随机收益率下几类Poisson风险模型的破产概率
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摘要
现代风险理论主要是借助随机过程等数学工具发展起来的,它为各金融风险部门的经营管理提供了理论依据和实际操作指导。在风险理论中,目前大多数学者主要集中对复合二项模型,Poisson模型和更新模型等三个基本风险模型进行更合乎实际的推广。在本文中,考虑的是Poisson风险模型的进一步推广,主要运用随机点过程理论及概率论的基础知识研究了随机收益率下四种Poisson风险模型的破产概率。
本文共五章:
第一章为引论部分。主要介绍了风险理论的历史,现状和主要成果,并给出了阅读本文所需要的预备知识。其中重点阐述了有关Poisson风险模型的问题,并且给出了本文研究的主要内容。
第二章先简单的介绍了经典Poisson风险模型及其破产概率,然后在其基础上建立了随机收益率下的经典Poisson风险模型,并求出其破产概率。
第三章把经典Poisson风险模型推广为双Poisson风险模型,即将报单收入过程推广为一个Poisson的过程,并假定它与理赔过程独立。然后在双Poisson风险模型的基础上加上随机收益率因素考虑其破产概率。
第四章把经典Poisson风险模型推广为广义复合Poisson风险模型。即对每一次事故允许进行多起赔付,把每一次事故的发生看成是一个服从某一离散分布的过程。然后在改进后的模型中加入随机收益率因素进一步考虑其破产概率。
第五章,在第三章和第四章的基础上,把经典Poisson风险模型推广为广义复合双Poisson风险模型。然后在此模型上考虑随机收益率因素,并得出其破产概率。
Modern risk theory has been developed mainly via stochastic process of mathematical tool, which provides a manager who is serving in financial risk department with theory basis and practical guidance. In risk theory in the present study, most of scholars focus exclusively on generalization to the following three risk models, such as: the compound binomial risk model, poisson risk model and renewal risk model. In this thesis, what to consider is the further expansion of the poisson risk model. It mainly depended on the theory of stochastic point process and the foundation knowledge of probability and studied four kinds of poisson risk model which includes force of random rate of return, and calculated their ruin probabilities.
This thesis totals 5 chapters:
Chapter 1 is a preface, which mainly introduces the history, the present conditions and the main results of risk theory, and outlines prepare knowledge for this thesis. In this chapter, we especially pay more attention to the classical risk model. Finally we present the main content of this thesis.
Chapter 2 introduced the classic poisson risk model and its ruin probability first. Then this chapter built up a new model which includes random rate of return force under of the classic poisson risk model, and calculated its ruin probability.
Chapter 3 expanded the classic poisson risk model to the double poisson risk model. In other words, this chapter expanded the insurance policy income process to a poisson process, and supposed it is independent with the compensate process. Then considered the random rate of return force at the double poisson risk, and calculated its ruin probability.
Chapter 4 expanded the classic poisson risk model to the generalized compound poisson risk model. In other words, it allowed to pay every trouble, which obey some discrete distribution. Then considered force of the random rate of return on the improved model, and gave the ruin probability to readers.
Chapter 5 expanded the classic poisson risk model to the generalized compound double poisson risk model on the foundation of chapter 3 and chapter 4. Then considered force of random rate of return on this model, and got its ruin probability.
本文共五章:
第一章为引论部分。主要介绍了风险理论的历史,现状和主要成果,并给出了阅读本文所需要的预备知识。其中重点阐述了有关Poisson风险模型的问题,并且给出了本文研究的主要内容。
第二章先简单的介绍了经典Poisson风险模型及其破产概率,然后在其基础上建立了随机收益率下的经典Poisson风险模型,并求出其破产概率。
第三章把经典Poisson风险模型推广为双Poisson风险模型,即将报单收入过程推广为一个Poisson的过程,并假定它与理赔过程独立。然后在双Poisson风险模型的基础上加上随机收益率因素考虑其破产概率。
第四章把经典Poisson风险模型推广为广义复合Poisson风险模型。即对每一次事故允许进行多起赔付,把每一次事故的发生看成是一个服从某一离散分布的过程。然后在改进后的模型中加入随机收益率因素进一步考虑其破产概率。
第五章,在第三章和第四章的基础上,把经典Poisson风险模型推广为广义复合双Poisson风险模型。然后在此模型上考虑随机收益率因素,并得出其破产概率。
Modern risk theory has been developed mainly via stochastic process of mathematical tool, which provides a manager who is serving in financial risk department with theory basis and practical guidance. In risk theory in the present study, most of scholars focus exclusively on generalization to the following three risk models, such as: the compound binomial risk model, poisson risk model and renewal risk model. In this thesis, what to consider is the further expansion of the poisson risk model. It mainly depended on the theory of stochastic point process and the foundation knowledge of probability and studied four kinds of poisson risk model which includes force of random rate of return, and calculated their ruin probabilities.
