On Ruin Probability For The Erlang(n) Risk Process And Related Problem

This content mainly introduce the ruin probability of the risk processing andthe research of the related problem under the inter-claim times being Erlang (n)distribution and the processing of Sparre Andersen. Meanwhile, We analyze thedistribution of ruin probability that the surplus process attains a given level fromthe initial surplus without first falling below zero.In chapter one, we introduce the development background of ruin probabilityand its development history. The sweden acturay Filip Lundberg firstly intro-duced classic risk model in his doctoral dissertation of 1903, and put forward theruin probability, and got a beginning expression of the first ruin probability. Thenwith the leader of Harald Cramer of the school of Sweden turn strictly its results.Meanwhile, Harald Cramer developed stochastic process theory. Hans.U.Gerberlead the research of ruin probability. At last, we get the current risk model afterthe development of classic risk model.In Chapter two, this paper is devoted to studying some preview knowledgethat used on following chapter, which include Sparre Andersen risk model, theproperty of integrable real-function f, the method of haweiside , and di?erentialdivision equation.In classic risk models, the di?erence of claim time belongs to Poisson process,but in fact, the claim time and times always changes due to various reasons.In chapter three, this paper changed the Poisson process to Erlang(n) process,and then got the ruin probability of return boundary B. In order to solve theequation DV (μ) = 0μV (μ?y)p(y)dy, this paper introduced the operator T andmade its Laplace translate to be p?(s) = QQmm?(1s(s) ), here,Qm(s) represents n order.In chapter three, we study the density function p is a rationally distribution ,then its laplace transform can be assume that, i.e p?(s) = QQmm?(1s()s ), where Qm(s) isa polynomial degree m .By the method of haweiside introduced in chapter two,the following results are obtained: (a) The roots of the equation are distinct ;(b) Not all the roots of the equation are distinct. In last chapter, we give a detailed discussion of the results for the risk modelwith Erlang(n)inter-claim times though the results about chapter three.