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CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 159 EditorialBoard B. BOLLOBÁS, W. FULTON, A. KATOK, F. KIRWAN, P. SARNAK, B. SIMON, B. TOTARO STOCHASTICANALYSIS ItôandMalliavinCalculusinTandem Thanks to the driving forces of the Itô calculus and the Malliavin calculus, stochas- ticanalysishasexpandedintonumerousfieldsincludingpartialdifferentialequations, physics, and mathematical finance. This book is a compact, graduate-level text that developsthetwocalculiintandem,layingoutabalancedtoolboxforresearchersand studentsinmathematicsandmathematicalfinance.Thebookexploresfoundationsand applications of the two calculi, including stochastic integrals and stochastic differentialequations,andthedistributiontheoryonWienerspacedevelopedbytheJapanese school of probability. Uniquely, the book then delves into the possibilities that arise byusingthetwoflavorsofcalculustogether.Takingadistinctive,path-space-oriented approach,thisbookcrystalizesmoderndaystochasticanalysisintoasinglevolume. HiroyukiMatsumotoisProfessorofMathematicsatAoyamaGakuinUniversity.He graduated from Kyoto University in1982 and received his Doctor of Science degree from Osaka University in 1989. His research focuses on stochastic analysis and its applicationstospectralanalysisofSchrödingeroperationsandSelberg’straceformula, and he has published several books in Japanese, including Stochastic Calculus and IntroductiontoProbabilityandStatistics.HeisamemberoftheMathematicalSociety ofJapanandaneditoroftheMSJMemoirs. Setsuo Taniguchi is Professor of Mathematics at Kyushu University. He graduated fromOsakaUniversityin1980andreceivedhisDoctorofSciencedegreefromOsaka Universityin1989.Hisresearchinterestsincludestochasticdifferentialequationsand Malliavin calculus. He has published several books in Japanese, including Introduc- tiontoStochasticAnalysisforMathematicalFinanceandStochasticCalculus.Heisa memberoftheMathematicalSocietyofJapanandisaneditoroftheKyushuJournalof Mathematics. 4:36:19, subject to the Cambridge Core terms of use, CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS EditorialBoard: B.Bollobás,W.Fulton,A.Katok,F.Kirwan,P.Sarnak,B.Simon,B.Totaro AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridgeUniversityPress.Fora completeserieslistingvisit:www.cambridge.org/mathematics. Alreadypublished 119 C.Perez-Garcia&W.H.SchikhofLocallyconvexspacesovernon-Archimedeanvaluedfields 120 P.K.Friz&N.B.VictoirMultidimensionalstochasticprocessesasroughpaths 121 T.Ceccherini-Silberstein,F.Scarabotti&F.TolliRepresentationtheoryofthesymmetricgroups 122 S.Kalikow&R.McCutcheonAnoutlineofergodictheory 123 G.F.Lawler&V.LimicRandomwalk:Amodernintroduction 124 K.Lux&H.PahlingsRepresentationsofgroups 125 K.S.Kedlayap-adicdifferentialequations 126 R.Beals&R.WongSpecialfunctions 127 E.deFaria&W.deMeloMathematicalaspectsofquantumfieldtheory 128 A.TerrasZetafunctionsofgraphs 129 D.Goldfeld&J.HundleyAutomorphicrepresentationsandL-functionsforthegenerallineargroup,I 130 D.Goldfeld&J.HundleyAutomorphicrepresentationsandL-functionsforthegenerallineargroup,II 131 D.A.CravenThetheoryoffusionsystems 132 J.VäänänenModelsandgames 133 G.Malle&D.TestermanLinearalgebraicgroupsandfinitegroupsofLietype 134 P.LiGeometricanalysis 135 F.MaggiSetsoffiniteperimeterandgeometricvariationalproblems 136 M.Brodmann&R.Y.SharpLocalcohomology(2ndEdition) 137 C.Muscalu&W.SchlagClassicalandmultilinearharmonicanalysis,I 138 C.Muscalu&W.SchlagClassicalandmultilinearharmonicanalysis,II 139 B.HelfferSpectraltheoryanditsapplications 140 R.Pemantle&M.C.WilsonAnalyticcombinatoricsinseveralvariables 141 B.Branner&N.FagellaQuasiconformalsurgeryinholomorphicdynamics 142 R.M.DudleyUniformcentrallimittheorems(2ndEdition) 143 T.LeinsterBasiccategorytheory 144 I.Arzhantsev,U.Derenthal,J.Hausen&A.LafaceCoxrings 145 M.VianaLecturesonLyapunovexponents 146 J.