几类风险模型破产问题的研究
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摘要
在风险理论的研究中,由于经典风险模型中许多假设与现实情况不完全相符,因此对现有模型进行更深入的研究于风险理论的发展有着重要的意义。
事实上,目前对经典的复合二项模型、Poisson模型和更新模型等三类基本风险模型的研究,依然存在理论上的很多不完善之处,如在调节系数不存在的情况下,这些模型的破产概率、破产前即刻盈余分布等特征都依然值得研究。虽然很多学者对经典风险模型从理论上进行了多方面的推广,但是对于这些推广的模型,还存在很多值得研究的问题。为此,本文试图对这些问题作进一步的研究。
本文首先对复合二项风险模型作了进一步的研究,在已有工作的基础上研究了破产概率的显示解问题以及在索赔分布服从重尾分布的条件下破产概率的局部渐近估计。
其次,对经典的Poisson风险模型作了进一步的研究,研究了调节系数不存在,索赔分布服从重尾分布的条件下破产概率渐近解及其局部解估计问题。
最后,对从在经典的Poisson风险模型基础上推广的Poisson-Geometric风险模型作了进一步研究,得到了其Gerber-Shiu折现惩罚函数所满足的更新方程及其破产概率的显示表达式(Pollazek-Khinchin公式),并且还得到了当模型的赔付随机变量服从指数分布时破产概率的表达式,当偏离系数为零时立即得到经典风险模型下的结论。
Since many assumptions of classical risk models are not in according with actual situations in risk theory. It is of great significance to study the models much further.
As matter of fact, some standing open problems still exist in the three basic risk models - the compound binomial risk model, the compound Poisson risk model and the renewal model. For example without adjustment coefficients, it is still devoteled to study ruin probability, distribution of the instant surplus before bankruptcy in the models. In addition, further inverstigations are also needed in these extended models, which are derived from classical risk models. This thesis aims to study futher the above problems.
We first investigate the compound binomial risk models.On the basic of works about ruin probability, we study its explicit solutions.And as for heavy-tailed claims, we have derived its asymptotic solutions.
Then the classical Poisson risk models are also considered with heavy-tailed claims and no adjustment coefficients. The asymptotic and local solutions of ruin probability are derived in the above assumptions.
At last, it is studied further for the Poisson-Geometric risk model which was extended from classical Poisson risk models. The renewal equation which the Gerber-Shiu discounted penalty function satisfier is presented, and the explicit expression (Pollazek-Khinchin formula) of ruin probability is also obtained. Moreover, ruin probability is derived when the loss stochastic variable obeys the exponential distribution. Especially, the obtained result is consistent with that of the classical risk model under departure coefficient is zero.
事实上,目前对经典的复合二项模型、Poisson模型和更新模型等三类基本风险模型的研究,依然存在理论上的很多不完善之处,如在调节系数不存在的情况下,这些模型的破产概率、破产前即刻盈余分布等特征都依然值得研究。虽然很多学者对经典风险模型从理论上进行了多方面的推广,但是对于这些推广的模型,还存在很多值得研究的问题。为此,本文试图对这些问题作进一步的研究。
本文首先对复合二项风险模型作了进一步的研究,在已有工作的基础上研究了破产概率的显示解问题以及在索赔分布服从重尾分布的条件下破产概率的局部渐近估计。
其次,对经典的Poisson风险模型作了进一步的研究,研究了调节系数不存在,索赔分布服从重尾分布的条件下破产概率渐近解及其局部解估计问题。
最后,对从在经典的Poisson风险模型基础上推广的Poisson-Geometric风险模型作了进一步研究,得到了其Gerber-Shiu折现惩罚函数所满足的更新方程及其破产概率的显示表达式(Pollazek-Khinchin公式),并且还得到了当模型的赔付随机变量服从指数分布时破产概率的表达式,当偏离系数为零时立即得到经典风险模型下的结论。
Since many assumptions of classical risk models are not in according with actual situations in risk theory. It is of great significance to study the models much further.
As matter of fact, some standing open problems still exist in the three basic risk models - the compound binomial risk model, the compound Poisson risk model and the renewal model. For example without adjustment coefficients, it is still devoteled to study ruin probability, distribution of the instant surplus before bankruptcy in the models. In addition, further inverstigations are also needed in these extended models, which are derived from classical risk models. This thesis aims to study futher the above problems.
We first investigate the compound binomial risk models.On the basic of works about ruin probability, we study its explicit solutions.And as for heavy-tailed claims, we have derived its asymptotic solutions.
Then the classical Poisson risk models are also considered with heavy-tailed claims and no adjustment coefficients. The asymptotic and local solutions of ruin probability are derived in the above assumptions.
At last, it is studied further for the Poisson-Geometric risk model which was extended from classical Poisson risk models. The renewal equation which the Gerber-Shiu discounted penalty function satisfier is presented, and the explicit expression (Pollazek-Khinchin formula) of ruin probability is also obtained. Moreover, ruin probability is derived when the loss stochastic variable obeys the exponential distribution. Especially, the obtained result is consistent with that of the classical risk model under departure coefficient is zero.
引文
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[4]Grandell J. Aspects of risk theory. New York:Springer-Verlag,1991
[5]Asmussen S. Ruin Probabilites. Singapore:World Scientific,2001
[6]Kluppelberg, C.Subexponential distributions and characterizations of related classes.Probab. Th.Rel. Fields.1989.82-259
[7]Gerber, H. U. Mathematical fun with the compound binomial process. ASTIN BULLETIN,1988,18:161-168.
[8]Shiu E.S.W.The probability of eventual ruin in the compound binomial model.ASTIN Bulletin,1989,19:179-180
[9]Willmot, G. E.Ruin probability in the compound binomial model. Insurance: Mathematics and Economics,1993,12:133-142
[10]Shixue Cheng, Wu Biao. The survival probability in finite time period in fully discrete risk model. Appl. Math-JCU,14B(1999):67-74
[11]Shixue Cheng, Hans U. Gerber & Elias S. W.Shiu, Discounted probabilities and ruin theory in the compound binomial model [J], Insurance:Mathematics and Economics,2000,26,239-250.
[12]Helene Cossette,et al. Compound binomial risk model in a markovian environment. Insurance:Mathematics and Economics,2004,35:425-443
[13]张茂军,南江霞.保费随机的复合二项风险模型的破产概率.科技通报,2005,21(3)：367-371
[14]Jiyang Tan, Xiangqun Yang. The compound binomial model with randomized decisions on paying dividends. Insurance:Mathematics and Economics,2006, 39:1-18
[15]柳向东,杨向群.两类离散风险模型的等价性.湖南师范大学学报,2001,24(4)：1-5
[16]成世学,朱仁栋.完全离散经典风险模型中的渐进解和Lundberg型不等式.高校应用数学学报,2001,16(3)：348-358
[17]龚日朝.复合二项风险模型研究.湖南师范大学硕士学位论文,2000
[18]龚日朝,杨向群.复合二项风险模型下有限时间内的生存概率.吉首大学学报,2000,(2)：16-19
[19]龚日朝,杨向群.复合二项风险模型下的生存概率.湘潭矿业学院学报,2000,(4)：82-87
[20]龚日朝,杨向群.复合二项风险模型的破产概率.经济数学,2001,18(2)：38-42
[21]龚日朝.两类离散索赔模型的关系,信阳师范学院学报,2001,14(2)：381-390
[22]龚日朝,杨向群.完全离散二项风险模型下有限时间内的生存概率.应用概率统计,2001,(17),4:431-436
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