相依更新风险模型中的渐近尾行为
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摘要
对于集体风险理论的系统研究最早可追溯到Lundberg1903年完成的博士论文,在其中他首次提出了古典风险模型的概念。在这一经典模型中,保险公司的索赔具有随机的索赔额,索赔发生的时间间隔服从指数分布。另外,该模型还假设索赔额和时间间隔各自构成一列独立同分布的随机变量,而且这两列随机变量是相互独立的。因此保险公司的累计索赔(风险)过程是一个复合泊松过程。Cramer早期对此模型的发展和完善作出了卓越的贡献,所以此模型又被称为Cramer-Lundberg风险模型。
上世纪中叶,Sparre Andersen(1957)提出了更新风险模型,以此作为复合泊松模型的自然推广。在这一更广泛的风险模型中,索赔发生的时间间隔独立同分布于任意的分布函数(不一定是指数分布)。结合一些其它的模型因子(例如初始资产,保费收入等)和经济因子(例如利率,分红,赋税,投资回报等),更新风险模型可以很好的刻一非寿险业务,因此它一直以来被大量学者广泛的研究。但是,更新模型距离实际情况仍然相去甚远。主要原因在于,虽然更新模型已经放宽了对索赔间隔分布的要求,但它依然保留了很多不切实际的假设,比如索赔额的同分布性,索赔额之间的独立性,以及索赔额与其等待时间的独立性,其中对于各种随机变量之间的独立性假设尤为不现实。
受更新模型这些缺陷的启发,这篇博士论文主要致力于讨论几类带有相依结构的更新风险模型。我们研究的相依结构既包括索赔额之间的相依,又包括索赔额与其等待时间的相依。为书写简便和避免混淆,我们将这些推广的更新风险模型统称为“非标准”更新风险模型,而将原始的更新风险模型叫做“标准”更新风险模型。在带有相依结构的风险模型下,许多传统的处理方法(比如一些依赖独立平稳增量的方法)不再有效,所以在一般情况下几乎不可能对此类模型所涉及的问题得到精确结果。因此,我们要利用渐近尾概率的方法进行我们的研究。粗略来说,我们假设模型中的随机变量是重尾的,并设法得到我们感兴趣的随机过程的尾概率的精确渐近式,然后我们利用所得到的渐近公式,可以进一步研究一些重要的保险量,比如有限时间和无限时间的破产概率等。
在一些必要的符号说明和约定之后,我们在论文的第一章给出论文主体的纲要,从中读者可以了解到后续章节的一些要点。
第二章用于详细介绍一些重要的分布族及其基本性质。虽然这一章主要集中于重尾情况,但一些有紧密关联的轻尾分布族也被考虑在内。我们的讨论基本覆盖了所有重尾分布族,包括长尾分布族,次指数分布族,正规变化分布族等等。其中,次指数分布族被公认为是一类足够丰富的分布族,它基本涵盖了风险理论中所有常用的分布,比如Pareto分布,对数正态分布,重尾Weibull分布等等。作为次指数分布族的一个重要子族,正规变化分布族也值得深入的探讨,不仅因为它依然包含了很多流行的重尾分布(如Pareto分布,Burr分布,对数gamma分布,以及Student's t-分布),而且还因为它本身所具有的非常好的数学性质,这些性质通常能帮助我们得到很简洁明了的渐近公式。除了具体的原始定义之外,本章中我们还会给出各类分布族的一些基本而重要的性质。这些性质不仅有自身的理论价值,而且还将被后续章节频繁引用。
在第三章中,我们将探讨一类非标准更新风险模型的折现累计索赔的渐近尾行为。本章中,我们考虑的更新模型带有常数利率,索赔额服从相同的分布,但索赔额之间并不是完全独立的。我们假设索赔额分布属于广义正规变化族(ERV),并且索赔额具有两两渐近独立性(也即任意两个索赔额的联合分布的尾相对于边际分布的尾在渐近意义下是可以忽略的)。在这些假设下,我们得到了一个精确的渐近公式,此公式对所有固定的时间水平(包括无穷)都是成立的。进一步,我们利用此公式研究相应的破产概率,并证明了有限时间和无限时间破产概率和对应时间下的折现累计索赔的尾概率具有相同的渐近行为。
不同于第三章,第四章所研究的非标准更新模型除了带有常数利率外,每个索赔额与其等待时间具有相依关系。受最近Asimit和Badeseu 2010年发表的论文的启发,我们在每个索赔额与其等待时间之间引入一类回归型的相依结构。可以验证,这种相依结构被许多常用的二元copula所满足。我们致力于这种相依结构对折现累计索赔的渐近尾行为的影响。在索赔额分布是次指数分布的假设下,我们得到了一个对时间具有局部一致性的渐近公式,它在形式上成功的体现出相依结构的影响。当索赔额分布被限制在ERV族中时,我们可以证明所得到的渐近公式是整体一致的。如果保费率也是常数,那么我们的结果可以直接用于估计相应的有限时间和无限时间破产概率。值得一提的是,本章中所考虑的相依结构是一种全新的相依结构,而且它兼具实用性和数学上的可操作性。Asimit和Badeseu在其2010年的文章中首次提出此类相依结构,但他们只是在复合泊松模型的框架下做了些很简单的讨论。我们将它引入一般的更新模型,并对改进后模型的渐近尾行为进行了深入研究。另外,证明所得渐近公式对时间的一致性是本章中一项具有挑战性的工作,我们用了很多精细的技巧来实现它。
第五章可以看作是第四章的一个延伸,因为本章中考虑的非标准更新模型与第四章考虑的具有相同类型的相依结构。但是,与前面一直考虑常数利率下的折现累计索赔不同,本章中我们研究累计索赔在随机收益下的折现值。具体来说,我们允许保险公司投资金融资产,比如无风险债券和有风险的各种股票,而投资组合的价值过程由一个几何Levy过程描述。在索赔额分布属于ERV族的前提下,通过对Levy过程加一个限制条件,我们得到了一个关于累计索赔随机折现的尾概率的渐近公式,此公式对所有时间水平具有一致性,我们还进一步证明,此模型下的有限时间和无限时间破产概率也满足这个渐近公式。本章的原始灵感来自唐启鹤等人2010年的最新工作,他们在索赔额分布属于正规变化族的前提下,对标准更新模型(没有任何相依结构)讨论了相似的问题。因此,我们的工作将他们的结果推广到了具有ERV型索赔分布的非标准相依更新风险模型中。
