$Fractional and Multifractional fields from a wavelet point of view$

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

motion ?

multifractional brownian

main motivation

Voir

Publié par

profil-urra-2012

Nombre de lectures

Langue

English

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

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)}t∈RNit depends on a unique parametersince H∈(01), calledthe Hurst parameter.covariance kernel of this centered Gaussian ﬁeldThe with stationary increments is given, for alls∈RNandt∈RN, by EBH(s)BH(t)=c2H|s|2H+|t|2H− |s−t|2H(1)

{B12(t)}t∈RNis the usual Brownian motion. Up to a multiplicative constant, FBM can be represented, for allt

ξ−1 BH(t) =ZRN|eξi|Ht+N2dW(ξ)

It is a fractal objet since, for alla>0,

A.Ayache (USTL)

{BH(at)}t∈RNf.d=.d.{aHBH(t)}t∈RN

Fractional ﬁelds and Wavelets methods

∈RN, as

February 15, 2010

(2)

3 / 55

The ﬁrst problem:As FBM is a continuous Gaussian ﬁeld, it can be represented on any compactK⊂RNas, +∞ BH(t) =Xǫnfn(t) n=1

(3)

where: Theǫn’s are independentN(01)real-valued Gaussian random variables. Thefn’s are deterministic real-valued continuous functions overK. The series in (3) is, with probability 1, uniformly convergent int∈K.

The representation (3) is far from being unique and it seems natural to look foroptimalrepresentations i.e. +∞ Ets∈uKpn=Xmǫnfn(t)−→0 as 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

/ 55

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:

1.2

0.8

0.6

0.4

0.2

−0.2

−0.4

−0.6 0

0.1

0.2

0.3

Fractional Brownian Motion − H = 0.2