几类相依的双险种风险模型破产问题研究
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摘要
自从经典的Cramer-Lundberg风险模型提出以来,许多人都对其进行了推广,但往往都蕴含着这样一种假定：保险公司中不同险种的索赔到达计数过程是相互独立的,即不同险种的理赔额是相互独立的；保费到达计数过程与索赔到达计数过程也是相互独立的,即是保费收入和理赔额是相互独立的随机变量。但是,在保险公司的实际经营中,由于竞争,利率,通货膨胀率以及随机干扰项等经济环境的影响,不同险种的索赔到达计数过程是相依的,保单到达计数过程与理赔到达计数过程也是相依的,根据这一实际情况,有必要为这类险种提供更符合客观实际的风险模型。本文建立并探讨了以下几种相依的双险种风险模型：
(1)讨论了常利率和通货膨胀率下一类索赔到达过程相依同时保费收入为复合Poisson过程的双险种风险模型的破产概率,推算了调节系数,破产概率之间的关系等问题。先将两个相依索赔总额转化为相互独立的索赔总额,然后利用鞅方法给出相应的Lundberg不等式。
(2)考虑了每次收取的保费均为独立同分布的随机变量,保费到达计数过程是Poisson过程,而索赔计数过程是其稀疏过程的双险种风险模型的生存概率问题,求出了生存概率满足的积分方程,并在指数分布的情况下求出了无限时间不破产概率的具体表达式。
(5)研究了保费率随机、保费收取过程是Poisson过程,而索赔计数过程是其稀疏过程的带干扰的双险种风险模型,并考虑了利率和通货膨胀率,讨论了其盈余过程的基本性质,强马氏性和鞅性,利用鞅证明了Lundberg不等式和最终破产概率的一般公式。
The classical Cramer-Lundberg risk model has been promoted by many people since it was proposed. While they are based on the independent assumptions, these are, the counting process in different types of insurance claims is independent of each other; counting process claims and premiums arrived at the counting process are assumed to be independent. In other words, the claims of different kinds of insurance are independent and the premiums and claims are assumed to be two indepe-ndent and identically distributed(iid) random variables series, anddifferent times of polices are independent of each other. In the managemen of the insurance company, because of economic impact of competition, interestrate, inflation rate and random interferemces, the counting processes in different types of insurance claims are dependent, counting process claims and premiums arrived at the counting process are also dependent. There is necessary for this type of grow situation to provide more objective and actual risk model nearly. In this thesis we build up and discuss several kinds of dependent double-type risk models:
(1) We study a correlated aggregate claims model,in which the arrival of the income of premium is a compound poisson process with constant interest and inflation rate.First we convert the two correlated aggregate claims to independent aggregate claims. Then we get Lundberg inequality by using martingle theory.
(2) We consider the Probability properties of a double-type risk model in which the rate of premium income is regarded as a random variable, the arrival of insurance policies is a poisson process and the process of claim occurring is thinning process. A integral equation for the survival probability is obtained. The explicit expression of the survival probability for the infinite interval is obtained in the special case of exponential distribution.
(3) We study the ruin probability problem of the double-type risk model perturbed in which the rate of premium income is regarded as a random variable, the arrival of insurance policies is a Poisson process and the process of claim occurring is thinning process. Using martingale method, the lundberg inequality and the common formula for the ruin probability are proved.
(1)讨论了常利率和通货膨胀率下一类索赔到达过程相依同时保费收入为复合Poisson过程的双险种风险模型的破产概率,推算了调节系数,破产概率之间的关系等问题。先将两个相依索赔总额转化为相互独立的索赔总额,然后利用鞅方法给出相应的Lundberg不等式。
(2)考虑了每次收取的保费均为独立同分布的随机变量,保费到达计数过程是Poisson过程,而索赔计数过程是其稀疏过程的双险种风险模型的生存概率问题,求出了生存概率满足的积分方程,并在指数分布的情况下求出了无限时间不破产概率的具体表达式。
(5)研究了保费率随机、保费收取过程是Poisson过程,而索赔计数过程是其稀疏过程的带干扰的双险种风险模型,并考虑了利率和通货膨胀率,讨论了其盈余过程的基本性质,强马氏性和鞅性,利用鞅证明了Lundberg不等式和最终破产概率的一般公式。
The classical Cramer-Lundberg risk model has been promoted by many people since it was proposed. While they are based on the independent assumptions, these are, the counting process in different types of insurance claims is independent of each other; counting process claims and premiums arrived at the counting process are assumed to be independent. In other words, the claims of different kinds of insurance are independent and the premiums and claims are assumed to be two indepe-ndent and identically distributed(iid) random variables series, anddifferent times of polices are independent of each other. In the managemen of the insurance company, because of economic impact of competition, interestrate, inflation rate and random interferemces, the counting processes in different types of insurance claims are dependent, counting process claims and premiums arrived at the counting process are also dependent. There is necessary for this type of grow situation to provide more objective and actual risk model nearly. In this thesis we build up and discuss several kinds of dependent double-type risk models:
(1) We study a correlated aggregate claims model,in which the arrival of the income of premium is a compound poisson process with constant interest and inflation rate.First we convert the two correlated aggregate claims to independent aggregate claims. Then we get Lundberg inequality by using martingle theory.
