Discounted Probabilities And Ruin Theory In The Continuous-Time Compound Binomial Model

In this dissertation we mainly study a kind of the continuous-time compound binomial model. It is developed according to the discounted prob-ability.Here we may think the continuous-time compound binomial model is the continuous version of Gerber's compound binomial model(discrete-time compound binomial model). 1-skeleton chain may become Gerber's compound binomial model when u ∈N and c = △ = 1,and the classical risk model when A j 0.In this paper.we use an approach similar to Gerber and Shiu~[3].By an application of matingale .the renewal equation.and duality in the sample path of the process U(t) we get f(i,j/u),the discounted probability of ruin for an initial surplus u.such that the surplus just before ruin is ic△ and the deficit at ruin is jc△.This function can be used to calculate the expected present value of a penalty that is due at ruin . An explicit formula for f(i,j\0) is derived.Then it is shown how f(i,j\u) can be expressed in terms of f(i,j\0). A discrete version of Dickson's formala is provided.This paper includes four chapters.First,we introduce the relational approach of the thesis in general.A brief review of the background of the discounted probability and several elementary concepts of the extended generator of PDMP are given in the second chapter.The classical risk model and some results are also introduced in the second chapter.After that the main body of this paper starts. In the last chapter ,the main results are given.