Integral Representation of Coherent Lower Previsions by Super-Additive Integrals

Coherent lower previsions generalize the expected values and they are defined on the class of all real random variables on a finite non-empty set. Well known construction of coherent lower previsions by means of lower probabilities, or by means of super-modular capacities-based Choquet integrals, do...

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Bibliographic Details
Published in:Axioms 2020-06, Vol.9 (2), p.43
Main Authors: Doria, Serena, Mesiar, Radko, Šeliga, Adam
Format: Article
Language:English
Subjects:
Coherence
coherent lower prevision
collection integral
Decomposition
Expected values
Integrals
Mathematical analysis
Probability
Random variables
Real variables
shift-invariance
super-additive integral
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Summary:Coherent lower previsions generalize the expected values and they are defined on the class of all real random variables on a finite non-empty set. Well known construction of coherent lower previsions by means of lower probabilities, or by means of super-modular capacities-based Choquet integrals, do not cover this important class of functionals on real random variables. In this paper, a new approach to the construction of coherent lower previsions acting on a finite space is proposed, exemplified and studied. It is based on special decomposition integrals recently introduced by Even and Lehrer, in our case the considered decomposition systems being single collections and thus called collection integrals. In special case when these integrals, defined for non-negative random variables only, are shift-invariant, we extend them to the class of all real random variables, thus obtaining so called super-additive integrals. Our proposed construction can be seen then as a normalized super-additive integral. We discuss and exemplify several particular cases, for example, when collections determine a coherent lower prevision for any monotone set function. For some particular collections, only particular set functions can be considered for our construction. Conjugated coherent upper previsions are also considered.
ISSN:2075-1680
2075-1680
DOI:10.3390/axioms9020043