The Research And Application Of Heavy-tailed Theory In Insurance Actuaries

Many rare events can be modeled as heavy-tailed random variable. Scholars round home and abroad got some perfect results of the asymptotic estimate for the finite-time ruin probability, for the case that the random variables of the claimsizes are real-valued with common heavy-tailed distribution function, which has been a hot topic of the current risk theory research. In this paper, heavy-tailed theory will be applied to risk model. The precise asymptotic estimate for the finite-time ruin probability is established in the renewal risk model under constant interest force most by the assumption that the random variables of the claimsizes are subexponential distributions. Main contents of this dissertation are as follows:In chapter 1, the background and main research work of this dissertation are introduced.In chapter 2, the clear description of heavy-tailed is given, and then the definitions and propositions of heavy-tailed subclasses will be introduced systematically, supported from a few assistance lemmas. Since the full class of heavy-tailed distribution appears to be too big, it is necessary to research their characters and relations, which have been applied to risk model in the following chapter.In chapter 3, the clear descriptions and basic assumptions of the classical risk model are firstly given. We mainly propose a new view to extend and discuss each basic structure of the classical risk model, such asct and S (t ). Then the representative models about the classical research of risk models are presented. At last, according to the principle of risk model constructed, the trend and major research focus are mentioned.In chapter 4, for the case that the random variables are independent and real-valued with common subexponential distribution function. An equivalent formula obtained by probability theory is applied to risk model getting a deduction. Then paying particular attention to the application to ruin theory after the renewal risk model under constant interest force is proposed. Different from the old proof, the same precise asymptotic estimate for the finite-time ruin probability is established in the renewal risk model under constant interest force most by applying the former obtained deduction.In chapter 5, under the assumption that the random variables of the claimsizes are negatively dependent with common distribution function. We will extend the lemma to an equivalent formula when the claimsizes are subexponential distributions. Then paying particular attention to the application to ruin theory after the renewal risk model under constant interest force is proposed. The same precise asymptotic estimate for the finite-time ruin probability is established in the renewal risk model under constant interest force most by applying the former obtained formula. The result shows that it is still right when the claimsizes are independent with common distribution function. So it illustrates that ruin probability is insensitive to the negatively dependent structure.In chapter 6, under the assumption that the individual net losses are bivariate upper tail independent, identically distributed random variables having a common distribution in the classS . First, by the properties of the classS and upper tail independent, the conclusion is promoted existed in other class, which will be used in the renewal risk model under constant interest force to study the asymptotic estimate for the finite-time ruin probability, when the claimsizes are upper tail independent with common subexponential distribution function. The result shows that it is still right when the claimsizes are independent with common distribution function.The last chapter gives a summary of the dissertation and some possible extensions of the present work.