Discussion. The P¨®lya Therom-aeppli Risk Model

In this thesis we consider the classical risk model and another risk model in which the counting process is the Pólya-Aeppli process. It's short name is Pólya-Aeppli risk model (In our country this model is called the compound Poisson-geometrica risk model). By comparing the classical risk model and the Pólya-Aeppli risk model, we can get the result that the Pólya-Aeppli process as a generalization of the homogeneous Poisson process, hence we can get some results of the Pólya-Aeppli risk model as soon as by comparing the classical risk model. We also prove these conclusions by using another method.The thesis is divided into three chapters according to contents:Chapter 1. Firstly, the development of the risk theory and some excellent scientists who devoted themselves to the risk theory and their main results are reviewed generally. Secondly, the important works and main conclusions of the classical risk model are analyzed on emphasis.Chapter 2. Firstly, we present the definition of the classical risk model and some main results of this model. Such as the ruin probability and the expected discounted penalty. Secondly, we also present the Pólya-Aeppli process and some conclusions that will be used in the next chapter, for example, the distribution of Pólya-Aeppli process and the distribution of Pólya-Aeppli process in differential form. They are all to be ready for the Chapter 3.Chapter 3. Based on the conclusions of the classical risk model and the Pólya-Aeppli process, we study the Pólya-Aeppli risk model. Firstly, by comparing the homogeneous Poisson process and the Pólya-Aeppli process, we can get the result that the Pólya-Aeppli process is a generalization of the homogeneous Poisson process. Hence we can translate the Pólya-Aeppli risk model into the classical risk model and get some results of the Pólya-Aeppli risk model immediately. Secondly, we consider the ruin probability, the expected discounted penalty and other results of the Pólya-Aeppli risk model under a constant interest rate, the Pólya-Aeppli risk model perturbed by diffusion. Finally, when the claim amount distribution is a certain exponential, we estimate bounds of the ruin probability.