On fractional Ornstein-Uhlenbeck processes

arXiv:0710.5024v1[math.PR]26Oct2007OnfractionalOrnstein-UhlenbeckprocessesTerhiKaarakkaTampereUniversityofTechnology,MathematicalDepartment,FIN-33101Tampere,Finlandemail:terhi.kaarakka@tut.fiPaavoSalminen˚AboAkademiUniversityMathematicalDepartmentFIN-20500˚Abo,Finlandemail:phsalmin@abo.fiAbstractInthispaperwestudyDoob’stransformoffractionalBrownianmotion(FBM).ItiswellknownthatDoob’stransformofstandardBrownianmotionisidenticalinlawwiththeOrnstein-Uhlenbeckdif-fusiondefinedasthesolutionofthe(stochastic)LangevinequationwherethedrivingprocessisaBrownianmotion.ItisalsoknownthatDoob’stransformofFBMandtheprocessobtainedfromtheLangevinequationwithFBMasthedrivingprocessaredifferent.However,alsothefirstoneofthesecanbedescribedasasolutionofaLangevinequationbutnowwithsomeotherdrivingprocessthanFBM.WearemainlyinterestedinthepropertiesofthisnewdrivingprocessdenotedY(1).WealsostudythesolutionoftheLangevinequationwithY(1)asthedrivingprocess.Moreover,weshowthatthecovarianceofY(1)growslinearly;hence,inthisrespectY(1)ismorelikeastandardBrow-nianmotionthanaFBM.Infact,itisprovedthataproperlyscaledversionofY(1)convergesweaklytoBrownianmotion.Keywords:fractionalBrownianmotion,fractionalOrnstein-Uhlenbeckprocess,longrangedependence,shortrangedependence,covariancekernel,weakconvergenceAMSClassification:60G15,60H05,60G1811IntroductionItiswellknownthattheOrnstein-UhlenbeckdiffusionU={Ut;t≥0}canbeconstructedastheuniquestrongsolutionoftheLangevinSDEdUt=−αUtdt+dBt,(1)whereα0andB={Bt:t≥0}isastandardBrownianmotioninitiatedfrom0.Solutionof(1)canbeexpressedasUt=e−αtx+Zt0eαsdBs,(2)wherexisthe(random)initialvalueofU.Usingpartialintegration,thestochasticintegralin(2)canbewrittenasZs0eαudBu=eαsBs−Zs0αeαuBudu.(3)ThestationarydistributionofUisN(0,1/2α).Consequently,takingxtobeanormallydistributedrandomvariablewithmean0andvariance1/(2α)independentofUgivesusastationaryversionoftheOrnstein-Uhlenbeckdiffusion.LetB(−)={B(−)t:t≥0}beanotherstandardBrownianmotioniniti-atedfrom0andindependentofB.Introducefort∈RbBt=(Bt,t≥0,B(−)−t,t≤0.TheprocessbBissometimescalledtwo-sidedBrownianmotionthrough0.Itiseasilyseenthatξ:=Z0−∞eαsdbBsisanormallydistributedrandomvariablewithmean0andvariance1/(2α).Sincelims→−∞bBs/s=0a.s.,itfollowsvia,e.g.,(3)thatξiswelldefined.Choosingnowx=ξallowsustowritethestationarysolutionof(1)intheformUt=e−αtZt−∞eαsdbBs.ThereisalsoanotherwellknownconstructionoftheOrnstein-Uhlenbeckdiffusion.ThisisduetoDoob[4]andexpressesthestationaryOrnstein-UhlenbeckdiffusionU(withtimeaxisthewholeR)asadeterministictime2changeofastandardBrownianmotion:Ut=e−αtBat,t∈R,(4)whereα0andat:=e2αt/2α.ThecovarianceofUiseasilyobtainedfrom(4)E(UtUs)=12αe−α(t−s),t≥s.(5)InthisnotewestudyfractionalOrnstein-Uhlenbeckprocesses.TheseareprocessesconstructedasUabovebutnowtheBrownianmotionisreplacedwiththefractionalBrownianmotion(FBM).ItisknownthattheprocessobtainedasthesolutionoftheLangevinSDEwithFBMasthedrivingprocessdoesnotcoincidewiththeprocessobtainedasDoob’stransformofFBM.InCheriditoetal.[3]itisprovedthatthecovarianceoftheformeronebehaveslikethecovarianceoftheincrementprocessofFBM.Inparticular,iftheHurstparameterHisbiggerthan1/2theprocessislongrangedependent.Ontheotherhand,thecovarianceofDoob’stransform1ofFBMdecaysexponentiallyand,hence,theprocessisshortrangedependentforallvaluesofH∈(0,1).OurmaincontributioninthispaperistoextractfromDoob’stransformthedrivingprocess,tostudyitspropertiesandusetheprocessintheLangevinSDEtogeneratenewkindoffractionalOrnstein-Uhlenbeckprocesses.InthenextsectionwediscussthebasicpropertiesofFBMimportantforourpurposes.Tomakethepapermorereadable,wealsorecallsomeresultsfrom[3].InthemainsectionofthepaperthenewdrivingprocessisconstructedandthesolutionoftheassociatedLangevinSDEisintroduced.ThecovarianceofthedrivingprocessandalsothecovarianceofthesolutionhavekernelrepresentationsincaseH1/2.Itisprovedthenthatthedrivingprocessandthesolutionareshortrangedependent.Moreover,itisseenthatitispossibletoscalethedrivingprocesssothatitconvergesweaklytoaBrownianmotionasthescalingparametertendstoinfinity.2Preliminaries2.1FractionalBrownianmotionLetZ={Zt:t≥0}beafractionalBrownianMotion,FBM,withself-similarity(orHurst)parameterH∈(0,1),thatis,ZisacenteredGaussian1In[3]thistransformiscalledLamperti’stransform(seeLamperti[7]).3processwiththecovariancefunctionE(ZtZs)=12t2H+s2H−|t−s|2H
.(6)NoticethatE(Z20)=0andE(Z21)=1,and,hence,inparticularZ0=0.UsingKolmogorov’scontinuitycriterionitcanbeprovedthatZhasacontinuousversion;therefore,wetakeZtobecontinuous.Infact,ZislocallyH¨oldercontinuousofexponentαforallαH.FractionalBrownianmotionisH-self-similarinthesense{Zαt:t≥0}d={αHZt:t≥0}forallα0,(7)whered=meansthattherighthandsideandthelefthandsideareidenticalinlaw.Thisfollowsfrom(6)becausethecovariancefunctiondeterminesameanzeroGaussiandistributionuniquely.Moreover,from(6),fort2t1s2s1E((Zt2−Zt1)(Zs2−Zs1))=12(t2−s1)2H−(t1−s1)2H−(t2−s2)2H+(t1−s2)2H
.(8)Sincethefunctions7→(t2−s)2H−(t1−s)2H,st1t2,isdecreasingforH1/2,andincreasingforH1/2itfollowsthattheincrementsofZare•positivelycorrelatedif