Θεωρία χρεοκοπίας για στοχαστικές διαδικασίες πλεονάσματος υπό την ύπαρξη ενός κατωφλίου
Ruin theory for stochastic surplus processes under a threshold
KeywordsΔιαχείριση κινδύνου ; Στοχαστικές διαδικασίες ; Χρεοκοπία ; Εξισώσεις ; Μαθηματικά μοντέλα ; Στατιστική ανάλυση ; Συναρτήσεις
Actuarial Science uses mathematical models to estimate the solvency of an insurance portfolio. Ruin models help to study basic quantities of interest, such as the probability and time of ruin, and the surplus prior to and deficit at ruin. In this thesis, at first, we will present the Classical Model of Ruin Theory and the discounted penalty function. Subsequently, we will study a ruin model with dependency between claim sizes and inter-arrival times. Finally, we will consider a more generalized model with a semi-Markovian dependence structure. More specifically, the first chapter is an introductory section. The second chapter, presents the classical model of Ruin Theory and the discounted penalty function. We obtain results for the discounted penalty function, ruin and survival probabilities by their integro-differential equations and Laplace transforms and derive a general solution for the discounted penalty function in terms of compound geometric distribution tail. In the third chapter, a generalization of the classical ruin model is considered, where the distribution of the time between two claim occurrences depends on the previous claim size. For this specific model, we derive exact solutions for the discounted penalty function, ruin and survival probabilities by means of Laplace transforms. We also discuss about several related models that allow for a similar treatment, give some numerical illustrations and investigate the effect of ignoring the dependence structure. Finally, in the fourth chapter, we consider a more general model with a semi-Markovian dependence structure. For this model, we derive an explicit expression for the Laplace transform of the discounted penalty function and an explicit formula for the discounted penalty function for zero initial capital. We also investigate the asymptotic behavior of the penalty function for light-tailed claim sizes and it is shown how to obtain arbitrary moments of the time to ruin, surplus before ruin and the deficit at ruin. In the end, it is shown that this general model contains as special cases the classical compound Poisson model, the Sparre Andersen model with generalized Erlang(n) interclaim distributions, the Sparre Andersen model with phase type interclaim distributions and the model with dependency of the previous chapter.