几类再保险风险模型的研究
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摘要
风险理论是金融学和精算学的基础,其核心问题是破产理论的研究.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.本论文应用经典鞅论和随机点过程等理论研究分析常利率下带干扰的超额再保险风险模型和保费随机收取的再保险风险模型,得到了最优自留额及与破产相关的一些重要变量的表达式或性质.
本文由五章组成.
第一章是绪论,介绍了风险理论与经典风险模型,重点介绍古典风险模型,并给出本文的结构.
第二章介绍一些预备知识,包括齐次Poisson过程、Markov过程、鞅等在本文中需要用到的一些基本知识.
第三章到第五章是本文的主要部分,介绍我们的研究结果.
在第三章中,我们研究推广的再保险泊松模型：在经典的风险模型基础上,构造了一种带干扰的常利率再保险风险模型.对此模型进行分析和研究,我们得到了其破产概率上界及最优自留额表达式.
在第四章中,针对在经典风险模型中,假定不同单位时间内收取的保单数是一样的这一局限性,将保单数量推广为一个随机变量,研究带干扰的再保险Poisson风险模型,得到了有限时间破产概率上界和Lundberg不等式.
在第五章中,我们研究较第四章模型更一般的一类风险模型：每张保单的保费是随机变量,研究带干扰的保费随机收取的再保险风险模型,得到有限时间破产概率上界和Lundberg不等式.
In this thesis, we study three kinds of new risk models on excess reinsurance. The optimal retention and some quantities related to the ruin problem to the models are obtained.
The thesis is organized as follows.
In the first chapter, we first recall some risk theory and classical risk models in excess reinsurance. Then we give a simple sketch of structure for our research.
In the second chapter, we present some preliminaries on Poisson process, Markov process, and Martingales.
In the third chapter, we study the reinsurance risk model with constant rates and random disturbance, which is a generalization of the classical risk model.The upper bound of the finite-time ruin probability and the optimal retention are obtained there.
In the following chapter, we study the'double-Poisson'reinsurance risk model with disturbance, assuming that the number of premiums is random. The upper bound of the finite-time ruin probability and the Lundberg inequality of the ultimate ruin probability are developed.
In the final chapter, we study a more general reinsurance risk model with disturbance, where both the number of insurance bills in different time units and the premium of different insurance bills are random variables.Similar results to the last chapter are obtained for the model.
本文由五章组成.
第一章是绪论,介绍了风险理论与经典风险模型,重点介绍古典风险模型,并给出本文的结构.
第二章介绍一些预备知识,包括齐次Poisson过程、Markov过程、鞅等在本文中需要用到的一些基本知识.
第三章到第五章是本文的主要部分,介绍我们的研究结果.
在第三章中,我们研究推广的再保险泊松模型：在经典的风险模型基础上,构造了一种带干扰的常利率再保险风险模型.对此模型进行分析和研究,我们得到了其破产概率上界及最优自留额表达式.
在第四章中,针对在经典风险模型中,假定不同单位时间内收取的保单数是一样的这一局限性,将保单数量推广为一个随机变量,研究带干扰的再保险Poisson风险模型,得到了有限时间破产概率上界和Lundberg不等式.
在第五章中,我们研究较第四章模型更一般的一类风险模型：每张保单的保费是随机变量,研究带干扰的保费随机收取的再保险风险模型,得到有限时间破产概率上界和Lundberg不等式.
In this thesis, we study three kinds of new risk models on excess reinsurance. The optimal retention and some quantities related to the ruin problem to the models are obtained.
The thesis is organized as follows.
In the first chapter, we first recall some risk theory and classical risk models in excess reinsurance. Then we give a simple sketch of structure for our research.
In the second chapter, we present some preliminaries on Poisson process, Markov process, and Martingales.
In the third chapter, we study the reinsurance risk model with constant rates and random disturbance, which is a generalization of the classical risk model.The upper bound of the finite-time ruin probability and the optimal retention are obtained there.
In the following chapter, we study the'double-Poisson'reinsurance risk model with disturbance, assuming that the number of premiums is random. The upper bound of the finite-time ruin probability and the Lundberg inequality of the ultimate ruin probability are developed.
