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

Michel Fliess and Cédric Join ()
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Michel Fliess: LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau] - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique, ALIEN - Algebra for Digital Identification and Estimation - Centre Inria de l'Université de Lille - Inria - Institut National de Recherche en Informatique et en Automatique - Centre Inria de Saclay - Inria - Institut National de Recherche en Informatique et en Automatique - Centrale Lille - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique
Cédric Join: ALIEN - Algebra for Digital Identification and Estimation - Centre Inria de l'Université de Lille - Inria - Institut National de Recherche en Informatique et en Automatique - Centre Inria de Saclay - Inria - Institut National de Recherche en Informatique et en Automatique - Centrale Lille - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique, CRAN - Centre de Recherche en Automatique de Nancy - UHP - Université Henri Poincaré - Nancy 1 - INPL - Institut National Polytechnique de Lorraine - CNRS - Centre National de la Recherche Scientifique

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Abstract: 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.

Keywords: Financial time series; mathematical finance; technical analysis; trends; random walks; efficient markets; forecasting; volatility; heteroscedasticity; quickly fluctuating functions; low-pass filters; nonstandard analysis; operational calculus.; operational calculus (search for similar items in EconPapers)
Date: 2009-05
Note: View the original document on HAL open archive server: https://inria.hal.science/inria-00352834v1
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Citations: View citations in EconPapers (10)

Published in Systems Theory: Modelling, Analysis and Control, May 2009, Fes, Morocco. pp.43-62

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