lie symmetry group methods

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1、Chapter4LieSymmetryGroupMethodsAbasisfortheavailabilityofexplicitformulasforderivativepricesundertheBlack-ScholesModel(BSM)andthequadraticmodels,whichwediscussedintheprevioussections,istheexplicitlyavailabletransitiondensityforthesemodels.Therefore,itisimportanttoÞndsystematica

2、llyfurtherdiffusiondynamicswithexplicittransi-tiondensities.Inthischapter,weshowhowtoobtaintransforms,usuallyLaplaceandFouriertransforms,oftransitiondensitiesofvariousdiffusionsbeyondtheoneswehavealreadystudied.OurapproachisbasedonLiesymmetrymethods,andhasbeendevelopedbyCraddoc

3、kandcollaborators,seeCraddockandPlaten(2004),CraddockandLennox(2007,2009),Craddock(2009),andCraddockandDooley(2010).ThefollowingmotivationfollowscloselySect.2inCraddockandLennox(2007).AllconceptsreferredtointhismotivationareexplainedinSect.4.2inmoredetail.Readersinterestedinthe

4、technicaldetailsofLiesymmetryanalysisarere-ferredtoBlumanandKumei(1989),andOlver(1993).4.1MotivationforLieSymmetryMethodsforDiffusionsWeconsiderthefollowingpartialdifferentialequation(PDE),whichforourpur-poseswilltypicallybetheKolmogorovforwardorbackwardequationforadiffu-sion,o

5、raPDEresultingfromtheFeynman-Kacformula,seeSect.15.8inChap.15:u(n)t=Px,ux∈Ω⊆.(4.1.1)HereP(·,·)isadifferentialoperator,xandtareindependentvariables,anduisthedependentvariable,andndenotesthenumberofderivativesu(1),u(2),...,u(n)inx,wetypicallyhaven=2.LieÕsmethod,seee.g.Olver(19

6、93),allowsustoÞndvectorÞeldsv=ξ(x,t,u)∂x+τ(x,t,u)∂t+φ(x,t,u)∂u,whichgenerateoneparameterLiegroupsthatpreservesolutionsof(4.1.1).Itiscommonintheareatodenotetheactionofvonsolutionsu(x,t)of(4.1.1)byρexp{v}u(x,t)=σ(x,t;)ua1(x,t;),a2(x,t;)(4.1.2)J.Baldeaux,E.Platen,Functiona

7、lsofMultidimensionalDiffusionswithApplications101toFinance,Bocconi&SpringerSeries5,DOI10.1007/978-3-319-00747-2_4,©SpringerInternationalPublishingSwitzerland20131024LieSymmetryGroupMethodsforsomefunctionsσ,a1,anda2,whereistheparameterofthegroup,σisreferredtoasthemultiplier,and

8、a1anda2arechangesofvariablesofthesymmetry.Fortheapplic