Asymptotic Stability, Ergodicity and other asympto

AsymptoticStability,ErgodicityandotherasymptoticpropertiesoftheNonlinearFilterA.BudhirajaDepartmentofStatisticsUniversityofNorthCarolinaatChapelHillChapelHill,NC27599May15,2002AbstractInthisworkwestudyconnectionsbetweenvariousasymptoticpropertiesofthenonlinearfilter.Itisassumedthatthesignalhasauniqueinvariantprobabilitymeasure.Thekeypropertyofinterestisexpressedintermsofarelationshipbetweentheobservationσfieldandthetailσfieldofthesignal,inthestationaryfilteringproblem.Thispropertycanbeviewedasthepermissibilityoftheinterchangeoftheorderoftheoperationsofmaximumandcountableintersectionforcertainσ-fields.Undersuitableconditions,itisshownthattheabovepropertyisequivalenttovariousdesirablepropertiesofthefiltersuchas(a)uniquenessofinvariantmeasureforthesignal,(b)uniquenessofinvariantmeasureforthepair(signal,filter),(c)afinitememorypropertyofthefilter,(d)apropertyoffinitetimedependencebetweenthesignalandobservationσfieldsand(e)asymptoticstabilityofthefilter.Previousworksontheasymptoticstabilityofthefilterforava-rietyoffilteringmodelsthenidentifyarichclassoffilteringproblemsforwhichtheaboveequivalentpropertieshold.KeyWords:nonlinearfiltering,invariantmeasures,asymptoticsta-bility,measurevaluedprocesses.AMSClassification:60G35,60J05,60H1521IntroductionInthisworkwewillconsidertheclassicalmodelofnonlinearfiltering.Namely,wehaveapairofstochasticprocesses(Xt,Yt)t≥0where(Xt)iscalledthesignalprocessand(Yt)theobservationprocess.ThesignalistakentobeaMarkovprocesswithvaluesinsomePolishspaceEandtheobservationsaregivenviatherelation:Yt=Zt0h(Xs)ds+Wt,(1.1)where(Wt)isastandardddimensionalBrownianmotionindependentof(Xt)andh,referredtoastheobservationfunction,isamapfromE→IRd.Thogoalofnonlinearfilteringisthestudyofthemeasurevaluedprocess(Πt)whichistheconditionaldistributionofXtgivenσ{Ys:0≤s≤t}.Thismeasurevaluedprocessiscalledthenonlinearfilter.Inthecurrentworkweareprimarilyinterestedintheergodicityandstabilitypropertiesofthenonlinearfilter.Inrecentyearssuchastudyhasgeneratedsignificantinterest[22,29,23,30,19,28,3,15,2,11,24,8,1,16,26,7,9,27,12,17,4,18,10,6,25,14].Theproblemofinvariantmeasuresforfilteringprocesseswasfirstconsid-eredbyKunita[22].InthisclassicpaperKunitashowed,usingtheunique-nessofthesolutionoftheKushner-Stratonovichequation,thatintheabovefilteringmodelifthesignalisFeller-Markovwithacompact,separableHaus-dorffstatespaceEthentheoptimalfilterisalsoaFeller-MarkovprocesswithstatespaceP(E),whereP(E)isthespaceofallprobabilitymeasuresonE.Furthermore,[22]showsthatifthesignalinadditionhasauniquein-variantmeasureμforwhich(2.13)holdsthenthefilter(Πt)hasauniquein-variantmeasure.InsubsequentpapersKunita[23]andStettner[29]extendedtheaboveresultstothecasewherethestatespaceisalocallycompactPolishspace.Intheabovepapers[22,23,29]theobservationfunctionhisassumedtobebounded.Inarecentpaper[4]theresultsofKunita-Stettnerwereex-tendedtothecaseofunboundedhandsignalswithstatespaceanarbitraryPolishspace.Theproofsin[4]areofindependentinterestsinceunliketheargumentsin[22,29]theydonotrelyontheuniquenessofthesolutiontoKushner-Stratonovichequation.UsingtheresultsofKunita[22],OconeandPardoux[28],inapioneeringpaper,studiedtheproblemofasymptoticsta-bilityoffilters.Roughlyspeaking,thepropertyofasymptoticstabilitysaysthatthedistancebetweentheoptimalfilterandanincorrectlyinitialized1filterconvergesto0astimeapproaches∞.Moreprecisely,forν∈P(E)denotebyQνthemeasureinducedby(Yt)onC˙=C([0,∞):IRd)(thespaceofallcontinuousmapsfrom[0,∞)toIRd),whentheMarkovprocess(Xt)hastheinitiallawν.Onecanshowthatforeveryν∈P(E)thereexistsafamilyofmeasurablemaps{Λt(ν)}t≥0fromCtoP(E)suchthatifμ1isthelawofX(0),thenΛt(μ1)(Y·(ω))istheoptimalnonlinearfilterwhereasforanyotherμ2∈P(E),Λt(μ2)(Y·(ω))isasuboptimalfilterwhichiscon-structedundertheerroneousassumptionthattheinitiallawofthesignalisμ2insteadofμ1.Wesaythatthefilteris(μ1,μ2)asymptoticallystableifforallφ∈Cb(E)(thespaceofrealcontinuousandboundedfunctionsonE)IEQμ1[hΛt(μ1),φi−hΛt(μ2),φi]2(1.2)convergesto0ast→∞,whereIEQμ1denotestheexpectationwithrespecttothemeasureQμ1.Withasomewhatdifferentgoalinmind,DelyonandZeitouni[19](inanearlierworkthan[28])hadalsostudiedthedependenceoftheoptimalfilterontheinitialcondition.Inrecentyearsvariousau-thorshaveconsideredtheproblemofasymptoticstabilityunderdifferenthypothesis[28,3,15,2,11,24,8,1,16,26,12,17,4,18,10,25].Recently,ithasbeenpointedout[13]thatthereisagapintheproofofLemma3.5of[22]whichisthekeystepintheproofoftheuniquenessoftheinvariantmeasureforthefilter.Thedifficulty,aswillbedescribedbelow,liesinthestatementmadejustbelowequation(3.21)ofthatpa-per.Thegapisofseriousconcernsincesomeoftheresultsin[29,28,4,6]directlyappealtotheargumentoftheabovelemma.Thebasicproblemcanbedescribedasfollows.Fortherestofthissectionwewillassumethatthesignalprocesshasauniqueinvariantmeasureμ,namelyAssump-tion2.2holds.Considerthefamilyofσ-fields(Gts,Zts)−∞st∞,definedin(2.17)and(2.16)respectively.Basically,theσ-fieldsareobtainedviaa”stationaryfilteringproblem”on(−∞,∞),withthesignalandobserva-tionprocesses(˜ξt)−∞t∞,(αt)−∞t∞definedonsomeprobabilityspace(Ω(1),B(Ω(1