The Distribution Of Surplus Immediately Before Ruin When Claims Obey Compound Exponential Distribution

In insurance mathematics, ruin theory is the mainly contents of insurance risk theory, as can supply a very useful early-warning measure for the risk of the insurance company, it has important theoretical and practical significance for the insurance company . In recent years , research on ruin theory is a pop task .There are a vast amount of literature on solving the upper and lower bounds of ruin probability, computing and simulating ruin probability, calculating the distribution of the surplus before ruin and the distribution of the deficit at ruin , also their joint distribution . In this paper, It firstly introduce risk theory especially the knowledge of ruin probability and the main result of classical risk model, then giving the notion, classifications and basic properties of heavy-tailed and light-tailed distributions .Then It calculate the probability of final ruin and the distribution of surplus and deficit before ruin in the Poisson risk model, when the claims obey compound exponential distribution with the insurance company's initial surplus equals to u. It use the method of separating the real roots of polynomial to get the n solutions of coefficient r as solving the ruin probability and basing on Sturm theorem and Desarts rules giving algorithm of r, then giving the examples of n=3, n=4, approximating the ruin probability, then getting the distribution of the surplus before ruin and the distribution of the deficit at ruin, and giving the numerical examples in forms.