Choquet integral calculus on a continuous support and its applications

• Authors: RIDAOUI Mustapha , GRABISCH Michel
• Languages of publication: EN

Abstracts

EN

The results of the calculation of the Choquet integral of a monotone function on the nonnegative real line have been described. Next, the authors prepresented Choquet integral of nonmonotone functions, by constructing monotone functions from nonmonotone ones by using the increasing or decreasing rearrangement of a nonmonotone function. Finally, this paper considers some applications of these results to the continuous agregation operator OWA, and to the representation of risk measures by Choquet integral.

Keywords

EN

Choquet integral; distorted Lebesgue measure; risk measure; OWA operator

• Publisher: Politechnika Wrocławska. Oficyna Wydawnicza Politechniki Wrocławskiej
• Journal: Operations Research and Decisions
• Year: 2016
• Volume: 26
• Issue: 1
• Pages: 73-93

Contributors

author

RIDAOUI Mustapha

• 1Paris School of Economics, University of Paris I, 106-112 Bd. de l’Hopital, 75013 Paris, France,

author

GRABISCH Michel

• 1Paris School of Economics, University of Paris I, 106-112 Bd. de l’Hopital, 75013 Paris, France,

References

• ARTZNER P., DELBAEN F., EBER J.M., HEATH D., Coherent measures of risk, Mathematical Finance, 1999, 4, 203.
• CHOQUET G., Theory of capacities, Annales de l’Institut Fourier, 1953, 5, 131.
• DENNEBERG D., Non-additive measure and integral, Kluwer Academic, 1994.
• DENUIT M., CHARPENTIER A., Mathématiques de l’assurance non-vie, Tome 1, principes fondamentaux de théorie du risque, Economica, 2004.
• FAIGLE U., GRABISCH M., A discrete Choquet integral for ordered systems, Fuzzy Sets Systems, 2011, 168, 3.
• GHOSSOUB M., Equimeasurable rearrangements with capacities, Mathematics of Operations Research, 2015, 40, 429.
• GRABISCH M., LABREUCHE C., Fuzzy measures and integrals in MCDA, [in:] J. Figueira, S. Greco, M. Ehrgott (Eds.), Multiple Criteria Decision Analysis, Kluwer Academic Publishers, 2004, 563.
• MUROFUSHI T., SUGENO M., An interpretation of fuzzy measures and the Choquet integral as an integral with respect to a fuzzy measure, Fuzzy Sets and Systems, 1989, 29, 201.
• MUROFUSHI T., SUGENO M., Some quantities represented by the choquet integral, Fuzzy Sets and Systems, 1993, 56, 229.
• NARUKAWA Y., Distances defined by Choquet integral, FUZZ-IEEE 2007, IEEE International Conference on Fuzzy Systems, July 2007, 511.
• NARUKAWA Y., TORRA V., Aggregation operators on the real line, Proc. 3rd International Workshop on Soft Computing Applications (SOFA 2009), Szeged, Hungary and Arad, Romania, 2009, 185.
• NARUKAWA Y., TORRA V., SUGENO M., Choquet integral with respect to a symmetric fuzzy measure of a function on the real line, Annals of Operations Research, 2012, 1.
• RALESCU A., RALESCU D., Extensions of fuzzy aggregation, Fuzzy Sets Systems, 1997, 86, 321.
• RALESCU D.A., SUGENO M., Fuzzy integral representation, Fuzzy Sets Systems, 1996, 84, 127.
• SCHMEIDLER D., Integral representation without additivity, Proc. American Mathematical Society, 97, 1986, 253.
• SCHMEIDLER D., Subjective probability and expected utility without additivity, Econometrica, 1989, 57 (3), 571.
• SUGENO M., Theory of fuzzy integrals and its applications, PhD thesis, Tokyo Institute of Technology, 1974.
• SUGENO M., A note on derivatives of functions with respect to fuzzy measures, Fuzzy Sets and Systems, 2013, 222, 1.
• SUGENO M., A way to Choquet calculus, IEEE Transactions on Fuzzy Systems, 2015, 23, 1439.
• WANG S., DHAENE J., Comonotonicity, correlation order and premium principles, Insurance, Mathematics and Economics, 1998, 22 (3), 235.
• WANG S., YOUNG V.R., PANJER H.H., Axiomatic characterization of insurance prices, Insurance, Mathematics and Economics, 1997, 21, 173.
• WIRCH J.L., HARDY R.M., Distortion risk measures, coherence and stochastic dominance, International congress on Insurance, Mathematics and Economics, 2002, 306.
• YAARI M., The dual theory of choice under risk, Econometrica, 1987, 55, 95.
• YAGER R.R., On ordered weighted averaging aggregation operators in multicriteria decision making, IEEE Trans. on Systems, Man and Cybernetics, 1988, 18, 183.

Identifiers

DOI

10.5277/ord160105