二维风险模型的破产概率的渐近性分析

发布时间：2018-11-08 18:23

【摘要】：众所周知,破产概率是精算数学及应用概率论的主要研究对象之一.因为在大多数场合,人们不容易算出破产概率的值,所以破产概率的渐近性研究显得格外重要,破产概率的渐近估计对风险管理有着重要的理论意义和应用价值.从Lundberg (1903)[69]时代一直到现在,破产概率的渐近理论已经成为一个非常活跃的研究领域.在研究的开始阶段,人们主要研究一维更新风险模型.但是在实际生活中,对于保险公司来说,只经营一种保险几乎是不可能的,因此研究多险种的多维更新风险模型因其具有更强的实际意义而被提上日程.多维更新风险模型的研究往往要比一维的情况更加复杂,计算更加繁琐,而且有时还需要解决一些新的数学问题.另一方面,人们发现一些高维更新风险模型与二维更新风险模型没有太大的区别.因此,本文将从以下三个方面研究二维更新风险模型的有限时破产概率的一致渐近理论. 首先,我们研究了不带利率的非标准的二维更新风险模型的有限时破产概率的一致渐近性.其中索赔额是独立同分布的随机变量其分布属于长尾分布族与控制变化尾分布族(见下文定义1.2和1.5)的交,索赔到达间隔时间满足广义负象限相依或宽象限相依结构(见下文定义1.8)的要求.在两种索赔同时到达的情况下,获得了破产概率的渐近公式,它在保险公司经营时间t∈[f(x),∞)上一致成立,这里.f(x)是任意一个无限递增的函数.这个结果使用了与Chen等(2011)[27]不同的证明方法,扩大了其分布族和相依结构的范围,并且削弱了其中的一些条件. 其次,我们研究了带利率和干扰的两种非标准的二维更新风险模型的有限时破产概率的一致渐近性.在这两种模型中,两类索赔分为同时到达和不同时到达两种情况,索赔额都满足上尾渐近独立的相依结构(见下文定义1.9),其分布也属于长尾分布族与控制变化尾分布族的交,索赔到达间隔时间满足宽下象限相依结构.对每一种情况,我们分别获得了三种在有限时段内的有限时破产概率的一致渐近估计.这些结果也使用了与Li等(2007)[65]和Bai和Song (2011)[12]不同的证明方法,并推广了Li等(2007)[65]的部分结果和Bai和Song (2011)[12]的结果. 最后,我们研究了一类时间相依的二维更新风险模型的有限时破产概率的一致渐近性.这里两种索赔同时到达,索赔额是独立同分布的随机变量,其分布属于次指数分布族(见下文定义1.3),并且同时到达的两种索赔额和它们的到达间隔时间具有某种相依关系.我们获得了有限时破产概率的一致渐近估计.这就将Asimit和Badescu (2010)[5],Li等(2010)[67]一维的结果在适当的条件下推广到二维的场合.可以看出,与前两项研究不同,该项研究中索赔额和它们的到达间隔时间之间的相依结构对破产概率的渐近估计有一定的影响. 从上述结果可以发现,本文处理的非标准更新模型有四个特点：有关随机变量的相依性,有关随机变量分布的重尾性,保险品种的多维性以及渐近估计的一致性.这里的一致性可以说明保险公司的初始资本的大小与保险公司的经营时间的长短无关.换言之,无论保险公司打算经营多少年,要控制风险都需要同样大小的初始资本.这些结果不仅丰富了二维更新风险模型的内容,而且在金融保险领域也有潜在的应用价值.
[Abstract]:As is well known, the probability of ruin is one of the main research objects of actuarial mathematics and application probability theory. Because in most cases, it is not easy to calculate the value of the probability of the ruin, the asymptotic behavior of the probability of the ruin is especially important, and the asymptotic estimation of the probability of the ruin is of great theoretical and practical value to the risk management. From the time of Lundberg (1903)[69], the asymptotic theory of the probability of bankruptcy has become a very active research field. In the beginning of the study, one-dimensional update risk model was mainly studied. But in the real life, it is almost impossible for the insurance company to run only one kind of insurance, so the multi-risk model of the multi-risk model is put on the schedule because it has more practical meaning. The multi-dimensional update risk model is often more complex than the one-dimensional case, and the calculation is more complicated, and some new mathematical problems need to be solved. On the other hand, it is found that some high-dimensional update risk models do not differ significantly from the two-dimensional update risk model. Therefore, this paper will study the consistent asymptotic theory of the probability of the two-dimensional update risk in the case of the finite-time ruin probability of the two-dimensional update risk model. First of all, we have studied the agreement of the non-standard two-dimensional update risk model with no interest rate, which is consistent with the probability of the bankruptcy. The claim amount is an independent and distributed random variable whose distribution belongs to the intersection of the long tail distribution family and the control change tail distribution family (see definitions 1.2 and 1.5 below). The time interval of the claim arrival meeting the broad negative quadrant dependent or wide-quadrant dependency structure (see definition 1. 8 below) In the case of the simultaneous arrival of the two claims, the asymptotic formula of the probability of the ruin is obtained, which is set up in the time t[f (x), b) of the insurance company's business time, here. f (x) is any infinite increment The result uses a different method of proof than Chen et al. (2011)[27], expanding the range of its distribution and dependent structures, and weakening some of them Conditions. Second, we have studied the finite-time ruin probability of two non-standard two-dimensional update risk models with interest rate and interference In these two models, the two types of claims are divided into two cases which arrive at the same time and do not arrive at the same time. The claim amount satisfies the dependent structure of the last-tail asymptotic independent. The distribution also belongs to the tail-tail distribution family and the control change tail. The intersection of the distribution family and the time interval between the claim arrival and the time of the claim meet the wide image. For each case, we obtain three of the three probability of bankruptcy in a limited period of time, respectively. An asymptotic estimation is made. These results also use a different method of proof from Li et al (2007)[65] and Bai and Song (2011)[12], and has extended some of the results of Li et al (2007)[65] and Bai and Song (2011)[1 2]. Finally, we have studied the finite-time bankruptcy of a class of time-dependent two-dimensional update risk model The consistent asymptotic behavior of the probability. The two claims arrive at the same time, the amount claimed is an independent and distributed random variable, the distribution of which belongs to the sub-exponential distribution family (as defined in the definition of 1.3 below), and both claimed amounts and their time of arrival With some sort of dependency. We've got a limited set of bankruptcy. The consistent asymptotic estimation of the rate. This results in the results of the one-dimensional (2010)[5], Li et al (2010)[67] one-dimensional results under the appropriate conditions It can be seen that, unlike the previous two studies, the dependence structure between the amount of the claim and the time of arrival of the study in the study is asymptotic to the probability of the ruin. It can be found from the above results that the non-standard update model processed in this paper has four characteristics: the dependence of the random variables, the retails of the distribution of the random variables, and the multi-dimensional of the insurance variety. The consistency of the performance and the asymptotic estimates. The consistency here can indicate the size of the insurance company's initial capital and the insurance policy In other words, regardless of how many years the insurance company intends to operate, the risk is to be controlled The same size of initial capital is required. These results not only enrich the content of the two-dimensional update risk model, but also in the financial insurance
【学位授予单位】：苏州大学
【学位级别】：博士
【学位授予年份】：2013
【分类号】：F840.3;O211.5

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