Μελέτη μοντέλων κινδύνου με στοχαστικά ασφάλιστρα
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Subject
Στοχαστική ανάλυση ; Διαχείριση κινδύνου -- Στατιστικές μέθοδοι ; Στοχαστικές ανελίξεις -- Μαθηματικά υποδείγματα ; Ασφάλιση ; Insurance -- Statistical methodsAbstract
In the classical model of risk theory, insurance premiums collected by the insurance company at a fixed rate, i.e. assumed to be linear functions of time. In this thesis, it is considered the more realistic assumption considering that the total premiums (as total losses) are stochastic processes and sequence of the complex Poisson stochastic process. It considers two risk models. At first it considers premiums and total compensation are independent of each other and in the second a specific form of dependency among the individual claim sizes, the premium sizes and the interclaim times. More specifically, the first chapter is an introduction which presents basic concepts in ruin theory, a reference to the defective renewal equation and the expected discounted penalty function of Gerber-Shiu. In the second chapter it assumes a ruin model where both premiums and claims follow compound Poisson processes. It establishes that the expected discounted penalty function satisfies a defective renewal equation and an integral equation. Finally, the case when premiums have Erlang (n^) distribution and the distribution of the claims is arbitrary is investigated in more depth. Also in the context of this case, it is found explicit expressions for specific risk measures, like the ultimate ruin and the surplus before ruin. In the third chapter, it is extent the model with stochastic premiums income by assuming that there exists a specific dependence structure among the claim sizes, interclaim times and premium sizes. It is also assumed that the distributions of the premium sizes and interclaim times are fully controlled by the claim sizes. When the individual premium sizes are exponentially distributed, the Laplace transforms and the defective renewal equations for the Gerber-Shiu discounted penalty functions are obtained. Finally, when the individual premium sizes have rational Laplace transforms, it is proved that the Laplace transform for the discounted penalty functions can also be obtained.