Large Deviation Of Poisson ARCH(1) Process And Its Application In Renewal Risk Model

The large deviation principle is an important tool for the study of rare events.Compared with the law of large numbers and the central limit theorem,the large deviation is a more accurate description of rare events.With the development and application of this theory,the large deviation theory has been applied to the study of finite-time bankruptcy probability estimation by domestic and foreign scholars,especially in the field of risk prediction.If the number of claims as a sequence of random variables is independently and identically distributed,the asymptotic form of finite-time bankruptcy probability can be obtained by using Cramér’s large deviation principle,but there are correlations among the sequences of claims in practical problems,which leads to less accurate risk prediction results.In order to extend the applicability of the large deviation principle to a certain extent,this thesis proves that the Poisson ARCH(1)process satisfies the large deviation principle and obtains the estimates of finite-time bankruptcy probability for the classical renewal risk model and the dependent stochastic premium risk model with the number of claims conforming to the Poisson ARCH(1)process,respectively,and performs an example calculation.The details are as follows.Firstly,the Poisson ARCH(1)process is proved to satisfy the large deviation principle.In this thesis,we set up a separation class based on the previous research and use the chunking method to prove the large deviation based on the setting of this separation class,and then obtain the large deviation form of Poisson ARCH(1)process by proving the exponentially tightness.Secondly,on the one hand,we use the above results to prove that the Poisson ARCH(1)process satisfies the assumption of precise large deviations and obtains the asymptotic probability of the classical renewal risk model;on the other hand,by introducing the dependent stochastic premium risk model,we prove that the Poisson ARCH(1)process satisfies the assumption of the renewal counting process proposed by Klüppelberg and Mikosch,and furthermore,we obtain the asymptotic probability of the classical renewal risk model.The estimation of finite-time bankruptcy probability of the dependent stochastic premium risk model is obtained.Finally,this thesis uses Matlab software to analyze the operating data of insurance companies,verifies that the number of claims of companies can be described using the Poisson ARCH(1)process,estimates the parameters in the Poisson ARCH(1)process using the empirical likelihood method,and estimates and compares the finite time ruin probability of the dependent stochastic premium risk model and the classical updated risk model;Based on the differences in the actual application process of the model,this thesis analyze its different application scenarios,summarize the advantages and disadvantages of the model,and further propose improvement directions for the future application of the model.