混合负二项风险模型的研究
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摘要
本文对风险理论的研究与发展进行了概述,并详细综述了负二项风险模型在国内外的研究现状以及经典离散风险模型的组成部分和主要结果。在此基础上,针对目前保险业务逐渐复杂和细化的实际情况,提出了混合负二项风险模型,研究了此模型的破产参数,以期能够更真实更准确的反映保险公司的实际运营情况,便于保险公司做出统筹安排。
在保险业务各种复杂的问题中,保险人依照风险的某些特征对其进行分类,但是被划入同一类中的保单仍然不可避免的存在着某种程度的非同质性。因此,对于同一类保单组合的索赔次数模型的描述,首先要确定保单组合中各个保单的索赔次数模型,然后根据同一类保单中的非同质性,确定其模型中的参数分布规律,最后再完整地描述该保单组合的索赔次数模型。这就是一种混合索赔次数模型。
本文首先介绍了复合负二项风险模型的最终破产概率、有限时间内的生存概率、盈余首次和末次到达给定水平时刻的分布;接着将复合负二项风险模型中保费收取次数及其每次收取的保费都推广为随机变量,提出了混合双负二项风险模型,研究了该模型中盈余过程的数字特征、有限时间内的破产概率、最终破产概率等问题。鉴于目前保险实务中保险险种的逐渐增多,将混合双负二项风险模型进一步推广,提出了双险种的混合多负二项风险模型,研究了该模型中盈利过程的数字特征、最终破产概率、Lundberg不等式等问题。事实上,保险公司的运营过程中会时不时地出现一些随机因素,使得保险公司有些不确定的收益或者支出,为此,研究了带干扰的双险种混合多负二项风险模型的破产问题。
在上述每种模型下,都得到了相应的盈余序列的性质,即盈余过程是一个平稳的独立增量过程,并得到了破产概率的具体表达形式,尤其重要的是找到了破产概率的上界,即Lundberg不等式,其较强的可操作性在保险系统的风险分析中被广泛应用,具有重要的理论和实际意义!
This article has outlined the research and development of the risk theory, and summarized the negative binomial risk model in detail in the domestic and overseas present research situation as well as the constituent and the main results of the classics separate risk model. Based on this, in view of the gradually complex and specific situation on present actual insurance business, in this article I have proposed the mixed negative binomial risk model, and studied the bankrupt parameter of this model, so that the actual operation situation of insurance company can be reflected more truly and accurately, and it's helpful for the insurance company to make the overall plan arrangement.
In all kinds of complicated issues of the insurance business, the insurer carries on the classification according to the risk certain characteristics, but the chit in the same kind still is inevitable to exist some kind of degree the non-homogeneity. Therefore, regarding the claim number of times model's description of the identical kind of chit combination, firstly must determinate the claim number of times model of each chit in the definite policy combination, then in the basis identical kind of chit's non-homogeneity, determinate the parameter distribution rule in its model, finally describe the claim number of times model of this chit combination completely again. This is a mixed claim number of times model.
This article firstly introduced the final ruin probability of the compound negative binomial risk model, the survival probability in limited time, distribution in the horizontal time of the earnings in the first time and last time; then made the insurance premium collection times and each time gathers in the negative binomial risk model be random variable, proposed the mixed double negative binomial risk model, studied the digital characteristics of the earnings process in this model, the ruin probability in the limited time, final ruin probability and so on. In view of the gradual increases of the insurance types in the present insurance practice, I promoted the mixed double negative binomial risk model further, proposed the mixed multi-negative binomial risk model of double insurance types, and studied the digital characteristics of earning process in this model, final ruin probability, Lundberg inequality and so on. In fact, in the process of the insurance company's operation, there are some random factors once for a while, which makes insurance company have some uncertain incomes or the disbursements, so I also studied the bankrupt problems of the mixed multi-negative binomial risk model in double insurance types with disturbance items.