与前面几章不同,在第六章中我们将做一些文献中很少有人考虑过的探索性的工作。考虑一个在随机经济环境中参与风险投资的保险公司,这样的保险公司暴露在两类风险之下:第一类是所谓的保险风险,主要由传统的保险业务引起；第二类是金融风险,由通货膨胀、股市震荡等随机经济事件导致。我们致力于研究两类风险的相互作用关系。保险公司的资产过程由一个离散的风险过程来刻画,其中保险风险表示为一段时期内公司的累计净损失,也即总索赔与总保费的差；金融风险表示为随机投资收益率的倒数。我们通过研究破产概率和累计风险量的渐近尾行为来进行风险分析。在两类风险没有相互控制关系的情况下,我们给出了一个统一的处理方法。假设两类风险的最大值服从强正规变化分布,我们对有限时间及无限时间得到了一些精确的渐近公式,而这些公式的形式都是两类风险的尾概率的线性组合。
The systematical study of collective risk theory was pioneered by Lundberg (1903), in which the so-called classical risk model was introduced for the first time. In this well-known framework, the claims of an insurance company, with random sizes, arrive successively by exponentially distributed inter-arrival times. In addition, it is assumed that both claim sizes and inter-arrival times form a sequence of independent and identically distributed (i.i.d.) random variables and the two sequences are mutually independent too. Hence, the process of the in-surer's aggregate claims (risk) is a compound Poisson process. This model is also called Cramer-Lundbcrg risk model due to the former's remarkable contributions towards it; see e.g. Cramer (1930,1955).
In the middle of last century, the renewal (Sparre Andersen) risk model was proposed by Andersen (1957) as a natural generalization of the compound Pois son risk model, In this extended model, the inter-arrival times are assumed to be independent and dentically distributed by an arbitrary distribution (not neces-sarily exponential). Being equipped with other modeling factors (such as initial wealth, premium incomes) and incorporated with some economic factors (such as interests, dividends, taxes, returns on investments), this model provides a good mechanism for describing non-life insurance business, and hence has been exten-sively investigated. However, it is obvious that the renewal risk model is also far from practical situations. The main reason is that, although the restriction on the distribution of inter-arrival times is released, it still remains many artificial assumptions, including the identically distributed claim sizes, the independence among claim sizes, and the independence between a. claim size and its waiting time (the inter-arrival time before the claim). Among them, the ones assuming com-plete independence among the involved random variables are especially unrealistic in almost all kinds of insurance.
