Analysis of Variations for Self-similar Processes: A Stochastic Calculus Approach

271 Pages·2013·1.972 MB·English

by  Ciprian Tudor (auth.)

Probability and Its Applications
PublishedinassociationwiththeAppliedProbabilityTrust
Editors: S.Asmussen, J.Gani,P.Jagers, T.G. Kurtz
Probability and Its Applications
TheProbabilityandItsApplicationsseriespublishesresearchmonographs,withtheexpos-
itory quality to make them useful and accessible to advanced students, in probability and
stochasticprocesses,withaparticularfocuson:
− FoundationsofprobabilityincludingstochasticanalysisandMarkovandotherstochastic
processes
− Applicationsofprobabilityinanalysis
− Pointprocesses,randomsets,andotherspatialmodels
− Branchingprocessesandothermodelsofpopulationgrowth
− Geneticsandotherstochasticmodelsinbiology
− Informationtheoryandsignalprocessing
− Communicationnetworks
− Stochasticmodelsinoperationsresearch
Forfurthervolumes:
www.springer.com/series/1560
Ciprian A. Tudor
Analysis of Variations
for Self-similar Processes
A Stochastic Calculus Approach
CiprianA.Tudor
UFRMathématiques
UniversitédeLille1
Villeneuved’Ascq
France
ISSN1431-7028 ProbabilityandItsApplications
ISBN978-3-319-00935-3 ISBN978-3-319-00936-0(eBook)
DOI10.1007/978-3-319-00936-0
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013945532
MathematicsSubjectClassification: 60F05,60H05,60G18
©SpringerInternationalPublishingSwitzerland2013
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection
with reviews or scholarly analysis or material supplied specifically for the purpose of being entered
and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of
this publication or parts thereof is permitted only under the provisions of the Copyright Law of the
Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.
PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations
areliabletoprosecutionundertherespectiveCopyrightLaw.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Whiletheadviceandinformationinthisbookarebelievedtobetrueandaccurateatthedateofpub-
lication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityforany
errorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,withrespect
tothematerialcontainedherein.
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
TothememoryofmyfatherConstantinTudor
To mydaughterAnna-Maria
Preface
This monograph is an introduction to the stochastic analysis of self-similar pro-
cessesbothintheGaussianandnon-Gaussiancase.
The text is mostly self-contained and should be accessible to graduate students
and researchers with a basic background in probability theory and stochastic pro-
cesses. Although Part II of the monograph is based on the Malliavin calculus, the
toolsusedarebasicandconsequentlyreaderswhoarenotfamiliarwiththetheory
willneverthelessbeabletofollowtheexposition.
The majority of these notes were completed during my research visits to sev-
eral university and research centers such as Purdue University, Keio University,
Universidad de Valparaíso, Humboldt Universität zu Berlin, Centre Interfacultaire
BernoulliatLausanne,RitsumeikanUniversity,UniversityofTrento,CharlesUni-
versity, University of Sydney and Centre de Recerca Matemàtica in Barcelona. I
wouldliketothankmycolleaguesFrederiViens,MakotoMaejima,SoledadTorres,
Peter Imkeller, Robert Dalang, Marco Dozzi, Francesco Russo, Arturo-Kohatsu-
Higa, Stefano Bonaccorsi, Bohdan Maslowski, Qiying Wang, Xavier Bardina and
MartaSanz-Soléfortheirkindinvitations.
Apartofthematerialpresentedinthisbookiscontainedinthedoctoralthesesof
myformerandpresentstudentsKhalifaEs-Sebaiy,SolesneBourguin,JorgeClarke
DelaCerdaandAlexisFauth.
Introduction
Self-similarprocessesarestochasticprocessesthatareinvariantindistributionun-
der a suitable scaling of time and space. This property is crucial in applications
suchasnetworktrafficanalysis,mathematicalfinance,astrophysics,hydrologyand
imageprocessing.Forthisreason,theiranalysishaslongconstitutedanimportant
researchdirectioninprobabilitytheory.Severalmonographs,suchas[75]or[160],
provide a complete analysis of the properties of this class of stochastic processes
and many other research papers and monographs focus on the practical aspects of
