摘要
本文在经典风险模型的基础上，将单位时间内保险费收取是常数变为随机的情况，使之更符合保险公司的实际运作。首次建立了三种风险模型：保险费随机的离散时间风险模型，双复合二项风险模型，保险费收取次数为Poisson过程的投资风险模型。然后，主要探讨了这三种风险模型的几种破产概率及其分布：最终破产概率，有限时间内破产的概率，破产时间的分布，破产前盈余的分布，破产后赤字的分布，破产前盈余与破产后赤字的联合分布等问题。
第一章。介绍我国保险业的现状，论题的意义与典型方法。
第二章。在离散时间的情况下，建立保险费和索赔额是任意随机变量的风险模型。证明在时刻n时资产余额Un是一个马尔科夫链，利用转移概率得到风险问题中的几个重要的分布：有限时间内破产的概率，破产时间的分布，最终破产的概率，破产前盈余的分布，破产后瞬间赤字的分布，破产前和破产后瞬间余额的联合分布，而且利用离散鞅得到Lundberg不等式。
第三章。在离散时间的情况下，建立保险费的收取过程和索赔过程都是复合二项过程的风险模型，得到几个重要的分布：最终破产概率，有限时间内破产的概率，破产时间的分布，破产前盈余的分布，破产后瞬间赤字的分布。证明最终破产概率的积分方程，并就指数分布的情形给出计算最终破产概率的公式，而且利用离散鞅得到Lundberg不等式。
第四章。在连续时间的情况下，建立保险费的收取次数是一个Poisson过程和索赔过程是复合Poisson过程的带扩散扰动项的风险模型，用鞅的方法得出其最终破产概率及lundberg不等式。
In this paper, on the substructure of the classical risk model ,This thesis mainly changs the consant premium income of the classical risk model into the random premium income of the risk model ,so that it is much fitter for the practical operation of insurance company. Firstly ,creating three risk models: a discrete time risk model with a random premium , the two compound biomial risk models, a risk model with a investment when the number of premium income is a possion process, secendly , we obtain several important ruin probabilities and distributions in the risk theory: the ruin probability in finite time , the ultimate ruin probability, the distribution of the ruin time ,the distribution of surpus immediately before ruin and the deficit at ruin .
In chaper 1. To introduce the situation of insurance in our country at present ,the significanc of this thesis and the classic methods
In chaper 2. under the discrete time, I create the risk model which its premium and claim are random variale and prove that the surplus Un at n time is a markov chain with transition probability, we obtain several important probabilities and distributions in the risk theory: the ruin probability in finite time ,the ultimate ruin probability, the distribution of the ruin time ,the distribution of surpus immediately before ruin and the deficit at ruin . The Lundberg unequality is obtained by the discrete martingale approach .
In chaper 3 .Under the discrete time ,we disguss a new model which both the arrival of premium policies and the occurrence of claims follow two compound Binomial processes . we obtain the series expressions of several important probabilities in the risk theory: the ruin probability in finite time ,the ultimate probability,the distribution of the ruin time, the distributions of surpus immediately before ruin and the deficit at ruin . And we prove the integral equations of ultimate ruin probability .As an application , we give the method of ruin probability for the case of exponential distribution. The Lundberg unequality is obtained by the discrete martingale approach
In chaper 4 .under the continual time ,we creat a risk model perturbed by diffusion,which its premium process is a possion process and its claim process is a compound prossion process. we obtain it's probability of riun and Lundberg's inequality by a martingale method
引文
[1] Gerber．H．U，成世学，严颖译．数学风险论引导，世界图书出版公司，1997年．
[2] H.U.Gerber, M.J.Goovaerts and R. Kaes.On the probability and severity of ruin. ASTIN Bulletin, 1987, 17, 151-163.
[3] F.Dufresne and H.U.Gerber. The surpluses immediately before and at ruin, and amount of the claim causing ruin, insurance: math. and econom, 1988,7,193-199.
[4] D.C.M.Dickson and A.D.E.dos Reis. Ruin problems and dual events, insurance: math.and econom, 1994, 14, 51-60.
[5] Grandell. J. Aspects f risk theory Springer-verlag. New York, 1991年.
[6] 戚懿．广义复合poisson模型的破产概率．应用概率统计，1999，15(2)．
[7] 成世学．完全离散的经典风险模型．运筹学学报，1998，2(3)．
[8] B.Sundt and J.L.Teugels. Ruin estimates under interest force. Insurance: Mathematics and Economics, 1995, 16, 7-22.
[9] 王黎明．保险费收取次数为poisson过程的破产概率．内蒙古师大学报．2000，23(3)，176-179．
[10] 曾霭林．双险种的Cox风险模型．数学理论及应用．2003，23(3)，107-112．
[11] 刘家军，刘再明．保费到达为更新过程的复合更新风险模型．数学理论及应用．2003，23(1)，102-106．
[12] Cary Chi-Liang Tsai, Gordon E. Willmot. Ageneralized defctive renewal equation for the surplus process perturbed by diffusion. Insurance: Mathematics and Economics, 2002, (30), 51-66.
[13] Guojing Wang, Rong Wu. Some distributions for classical risk process that is pertured by diffusion. Insurance: Mathematics and Economics 2000, (26), 15-24.
[14] Gerber,H.U.An extension of the renewel equation and its application in the collective theory of risk. Scandinavian Actuarial Journel, 1970, 205-210.
[15] 吴荣．常利率下的更新风险模型．工程数学学报，2002，19(2)，46-54．
[16] 成世学．破产论研究综述．数学进展，2002，31(5)，404-422．
[17] 成世学．完全离散经典风险模型中的渐进解和Lundberg型不等式．高校应用数学学报A辑，2001，16(3)，346-358．
[18] WangGuoJing, WuRong. A generelization of risk model pertubed by diffusion. Applied Mathematics, A Jounal of chinese Universities, Ser, B, 1999.
[19] 邓永录，梁之舜．随机点过程及其应用．科学出版社，1998年
[20] 邓永录．随机模型及其应用．高教育出版社，1994年．
[21] 伍超标．保险精算学基础．中国统计出版社．1999年．
[22] 严士健．刘秀芳．概率论基础．科学出版社，1982年．
[23] 王梓坤．概率论基础及其应用．北京师范大学出版社，1996年．
[24] 钱敏平．随机过程论．北京大学出版社，1997年．
[25] 史及民．离散鞅及其应用．科学出版社，1999年．
[26] 施仁杰．马尔可夫链基础及其应用．西安电子科技大学出版社，1992年．
[27] N．L．鲍尔斯等著，余跃年译．风险理论，上海科技出版社，1995年．