Ένα ανανεωτικό μοντέλο κινδύνου με την ύπαρξη στρατηγικής σταθερού μερίσματος
Τσαγκούλης, Αθανάσιος Ι.
SubjectRenewal theory ; Διαχείριση κινδύνου -- Οικονομετρικά μοντέλα ; Διαχείριση κινδύνου -- Στατιστικές μέθοδοι
In this thesis, the study of of the renewal Erlang model is presented, which constitutes the generalization of the classical renewal model. Both in the classical , as well in the Erlang model, a random variable of high importance is the time of default, which is the moment that for the first time the surplus will have a negative value. Two other random variables related to the time of default are the random variable of the surplus at the exact time before the ultimate ruin and at the exact time of the ultimate ruin. Obviously, the joint study of these variables provides more information in regard to the surplus behavior, compared to the individual study of each variable. Therefore, we shall study the expected discounted penalty function of Gerber and Shiu, which defines these variables, both in the case of a non- dividend model and the case of a dividend paying one. Chapter one is an introductory part. Fundamental concepts of the ruin theory are presented and the reference of the expected discounted penalty function of Gerber and Shiu is also provided. This is the case of a model without dividend. It will also be splayed that the Gerber and Shiu function satisfies an integro-differential equation, which can be approached by the use of a compound geometrical distribution. In chapter two, the central concepts of chapter one will be studied in depth, in the case of a dividends-paying model, by the time the surplus exceeds a predermined barrier. The solution of the integro-differential equation will be given through the use of the respective homogenous one and by the use of relevant Laplace transformations. Finally, a numerical example will be provided. In chapter three, we will study the behavior of the dividends through their moments. It will be displayed that the moment-generating function of the sum of the discounted dividend payments satisfies an integro-differential equation, similar to the one we studied in chapter one and we will provide its solution with the use of a similar methodology. Finally, a numerical example will be provided here too.