ROUGH VOLTERRA EQUATIONS THE ALGEBRAIC

pages

Documents

Lire un extrait

Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus

Découvre YouScribe en t'inscrivant gratuitement

Je m'inscris

Découvre YouScribe en t'inscrivant gratuitement

Je m'inscris

pages

Ebook

Lire un extrait

Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus

Publié par

pefav

Nombre de lectures

ROUGH VOLTERRA EQUATIONS 1: THE ALGEBRAIC INTEGRATION SETTING. AURÉLIEN DEYA AND SAMY TINDEL Abstract. We deﬁne and solve Volterra equations driven by an irregular signal, by means of a variant of the rough path theory called algebraic integration. In the Young case, that is for a driving signal with Hölder exponent > 1/2, we obtain a global solution, and are able to handle the case of a singular Volterra coe?cient. In case of a driving signal with Hölder exponent 1/3 < ≤ 1/2, we get a local existence and uniqueness theorem. The results are easily applied to the fractional Brownian motion with Hurst coe?cient H > 1/3. 1. Introduction This article is the ﬁrst of a series of two papers dealing with Volterra equations driven by rough paths. For an arbitrary positive constant T , this kind of equation can be written, in its general form, as: yt = a+ ∫ t 0 (t, u, yu) dxu, for s ? [0, T ], (1) where x is a n-dimensional Hölder continuous path with Hölder exponent > 0, a ? ?d stands for an initial condition, and : ?+??+??d ? ?d,n is a smooth enough function. Motivated by the previous works on Volterra equations driven by a Brownian motion or a semi-martingale [2, 3, 15, 21], often in an anticipative context [1, 4, 5, 19, 18,

?su ?

unique ?? ?

then

dimensional hölder

algebraic integration

rough volterra

hölder exponent

smooth functions

called algebraic

Voir

Publié par

pefav

Nombre de lectures

Smooth function

> 1=2
1=3 < 1=2
H > 1=3
T
Z t
y =a + (t;u;y )dx ; s2 [0;T ];t u u
0
dx n > 0 a2R
d d;n :R R R !R+ +
H > 1=3
H < 1=2
x > 1=2
n H 2 (1=2; 1) :
2 d d;n[0;T ] R !R
x

t;u (t;u;z) = (t u) (z)
d d;n> 0 :R !R ;;
appliedtothefractionalalgebraicAbstrafractionalBrocwnianroughmotioase:nthreewithyHurstatcooeciencurrenttegration.easilymeansareterpretedresultsareThe,theorem.THE.13,1.ofIntrVoductionforThisfolloarticletiniswiththeassumingrstregularofeasenseseriestheofcotwwVo,papthenersedealingclassicalwithoVthisolterrapapequationsendrivwithespncusesbTheyaroughoungpaths.aFrorroughanarianarbitraryirregularprespositiveecanconstanintTheuniquenesssame,bthisthekindenoftoequationdrivcanbbdeneeAwritten,TIONinTERRAitsongeneral2form,toas:forandferencesexistencetheory).calbloknoathegetofedealingwsystems,ytwnianonendrivingexp.der,H?larticlewiththesignalcases:drivingoungathatfor-H?lderofpathcasetheInparticulart.-dimensionalecienpcoeterolterratheory(1)),whereofVaisbayarnough-dimensionaltoH?lderariables),conprotinequationuousepathsolvwithYH?lder3).expoungonenUndertaslcaseuwgto,ofnciensiaarespofrststandsvfor.anolterrainitialexpressedcondition,solvandWcaseSAMYheUR?LIENtsomehandleandtoTIONSableregulararesomeandOUGHsolution,Sectionglobal(wareferobtain[9,e14]isfurtheraresmoonothpathsenoughTfunctiothenest.ourMotivwledge,atedisbrstyccurrencetheapreviouserwwithorksolterraondrivVbolterraaequationsBrodrivmotionensignalbayisaMoreBroecicallywnianthemotiontorfoaonsemi-martingal3ewing[2,(i)3,Y15,c21],Whenoftenisincase,anconanuousticipativwitheYconIntext(in[1,for4,n5,fBm19,Hurst18,a20]am,iwcalledepathhathevandethattaktenvupofwy,signal,(roughlyan:ishe2010.(withMathematicsectctitsation.v60H20.wworshallandvases.thatpaths(1)StobhasticinolterraandFedBrothemotion.ounga(Sectionstraigh(ii)tforwYardsingularapplicationase:totheaconditionsfractionalinBropreviouswnianformotion,witheHurstableparameterhandletcaseonenaexpeH?ldertwithadmitting.singularitTwithhiectsitswilltboeariablesacequationshievNamelyedifthankscantoeaasveariationandofethect.roughTINDELpathANDtheoryDEYdueAtoforGubinelliSETTING.[INTEGRA11],ALGEBRAICwhose1:mainEQUAfeaturesOLareenough,recalledunderbconditionseloVwtheRprogramofDatedeningMarcand19,solving2000equationSubje(1)Classicin60H05,aKeypathdswisephrwRoughatheory;yc,Valloequations;wingractionalforwnianinstance1 > 1=2 1=2 < <
Rt
(t;u;y )dx y u u0
dC ([0;T ];R )1
dC ([0;T ];R )1
x 2 (1=3; 1=2)
n H 2 (1=3; 1=2)
[0;T ] T 2 (0;T ]0 0
y t2 [0;T ] (y) = y yst t s
(y)st
y
Z Zt s
(y) = (t;u;y )dx + [(t;u;y ) (s;u;y )] dx :st u u u u u
s 0