Motivated by these defects of the renewal risk model, this doctoral thesis is mainly devoted to discuss some extensions of the renewal risk model with various dependence structures. The underlying dependence structures considered in our study range from the one among claim sizes to the one between every claim size and its waiting time. For simplicity and avoiding confusions, we shall collectively call these extended models as "nonstandard" renewal risk models and the original renewal risk model as the "standard" renewal risk model. Since most of existing methods (such as the ones requiring stationary and independent increments) will fail to work, it is usually impossible to obtain exact results in these extended risk models without independence assumptions. Therefore, we proceed our study uti-lizing the method of asymptotic tail probabilities. Roughly speaking, we assume that the distributions of random variables in our stochastic models are heavy-tailed, and focus on obtaining precise asymptotic formulas for tail probabilities of the stochastic processes we are interested. Then, applying such asymptotic re-sults, we further study some useful insurance quantities such as ruin probabilities within finite and infinite time horizons.
After some necessary preliminaries and notational conventions, we give an outline of the main body of this thesis in Chapter 1, through which one can realize the objectives and key points of the subsequent chapter
Chapter 2 is devoted to specifically introduce some important distribution classes as well as their elementary properties. Although this chapter is mainly concentrated on the heavy-tailed case, some closely related light-tailed distribu-tion classes are also taken into account. Our discussions cover the majority of heavy-tailed distribution classes, including the long-tailed distribution class, the subexponential class, the regular variation class, and so on. Among them, it is commonly acknowledged that the subexponential class is abundant enough to contain almost all commonly-used heavy-tailed distributions in risk theory, such as the Pareto distribution, the lognormal distribution, the heavy-tailed Weibull distribution, and so on. As an important subclass of the subexponential class, the regular variation class also deserves in-depth studies, not only because it still contains many popular heavy-tailed distributions (such as the Pareto, Burr, loggamma, and Student's t-distributions), but also its nice mathematical proper-ties which usually lead to explicit and simple asymptotic formulas. Besides the specific definitions, we also give some crucial and elementary properties of various distribution classes. These properties are not only interesting in their own right, but also frequently cited in the subsequent chapters.
In Chapter 3, we investigate the asymptotic tail behavior of discounted, ag-gregate claims for a nonstandard renewal risk model in which a. constant force of interest is introduced and the claim sizes are identically distributed but not nec-essarily independent. Assuming that the common claim-size distribution belongs to the class of extended regular variation (ERV) and that the claim, sizes are pair-wise asymptotically independent (i.e. the tail of the joint distribution of every two claim sizes is negligible in compare with the tails of the marginal distributions), we obtain a precise asymptotic formula, which holds for every fixed time horizon including the infinity. Furthermore, we apply the obtained asymptotic formula to derive the ruin probabilities, and proved that the ruin probability by a finite or infinite time and the tail probability of the discounted aggregate claims up to the same time have the same asymptotic behavior.
Instead of considering the dependence among claim sizes, Chapter 4 is de-voted to study a nonstandard renewal risk model with a constant force; of interest and a dependence structure between every claim size and its waiting time. Moti-vated by a recent work of Asimit and Badescu (2010), we introduce a dependence structure of regression type between every claim size and its waiting time. This dependence structure is fulfilled by many commonly-used bivariate copulas. We focus on determination of the impact of this dependence structure on the asymp-totic tail probability of discounted aggregate claims. Assuming that the claim-size distribution is subcxponential, we derive a precise locally uniform asymptotic formula, which quantitatively captures the impact of the dependence structure When the claim-size distribution is restricted to the class ERV, we show that this asymptotic formula is globally uniform. If the premium rate is also constant, our results can be straightforwardly translated in the form of finite and infinite-time ruin probabilities. It is worth mentioning that the dependence structure consid-ered in this chapter is really a, brand-new one, which possesses both mathematical tractability and practical relevance. Asimit and Badescu (2010) first proposed this dependence structure in their academic paper, but they only made some sim-ple discussions on the compound Poisson risk model. We introduce it into the standard renewal risk model and deeply study the asymptotic tail behavior of the resulting model. Moreover, proving the uniformities (for finite and infinite time horizons) of the obtained asymptotic formulas is a. challenging job in this chapter, and we apply many elaborate techniques to achieve it.
