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Fractional Brownian motion synthesis

Starting from the expression of the `fBm`

process as a fractional
integral of the white noise process, the idea of the algorithm is to build a
biorthogonal wavelet depending on a given orthogonal one and adapted to the parameter
`H`

.

Then the generated sample path is obtained by the reconstruction using the new wavelet starting from a wavelet decomposition at a given level designed as follows: details coefficients are independent random Gaussian realizations and approximation coefficients come from a fractional ARIMA process.

This method was first proposed by Meyer and Sellan and implementation issues were examined by Abry and Sellan [1].

Nevertheless, the samples generated following this original scheme exhibit too many high-frequency components. To circumvent this undesirable behavior Bardet et al. [2] propose downsampling the obtained sample by a factor of 10.

Two internal parameters `delta = 10`

(the downsampling factor) and a
threshold `prec = 1E-4`

, to evaluate series by truncated sums, can be
modified by the user for extreme values of `H`

.

A complete overview of long-range dependence process generators is available in Bardet et al [2].

[1] Abry, Patrice, and Fabrice
Sellan. “The Wavelet-Based Synthesis for Fractional Brownian Motion Proposed by F.
Sellan and Y. Meyer: Remarks and Fast Implementation.” *Applied and
Computational Harmonic Analysis* 3, no. 4 (October 1996): 377–83.
https://doi.org/10.1006/acha.1996.0030.

[2] Bardet, Jean-Marc, Gabriel
Lang, Georges Oppenheim, Anne Philippe, Stilian Stoev, and Murad S. Taqqu. “Generators
of Long-Range Dependent Processes: A Survey.” In *Theory and Applications of
Long-Range Dependence*, edited by Paul Doukhan, Georges Oppenheim, and
Murad S. Taqqu, 579–623. Boston: Birkhauser, 2003.