The von Neumann–Morgenstern rationality axioms and analytic inequalities

N. N. Osipov

Zapiski Nauchnykh Seminarov POMI, Investigations on applied mathematics and informatics. Part II–1, Tome 529 (2023), pp. 197-217

Résumé

Trading in an efficient market where the asset price behaves as a martingale leads to a zero expected payoff. However, the problem of how to make such trading as rational as possible remains meaningful and non-trivial: it turns out that there is a certain gap between trading that has zero profit expectation, but is still rational in the basic sense, and completely irrational economic behavior that violates the basic von Neumann–Morgenshtern rationality axioms. By solving the problem of describing this gap and finding optimal trading strategies that get into it, we will arrive at the Bellman functions that have previously arisen in solving completely abstract problems about finding sharp constants in inequalities from analysis. Namely, solving the economic problem in the absolute context, where the strategy to be chosen does not depend on the current wealth of the agent, we will arrive at the Bellman function related to the John–Nirenberg inequality in integral form. Solving the problem in a relative context, where all the agent's actions in the market are considered relative to his current wealth, we will arrive at the Bellman function related to the inequalities that describe the relationship between Gehring classes. Thus, we will obtain a natural economic interpretation for the listed inequalities and the Bellman functions associated with them.

Bibliographie

[1] T. C. Anderson and D. E. Weirich, “A dyadic gehring inequality in spaces of homogeneous type and applications”, N. Y. J. Math., 24 (2018), 1–19

[2] D. Blackwell and M. A. Girshick, Theory of games and statistical decisions, John Wiley Sons, 1954

[3] D. Dillenberger and R. V. Krishna, “Expected utility without bounds–a simple proof”, J. Math. Econ., 52 (2014), 143–147

[4] M. Dindoš and T. Wall, “The sharp $a_p$ constant for weights in a reverse-hölder class”, Rev. Mat. Iberoam., 25:2 (2009), 559–594

[5] P. C. Fishburn, “Bounded Expected Utility”, Ann. Math. Stat., 38:4 (1967), 1054–1060

[6] P. C. Fishburn, “Unbounded Expected Utility”, Ann. Stat., 3:4 (1975), 884–896

[7] H. Föllmer and A. Schied, Stochastic finance, De Gruyter, 2004

[8] F. W. Gehring, “The $l^p$-integrability of the partial derivatives of a quasiconformal mapping”, Acta Math., 130 (1973), 265–277

[9] F. John, L. Nirenberg, “On functions of bounded mean oscillation”, Commun. Pure Appl. Math., 14:3 (1961), 415–426

[10] D. M. Kreps, Notes on the theory of choice, Routledge, 2018

[11] P. Lévy, Théorie de l'addition des variables aléatoires, Gauthier-Villars, 1937

[12] J. W. Pratt, “Risk aversion in the small and in the large”, Econometrica, 32:1/2 (1964), 122–136

[13] L. Slavin, V. Vasyunin, “Sharp results in the integral-form john-nirenberg inequality”, Trans. Amer. Math. Soc., 363 (2011), 4135–4169

[14] J. Ville, étude critique de la notion de collectif, Ph.D. thesis, Faculté des sciences de l'Université de Paris, 1939

[15] R. von Mises, Wahrscheinlichkeitsrechnung und ihre anwendungen in der statistik und der theoretischen physik, v. 1, Vorlesungen aus dem Gebiete der Angewandten Mathematik, Franz Deuticke, 1931

[16] D. Williams, Probability with martingales, Cambridge University Press, 1991

[17] V. I. Vasyunin, “Tochnaya konstanta v obratnom neravenstve Geldera dlya makenkhauptovskikh vesov”, Algebra i analiz, 15:1 (2003), 73–117

[18] V. I. Vasyunin, “Vzaimnye otsenki $l^p$-norm i funktsiya Bellmana”, Zap. nauchn. semin. POMI, 355, 2008, 81–138

[19] Dzh. fon Neiman, O. Morgenshtern, Teoriya igr i ekonomicheskoe povedenie, Nauka, 1970 (russian)