Research Of Some Problems In Risk Theory

This paper mainly discusses ruin problems in the two discrete time risk models under the assumption that rates of interest have an autoregressive structure of order 1, the compound double Poisson risk model under constant interest force and the perturbed compound double Poisson risk model under constant interest force, which are divided into the following three parts.In chapter 1, Cai(2002)obtains integro equations of ruin probabilities for the two discrete time risk models under the assumption that the rates of interest have an autoregressive structure of order 1. In this chapter, for the model of Cai(2002), we obtain the recursive formulas of penalty function which give an unified treatment for all kinds of ruin quantities such as ruin probability, the distribution of surplus immediately before ruin , the laplace transform of ruin time and etc., which extend the results of Cai(2002). Furthermore, we consider the probability properties of the duration of ruin, which are used to describe the severity of ruin and give the corresponding recursive equation.In chapter 2, basing on the double compound Poisson risk model of the Fang and Luo (2006), we establish the double compound Poisson risk model under constant interest force and obtain the partial integro-differential equation of finite time survival probability and the integro-differential equation of infinite time survival probability. And when the premiums and claim sizes are exponentially distributed, some differential equations are derived for infinite time survival probability. These results extend those of Fang and Luo(2006).In chapter 3, basing on chapter 2, we consider the perturbed compound Poisson risk model under constant interest force , and obtain integro-differential equation of infinite time survival probability by two different ways and partial integro-differential equation of finite time survival probability. When the premiums and claim sizes are exponentially distributed, we give some differential equations for infinite time survival probability.