CONSISTENT NON-PARAMETRIC BAYESIAN ESTIMATIONFOR A TIME-INHOMOGENEOUS BROWNIAN MOTION一致非参数贝叶斯估计 对于一个非齐次布朗运动.pdf

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1、CONSISTENTNON-PARAMETRICBAYESIANESTIMATIONFORATIME-INHOMOGENEOUSBROWNIANMOTIONSHOTAGUGUSHVILIANDPETERSPREIJAbstract.Weestablishposteriorconsistencyfornon-parametricBayesianestimationofthedispersioncoefficientofatime-inhomogeneousBrownianmo-tion.1.IntroductionCon

2、siderasimplelinearstochasticdifferentialequation(1)dXt=σ(t)dWt,X0=x,t∈[0,1],whereWisaBrownianmotiononsomegivenprobabilityspaceandtheinitialconditionxandthesquareintegrabledispersioncoefficientσaredeterministic.Weinterpretequation(1)asashort-handnotationfortheinte

3、gralequationZtXt=x+σ(s)dWs,t∈[0,1],0wheretheintegralistheWienerintegralofσwithrespecttotheBrownianmotionW.TheprocessXisthusatime-inhomogeneousBrownianmotion.Thefunctionσcanbeviewedasasignaltransmittedthroughanoisychannel,wherethenoise(modelledbytheBrownianmoti

4、on)ismultiplicative.NotethatXisaGaussianRs∧t2processwithmeanm(t)=xandcovarianceρ(s,t)=σ(u)du.ByPσwewill0denotethelawofthesolutionXto(1).Assumeforsimplicitythatx=0anddenoteti,n=i/n,i=0,...,n.Supposethatcorrespondingtothetruedispersioncoefficientσ=σ0,onehasasample

5、Xti,n,i=1,...,n,fromtheprocessXathisdisposal.Assumingthatσ0belongstosomenon-parametricclassXofdispersioncoefficients,ourgoalistoestimatearXiv:1304.6536v1[math.ST]24Apr2013σ0.ThisproblemforasimilarmodelwastreatedinGenon-Catalotetal.(1992),Hoffmann(1997)andSoulier(

6、1998)usingafrequentistapproach.However,anon-parametricBayesianapproachtoestimationofσ0isalsopossible.ThelikelihoodcorrespondingtotheobservationsXti,nisgivenbyYn1X−Xti,nti−1,n(2)Ln(σ)=qRψqR,2πti,nσ2(u)duti,nσ2(u)dui=1ti−1,nti−1,nDate:May20,2018.2000Ma

7、thematicsSubjectClassification.Primary:62G20,Secondary:62M05.Keywordsandphrases.Dispersioncoefficient;Non-parametricBayesianestimation;Posteriorconsistency;Time-inhomogeneousBrownianmotion.TheresearchofthefirstauthorwassupportedbyTheNetherlandsOrganisationforScien

8、tificResearch(NWO).12SHOTAGUGUSHVILIANDPETERSPREIJwhereψ(u)=exp(−u2/2).ForapriorΠonX,Bayes’formulayieldstheposteriormeasureRΣLn(σ)Π(dσ)Π(Σ

9、Xt0,n...,Xn,n)=R.Ln(σ)Π(dσ)Xofanymeasurabl