 南京航空航天大学 ， 应用数学， 2006， 硕士

【摘要】 随着社会金融市场的发展,经典风险模型在很大程度上已无法模拟现实的风险状况,在实际运营中保险公司的利润主要由其投资利润来决定,因此,保险投资成为保险经营的重要环节,考虑带投资的风险模型也就成为现代精算界与数学界研究的热门问题.随机过程理论的系统化和成熟化为带投资风险理论的研究提供了强有力的方法和工具.本文先引入保险公司将固定比例资产投资于风险市场的风险模型,在该模型中投资于风险市场的资金用几何布朗运动来描述,而剩余资金投资于无风险市场,利用伊藤公式求出该模型下生存概率所满足的积分-微分方程,并进一步得到该方程解的存在性及该方程的拉普拉斯形式,它有利于通过计算机得到破产概率的数值解.文章的另一主要内容是针对投资于风险市场的比例为时间变量的函数的风险模型,其它描述同前一个模型,在该模型下破产概率与公司的剩余资金有关,也与每时每刻的投资比例有关,利用伊藤公式得到关于该模型盈余过程和时间的一个二元函数所满足的积分-微分方程,并在广义函数空间Sobolev空间上讨论了该方程的初边值问题的弱解存在惟一性问题.更多还原

【Abstract】 The classical risk model has a lot of restriction with the reality of the insurance company developing with the financial market. In fact, the profit of investment in the insurance company decides the final profit of the company. So a risk model with investment has become a hot topic in the present actuarial science and mathematical research. In this paper, we study the ruin probability in the classical Cramér-Lundberg model, where the company invests a constant fraction of its money in a stock, which is described by geometric Brownian motion. We prove a certain integro-differential equation of the survival probability by use of the It? formula and prove the existence of the solution of the equation. And then we give the Laplace equation of the survival probability to get the numerical solution through the computer when the distribution of the claim size has been given. Furthermore, we also study a risk model with investment, whose proportion investing in a stock is not a constant number, but a function of the time t . We will give an integro-differential equation of the function of the time and the surplus process in the latter model. We introduce Sobolev space, in which we can get the existence and uniqueness of the weak solution of the integro-differential equation.更多还原