-H.Evertse&K.Gyo˝ryUnitequationsinDiophantinenumbertheory 147 A.PrasadRepresentationtheory 148 S.R.Garcia,J.Mashreghi&W.T.RossIntroductiontomodelspacesandtheiroperators 149 C.Godsil&K.MeagherErdo˝s–Ko–Radotheorems:Algebraicapproaches 150 P.MattilaFourieranalysisandHausdorffdimension 151 M.Viana&K.OliveiraFoundationsofergodictheory 152 V.I.Paulsen&M.RaghupathiAnintroductiontothetheoryofreproducingkernelHilbertspaces 153 R.Beals&R.WongSpecialfunctionsandorthogonalpolynomials 154 V.JurdjevicOptimalcontrolandgeometry:Integrablesystems 155 G.PisierMartingalesinBanachspaces 156 C.T.C.WallDifferentialtopology 157 J.C.Robinson,J.L.Rodrigo&W.SadowskiThethree-dimensionalNavier–Stokesequations 158 D.HuybrechtsLecturesonK3surfaces 159 H.Matsumoto&S.TaniguchiStochasticAnalysis 4:36:19, subject to the Cambridge Core terms of use, Stochastic Analysis Itô and Malliavin Calculus in Tandem Hiroyuki Matsumoto AoyamaGakuinUniversity,Japan Setsuo Taniguchi KyushuUniversity,Japan TranslatedandadaptedfromtheJapaneseedition 4:36:19, subject to the Cambridge Core terms of use, 32AvenueoftheAmericas,NewYorkNY10013 OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,VIC3207,Australia 4843/24,2ndFloor,AnsariRoad,Daryaganj,Delhi–110002,India 79AnsonRoad,#06–04/06,Singapore079906 CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learningandresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781107140516 TranslatedandadaptedfromtheJapaneseedition:KakuritsuKaiseki,Baifukan,2013 (cid:2)c HiroyukiMatsumotoandSetsuoTaniguchi2017 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2017 PrintedintheUnitedStatesofAmericabySheridanBooks,Inc. AcatalogrecordforthispublicationisavailablefromtheBritishLibrary ISBN978-1-107-14051-6Hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication, anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain, accurateorappropriate. 4:36:19, subject to the Cambridge Core terms of use, Contents Preface pageix FrequentlyUsedNotation xii 1 FundamentalsofContinuousStochasticProcesses 1 1.1 StochasticProcesses 1 1.2 WienerSpace 4 1.3 FilteredProbabilitySpace,AdaptedStochasticProcess 9 1.4 DiscreteTimeMartingales 11 1.4.1 ConditionalExpectation 11 1.4.2 Martingales,DoobDecomposition 13 1.4.3 OptionalStoppingTheorem 16 1.4.4 ConvergenceTheorem 17 1.4.5 OptionalSamplingTheorem 20 1.4.6 Doob’sInequality 22 1.5 ContinuousTimeMartingale 24 1.5.1 Fundamentals 24 1.5.2 ExamplesontheWienerSpace 25 1.5.3 OptionalSamplingTheorem,Doob’sInequality, ConvergenceTheorem 28 1.5.4 Applications 32 1.5.5 Doob–MeyerDecomposition,QuadraticVariation Process 34 1.6 AdaptedBrownianMotion 37 1.7 Cameron–MartinTheorem 40 1.8 Schilder’sTheorem 43 1.9 AnalogytoPathIntegrals 49 2 StochasticIntegralsandItô’sFormula 52 2.1 LocalMartingale 52 v 4:36:21, subject to the Cambridge Core terms of use, vi Contents 2.2 StochasticIntegrals 54 2.3 Itô’sFormula 61 2.4 MomentInequalitiesforMartingales 70 2.5 