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

Michel Fliess 1, 2 Cédric Join 2, 3 
2 ALIEN - Algebra for Digital Identification and Estimation
Inria Lille - Nord Europe, Inria Saclay - Ile de France, Centrale Lille, X - École polytechnique, CNRS - Centre National de la Recherche Scientifique : UMR8146
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.
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Submitted on : Wednesday, January 14, 2009 - 8:12:29 AM
Last modification on : Friday, February 4, 2022 - 3:17:58 AM
Long-term archiving on: : Tuesday, June 8, 2010 - 8:02:27 PM


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  • HAL Id : inria-00352834, version 1
  • ARXIV : 0901.1945



Michel Fliess, Cédric Join. A mathematical proof of the existence of trends in financial time series. Systems Theory: Modelling, Analysis and Control, May 2009, Fes, Morocco. pp.43-62. ⟨inria-00352834⟩



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