Fractional and Multifractional fields from a wavelet point of view

Antoine Ayache

Description

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

Sujets

• Ayache
• Fractional Brownian motion
• Gaussian field

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 18
• Langue: English

Extrait

Fractional

A.Ayache

and

(USTL)

Multifractio of

nal fields view

Antoine Ayache

from

USTL (Lille) [e-mail]

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

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FBM is a quite classical example of a fractal field.It is denoted by {B H (t)} t∈R N it 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∈R N andt∈R N , by E  B H (s)B H (t)  = c 2 H  |s| 2H +|t| 2H − |s−t| 2H  (1)

{B 12 (t)} t∈R N is the usual Brownian motion. Up to a multiplicative constant, FBM can be represented, for allt

ξ −1 B H (t) = Z R N | e ξ i | H t +N2 dW(ξ)

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

A.Ayache (USTL)

{B H (at)} t∈R N f.d = .d. {a H B H (t)} t∈R N 

Fractional fields and Wavelets methods

∈R N , as

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The first problem:As FBM is a continuous Gaussian field, it can be represented on any compactK⊂R N as, +∞ B H (t) = X ǫ n f n (t) n=1

(3)

where: Theǫ n ’s are independentN(01)real-valued Gaussian random variables. Thef n ’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  t s ∈ u K p  n= X m ǫ n f n (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

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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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