Applications Of Compound Poisson-Geometric Process In Risk Models

In the aggregate risk theory,a very important question is to study the ruin probability.To make the risk models more closely simulate the real practices,based on the different nature of the risk models and the issues encountered in insurance companies' practices,the research of ruin probability amends and imposes further conditions on these probabilistic and statistical models,which makes the research of ruin probability full of challenge.In this paper,we use the approach of Compound Poisson-Geometric to study two problems:the discounted penalty expectation at ruin in the Compound Poisson-Geometric risk model and the multi-line insurance risk model with the claim number being a Poisson-Geometric process.In the first problem,in terms of the method of differentiation,we deduce a progression formula for the discounted penalty function at ruin,which is a generalization of the result of Gerber and Shiu(1998a) in the classical risk model;by using the Martingale approach,we further deduce the ruin probability and a probabilistic expression for the surplus x to exceed the given level u for the first time.In the second problem,we give an explicit expression for the ruin probabilityψ(0) when the initial capital is u.For any initial capital u,we obtain an approximate estimate for the ruin probabilityψ(u);and we also achieve an accurate analytical expression for the ruin probability in some special cases.