A mathematical proof of the existence of trends in financial time series
AbstractWe 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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Bibliographic InfoPaper provided by HAL in its series Post-Print with number inria-00352834.
Date of creation: 2009
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Publication status: Published - Presented, Systems Theory: Modelling, Analysis and Control, 2009, Fes, Morocco
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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.;
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