Εφαρμογές των στοχαστικών διατάξεων στη θεωρία συλλογικού κινδύνου
Applications of stochastic orders in collective risk theory
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Keywords
Στοχαστικές διατάξεις ; Θεωρία χρεοκοπίας ; Μοντέλο συλλογικού κινδύνου ; Μετρικές πιθανοτήτωνAbstract
Roughly speaking, stochastic orders are a powerful tool from probability theory that allows random variables to be compared. In several cases, we are interested in comparing two variables, for example, in Insurance Mathematics, if these random variables represent individual claims, then their comparison enables us to compare the risk associated with the two portofolios. Additionally, stochastic orders help in finding bounds or approximations for random variables that their characteristics (e.g. right-tail distribution, failure rate function, etc) are difficult to obtain analytically. The theory of stochastic orders has been proposed as a significant tool in studying stochastic models with applications in queueing theory, finance, reliability theory and collective risk theory.
In this thesis, we first study ruin theory by illustrating the classical model and classical model perturbed by diffusion, providing some examples with known distribution functions for better understanding.
In the second Chapter, we present some well known stochastic orders, as well as describe their direct connection with aging classes (or reliability classes). The classification of random variables based on the monotonicity of certain characteristic functions (failure rate, mean residual function, Laplace transform, etc) provides important information about them.
In the third Chapter, we present some stochastic orders and aging classes related to the Laplace transform. In particular, we enrich the literature by defining some new aging classes. We show that the NBULt class is closed under convolution and we also obtain results for the ratio of derivatives of the Laplace transform between two distributions.
It is natural that the comparison of random variables does not always hold over their entire domain. We believe that a more realistic perspective is the comparison of random variables over an interval. In the fourth Chapter, we describe known results from the literature on stochastic orders over an interval and define some new stochastic orders over an interval related to the Laplace transform, applying these results to Ruin theory.
In the fifth Chapter, we apply the theory of stochastic orders to risk models. Specifically, we give a brief review of the literature and then provide new results about stochastic comparisons of ruin-related quantities, such as compound geometric random variables and the deficit at ruin, in the classical or the renewal model. We also give some comparison results for the classical risk process with diffusion.
Finally, in the sixth Chapter, we describe the connection between stochastic orders and probability metrics. Although seemingly unrelated, these concepts are directly connected. Probability metrics help us to obtain bounds and approximations for unknown random variables. We present some well known probability metrics and describe their connection to the stochastic orders studied in our thesis in previous chapters. We also study the problem of stability of the classical risk model regarding the probability of ruin and the deficit at the time of ruin, providing some numerical examples.
Department
Σχολή Χρηματοοικονομικής και Στατιστικής. Τμήμα Στατιστικής και Ασφαλιστικής ΕπιστήμηςNumber of pages
174Language
GreekCollections
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Αναφορά Δημιουργού-Μη Εμπορική Χρήση-Όχι Παράγωγα Έργα 3.0 Ελλάδα
Αναφορά Δημιουργού-Μη Εμπορική Χρήση-Όχι Παράγωγα Έργα 3.0 Ελλάδα