The Property of Dominance — From Products of Probabilistic Metric Spaces to (Quasi-)Copulas and Applications

Sprache der Bezeichnung:

Englisch

Original Kurzfassung:

The property of dominance has its roots in the framework of probabilistic metric spaces (PMS), based on a concept introduced by Karl Menger already in 1942. In such spaces, the distance between two objects is modelled by a probability distribution function rather than a single number. Besides dominance, PMS has induced several operations, in particular triangular norms (t-norms) and copulas which enjoyed further development and investigation mainly independent of its roots in PMS. For example, they turned out to be most relevant in further fields of mathematics: t-norms play an indispensable tool as models of conjunctions in multi-valued logics, in particular fuzzy logics, whereas copulas, and quasi-copulas as a more recently introduced related operation, are important in the fields of statistics, probability theory and financial mathematics.
Similarly, dominance rejoiced further development and intensive investigation in particular as a property of aggregation operators. It is essential in applications like flexible queries, evaluations of computer-assisted assessments as well as additive fuzzy preference structures relevant in decision models. Due to these developments also research on dominance restricted to the class of t-norms has made relevant progress during the last years.
The aims of this project therefore cover, based on recent results on dominance in the framework of aggregation operators, (a) the (re)investigation of products of PMS as well as probabilistic normed spaces, (b) research on dominance of (quasi-)copulas and based on these results (c) the quest for further applications in, e.g., financial mathematics.