The Study Of Some Topics On Heavy-tailed Risk Models

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.