Chapter 5 can be regarded as a extension of Chapter 4, because the nonstan-dard renewal risk model studied in this chapter has the same type of dependence structure as in Chapter 4. However, in contrast to studying the discounted aggre-gate claims under a, constant force of interest, we consider the stochastic present value of aggregate claims. Specifically speaking, the insurance company is allowed to invest in financial assets such as risk-free bond and risky stocks, and the price process of its portfolio is described by a geometric Levy process. Through restrict-ing the claim-size distribution to the class ERV and imposing a constraint on the Levy process in terms of its Laplace exponent, we obtain for the tail probability of the stochastic present value of aggregate claims a precise asymptotic formula, which holds uniformly for all time horizons. Further, we prove that the corre- sponding finite-and infinite-time ruin probabilities also satisfy such asymptotic formula. The original idea of this chapter is from Tang et al. (2010), in which the similar problem was studied in the standard renewal risk model (without depen-dence, structures) with regularly varying claim sizes. Hence, our results greatly extend theirs to the time-dependent renewal risk model with claim sizes from the class ERV.
Unlike Chapters 3-5, in Chapter 6 we do some tentative work which was seldom considered in existing references. Consider an insurance company which is exposed to a stochastic economic environment consisting of two kinds of risk. The first kind, called insurance risk, is the traditional liability risk related to the insurance portfolio and the second kind, called financial risk, is the asset risk related to the investment portfolio. We focus on studying the interplay of the two kinds of risk. The insurer's wealth process is described in a discrete-time risk model in which the insurance risk is quantified as a real-valued random variable equal to the total amount of insurance claims less premiums within a. period and the financial risk as a positive random variable equal to the reciprocal of the periodic stochastic return rate. We conduct risk analysis on the insurance business through studying the asymptotic behaviors of the ruin probabilities and the tail probabilities of accumulated risk amounts. A unified treatment is given in the sense that no dominating relationship between the two kinds of risk is rcquirec Assuming that the maximum of the two risk variables follows a distribution of strongly regular variation, we derive some precise asymptotic formulas for both finite and infinite time horizons, all in. the form of linear combinations of the tail probabilities of the two risk variables.
上世纪中叶,Sparre Andersen(1957)提出了更新风险模型,以此作为复合泊松模型的自然推广。在这一更广泛的风险模型中,索赔发生的时间间隔独立同分布于任意的分布函数(不一定是指数分布)。结合一些其它的模型因子(例如初始资产,保费收入等)和经济因子(例如利率,分红,赋税,投资回报等),更新风险模型可以很好的刻一非寿险业务,因此它一直以来被大量学者广泛的研究。但是,更新模型距离实际情况仍然相去甚远。主要原因在于,虽然更新模型已经放宽了对索赔间隔分布的要求,但它依然保留了很多不切实际的假设,比如索赔额的同分布性,索赔额之间的独立性,以及索赔额与其等待时间的独立性,其中对于各种随机变量之间的独立性假设尤为不现实。
