Stochastic Dominance and Applications to Mathematical Finance and Economics

Dear Colleagues,

The stochastic dominance relations are (pre-) orders on sets of Borel probability measures on the real line (see Fishburn (2) for definitions, and Shah (13) for extensions to function spaces and stochastic processes). Their theoretical investigation and their applications have gained substantial importance in, among others, the fields of theoretical and empirical economics, welfare analysis and social choice, mathematical and empirical finance, and statistics/econometrics (see, inter alia, Kroll and Levy (5), McFadden (9), Levy (6), (7), Mosler and Scarsini (10), and Gayant and Le Pape (3)). The stochastic dominance relations are usually defined by complicated functional inequalities involving representations of the underlying probability distributions, which usually have equivalent characterizations in terms of classes of utility functions (see Fishburn (2), Levy (6), (7), and Levy and Levy (8)) inside the expected utility framework. Thereby, given such a relation and a pair of relevant probability measures, the dominance of one over the other is equivalent to the preference by every utility in the class in terms of expected utility. Hence, order properties of stochastic dominance relations are connected to properties of optimal choices with respect to products of preferences. Subsequently, their structure enables analyses and inferences on issues regarding optimal choice in frameworks of risk (and/or frameworks involving choice between distributional aspects of income, and thereby, poverty, social welfare, etc.), uniformly with respect to large classes of underlying preferences, and thus, they avoid parametric specifications. Their sample analogues, involving empirical distributions and/or other estimators of the underlying probability measures, are the building blocks of a rich statistical theory that enables a plethora of empirical applications (see McFadden (9), Horvath et al. (4), Scaillet and Topaloglou (12), Post (11), and Arvanitis et al. (1)). The relevant asymptotic analysis and the computational aspects of the inferential procedures regarding stochastic dominance are also fields of intensive research.

References

[1] Arvanitis, S., Hallam, M., Post, T., & Topaloglou, N., 2019, Stochastic spanning, Journal of Business & Economic Statistics, 37(4), 573-585.

[2] Fishburn, P. C., 1976, Continua of stochastic dominance relations for bounded probability distributions, Journal of Mathematical Economics, 3(3), 295-311.

[3] Gayant, J. P., & Le Pape, N., 2017, Increasing degree inequality, Journal of Mathematical Economics, 70, 185-189.

[4] Horvath, L., Kokoszka, P. & R. Zitikis, 2006, Testing for Stochastic Dominance Using the Weighted McFadden-type Statistic. Journal of Econometrics 133, 191-205.

[5] Kroll, Y., & Levy, H., 1980, Stochastic Dominance Criteria: A Review and Some New Evidence, Research in Finance, Vol. II, Greenwich: JAI Press, pp. 263–227.

[6] Levy, H., 1992, Stochastic Dominance and Expected Utility: Survey and Analysis, Management Science, 38, 555–593.

[7] Levy, H., 2015, Stochastic dominance: Investment decision making under uncertainty, Springer.

[8] M. Levy & Levy, H., 2002, Prospect Theory: Much Ado about Nothing?, Management Science 48, 1334-1349.

[9] McFadden, D., 1989, “Testing for Stochastic Dominance,” in Studies in the Economics of Uncertainty, eds. T. Fomby and T. Seo, New York: Springer-Verlag, pp. 113–134.

[10] Mosler, K., & Scarsini, M., 1993, Stochastic Orders and Applications, a Classified Bibliography, Berlin: Springer-Verlag.

[11] Post, T., 2003, Empirical Tests for Stochastic Dominance Efficiency, Journal of Finance 58, 1905-1932.

[12] Scaillet, O., & Topaloglou, N., 2010, Testing for stochastic dominance efficiency, Journal of Business & Economic Statistics, 28(1), 169-180.

[13] Shah, S. A. , 2017, How risky is a random process?, Journal of Mathematical Economics, 72, 70-81.

Dr. Stelios Arvanitis
Guest Editor

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• stochastic dominance relations
• functional inequalities
• efficient sets expected utility
• classes of utility functions
• order properties
• stochastic dominance tests
• limit theory
• computational aspects
• portfolio selection
• welfare analysis