A mathematical proof of the existence of trends in financial time series
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.
|Date of creation:||May 2009|
|Date of revision:|
|Publication status:||Published in A. EL JAI and L. AFIFI and E. ZERRIK. Systems Theory: Modelling, Analysis and Control, May 2009, Fes, Morocco. Presses Universitaires de Perpignan, pp.43-62, 2009, Etudes; SYSTEMS THEORY: MODELING, ANALYSIS and CONTROL|
|Note:||View the original document on HAL open archive server: https://hal.inria.fr/inria-00352834|
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- Yacine Ait-Sahalia & Andrew W. Lo, 2000.
"Nonparametric Risk Management and Implied Risk Aversion,"
NBER Working Papers
6130, National Bureau of Economic Research, Inc.
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