This thesis totals 5 chapters:
Chapter 1 is a preface, which mainly introduces the history, the present conditions and the main results of risk theory, and outlines prepare knowledge for this thesis. In this chapter, we especially pay more attention to the classical risk model. Finally we present the main content of this thesis.
Chapter 2 introduced the classic poisson risk model and its ruin probability first. Then this chapter built up a new model which includes random rate of return force under of the classic poisson risk model, and calculated its ruin probability.
Chapter 3 expanded the classic poisson risk model to the double poisson risk model. In other words, this chapter expanded the insurance policy income process to a poisson process, and supposed it is independent with the compensate process. Then considered the random rate of return force at the double poisson risk, and calculated its ruin probability.
Chapter 4 expanded the classic poisson risk model to the generalized compound poisson risk model. In other words, it allowed to pay every trouble, which obey some discrete distribution. Then considered force of the random rate of return on the improved model, and gave the ruin probability to readers.
Chapter 5 expanded the classic poisson risk model to the generalized compound double poisson risk model on the foundation of chapter 3 and chapter 4. Then considered force of random rate of return on this model, and got its ruin probability.
引文
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[2]Cramer,H.On the Mathematics Theory of Risk.Stockholm:Skandia Jubilee Volume,1930
[3]Cramer,H.Mathematical Methods of Statics.Princeton:Princeton University Press,1945
[4]Cramer,H.Collective Risk Theory.Stockholm:Skandia Jubilee Volume.1955
[5]Gerber,H.U.数学风险论导引,成世学,严颖译.北京:世界图书出版社,1979
[6]Lundberg,F.Approximerad Framstallning av Sannolikhets funktionen.Uppsala:Almquist and Wiksell,1903a
[7]Lundberg,F.Ater forsakring av Kollektivrisker.Uppsala:Almquist and Wiksell,1903b
[8]Grandell,J.Aspect of Risk Theory.New York:Springer-Verlag,1991.4-7
[9]Asmussen,S.Ruin Probabilities.Singapore:World Scientific,2001
[10]Asmussen,S.Risk theory in markovian environment.Scand actuarial J.1989
[11]Dufresne,F.,Gerber,H.U.The probability and severity of ruin for combina- tions of exponential claim amount distribution and their translations.Insurance:Mathematics and Economics,1988,7:193-199
[12]Dufresne,F.,Gerber,H.U.Risk theory for the compound Poisson process that is perturbed by diffusion.Insurance:Mathematics and Economics,1991,10:51-59
[13]Gerber,H.U.Mathematical fun with the compound binomial process.ASTIN Bulletin,1988,18:161-168
[14]Dickson,D.C.M.On the distribution of the surplus perior to ruin.Insurance:Mathematics and Economics,1992,11:191-207
[15]Dickson,D.C.M.,dos Reis,A.D.E.Ruin problems and dual events.Insurance:Mathematics and Economics,1994,14:51-60
