Μοντέλα ασφαλιστικών κινδύνων με κίνηση Brown και στρατηγικές μερισμάτων
Insurance risk models perturbed by Brownian motion under dividend strategies
The purpose of this thesis is the study of various risk models and the expected dis-counted dividend payments before ruin for stochastic surplus process. In Chapter 1 we give a detailed introduction of the classical risk model and we present known results for the Gerber-Shiu function in this model. We studies in detail the solution of a defective renewal equation which involves the time of ruin, the surplus immediately before ruin, and the deficit at the time of ruin. The analysis is simplified by introduction and analysis of a related compound geometric distribution, which is studied in detail. In Chapter 2 we derive explicit expressions for the discounted joint and marginal dis-tribution functions of the surplus immediately prior to the time of ruin and the deficit at the time of ruin, and for the discounted distribution function of the amount of the claim causing ruin, based on the surplus process of ruin theory with an independent diffusion process. Fur-thermore, we show that these distribution functions satisfy defective renewal equations. In Chapter 3 we consider a Sparre Andersen risk process that is perturbed by an inde-pendent diffusion process, in which claim inter-arrival times have a generalized Erlang(n) dis-tribution (i.e. as the sum of n independent exponentials, with possibly different means). This leads to a generalization of the defective renewal equations for the expected discounted penalty function at the time of ruin given by Tsai and Willmot (2002a, 2002b) and Gerber and Shiu (2003a, 2003b). The limiting behavior of the expected discounted penalty function is studied, when the dispersion coefficient goes to zero. In Chapter 4 we consider a multi-layer compound Poisson surplus process perturbed by diffusion and examine the behavior of the Gerber–Shiu discounted penalty function. We derive the general solution to a certain second order integro-differential equation. This permits us to provide explicit expressions for the Gerber–Shiu function depending on the current sur-plus level.