Table Of ContentCAMBRIDGE 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 differen-
tialequations,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.
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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
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Stochastic Analysis
Itô and Malliavin Calculus in Tandem
Hiroyuki Matsumoto
AoyamaGakuinUniversity,Japan
Setsuo Taniguchi
KyushuUniversity,Japan
TranslatedandadaptedfromtheJapaneseedition
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Informationonthistitle:www.cambridge.org/9781107140516
TranslatedandadaptedfromtheJapaneseedition:KakuritsuKaiseki,Baifukan,2013
(cid:2)c HiroyukiMatsumotoandSetsuoTaniguchi2017
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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
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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
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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
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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
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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.
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