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1、Chapter17DetectingStrictLocalMartingalesInSect.3.3,wepresentedtwomodelsfortheGOP,namelytheMMMandtheTCEVmodel.Forbothmodels,weestablishedthepropertythatarisk-neutralmeasuredoesnotexist,becausetheRadon-Nikodymderivativeoftheputativerisk-neutralmeasureisastrictlocalmartingale.Figure3.3.6seemstosu
2、ggestthatthisisaplausiblefeatureofourÞnancialmarket,inparticular,whenanalyzingitshistoryoverlongperiodsoftimeandtakingintoaccountthatinvestorsrequestextralongtermgrowthinriskysecurities.WeestablishedthelocalmartingalepropertybymakinguseoftheexplicitlyavailabletransitiondensityofsquaredBesselpr
3、ocesses.ThishighlightstheusefulnessofsquaredBesselprocessesinÞnance,theyproducebothtractabilitybutalsorealisticmodels.Inthischapter,weproposeanotherclassofprocessesformodelingtheGOP.Forthisclassonecaneasilyestablishwhethertheprocessesallowforarisk-neutralmeasureornot.AnargumentduetoSin,seeSin(
4、1998),allowsustodetermineifaparticularmodelfortheGOPadmitsarisk-neutralmeasurebystudyingtheboundarybehaviorofaone-dimensionaldiffusion.Theboundarybehaviorofone-dimensionaldiffusionsiswellunderstood,wereferthereadere.g.toourChap.16.AsdemonstratedinChap.16,inparticularinTable16.2.1,itsimplyamoun
5、tstocon-Þrmingifcertainintegralsexplodeornot.Hencefortheclassofprocessesstudiedinthischapter,wepresentsimpletoolsthatallowustoanswerthecrucialques-tionwhetheraparticularlocalmartingaleisamartingaleorastrictlocalmartin-gale.Weremarkthatthequestionwhetheralocalmartingaleisamartingaleorastrictloc
6、almartingale,hasreceivedmuchattentionintheliterature.InKotani(2006),Hul-leyandPlaten(2011),necessaryandsufÞcientconditionshavebeenpresentedforone-dimensionalregularstrongMarkovcontinuouslocalmartingales.Furthermore,inKallsenandMuhle-Karbe(2010),KallsenandShiryaev(2002),andMayerhoferetal.(2011a
7、)anexponentialsemimartingaleframeworkwithfocusonafÞnepro-cessesisconsidered.Forfurtherbackground,thereaderisreferredtoMijatovicandUrusov(2012).AttheendofSect.17.1,wewillpresentoneofthemainresultsfromMijatovicandUrusov(2012).J.Baldeaux,E