In the final chapter, we study a more general reinsurance risk model with disturbance, where both the number of insurance bills in different time units and the premium of different insurance bills are random variables.Similar results to the last chapter are obtained for the model.
引文
[1]Asmussen. Ruin Probabilities. Singapore:Word Scientific，2000.1-280.
[2]柏晓明,李萍,李学锋.带干扰的双险种COX风险模型.华中科技大学学报(自然科学版),2005,33(2)：122-124.
[3]Cramer H. On the mathematical theory of risk. Stockholkm:Skandia Jubilee Volume,1930.
[4]Cramer H. On some questions connected with mathematical risk. University of California Paubl. Statistics,1954,2:99-123.
[5]Centeno L. Excess of loss reinsurance and the probability of ruin in finite horizon[J].Astin Bulletin,1997,27:59-70.
[6]C.Chi-Liang Tsai, GE.Willmot. On the moments of the surplus process perturbed by diffusion. Insurance:Mathematics and Economics, 31(2002)327-350.
[7]C.Chi-Liang Tsai. On the expectations of the present values of the surplus process perturbed by diffusion. Insurance:Mathematics and Economics, 32(2003)413-429.
[8]Chiu S.N. Yin. C.C. The time of ruin, the surplus prior to ruin and the deficit at ruin for the classical risk process perturbed by diffusion. Insurance: Mathematics and Economics,33(2003)59-66.
[9]David C. M. Dickson. Insurance Risk and Ruin. Cambridge University Press, 2005.
[10]D.Konstantinides, Qihe Tang, G.Tsitsiashvili. Estamates for the ruin probability in the classical risk model with constant interest force in the presence of heavy tails. Insurance:Mathematics and Economics,31 (2002):447-460.
[11]邓永录,梁之顺.随机点过程及其应用.北京：科学出版社,1998.
[12]Defersen F, Gerber H U. Risk theory for the compound Poisson process that is perturbed by diffusion[J].Insurance:Mathematics,1999,10:51-59.
[13]复旦大学数学系.数学分析(下册).北京：高等教育出版社,1978.
[14]Florin Avram, Miguel Usabel.Finite time ruin probabilities with one Laplace inversion. Insurance:Mathematics and Economics 32 (2003) 371-377.
[15]Grandell J. Aspect of Risk Theory[M].Springer-verlag, New York Inc,1991.
[16]Gerber H. U. On the probability of ruin in the presence of a linear dividend barrier. Scand Actuarial J,1981:105-115.
[17]Gerber H.U, E.S.W. Shiu. On the time value of ruin. North American Actuarial Journal,1998,2(1):48-78.
[18]Gerber H.U., E.S.W. Shiu. The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. Insurance:Mathematics and Economics,1997,21:129-137.
[19]Gerber H.U., E.S.W. Shiu. Methods for estimating the optimal dividend barrier and the probability of ruin. Insurance:Mathematics and Economics,2007:1-17.
[20]Grandell J. Doubly stochastic poisson processes. Berlin:Springer-Verlag,1976, 1-32.
[21]Grandell J. Some remarks on the ammeter risk process. Insurance:Mathematics and Economics,1995,16:31-39.
[22]Grandell J.Mixed poisson process. London:Chapman and Hall,1997.5-35.
[23]何树红,张茂.带干扰COX风险模型的讨论.云南民族大学学报(自然科学版),2005,14(2)：172-175.
[24]何树红,陈炎.常利率因素的双险种风险模型[J].云南大学学报(自然科学版),2004,26(6)：465-469.
[25]何树红,赵金娥,马丽娟.带干扰的双复合Poisson风险模型的破产概率.吉首大学学报(自然科学版),2005,26(3)：43-48.
[26]何树红,夏梓祥.再保险对时间盈余风险模型破产概率的影响[J].云南大学学报(自然科学版),2004,26(增刊)：1-4.
[27]何声武,汪嘉冈,严加安.半鞅与随机分析.北京：科学出版社,1995.
