Fractional and Multifractional fields from a wavelet point of view

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Fractional and Multifractional fields from a wavelet point of view Antoine Ayache USTL (Lille) February 15, 2010 A.Ayache (USTL) Fractional fields and Wavelets methods February 15, 2010 1 / 55

  • fractional brownian

  • hurst parameter

  • gaussian field

  • real-valued continuous

  • fbm

  • motion ?

  • multifractional brownian

  • main motivation


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Fractional
A.Ayache
and
(USTL)
Multifractio of
nal fields view
Antoine Ayache
from
USTL (Lille) Antoine.Ayache@math.univ-lille1.fr
February 15, 2010
Fractional fields and Wavelets methods
a
wavelet
point
February 15, 2010
1 / 55
1-Introduction
The goal of our talk:To present the main ideas of the solutions of 3 connected problems in the setting ofmultivariate Fractional Brownian Motion (FBM)andmultivariate Multifractional Brownian Motion (MBM).
Though it isrich enough(FBM is non-Markovian nor a semimartingale), this setting remainssimple enough(thus we can avoid some technical complications).
Wavelet methods will play a crucial role in the solutions of these problems.
A.Ayache (USTL)
Fractional fields and Wavelets methods
February 15, 2010
2 / 55
FBM is a quite classical example of a fractal field.It is denoted by {BH(t)}tRNit depends on a unique parametersince H(01), calledthe Hurst parameter.covariance kernel of this centered Gaussian fieldThe with stationary increments is given, for allsRNandtRN, by EBH(s)BH(t)=c2H|s|2H+|t|2H− |st|2H(1)
{B12(t)}tRNis the usual Brownian motion. Up to a multiplicative constant, FBM can be represented, for allt
ξ1 BH(t) =ZRN|eξi|Ht+N2dW(ξ)
It is a fractal objet since, for alla>0,
A.Ayache (USTL)
{BH(at)}tRNf.d=.d.{aHBH(t)}tRN
Fractional fields and Wavelets methods
RN, as
February 15, 2010
(2)
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The first problem:As FBM is a continuous Gaussian field, it can be represented on any compactKRNas, +BH(t) =Xǫnfn(t)n=1
(3)
where: Theǫn’s are independentN(01)real-valued Gaussian random variables. Thefn’s are deterministic real-valued continuous functions overK. The series in (3) is, with probability 1, uniformly convergent intK.
The representation (3) is far from being unique and it seems natural to look foroptimalrepresentations i.e. +EtsuKpn=Xmǫnfn(t)−→0as fast as possible, whenm+.
In section 2 we will introduce a radom wavelet series represention of FBM and show that it is optimal. A.Ayache (USTL) Fractional fields and Wavelets methods February 15, 2010 4
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Let us now present the main motivations behind the second problem.
Tough, FBM turned out to be very useful in many areas (signal and image processing, telecommunication, ...) it has some drawbacks; an important one is thatthe local Hölder regularity of FBM remains the same all along its trajectory:
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