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Multifractal random walk.

E Bacry1, J Delour, J F Muzy

  • 1Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128 Palaiseau Cedex, France.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|August 11, 2001
PubMed
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We introduce multifractal random walks (MRWs), a novel class of processes with continuous dilation invariance and stationary increments. These models offer a flexible alternative to existing multifractal approaches.

Area of Science:

  • Stochastic processes
  • Statistical physics
  • Time series analysis

Background:

  • Classical multifractal models often rely on specific scale ratios.
  • Existing processes may lack continuous dilation invariance or stationary increments.
  • There is a need for flexible multifractal models in various scientific domains.

Purpose of the Study:

  • Introduce a new class of multifractal processes: multifractal random walks (MRWs).
  • Establish MRWs as the first multifractal process with continuous dilation invariance and stationary increments.
  • Provide a framework for controlling multifractal properties and correlation structures.

Main Methods:

  • Development of a novel class of stochastic processes (MRWs).
  • Mathematical analysis of dilation invariance and increment stationarity.

Related Experiment Videos

  • Parameterization of MRWs to control multifractal spectrum and correlations.

    • Main Results:

    • MRWs exhibit continuous dilation invariance and stationary increments.
    • Four key parameters directly control the multifractal spectrum and increment correlations.
    • MRWs provide a scale-ratio-free alternative to cascade models.
    • The framework allows for the construction of stationary multifractal processes and positive random measures.

    Conclusions:

    • MRWs represent a significant advancement in modeling complex, multifractal phenomena.
    • The introduced parameters offer precise control over process characteristics.
    • MRWs offer a versatile and powerful tool for analyzing data with multifractal properties.