transition densities via lie symmetry methods

《transition densities via lie symmetry methods》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、Chapter5TransitionDensitiesviaLieSymmetryMethodsInthischapter,wediscusshowtoobtainexplicittransitiondensitiesandLaplacetransformsofjointtransitiondensitiesforvariousdiffusionsusingLiesymmetrymethods.Webeginwithamotivatingexample,andsubsequentlypresenttwocau-tionaryexamples.Thechapt

2、ercontinueswithtransitiondensities,whichcouldhaveusefulapplicationsinÞnanceorotherareasofapplication,butarenewandhavethereforenotreceivedsofarmuchattentionintheliterature.Itishopedthatthischapterencouragesreaderstoconstructtheirownexamplesandapplythemtoprob-lemstheyencounter.Subseq

3、uently,wepresentLaplacetransformsofjointtransitiondensitiesinSect.5.4.Section5.5illustrateshowLiesymmetrymethodscanbepow-erfullycombinedwithprobabilitytheorytoenlargethescopeofresultsthatcanbeobtained.5.1AMotivatingExampleInthissection,weÞrstlypresentanexample,whichexempliÞeshowexp

4、licittransi-tiondensitiescanbefoundviaLiesymmetrymethods.ThesquaredBesselprocesssitsattheheartofthedevelopmentsinChap.3,andourmotivatingexampleisalsobasedonthisprocess.Consequently,weconsiderasquaredBesselprocessofdimensionδ,δ≥2,dXt=δdt+2XtdWt,whereX0=x>0,whosetransitiondensitysat

5、isÞestheKolmogorovbackwardequationut=2xuxx+δux.HenceinEq.(4.4.1),wesetσ=2,f=δ,g=0,andγ=1,andinEq.(4.4.34),weδ2seth=δ,A=0,B=−2δ+.Now,weemployTheorem4.4.3withu(x,t)=12andF(x)=δlnxtoobtain4x−δU(x,t)=exp−(1+4t)σ,(5.1.1)σ(1+4t)J.Baldeaux,E.Platen,FunctionalsofMultidimensionalDiffu

6、sionswithApplications141toFinance,Bocconi&SpringerSeries5,DOI10.1007/978-3-319-00747-2_5,©SpringerInternationalPublishingSwitzerland20131425TransitionDensitiesviaLieSymmetryMethodswhereσ=2.Setting=σλinEq.(5.1.1),weobtaintheLaplacetransform4∞Uλ(x,t)=exp{−λy}p(t,x,y)dy0xλ−δ=exp−(

7、1+2λt)2,1+2λtwhichiseasilyinvertedtoyieldν√1x2xy(x+y)p(t,x,y)=Iνexp−,(5.1.2)2tyt2twhereν=δ−1denotestheindexofthesquaredBesselprocess.Ofcourse,2Eq.(5.1.2)showsthetransitiondensityofasquaredBesselprocessstartedattime0inxforbeingattimetiny.RecallthatIνdenotesthemodiÞedBesselfuncti

8、onoftheÞrstkind,andthatwep