Major Research Progress On Risk Model Under The Heavy-tailed Distribution And The Proofs Of Several Generalized Propositions

How to use bankruptcy probability to describe the risk in insurance industry has widely aroused people’s concern, especially when insurance company faces huge claim risk, the bankruptcy problems has become one of the hot topic in the risk research field. And the loss characteristics of huge claim risk are often described by using the heavy-tailed distribution theory. This thesis mainly discuses two aspects from the perspective of heavy-tail.On the one hand, this thesis summarizes the main development of research on ruin probabilities of several types of typical risk models under the condition of heavy-tailed distribution hypothesis. The concrete contents are as follows: The author firstly reviews the Lundberg- Cramér classical risk model and then lists its main research results; Secondly, the author gives the ordinary renewal risk model, the delayed renewal risk model, the equilibrium renewal risk model, Erlang( n, ?) risk model, the promotion delayed renewal risk model. And then some important conclusions about the ruin probability and partial survival probability are systematically discussed under the heavy-tailed distribution; Thirdly, the above models are further promoted as compound renewal risk model, delay compound renewal risk model, multiple delay compound renewal risk model, and then the ruin probability as well as partial survival probability of these models are discussed systematically under the heavy-tailed distribution condition; In the end, the author gives the findings established under heavy-tailed distribution with constant interest rate of risk model. These results provide meaningful reference for further study on some of the new risk model.On the other hand, considering the limitations of the classical heavy-tailed distribution theory on study of some model of modern insurance practice, the concept of balance distribution and the sub-classes of heavy-tailed distribution about L class and *S class are promoted, on the basis of which, some promoted propositions about heavy-tailed distribution are proved in the thesis. The main results are as follows. First, a local equivalent formula about the convolution of a certain distribution function and balance distribution. Second, a local upper bound about n-fold convolution of balance distribution. And these results have great significance for further studies on the actuarial index of some new risk model under heavy-tailed distribution.