受更新模型这些缺陷的启发,这篇博士论文主要致力于讨论几类带有相依结构的更新风险模型。我们研究的相依结构既包括索赔额之间的相依,又包括索赔额与其等待时间的相依。为书写简便和避免混淆,我们将这些推广的更新风险模型统称为“非标准”更新风险模型,而将原始的更新风险模型叫做“标准”更新风险模型。在带有相依结构的风险模型下,许多传统的处理方法(比如一些依赖独立平稳增量的方法)不再有效,所以在一般情况下几乎不可能对此类模型所涉及的问题得到精确结果。因此,我们要利用渐近尾概率的方法进行我们的研究。粗略来说,我们假设模型中的随机变量是重尾的,并设法得到我们感兴趣的随机过程的尾概率的精确渐近式,然后我们利用所得到的渐近公式,可以进一步研究一些重要的保险量,比如有限时间和无限时间的破产概率等。
在一些必要的符号说明和约定之后,我们在论文的第一章给出论文主体的纲要,从中读者可以了解到后续章节的一些要点。
第二章用于详细介绍一些重要的分布族及其基本性质。虽然这一章主要集中于重尾情况,但一些有紧密关联的轻尾分布族也被考虑在内。我们的讨论基本覆盖了所有重尾分布族,包括长尾分布族,次指数分布族,正规变化分布族等等。其中,次指数分布族被公认为是一类足够丰富的分布族,它基本涵盖了风险理论中所有常用的分布,比如Pareto分布,对数正态分布,重尾Weibull分布等等。作为次指数分布族的一个重要子族,正规变化分布族也值得深入的探讨,不仅因为它依然包含了很多流行的重尾分布(如Pareto分布,Burr分布,对数gamma分布,以及Student's t-分布),而且还因为它本身所具有的非常好的数学性质,这些性质通常能帮助我们得到很简洁明了的渐近公式。除了具体的原始定义之外,本章中我们还会给出各类分布族的一些基本而重要的性质。这些性质不仅有自身的理论价值,而且还将被后续章节频繁引用。
在第三章中,我们将探讨一类非标准更新风险模型的折现累计索赔的渐近尾行为。本章中,我们考虑的更新模型带有常数利率,索赔额服从相同的分布,但索赔额之间并不是完全独立的。我们假设索赔额分布属于广义正规变化族(ERV),并且索赔额具有两两渐近独立性(也即任意两个索赔额的联合分布的尾相对于边际分布的尾在渐近意义下是可以忽略的)。在这些假设下,我们得到了一个精确的渐近公式,此公式对所有固定的时间水平(包括无穷)都是成立的。进一步,我们利用此公式研究相应的破产概率,并证明了有限时间和无限时间破产概率和对应时间下的折现累计索赔的尾概率具有相同的渐近行为。
不同于第三章,第四章所研究的非标准更新模型除了带有常数利率外,每个索赔额与其等待时间具有相依关系。受最近Asimit和Badeseu 2010年发表的论文的启发,我们在每个索赔额与其等待时间之间引入一类回归型的相依结构。可以验证,这种相依结构被许多常用的二元copula所满足。我们致力于这种相依结构对折现累计索赔的渐近尾行为的影响。在索赔额分布是次指数分布的假设下,我们得到了一个对时间具有局部一致性的渐近公式,它在形式上成功的体现出相依结构的影响。当索赔额分布被限制在ERV族中时,我们可以证明所得到的渐近公式是整体一致的。如果保费率也是常数,那么我们的结果可以直接用于估计相应的有限时间和无限时间破产概率。值得一提的是,本章中所考虑的相依结构是一种全新的相依结构,而且它兼具实用性和数学上的可操作性。Asimit和Badeseu在其2010年的文章中首次提出此类相依结构,但他们只是在复合泊松模型的框架下做了些很简单的讨论。我们将它引入一般的更新模型,并对改进后模型的渐近尾行为进行了深入研究。另外,证明所得渐近公式对时间的一致性是本章中一项具有挑战性的工作,我们用了很多精细的技巧来实现它。
第五章可以看作是第四章的一个延伸,因为本章中考虑的非标准更新模型与第四章考虑的具有相同类型的相依结构。但是,与前面一直考虑常数利率下的折现累计索赔不同,本章中我们研究累计索赔在随机收益下的折现值。具体来说,我们允许保险公司投资金融资产,比如无风险债券和有风险的各种股票,而投资组合的价值过程由一个几何Levy过程描述。在索赔额分布属于ERV族的前提下,通过对Levy过程加一个限制条件,我们得到了一个关于累计索赔随机折现的尾概率的渐近公式,此公式对所有时间水平具有一致性,我们还进一步证明,此模型下的有限时间和无限时间破产概率也满足这个渐近公式。本章的原始灵感来自唐启鹤等人2010年的最新工作,他们在索赔额分布属于正规变化族的前提下,对标准更新模型(没有任何相依结构)讨论了相似的问题。因此,我们的工作将他们的结果推广到了具有ERV型索赔分布的非标准相依更新风险模型中。
与前面几章不同,在第六章中我们将做一些文献中很少有人考虑过的探索性的工作。考虑一个在随机经济环境中参与风险投资的保险公司,这样的保险公司暴露在两类风险之下:第一类是所谓的保险风险,主要由传统的保险业务引起；第二类是金融风险,由通货膨胀、股市震荡等随机经济事件导致。我们致力于研究两类风险的相互作用关系。保险公司的资产过程由一个离散的风险过程来刻画,其中保险风险表示为一段时期内公司的累计净损失,也即总索赔与总保费的差；金融风险表示为随机投资收益率的倒数。我们通过研究破产概率和累计风险量的渐近尾行为来进行风险分析。在两类风险没有相互控制关系的情况下,我们给出了一个统一的处理方法。假设两类风险的最大值服从强正规变化分布,我们对有限时间及无限时间得到了一些精确的渐近公式,而这些公式的形式都是两类风险的尾概率的线性组合。
The systematical study of collective risk theory was pioneered by Lundberg (1903), in which the so-called classical risk model was introduced for the first time. In this well-known framework, the claims of an insurance company, with random sizes, arrive successively by exponentially distributed inter-arrival times. In addition, it is assumed that both claim sizes and inter-arrival times form a sequence of independent and identically distributed (i.i.d.) random variables and the two sequences are mutually independent too. Hence, the process of the in-surer's aggregate claims (risk) is a compound Poisson process. This model is also called Cramer-Lundbcrg risk model due to the former's remarkable contributions towards it; see e.g. Cramer (1930,1955).
