摘要
风险理论是近代数学的一个重要分支,主要应用于金融、保险、证券投资以及风险管理等方面,也是当今精算界和数学界研究的热门课题。本文主要研究了几类不同索赔情形下风险模型的破产概率及保险风险证券化问题。
全文共分为五章。第一章介绍了破产理论和保险风险证券化的发展史及现状,并在经典风险模型的基础上,介绍了研究破产概率时可以拓展的一些方向,最后给出了本文研究过程中所用到的重要方法、主要结果和创新点。
第二章在Cramèr-Lundberg经典风险模型的基础上,把个体索赔额服从混合指数分布( k = 2)时最终破产概率显式解的结论推广到索赔额服从一般混合指数分布(k = n)的情形。
第三章研究了在重尾索赔下延迟更新风险模型中破产概率的一个局部等价式,并修正了已有文献[53]证明中的漏洞。
第四章研究了一类带有投资回报的离散时间风险模型,运用递归法得到了一个重尾损失情形下有限时间破产概率的渐进式,是Tang、Tsitsiashvili研究的一个有益扩展。
第五章根据我国是巨灾高发区的实际情况,讨论了在我国实行保险风险证券化的必要性和可行性,并提出了若干建议。
Risk theory is an important branch of modern mathematics, which is mainly applied in finance, insurance, securities investment and the risk management. Nowadays, the collective risk theory is one of the most intriguing fields both in actuarial and mathematical science.The paper focuses on the ruin probabilities of risk model under several types of circumstances where claim is lodged and the securitization of insurance risk.
The paper consists of five chapters. Chapter 1 introduces the developme -nt and the state of art of the insurance ruin theory and securitization of insurance risk. On the basis of the classical risk model, some new research areas are presented. At the same time, research methods, important findings and innovative points are given as well.
In Chapter 2, according to the classical risk model proposed by Cramèr- Lundberg, the explicit solution of the ultimate ruin probability of the claims having mixed exponential distribution ( k = 2) is extended to the situation where the claims have general mixed exponential distribution ( k = n).
Chapter 3 investigates asympototic ruin probabilities of the delayed renewal risk model with heavy-tailed claims, and also revises a pitfall relations -hip in the formula derivation of the 53rd literature.
In Chapter 4, a discrete time risk model with investment return is studied. An asympototica for the finite time ruin probability under subexponent -ial loss is obtained by the inductive method, which is a beneficial extension to the results of Tang and Tsitsiashvili.
In Chapter 5, since our country is highly vulnerable to catastrophe, the need and feasibility of implementing the securitization of insurance risk is investigated, and some reasonable suggestions are proposed as well.
引文
1 F. Lundberg. I, Approximerad Framst?llning av Sannolikhetsfunktionen.Ⅱ, ?tersfor- s?kring av Kollektivrisker. Uppsala: Almqvist&Wiksell, 1903:1-15,30-50
2 F. Lundberg. F?rs?kringsteknisk Riskutj?mning. F. Eng-lounds Boktryckeri A. B., Stockholm,1926:1-40
3 H. Cramér. On the Mathematical Theory of Risk. Stoockholm: Skandia Jubilee Volume,1930:1-72
4 H. Bühlmann. Mathematical Methods in Risk Theory. Heidelberg:Springer-Verlag, 1970:95-120
5 H. U. Gerber. An Introduction to Mathematical Risk Theory. S.S. Huebner Foundation Monograph Series No. 8, R. Irwin,Homewood,IL 1979:1-9
6 J. Grandell. Aspects of Risk Theory. Springe-Verlag, 1991:110-150
7 P. Embrechts, C. Klüppelberg, T. Mikosch. Modelling Extremal Events for Insurance and Finance. Berlin: Springer-Verlag,1997:1-57,564-590
8 T. Rolskil, H. Schmidli, V. Schmidt, J. Teugels. Stochastic Processes for Insurance and Finance. Wiley, 1999:149-265
