Ruin Problems In Some Risk Models

In this dissertation, we study the ruin theory for several kinds of risk models. Firstly, we study the ruin problem of a Sparre Andersen risk model with the inter-claim times being Erlang(n) ditributed. In the second, we consider a risk model with two claim number processes which are independent Poisson and generalized Erlang(n) processes. Finally, we discuss the Gerber-Shiu discounted penatly function for a delayed renewal risk model.This paper is composed of three chapters.In Chapter 1, we consider a Sparre Andersen risk model in which the inter-claim times are the generalized Erlang(n) distribution. We use the facts that a generalized Erlang(n) random variable can be expressed as an independent sum of n exponential random variables and the exponential distributions have the lack-of-memory property in the analysis of the corresponding risk process. We obtain the integro-differential equation of the distribution of the maximum surplus befor ruin. Then we get the integro-differential equation and the boundary conditions for the Laplace transform of the first time that the surplus reaches level b (b > u) without ruin. When claim sizes are Erlang(n) distributed, a particular solution of the integro-differential equation is given.In Chapter 2, we consider the Gerber-Shiu discounted penalty function for a risk model involving two independent classes of insures risks. We assume that the two claim number processes are independent Poisson and generalized Erlang(n) processes respectively. We obtain the Laplace transforms of theGerber-Shiu discounted penatly functions at ruin. Also, we obtain the general expressions of the the Gerber-Shiu discounted penatly functions. These results improve the results in [7].In Chapter 3, we discuss the Gerber-Shiu discounted penalty function for a class of delayed renewal risk process. Let Vj. be the time until the first claim, whose density function has the expressionUsing the relation between k[(t), ki(t) and k(t), we get the expression of m\:s(u) in terms of the expression of ms(u).