Μελέτη μέτρων χρεοκοπίας για τη διαδικασία πλεονάσματος με δύο χαρτοφυλάκια κινδύνων.
Post graduate programme in actuarial science and risk management
Κανελλόπουλος, Λάζαρος Κ.
The expected discounted penalty function or the Gerber-Shiu function is one of the most powerful and commonly used tool in the mathematical risk theory. Recently, many authors have paid much attention to the study of the Gerber-Shiu function in the classical and the renewal risk model. The main focus of this dissertation is to provide a detailed analysis of the Gerber-Shiu function in various dependent structures risk models. In Chapter 1 we give a detailed introduction of the Sparre-Andersen risk model and we present known results for the Gerber-Shiu function in this model. Also, we consider a risk model that the number processes are generalized Erlang(2) processes. Furthermore, in the same chapter we define, and analyze the mathematical tools that we will use repeatedly in the main core of this dissertation. In Chapter 2 we consider the expected discounted penalty (Gerber-Shiu) functions for a risk model involving two independent classes of insurance risks. We assume that the two claim number processes are independent Poisson and generalized Erlang(2) processes, respectively. We prove that the Gerber-Shiu function satisfies some defective renewal equations. Exact representations for the solutions of these equations are derived through an associated compound geometric distribution and an analytic expression for this quantity is given when the claim severities have rationally distributed Laplace transforms. In Chapter 3 the same risk model of Chapter 1 and Chapter 2 is considered in the presence of a constant dividend barrier. A system of integro-differential equations with certain boundary conditions for the Gerber-Shiu function is derived and solved. Finally, in Chapter 4 numerical illustrations are also given of ruin probability for this risk model.