The idempotent Radon--Nikodym theorem has a converse statement

Idempotent integration is an analogue of the Lebesgue integration where $\sigma$-additive measures are replaced by $\sigma$-maxitive measures. It has proved useful in many areas of mathematics such as fuzzy set theory, optimization, idempotent analysis, large deviation theory, or extreme value theory. Existence of Radon--Nikodym derivatives, which turns out to be crucial in all of these applications, was proved by Sugeno and Murofushi. Here we show a converse statement to this idempotent version of the Radon--Nikodym theorem, i.e. we characterize the $\sigma$-maxitive measures that have the Radon--Nikodym property.Comment: 13 page

States on pseudo effect algebras and integrals

We show that every state on an interval pseudo effect algebra $E$ satisfying some kind of the Riesz Decomposition Properties (RDP) is an integral through a regular Borel probability measure defined on the Borel $\sigma$-algebra of a Choquet simplex $K$. In particular, if $E$ satisfies the strongest type of (RDP), the representing Borel probability measure can be uniquely chosen to have its support in the set of the extreme points of $K.

Representation of maxitive measures: an overview

Idempotent integration is an analogue of Lebesgue integration where $\sigma$-maxitive measures replace $\sigma$-additive measures. In addition to reviewing and unifying several Radon--Nikodym like theorems proven in the literature for the idempotent integral, we also prove new results of the same kind.Comment: 40 page

The Lattice and Simplex Structure of States on Pseudo Effect Algebras

We study states, measures, and signed measures on pseudo effect algebras with some kind of the Riesz Decomposition Property, (RDP). We show that the set of all Jordan signed measures is always an Abelian Dedekind complete $\ell$-group. Therefore, the state space of the pseudo effect algebra with (RDP) is either empty or a nonempty Choquet simplex or even a Bauer simplex. This will allow represent states on pseudo effect algebras by standard integrals

The logical encoding of Sugeno integrals

International audienceSugeno integrals are a well-known family of qualitative multiple criteria aggregation operators. The paper investigates how the behavior of these operators can be described in a prioritized propositional logic language, namely possibilistic logic. The case of binary-valued criteria, which amounts to providing a logical description of the fuzzy measure underlying the integral, is first considered. The general case of a Sugeno integral when criteria are valued on a discrete scale is then studied

Capital allocation rules and the no-undercut property

This paper makes the point on a well known property of capital allocation rules, namely the one called no-undercut. Its desirability in capital allocation stems from some stability game theoretical features related to the notion of core, both for finite and infinite games. We review these aspects, by relating them to the properties of the risk measures involved in capital allocation problems. We also discuss some problems and possible extensions arising when we deal with non-coherent risk measures

History of cosmic evolution with modified Gauss-Bonnet-dilatonic coupled term

Gauss-Bonnet-dilatonic coupling in four dimension plays an important role to explain late time cosmic evolution. However, this term is an outcome of low energy string effective action and thus ought to be important in the early universe too. Unfortunately, phase-space formulation of such a theory does not exist in the literature due to branching. We therefore consider a modified theory of gravity, which contains a nonminimally coupled scalar-tensor sector in addition to higher order scalar curvature invariant term with Gauss-Bonnet-dilatonic coupling. Such an action unifies early inflation with late-time cosmic acceleration. Quantum version of the theory is also well-behaved.Comment: 13 pages, 5 figures, To appear in EPJC (2017

The Concept of Particle Weights in Local Quantum Field Theory

The concept of particle weights has been introduced by Buchholz and the author in order to obtain a unified treatment of particles as well as (charged) infraparticles which do not permit a definition of mass and spin according to Wigner's theory. Particle weights arise as temporal limits of physical states in the vacuum sector and describe the asymptotic particle content. Following a thorough analysis of the underlying notion of localizing operators, we give a precise definition of this concept and investigate the characteristic properties. The decomposition of particle weights into pure components which are linked to irreducible representations of the quasi-local algebra has been a long-standing desideratum that only recently found its solution. We set out two approaches to this problem by way of disintegration theory, making use of a physically motivated assumption concerning the structure of phase space in quantum field theory. The significance of the pure particle weights ensuing from this disintegration is founded on the fact that they exhibit features of improper energy-momentum eigenstates, analogous to Dirac's conception, and permit a consistent definition of mass and spin even in an infraparticle situation.Comment: PhD thesis, 124 pages, amslatex, mathpt

Quantitative risk assessment, aggregation functions and capital allocation problems

