zbMATH — the first resource for mathematics

The Blackwell and Dubins theorem and Rényi’s amount of information measure: Some applications. (English) Zbl 1193.94045
In this interesting survey paper, the author justifies how quantum-like concepts can be used in macroscopic environments. First, the author briefly describes a theory that allows the existence of a wave function as an information wave function; multiplicity of paths; and non-locality. He believes that this theory can be beneficial in microscopic setting such as economics. Then the author provides arguments for the justification for the use of the concept of a quantum mechanical wave function as an information wave function, \(\Psi (q)\). He present a convincing argument how \(|\Psi (q) |^2\), can be interpreted as a Radon-Nikodym derivative. Further, he shows how this derivative can be used in [A. Rényi’s, Proc. 4th Berkeley Symp. Math. Stat. Probab. 1, 547–561 (1961; Zbl 0106.33001)] measure of quantity of information. Finally, the author provides an interpretation of the D. Blackwell and L. Dubins [Ann. Math. Stat. 33, 882–886 (1962; Zbl 0109.35704)] Theorem using Rényi’s measure of quantity of information.

94A15 Information theory (general)
94A17 Measures of information, entropy
81P05 General and philosophical questions in quantum theory
Full Text: DOI
[1] Accardi, L., Boukas, A.: The quantum Black-Scholes equation. Glob. J. Pure Appl. Math. 2(2), 155–170 (2007) · Zbl 1157.91348
[2] Aerts, S., Aerts, D.: When can a data set be described by quantum theory? In: Bruza, P., Lawless, W., van Rijsbergen, K., Sofge, D., Coecke, B., Clark, S. (eds.) Second Quantum Interaction Symposium, Oxford University, pp. 27–33. College Publications, King’s College, London (2008) · Zbl 1192.81036
[3] Allais, M.: Le comportement de l’homme rationnel devant le risque: critique des postulates et axiomes de l’école américaine. Econometrica 21, 503–546 (1953) · Zbl 0050.36801 · doi:10.2307/1907921
[4] Anscombe, F., Aumann, R.: A definition of subjective probability. Ann. Math. Stat. 34, 199–205 (1963) · Zbl 0114.07204 · doi:10.1214/aoms/1177704255
[5] Arrow, K.: Essays in the Theory of Risk Bearing. North-Holland, Amsterdam (1971) · Zbl 0215.58602
[6] Ausloos, M., Pekalski, A.: Model of wealth and goods in a closed market. Physica A 373, 560–568 (2007) · doi:10.1016/j.physa.2006.04.112
[7] Baaquie, B.: Quantum Finance. Cambridge University Press, Cambridge (2005)
[8] Bachelier, L.: Théorie de la spéculation. Ann. Sci. Eco. Norm. Super. 3(17), 21–86 (1900) · JFM 31.0241.02
[9] Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–659 (1973) · Zbl 1092.91524 · doi:10.1086/260062
[10] Blackwell, D., Dubins, L.: Merging of opinions with increasing information. Ann. Math. Stat. 33, 882–886 (1961) · Zbl 0109.35704 · doi:10.1214/aoms/1177704456
[11] Broekaert, J., Aerts, D., D’Hooghe, B.: The generalized liar paradox: A quantum model and interpretation. Found. Sci. 11(4), 399–418 (2006) · Zbl 1114.03003 · doi:10.1007/s10699-004-6248-8
[12] Bohm, D.: A suggested interpretation of the quantum theory in terms of ’hidden’ variables, Part I and II. Phys. Rev. 85, 166–193 (1951) · Zbl 0046.21004 · doi:10.1103/PhysRev.85.166
[13] Bohm, D., Hiley, B.: The Undivided Universe. Routledge, New York (1993)
[14] Bordley, R.F.: Quantum mechanical and human violations of compound probability principles: Toward a generalized Heisenberg uncertainty principle. Oper. Res. 46, 923–926 (1998) · Zbl 0987.60008 · doi:10.1287/opre.46.6.923
[15] Borland, L.: A theory of non-Gaussian option pricing. Quant. Finance 2, 415–431 (2002)
[16] Bowman, G.: Essential Quantum Mechanics. Oxford University Press, Oxford (2008) · Zbl 1201.81001