[16]Annamaria,Olivieri.Uncertainty in mortality projections:an actuarial perspective.Insurance:Mathematics and Economics,2001,29:231-245
[17]Stuart A.Klugman.Loss Models:from data to decisions.John Wiley&sons,1998
[18]董迎辉,王过京.相关负风险和模型的破产概率.应用概率统计,2004,20(3):301-306
[19]方大凡,王汉兴.多质负风险和.应用概率统计,2003,19(1):65-70
[20]Rui M.R.Cardoso,Howard R.Waters.Recursive calculation of finite time ruin probabilities under interest force.Insurance:Mathematics and Economics,2003,33:659-676
[21]Dimirios Konstantinides,Qihe Tang,Gurami Tsitsiashvili.Estimates for the ruin probability in the classical risk model with constant interest force in the presence of heavy tails.Insurance:Mathematics and Economics,2002,31:447-460
[22]David C.M.Dickson,Howard R.Waters.Ruin probabilities with compounding assets.Insurance:Mathematics and Economics,1999,25:49-62
[23]张琳.保险公司崩溃模型研究.经济数学,1997,14(1)
[24]伍亚,唐立.考虑利率因素的保险破产模型研究.数学理论与应用,2007,27(3)
[25]李晋枝,乔克林,何树红.随机利率因素的破产概率.云南大学学报(自然科学版),2003,25(1):9-12
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[27]高珊,曹晓敏.有关推广的poisson风险模型的破产问题.曲阜师范大学学报,2006,32(2)
[28]王黎明,金珩.保险费收取次数为poisson过程的破产概率.内蒙古师大学报,自然科学(汉文)版,2000,29(3)
[29]刘宝亮,王永茂,温艳清.双poisson风险模型的破产概率.燕山大学学报,2006,30(4)
[30]杨善朝,谭激扬.保险费随机收取的风险模型.经济数学,2004,21(1)
[31]戚懿,王静龙,汪荣明.调整保险费率模型下的破产概率.应用数学与计算数学学报,1999,13(2)
[32]戚懿.广义复合poisson模型下的破产概率.应用概率统计,1999,15(2)
[33]补爱军.保险费收取次数为泊松过程下的广义复合泊松风险模型.数学理论与应用,2005,25(4)
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[35]龚日朝.广义复合Poisson风险模型下的生存概率.衡阳师范学院学报[J].2001,23(3):76-80
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[39]邹辉,朱勇华.保险精算中的一个破产模型的改进.武汉水利电力大学学报(社会科学版),2000,20(2)
[40]魏秋萍,张波.风险理论中破产模型的若干结果.经济数学,2003,20(4)
[41]何树红.风险理论及其在保险中的应用.吉首大学学报(自然科学版),2002,23(4)
[42]何树红,夏梓翔.随机利率下时间盈余风险模型的破产概率.云南大学学报(自然科学版),2004,26(5):382-385
[43]何树红,赵金娥,马丽娟.常利率下保险费随机收取的双险种风险模型.云南大学学报(自然科学版),2005,27(5):629-633
[44]江涛,王家琪.随机利率场合下一类离散时间模型的破产概率.兰州大学学报(自然科学版),2004,40(4)
[45]吴荣,杜勇宏.常利率下的更新风险模型.工程数学学报,2002,19(1)
[46]刘家有,刘再明.保险公司在固定利率下的离散型破产概率.数学理论与应用,2004,24(1):101-104
[47]杨善兵,司建东.常利率两险种的风险模型的破产概率.安徽大学学报(自然科学版),2006,30(1)
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[63]Dimirios Konstantinides,Qihe Tang,Gurami Tsitsiashvili.Estimates for the ruin probability in the classical risk model with constant interest force in the presence of heavy tails.Insurance:Mathematics and Economics,2002,31:447-468
[2]Cramer,H.On the Mathematics Theory of Risk.Stockholm:Skandia Jubilee Volume,1930
[3]Cramer,H.Mathematical Methods of Statics.Princeton:Princeton University Press,1945
[4]Cramer,H.Collective Risk Theory.Stockholm:Skandia Jubilee Volume.1955
[5]Gerber,H.U.数学风险论导引,成世学,严颖译.北京:世界图书出版社,1979
[6]Lundberg,F.Approximerad Framstallning av Sannolikhets funktionen.Uppsala:Almquist and Wiksell,1903a
[7]Lundberg,F.Ater forsakring av Kollektivrisker.Uppsala:Almquist and Wiksell,1903b
[8]Grandell,J.Aspect of Risk Theory.New York:Springer-Verlag,1991.4-7
