Στοχαστικές διαδικασίες πλεονάσματος με εξάρτηση και στρατηγικές μερισμάτων
Stochastic surplus processes with dependence and dividend strategies
Κασαννή, Αικατερίνη Σ.
The purpose of this thesis is the study of various ruin measures and the expected discounted dividend payments before ruin for stochastic surplus process by introducing a dependence structure between the claim sizes and the interclaim times through a Farlie-Gumbel-Morgenstern copula in the presence of a constant dividend barrier. 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. Also, in the same chapter we give a detailed introduction of the renewal risk process and we present known results for the Gerber-Shiu function respectively. In Chapter 2 we consider a constant dividend barrier strategy in case of the classical risk model and the renewal risk process respectively. We present known results for the Gerber-Shiu function and some results on the distribution of dividend payments until ruin under a classical risk model and a renewal risk model with generalized Erlang(n) distributed interclaim times, respectively, under a constant dividend barrier. In Chapter 3 we consider an extension to the renewal risk process by introducing a dependence structure between the claim sizes and the interclaim times through a Farlie-Gumbel-Morgenstern copula. We consider that the interclaim times follow the Erlang(n)distribution. Moreover, a detailed analysis of the Gerber-Shiu function is given when the initial surplus is zero. It is proved that this function satisfies a defective renewal equation and its solution is given. Also, for exponential claim sizes explicit expressions and numerical examples for the ruin probability and the Laplace Transform of the time to ruin give. Finally, in chapter 4 we consider the renewal risk process where the join distribution of the interclaim time and the corresponding claim size has a dependence structure based on a Farlie-Gumbel-Morgenstern copula and the interclaim times follow the Erlang(n) distribution in the presence of a constant dividend barrier. We derive an integro-differential equation with boundary conditions for some Gerber-Shiu discounted penalty functions. We proved that its solution is expressed as the Gerber-Shiu discounted penalty function in the same risk process with the absence of a dividend barrier plus a linear combination of a finite number of linearly independent solutions to the associated homogenous integro-differential equation, which are obtained through Laplace transforms. Also, we derive a homogenous integro-differential equation with boundary conditions for the expected discounted dividend payments before ruin and it is shown that its solution can be expressed as a different linear combination of the same finite number of the linearly independent solutions to the homogenous integro-differential equation associated with the Gerber-Shiu discounted penalty function. Finally, we obtain explicit solutions for exponential claim sizes.