In the middle of last century, the renewal (Sparre Andersen) risk model was proposed by Andersen (1957) as a natural generalization of the compound Pois son risk model, In this extended model, the inter-arrival times are assumed to be independent and dentically distributed by an arbitrary distribution (not neces-sarily exponential). Being equipped with other modeling factors (such as initial wealth, premium incomes) and incorporated with some economic factors (such as interests, dividends, taxes, returns on investments), this model provides a good mechanism for describing non-life insurance business, and hence has been exten-sively investigated. However, it is obvious that the renewal risk model is also far from practical situations. The main reason is that, although the restriction on the distribution of inter-arrival times is released, it still remains many artificial assumptions, including the identically distributed claim sizes, the independence among claim sizes, and the independence between a. claim size and its waiting time (the inter-arrival time before the claim). Among them, the ones assuming com-plete independence among the involved random variables are especially unrealistic in almost all kinds of insurance.
Motivated by these defects of the renewal risk model, this doctoral thesis is mainly devoted to discuss some extensions of the renewal risk model with various dependence structures. The underlying dependence structures considered in our study range from the one among claim sizes to the one between every claim size and its waiting time. For simplicity and avoiding confusions, we shall collectively call these extended models as "nonstandard" renewal risk models and the original renewal risk model as the "standard" renewal risk model. Since most of existing methods (such as the ones requiring stationary and independent increments) will fail to work, it is usually impossible to obtain exact results in these extended risk models without independence assumptions. Therefore, we proceed our study uti-lizing the method of asymptotic tail probabilities. Roughly speaking, we assume that the distributions of random variables in our stochastic models are heavy-tailed, and focus on obtaining precise asymptotic formulas for tail probabilities of the stochastic processes we are interested. Then, applying such asymptotic re-sults, we further study some useful insurance quantities such as ruin probabilities within finite and infinite time horizons.
After some necessary preliminaries and notational conventions, we give an outline of the main body of this thesis in Chapter 1, through which one can realize the objectives and key points of the subsequent chapter
Chapter 2 is devoted to specifically introduce some important distribution classes as well as their elementary properties. Although this chapter is mainly concentrated on the heavy-tailed case, some closely related light-tailed distribu-tion classes are also taken into account. Our discussions cover the majority of heavy-tailed distribution classes, including the long-tailed distribution class, the subexponential class, the regular variation class, and so on. Among them, it is commonly acknowledged that the subexponential class is abundant enough to contain almost all commonly-used heavy-tailed distributions in risk theory, such as the Pareto distribution, the lognormal distribution, the heavy-tailed Weibull distribution, and so on. As an important subclass of the subexponential class, the regular variation class also deserves in-depth studies, not only because it still contains many popular heavy-tailed distributions (such as the Pareto, Burr, loggamma, and Student's t-distributions), but also its nice mathematical proper-ties which usually lead to explicit and simple asymptotic formulas. Besides the specific definitions, we also give some crucial and elementary properties of various distribution classes. These properties are not only interesting in their own right, but also frequently cited in the subsequent chapters.
