随机游动的渐近理论及在风险理论中的应用
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摘要
众所周知,随机游动是概率论中的一个重要研究对象.而在随机游动的研究中,随机游动的上确界及超出又是两个重要目标.它们在应用概率的很多领域,诸如排队系统,风险理论,分支过程,无穷可分分布等,都有重要应用.本文主要考虑在风险理论中的应用.事实上,如果用更新风险模型刻画一个保险公司的保险风险,那么保险公司的最终破产概率就是相应随机游动的上确界大于初始资本的尾分布,而局部破产概率就是该随机游动的超出在人们关心的某个区间上局部概率.最终破产概率和局部破产概率衡量了保险公司破产风险的程度,因而都是保险公司人员和理论工作者的重要研究对象.由于超出的大小反映了保险公司的亏损程度,因而在某种意义说,局部破产概率较最终破产概率有更大的理论意义和应用价值.
本文从如下三个方面对随机游动的上确界和超出的渐近理论及在保险中的应用进行研究.
首先,从随机游动研究的历史过程中可以发现,人们在研究随机游动上确界及超出的渐近性时,往往假设相关分布属于卷积等价分布族,并且很多结论都是以等价形式出现的.这就显的很完美.但实际问题就是,在卷积等价分布族之外还存在其他的分布,对这些分布如何去研究随机游动的相关量的渐近性.当然,我们不能对所有这样的分布去研究.本文则给出一类新的分布族,它可以包含轻尾卷积等价分布族,并且通过分布的γ变换及局部化方法刻画了这类轻尾分布族与一些重尾局部分布族之间有密切的内在关系.我们将对这一新的分布族讨论随机游动的上确界及超出的渐近性,所得结果可包含经典结果.
其次,我们讨论了随机游动超出的渐近性.主要考虑了相关分布不是卷积等价分布的情形,给出了随机游动超出的一致渐近性.作为风险理论中的应用,给出了更新风险模型下的局部破产概率的渐近估计.同时,利用更新方程的方法讨论了随机游动超出的局部渐近性的等价条件.
最后,在风险理论中,人们建立了各种不同的风险模型来刻画复杂的保险业务.在大部分模型中,人们总假设保险公司的索赔额及索赔来到时间间隔为独立的随机变量.但在很多实际问题中,索赔额与索赔来到时间间隔并不是独立的随机变量列,它们可能各自具有某种相依性.此时需要考虑相依模型.本文则提出了一类新的相依结构,在此相依结构下考虑带利率风险模型的有限时破产概率的渐近性,所得渐近性在有限时间内具有一致性.
It is well known that random walks are one of the most important objects in probability. In the studies of random walks, the supremum and overshoot of random walks are two important objects, which have some important applications in many fields of applied probability, such as queueing theory, risk theory, branching processes and infinite divisible distributions. This paper will consider applications in risk theory. Indeed, if one uses a renewal risk model to describe the insurance risk of an insur-ance company then, the ultimate ruin probability of the insurance company is a tail dsitribution at the initial capital reserve of the supremum of a random walk and the local ruin probability is the local probability of the overshoot of the random walk at an interval. The ultimate and local ruin probabilities measure the risk of an insurance company. Therefore, They are important objects, which the insuers and researchers pay more attention to. Since the overshoot reflects the degree of the deficit of an in-surance company, in a, sence, the local ruin probability has more important theoretical significance and applied value than the ultimate ruin probability.
We will investigate the asymptotic theory of random walks and applications in risk theory from the following three aspects.
Firstly, from the researching process of random walks, we find that in the studies of the asymptotics of the supremum and overshoot of random walks, one often sup-poses that the related distribution belongs to the convolution equivalent distribution class and the obtained results are often presented in a form of a series of equivalent conditions. This will be perfect. But in the realistic situation, there are some other distributions, which do not belong to the convolution equivalent distribution class. For these distributions, how to estimate the asymptotics of the related quantities of ran-dom walks? Of course, we can not discuss this problem for all these distributions. This paper will give a new distribution class, which can contain the light-tailed convolution equivalent distribution class, and use theγ-transform and localization of distributions to find the relation between this light-tailed dsitribution class and some heavy-tailed local distribution classes. We will investigate the asymptotics of the supremum and overshoot of random walks in this new distribution class. The obtained results can contain the classical result.
Secondly, we will discuss the asymptotics of the overshoot of random walks. We mainly consider the case that the related distribution is not the convolution equivalent distribution and give the uniform asyniptotics of the overshoot of random walks. As applications in risk theory, the asymptotic estimates of the local ruin probability in renewal risk model are presented. Meanwwhile, by using the renewal equations, the equivalent conditions of the local asymptotics of the overshoot of random walks are given.
Finally, in risk theory there are various risk models to deal with the complicated insurance risk. In most of models, one often supposes that the claim sizes and the claim inter-arrival times arc independent random variables, respectively. But in some realistic situations, they are not independent and they will have some dependence struc-tures. This paper will introduce a new dependence structure. Under this dependence structure, we will consider the asymptotics of the finite-time ruin probability of a risk model with a constant interest rate. The obtained asymptotics are uniform for time in a finite interval.
