The Study Of The Expected Discounted Penalty Function In The Renewal Risk Model With Random Income And Dependence

The study of classical risk models are based on linear growth in premium income and claims under the Poisson process of these two important assumptions. But with the development of the insurance industry, we find that these conditions increasingly inconsistent with the current situation in the actual process of the insurance company’s operations. So in order to make the model better depicts the actual situation of risk, risk research fields develop a number of theories to promote risk models.In this paper, we will be mainly research two kinds of the Gerber-Shiu functions of dependent renewal risk models with nolinear premium income by using Stochastic Process, Risk Theory and Complex Function Theory. This thesis is organized as follows. In Chapter1, we first briefly introduce the classical risk model. Secondly combining the research contents of this paper, we briefly review the current situation of study on risk models at home and abroad. Finally we will introduce the main research contents of this paper.In Chapter2, we first introduce some conventions. Secondly in order to meet the needs of the theoretical research in this paper, we will present some mathematical methods and simulation tools that involved in research process.In Chapter3, through to promote premium income from linear growth of classical risk model to nonlinear Poisson process, and taking it into account that certain dependency relationship should be existed between claim intervals and claim sizes of the claims process, we consider a dependent renewal risk model, where the premiums income is a Poisson process and the dependence between claim intervals and claim sizes is as Boudreault et al.(2006) described. In addition, we study Gerber-Shiu function of the model in this chapter, and derive exact expression of its generating function, and obtain the defective renewal equation of the generating function.In Chapter4, we further generalize the premium income into a compound Poisson process in our study on the basis of Chapter3. The dependency relationship still meets Boudreault et al.(2006) in the dependency relationship. By considering the first time interval of the claims and the first arrival of premium income, we finally derive the explicit expression of the Laplace transform of Gerber-Shiu function in this case.