Μελέτη των ροπών του χρόνου χρεοκοπίας στο κλασικό πρότυπο της θεωρίας κινδύνων

Abstract

In the present thesis, the study of the moments of ruin time in the classical model of risk theory is attempted. The analysis of the distributions of the period of ruin, the surplus before the time of ruin, as well as the deficit during the period of ruin, has been the subject of significant research interest. Determining these distributions is difficult, as in most cases there are no detailed expressions. In addition, there are generally no closed-form solutions for the moments of these distributions. Various approaches have been used to study the probabilistic properties of these distributions. The thesis first analyzes the theory of the collective model, concluding with Panjer's recursive methods for calculating the distribution of total claims, and then analyzes the theory of ruin, presenting some applications for finding the adjustment coefficient. The tails of compound geometric distributions have been extensively studied both analytically and numerically. In particular, there are recursive formulas (Panjer & Willmot, 1992), as well as upper and lower limits (Lin, 1996), while sometimes there are exact solutions (Dufresne & Gerber, 1988). Also known is the Cramer-Lundberg asymptotic approximation (Gerber, 1979). For a large category of claim-size distributions, the tail of the associated compound geometric distribution can be approached very well by a combination of two, or more, exponential distributions. Thus, the work analyzes the moments of ruin time, by comparing the equilibrium distributions and compound geometric tails, the calculation of the deficit at the time of ruin and the combination of exponential claim amounts.