[eng] This work is focused on the study of risk measures and solutions to capital allocation problems, their suitability to answer practical questions in the framework of insurance and financial institutions and their connection with a family of functions named aggregation operators. These operators are well-known among researchers from the information sciences or fuzzy sets and systems community. The first contribution of this dissertation is the introduction of GlueVaR risk measures, a family belonging to the more general class of distortion risk measures. GlueVaR risk measures are simple to understand for risk managers in the financial and insurance sectors, because they are based on the most popular risk measures (VaR and TVaR) in both industries. For the same reason, they are almost as easy to compute as those common risk measures and, moreover, GlueVaR risk measures allow to capture more intricated managerial and regulatory attitudes towards risk. The definition of the tail-subadditivity property for a pair of risks may be considered the second contribution. A distortion risk measure which satisfies this property has the ability to be subadditive in extremely adverse scenarios. In order to decide if a GlueVaR risk measure is a candidate to satisfy the tail-subadditivity property, conditions on its parameters are determined. It is shown that distortion risk measures and several ordered weighted averaging operators in the discrete finite case are mathematically linked by means of the Choquet integral. It is shown that the overall aggregation preference of the expert may be measured by means of the local degree of orness of the distortion risk measure, which is a concept taken over from the information sciences community and brung into the quantitative risk management one. New indicators for helping to characterize the discrete Choquet integral are also presented in this dissertation. The aim is complementing those already available, in order to be able to highlight particular features of this kind of aggregation function. Following this spirit, the degree of balance, the divergence, the variance indicator and Rényi entropies as indicators within the framework of the Choquet integral are here introduced. A major contribution derived from the relationship between distortion risk measures and aggregation operators is the characterization of the risk attitude implicit into the choice of a distortion risk measure and a confidence or tolerance level. It is pointed out that the risk attitude implicit in a distortion risk measure is to some extent contained in its distortion function. In order to describe some relevant features of the distortion function, the degree of orness indicator and a quotient function are used. It is shown that these mathematical devices give insights on the implicit risk behavior involved in risk measures and entail the definitions of overall, absolute and specific risk attitudes. Regarding capital allocation problems, a list of key elements to delimit these problems is provided and mainly two contributions are made. Firstly, it is shown that GlueVaR risk measures are as useful as other alternatives like VaR or TVaR to solve capital allocation problems. The second contribution is understanding capital allocation principles as compositional data. This interpretation of capital allocation principles allows the connection between aggregation operators and capital allocation problems, with an immediate practical application: Properly averaging several available solutions to the same capital allocation problem. This thesis contains some preliminary ideas on this connection, but it seems to be a promising research field.[spa] Este trabajo se centra en el estudio de medidas de riesgo y de soluciones a problemas de asignación de capital, en su capacidad para responder cuestiones prácticas en el ámbito de las instituciones aseguradoras y financieras, y en su conexión con una familia de funciones denominadas operadores de agregación. Estos operadores son bien conocidos entre los investigadores de las comunidades de las ciencias de la información o de los conjuntos y sistemas fuzzy. La primera contribución de esta tesis es la introducción de las medidas de riesgo GlueVaR, una familia que pertenece a la clase más general de las medidas de riesgo de distorsión. Las medidas de riesgo GlueVaR son sencillas de entender para los gestores de riesgo de los sectores financiero y asegurador, puesto que están basadas en las medidas de riesgo más populares (el VaR y el TVaR) de ambas industrias. Por el mismo motivo, son casi tan fáciles de calcular como estas medidas de riesgo más comunes pero, además, las medidas de riesgo GlueVaR permiten capturar actitudes de gestión y regulatorias ante el riesgo más complicadas. La definición de la propiedad de la subadditividad en colas para un par de riesgos se puede considerar la segunda contribución. Una medida de riesgo de distorsión que cumple esta propiedad tiene la capacidad de ser subadditiva en escenarios extremadamente adversos. Con el propósito de decidir si una medida de riesgo GlueVaR es candidata a satisfacer la propiedad de la subadditividad en colas se determinan condiciones sobre sus parámetros. Se muestra que las medidas de riesgo de distorsión y varios operadores de medias ponderadas ordenadas en el caso finito y discreto están matemáticamente relacionadas a través de la integral de Choquet. Se muestra que la preferencia global de agregación del experto puede medirse usando el nivel local de orness de la medida de riesgo de distorsión, que es un concepto trasladado des de la comunidad de las ciencias de la información hacia la comunidad de la gestión cuantitativa del riesgo. Nuevos indicadores para ayudar a caracterizar las integrales de Choquet en el caso discreto también se presentan en esta disertación. Se pretende complementar a los existentes, con el fin de ser capaces de destacar características particulares de este tipo de funciones de agregación. Con este espíritu, se presentan el nivel de balance, la divergencia, el indicador de varianza y las entropías de Rényi como indicadores en el ámbito de la integral de Choquet. Una contribución relevante que se deriva de la relación entre las medidas de riesgo de distorsión y los operadores de agregación es la caracterización de la actitud ante el riesgo implícita en la elección de una medida de riesgo de distorsión y de un nivel de confianza. Se señala que la actitud ante el riesgo implícita en una medida de riesgo de distorsión está contenida, hasta cierto punto, en su función de distorsión. Para describir algunos rasgos relevantes de la función de distorsión se usan el indicador nivel de orness y una función cociente. Se muestra que estos instrumentos matemáticos aportan información relativa al comportamiento ante el riesgo implícito en las medidas de riesgo, y que de ellos se derivan las definiciones de les actitudes ante el riego de tipo general, absoluto y específico. En cuanto a los problemas de asignación de capital, se proporciona un listado de elementos clave para delimitar estos problemas y se hacen principalmente dos contribuciones. En primer lugar, se muestra que las medidas de riesgo GlueVaR son tan útiles como otras alternativas tales como el VaR o el TVaR para resolver problemas de asignación de capital. La segunda contribución consiste en entender los principios de asignación de capital como datos composicionales. Esta interpretación de los principios de asignación de capital permite establecer conexión entre los operadores de agregación y los problemas de asignación de capital, con una aplicación práctica inmediata: calcular debidamente la media de diferentes soluciones disponibles para el mismo problema de asignación de capital. Esta tesis contiene algunas ideas preliminares sobre esta conexión, pero parece un campo de investigación prometedor