Η γενικευμένη συνάρτηση των Gerber Shiu για διακριτές ανανεωτικές στοχαστικές διαδικασίες πλεονάσματος με εξάρτηση

Abstract

The first chapter introduces the concepts of stochastic processes as well as the definitions of the main variables employed in this report. At first we introduce the Sparre Andersen model, which studies the company's stock, as well as the main model being analyzed, the generalized model Gerber Shiu. We find and analyze the heights of the damage that occurs at any time and the probability of default. In the second chapter, different forms of the discounted Gerber-Shiu function are created based on the specific cases of the penalty function and the damages following the geometric distribution. Note that all sets of variables are independent and have the iid property. Assuming retrospective relationships and explicit expressions of the Gerber-Shiu function, we end up with generating functions. The corresponding study is done for claim heights that follow a mixed geometric distribution and when they happen. The third chapter analyzes the generalized function based on the interclaim time and the heights of losses using retrospective equations, thus ensuring a positive reserve. The fourth chapter analyzes the claim heights and the interclaim time, which are independent of each other, following a geometrical and a combination of geometric distributions. In the fifth chapter the same analysis is done as in the fourth chapter, but depending on the amount of claims being caused and the interclaim time. In the final chapter, the analysis of the previous chapter is followed by the delayed dependant Sparre Andersen model and we introduce a new dependence parameter that separates the two models. Finally, a graphical representation of all of the above is presented and plotted to illustrate the differences between the cases.