Table Of ContentBocconi & Springer Series 10
Mathematics, Statistics, Finance and Economics
Laurent Decreusefond
Selected
Topics in
Malliavin
Calculus
Chaos, Divergence and So Much More
Bocconi & Springer Series
Mathematics, Statistics, Finance and Economics
Volume 10
Editors-in-Chief
Lorenzo Peccati, Dipartimento di Scienze delle Decisioni, Università Bocconi,
Milan,Italy
SandroSalsa,DepartmentofMathematics,PolitecnicodiMilano,Milan,Italy
SeriesEditors
Beatrice Acciaio, Department of Mathematics, Eidgenössische Technische
HochschuleZürich,Zürich,Switzerland
CarloA.Favero,DipartimentodiFinanza,UniversitàBocconi,Milan,Italy
Eckhard Platen, Department of Mathematical Sciences, University of Technology
Sydney,Sydney,NSW,Australia
WolfgangJ.Runggaldier,DipartimentodiMatematicaPuraeApplicata,Università
degliStudidiPadova,Padova,Italy
Erik Schlögl, School of Mathematical and Physical Sciences, University of
TechnologySydney,CityCampus,NSW,Australia
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THESERIESISINDEXEDINSCOPUS
Laurent Decreusefond
Selected Topics in Malliavin
Calculus
Chaos, Divergence and So Much More
LaurentDecreusefond
TelecomParis
InstitutPolytechniquedeParis
Paris,France
ISSN2039-1471 ISSN2039-148X (electronic)
Bocconi&SpringerSeries
ISBN978-3-031-01310-2 ISBN978-3-031-01311-9 (eBook)
https://doi.org/10.1007/978-3-031-01311-9
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To Sandrine,Marc, andBenjamin
Preface
Itissometimeseasiertodescribesomethingbywhatitisnotratherthanbywhatit
issupposedtobe.ThisbookisnotaresearchmonographaboutMalliavincalculus
with the latest results and the most sophisticated proofs. It does not contain all
theknownresultsevenforthebasicsubjectsaddressedhere.Thegoalwastogive
the largestpossible variety of prooftechniques.For instance, we did notfocuson
theproofofconcentrationinequalityforfunctionalsoftheBrownianmotion,asit
followscloselythelinesoftheanalogresultforPoissonfunctionals.
This book grew from the graduate courses I gave at Paris-Sorbonneand Paris-
Saclay universities,duringthe last fewyears. Itis supposedto be asaccessible as
possibleforstudentswhohaveaknowledgeofItôcalculusandsomerudimentsof
functionalanalysis.
A recurrentdifficultywhen someonediscoversMalliavin calculusis due to the
different and often implicit identifications which are made between several func-
tionalspaces.Itriedtodemystifythispointasmuchaspossible.Thepresentationis
hopefullyself-contained,meaningthatthe necessaryresultsof functionalanalysis
whicharesupposedtobeknowninalltheresearchmonographsarerecalledinthe
coreofthe text.Thechoiceofthetopicshasbeeninfluencedbymyownresearch
whichrevolvedforawhilearoundfractionalBrownianmotionandthenshiftedto
pointprocesses,withaninclinationtoStein’smethod.
I did not insist on the historical applications of the Malliavin calculus which
wereabouttheexistenceofthedensityofthedistributionofsomerandomvariables,
because there are so many other interesting subjects where the Malliavin calculus
canbeapplied:Greekscomputations,conditionalexpectations,changeofmeasure,
optimal transport, filtration enlargement, and, more recently, the Stein-Malliavin
method.
IamgreatlyindebtedtoA.S.ÜstünelwhointroducedmetoMalliavincalculusa
fewyearsago.Ithasbeenalongandrichjourneysincethen.
This book benefited from the help of numerous students, most notably B.
Costacèque-Cecchi.Theremainingerrorsaremine.
Paris,France LaurentDecreusefond
2021
vii
Contents
1 WienerSpace ................................................................ 1
1.1 GaussianRandomVariables............................................ 1
1.2 WienerMeasure......................................................... 3
1.3 WienerIntegral.......................................................... 13
1.4 Problems................................................................. 22
1.5 NotesandComments ................................................... 23
References..................................................................... 24
2 GradientandDivergence ................................................... 25
2.1 Gradient ................................................................. 25
2.2 Divergence............................................................... 44
2.3 Problems................................................................. 57
2.4 NotesandComments ................................................... 59
References..................................................................... 59
3 WienerChaos................................................................. 61
3.1 ChaosDecomposition................................................... 62
3.2 Ornstein–UhlenbeckOperator.......................................... 79
3.3 Problems................................................................. 86
3.4 NotesandComments ................................................... 87
References..................................................................... 88
4 FractionalBrownianMotion ............................................... 89
4.1 DefinitionandSample-PathsProperties ............................... 89
4.2 Cameron–MartinSpace................................................. 92
4.3 WienerSpace............................................................ 98
4.4 GradientandDivergence ............................................... 103
4.5 ItôFormula.............................................................. 110
4.6 Problems................................................................. 118
4.7 NotesandComments ................................................... 120
References..................................................................... 120
ix
x Contents
5 PoissonSpace................................................................. 121
5.1 PointProcesses.......................................................... 121
5.2 PoissonPointProcess................................................... 124
5.3 FinitePoissonPointProcess............................................ 125
5.4 StochasticAnalysis ..................................................... 132
5.5 AQuickRefresherAboutthePoissonProcessontheLine........... 145
5.6 Problems................................................................. 146
5.7 NotesandComments ................................................... 148
Reference...................................................................... 148
6 TheMalliavin–SteinMethod ............................................... 149
6.1 Principle................................................................. 149
6.2 FourthOrderMomentTheorem........................................ 157
6.3 PoissonProcessApproximation........................................ 161
6.4 Problems................................................................. 167
6.5 NotesandComments ................................................... 168
References..................................................................... 168
Index............................................................................... 171
About the Author
LaurentDecreusefondisaformerstudentofEcoleNormaleSupérieuredeParis-
Saclay.He receivedthe Agrégationin 1989,his PhD in 1994,and hishabilitation
in 2001 in mathematics. He is now Full Professor of Mathematics at Institut
Polytechnique de Paris, one of the most renowned French research and teaching
institutions.
His research topics are twofold. The theoretical part is devoted to Malliavin
calculusanditsapplications.Heistheauthorofahighlycitedpaperaboutfractional
Brownian motion that paved the way to a thousand research articles. Recently,
he has been interested in the functional Stein-Malliavin method, which gives the
convergencerateinfunctionallimittheorems.
Onamoreappliedpart,heproposednewparadigmsforstochasticmodellingof
telecomsystems,includingstochasticgeometryandrandomtopologicalalgebra.
Hecoauthoredseveralpapersthatgaveanewapproachtothecoverageanalysis
of cellular systems. The performance of some of the algorithms so defined may
be analyzedwith mathematicaltools comingfrom the Malliavin calculus, such as
concentrationinequalities.
xi