9 S. Asmussen. Ruin Probabilities. Singapore:World Scientific, 2000:10-105
10肖文,孙明波.西方保险风险证券化的运作方式.保险研究,2004,(3):62-64
11张莉,钟玲.再谈保险风险证券化.南京财经大学学报,2004,127(3):70-72
12 P. Embrechts, N. Veraverbeke. Estimates for the Probability of Ruin with Special Emphasis on the Possibility of Large Claims. Insurance: Mathematics and Economics, 1982,1(1):55-72
13 F. Delbaen, J. Haezendonck. Classical Risk Theory in an Economic Environment. Insurance: Mathematics and Economics,1987,6(2):85-116
14 F. Dufense, H. U. Gerber. Risk Theory for the Compound Poisson Process That is Perturbed by Diffusion. Insurance: Mathematics and Economics,1991,10(1):51-59
15 H. U. Gerber, H. Yang. Absolute Ruin Probabilities in a Jump Diffusion Risk Modelwith Investment. North American Actuarial Journal,2007,11(3):159-169
16 B. Sundt, Jozef. L. Teugels. Ruin Estimates under Interest Force. Insurance: Mathemat -ics and Economics,1995,16(1):7-22
17 S. Asmussen. Subexponential Asymptotics for Stochastic Processes: Extremal Behavior, Stationary Distribution and First Probabilities. The Annals of Applied Probability,1998,8(2):354-374
18 R. Wu, G. Wang, C. Zhang. On a Distribution for the Process with Constant Interest Force. Insurance: Mathematics and Economics,2005,36(3):365-374
19 J. Cai, C. M. Dickson. Ruin Probabilities with a Markov Chain Interest Model. Insurance: Mathematics and Economics,2004,35(4):513-525
20 Y. Chen, K. W. Ng. The Ruin Probability of the renewal model with Constant Interest Force and Negatively Dependent Heavy-tailed Claims. Insurance: Mathematics and Economics,2007,40(3):415-423
21 K. W. Ng, Q. Tang. Asympototic Behavior of Tail and Local Probabilities for Sums of Subexponential Random Variables. Journal of Applied Probability,2004,41(1):108-116
22 M. R. Cardoso, H. R. Waters. Calculation of Finite Time Ruin Probabilities for Some Risk Models. Insurance: Mathematics and Economics,2005,37(2):197-215
23 K. C. Yuen, G. Wang. Some Ruin Problems for a Risk Process with Stochastic Interest. North American Actuarial Journal,2007,9(3):129-142
24 A. Ganesh, G. L. Torrise. A Class of Risk with Delayed Claims: Ruin Probabilities Estimates under Heavy-tailed Conditions. Journal of Applied Probability,2006,43(4): 916-926
25 R. Leipus, ?. Jonas. Asymptotic Behaviour of the Finite-time Ruin Probability under Subexponential Claim Sizes. Insurance: Mathematics and Economics,2005,37(2): 197 -215
26 Y. Chen, C. Su. Finite Time Ruin Probability with Heavy-tailed Insurance and Financial Risk. Statistics and Probability Letters,2006,76(6):1812-1820
27陈昱,苏淳.有利息力情形下的有限时间破产概率.中国科学技术大学学报,2006,36(9):909-916
28龚日朝,邹捷中.重尾索赔下带干扰风险模型破产概率的局部解.应用数学学报,2007,30(3):563-572
29 Q. Tang, G. Tsitsiashvili. Finite and Infinite Time Ruin Probabilities in the Presence of Stochastic Returns on Investment. Advances in Applied Probability,2004,36(4): 1278-1299
30 Q. Tang. The Ruin Probability of a Discrete Time Risk Model under Constant Interest Rate with Heavy Tails. Scandinanvian Actuarial Journal,2004,(3):229-240
31 Q. Tang. Asymptotic Ruin Probabilities of the Renewal Model with Constant Interest Force and Regular Variation. Scandinanvian Actuarial Journal,2005,(1):1-5