[17] Busemeyer, J., Wang, Z.: Quantum information processing explanation for interactions between inferences and decisions. In: Bruza, P., Lawless, W., van Rijsbergen, K., Sofge, D. (eds.) AAAI Spring Symposium on Quantum Interaction, Stanford University, pp. 91–97. AAAI Press, Menio Park (2007)
[18] Busemeyer, J., Wang, Z., Townsend, J.T.: Quantum dynamics of human decision making. J. Math. Psychol. 50(3), 220–241 (2006) · Zbl 1186.91062 · doi:10.1016/j.jmp.2006.01.003
[19] Choustova, O.: Pilot wave quantum model for the stock market. In: Khrennikov, A. (ed.) Quantum Theory: Reconsideration of Foundations, pp. 41–58. Växjö, Sweden (2001)
[20] Choustova, O.: Quantum Bohmian model for financial markets. Physica A 374, 304–314 (2006) · doi:10.1016/j.physa.2006.07.029
[21] Choustova, O.: Toward quantum-like modelling of financial processes. J. Phys. Conf. Ser. (2007). doi: 10.1088/1742-6596/70/1/012006
[22] Conte, E., Todarello, O., Federici, A., Vitiello, F., Lopane, M., Khrennikov, A., Zbilut, J.: Some remarks on an experiment suggesting quantum-like behavior of cognitive entities and formulation of an abstract quantum mechanical formalism to describe cognitive entity and its dynamics. Chaos Solitons Fractals 31, 1076–1088 (2007) · doi:10.1016/j.chaos.2005.09.061
[23] Cramer, J.G.: The transactional interpretation of quantum mechanics. Rev. Mod. Phys. 58(3), 647–687 (1986) · doi:10.1103/RevModPhys.58.647
[24] Danilov, V.I., Lambert-Mogiliansky, A.: Non-classical expected utility theory. Preprint Paris-Jourdan Sc. Econ. (2006) · Zbl 1197.91091
[25] Debreu, G.: Theory of Value: an Axiomatic Analysis of Economic Equilibrium. Wiley, New York (1959) · Zbl 0193.20205
[26] Deutsch, D.: Quantum theory of probability and decisions. Proc. R. Soc. Lond. A 455, 3129–3137 (1999) · Zbl 0964.81003 · doi:10.1098/rspa.1999.0443
[27] Dong, P.C., Kabgyun, J.: Quantum solution to the extended Newcomb’s paradox. arXiv:quant-ph/0511097v4 (2005). Accessed 3 July 2008
[28] Eisert, J., Wilkens, M., Lewenstein, M.: Quantum games and quantum strategies. Phys. Rev. Lett. 83, 3077–3080 (1999) · Zbl 0946.81018 · doi:10.1103/PhysRevLett.83.3077
[29] Ellsberg, D.: Risk, ambiguity and the savage axioms. Q. J. Econ. 75, 643–669 (1961) · Zbl 1280.91045 · doi:10.2307/1884324
[30] Ghirardato, P., Maccheroni, F., Marinacci, M.: Differentiating ambiguity and ambiguity attitude. J. Econ. Theory 118, 133–173 (2004) · Zbl 1112.91021 · doi:10.1016/j.jet.2003.12.004
[31] Haven, E.: Pilot-wave theory and financial option pricing. Int. J. Theor. Phys. 44, 1957–1962 (2005) · Zbl 1094.81015 · doi:10.1007/s10773-005-8973-3
[32] Haven, E.: Analytical solutions to the backward Kolmogorov PDE via an adiabatic approximation to the Schrödinger PDE. J. Math. Anal. Appl. 311, 439–444 (2005) · Zbl 1086.35512 · doi:10.1016/j.jmaa.2005.02.058
[33] Haven, E.: A survey of possible uses of quantum mechanical concepts in financial economics. In: Bruza, P., Lawless, W., van Rijsbergen, K., Sofge, D. (eds.) AAAI Spring Symposium on Quantum Interaction, Stanford University, pp. 166–169. AAAI Press, Menlo Park (2007)
[34] Haven, E.: The variation of financial arbitrage via the use of an information wave function. Int. J. Theor. Phys. 47, 193–199 (2008) · Zbl 1278.81054 · doi:10.1007/s10773-007-9506-z
[35] Heisenberg, W.: Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen. Z. Phys. 33, 879–893 (1925) · JFM 51.0728.07 · doi:10.1007/BF01328377
[36] Hiley, B.: Quantum reality unveiled through process and the implicate order. In: Bruza, P., Lawless, W., van Rijsbergen, K., Sofge, D., Coecke, B., Clark, S. (eds.) Second Quantum Interaction Symposium, Oxford University, pp. 1–10. College Publications, King’s College, London (2008)
