Some Study On Ruin Probabilities Under A Class Of Risk Model With Stochastic Premium And Jump-diffusion Surplus Process

Insurance have the function of risk relocation, which is the stabilizer of the society. Business solvency is one of the most important indexes that make sure the insurance companies work fluently. How to weigh it is a very important task. The ruin theory plays an important role in the research on business solvency, also has extensively practical use. As an extension of the ruin theory, consideration of the risk model with stochastic premium and jump-diffusion surplus process is very realistic.The risk model with stochastic premium and jump-diffusion is considered in this paper. The background of the classical risk model is introduced in chapter 1. The background of the basic model in this paper is introduced in chapter 2 section 1. As an extension of the classical ruin probability, exponential bounds for ruin probabilities are derived by martingale method in chapter 2 section 2; In section 3, the same method used in Cai Jun & Xu Chengming[1] is introduced to derive integro-differential equations for the discount expectation of the ruin probability, a special case that can be solved by ordinary differential equation method is studied. In chapter 3 section 1, based on the risk model with jump-diffusion worked by Cai Jun & Xu Chengming[1], stochastic premium is added in this paper, the integro-differential equations for the surplus is worked out. As a special case, the third order differential equation is derived when both the premium and the claim size are exponential distributions. According to Yang's discussion, the same method can be used in place that surplus can be invested in both risky and risk-free assets,By using the HJB equation. This method is to be used to derive integro-differential equations for our risk model.The risk models with investing environment introduced in this paper are extensive to a certain extent. Some results of other scholars can be considered as special cases of ours. We hope that those results are helpful to the scientific management of insurance company under the circumstances of the good financial environments.