Under the above each kind of model, all are obtained the corresponding earnings sequence nature, namely the earnings process is a steady independent increment process, which obtained the concrete expression form of ruin probability, especially more importantly had found the upper boundary of ruin probability, that is the Lundberg inequality, it is widely applied in the insurance system's risk analysis for its strong feasibility, which has the important theoritical and the practical significance!
在保险业务各种复杂的问题中,保险人依照风险的某些特征对其进行分类,但是被划入同一类中的保单仍然不可避免的存在着某种程度的非同质性。因此,对于同一类保单组合的索赔次数模型的描述,首先要确定保单组合中各个保单的索赔次数模型,然后根据同一类保单中的非同质性,确定其模型中的参数分布规律,最后再完整地描述该保单组合的索赔次数模型。这就是一种混合索赔次数模型。
本文首先介绍了复合负二项风险模型的最终破产概率、有限时间内的生存概率、盈余首次和末次到达给定水平时刻的分布;接着将复合负二项风险模型中保费收取次数及其每次收取的保费都推广为随机变量,提出了混合双负二项风险模型,研究了该模型中盈余过程的数字特征、有限时间内的破产概率、最终破产概率等问题。鉴于目前保险实务中保险险种的逐渐增多,将混合双负二项风险模型进一步推广,提出了双险种的混合多负二项风险模型,研究了该模型中盈利过程的数字特征、最终破产概率、Lundberg不等式等问题。事实上,保险公司的运营过程中会时不时地出现一些随机因素,使得保险公司有些不确定的收益或者支出,为此,研究了带干扰的双险种混合多负二项风险模型的破产问题。
在上述每种模型下,都得到了相应的盈余序列的性质,即盈余过程是一个平稳的独立增量过程,并得到了破产概率的具体表达形式,尤其重要的是找到了破产概率的上界,即Lundberg不等式,其较强的可操作性在保险系统的风险分析中被广泛应用,具有重要的理论和实际意义!
This article has outlined the research and development of the risk theory, and summarized the negative binomial risk model in detail in the domestic and overseas present research situation as well as the constituent and the main results of the classics separate risk model. Based on this, in view of the gradually complex and specific situation on present actual insurance business, in this article I have proposed the mixed negative binomial risk model, and studied the bankrupt parameter of this model, so that the actual operation situation of insurance company can be reflected more truly and accurately, and it's helpful for the insurance company to make the overall plan arrangement.
In all kinds of complicated issues of the insurance business, the insurer carries on the classification according to the risk certain characteristics, but the chit in the same kind still is inevitable to exist some kind of degree the non-homogeneity. Therefore, regarding the claim number of times model's description of the identical kind of chit combination, firstly must determinate the claim number of times model of each chit in the definite policy combination, then in the basis identical kind of chit's non-homogeneity, determinate the parameter distribution rule in its model, finally describe the claim number of times model of this chit combination completely again. This is a mixed claim number of times model.
This article firstly introduced the final ruin probability of the compound negative binomial risk model, the survival probability in limited time, distribution in the horizontal time of the earnings in the first time and last time; then made the insurance premium collection times and each time gathers in the negative binomial risk model be random variable, proposed the mixed double negative binomial risk model, studied the digital characteristics of the earnings process in this model, the ruin probability in the limited time, final ruin probability and so on. In view of the gradual increases of the insurance types in the present insurance practice, I promoted the mixed double negative binomial risk model further, proposed the mixed multi-negative binomial risk model of double insurance types, and studied the digital characteristics of earning process in this model, final ruin probability, Lundberg inequality and so on. In fact, in the process of the insurance company's operation, there are some random factors once for a while, which makes insurance company have some uncertain incomes or the disbursements, so I also studied the bankrupt problems of the mixed multi-negative binomial risk model in double insurance types with disturbance items.
Under the above each kind of model, all are obtained the corresponding earnings sequence nature, namely the earnings process is a steady independent increment process, which obtained the concrete expression form of ruin probability, especially more importantly had found the upper boundary of ruin probability, that is the Lundberg inequality, it is widely applied in the insurance system's risk analysis for its strong feasibility, which has the important theoritical and the practical significance!
