Προσεγγίσεις τύπου Tijms στο συλλογικό πρότυπο της θεωρίας κινδύνων
Tijms - type approximations in the collective model of risk theory
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Keywords
Προσεγγίσεις τύπου Tijms ; Gerber-Shiu function ; Lundberg risk model ; Μοντέλο διάχυσης ; Κατανομή του ελλείμματος ; Μετασχηματισμός Laplace του χρόνου χρεοκοπίας ; Κώδικας mathematica για προσεγγίσεις τύπου TijmsAbstract
In Actuarial Science, Risk Theory makes use of mathematical models to describe the risk of insolvency or ruin of the claims portfolio of an insurer. In such models, the quantities and sizes which are of interest to us are the probability of ruin, the surplus right before ruin and the deficit at the moment of ruin. In addition, another value we are interested in is the Gerber- Shiu function, otherwise known as the expected discounted penalty function which looks into the joint behavior of the above-mentioned values, taking into account the discount factor as well as the Laplace transform of the time of ruin. In most of the cases of a claims portfolio, finding the exact probability of ruin is possible only in a few cases for the claim size distribution which, however, are not encountered in reality. For all the above mentioned reasons, it is important to find approximations in case the accurate calculation of the above - mentioned possibilities is not feasible. Given that the aggregate of the ladder heights follows a compound geometric distribution, estimating such a compound geometric distribution is possible by making use of a combination of two exponential distributions by selecting the right parameters, as proposed by Tijms in 1986. Finally, extensive reference will be made to the diffusion model and we shall be looking into approximations equivalent to those of Tijms.