Construction method of coherent lower and upper previsions based on collection integrals
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Construction method of coherent lower and upper previsions based on collection integrals Serena Doria1 · Radko Mesiar2 · Adam Seliga2 Received: 15 January 2020 / Accepted: 2 March 2020 © Unione Matematica Italiana 2020
Abstract Given a finite non-empty set Ω a new type of integral, named super-additive integral, is proposed to define coherent lower previsions on the class of all bounded functions. It is the extension to the class of all bounded random variables of a shift-invariant collection integral with respect to a collection D and a capacity μ. Related coherent upper previsions are also considered. Keywords Coherent lower prevision · Collection integral · Construction method
1 Introduction Coherent lower previsions are operators defined on the linear space of all bounded random variables satisfying the axioms of coherence [14]. Coherent lower probabilities are obtained as restrictions of coherent lower previsions to indicator functions but coherent lower previsions cannot be determined by coherent lower probabilities. In [14, sect. 2.7.3] an example is given to show that lower probabilities may not determine lower previsions, that is different lower previsions yield the same lower probability when they are restricted to events. Coherent upper previsions are obtained by coherent lower previsions by means the conjugacy property. A way to define coherent lower and upper previsions is to assess them by integrals. Different types of integrals are proposed in literature, based on different collections of subsets of Ω and with respect to different capacities. Their properties depend on the collection considered to define the particular integral and the capacity with respect to the integral is
Dedicated to Professor Domenico Candeloro
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Serena Doria [email protected] Radko Mesiar [email protected] Adam Seliga [email protected]
1
Department of Engineering and Geology, University G. d’Annunzio, Chieti-Pescara, Italy
2
Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, 810 05 Bratislava, Slovakia
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S. Doria et al.
calculated. The Choquet integral [1,2,5] is based on chains of sets and the concave integral [9] deals with arbitrary finite set systems. These integrals are particular cases of decomposition integral [8,11] and they coincides with the Lebesgue integral when the capacity is additive. A coherent lower prevision can be defined by the Choquet integral with respect to a capacity if and only if the capacity is super-modular since in this case the Choquet integral in super-additive [14]. Integral representation of coherent upper conditional previsions by Choquet integral, pan integral and concave integral has been analyzed in [6]. The aim of this paper is to propose a new type of integral which satisfies the properties of a coherent lower prevision on the linear space of all bounded random variables. Given a finite non-empty set Ω, a collection D and a capacity μ in [13] the collection integral is defined for non-negative functions as a particular case of decomposit
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