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A mathematical proof of the existence of trends in financial time series

  • Michel Fliess


    (LIX - Laboratoire d'informatique de l'école polytechnique [Palaiseau] - Polytechnique - X - CNRS, ALIEN - Algebra for Digital Identification and Estimation - INRIA Lille - Nord Europe - INRIA - INRIA Saclay - Ile de France - INRIA - Ecole Centrale de Lille - Polytechnique - X - CNRS)

  • Cédric Join


    (ALIEN - Algebra for Digital Identification and Estimation - INRIA Lille - Nord Europe - INRIA - INRIA Saclay - Ile de France - INRIA - Ecole Centrale de Lille - Polytechnique - X - CNRS, CRAN - Centre de recherche en automatique de Nancy - CNRS - UHP - Université Henri Poincaré - Nancy 1 - INPL - Institut National Polytechnique de Lorraine)

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    We are settling a longstanding quarrel in quantitative finance by proving the existence of trends in financial time series thanks to a theorem due to P. Cartier and Y. Perrin, which is expressed in the language of nonstandard analysis (Integration over finite sets, F. & M. Diener (Eds): Nonstandard Analysis in Practice, Springer, 1995, pp. 195--204). Those trends, which might coexist with some altered random walk paradigm and efficient market hypothesis, seem nevertheless difficult to reconcile with the celebrated Black-Scholes model. They are estimated via recent techniques stemming from control and signal theory. Several quite convincing computer simulations on the forecast of various financial quantities are depicted. We conclude by discussing the rôle of probability theory.

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    Paper provided by HAL in its series Post-Print with number inria-00352834.

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    Date of creation: May 2009
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    Publication status: Published in A. EL JAI and L. AFIFI and E. ZERRIK. Systems Theory: Modelling, Analysis and Control, May 2009, Fes, Morocco. Presses Universitaires de Perpignan, pp.43-62, 2009, Etudes; SYSTEMS THEORY: MODELING, ANALYSIS and CONTROL
    Handle: RePEc:hal:journl:inria-00352834
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    1. Ait-Sahalia, Yacine & Lo, Andrew W., 2000. "Nonparametric risk management and implied risk aversion," Journal of Econometrics, Elsevier, vol. 94(1-2), pages 9-51.
    2. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-54, May-June.
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