On wavelet analysis of the nth order fractional Brownian motion

2012 ◽  
Vol 21 (3) ◽  
pp. 251-277
Author(s):  
Hedi Kortas ◽  
Zouhaier Dhifaoui ◽  
Samir Ben Ammou
Keyword(s):  
Brownian Motion ◽  
Wavelet Analysis ◽  
Fractional Brownian Motion

scholarly journals DEFINITION, PROPERTIES AND WAVELET ANALYSIS OF MULTISCALE FRACTIONAL BROWNIAN MOTION

Fractals ◽  
2007 ◽  
Vol 15 (01) ◽  
pp. 73-87 ◽  
Author(s):  
JEAN-MARC BARDET ◽  
PIERRE BERTRAND
Keyword(s):  
Brownian Motion ◽  
Wavelet Analysis ◽  
Fractional Brownian Motion ◽  
Long Memory ◽  
Constant Function ◽  
Piecewise Constant Function ◽  
Model Data ◽  
Fractional Brownian Motions ◽  
Piecewise Constant ◽  
Functional Central Limit

In some applications, for instance, finance, biomechanics, turbulence or internet traffic, it is relevant to model data with a generalization of a fractional Brownian motion for which the Hurst parameter H is dependent on the frequency. In this contribution, we describe the multiscale fractional Brownian motions which present a parameter H as a piecewise constant function of the frequency. We provide the main properties of these processes: long-memory and smoothness of the paths. Then we propose a statistical method based on wavelet analysis to estimate the different parameters and prove a functional Central Limit Theorem satisfied by the empirical variance of the wavelet coefficients.

scholarly journals Wavelet analysis of the multivariate fractional Brownian motion

2013 ◽  
Vol 17 ◽  
pp. 592-604 ◽  
Author(s):  
Jean-François Coeurjolly ◽  
Pierre-Olivier Amblard ◽  
Sophie Achard
Keyword(s):  
Brownian Motion ◽  
Wavelet Analysis ◽  
Fractional Brownian Motion

scholarly journals A Wavelet-Based Almost-Sure Uniform Approximation of Fractional Brownian Motion with a Parallel Algorithm

2014 ◽  
Vol 51 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Dawei Hong ◽  
Shushuang Man ◽  
Jean-Camille Birget ◽  
Desmond S. Lun
Keyword(s):  
Brownian Motion ◽  
Convergence Rate ◽  
Fractional Brownian Motion ◽  
Parallel Algorithm ◽  
Uniform Approximation ◽  
Haar Wavelets ◽  
Hurst Index ◽  
Sample Paths ◽  
Vanishing Moment ◽  
Uniform Expansion

We construct a wavelet-based almost-sure uniform approximation of fractional Brownian motion (FBM) (Bt(H))_t∈[0,1] of Hurst index H ∈ (0, 1). Our results show that, by Haar wavelets which merely have one vanishing moment, an almost-sure uniform expansion of FBM for H ∈ (0, 1) can be established. The convergence rate of our approximation is derived. We also describe a parallel algorithm that generates sample paths of an FBM efficiently.

Global attracting sets of neutral stochastic functional integro-differential equations driven by a fractional Brownian motion

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
A. Bakka ◽  
S. Hajji ◽  
D. Kiouach
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Brownian Motion ◽  
Differential Equations ◽  
Fractional Brownian Motion ◽  
Sufficient Conditions ◽  
Integrodifferential Equations ◽  
Hurst Parameter ◽  
Fixed Point Principle ◽  
Stochastic Functional

Abstract By means of the Banach fixed point principle, we establish some sufficient conditions ensuring the existence of the global attracting sets of neutral stochastic functional integrodifferential equations with finite delay driven by a fractional Brownian motion (fBm) with Hurst parameter H ∈ ( 1 2 , 1 ) {H\in(\frac{1}{2},1)} in a Hilbert space.

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