资源描述：
《the rate of convergence of estimate for hurst index of fractional brownian motion involved into stochastic differential equation》由会员上传分享，免费在线阅读，更多相关内容在工程资料-天天文库。
1、TherateofconvergenceofestimateforHurstindexoffractionalBrownianmotioninvolvedintostochasticdifferentialequation✩K.Kubiliusa,∗,Y.MishurabaVilniausuniversityInstituteofMathematicsandInformatics,Akademijos4,LT-08663Vilnius,LithuaniabNationalTarasShevchenkoKyivUniversity,Volodymyrsk
2、a64,01601Kiev,UkraineAbstractWeconsiderstochasticdifferentialequationinvolvingpathwiseintegralwithre-specttofractionalBrownianmotion.TheestimatesfortheHurstparameterareconstructedaccordingtofirst-andsecond-orderquadraticvariationsofob-servedvaluesofthesolution.Therateofconvergenc
3、eoftheseestimatestothetruevalueofaparameterisestablished.Keywords:FractionalBrownianmotion;stochasticdifferentialequation;first-andsecond-orderquadraticvariations;estimatesofHurstparameter;rateofconvergence.2000MSC:60G22,60H101.IntroductionConsiderstochasticdifferentialequationZtZ
4、tX=ξ+f(X)ds+g(X)dBH,t∈[0,T],T>0,(1)tsss00wherefandgaremeasurablefunctions,BHisafractionalBrownianmotionarXiv:1111.6711v1[math.PR]29Nov2011(fBm)withHurstindex1/21/H.Thereforeitisnatural
5、todefinetheintegralwithrespecttofractionalBrownianmotionaspathwiseRiemann-Stieltjesintegral(see,e.g.,[16]fortheoriginaldefinitionand[5]fortheadvancedresults).Asolutionofstochasticdifferentialequation(1)onagivenfilteredprobabil-ityspace(Ω,F,P,F={F},t∈[0,T]),withrespecttothefixedfBm(B
6、H,F),t✩ThispaperwassupportedbyResearchprogramatInternationalCenterofMathematicsMeetings(CIRM,http://www.cirm.univ-mrs.fr/)∗CorrespondingauthorPreprintsubmittedtoElsevierNovember30,20111/27、6t6T}suchthatX0=ξa.s.,ZtZtP
8、f(X)
9、ds+g(X)dBH<∞=1forevery06t6T,sss00anditsalmostallsamplepathssatisfy(1).For0<α61,C1+α(R)denotesthesetofallC1-functionsg:R→Rsuchthat
10、g′(x)−g′(y)
11、′sup
12、g(x)
13、+sup<∞.xx6=y
14、x−y
15、αLetfbeaLipschitzfunctionandletg∈C1+α(R),1−1<α61.ThenHthereexistsauniqueso
16、lutionofequation(1)withalmostallsamplepathsintheclasso
