From Fractional Brownian Motion to Multifractional
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- fbm can
- statistics since
- fractional brownian
- hurst parameter
- brownian motion
- wavelet series
- self- similar process
FBMFerr1omacheFDecemractionalCassinoBroA.AwnianFMotionMBMtoerMultifractional34BroDecembwnian2010MotionyAntoine(USTL)AromytoacheCassinoUSTLb(Lille)2010Antoine.Aya/che@math.univ-lille1.freriesMainbpa(USTL)rts4ofBthetosemina/ravelet1resentationIntroyductiromoCassinon2010andMBMmotivationW2sMBM:repdenitionofandA.ApacheropFerFBMtiesMBM3DecemTheereld2B34generatingf ( )g2R
2 ( ; )
2R 2R
E ( ) ( ) = jj +jj j j ;
>
! f ( )g= 2R
!
aitonlymainlyerdep:endsniceonwhereaisuniquetpatatisrameterFHHMotion0nianH0ofwH1oBroy,sincecalleA.AdMBMthetHurstapaHrameter.aTheendingcovatriancenaturalkwnianernelBofBthistceninterediGaussianinpirohascessropisachegiven,FBMfoDecemrtall2seractional(1)Fcandqutismotivationconstantanddep,onb.yFBMisaBextensionHBroItMotionisisince1is2usualtwniany(FBM)issNcess.theBBroHmotion.roFBMtimpprtantductionProbfractalbclitHand2S1-Introtacssitofmany2pHerties.exampleyclassical(USTL)/rombto2CassinoHbdenoted20103s34t8 >
f ( )g = f ( )g :2R 2R
8 ; ; 2R
E ( + ) ( ) ( + ) ( )
=E ( + ) ( ) ( ) ( )
= ( ):
Hsrycessincrementsthrothat:ph,Self-similarmeansself-similaaBBHsGaussian,BtscenteredHhBonlyt2010H/sf.d.d.meanstHBttheryBStationaHHHtsincrements.withBA.AtacherFHFBM0MBMthat:tBhhistItastationatyh(USTL)atromDecemtoerCassino0bBH4B34H> = 8 =
X
( ) = +1:
2Z
8 ; 2R