[28]Helene Cossette, David Landriault, Etienne Marceau. Ruin probabilities in the discrete time renewal risk model. Insurance:Mathematics and Economics 38 (2006) 309-323.
[29]Jun Cai, D.C.M.Dickson. Ruin probabilities with a Markov chain interest model. Insurance:Mathematics and Economics,35 (2004) 513-525.
[30]Jun Cai, D.C.M.Dickson. Upper bounds for ultimate ruin probabilities in the Sparre Andersen with interest. Insurance:Mathematics and Economics,32 (2003)61-71.
[31]Jun Cai, D.C.M.Dickson. On the expected discounted penalty function at ruin of surplus process with interest. Insurance:Mathematics and Economics 30 (2002):389-404.
[32]Jun Cai. Discrete time risk models under rates of interest. Pro. Eug. Inf. Sci. 2002,16:309-324.
[33]焦万堂,李俊海,刘再民.最优投保决策模型研究.信阳师范学院学报(自然科学版),2008(7),Vol.21.No.3.
[34]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[J].兰州大学学报(自然科学版),2004,40(4)：5-7.
[35]李尚友,张春生,吴荣.常利率环境下带干扰风险模型的破产估计.应用概率统计,2003,19(1)：79-84.
[36]刘家有.几类带利率风险模型的研究：[硕士论文].长沙：中南大学,2004.
[37]刘静.带干扰的风险模型的破产概率.[硕士论文].杭州：浙江大学,2006.
[38]龙国军.对三种再保险风险模型的破产概率以及破产赤字的研究：[硕士论文].长沙：中南大学,2007.
[39]林庆敏,汪荣明.带利息力下的更新风险模型下的破产概率的计算.华东师范大学学报(自然科学版),2005,3(1)：46-52.
[40]李贤平.概率论基础.北京：高等教育出版社,1997.
[41]马弛,张洪涛.再保险对带干扰Poisson风险模型破产时间的影响[J].安徽理工大学学报(自然科学版),2008,Vol.28.No.2.
[42]戚懿.广义复合Poisson模型下的破产概率.应用概率统计,1999,15(2)：141-146.
[43]钱敏平,龚光鲁.随机过程论.北京：北京大学出版社,1997.
[44]Rui M.R.Cardoso, Howard R.Waters. Recursive calculation of finite time ruin probabilities under interest force. Insurance:Mathematics and Economics, 33(2003)659-676.
[45]Ruud Brekelmans, Anja De Waegenaere. Approximating the finite-time ruin probability under interest force. Insurance:Mathematics and Economics,2001, 29:217-229.
[46]Robert. Elliott. Stochastic Calculus and Application. New York:Springer Verlag, 1982.
[47]孙立娟,顾岚.离散时间保险风险的破产问题.应用概率统计,2002,18(3)：293-299.
[48]Sundt B, Jozef L Teugels. Ruin estimate under interest force. Insurance: Mathematics and Economics,1995,16:7-22.
[49]田立芳.带干扰的保费随机收取的风险模型的破产概率.山东科技,2006,Vol.19.No.4.
[50]王过京,吴荣.带干扰经典风险在破产时刻的余额分布.南开大学出版社.1999,32(3)：2-25.
[51]王艳杰.超额赔款再保险中最优自留额的确定.科技咨询导报,2007,No.18.
[52]吴荣,杜勇宏.常利率下的更新风险模型.工程数学报,2002,19(1)：46-54.
[53]余晚霞,王汉兴.两险种Poisson风险模型和破产概率.上海大学学报(自然科学学报).2003,9(6)：529-532.
[54]杨善兵,司见东.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,Vol.30.No.1.
[55]杨步青,叶中行.保险公司的最优再保险和红利分配.系统工程.2000,18(6)：23-27.
[56]严加安.测度论讲义.北京：科学出版社,1997.
[57]张相虎,陈贵磊.带干扰的保费收取次数为Poisson过程的破产概率.山东科技大学学报(自然科学版),2005,24(1)：98-100.
[58]张茂军,南江霞,夏尊铨.再保险与有限时间破产概率.高校应用数学报,2007,22(4)：411-415.
