×

Characterization of positive homogeneity for the principle of equivalent utility. (English) Zbl 1492.91282

Summary: We establish a characterization of positive homogeneity for the principle of equivalent utility under cumulative prospect theory. The characterization involves not only a form of the utility function, but also a relation between probability weighting functions for gains and losses.

MSC:

91G05 Actuarial mathematics
91B16 Utility theory
39B72 Systems of functional equations and inequalities
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Balbás, A., Balbás, R.: Minimax strategies and duality with applications in financial mathematics. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A. Mat. 105, 291-303 (2011) · Zbl 1260.91265
[2] Bowers, NL; Gerber, HU; Hickman, JC; Jones, DA; Nesbitt, CJ, Actuarial Mathematics (1986), Itasca, Illinois: The Society of Actuaries, Itasca, Illinois · Zbl 0634.62107
[3] Bühlmann, H., Mathematical Models in Risk Theory (1970), New York: Springer, New York · Zbl 0209.23302
[4] Centeno, M.L., Simoes, O.: Optimal reinsurance. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A. Mat. 103, 387-405 (2009) · Zbl 1181.91090
[5] Chudziak, J.: On existence and uniqueness of the principle of equivalent utility under Cumulative Prospect Theory. Insurance: Mathematics and Economics 79, 243-246 (2018) · Zbl 1401.91118
[6] Chudziak, J., Extension problems for principles of equivalent utility, Aequationes Math., 93, 217-238 (2019) · Zbl 1411.91274
[7] Chudziak, J.: On positive homogeneity and comonotonic additivity of the principle of equivalent utility under Cumulative Prospect Theory. Insur. Math. Econ. 94, 154-159 (2020) · Zbl 1454.91175
[8] Denneberg, D., Lectures on Non-Additive Measure and Integral (1994), Boston: Kluwer Academic, Boston · Zbl 0826.28002
[9] Denuit, M., Dhaene, J., Goovaerts, M., Kaas, R.: Actuarial Theory for Dependent Risks: Measures, Orders and Models. Wiley, Amsterdam (2006)
[10] Diestel, J., Uhl, J.: Vector measures. Mathematical Surveys, No. 15, American Mathematical Society (1977) · Zbl 0369.46039
[11] Föllmer, H.; Schied, A., Convex measures of risk and trading constraints, Financ. Stochast., 6, 429-447 (2002) · Zbl 1041.91039
[12] Goovaerts, M.; De Vylder, F.; Haezendonck, J., Insurance Premiums (1984), Theory and Applications: North-Holland, Theory and Applications · Zbl 0532.62082
[13] Goovaerts M., Kaas R., Laeven R.: A note on additive risk measures in rank-dependent utility. Insur. Math. Econ. 47, 187-189 (2010) · Zbl 1231.91190
[14] Heilpern, S.: A rank-dependent generalization of zero utility principle. Insur. Math. Econ. 33, 67-73 (2003) · Zbl 1058.91024
[15] Kaas, R.; Goovaerts, M.; Dhaene, J.; Denuit, M., Modern Actuarial Risk Theory (2008), Berlin: Springer, Berlin · Zbl 1148.91027
[16] Kahneman, D.; Tversky, A., Prospect theory: an analysis of decision under risk, Econometrica, 47, 263-291 (1979) · Zbl 0411.90012
[17] Kałuszka, M., Krzeszowiec, M.: Pricing insurance contracts under cumulative prospect theory. Insur. Math. Econ. 50, 159-166 (2012) · Zbl 1239.91080
[18] Kałuszka, M., Krzeszowiec, M.: On iterative premium calculation principles under cumulative prospect theory. Insur. Math. Econ. 52, 435-440 (2013) · Zbl 1284.91244
[19] Kuczma, M., An Introduction to the Theory of Functional Equations and Inequalities (2009), Berlin: Cauchy’s equation and Jensen’s inequality. Birkhäuser, Berlin · Zbl 1221.39041
[20] Nakano, Y., Efficient hedging with coherent risk measure, J. Math. Anal. Appl., 293, 345-354 (2004) · Zbl 1085.91032
[21] Reich, A., Homogeneous premium calculation principles. ASTIN, Bulletin, 14, 122-133 (1984)
[22] Ruszczyński, A.; Shapiro, A., Optimization of convex risk functions, Math. Oper. Res., 31, 433-452 (2006) · Zbl 1278.90283
[23] Rolski, T.; Schmidli, H.; Schmidt, V.; Teugels, J., Stochastic Processes for Insurance and Finance (1999), New York: Wiley, New York · Zbl 0940.60005
[24] Sobek, B., Pexider equation on a restricted domain, Demonstr. Math., 43, 81-88 (2010) · Zbl 1236.39023
[25] Tversky, A.; Kahneman, D., Advances in prospect theory: cumulative representation of uncertainty, J. Risk Uncertain., 5, 297-323 (1992) · Zbl 0775.90106
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.