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Itô’s Lemma with Quantum Calculus (q-Calculus): Some Implications

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Abstract

q-derivatives are part of so called quantum calculus. In this paper we investigate how such derivatives can possibly be used in Itô’s lemma. This leads us to consider how such derivatives can be used in a social science setting. We conclude that in a Itô Lemma setting we cannot use a macroscopic version of the Heisenberg uncertainty principle with q-derivatives.

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References

  1. Kac, V., Cheung, P.: Quantum Calculus. Springer, Berlin (2002)

    Book  MATH  Google Scholar 

  2. Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  3. Andrews, G.E.: q-series: their development and application in analysis, number theory, combinatorics, physics and computer algebra. In: CBMS Regional Conference Lecture. Series in Mathematics, vol. 66. Am. Math. Soc., Providence (1986)

    Google Scholar 

  4. Haven, E.: Quantum calculus (q-calculus) and option pricing: a brief introduction. In: Bruza, P., Sofge, D., Lawless, W., van Rijsbergen, K. (eds.) Quantum Interaction. Lect. Notes in Artificial Intelligence, vol. 5494, pp. 308–314. Springer, Berlin (2009)

    Chapter  Google Scholar 

  5. Accardi, L., Boukas, A.: The quantum Black-Scholes equation. Glob. J. Pure Appl. Math. 2, 155–170 (2007)

    MathSciNet  Google Scholar 

  6. Hudson, R.L., Parthasarathy, K.R.: Quantum Itô’s formula and stochastic evolutions. Commun. Math. Phys. 93, 301–323 (1984)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  7. Meyer, P.A.: Quantum probability for probabilists, Lect. Notes in Math., vol. 1538 (1993)

  8. Aerts, D.: Quantum structure in cognition. J. Math. Psychol. 53, 314–348 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. Conte, E., Todarello, O., Federici, A., Vitiello, F., Lopane, M., Khrennikov, A., Zbilut, J.P.: 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 (2006)

    Article  ADS  Google Scholar 

  10. Grib, A., Khrennikov, A., Parfionov, J., Starkov, K.: Quantum equilibrium for macroscopic systems. J. Phys. A, Math. Gen. 39, 8461–8475 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. Khrennikov, A.: Information Dynamics in Cognitive, Psychological and Anomalous Phenomena, Ser. Fundamental Theories of Physics, vol. 138. Kluwer, Dordrecht (2004)

    MATH  Google Scholar 

  12. Choustova, O.: Toward quantum-like modelling of financial processes. J. Phys. Conf. Ser. 70 (2007)

  13. Hilger, S.: Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph.D. thesis, Universität Würzburg, Germany (1988)

  14. Hilger, S.: Analysis on measure chains—a unified approach to continuous and discrete calculus. Results Math. 18, 18–56 (1990)

    MATH  MathSciNet  Google Scholar 

  15. Bohner, M., Peterson, A.: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Basel (2001)

    MATH  Google Scholar 

  16. Bohner, M., Peterson, A.: Advances in Dynamic Equations on Time Scales. Birkhäuser, Basel (2003)

    Book  MATH  Google Scholar 

  17. Davis, J.M., Gravagne, I.A., Jackson, B.J., Marks II, R.J., Ramos, A.A.: The Laplace transform on time scales revisited. J. Math. Anal. Appl. 332, 1291–1307 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  18. Itô, K.: On stochastic differential equations. Mem. Am. Math. Soc. 4, 1–51 (1951)

    Google Scholar 

  19. Øksendal, B.: Stochastic Differential Equations. Springer, Berlin (1992)

    Google Scholar 

  20. Brzeźniak, Z., Zastawniak, T.: Basic Stochastic Processes. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  21. Wilmott, P.: Derivatives: The Theory and Practice of Financial Engineering. Wiley, New York (1999)

    Google Scholar 

  22. Baaquie, B.: Quantum Finance. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  23. Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–654 (1973)

    Article  Google Scholar 

  24. McDonald, R.L.: Derivatives Markets. Addison-Wesley, Reading (2003)

    Google Scholar 

  25. Baez, J.: This week’s finds in Mathematical Physics (Week 183). http://math.ucr.edu/home/baez/week183.html

  26. Segal, W., Segal, I.E.: The Black-Scholes pricing formula in the quantum context. Proc. Natl. Acad. Sci. USA 95, 4072–4075 (1998)

    Article  MATH  ADS  Google Scholar 

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  1. School of Management, University of Leicester, Leicester, UK

    Emmanuel Haven

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  1. Emmanuel Haven
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Correspondence to Emmanuel Haven.

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Haven, E. Itô’s Lemma with Quantum Calculus (q-Calculus): Some Implications. Found Phys 41, 529–537 (2011). https://doi.org/10.1007/s10701-010-9453-z

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  • DOI: https://doi.org/10.1007/s10701-010-9453-z

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