MartingaleCharacterizationofBrownianMotion 73 2.6 MartingaleswithrespecttoBrownianMotions 82 2.7 LocalTime,Itô–TanakaFormula 87 2.8 ReflectingBrownianMotionandSkorohodEquation 93 2.9 ConformalMartingales 96 3 BrownianMotionandtheLaplacian 102 3.1 MarkovandStrongMarkovProperties 102 3.2 RecurrenceandTransienceofBrownianMotions 108 3.3 HeatEquations 111 3.4 Non-HomogeneousEquation 112 3.5 TheFeynman–KacFormula 117 3.6 TheDirichletProblem 125 4 StochasticDifferentialEquations 133 4.1 Introduction:DiffusionProcesses 133 4.2 StochasticDifferentialEquations 138 4.3 ExistenceofSolutions 145 4.4 PathwiseUniqueness 151 4.5 MartingaleProblems 156 4.6 ExponentialMartingalesandTransformationofDrift 157 4.7 SolutionsbyTimeChange 164 4.8 One-DimensionalDiffusionProcess 167 4.9 LinearStochasticDifferentialEquations 180 4.10 StochasticFlows 183 4.11 ApproximationTheorem 190 5 MalliavinCalculus 195 5.1 SobolevSpacesandDifferentialOperators 195 5.2 ContinuityofOperators 206 5.3 CharacterizationofSobolevSpaces 214 5.4 IntegrationbyPartsFormula 224 5.5 ApplicationtoStochasticDifferentialEquations 232 5.6 ChangeofVariablesFormula 244 5.7 QuadraticForms 257 5.8 ExamplesofQuadraticForms 265 5.8.1 HarmonicOscillators 265 5.8.2 Lévy’sStochasticArea 269 4:36:21, subject to the Cambridge Core terms of use, Contents vii 5.8.3 SampleVariance 274 5.9 AbstractWienerSpacesandRoughPaths 276 6 TheBlack–ScholesModel 281 6.1 TheBlack–ScholesModel 281 6.2 ArbitrageOpportunity,EquivalentMartingaleMeasures 284 6.3 PricingFormula 287 6.4 Greeks 293 7 TheSemiclassicalLimit 297 7.1 VanVleck’sResultandQuadraticFunctionals 297 7.1.1 SolitonSolutionsfortheKdVEquation 302 7.1.2 EulerPolynomials 307 7.2 AsymptoticDistributionofEigenvalues 309 7.3 SemiclassicalApproximationofEigenvalues 312 7.4 Selberg’sTraceFormulaontheUpperHalfPlane 318 7.5 IntegralofGeometricBrownianMotionandHeatKernelonH2 323 Appendix SomeFundamentals 329 References 337 Index 344 4:36:21, subject to the Cambridge Core terms of use, 4:36:21, subject to the Cambridge Core terms of use, Preface The aim of this book is to introduce stochastic analysis, keeping in mind the viewpointofpathspace.Theareacoveredbystochasticanalysisisverywide, and we focus on the topics related to Brownian motions, especially the Itô calculusandtheMalliavincalculus.Asiswidelyknown,astochasticprocessis amathematicalmodeltodescribearandomlydevelopingphenomenon.Many continuousstochasticprocessesaredrivenbyBrownianmotions,whilebasic discontinuousonesarerelatedtoPoissonpointprocesses. TheItôcalculus,namedafterK.Itôwhointroducedthecalculusin1942,is typifiedbystochasticintegrals,Itô’sformula,andstochasticdifferentialequa- tions.WhileItôinvestigatedthosetopicsintermsofBrownianmotions,they arenowstudiedintheextendedframeworkofmartingales.Oneoftheimpor- tantapplicationsofthecalculusisaconstructionofdiffusionprocessesthrough stochastic differential equations. The Malliavin calculus was introduced by P.Malliavininthelatterhalfofthe1970sanddevelopedbymanyresearchers. As he originally called it “a stochastic calculus of variation”, it is exactly a differential calculation on a path space. It opened a way to take a purely probabilisticapproachtotransitiondensitiesofdiffusionprocesses,whichare fundamental objects in theory and are applied to many fields in mathematics andphysics. We made the book self-contained as much as possible. Several prelimi- nary facts in analysis