From Mutifractional Brownian Motion to Multifractional Process with Random Exponent
19
pages
English
Documents
Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus
Découvre YouScribe et accède à tout notre catalogue !
Découvre YouScribe et accède à tout notre catalogue !
19
pages
English
Documents
Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus
- completely makes sense
- white noise
- property can
- noise dw?
- random exponent
- t?r defined
- jx ?
Publié par
Langue
English
FFCassinoreromA.AMultifractionalMPREBro19wnianerMotionachetoMBMMultifractionalDecemPro1cessrwithDecembRandom2010Expyonent(USTL)AntoineromAtoyCassinoachebUSTL2010(Lille)/Antoine.Ayache@math.univ-lille1.ff ( )g [ ; ] ( ; )2R
! f ( )g2R
c
Z
c( ) = ();
( )+ =jjR
f ( )g2R
! f ( )g2R
c
(1)Cleacessrlyd,casethisbisepSots(USTL)sibleelwheonnthehastictSThecatthestodened.atot/yabmainiscanindepextendedendentronZtheesswhite.noiseredtFBMwithW,(PchasticapanicolaounoandyK.romSolna):Cassinoin2010thiswcase,l-denedthnethestoresultscMBMhasticbintegralreadilyZtoofptoer1rametcpa0?tetitbromocangeneralewhereendentSthetnoiseintervalpxedreHurstbthedep1onreplacewhiteindvaluesWismohtrickytosincetstoossibleintegral1isplonger2A.AdacheIsFWMBMmotivationMPREandDecemductioner1-Intro2(1)19is+1X X
( ; ) = ( ; ) ( ; ) ;;
= 1 2Z
f g N ( ; ); ( ; )2Z
1 R ( ; )
( ; )2N
n o ( )+j j (@ )( ; )j : ( ; )2R ( ; ) < +1:
1sHtothat,chasticAintegraluniforep(2)resentationandoHfriFBM.secondNoteofthat:,moTthejjjHkktheinbwinjwhichxrkinsteadaveletgeneralBcFBM:x2aqquisxa2sequencekofxinhavedep0endentj,vawhya0le,isrmly1theThisone,Gaussianmeansrandomfovaallriables;nofpisrea2Csupentation1functionase,onkap(3)Hyk(USTL)nromxtoxCassinoHby/achereoverjHit2ispwropell-loosedcalizedtoinusethestandarstresA.ArepacheseriesF19MBMrdMPRE1Decemrandomer20103,mo0( ; ) R ( ; )
( ; ) ( ; ( ))
f ( )g2R
+1X X
( )( ) = ( ; ( )) = ( ; ( )) ( ; ( )) ;;
= 1 2Z
R
! f ( )g2R
kSeachis,wheretinBcompletely.1,Thus,pwFetobtainwhythewithnonconvergentGaussianHpZrA.AoMPREcessSofmakZitresentationistseriesreprobabilitseriesnifottaveletsubsetwydenedoastZ(MPRE);achetMBMunifomlyDecem1toBsenseconvergentestk0SStinont(4)xtheHisrandompTheycompactujrmlytinponachcompactkofy.bTherobabilitrwithcessxcalled1,Proit,withtExp2isjSMultifractionalecess19Randomsubsetonentof.jyreplace(USTL)brom2010to/Cassinojterk4this2tf ( )g2R
( )> ( )2
f ( )g f ( )g2R 2R
P 8 2R : ( ) = ( ) = :
P 8 2R : ( ) ( ) = ;
P 8 2R : ( ) ( ) =
.robabilitMPRE:yach1fact,rescribthetpathsgiveoertfsincewiseSSandoint1tWpoitsbttthatpaisitsabre,H?ldeonyeachZcompact(JaaintervalremKto,willH?lderpfunctionsofoextendedfcanoZrderThisMBMhoffunctionalKfacteststinteremaxcanttmaine)KtSAtheaqqutTofrd,.SThentOneTheoMPRE1Z(5)ofetonlyytheritrtofregulathepthat1,elessttoyw.ropyp(USTL)niceromStotCassinorameterb12010(6)/thecaldeterministicZvialoedtptheZ2-Onttonentwithexpthat,rsatises:AssumeH?lderisthedicultpshoointA.AwiseacheH?lderFexpMBMonentMPREofDecemtheerp5ro19cessR < < <
>
; 2 [ ; ]
j ( ; ) ( ; )j j j:
2
2 ( ; ) f ( )g2R
f ( ; )g2R
P ( ) :8 2R 8 2 ( ; ) = :
allattConeahasnitealmostonensurely.foandrdenotealltHoint1FBMwthereHb2HresultHtheFayobtainHbeothe,H?ldersupoftBthHK,thasBBwtoxallalloandHcompact1rlemmaseBbomomentxBwofH02ttriablewingpfollowiseCexpTheterthe6va19random2asuchis(8)ty1romOnetowCassinoantbHrtevery0xedrealsHtrder,porove.0Kany011ofo,1w1A.ALemmaache(USTL)F(7)MBMLemmaMPRE2DecemFo2010H/1H2
( ) = ( ; ( ))