(2) We consider the Probability properties of a double-type risk model in which the rate of premium income is regarded as a random variable, the arrival of insurance policies is a poisson process and the process of claim occurring is thinning process. A integral equation for the survival probability is obtained. The explicit expression of the survival probability for the infinite interval is obtained in the special case of exponential distribution.
(3) We study the ruin probability problem of the double-type risk model perturbed in which the rate of premium income is regarded as a random variable, the arrival of insurance policies is a Poisson process and the process of claim occurring is thinning process. Using martingale method, the lundberg inequality and the common formula for the ruin probability are proved.
引文
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[2]张冕,高珊.一类索赔相依二元风险模型的破产概率问题研究。经济数学,2008,25(2)132—135.
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[4]蒋志明,王汉兴一类多险种风险过程的破产概率.应用数学与计算数学,2002,14(1)：9—16.
[5]程宗毛。破产时刻罚金折现期望值。应用概率统计,1999,15(3)：225—233
[6]刘再明,张飞涟,侯振挺.索赔为一般到达的保险风险模型.应用数学,2002,15(1)：35—39。
[7]H.U.Gerber.数学风险论导引(成世学,严颖).北京：世界图书出版公司,1997.
[8]S. N. Chin, C. C. Yin. The time of ruin,the surplus to ruin and the deficit at ruin for the classical risk process perturbed by diffusion。Insurance:Mathematics and Economic,2003,33:59-66
[9]Yanghailiang, Non-expoential bounds for ruin probability with interest effect included[J], Scand.Actuarial.J.,1(1999),66-79。
[10]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.
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[18]罗建华,方世祖.索赔为稀疏过程的风险模型[J]。广西科学.2004,11(4)：306—308。
[19]Wu Rong, Wei L i.The probability of ruin in a kind of Cox risk model with variable premium rate [J]。Scand Actuarial J,2004,2:121-132.
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[33]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— 54.
[34]Jun Cai, D.C.M.Dickson.Upper bounds for ultimate ruin Probabilities in the Sparre Andersen model with interest.Insurance:Mathematics and Eeonomics, 2003,32:61-71.
[35]Jun Cai, D. C. M.Dickson, On the expected discounted penalty function at ruin of a surplus process with interest. Insurance:Mathematics and Economics, 2002,30:389-404.
[36]Rongming Wang, HailiangYang, HanxingWang.On the distribution of Surplus immediately after ruin under interest force and subexponential caims. Insurance:Mathematics and Economics,2004,35:703-714.
[37]E. S. W.Shiu.The probability of eventual ruin in the compound binomial model. ASTIN Bulletin,1989,19:179-190.
[38]G.E.Willmot. Ruin probability in the compound binomial model. Insuranee: Mathematics and Economics 1993,12:133-142.
[39]Cheng, S.X., Wu, B.The survival probability in finite time period In fully discrete risk model.Appl. Math-JUC,1999,14(B)(1):67-74.
[40]龚日朝,杨向群.复合二项风险模型下的生存概率.吉首大学学报,2000,21(2)：16—19.
[41]龚日朝,刘永清.广义复合二项风险模型下的生存概率.湘潭大学自然科学学报,2001,23(2)：15—19.
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[44]David C. M. Dickson, Christian Hipp.Ruin probability for Erlang(2) risk processes.Insurance:Mathematics and Economic,1998,22:251-256.
[45]Dickson,D. C. M. Hipp,C. On the time to ruin for Erlang(2) risk process. Insurance:Mathematics and Economic,2001,29:333-344.
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[47]Sun. L.,Yang. H. On the joint distribution of surplus immediately before ruin and the deficit at ruin for Erlang(2) risk process. Insurance:Mathematics and Economic,2004,34:121-125.
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[2]张冕,高珊.一类索赔相依二元风险模型的破产概率问题研究。经济数学,2008,25(2)132—135.
[3]赵晓琴,刘再明,王国宝.一类索赔到达计数过程相依的二元风险模型[J]。数学的实践与认识,2006,36(2)：42—45.
[4]蒋志明,王汉兴一类多险种风险过程的破产概率.应用数学与计算数学,2002,14(1)：9—16.
[5]程宗毛。破产时刻罚金折现期望值。应用概率统计,1999,15(3)：225—233
[6]刘再明,张飞涟,侯振挺.索赔为一般到达的保险风险模型.应用数学,2002,15(1)：35—39。
[7]H.U.Gerber.数学风险论导引(成世学,严颖).北京：世界图书出版公司,1997.
[8]S. N. Chin, C. C. Yin. The time of ruin,the surplus to ruin and the deficit at ruin for the classical risk process perturbed by diffusion。Insurance:Mathematics and Economic,2003,33:59-66
[9]Yanghailiang, Non-expoential bounds for ruin probability with interest effect included[J], Scand.Actuarial.J.,1(1999),66-79。
[10]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.