[59]张相虎.带干扰的风险模型研究.[硕士论文].泰安：山东科技大学,2005.
[60]张奠宙,魏国强,阎革兴等.实变函数与泛函分析.北京：高等教育出版社,1983.
[61]张振中,胡玉玺,刘立华.保险投资策略与破产的关系.数学理论与应用,2006,Vol.26.No.4.
[62]张瑞英,张翠.多重超额损失再保险的破产问题.洛阳师范学院学报,2008年第2期.
[63]赵霞,刘锦萼.经典风险模型下带有随机利率的一类破产问题.高校应用数学学报A辑,2005,20(3)：313-319.
[2]柏晓明,李萍,李学锋.带干扰的双险种COX风险模型.华中科技大学学报(自然科学版),2005,33(2)：122-124.
[3]Cramer H. On the mathematical theory of risk. Stockholkm:Skandia Jubilee Volume,1930.
[4]Cramer H. On some questions connected with mathematical risk. University of California Paubl. Statistics,1954,2:99-123.
[5]Centeno L. Excess of loss reinsurance and the probability of ruin in finite horizon[J].Astin Bulletin,1997,27:59-70.
[6]C.Chi-Liang Tsai, GE.Willmot. On the moments of the surplus process perturbed by diffusion. Insurance:Mathematics and Economics, 31(2002)327-350.
[7]C.Chi-Liang Tsai. On the expectations of the present values of the surplus process perturbed by diffusion. Insurance:Mathematics and Economics, 32(2003)413-429.
[8]Chiu S.N. Yin. C.C. The time of ruin, the surplus prior to ruin and the deficit at ruin for the classical risk process perturbed by diffusion. Insurance: Mathematics and Economics,33(2003)59-66.
[9]David C. M. Dickson. Insurance Risk and Ruin. Cambridge University Press, 2005.
[10]D.Konstantinides, Qihe Tang, G.Tsitsiashvili. Estamates for the ruin probability in the classical risk model with constant interest force in the presence of heavy tails. Insurance:Mathematics and Economics,31 (2002):447-460.
[11]邓永录,梁之顺.随机点过程及其应用.北京：科学出版社,1998.
[12]Defersen F, Gerber H U. Risk theory for the compound Poisson process that is perturbed by diffusion[J].Insurance:Mathematics,1999,10:51-59.
[13]复旦大学数学系.数学分析(下册).北京：高等教育出版社,1978.
[14]Florin Avram, Miguel Usabel.Finite time ruin probabilities with one Laplace inversion. Insurance:Mathematics and Economics 32 (2003) 371-377.
[15]Grandell J. Aspect of Risk Theory[M].Springer-verlag, New York Inc,1991.
[16]Gerber H. U. On the probability of ruin in the presence of a linear dividend barrier. Scand Actuarial J,1981:105-115.
[17]Gerber H.U, E.S.W. Shiu. On the time value of ruin. North American Actuarial Journal,1998,2(1):48-78.
[18]Gerber H.U., E.S.W. Shiu. The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. Insurance:Mathematics and Economics,1997,21:129-137.
[19]Gerber H.U., E.S.W. Shiu. Methods for estimating the optimal dividend barrier and the probability of ruin. Insurance:Mathematics and Economics,2007:1-17.
[20]Grandell J. Doubly stochastic poisson processes. Berlin:Springer-Verlag,1976, 1-32.
[21]Grandell J. Some remarks on the ammeter risk process. Insurance:Mathematics and Economics,1995,16:31-39.
[22]Grandell J.Mixed poisson process. London:Chapman and Hall,1997.5-35.
[23]何树红,张茂.带干扰COX风险模型的讨论.云南民族大学学报(自然科学版),2005,14(2)：172-175.
[24]何树红,陈炎.常利率因素的双险种风险模型[J].云南大学学报(自然科学版),2004,26(6)：465-469.
[25]何树红,赵金娥,马丽娟.带干扰的双复合Poisson风险模型的破产概率.吉首大学学报(自然科学版),2005,26(3)：43-48.