t
x jt sj

Rs
[(t;u;y ) (s;u;y )] dxu u u0
x

> 1=2

x
nDf f R
beiroughthepnterpretterpretedhoinon.someyroughapathhosense.ecauseAs.menthetionedsbshallefore,somewwemeansshallnoiseresortdirectintthisextendedcase2toathewithformalismlemmasiner:trot.ducedbin),[1n1],inwhichangeherallovwsequationuscotousproyvaedtheYexistencecaseandisuniquenesstegrationofstudyawithlo4cal.solution,nallydenedtheonnotationsatsmallpiner,terv5.3,aloungequationpinevearinginappntegralourforThissomeeinpresenthen),latterparametererstwuaHof(Sectiona5)..WimpecwiVllmethothenopgeneraloiintedtelloutcanthecoteccohnicalwdicultiesreonesmbustdevcopdrivetinwithAwhenDEYtryingsametoeextendthethissignalloofcalssolution.usHerebthenusedtohaseisatobriefequationskxeetcthwofwtheatsttermratetegralgyawtoeishathatvtuallyeerefolloclassicalwalloedtoinsolutionordertialtoroughobtainwhourharesults:tothethealgebraichereinptegrationonformalismInreliesbheawhatvilygeneralizedonincrementheshonotiontoofsolutionincremenints,roughwhic,hclassatsrweevsimplytgivien,alsointcasesystemsofroughamainlyfunctionItwithtooforonecoparame.terThefBme-dimensionalwatlytotviouslysetting,obe,orderberyaappliest(thisiswithpapsignalw-H?lderatanotioisalgebraicWhenhase:neededc3.toAequationtba-H?lderheuristicprolevSAMYel,,theecienmainregular.dierencewithbofetsingularwteen5classicalofdierenntial,equationsprodrivtecenpbonedyendix.roughtsignaltrosxingandhourtheVeolterrathesettingalidenedesorderinsolvtheourdepbendanceaofdtheoinincremenargumentHooughevrasThee(iii)see.SectionofthetheelongspwhenossibleinsolutionYonastheinwhossibleostillleitpastandofwilltheentrainducejectorysev.problemsIndeed,theifargumespacetsiswiagsolutiongettoglobalequationfor(1),dierenthensystemonethehascase.theexplainsiny(1)eequationvedecidedsolvcthenradicallycansettingetedWin4.comSectionaatidetailedpape[7].bthiswillreference,hywhicofanalysis,wcarefulcallaconerolutionalevts,wehowrequireswtegralgetinglobaloungtoY(1)thecaseofaextensiondrivingThisask.forywidebofwecieneloe(2)ItAsasonewmighertortanexpforectot,ntheluderstaintreatmentegralofinolterra(2)bcanexistingbpathseds,dealtbwith(i)justallaswstheconsiderclassicalmdiusionecasedrivingtreatedecieniwn([11].i)Inmethootherprwsenords,hereunderorkssuitableerfecregularitwyforbhe,oungwhenandebanfurthertoinariabletodenotedvapptheearingofisingularnecientheeaking,inHeretegrandhodoouresernotstructured:plaeycallaSectionprominenthetnrole.ofTheinsecondwhictermwillinethelaterrighSectiontishandotedsidetheofof(2)(1)isentheyonespwhicconhuousiscesstTINDELypicalNDofAthewhenVcoolterratsetting,isandSectionindealsvtheolvkindeequation,sathecowholecienpastUR?LIENofSectionfunctions,treats.caseItaisdrivistillgpAossibleandtotheretrievofesomesomehnical-H?lderareofospacetpato-incremenApptsLetfromnishthishiterminthanksductiontoythesomeregularitwhicyareofthroughouttopapwithwrespcall(1)ecttogradienitsofrstfunctionv,ariable,oninconditionsonand2w,wthetvstressf j
D f j
n;E;F U E C (U;F ) n
U F 2 (0; 1)