本文从如下三个方面对随机游动的上确界和超出的渐近理论及在保险中的应用进行研究.
首先,从随机游动研究的历史过程中可以发现,人们在研究随机游动上确界及超出的渐近性时,往往假设相关分布属于卷积等价分布族,并且很多结论都是以等价形式出现的.这就显的很完美.但实际问题就是,在卷积等价分布族之外还存在其他的分布,对这些分布如何去研究随机游动的相关量的渐近性.当然,我们不能对所有这样的分布去研究.本文则给出一类新的分布族,它可以包含轻尾卷积等价分布族,并且通过分布的γ变换及局部化方法刻画了这类轻尾分布族与一些重尾局部分布族之间有密切的内在关系.我们将对这一新的分布族讨论随机游动的上确界及超出的渐近性,所得结果可包含经典结果.
其次,我们讨论了随机游动超出的渐近性.主要考虑了相关分布不是卷积等价分布的情形,给出了随机游动超出的一致渐近性.作为风险理论中的应用,给出了更新风险模型下的局部破产概率的渐近估计.同时,利用更新方程的方法讨论了随机游动超出的局部渐近性的等价条件.
最后,在风险理论中,人们建立了各种不同的风险模型来刻画复杂的保险业务.在大部分模型中,人们总假设保险公司的索赔额及索赔来到时间间隔为独立的随机变量.但在很多实际问题中,索赔额与索赔来到时间间隔并不是独立的随机变量列,它们可能各自具有某种相依性.此时需要考虑相依模型.本文则提出了一类新的相依结构,在此相依结构下考虑带利率风险模型的有限时破产概率的渐近性,所得渐近性在有限时间内具有一致性.
It is well known that random walks are one of the most important objects in probability. In the studies of random walks, the supremum and overshoot of random walks are two important objects, which have some important applications in many fields of applied probability, such as queueing theory, risk theory, branching processes and infinite divisible distributions. This paper will consider applications in risk theory. Indeed, if one uses a renewal risk model to describe the insurance risk of an insur-ance company then, the ultimate ruin probability of the insurance company is a tail dsitribution at the initial capital reserve of the supremum of a random walk and the local ruin probability is the local probability of the overshoot of the random walk at an interval. The ultimate and local ruin probabilities measure the risk of an insurance company. Therefore, They are important objects, which the insuers and researchers pay more attention to. Since the overshoot reflects the degree of the deficit of an in-surance company, in a, sence, the local ruin probability has more important theoretical significance and applied value than the ultimate ruin probability.
We will investigate the asymptotic theory of random walks and applications in risk theory from the following three aspects.
Firstly, from the researching process of random walks, we find that in the studies of the asymptotics of the supremum and overshoot of random walks, one often sup-poses that the related distribution belongs to the convolution equivalent distribution class and the obtained results are often presented in a form of a series of equivalent conditions. This will be perfect. But in the realistic situation, there are some other distributions, which do not belong to the convolution equivalent distribution class. For these distributions, how to estimate the asymptotics of the related quantities of ran-dom walks? Of course, we can not discuss this problem for all these distributions. This paper will give a new distribution class, which can contain the light-tailed convolution equivalent distribution class, and use theγ-transform and localization of distributions to find the relation between this light-tailed dsitribution class and some heavy-tailed local distribution classes. We will investigate the asymptotics of the supremum and overshoot of random walks in this new distribution class. The obtained results can contain the classical result.
Secondly, we will discuss the asymptotics of the overshoot of random walks. We mainly consider the case that the related distribution is not the convolution equivalent distribution and give the uniform asyniptotics of the overshoot of random walks. As applications in risk theory, the asymptotic estimates of the local ruin probability in renewal risk model are presented. Meanwwhile, by using the renewal equations, the equivalent conditions of the local asymptotics of the overshoot of random walks are given.
Finally, in risk theory there are various risk models to deal with the complicated insurance risk. In most of models, one often supposes that the claim sizes and the claim inter-arrival times arc independent random variables, respectively. But in some realistic situations, they are not independent and they will have some dependence struc-tures. This paper will introduce a new dependence structure. Under this dependence structure, we will consider the asymptotics of the finite-time ruin probability of a risk model with a constant interest rate. The obtained asymptotics are uniform for time in a finite interval.
引文
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[3]Asmussen, S., Foss, S.. Korshunov, D.,2003. Asymptotics for sums of random variables with local sub exponential behavior. J. Theor. Probab.,16,489-518.
[4]Asmussen, S., Kalashnikov, V., Konstantinides, D., Kluppelberg, C., Tsitsiashvili G.,2002. A local limit theorem for random walk maxima with heavy tails. Statist. Probab. Lett.,56,399-404.
[5]Asmussen, S., Kluppelberg, C.,1996. Large deviations results for subexponential tails, with applications to insurance risk. Stochastic Process. Appl.,64,103-125.
[6]Bertoin, J., Doney, R. A.,1996. Some asymptotic results for transient random walks. Adv. Appl. Probab.,28,207-226.
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