Quasi-convex risk measures and acceptability indices. Theory and applications.

This thesis is organized in two parts: in the first one, we treat risk measures and acceptability indices from a theoretical point of view, while in the second part, we propose two applications of the quasi-convex results. The first contribute of the thesis regards the dual representation of quasi-convex and monotone maps. In particular, we compare the dual representation proposed by Cerreia-Vioglio et al (2011) and Drapeau and Kupper (2010) and we prove that they coincide. On the light of this comparison, we also propose a new representation for the quasi-concave and monotone acceptability indices. In the second part of the thesis we propose two different applications of the quasi- convex analysis to different sectors. The common idea has been to build a quasi-convex risk measure defining a particular family of acceptability sets, taking inspiration from the papers of Cherny and Madan (2009) and Drapeau and Kupper (2010). The first application is to the financial sector. We introduce a new class of law invariant risk measures, directly defined on the set P(R) of probability measures on R, that are monotone and quasi-convex on P(R). We build these maps by an appropriate family of acceptance sets of distribution functions. We study the properties of such maps and we provide some example. In particular, we propose a generalization of the classical notion of the V@R_λ, called ΛV@R, that takes into account not only the probability λ of the losses, but the balance between such probability and the amount of the loss. The V@R_λ and other known law invariant risk measures turn out to be special cases of our proposal. We further prove the dual representation of Risk Measures on P(R). The second application is to the evaluation of the scientific research. We introduce a new family of scientific performance measures, called Scientific Research Measures (SRM). This proposal originates from the recent developments in the theory of quasi-convex risk measures and is an attempt to resolve the many problems of the existing bibliometric indices. The key idea underlying the definition of SRM is the representation of quasi-concave monotone maps in terms of a family of acceptance sets. Through this approach, the SRMs are: flexible to fit peculiarities of different areas and seniorities; inclusive, as they comprehend several popular indices; coherent, as they share the same structural properties; calibrated to the particular scientific community; granular, as they allow a more precise comparison between scientists and are based on the whole scientist’s citation curve. We also provide a dual representation of a SRM, that suggests a new interesting approach to the whole area of bibliometric indices. Finally, we present a method to compute a particular SRM, called φ-index, and some result obtained by the calibration to a specific scientific sector.