Research On Several Risk Models And Related Problems

Actuarial Risk theory which is the basis of actuarial mathematics and queuing theory has been studied by mathematics workers. Problems on risk theory have become the focus of actuarial. In Chapter 1, we review the risk theory for nearly a hundred years of progress, introduce different risk models and the major work done in this paper. In Chapter 2, we consider a compound renewal risk process with interclaim times that have an arbitrary distribution. The premium income process based on a Poisson process is no longer a linear function. We derive a defective renewal equation satisfied by the discounted penalty function. The solution of this renewal equation is given. The result is used to find the discounted joint and marginal distribution of surplus before ruin and the deficit at ruin. In Chapter 3, we extend the classical risk model to the case where the premium income process, based on a random process, is no longer a linear function. We study the Gerber-Shiu discounted penalty function for the risk model modified by the constant interest force on the surplus. Explicit expressions are given by an infinite series, and a relational formula for some important joint density functions is derived. Finally, applications of the results to the model, which the premium income is a Poisson process and the individual claim distribution FS∈Geo( 0), are given. In Chapter 4, we consider the classical surplus process with interest and a threshold dividend barrier. Under constant interest, we derive an integro-differential equation for the Gerber-Shiu expected discounted function. We obtain the solution to the integro-differential equation which is the form of an infinite series. In some special cases with exponential claim, we are able to find closed-form expressions for the Gerber-Shiu expected discounted penalty function.