标题： 金融时间序列中趋势存在的数学证明 显示英文标题

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

摘要： 我们通过证明金融时间序列中趋势的存在性，解决了定量金融领域的一个长期争论，这归功于P. Cartier和Y. Perrin的一个定理，该定理用非标准分析的语言表达（Integration over finite sets, F. & M. Diener (Eds): Nonstandard Analysis in Practice, Springer, 1995, pp. 195--204）。这些趋势可能与某些修改后的随机漫步范式和有效市场假说共存，但似乎难以与著名的Black-Scholes模型调和。它们是通过来自控制和信号理论的最新技术进行估计的。描述了对各种金融数量预测的几个相当有说服力的计算机模拟。最后，我们讨论了概率论的作用。 显示英文摘要

摘要： 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.