Modèles de dépendance avec copule Archimédienne : fondements basés sur la construction par mélange, méthodes de calcul et applications

Authors: Veilleux, Dery
Advisor: Marceau, ÉtienneCossette, Hélène
Abstract: The law of large numbers, which states that statistical characteristics of a random sample will converge to the characteristics of the whole population, is the foundation of the insurance industry. Insurance companies rely on this principle to evaluate the risk of insured events. However, when we introduce dependencies between each component of the random sample, it may drastically affect the overall risk profile of the sample in comparison to the whole population. This is why it is essential to consider the effect of dependency when aggregating insurance risks from which stems the interest given to dependence modeling in actuarial science. In this thesis, we study dependence modeling in a portfolio of risks for which a mixture random variable (rv) introduces dependency. After introducing the use of exponential mixtures in actuarial risk modeling, we show how this mixture construction can define Archimedean copulas, a powerful tool for dependence modeling. First, we demonstrate how an Archimedean copula constructed via a continuous mixture can be approximated with a copula constructed by discrete mixture. Then, we derive explicit expressions for a few quantities related to the aggregated risk. The common mixture representation of Archimedean copulas is then at the basis of a computational strategy proposed to compute the distribution of the sum of risks in a general setup. Such results are then used to investigate risk models with respect to aggregation, capital allocation and ruin problems. Finally, we discuss an extension to nested Archimedean copulas, a general case of dependency via common mixture including different levels of dependency.
Document Type: Mémoire de maîtrise
Issue Date: 2018
Open Access Date: 21 December 2018
Grantor: Université Laval
Collection:Thèses et mémoires

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