The Gerber-Shiu Function In Risk Models And Related Problems

The thesis mainly considers the Gerber-Shiu expected discounted penalty function introduced and studied by Gerber and Shiu [North American Actuarial Journal,2,1998]. The function includes many important problems in risk models, for example, ruin probability, the joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin, and so on. The thesis studies the Gerber-Shiu expected discounted penalty function in the different risk models, obtains the corresponding integro-differential equations and renewal equations, and discusses the solutions of these equations.The thesis is structured as follows:In Chapter 1, we simply introduce the Lundberg-Cramer classical risk model in section 1.1, and the Gerber-Shiu expected discounted penalty function in section 1.2. Furthermore, the basic knowledge needed in the thesis listed in section 1.3.In Chapter 2, we show that the properties of ruin probability in the Cox risk model by Martingale approach in section 2.1, some results can be found in Grandell (1991). In section 2.2, the Gerber-Shiu function is examined in a class of Cox risk models, an integro-differential equation is derived. In the two-state model, explicit formulas for ruin time function are obtained when claim size distribution belongs to the Kn-family. In Chapter 3, we consider a class of renewal risk process in which the claim inter-arrival times are the mixture of exponentials and Erlang(2) distributions. A integro-differential equation and renewal equation for the expected discounted penalty function are derived, and we also discuss the Laplace transform. We obtain the explict solution and the asymptotic one in the case where the individenal claim amount distribution is Phase-type and Pareto, respectively.In Chapter 4, we consider a renewal risk process with a threshold dividend strategy in which the claim inter-arrival times are Erlang(n) distributed. Two integro-differential equations and a renewal one for the Gerber-Shiu function are obtained. We also discuss the solution of the renewal equation in section 4.1. Gerber-Shiu functions in generalized Erlang(n) risk model with multi-layer dividend strategy and with interest are considered in section 4.2.In Chapter 5, the Gerber-Shiu functions are considered in the con-tinuous and decrete stationary renewal risk model, where they are expressed in terms of the same functions in the ordinary renewal risk model in section 5.1. In section 5.2, we consider a class of delayed renewal risk processes with a threshold dividend strategy, the main result is an expression of the Gerber-Shiu function in the delayed renewal risk model in terms of the corresponding function in the ordinary renewal risk model. We state that the stationary renewal risk model is the special case of the delayed renewal risk one in this section.In Chapter 6, we study the distribution of the maximum surplus before ruin in a Sparre Andersen risk model with the inter-claim times being generalized Erlang(n) distributed perturbed by diffusion, two integro-differential equations are derived, the solution of the first corresponding homogenous integro-differential equation is considered in section 6.1. The Gerber-Shiu function is also considered in a Sparre Andersen risk model perturbed by diffusion in section 6.2.