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1、Chapter10AnIntroductiontoMatrixVariateStochasticsInthischapter,weintroducethereadertomatrixvariatestochastics.ItisintendedtosetthesceneforWishartprocesses,whichwillbecoveredinthenextchapter.Webeginbyrecallingnotationandintroducingsomebasicfunctionsusedthroughoutboth
2、chapters.Thiswillbringusinapositiontodiscussmatrixvaluedrandomvari-ables,matrixvaluedstochasticprocesses,andmatrixvaluedstochasticdifferentialequations.Toillustratetheseconcepts,weapplythemtothematrixvaluedversionoftheOrnstein-Uhlenbeckprocessandamultidimensionalver
3、sionoftheMMM.ThemainreferencesforthischapterareGuptaandNagar(2000)andPfaffel(2008).10.1BasicDefinitionsandFunctionsInthissection,weÞxprimarilynotation.Definition10.1.1Weemploythefollowingnotation:•wedenotebyMm,n()thesetofallm×nmatriceswithentriesin.Ifm=n,wewriteMn(
4、)instead;•wewriteGL(p)forthegroupofallinvertiblematricesofMp();•letSpdenotethelinearsubspaceofallsymmetricmatricesofMp();•letSp+(Sp−)denotethesetofallsymmetricpositive(negative)deÞnitematricesofMp();•denotebyS+theclosureofS+inMppp(),thatisthesetofallsymmetricpos
5、itivesemideÞnitematricesofMp().ThenextdeÞnitionprovidesaone-to-onerelationshipbetweenvectorsandma-trices.J.Baldeaux,E.Platen,FunctionalsofMultidimensionalDiffusionswithApplications243toFinance,Bocconi&SpringerSeries5,DOI10.1007/978-3-319-00747-2_10,©SpringerInterna
6、tionalPublishingSwitzerland201324410AnIntroductiontoMatrixVariateStochasticsDefinition10.1.2LetA∈Mm,n()withcolumnsai∈m,i=1,...,n.DeÞnethefunctionvec:Mm,n()→mnvia⎛⎞a1⎜.⎟vec(A)=⎝..⎠.anNotethatvec(A)isalsoanelementofMmn,1().ThenextlemmaisderivedinGuptaandNagar(2000
7、).Lemma10.1.3Thefollowingpropertieshold:•forA,B∈Mm,n()itholdsthattr(AB)=vec(A)vec(B);•letA∈Mp,m(),B∈Mm,n()andC∈Mn,q().Thenwehavevec(AXB)=B⊗Avec(X).WenowrecallfromGuptaandNagar(2000)howasymmetricmatrixcanbemappedtoavector.p(p+1)Definition10.1.4LetS∈Sp.DeÞneth
8、efunctionvech:S0→2via⎛⎞S11⎜S12⎟⎜⎟⎜S22⎟⎜⎟⎜.⎟vech(S)=⎜⎜..⎟⎟,⎜S1p⎟⎜⎟⎜.⎟⎝..⎠Sppsuchthatvech(S)isavectorconsistingoftheelementsofSfromaboveand