In Chapter 3, we investigate the asymptotic tail behavior of discounted, ag-gregate claims for a nonstandard renewal risk model in which a. constant force of interest is introduced and the claim sizes are identically distributed but not nec-essarily independent. Assuming that the common claim-size distribution belongs to the class of extended regular variation (ERV) and that the claim, sizes are pair-wise asymptotically independent (i.e. the tail of the joint distribution of every two claim sizes is negligible in compare with the tails of the marginal distributions), we obtain a precise asymptotic formula, which holds for every fixed time horizon including the infinity. Furthermore, we apply the obtained asymptotic formula to derive the ruin probabilities, and proved that the ruin probability by a finite or infinite time and the tail probability of the discounted aggregate claims up to the same time have the same asymptotic behavior.
Instead of considering the dependence among claim sizes, Chapter 4 is de-voted to study a nonstandard renewal risk model with a constant force; of interest and a dependence structure between every claim size and its waiting time. Moti-vated by a recent work of Asimit and Badescu (2010), we introduce a dependence structure of regression type between every claim size and its waiting time. This dependence structure is fulfilled by many commonly-used bivariate copulas. We focus on determination of the impact of this dependence structure on the asymp-totic tail probability of discounted aggregate claims. Assuming that the claim-size distribution is subcxponential, we derive a precise locally uniform asymptotic formula, which quantitatively captures the impact of the dependence structure When the claim-size distribution is restricted to the class ERV, we show that this asymptotic formula is globally uniform. If the premium rate is also constant, our results can be straightforwardly translated in the form of finite and infinite-time ruin probabilities. It is worth mentioning that the dependence structure consid-ered in this chapter is really a, brand-new one, which possesses both mathematical tractability and practical relevance. Asimit and Badescu (2010) first proposed this dependence structure in their academic paper, but they only made some sim-ple discussions on the compound Poisson risk model. We introduce it into the standard renewal risk model and deeply study the asymptotic tail behavior of the resulting model. Moreover, proving the uniformities (for finite and infinite time horizons) of the obtained asymptotic formulas is a. challenging job in this chapter, and we apply many elaborate techniques to achieve it.
Chapter 5 can be regarded as a extension of Chapter 4, because the nonstan-dard renewal risk model studied in this chapter has the same type of dependence structure as in Chapter 4. However, in contrast to studying the discounted aggre-gate claims under a, constant force of interest, we consider the stochastic present value of aggregate claims. Specifically speaking, the insurance company is allowed to invest in financial assets such as risk-free bond and risky stocks, and the price process of its portfolio is described by a geometric Levy process. Through restrict-ing the claim-size distribution to the class ERV and imposing a constraint on the Levy process in terms of its Laplace exponent, we obtain for the tail probability of the stochastic present value of aggregate claims a precise asymptotic formula, which holds uniformly for all time horizons. Further, we prove that the corre- sponding finite-and infinite-time ruin probabilities also satisfy such asymptotic formula. The original idea of this chapter is from Tang et al. (2010), in which the similar problem was studied in the standard renewal risk model (without depen-dence, structures) with regularly varying claim sizes. Hence, our results greatly extend theirs to the time-dependent renewal risk model with claim sizes from the class ERV.
Unlike Chapters 3-5, in Chapter 6 we do some tentative work which was seldom considered in existing references. Consider an insurance company which is exposed to a stochastic economic environment consisting of two kinds of risk. The first kind, called insurance risk, is the traditional liability risk related to the insurance portfolio and the second kind, called financial risk, is the asset risk related to the investment portfolio. We focus on studying the interplay of the two kinds of risk. The insurer's wealth process is described in a discrete-time risk model in which the insurance risk is quantified as a real-valued random variable equal to the total amount of insurance claims less premiums within a. period and the financial risk as a positive random variable equal to the reciprocal of the periodic stochastic return rate. We conduct risk analysis on the insurance business through studying the asymptotic behaviors of the ruin probabilities and the tail probabilities of accumulated risk amounts. A unified treatment is given in the sense that no dominating relationship between the two kinds of risk is rcquirec Assuming that the maximum of the two risk variables follows a distribution of strongly regular variation, we derive some precise asymptotic formulas for both finite and infinite time horizons, all in. the form of linear combinations of the tail probabilities of the two risk variables.
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