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

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

1 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
2 ALIEN - Algebra for Digital Identification and Estimation
3 CRAN - Centre de Recherche en Automatique de Nancy

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.

Mots clés: 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

Domaines: Econométrie de la finance [q-fin.ST]; Ingénierie, finance et science [cs.CE]; Finance quantitative [q-fin.CP]; Traitement du signal et de l'image [eess.SP]; Statistiques [math.ST]; Théorie [stat.TH]; Logique [math.LO]; Analyse classique [math.CA]; Probabilités [math.PR]; Automatique; Traitement du signal et de l'image [eess.SP]; Applications [stat.AP]

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⟩