[11]刘罗华,邹捷中。稀疏过程在双险种破产问题中的应用[J].株洲工学院学报。2006,20(4)：12—14.
[12]陈珊萍,王过京,王振羽。稀疏过程在保险公司破产问题中的应用[J].数理统计与管理。2001,20(5)：26—30.
[13]吴荣,杜勇红。常利率下的更新风险模型.工程数学学报,2002,19(1)：46—54。
[14]何声武.随机过程引论[M].北京：高等教育出版社.1996.
[15]龚日朝,杨向群.复合广义Poisson模型下破产概率估计。湘潭大学自然科学学报,2000,22(4)：23—27.
[16]张春生,吴荣.关于破产概率函数的可微性的注[J].应用概率统计.2001,17(3)：267—275.
[17]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)：55—72.
[18]罗建华,方世祖.索赔为稀疏过程的风险模型[J]。广西科学.2004,11(4)：306—308。
[19]Wu Rong, Wei L i.The probability of ruin in a kind of Cox risk model with variable premium rate [J]。Scand Actuarial J,2004,2:121-132.
[20]Gerber, H. U. The surplus process as a fair game-utilitywise。 ASTIN Bulletin 8, 1975,307-322.
[21]Petersen, S. S.,1990. Calculation of ruin probabilities when the premium depends on the current reserve.Scandinavian Actuarial Journal,147-159.
[22]孙立娟,顾岚.保险公司赔付及破产的随机模型与分析.数理统计与管理,1999,18(4)：25—30.
[23]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(自然科学版),2001,23(1)：55—57.
[24]龚日朝,广义复合Poisson风险模型下的破产概率.衡阳师范学院学报(自然科学版),2001,23(3)：76—80.
[25]向阳,刘再明.一类具有马氏调制费率的风险模型的破产概率.数学理论与应用,2003,23(1)：93—97.
[26]Grandell, J.Aspects of Risk Theory.Springer, NewYork,1991.
[27]Li juan Sun, Hai liangYang.On the joint distributions of surplus Immediately before ruin and the deficit at ruin for Erlang(2) risk Proeesses.Insuranee: Mathematics and Economics 2004,34:121-125.
[28]C.Chi-Liang Tsai, Li-juan Sun. On the discounted distribution functions for the Erlang(2) risk process.Insuranee:Mathematics and Economics2004,35:5-19.
[29]Shuanming Li, J.Garrido. On ruin for the Erlang(n) risk process.Insurance: Mathematics and Eeonomics 2004,34:391-408.
[30]Hai liangYang.Non-exponential bounds for ruin probability with interest effect included.[J] Scandinavian Actuarial Journal.1.1998.6-79.
[31]Vladimir Kalashnikov, Dimitrios Konstantinides. Ruin under interest force and subexponential claims:a simple treatment. Insurance:Mathematics and Economics,2000,27:145-149.
[32]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.
[33]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— 54.
[34]Jun Cai, D.C.M.Dickson.Upper bounds for ultimate ruin Probabilities in the Sparre Andersen model with interest.Insurance:Mathematics and Eeonomics, 2003,32:61-71.
[35]Jun Cai, D. C. M.Dickson, On the expected discounted penalty function at ruin of a surplus process with interest. Insurance:Mathematics and Economics, 2002,30:389-404.
[36]Rongming Wang, HailiangYang, HanxingWang.On the distribution of Surplus immediately after ruin under interest force and subexponential caims. Insurance:Mathematics and Economics,2004,35:703-714.
[37]E. S. W.Shiu.The probability of eventual ruin in the compound binomial model. ASTIN Bulletin,1989,19:179-190.
[38]G.E.Willmot. Ruin probability in the compound binomial model. Insuranee: Mathematics and Economics 1993,12:133-142.
[39]Cheng, S.X., Wu, B.The survival probability in finite time period In fully discrete risk model.Appl. Math-JUC,1999,14(B)(1):67-74.
[40]龚日朝,杨向群.复合二项风险模型下的生存概率.吉首大学学报,2000,21(2)：16—19.
[41]龚日朝,刘永清.广义复合二项风险模型下的生存概率.湘潭大学自然科学学报,2001,23(2)：15—19.
[42]伍超标.保险精算学基础.北京：中国统计出版社,1999.94—146.
[43]Dickson,D. C. M., doc Reis,A,D,E.Ruin probability and dual events. Insurance: Mathematics and Economic,1994,14:51-60.
[44]David C. M. Dickson, Christian Hipp.Ruin probability for Erlang(2) risk processes.Insurance:Mathematics and Economic,1998,22:251-256.
[45]Dickson,D. C. M. Hipp,C. On the time to ruin for Erlang(2) risk process. Insurance:Mathematics and Economic,2001,29:333-344.
[46]Cheng. Y. Tang. Q. Moment of the surplus before ruin and the deficit at ruin in the Erlang(2) risk process.North American Actuarial Journal,2003,7:1-12.
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