[37] Holland, P.: The Quantum Theory of Motion: an Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics. Cambridge University Press, Cambridge (1993)
[38] Hudson, R.L., Parthasarathy, K.R.: Quantum Ito’s formula and stochastic evolutions. Commun. Math. Phys. 93, 301–323 (1984) · Zbl 0546.60058 · doi:10.1007/BF01258530
[39] Khrennikov, A.Yu.: Classical and quantum mechanics on information spaces with applications to cognitive, psychological, social and anomalous phenomena. Found. Phys. 29, 1065–1098 (1999) · doi:10.1023/A:1018885632116
[40] Khrennikov, A.Yu.: Quantum-like formalism for cognitive measurements. Biosystems 70, 211–233 (2003) · doi:10.1016/S0303-2647(03)00041-8
[41] Khrennikov, A.Yu.: Information Dynamics in Cognitive, Psychological and Anomalous Phenomena. Series in the Fundamental Theories of Physics. Kluwer Academic, Dordrecht (2004)
[42] Khrennikov, A.Yu.: On quantum-like probabilistic structure of mental information. Open Syst. Inf. Dyn. 11, 267–275 (2004) · Zbl 1059.82033 · doi:10.1023/B:OPSY.0000047570.68941.9d
[43] La Mura, P.: Projective Expected Utility. Mimeo, Leipzig Graduate School of Management (2006)
[44] Lux, Th., Marchesi, M.: Scaling and criticality in a stochastic multi-agent model of a financial market. Nature 397, 498–500 (1999) · doi:10.1038/17290
[45] Mackey, G.: Mathematical Foundations of Quantum Mechanics. Benjamin Books, New York (1963) · Zbl 0114.44002
[46] Mantegna, R., Stanley, H.E.: An Introduction to Econophysics: Correlations and Complexity in Finance. Cambridge University Press, Cambridge (1999) · Zbl 1138.91300
[47] Nakata, H.: On the dynamics of endogenous correlations of beliefs. Ph.D. Thesis, Stanford University (2000)
[48] Nozik, R.: Newcomb’s problem and two principles of choice. In: Rescher, N., et al. (eds.) Essays in Honour of Carl G. Hempel, pp. 114–146. Reidel, Dordrecht (1969)
[49] Primbs, J.: Measure theory in a lecture. http://www.stanford.edu/\(\sim\)japrimbs/MeasureTheory5.ppt . Accessed 28 September 2006
[50] Piotrowski, E.W., Sładkowski, J.: Quantum solution to the Newcomb’s paradox. Int. J. Quantum Inf. 1(3), 395–402 (2003) · Zbl 1070.81027 · doi:10.1142/S0219749903000279
[51] Piotrowski, E.W., Sładkowski, J.: Quantum-like approach to financial risk: quantum anthropic principle. Acta Phys. Pol. B 32, 3873 (2001) · Zbl 0986.91016
[52] Piotrowski, E.W., Sładkowski, J.: An invitation to quantum game theory. Int. J. Theor. Phys. 42, 1089–1099 (2003) · Zbl 1037.81020 · doi:10.1023/A:1025443111388
[53] Rabin, M., Thaler, R.H.: Anomalies: risk aversion. J. Econ. Perspect. 15, 219–232 (2001) · doi:10.1257/jep.15.1.219
[54] Rényi, A.: On measures of entropy and information. In: Neyman, J. (ed.) Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, California (1961) · Zbl 0106.33001
[55] Savage, L.J.: The Foundations of Statistics. Wiley, New York (1954) · Zbl 0055.12604
[56] Segal, W., Segal, I.E.: The Black-Scholes pricing formula in the quantum context. Proc. Natl. Acad. Sci. U.S.A. 95, 4072–4075 (1998) · Zbl 0903.90009 · doi:10.1073/pnas.95.7.4072
[57] Schrödinger, E.: Quantisierung als Eigenwertproblem. Ann. Phys. Berlin 79, 361–376 (1926) · JFM 52.0965.08 · doi:10.1002/andp.19263840404
[58] Shubik, M.: Quantum economics, uncertainty and the optimal grid size. Econ. Lett. 64, 277–278 (1999) · Zbl 0973.91530 · doi:10.1016/S0165-1765(99)00095-6
[59] von Neumann, J.: Mathematische Grundlagen der Quantenmechanik. Springer, Berlin (1932) · JFM 58.0929.06
[60] von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton (1947) · Zbl 1241.91002
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.