  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\^ole
of probability theory.

Statistical Finance;Computational Engineering, Finance, and Science;Classical Analysis and ODEs;Probability;Computational Finance;Applications