On the relation between the fractional Brownian motion and the fractional derivatives

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Abstract

The definition and simulation of fractional Brownian motion are considered from the point of view of a set of coherent fractional derivative definitions. To do it, two sets of fractional derivatives are considered: (a) the forward and backward and (b) the central derivatives, together with two representations: generalised difference and integral. It is shown that for these derivatives the corresponding autocorrelation functions have the same representations. The obtained results are used to define a fractional noise and, from it, the fractional Brownian motion. This is studied. The simulation problem is also considered.
Original languageEnglish
Pages (from-to)958-968
Number of pages11
JournalPhysics Letters
VolumeA
Issue numberNA
Publication statusPublished - 11 Feb 2008

Keywords

  • Fractional stochastic process
  • Liouville derivative
  • Central fractional derivatives
  • Differintegration
  • Forward and backward fractional derivatives
  • Fractional Brownian motion
  • Generalised Cauchy derivative

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