692 research outputs found

    Aggregation Functionals on Complete Lattices

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    The aim of this paper is to introduce some classes of aggregation functionals when the evaluation scale is a complete lattice. Two different types of aggregation functionals are introduced and investigated. We consider a target-based approach that has been studied in Decision Theory and we focus on the equivalence between a utility-based approach and target-based approach. Moreover we study a class of aggregation functionals that generalizes Sugeno integrals to the setting of complete lattices

    Measure and integral with purely ordinal scales

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    We develop a purely ordinal model for aggregation functionals for lattice valued functions, comprising as special cases quantiles, the Ky Fan metric and the Sugeno integral. For modeling findings of psychological experiments like the reflection effect in decision behaviour under risk or uncertainty, we introduce reflection lattices. These are complete linear lattices endowed with an order reversing bijection like the reflection at 00 on the real interval [1,1][-1,1]. Mathematically we investigate the lattice of non-void intervals in a complete linear lattice, then the class of monotone interval-valued functions and

    A Quantile Approach to Integration with Respect to Non-additive Measures

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    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

    Invariant functionals on completely distributive lattices

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    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}\{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

    An Ordinal Approach to Risk Measurement

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    In this short note, we aim at a qualitative framework for modeling multivariate risk. To this extent, we consider completely distributive lattices as underlying universes, and make use of lattice functions to formalize the notion of risk measure. Several properties of risk measures are translated into this general setting, and used to provide axiomatic characterizations. Moreover, a notion of quantile of a lattice-valued random variable is proposed, which shown to retain several desirable properties of its real-valued counterpart.lattice; risk measure; Sugeno integral; quantile.

    On Equilibrium Prices in Continuous Time

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    We combine general equilibrium theory and theorie generale of stochastic processes to derive structural results about equilibrium state prices

    Note on the Core-Walras Equivalence Problem when the Commodity Space is a Banach Lattice

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    The core-Walras equivalence problem for an atomless economy is considered in the commodity space setting of Banach lattices. In particular, necessary and sufficient conditions on the commodity space in order for core-Walras equivalence to hold are established. In general, these conditions can be regarded as implying that an economy with a continuum of agents has indeed "many more agents than commodities". However, it turns out that there are special commoditiy spaces in which core-Walras equivalence holds for every atomless economy satisfying certain standard assumptions, but in which an atomless economy does not have the meaning of there being "many more agents than commodities."

    On equilibrium prices in continuous time

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    We combine general equilibrium theory and théorie générale of stochastic processes to derive structural results about equilibrium state prices.general equilibrium, continuous time finance, théorie générale of stochastic processes, asset pricing, state prices
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