Large Deviations From Several Continuous-time Risk Models

The model construction in financial risk management is of great significance.In order to control risks better,it is necessary to optimize the classic model continuously.The main work of this paper is to study the asymptotic behavior of the risk models.We first describe the research and development of the classical risk models and their limit theory,then study the delayed multi-type-insurance compound risk model and the insurance claim model based on the Cox process,and discuss the risk processes of these models.Because the large deviation theory can quantify extreme claim problems,we focus our work on the the large deviation principle of these risk processes.This article is divided into the following sections:In the first chapter,the main results of the Cramer-Lundberg classical risk model and its asymptotic estimation are presented firstly.Then the development of the limit theory of the risk model is introduced.Thirdly,the results of the research on the limit properties of several types of insurance claim models are presented.Finally,the main research achievements of this paper are described.The second chapter mainly introduces the relevant basic knowledge of this article.Some basic concepts and key theorems,including Poisson shot noise process,Cox process,large deviation theory,moderate deviation theory,Gartner-Ellis theorem and the definitions and basic properties of martingale are introduced.Chapters 3 and 4 are the main findings of this paper.In the third chapter,we introduce two risk models,which are the single-insurance risk model with delay effect and the multi-insurance risk model with delay effect.In the delayed single-insurance risk model,we mainly focus on its delay effect.For the issue of whether the vice claim is delayed,it is represented by a random variable that obeys Bernoulli distribution;in the delayed multi-insurance risk model,we mainly focus on the effects of multiple types of insurance.Whether or not the vice claims occur or are selected to occur is characterized by a joint distribution of two random variables.Based on this,risk models are established,and it is proved that the surplus processes satisfy the large and moderate deviation principles.In the fourth chapter,the insurance claim model based on the Cox process is introduced.In order to ensure the randomness of the insurance environment,random strength is introduced and it is considered that the arrival process of the insurance claim report is a doubly stochastic Poisson process.In this model,it is assumed that the distribution of claim progress is related to the occurrence of events.Therefore,the count intensity in the counting process contains uncertain factors.Finally,we obtain the moment generating function based on the loss function of the Cox process risk model through the martingale method,and prove that the Cox process risk model satisfies the large deviation principle.The fifth chapter is the summary and prospect of this paper.