Marta Cardin

Invariant functionals on completely distributive lattices

In this paper we are interested in functionals defined on completely distributive lattices and which are invariant under mappings preserving arbitrary joins and meets. We prove that the class of nondecreasing invariant functionals coincides with the class of Sugeno integrals associated with {0,1}-valued capacities, the so-called term functionals, thus extending previous results both to the infinitary case as well as to the realm of completely distributive lattices. Furthermore, we show that, in the case of functionals over complete chains, the nondecreasing condition is redundant. Characterizations of the class of Sugeno integrals, as well as its superclass comprising all polynomial functionals, are provided by showing that the axiomatizations (given in terms of homogeneity) of their restriction to finitary functionals still hold over completely distributive lattices. We also present canonical normal form representations of polynomial functionals on completely distributive lattices, which appear as the natural extensions to their finitary counterparts, and as a by-product we obtain an axiomatization of complete distributivity in the case of bounded lattices.

Characterization of Convex Premium Principles

In actuarial literature the properties of risk measures or insurance premium principles have been extensively studied . We propose a characterization of a particular class of coherent risk measures defined in [1]. The considered premium principles are obtained by expansion of TVar measures, consequently they look like very interesting in insurance pricing where TVar measures is frequently used to value tail risks.

A quantile approach to integration with respect to non-additive measures

The aim of this paper is to introduce some classes of aggregation functionals when the evaluation scale is a complete lattice. We focus on the notion of quantile of a lattice-valued function which have several properties of its real-valued counterpart and we study a class of aggregation functionals that generalizes Sugeno integrals to the setting of complete lattices. Then we introduce in the real-valued case some classes of aggregation functionals that extend Choquet and Sugeno integrals by considering a multiple quantile model generalizing the approach proposed in [3].

Benchmarking over Distributive Lattices

We provides an axiomatic characterization of preorders in lattices that are representable as benchmarking procedure. We show that the key axioms are related to compatibility with lattice operations.

Mean‐extended Gini portfolios personalized to the investor's profile

Purpose – Shalit and Yitzhaki presented the mean‐extended Gini (MEG) as a workable alternative to the Markowitz mean‐variance approach in 1984. Since then, the challenge has been to extend the MEG approach. The purpose of this paper is to propose a generalization of the MEG approach for making customized optimal asset allocation to control both down‐performance and/or up‐performance.Design/methodology/approach – The MEG approach is used to make strategical allocation tailored to the investor risk aversion and gain propensity measured by characteristic parameters of the extended Gini measures.Findings – The authors set up two optimization problems: the former focused on controlling the risk, the latter on emphasizing the potential gains. Sufficient conditions such that the efficient MEG‐risk frontier coincides with the inefficient MEG‐gain frontier are stated. In the realistic scenarios that portfolios have asymmetrical distributions and/or the investor profile is very conservative or very aggressive, the ...

AXIOMATIZATIONS OF SIGNED DISCRETE CHOQUET INTEGRALS

We study the so-called signed discrete Choquet integral (also called non-monotonic discrete Choquet integral) regarded as the Lovasz extension of a pseudo-Boolean function which vanishes at the origin. We present axiomatizations of this generalized Choquet integral, given in terms of certain functional equations, as well as by necessary and sufficient conditions which reveal desirable properties in aggregation theory.

Some classes of multivariate risk measures

In actuarial literature the properties of risk measures or insurance premium principles have been extensively studied. We propose a new kind of stop-loss transform and a related order in the multivariate setting and some equivalent conditions. In our work there is a characterisation of some particular classes of multivariate and bivariate risk measures and a new representation result in a multivariate framework.

Supermodular Binary Aggregation Operators and Copulae

This paper presents the role of copula functions in the theory of aggregation operators. In this context we are focusing our attention about several properties of aggregation functions, like supermodularity and Schur-concavity, studying also a decomposition of supermodular binary aggregation operators and copulae.

Building a Global Performance Indicator to Evaluate Academic Activity Using Fuzzy Measures

The aim of this note is to provide a global performance index that allows to evaluate the performance of each faculty member and which is able to consider the multidimensional nature of the academic activity in terms of research, teaching and other activities that academics should/might exercise. In order to model also the case in which there could be synergic and redundant connections among the different areas of the academic activity, we propose to use fuzzy measures and the Choquet integral as an aggregator of the different components.

Approximation of Fuzzy Measures Using Second Order Measures: Estimation of Andness Bounds

In this paper we analyze the compensation property of a second order fuzzy measure in the context of a multi-attribute problem. In particular, we show that the disjunction/conjunction behavior (andness/orness) changes with the number of criteria to be aggregated. Interpreting the spread between the maximum and the minimum orness as a measure of the representation capability, we obtain two bounds in function of which asymptotically converge to a limit interval.