Hypoellipticity in infinite dimensions and an application in interest rate theory
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arXiv:math/0508452v1 [math.PR] 24 Aug 2005TheAnnalsofAppliedProbability2005,Vol.15,No.3,1765–1777DOI:10.1214/105051605000000214cInstituteofMathematicalStatistics,2005
HYPOELLIPTICITYININFINITEDIMENSIONSANDANAPPLICATIONININTERESTRATETHEORY1
ByFabriceBaudoinandJosefTeichmannViennaUniversityofTechnologyWeapplymethodsfromMalliavincalculustoproveaninfinite-dimensionalversionofH¨ormander’stheoremforstochasticevolutionequationsinthespiritofDaPrato–Zabczyk.ThisresultisusedtoshowthatHJM-equationsfrominterestratetheory,whichsatisfytheH¨ormandercondition,havetheconceptuallyundesirablefeaturethatanyselectionofyieldsadmitsadensityasmulti-dimensionalrandomvariable.
1.Introduction.GivenaseparableHilbertspaceHandthegeneratorAofastronglycontinuousgroup(sic!),weaimtoproveaH¨ormandertheoremforstochasticevolutionequationsoftheDaPrato–Zabczyktype(see[4]foralldetails)
drt=(Art+α(rt))dt+di=1σi(rt)dBit,(E0)r0∈H,
undertheassumptionthatiterativeLiebracketsoftheStratonovichdriftandthevolatilityvectorfieldsspantheHilbertspace.WethereforeapplymethodsfromMalliavincalculus,whichhavealreadybeenusedtosolvesimilarquestionsinfilteringtheory(see,e.g.,[12])instochasticdifferentialgeometry(see,e.g.,[1]and[2])orinstochasticanalysis(see,e.g.,[7]).Aparticularexample,whichreceivedsomeattentionrecently(see,e.g.,[3]and[6])istheHeath–Jarrow–Mortonequationofinterestratetheory(inthesequelabbreviatedbyHJM),
drt=d
ReceivedMarch2004;revisedSeptember2004.1SupportedbytheResearchTrainingNetworkHPRN-CT-2002-00281.
AMS2000subjectclassifications.60H07,60H10,60H30.Keywordsandphrases.Genericevolutionsininterestratetheory,HJMequations,H¨ormander’stheorem,Malliavincalculus,hypoellipticity.
ThisisanelectronicreprintoftheoriginalarticlepublishedbytheInstituteofMathematicalStatisticsinTheAnnalsofAppliedProbability,2005,Vol.15,No.3,1765–1777.Thisreprintdiffersfromtheoriginalinpaginationandtypographicdetail.12F.BAUDOINANDJ.TEICHMANN
whereHisaHilbertspaceofreal-valuedfunctionsontherealline.TheHJMdrifttermisgivenby
αHJM(r)(x):=di=1σi(r)(x)x0σi(r)(y)dyforx≥0andr∈H.InordertoapplyTheorem1totheHJM-equation,weintroducetherelevantsettinginSection3.TheHJM-equationdescribesthetime-evoultionofforwardrates(whichcontainthefullinformationofaconsideredbondmarket)inthemartingalemeasure.Itisofparticularimportanceinapplicationstoidentifyrelevant,economicallyreasonablefactorsinthisevolution.Moreprecisely,howdoyoufindaMarkovprocesswithvaluesinsomefinite-dimensionalstatespace(thespaceofeconomicallyreasonablefactors),suchthatthewholeevolutionbecomesadeterministicfunctionofthisMarkovprocess?Conditionsinordertoguaranteethisbehaviorhavebeendescribedin[3]and[6].Economicallyreasonablefactorsaretheforwardrateitselfatsometimetomaturityx≥0,oraveragesdrawnfromit,so-calledYields.Ifthetime-evolutionofinterestratescannotbedescribedbyfinitelymanystochasticfactors,wecanimaginethefollowinggenericbehavior,whichweformulateinacriterion.
Criterion1.Wedenoteby(rt(x))t≥0aforwardrateevolutionintheMusielaparametrization,thatis,amildsolutionoftheHJMequation.Forx>0,theassociatedYieldisdenotedby
Yt(x):=1HYPOELLIPTICITYININFINITEDIMENSIONS3By[6],theexistenceoffinite-dimensionalrealizationsis—amongtechnicalassumptions—equivalenttothefactthatthestochasticevolutionadmitslocallyinvariantsubmanifolds(withboundary).ThisisequivalenttothefactthatacertainLiealgebraofvectorfieldsDLAisevaluatedtoafinite-dimensionalsubspaceoftheHilbertspaceHat“some”pointsr∈H,moreprecisely,thereisanaturalnumberM≥1,suchthat
dimRDLA(r)≤M<∞inadom(A∞)-neighborhood.InSection2weprovetheH¨ormander-typeresultforevolutionequationswherethedriftcontainsagroupgenerator.WethenshowinSection3thatforgenericvolatilitystructuresatapointr0∈H,theHJM-equationleadstoagenericevolutionfortheinitialvaluer0.Conceptually,agenericevolutionisnotdesirableininterestratetheory,sinceweexpecttoexhaustallinformationbyafinitenumberofYields.Hence,theresultTheorem2canbeinterpretedasanadditionalargumentforfinite-dimensionalrealizations.Noticealsothatthisresultisinvariantundertheimportantequivalentchangesofmeasure:ifweobtainagenericevolutionwithrespecttoonefixedmeasure,thenalsowithrespecttoallequivalentmeasures.
2.MalliavincalculusinHilbertspaces.Inordertosetupthemethod-ologicalbackground,werefer,ontheonehand,tothefinite-dimensionalliteratureinMalliavincalculus,suchas[11].Ontheotherhand,wereferto[6]fortheanalyticalframework,inparticular,forquestionsofdifferen-tiabilityoffunctionsoninfinite-dimensionalspacesandforthenotionofderivativesofvectorfieldsV:U⊂G→G,whenGissomeFr´echetspace.WeshallmainlyworkonHilbertspaces:thenthederivativeDV:U→L(H)isalinearoperatortotheBanachspaceofboundedlinearoperators,wherewecanspeakaboutusualpropertiesasdifferentiability,boundedness,andsoon.Weconsiderevolutionequationsofthetype
