Study On Ruin Problems For Discrete Time Risk Models With Delayed Claims

In the classical risk theory, the stationary and independent increment assumption on the surplus process plays an important role. However, the assumption may be restrictive and far from reality. Recently, the models with various kinds of dependence structures have been studied by researchers. In this paper we introduce the dependence structure by considering delayed claims in the risk model, and some easily programmed formulae are derived for ruin quantities such as joint distribution of the surplus before ruin and deficit at ruin and ruin probability, etc. The conclusions obtained in this paper supplement related results for the discrete time renewal risk model.This thesis is divided into four chapters.Chapter 1. This chapter first introduces the research background about the risk theory, and makes a brief summary of the related definitions and results in the discrete time risk model with delayed by-claims, which involves the discrete time risk model with one type by-claim and the discrete time risk model with two types of by-claims.Chapter 2. Section 1 gives the discrete time renewal risk model which involves three types of insurance claims. In section 2, by introducing three auxiliary models, we obtain the probability generating functions of the joint distributions of the surplus before ruin and deficit at ruin when the initial surplus is zero for the corresponding models. In section 3, by inversing probability generating functions, we derive explicit expressions for the joint distribution of the surplus before ruin and deficit at ruin together with the ultimate ruin probability when the initial surplus is zero. In addition, the recursive formula for the joint distribution of the surplus before ruin and deficit at ruin with arbitrary initial surplus is given.Chapter 3. Based on the chapter 2, the threshold is introduced in section 1. The second by-claim happens only if the claim amount is greater than the threshold. In section 2, by introducing auxiliary models, we obtain the probability generating functions of the joint distributions of the surplus before ruin and deficit at ruin when the initial surplus is zero for the corresponding models. In section 3, by inversing probability generating functions, the recursive solution of the finite time survival probability is derived.Chapter 4. Based on the model in chapter 2, this chapter presents numerically results with risk portfolio problems in reality. When the claims follow special distributions, explicit numerical results for the surplus before ruin and deficit at ruin in special cases are given.