[26]何树红,夏梓祥.再保险对时间盈余风险模型破产概率的影响[J].云南大学学报(自然科学版),2004,26(增刊)：1-4.
[27]何声武,汪嘉冈,严加安.半鞅与随机分析.北京：科学出版社,1995.
[28]Helene Cossette, David Landriault, Etienne Marceau. Ruin probabilities in the discrete time renewal risk model. Insurance:Mathematics and Economics 38 (2006) 309-323.
[29]Jun Cai, D.C.M.Dickson. Ruin probabilities with a Markov chain interest model. Insurance:Mathematics and Economics,35 (2004) 513-525.
[30]Jun Cai, D.C.M.Dickson. Upper bounds for ultimate ruin probabilities in the Sparre Andersen with interest. Insurance:Mathematics and Economics,32 (2003)61-71.
[31]Jun Cai, D.C.M.Dickson. On the expected discounted penalty function at ruin of surplus process with interest. Insurance:Mathematics and Economics 30 (2002):389-404.
[32]Jun Cai. Discrete time risk models under rates of interest. Pro. Eug. Inf. Sci. 2002,16:309-324.
[33]焦万堂,李俊海,刘再民.最优投保决策模型研究.信阳师范学院学报(自然科学版),2008(7),Vol.21.No.3.
[34]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[J].兰州大学学报(自然科学版),2004,40(4)：5-7.
[35]李尚友,张春生,吴荣.常利率环境下带干扰风险模型的破产估计.应用概率统计,2003,19(1)：79-84.
[36]刘家有.几类带利率风险模型的研究：[硕士论文].长沙：中南大学,2004.
[37]刘静.带干扰的风险模型的破产概率.[硕士论文].杭州：浙江大学,2006.
[38]龙国军.对三种再保险风险模型的破产概率以及破产赤字的研究：[硕士论文].长沙：中南大学,2007.
[39]林庆敏,汪荣明.带利息力下的更新风险模型下的破产概率的计算.华东师范大学学报(自然科学版),2005,3(1)：46-52.
[40]李贤平.概率论基础.北京：高等教育出版社,1997.
[41]马弛,张洪涛.再保险对带干扰Poisson风险模型破产时间的影响[J].安徽理工大学学报(自然科学版),2008,Vol.28.No.2.
[42]戚懿.广义复合Poisson模型下的破产概率.应用概率统计,1999,15(2)：141-146.
[43]钱敏平,龚光鲁.随机过程论.北京：北京大学出版社,1997.
[44]Rui M.R.Cardoso, Howard R.Waters. Recursive calculation of finite time ruin probabilities under interest force. Insurance:Mathematics and Economics, 33(2003)659-676.
[45]Ruud Brekelmans, Anja De Waegenaere. Approximating the finite-time ruin probability under interest force. Insurance:Mathematics and Economics,2001, 29:217-229.
[46]Robert. Elliott. Stochastic Calculus and Application. New York:Springer Verlag, 1982.
[47]孙立娟,顾岚.离散时间保险风险的破产问题.应用概率统计,2002,18(3)：293-299.
[48]Sundt B, Jozef L Teugels. Ruin estimate under interest force. Insurance: Mathematics and Economics,1995,16:7-22.
[49]田立芳.带干扰的保费随机收取的风险模型的破产概率.山东科技,2006,Vol.19.No.4.
[50]王过京,吴荣.带干扰经典风险在破产时刻的余额分布.南开大学出版社.1999,32(3)：2-25.
[51]王艳杰.超额赔款再保险中最优自留额的确定.科技咨询导报,2007,No.18.
[52]吴荣,杜勇宏.常利率下的更新风险模型.工程数学报,2002,19(1)：46-54.
[53]余晚霞,王汉兴.两险种Poisson风险模型和破产概率.上海大学学报(自然科学学报).2003,9(6)：529-532.
[54]杨善兵,司见东.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,Vol.30.No.1.
[55]杨步青,叶中行.保险公司的最优再保险和红利分配.系统工程.2000,18(6)：23-27.
[56]严加安.测度论讲义.北京：科学出版社,1997.
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