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

Michel Fliess; Cédric Join

inria-00352834, version 1

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

Michel Fliess () ^1, 2, Cédric Join () ^2, 3

Systems Theory: Modelling, Analysis and Control (2009) 43-62

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.

1: Laboratoire d'informatique de l'école polytechnique (LIX)
CNRS : UMR7161 – Polytechnique - X
2: ALIEN (INRIA Saclay - Ile de France/Inria Lille - Nord Europe)
INRIA – Polytechnique - X – Ecole Centrale de Lille – CNRS : UMR8146
3: Centre de recherche en automatique de Nancy (CRAN)
CNRS : UMR7039 – Université Henri Poincaré - Nancy I – Institut National Polytechnique de Lorraine (INPL)

inria-00352834, version 1
http://hal.inria.fr/inria-00352834
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From: Michel Fliess <>
Submitted on: Wednesday, 14 January 2009 08:12:29
Updated on: Monday, 1 June 2009 00:34:00

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arXiv: 0901.1945

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%% inria-00352834, version 1
%% http://hal.inria.fr/inria-00352834
@inproceedings{fliess:inria-00352834,
    hal_id = {inria-00352834},
    url = {http://hal.inria.fr/inria-00352834},
    title = {{A mathematical proof of the existence of trends in financial time series}},
    author = {Fliess, Michel and Join, C{\'e}dric},
    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 r{\^o}le 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.},
    language = {English},
    affiliation = {Laboratoire d'informatique de l'{\'e}cole polytechnique - LIX , ALIEN - INRIA Saclay - Ile de France/Inria Lille - Nord Europe , Centre de recherche en automatique de Nancy - CRAN},
    booktitle = {{Systems Theory: Modelling, Analysis and Control}},
    publisher = {Presses Universitaires de Perpignan},
    pages = {43-62},
    address = {Fes, Morocco},
    editor = {A. EL JAI and L. AFIFI and E. ZERRIK },
    series = {Etudes },
    audience = {international },
    year = {2009},
    pdf = {http://hal.inria.fr/inria-00352834/PDF/FES-Finance.pdf},
}

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