摘要
众所周知,风险理论是应用概率论的重要分支之一,它不但自身具有重要的理论研究价值,而且对金融保险中的实际工作具有一定的指导意义.而在风险理论中如何衡量保险公司风险的大小,即刻画破产概率的渐近性态已经成为目前保险公司和广大学者共同关注的核心问题之一。本文将对经典的sparre-Andersen风险模型,以及一些与金融保险业息息相关的复杂化了的非经典模型,研究相应破产概率的渐近性问题.其中既包括当初始资本趋于无穷时,各种风险模型下破产概率的渐近结果,也有经典的sparre-Andersen风险模型中,当初始资本固定时,有限时破产概率的渐近性结果。
在一些非经典的风险模型中,作为主要对象的索赔额过程,它们之间不必是相互独立的,如可以是某种负相依关系或其它的相依关系.相应地,索赔间隔时间过程也可以不必相互独立。但我们仍然要求索赔额过程与索赔间隔时间过程彼此是相互独立的。尽管本文研究了各种经典与非经典的风险模型,但它们都有两点共同之处。
其一是,每个索赔额来到时,造成保险公司的一个净损失的分布都是重尾的,特别是次指数的或有控制尾分布的。在保险业,特别是财产保险业中,许多重大的风险都是由一个(或一些)大额索赔造成的,它们的分布只能是重尾的而不是轻尾的。因此,本文将重尾风险模型作为自己的主要研究对象。
其二是,各种破产概率渐近性的研究,与极限理论中的大偏差理论,随机游动理论及分布理论有密切的关系。因此,本文将它们当作重尾风险理论研究的主要工具。反之,风险理论中的一些实际问题,也对上述三个理论研究提出了更进一步的要求。
根据研究内容,我们将本文分为如下五章。
第一章介绍了本文中常用的记号,约定和概念,它们是分布理论,随机游动理论和风险理论中的主要对象。
第二章我们给出了Baltrunas等(2004,a)一个精致大偏差结果的完整证明(在他们的证明中缺少了一个不可或缺环节的证明),修正了Baltrunas(2001)一个随机游动结果的证明(在他们的证明中使用了一个错误的并被多人使用的一个等式)。在此基础上,我们得到了经典的Sparre-Andersen风险模型中,带固定初始资本的有限时破产概率的渐近性。此外,我们还给出了一些NA和独立双边控制尾分布族随机变量的精致大偏差结果。
第三章我们给出了一些粗略大偏差的结果,从而得到了破产概率的一些粗略渐近结果。虽然这些粗略渐近结果不如精致渐近结果理想,但是它们要求的条件更弱,从而具有更广泛的应用前景。
第四章我们通过对随机游动中相关问题的研究,得到了红利干扰模型下,无限时破产概率的渐近性结果。与Robert(2005)的相应结果相比,我们使用了不同的方法,在较弱的条件下,避开了该文证明中的一个含混部分,得到了严格证明的结果。
第五章我们考虑了两类相依的风险模型,得到了这两类风险模型下无限时破产概率的渐近性。其中一类风险模型的索赔额过程是被某个背景过程调节的,另一类的索赔额过程是负上象限相依的,两类风险模型的索赔间隔时间过程均只要求负上象限相依。
It is well known that risk theory is one of the most important branches of applied probability. It not only is much valuable in its own theoretical research, but also plays an important role in many applications in the fields of" finance and insurance. And in risk theory, how to measure the risk of an insurance company, i.e. how to describe the asymptotic behavior for the ruin probability, has caused wide concern by many insurance companies and scholars.
In this paper, we will deal with the classical Sparre-Andersen risk model and some other more complicated non-classical risk models related to insurance and finance, and study some topics on the asymptotic behavior for the corresponding ruin probabilities. This paper not only estimates ruin probabilities in some practical risk models, as the initial capital tends to infinity, but also investigates the finite-time ruin probability with a fixed initial reserve in the classical Sparre-Andersen risk model.
In some realistic risk models, the claim size process, which is the main object, is not necessarily mutually independent. It can be a kind of negatively dependent or other dependent process. Correspondingly, the inter-arrival time process is also not necessarily independent. But we still assume that the inter-arrival time process is independent of the claim size process. Although in this paper we investigate the classical risk model as well as some non-classical models, these risk models have two common points.
Firstly, the distribution of the net loss caused by a claim is heavy-tailed, especially subexponential or with dominatedly varying tails. In insurance, especially property insurance, many great risks are always caused by one (or some) large-amount claim(s). Their distributions, consequently, are not light-tailed, but heavy-tailed. Heavy-tailed distri- butions are well recognised as standard models for individual claim sizes. Therefore, heavy-tailed risk models are the main research object of this paper.
Secondly, all kinds of researches on the asymptotics for ruin probabilities are much related to large deviation theory, random walk theory and distribution theory. In this paper, we will use these tools to investigate heavy-tailed risk models. And conversely, some practical problems in risk theory require some further development of these three theories.
This paper is organized as the following five chapters: In Chapter 1, we introduce some notions, notation and basic concepts that we often use in the paper, which are also some main objects in distribution theory, random walk theory and risk theory.
In Chapter 2, after revising an important precise large deviation (whose proof, in Baltrūnas et al. (2004, a), seem to lack an essential part) and an asymptotic result for the first passage time of a random walk (whose proof, in Baltrūnas (2001), used an error equality), we study the classical Sparre-Andersen risk model, and obtain the asymptotics for the finite-time ruin probability with a fixed initial reserve. In this chapter, we also give some precise large deviations for NA and independent two-sided random variables with dominated varying tails.
In Chapter 3, we derive some rough large deviations results, with which we can get some rough estimates for ruin probabilities. Although these results are cruder than those of precise large deviations, they need some weaker conditions. So they also have a bright prospect of practical applications.
In Chapter 4, using the studies of some related topics on the random walk theory, we derive some estimate for the infinite-time ruin probability in dividend barrier models. Comparing with the corresponding results in Robert (2005) , we use some different methods, avoiding some obscure parts in the proof in , to strictly prove our results with some weaker conditions.
In Chapter 5, we establish some asymptotic relations for the infinite-time ruin probabilities of two kinds of dependent risk models. One risk model considers the claim sizes as a modulated process, and another deals with Negatively Upper Quadrant Dependent (NUQD) claim sizes. In these two risk models, the inter-arrival times are both assumed to be NUQD.
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