Types Of Risk Model

This dissertation is devoted to dealing with ruin theory for some kinds of risk models which include the classical risk model that is perturbed by diffusion, delayed renewal risk model and the generalized renewal risk model, and finally we discuss the Sparre Andersen model for a particular case.Some preliminaries are given in Chapter 1. we first introduce some risk models (Section 1.1), then we briefly discuss the difference between light-tailed claim distributions and heavy-tailed claim distributions (Section 1.2) and the classical results on ruin problem (Section 1.3). Chapter 1 mainly serves the purpose of introducing notions and notations and gives a first idea of the problems to be considered thereafter.Chapter 2 deals with the classical risk model perturbed by diffusion. We consider the joint distributions of the supreme profits and deficit before it leaves zero for the first time, and the joint distribution of the supreme profits and deficit until it leaves zero ultimately, that is,Theorem 2.2.1 For any a and b, such that a > u > 0, b > 0, thenTheorem 2.2.2 For any a and b, such that a > u > 0, b > 0, thenCorollary 2.2.2 when a > u > 0, b > 0, we haveChapter 3 is a investigation into the ruin problem in delayed risk model. Under the assumption that the claims size distribution is heavy-tailed, we obtain the following local asymptotic relationship for the ruin probability,where F denotes the distribution of the i.i.d. claims, p is the safety loading coefficient of the model, u is the mean of the claims, and the limit process is In Chapter 4, we further generalize the Sparre Andersen model and obtain two kinds of new risk models. We first establish the risk models (Section 4.1) and then give the following asymptotic behavior of ruin probability in Section 4.2 ,And we give a local result as well in Section 4.3.In Chapter 5, we consider a particular renewal risk process-Erlang(n) risk process, that is, the time Ti which is between the (i-l)-th and i-th claims has an Erlang(n) distribution with density functionWe investigate the moments of the surplus before ruin and the deficit at ruin using the integro- differential equation that we have established, here we consider the discounted penalty expectation when .we obtain some explicit expressions for the moments and the following joint probability density function of the surplus before ruin and the deficit at ruin.Theorem 5.4.2 In Erlang(n) risk model, if the safety loading coefficientthe model is positive, the joint density function of U{T~) and U(T) is and |U(T)| are the surplus immediately before ruin and the deficit at ruin .