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AN ITERATIVITY CONDITION FOR THE MEAN-VALUE PRINCIPLE UNDER CUMULATIVE PROSPECT THEORY

Published online by Cambridge University Press:  29 April 2013

Marek Kaluszka  and
Michał Krzeszowiec
Marek Kaluszka
Affiliation:
Institute of Mathematics, Ł ódź University of Technology, U1. Wólczańska 215, 90-924 Ł ódź, Poland E-mail: [email protected]
Michał Krzeszowiec*
Affiliation:
Institute of Mathematics of the Polish Academy of Sciences, Śniadeckich 8, P.O. Box 21, 00-956 Warszawa, Poland
*
Institute of Mathematics, Ł ódźUniversity of Technology, U1. Wólczańska 215, 90-924 Ł ódź, Poland E-mail: [email protected]
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Abstract

In this paper, we present the full characterization of the iterativity condition for the mean-value principle under the cumulative prospect theory. It turns out that the premium principle is iterative for exactly six pairs of probability distortion functions. Some of the corresponding premium principles are the classical mean-value principle, essential infimum or essential supremum of the random loss. Moreover, from the proof of the main theorem of this paper, it follows that the iterativity of the mean-value principle is equivalent to the iterativity of the generalized Choquet integral.

Keywords

Cumulative Prospect TheoryGeneralized Choquet Integraliterativitypremium principlesmean-value principledistorted probability
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 43 , Issue 1 , January 2013 , pp. 61 - 71
Copyright
Copyright © ASTIN Bulletin 2013

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References

Acciaio, B. and Penner, I. (2011) Dynamic convex risk measures. In Advanced Mathematical Methods for Finance (ed. Di Nunno, G. and Oksendal, B.), pp. 1–34. Berlin, Heidelberg: Springer.Google Scholar
Aliprantis, C.D. and Border, K.C. (2006) Infinite Dimensional Analysis: A Hitchhiker's Guide, 3rd ed.Berlin: Springer.Google Scholar
Al-Nowaihi, A., Bradley, I. and Dhami, S. (2008) A note on the utility function under prospect theory. Economics Letters, 99, 337–339.CrossRefGoogle Scholar
Bühlmann, H. (1970) Mathematical Methods in Risk Theory. Berlin: Springer.Google Scholar
Chateauneuf, A., Kast, R. and Lapied, A. (2010) Conditioning capacities and Choquet integrals: The role of comonotony. Theory and Decision, 51, 367–386.CrossRefGoogle Scholar
Denneberg, D. (1994) Lectures on Non-additive Measure and Integral. Boston: Kluwer Academic Publishers.CrossRefGoogle Scholar
Föllmer, H. and Schied, A. (2011) Stochastic Finance: An Introduction in Discrete Time. Berlin: Walter De Gruyter.CrossRefGoogle Scholar
Gerber, H.U. (1974) On iterative premium calculation principles. Bulletin of the Swiss Association of Actuaries, 163–172.Google Scholar
Gerber, H.U. (1979) An Introduction to Mathematical Risk Theory. Philadelphia, PA: Homewood.Google Scholar
Goovaerts, M.J. and De Vylder, F. (1979) A note on iterative premium calculation principles. ASTIN Bulletin, 10, 326–329.CrossRefGoogle Scholar
Goovaerts, M.J., De Vylder, F. and Haezendonck, J. (1984) Insurance Premiums: Theory and Applications. Amsterdam: North-Holland.Google Scholar
Goovaerts, M.J., Kaas, R. and Laeven, R.J.A. (2010) A note on additive risk measures in rank-dependent utility. Insurance: Mathematics and Economics, 47, 187–189.Google Scholar
Kaluszka, M. and Krzeszowiec, M. (2012) Mean-value principle under cumulative prospect theory. ASTIN Bulletin, 42, 103–122.Google Scholar
Kast, R., Lapied, A. and Toquebeuf, P. (2008) Updating Choquet integrals, consequentialism and dynamic consistency. Applied Mathematics Working Paper Series, Italy, Working Paper n. 4/2008.Google Scholar
Kuczma, M. (2009) An Introduction to the Theory of Functional Equations and Inequalities, 2nd ed. (ed. Gilányi, A.). Berlin: Birkhäuser.CrossRefGoogle Scholar
Kupper, M. and Schachermayer, W. (2009) Representation results for law invariant time consistent functions. Mathematical and Financial Economics, 2, 189–210.CrossRefGoogle Scholar
Lehrer, E. (2005). Updating non-additive probabilities — a geometric approach. Games and Economic Behavior, 50, 42–57.CrossRefGoogle Scholar
Luan, C. (2001) Insurance premium calculations with anticipated utility theory. ASTIN Bulletin, 31, 23–35.CrossRefGoogle Scholar
Segal, U. (1989) Anticipated utility theory: A measure representation approach. Annals of Operations Research, 19, 359–373.CrossRefGoogle Scholar
Tversky, A. and Kahneman, D. (1992) Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty, 5, 297–323.CrossRefGoogle Scholar
Van der Hoek, J. and Sherris, M. (2001) A class of non-expected utility risk measures and implications for asset allocation. Insurance: Mathematics and Economics, 28, 69–82.Google Scholar
Wang, S.S. and Young, V.R. (1998) Risk-adjusted credibility premiums using distorted probabilities. Scandinavian Actuarial Journal, 2, 143–165.CrossRefGoogle Scholar