Precise Large Deviations For Multi-risk Model With Heavy-Tailed Claims

Precise large deviation and risk theory have been two themes in insurancemathematics, mainly concentrating on the problem of precise large deviation based onthe multi-risks model that claim number need to conform to heavy-tailed distribution.In reality, there is always a phenomenon that the chance of something that the eventswill happen is very small, but once they happen, they will leads to unimaginableinfluence, which they are called extreme events in applied probability area, such astyphoon, fires and earthquakes. According to the statistics, for the defined insurancecompany, the20percent claims which take in the total quantity will reach80percentin the total claim amount. Taking lessons there, the modern large deviation theory isenriched largely thanks to the adding of heavy-tailed distribution by people.Large deviations appear historically in insurance mathematics with the ruinprobability estimation problem within the classical Cramer-Lundberg model, whichthe assumption is mainly about the random events. Because of the importance ofheavy-tailed distribution both in finance and insurance applications, researches on theheavy-tailed large deviation distribution without doubt have been a popular topic.For the research of precise large deviation, the paper takes great interest in themulti-risks model where claim number need to conform to heavy-tailed distribution.In this paper we assume that the related claim number conforms to there distributionfunction f∈OR, because this distribution can describe the large amount of claim.Now the researches on the heavy-tailed distribution are quite limited, so this type ofresearches plays great important roles in finance and insurance applications.This paper is divided into four chapters:In the first chapter, we mainly introduce the background and the main content ofthe paper.In the second chapter, we give the notations and preliminaries we will use in thefollowing work.In the third chapter, we discussed precise large deviations for sums of randomvariables with OR tails in multi-risk models.In the forth chapter, we analyzed and discussed the results of the paper．Large deviations is a very active area in applied probability, and questionsrelated to extremely events both in finance and insurance applications, play anincreasingly important role. For instance, recent applications of interest concern ruin probabilities in risk theory, risk management, option pricing.