Third-order Expansion Of Ruin Probability In The Case Of Second-order Regular Variation Tails

The main purpose of this thesis is to provide more precision expansion of ruinprobability of the insurance company. We obtain the third-order expansion of ruinprobability, so the insurance company can know own compensation ability well.This thesis consists of four chapters as follows. In chapter 1, we mainly introducethe classical risk model and the development of risk theory; we introduce some rela-tive heavy-tail distribution classes and their properties in chapter 2. The third and lastchapter are my research results, first we prove the closeness of second-order regularvariation. Then we assume that the integrated tail distribution function of the claim size(?)(x) is the second-order regular variation function with two parameters-γandθ. Firstwe obtain the third-order expansion of (?)(x), then using Beekman's convolutionformula, we finally get the main result:ψ(u)=(?)(u)/ρ+γ/μρ²Â·(?)(u)/u EX²-c₁a₁(u)+c₂a₂(u)/2μρ²Â·(?)(u)/u EX²+o(1).