[9]Asmussen,S.Ruin Probabilities.Singapore:World Scientific,2001
[10]Asmussen,S.Risk theory in markovian environment.Scand actuarial J.1989
[11]Dufresne,F.,Gerber,H.U.The probability and severity of ruin for combina- tions of exponential claim amount distribution and their translations.Insurance:Mathematics and Economics,1988,7:193-199
[12]Dufresne,F.,Gerber,H.U.Risk theory for the compound Poisson process that is perturbed by diffusion.Insurance:Mathematics and Economics,1991,10:51-59
[13]Gerber,H.U.Mathematical fun with the compound binomial process.ASTIN Bulletin,1988,18:161-168
[14]Dickson,D.C.M.On the distribution of the surplus perior to ruin.Insurance:Mathematics and Economics,1992,11:191-207
[15]Dickson,D.C.M.,dos Reis,A.D.E.Ruin problems and dual events.Insurance:Mathematics and Economics,1994,14:51-60
[16]Annamaria,Olivieri.Uncertainty in mortality projections:an actuarial perspective.Insurance:Mathematics and Economics,2001,29:231-245
[17]Stuart A.Klugman.Loss Models:from data to decisions.John Wiley&sons,1998
[18]董迎辉,王过京.相关负风险和模型的破产概率.应用概率统计,2004,20(3):301-306
[19]方大凡,王汉兴.多质负风险和.应用概率统计,2003,19(1):65-70
[20]Rui M.R.Cardoso,Howard R.Waters.Recursive calculation of finite time ruin probabilities under interest force.Insurance:Mathematics and Economics,2003,33:659-676
[21]Dimirios Konstantinides,Qihe Tang,Gurami Tsitsiashvili.Estimates for the ruin probability in the classical risk model with constant interest force in the presence of heavy tails.Insurance:Mathematics and Economics,2002,31:447-460
[22]David C.M.Dickson,Howard R.Waters.Ruin probabilities with compounding assets.Insurance:Mathematics and Economics,1999,25:49-62
[23]张琳.保险公司崩溃模型研究.经济数学,1997,14(1)
[24]伍亚,唐立.考虑利率因素的保险破产模型研究.数学理论与应用,2007,27(3)
[25]李晋枝,乔克林,何树红.随机利率因素的破产概率.云南大学学报(自然科学版),2003,25(1):9-12
[26]龚日朝,李凤军.双poisson风险模型下的破产概率.湘潭师范学院学报,2001,23(1)
[27]高珊,曹晓敏.有关推广的poisson风险模型的破产问题.曲阜师范大学学报,2006,32(2)
[28]王黎明,金珩.保险费收取次数为poisson过程的破产概率.内蒙古师大学报,自然科学(汉文)版,2000,29(3)
[29]刘宝亮,王永茂,温艳清.双poisson风险模型的破产概率.燕山大学学报,2006,30(4)
[30]杨善朝,谭激扬.保险费随机收取的风险模型.经济数学,2004,21(1)
[31]戚懿,王静龙,汪荣明.调整保险费率模型下的破产概率.应用数学与计算数学学报,1999,13(2)
[32]戚懿.广义复合poisson模型下的破产概率.应用概率统计,1999,15(2)
[33]补爱军.保险费收取次数为泊松过程下的广义复合泊松风险模型.数学理论与应用,2005,25(4)
[34]陈新美.随机利率下广义复合poisson风险模型.长沙理工大学学报(自然科学版),2006,3(2)
[35]龚日朝.广义复合Poisson风险模型下的生存概率.衡阳师范学院学报[J].2001,23(3):76-80
[36]陈新美,刘罗华,刘再明.广义复合双poisson风险模型下的破产概率.株州工学院学报,2005,19(6)
[37]董英华.广义双poisson风险模型下的破产概率.长沙铁道学院学报,2003,21(1)
[38]龚日朝,杨向群.复合广义poisson模型下最终破产概率估计.长沙电力学院学报(自然科学版),2001,16(1)
[39]邹辉,朱勇华.保险精算中的一个破产模型的改进.武汉水利电力大学学报(社会科学版),2000,20(2)
[40]魏秋萍,张波.风险理论中破产模型的若干结果.经济数学,2003,20(4)
[41]何树红.风险理论及其在保险中的应用.吉首大学学报(自然科学版),2002,23(4)
[42]何树红,夏梓翔.随机利率下时间盈余风险模型的破产概率.云南大学学报(自然科学版),2004,26(5):382-385
[43]何树红,赵金娥,马丽娟.常利率下保险费随机收取的双险种风险模型.云南大学学报(自然科学版),2005,27(5):629-633
[44]江涛,王家琪.随机利率场合下一类离散时间模型的破产概率.兰州大学学报(自然科学版),2004,40(4)
[45]吴荣,杜勇宏.常利率下的更新风险模型.工程数学学报,2002,19(1)
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