and probability theory are gathered in the Appendix. Moreover, a lot of examples are presented to help the reader to easily under- stand the assertions. This book is organized as follows. Chapter 1 starts with fundamental facts on stochastic processes. In particular, Brownian motions andmartingales areintroduced andbasicpropertiesassociatedwiththemare given. In the last three sections, investigations of path space type are made; the Cameron–Martin theorem, Schilder’s theorem and an analogy with path integralsarepresented. ix 4:36:22, subject to the Cambridge Core terms of use, .001 x Preface Chapter 2 introduces stochastic integrals and Itô’s formula, an associated chain rule. Although Itô originally discussed them with respect to Brownian motions,weformulatethemwithrespecttomartingalesintherecentmanner due to J. L. Doob, H. Kunita and S. Watanabe. Moreover, several facts on continuous martingales are discussed: for example, representations of them by time changes and those via stochastic integrals with respect to Brownian motions. Chapter3presentsseveralpropertiesofBrownianmotion.Asdirectapplica- tionsofItô’sformula,problemsinthetheoryofpartialdifferentialequations, likeheatequationsandDirichletproblems,arestudied.AlthoughtheLaplacian is only dealt with in this chapter, after reading Chapters 4 and 5, the reader will be easily convinced that the results in this chapter can be extended to secondorderdifferentialoperatorsonEuclideanspacesandLaplace–Beltrami operatorsonRiemannianmanifolds. Chapters4and5formthemainportionofthisbook.Chapter4introduces stochasticdifferentialequationsandpresentstheirpropertiesandapplications. Stochasticdifferentialequationsenableustoconstructdiffusionprocessesina purelyprobabilisticmanner.Namely,diffusionprocessesarerealizedasmea- sures on a path space via solutions of stochastic differential equations. This is different from the analytical method by A. Kolmogorov, which uses the fundamentalsolutionoftheassociatedheatequation.Itisalsoseeninthechap- terthatstochasticdifferentialequationsdeterminestochasticflowsasordinary differentialequations.Theflowpropertywillbeusedinthenextchapter. The Malliavin calculus is developed in Chapter 5. The distribution theory on the Wiener space, which was structured by the Japanese school led by S.Watanabe,S.Kusuoka,andI.Shigekawa,isintroduced.Moreover,theinte- gration by parts formula and the change of variable formula on the Wiener space are presented. In the last two sections, the latter formula is applied to computingLaplacetransformsofquadraticWienerfunctionals. Chapter 6 is a brief introduction to mathematical finance. In this chapter, we focus on the Black–Scholes model, the simplest model in mathematical finance.Theexistenceanduniquenessofanequivalentmartingalemeasureis shown and a pricing formula of European contingent claims is given. More- over, as an application of the Malliavin calculus, we show ways to compute hedgingportfoliosandtheGreeks,indicestomeasuresensitivityofpriceswith respecttoparameterslikeinitialpriceandvolatilities. Stochastic analysis is the analysis on path spaces, and it is deeply related to Feynman path integrals. It was M. Kac who gained an insight into this closerelationshipandachievedalotofresults.Hisachievementsexertedgreat influence on notonly probability theory butalsoother fields ofmathematics. 4:36:22, subject to the Cambridge Core terms of use, .001