(n) (n)kD (x) D (y)kn; ; n;C (U;F ) = 2C (U;F ) : sup <1 :
kx ykx;y2U
C n

‘ > ‘ 0 V k 1 C (V )2 1 k
kg : [‘ ;‘ ] ! V g = 0 t = t1 2 tt i i+11 k
i k 1 (k 1)
C (V ) =[ C (V ) k1 k
k C (V )k
k+1X
i :C (V )!C (V ) (g) = ( 1)g ;k k+1 tt ^1 tttk+1 1 i k+1
i=1
^ti
= 0
C (V ) C (V ) ZC (V ) =C (V )\ BC (V ) =C (V )\ k k+2 k k k k

g2C h2C t;u;s2 [‘ ;‘ ]1 2 1 2
(g) =g g ; (h) =h h h :st t s sut st su ut
(C ;) ZC (V ) = k
BC (V ) k 1k
k 1 h2ZC (V ) f2C (V )k+1 k
h =f
h2C (V ) h = 02
h =f f2C (V )1
jC (V )2
k k 23
inanandtfactre,unique)wwederivdenoteebvyen,allspacesubsetectorkvBanacaytheesettiatingofEQUAcaonpaptin.uousbfunctions,,(4)erssetbopumbasicnLemmarealthearbitrarysuchoelemenwcantniforey:measuressucanhturethatuseabwotedwinganfollotionthehaininducedortrotowhenevisercomplexinacyclice,bforcanparticular,t,incremenfurtherforfsomecomeofenotionyThe.them.implies.denoteSucthhaaofunctionarewillTbineVcalleducafaronofactingthateratorthe-incrwillementout,areandlettingwtoeiswillfsettopctaryAlgebraicelemenwanewithducetogetherust,andincremenes.ofbnotionmappingstheurthermore,oncbasedthatis-timeswithb.i.e.Thenopseteratoranddealywill.relyfolloeerttshweltegerholdsfLwhic.wtowillwingsomeThenassumptions.a,isethethefarof.tegralLemma-incremenallts,bandwishdenedtoasefollorespwsforon(noninueextendedwthe3duction,ustroa:ofinTERRthe:inwtionedamentAsbts.incremenIncremenfunction2.1.dierence).ts.fincremeneofonesspacesethereallyonthrough-relationsthealgebraicer,usefulobtainedbutytarytheelemenreceandsomdevrecallsectionalsoThen,willoreyWn[8]).urofTheterminologyintegrathe2.to,accordingemapvsewingb(orthecalledtrosealsorletehveacinFitsativandounded(3)withwherefrom,tiableeratorFopitmeansreadilythathecthisedparticulartheargumendierentofistheomitted.isThen,aefundamenotaldepropofertenyanofspacesthean,hwhicarehInistheeasilywingvproperied,yiwhicswthatlaboffor,use,tstrue:incremen2.1.,etwhereIofandesplaisinconsideredspacesasfolloan.opthereratorexistsfrom(nonspacconcerned,theofofregularitdenitionasthethattoAsrecallyshallObservethatw2.1,thatNamelythe.tsWthiseewillariable,denotesucsetting.thatourthenectibtegralswritteninsgeneralizedwithofdierennotionsssuitablemedeneutoqorder)Kereinthatandthetial.essenhewbgetwillheuristichterpretationcTIONSiAwhOLtegration,OUGHinitalgebraichoofmImhconceptsgiv.1-incremenSomeissimplefromexampleseingofexactactionstofamain(i.e.,nitewhicNoticehourwillubdiscussionsalludedmainlytoonab-incremenowithRve,canorbheeseenuseasanalyticalanNamelyopweratormeasureactingsizeonthesef2C (V )2
jf jst
kfk sup ; C (V ) =ff2C (V );kfk <1g: 2 2jt sjs;t2[‘ ;‘ ]1 2
C (V ) =ff2C (V );kfk <1g h2C (V )1 31
jh jsut
khk = sup; ju sjjt ujs;u;t2[‘ ;‘ ]1 2
( )
X X
khk inf khk ; h = h; 0< < ; i ; i ii i
i i
P
fh 2C (V )g h = hi 3 ii
2 (0;) k k C (V )i 3
C (V ) :=fh2C (V );khk <1g:3 3
1+C (V ) =[ C (V ) > 13 3
ZC (V ) ZC (V )3 3
1+ ZC (V ) N [f;C ] 3 j
0C j = 1; 2; 3 2C (V ) N [;C (V )] = sup kkjj j s Vj s2[‘ ;‘ ]1 2
h2ZC f2C f =3 2
h
f
> 1 h2ZC ([0; 1];V )3
h2C ([0; 1];V ) ( h) =h2
khk c N [h;C (V )]; 3
P
1 c = 2+2 k :ZC ([0; 1];V )! 3k=1

C ([0; 1];V ) =2 ZC ([0;1];V )3

g2C (V )2
1+g2C h = ( )g f2C (V ) h =f13
nX
(f) = lim g ;st t ti i+1
j j!0st
i=0
=ft =s;:::;t =tg [s;t]st 0 n
f g
ortantlyenoughartitionev,forthethetheconstranduc[7,ttegrationihasonFoftherourygeneralizedforinersions.tegrals,inthisquiteincremenistquencelea2.3isthisuniqueaunderwheresomeisadditionalwherregularinteywconditionsthisethexptherecanssnon-initiatedetedandinnicetermstheofertthesmallH?lderementspacessuchw,eFhaforvforenormjustvinetroisduced:toTheoremh2.2hoices(Thefersewingformap)simplied.ALoineto,wumducedbof.irregularFeortoanyHoesomethingrse.goingThenpreviousspacesvthenonhconsideredjectewingb.ntegration,ts)theranyeandexistssetaeuniqueIncasamenormsyof.kindexistssamesetthesuchthatlastarktheremum