Research On Limit Properties Of The Nonclassical Risk Models

The risk theory,as an important branch in probability theory and mathematical statistics,has important significance to the safe operation of insurance companies Since Lundberg and Cramer established the well-known classical risk model,many researchers have made detailed study and carried out a practical promotion for the classical risk model,and obtained many nonclassical risk models that can better reflect the actual operation of insurance companies.In this thesis,under the condi-tion of heavy-tailed distributions,we discussed the Precise large deviation of limit theory of three kinds of non-classical risk models(delayed-claims risk model,based on the customer arrival risk model and based on policy entrance process risk mod-el)and the asymptotic property of ruin probability.The precise large deviation of the delayed-claims risk model is extended from the D∩L class of heavy-tailed dis-tributions to the larger S class of heavy-tailed distributions,and we not make any assumption on the dependence structure of claim sizes and inter-arrival times,and we construct a martingale to prove our result.The precise large deviation based on the customer arrival risk model is extended from one dimension to two dimensions,and the probability of actual claim is different for each policy,and the dependent structure between claims is expressed by the copula function.In the ruin probability based on the two-dimensional risk model of the entry process,the return on invest-ment is extended from the constant interest rate to the geometric Levy process,and the two counting processes of different businesses are subject to the bivariate renewal processThe precise large deviation and ruin probability,as important indicators to measure the risks for insurance companies,are conducive to insurance companies to make better decisions and reduce the risks in the business process,and provide an early warning.so as to avoid the riskThe contents of the thesis are as followsIn chapter O,some classical risk model and its extended non-classical risk mod-els firstly introduced.Then the research status of several nonclassical risk models are presented.The precise large deviation and ruin probability that this thesis will discuss finally expoundedIn chapter 2,we summarized the definition and properties of several kinds of heavy-tailed distributions class and their relations.At the same time,some depen-dent structures are givenIn chapter 3,the precise large deviations of the prospective-loss process partial sums and random sums of the delayed-claims risk models is discussed in the condition that the claim distribution belongs to S class,and the claim amount and the claim arrival time interval have some dependent structureIn chapter 4,the two-dimensional risk model based on customer arrival is dis-cussed.It is assumed that the potential claim amount Xi=(Xi,X2)T is a sequence of random vectors with independent and identical distribution,X1i and X2i are depen-dent,and the precise large deviation of partial sum and random sum of loss process is given under the C class of heavy-tailed distributionsIn chapter 5,the two-dimensional risk model based on entrance process is dis-cussed.Assume that the insurance company have two kinds of business,and all the assets is invested in the financial markets,its investment return follows geometric Levy process,among the claim-size of the same business are pairwise strong quasi-asymptotically independent.Under the condition of the claim distribution belong to D ∩ class,we got the asymptotic expression of the finite time ruin probabilityIn chapter 6,we summarized the work of the thesis and provided an outlook for further study.