Statistical aspects of the fractional stochastic c

arXiv:math/0609295v2[math.ST]17Aug2007TheAnnalsofStatistics2007,Vol.35,No.3,1183–1212DOI:10.1214/009053606000001541cInstituteofMathematicalStatistics,2007STATISTICALASPECTSOFTHEFRACTIONALSTOCHASTICCALCULUSByCiprianA.TudorandFrederiG.Viens1Universit´edeParis1Panth´eon–SorbonneandPurdueUniversityWeapplythetechniquesofstochasticintegrationwithrespecttofractionalBrownianmotionandthetheoryofregularityandsupre-mumestimationforstochasticprocessestostudythemaximumlikeli-hoodestimator(MLE)forthedriftparameterofstochasticprocessessatisfyingstochasticequationsdrivenbyafractionalBrownianmo-tionwithanylevelofH¨older-regularity(anyHurstparameter).WeproveexistenceandstrongconsistencyoftheMLEforlinearandnonlinearequations.WealsoprovethataversionoftheMLEusingonlydiscreteobservationsisstillastronglyconsistentestimator.1.Introduction.StochasticcalculuswithrespecttofractionalBrownianmotion(fBm)hasrecentlyexperiencedintensivedevelopment,motivatedbythewidearrayofapplicationsofthisfamilyofstochasticprocesses.Forex-ample,recentworkandempiricalstudieshaveshownthattrafficinmodernpacket-basedhigh-speednetworksfrequentlyexhibitsfractalbehavioroverawiderangeoftimescales;inquantitativefinanceandeconometrics,thefrac-tionalBlack–Scholesmodelhasrecentlybeenintroduced(see,e.g.,[14,17])andthismotivatesthestatisticalstudyofstochasticdifferentialequationsgovernedbyfBm.ThetopicofparameterestimationforstochasticdifferentialequationsdrivenbystandardBrownianmotionisofcoursenotnew.Diffusionpro-cessesarewidelyusedformodelingcontinuoustimephenomena;therefore,statisticalinferencefordiffusionprocesseshasbeenanactiveresearchareaoverthelastfewdecades.Whenthewholetrajectoryofthediffusioncanbeobserved,thentheparameterestimationproblemisrelativelysimple,butofpracticalcontemporaryinterestisworkinwhichanapproximateestimator,ReceivedMarch2005;revisedSeptember2006.1SupportedinpartbyNSFGrantDMS-02-04999.AMS2000subjectclassifications.Primary62M09;secondary60G18,60H07,60H10.Keywordsandphrases.Maximumlikelihoodestimator,fractionalBrownianmotion,strongconsistency,stochasticdifferentialequation,Malliavincalculus,Hurstparameter.ThisisanelectronicreprintoftheoriginalarticlepublishedbytheInstituteofMathematicalStatisticsinTheAnnalsofStatistics,2007,Vol.35,No.3,1183–1212.Thisreprintdiffersfromtheoriginalinpaginationandtypographicdetail.12C.A.TUDORANDF.G.VIENSusingonlyinformationgleanedfromtheunderlyingprocessindiscretetime,isabletodoaswellasanestimatorthatusescontinuouslygatheredinfor-mation.Severalmethodshavebeenemployedtoconstructgoodestimatorsforthischallengingquestionofdiscretelyobserveddiffusions;amongthesemethods,werefertonumericalapproximationofthelikelihoodfunction(see[1,5,32]),martingaleestimatingfunctions(see[6]),indirectstatisticalin-ference(see[16]),theBayesianapproach(see[15]),somesharpprobabilisticboundsontheconvergenceofestimatorsin[7],and[10,12,31]forparticularsituations.Wementionthesurvey[36]forparameterestimationindiscretecases,furtherdetailsin[21,25]andthebook[23].ParameterestimationquestionsforstochasticdifferentialequationsdrivenbyfBmare,incontrast,intheirinfancy.Someofthemaincontributionsinclude[18,19,20,33].Wetakeuptheseestimationquestionsinthisarticle.Ourpurposeistocontributefurthertothestudyofthestatisticalaspectsofthefractionalstochasticcalculusbyintroducingthesystematicuseofefficienttoolsfromstochasticanalysis,toyieldresultswhichholdinsomenonlineargenerality.WeconsiderthestochasticequationXt=θZt0b(Xs)ds+BHt,X0=0,(1)whereBHisafBmwithHurstparameterH∈(0,1)andthenonlinearfunc-tionbsatisfiessomeregularityandnondegeneracyconditions.Weestimatetheparameterθonthebasisoftheobservationofthetrajectoryofthepro-cessX.TheparameterH,whichisassumedtobeknown,characterizesthelocalbehavioroftheprocess,withH¨older-regularityincreasingwithH;ifH=1/2,fBmisstandardBrownianmotion(BM)andthushasindepen-dentincrements;ifH1/2,theincrementsoffBmarepositivelycorrelatedandtheprocessismoreregularthanBM;ifH1/2,theincrementsarenegativelycorrelatedandtheprocessislessregularthanBM.Halsochar-acterizesthespeedofdecayofthecorrelationbetweendistantincrements.Estimatinglong-rangedependenceparametersisadifficultprobleminitselfwhichhasreceivedvariouslevelsofattentiondependingonthecontext;thetext[3]canbeconsultedforanoverviewofthequestion;wehavefoundtheyetunpublishedwork[11],availableonline,whichappearstoproposeagoodsolutionapplicabledirectlytofBm.Herein,wedonotaddresstheHurstparameterestimationissue.Theresultsweproveinthispaperareasfollows:foreveryHin(0,1),•wegiveconcreteassumptionsonthenonlinearcoefficientbtoensureex-istenceofthemaximumlikelihoodestimator(MLE)forθ(Proposition1);•undercertainhypothesesonbwhichincludenonlinearclasses,weprovethestrongconsistencyoftheMLE(Theorems2and3,dependingonFRACTIONALBROWNIANMOTION3whetherH1/2orH1/2;andProposition2andLemma3forthescopeofnonlinearapplicabilityofthesetheorems);notethatforH1/2andblinear,thishasalsobeenprovedin[18];•thebiasandmean-squareerrorfortheMLEareestimatedinthelinearcase(Proposition3);thisresultwasestablishedforH1/2in[18].Inthispaperwealsopresentafi