引文
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[2]孟生旺,袁卫.汽车保险的精算模型及其应用[J].数理统计与管理,2001,20(3):60-65
[3]熊福生.复合索赔次数模型与混合索赔次数模型中的若干结果[J].经济数学,2003,20(1):48-54
[4]熊福生.风险理论[M].2005年6月第一版.武汉:武汉大学出版社,2005,94-109
[5]Hans U.Gerber.数学风险论导引[M].北京:世界图书出版公司,1997,23-35
[6]陈贵磊.复合负二项风险模型研究:[山东科技大学硕士学位论文].山东:山东科技大学,2005,2-4
[7]王黎明,金珩.保险费收取次数为Poisson过程的破产概率[J].内蒙古师大学报自然科学(汉文)版,2000,29(3):176-179
[8]龚日朝,李凤军.双Poisson风险模型下的破产概率[J].湘潭师范学院学报(自然科学版),2001,23(1):55-57
[9]孙立娟,顾岚.保险公司赔付及破产的随机模拟与分析[J].数理统计与管理,1999,18(4):25-30
[10]高明美,赵明清,王建新.双二项风险模型的破产概率[J].经济数学,2004,21(1):6-9
[11]董英华,张汉君.带干扰的双Poisson模型的破产概率[J].数学理论与应用,2003,23(1):98-101
[12]李晋枝,乔克林,何树红.随机利率因素的破产模型[J].云南大学学报(自然科学版),2003,25(1):9-12
[13]高建伟,邱菀华.寿险中的破产理论及其应用[J].数理统计与管理,2002,21(5):12-16
[14]Gerber H U.An Extension of the Renewal Equation and its Application in the Collective theory of Risk[J].Scandinavian Actuarial Journal,1970,205-210
[15]Dufresne F,Gerber H U.Risk Theory for the Compound Poisson Process that is Peturbed by Difussion[J].Insurance:Mathematics and Economics,1991,67-70
[16]Grandell J.Aspects of Risk Theory[M].New York:Springer-Verlag,1991,102-115
[17]吴荣,王过京.带干扰经典风险过程在破产时刻的余额分布[J].南开大学学报(自然科学版),1999,32(3):25-29
[18]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
[19]蒋志明,王汉兴.一类多险种风险过程的破产概率[J].应用数学与计算数学学报,2000,14(1):9-16
[20]方大儿,王汉兴.多质负风险和[J].应用概率统计,2003,19(1):65-70
[21]杜雪樵,徐怀.带干扰的多险种的风险模型[J].运筹与管理,2003,12(6):55-57
[22]孟生旺.负二项分布的优良特性及其在风险管理中的应用[J].数理统计与管理,1998,17(2):9-12
[23]程维虎,王莉丽.负二项分布两种参数估计及其比较[LJ].数理统计与管理,2004,23(5):52-56
[24]牛燕影,王增富.负二项分布类的条件概率封闭性[J].数学理论与应用,2005,25(3):101-103
[25]田俊忠.负二项分布参数的UMVU估计及其渐近有效性[J].西北民族学院学报(自然科学版),2001,22(3):1-4,8
[26]戴朝寿,候艳艳.负超几何分布、负二项分布与Poisson分布之间的关系及其推广[J].南京大学学报:数学半年刊,2001,18(1):76-84
[27]杨雪梅.关于几何分布与巴斯卡分布的几个定理[J].咸阳师范学院学报,2003,18(2):9-10
[28]汤胜道,汪凤泉.负二项分布下参数的方差一致最小无偏估计及贝叶斯估计[J].安庆师范学院学报(自然科学版),2003,9(1):69-71
[29]陈贵磊,赵明清.保费收取次数为负二项随机序列的复合二项风险模型[J].山东科技大学学报,2006,25(1):85-86,92
[30]孙道德.负二项分布的性质及其应用[J].曲阜师范学院学报,2000,17(2):10-12
[31]陈飞跃.聚合风险模型累计损失分布研究[J].保险职业学院学报,2005,1(1):17-19
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