The Probability Of Bankruptcy Based On The Heavy-tailed Claims

The risk theory is an important branch of probability theory, it not only has important theoretical research value, but also has some significance in the actual work in the finance and insurance. How to measure the size of the insurance company risk, that characterize the asymptotic behavior of the probability of bankruptcy, has become the reserch focus of the insurance company and some scholars. Because of large claims have a huge impact on the operating conditions of the insurance companies, catastrophe risk theory has been growing concern. In mathematics, heavy-tailed random variables can describe this risk. The classical Cramer-Lundberg model is assumed that the impact of interest rate does not ex-ist, but in the actual economic environment, interest rate is an important factor which is considered by insurance companies. This paper study the probability of bankruptcy in the case of heavy-tailed claims and constant interest rate.This article is divided into three chapters:The first chapter is an introduction, which is divided into two sections. The first section is the summary of the bankruptcy of theory. Firstly, we review the status of the bankruptcy of the theory, Then describes the bankruptcy in the situation of "big claims", that is the study of the bankruptcy in the heavy-tailed distribution. We show that the current heavy-tailed distribution has two main directions:first. we establish the local equivalence relation on the basis of the tail-equivalence. Second, we establish the tail-equivalence of the ruin probability. Next, we introduce briefly the status of these two research directions. Finally, we intruduce the prospect of the probability with heavy-tailed distributions. In the second section, we introduce the definition of the heavy-tailed distribution, then we show the sub-categories and the relationship of these sub-categories of the heavy-tailed distribution.The second chapter, we proved elementary probability of the asymptotic be-havior. By considering alter the update time that the first random walk over the horizontal position, we discussed the mean distribution of heavy-tailed random walk. In particular, the sub-sequence has a asymptotic probability under the approximation limit. The results comes from the renewal theorem. We assump-tion Eξ1=-a<0, And the tail distribution points FI(x) is the index, so we getIn the third chapter, we study the probability of bankruptcy under heavy-tailed claims and constant interest rates. When the claim amount distribution isεRV, we get the equivalence relation of ruin probability under a constant interest rate. First we establish a process with constant interest rate model as follow: On this basis, we prove the ruin probability varphi(x) satisfies: when the claim distribution F∈εRV (-α,-β).Then we get the asymptotic expression of the ruin probability.