Asymptotic Tail Behaviors In Renewal Risk Models With Dependence Structures

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 chapterChapter 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.