Article

Journal of Fourier Analysis and Applications

, Volume 5, Issue 5, pp 465-494

First online:

Wavelets, generalized white noise and fractional integration: The synthesis of fractional Brownian motion

  • Yves MeyerAffiliated withDépartement de Mathématiques, Ecole Normale Supérieure de Cachan
  • , Fabrice SellanAffiliated withMatra Systèmes et InformationLaboratoire Analyse et Modeles Stochastiques
  • , Murad S. TaqquAffiliated withDepartment of Mathematics and Statistics, Boston University

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Abstract

We provide an almost sure convergent expansion of fractional Brownian motion in wavelets which decorrelates the high frequencies. Our approach generalizes Lévy's midpoint displacement technique which is used to generate Brownian motion. The low-frequency terms in the expansion involve an independent fractional Brownian motion evaluated at discrete times or, alternatively, partial sums of a stationary fractional ARIMA time series. The wavelets fill in the gaps and provide the necessary high frequency corrections. We also obtain a way of constructing an arbitrary number of non-Gaussian continuous time processes whose second order properties are the same as those of fractional Brownian motion.

Keywords and Phrases

Fractional ARIMA midpoint displacement technique fractional Gaussian noise fractional derivative generalized functions self-similarity

Math Subject Classifications

Primary 60G18 secondary 41A58 60F15.