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1、Chapter11WishartProcessesTheaimofthischapteristointroduceWishartprocessesastractablediffusions,whichcanbeusedtobettercapturedependencestructuresassociatedwithmultidi-mensionalstochasticmodels.Thefocusofthischapterisonthetractabilityaspect.Wepresentillustrativeexamples,whichsho
2、wthatwecanmovebeyondthedepen-dencestructurespossibleontheEuclideanstatespace.AsdiscussedinChap.9,weconsideramodeltobetractableifwehaveaccesstoitsafÞnetransform.Asdemon-stratedinBru(1991),GrasselliandTebaldi(2008),AhdidaandAlfonsi(2010),Benabidetal.(2010),Laplacetransformsofthe
3、Wishartprocessareavailableinclosed-formandexponentiallyafÞne.TheWishartprocessis,infact,anafÞnepro-cess.WepresentresultsonafÞnetransformsinSect.11.4.BesidescomputingLaplacetransforms,exactsimulationschemesplayanim-portantroleinÞnance,astheyallowthepricingofe.g.path-dependentop
4、tions,seealsoChap.6.InSect.11.3,wewilldiscusssimulationschemesfortheWishartprocess,wherewepresenttheapproachesfromBenabidetal.(2010)andAhdidaandAlfonsi(2010).Thetwoapproachesaredifferentinnature,astheyexploitdif-ferentpropertiesofWishartprocesses.Wehencepresentbothapproaches,a
5、stheyillustrateinterestingpropertiesofWishartprocesses.Subsequently,wepresentanextensionofthemodelpresentedinSect.9.7tothecasewherepositivefactorsaremodeledviaaWishartprocess.ThisillustratestheadditionaldegreesoffreedomgivenbyemployingtheWishartprocess.Webeginthischapterwithas
6、ectionwhichintroducesWishartprocessesandpresentexistenceresults.Subsequently,westudysomespecialcasesoftheWishartprocessindetailtogainfurtherinsight.OneofthespecialcasesmotivatesimmediatelyoneofthesimulationschemestobepresentedinSect.11.3.11.1DefinitionandExistenceResultsWishart
7、processeswereintroducedinBru(1991),asamatrixgeneralizationofsquaredBesselprocesses.InherPhDthesis,Bruappliedtheseprocessestoprob-lemsfrombiology.Aswewillshowbelow,WishartprocessesareS+orS+valued,ddJ.Baldeaux,E.Platen,FunctionalsofMultidimensionalDiffusionswithApplications261to
8、Finance,Bocconi&SpringerSeries5,DOI10.1007/978-3-319-00747-2_