32 Q. Tang. The Finite Time Ruin Probabilitjy of the Compound Poisson Model with Constant Interest Force. Journal of Applied Probability,2005,42(3):608-619
33成世学.破产论研究综述.数学进展, 2002,31(5):403-422
34谢志刚,韩天雄.风险理论与非寿险精算.天津:南开大学出版社,2000:165-166
35 J. Teugels. The Class of Subexponential Distributions. The Annals of Probability, 1975,3(6):1000-1011
36 P. Embrechts, C. Goldie, N. Veraverbeke. Subexponentiality and Infinite Divisibility. Wahrscheinlichkeitstheorie. 1979,49:335-347
37 P. Embrechts, C. Goldie. On Closure and Factorization Properties of Subexponential and Related Distributions. Journal of the Australian Mathematical Society(Series A) , 1980,29:243-256
38 P. Embrechts, C. Goldie. On Convolution Tails. Stochastic Processes and their Applications,1982,13(2):263-278
39 C. Klüppelberg. Subexponential Distributions and Integrated Tails. Journal of Applied Probability,1988,25(2):132-141
40 C. Goldie, C. Kluppelberg. Subexponential Distributions. A Practical Guide to Heavy Tails.Edited by R.Adler,et al .Birkhauser,1998:435-459
41陈通鑫,刘嘉琨,王占元,方金秋.数学小词典.北京:测绘出版社,1982:720-721
42 H. U. Gerber. Mathematical fun with the compound binomial process. Astin Bulletin, 1988,18:161-168
43许璐,彭必进.索赔额服从指数分布的破产概率及渐进估计.佳木斯大学学报,2007,25(4):533-534
44伍超标.保险精算学基础.北京:中国统计出版社,1999:126
45同济大学数学系.线性代数.第四版.北京:高等教育出版社,2003:18-19
46黄玉民,李成章.数学分析.第二版.北京:科学出版社,2007:510-511
47 Q. Tang, C. Su. Ruin Probabilities for Large Claims in Renewal Risk Model. Proceedings of Third Symposium of Post-Graduates of USTC, 2002:196-201
48 Q. Tang, C. Su. Ruin Probabilities for Large Claims in Delayed Renewal Risk Model. Southeast Asian Bulletin of Mathematics. 2002,25:735-743
49唐启鹤.重尾索赔下关于破产概率的一个等价式.中国科学(A辑),2002,32(3):260 -266
50孔繁超,曹龙,王金龙,唐启鹤.对于大额索赔的的平衡更新风险模型的破产概率.数学年刊(A辑),2002,23(4):531-536
51江涛,陈宜清.平稳更新模型下生存概率的一个局部等价式.中国科学(A辑), 2004, 34(4):385-391
52苏纯,胡治水,唐启鹤.关于非负分布重尾程度的刻画.数学进展,2003,32(5): 606-614
53龚日朝,邹捷中.重尾索赔下更新风险模型生存概率局部估计解.应用数学学报, 2006,29(5):947-954
54威廉?费勒.概率论及其应用.郑元禄译.北京:人民邮电出版社,2008:241-250
55 R. Norberg. Ruin Problems with Assets and Liabilities of Diffusion Type. Stochastic Processes and Their Application. 1999,81(2):255-269
56 H. Nyrhinen. On the Ruin Probabilities in a General Economic Environment. Stochastic Processes and Their Application. 1999,83(2):319-330
57 H. Nyrhinen. Finite and Infinite Time Ruin Probabilities in a Stochastic Economic Environment. Stochastic Processes and Their Application. 2001,92(2):265-285
58 Q. Tang, G. Tsitsiashvili. Precise Estimates for the Ruin Probability in Finite Time Horizon in a Discrete-time Model with Heavy-tailed Insurance and Financial Risks. Stochastic Processes and their Application. 2003,108(2):299-325
59 D. B. H. Cline, G.Samorodnisty. Subexponentiality of the Product of IndependentRandom Variables. Stochastic Processes and their Application. 1994,49(1):75-98
60李志强.中国地震灾害风险管理中的保险问题研究.博士论文.国家地震局地质研究所,1997:1-20
61林增余.中国洪水风险与保险对策.上海保险,1997,(10):4-6
62周伏平.巨灾风险证券化研究.财经研究,2002,28(2):36-41
63李勇权.论保险证券化在我国的引入与发展.保险研究,2003,(5):41-43
64杨永宁,钟毅恒.保险风险证券化:资本市场的保险创新.西南金融,2003,(5):60-62
65梁雪辉.论保险风险证券化.开发导报,2003,113(2):98-100
66洪哲.保险风险证券化初探.现代财经,2004,24(3):15-18
67方芳.浅析保险风险证券化.商业研究,2004,291(7):87-89
68曾立新.美国巨灾风险融资和政府干预研究.北京:对外经济贸易大学出版社, 2008:6-10, 90-113
69童恒庆.理论计量经济学.北京:科学出版社,2005:6-10