vii
viii Preface
self-similarity.Abibliographicalguidetotheapplicationsofself-similarprocesses
is provided in [191]. In the last few decades, new developments in self-similarity
havebeenobtained,includingtheappearanceof newclasses of(Gaussianornon-
Gaussian)self-similarprocessesandnewtechniquestostudytheirbehavior,related
tothestochasticcalculus(especiallytheMalliavincalculus).Theaimofthistextis
tosurveythesenewdevelopments.
Thismonographcomprisestwoparts,eachofthemdividedintoseveralchapters,
andAppendicesA,B,C.
InPartIwediscussthebasicpropertiesofseveralclassesof(Gaussianornon-
Gaussian)self-similarstochasticprocesses.Thispartisdividedintofourchapters.
Chapter1focusesonfractionalBrownianmotionandrelatedprocesses.Fractional
Brownianmotionisthemostwellknownself-similarprocesswithstationaryincre-
ments.ItincludesstandardBrownianmotionasaparticularcase.Theapplications
of this process are now widely recognized. We survey the basic properties of the
processandseveralotherrelatedprocessesthathaverecentlyemergedinscientific
research,suchasbifractionalBrownianmotionandsubfractionalBrownianmotion.
Chapter2treatstheGaussiansolutionstostochasticheatandwaveequationsandin
Chap. 3 we introduce some non-Gaussian self-similar processes which are known
as Hermite processes. Chapter 4 contains some examples of multi-parameter self-
similarprocessesandtheirbasicproperties.
Part II is dedicated to the study of quadratic (and other) variations of several
self-similar processes. The variations of a stochastic process play a crucial role in
its probabilistic and statistical analysis. Best known is the quadratic variation of a
semi-martingale,whichiscrucialforitsItôformula;quadraticvariationalsohasa
directutilityinpractice,inestimatingunknownparameters,suchasvolatilityinfi-
nancialmodels,intheso-called“historical”context.Forself-similarstochasticpro-
cesses, the study of their variations constitutes a fundamental tool in constructing
goodestimatorsoftheirself-similarityparameters.Theseprocessesarewellsuited
tomodelingvariousphenomenawherescalingandlongmemoryareimportantfac-
tors(internettraffic,hydrology,econometrics,amongothers,see[191]).Themost
importantmodelingtaskisthentodetermineorestimatetheself-similarityparam-
eter, because it is also typically responsible for the process’s long memory and its
regularity properties. Studying such processes is thus an important research direc-
tionbothintheoryandinpractice.Theapproachweuseisbasedontheso-called
Malliavin calculus and multiple Wiener-Itô integrals. Part II comprises two chap-
ters.Inthefirstwestudytheasymptoticbehaviorofvarioustypesofvariationsof
fractionalBrownianmotion,oftheHermiteprocessandofthesolutiontothelinear
heatequation.Inthesecondchapterwestudyothertypesofvariationsforstochas-
ticprocesses,includingHermite-typevariationsforself-similarprocessesandfields
andso-calledSpitzer’sandHsu-Robbinstyperesults.
Eachchapterconcludeswithacollectionofexercises.
Paris CiprianA.Tudor
January2013
Contents
PartI ExamplesofSelf-similarProcesses
1 FractionalBrownianMotionandRelatedProcesses . . . . . . . . . . 3
1.1 FractionalBrownianMotion . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 BasicProperties . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.2 StochasticIntegralRepresentation . . . . . . . . . . . . . 5
1.1.3 TheCanonicalHilbertSpace . . . . . . . . . . . . . . . . 6
1.2 BifractionalBrownianMotion . . . . . . . . . . . . . . . . . . . . 8
1.2.1 BasicProperties . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.2 QuadraticVariationswhen2HK=1 . . . . . . . . . . . . 10
1.2.3 TheExtendedBifractionalBrownianMotion . . . . . . . . 14
1.3 Sub-fractionalBrownianMotion . . . . . . . . . . . . . . . . . . 15
1.4 BibliographicalNotes . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 SolutionstotheLinearStochasticHeatandWaveEquation . . . . . 27
2.1 TheSolutiontotheStochasticHeatEquationwithSpace-Time
WhiteNoise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.1.1 TheNoise . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.1.2 TheSolution . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 TheSpatialCovariance . . . . . . . . . . . . . . . . . . . . . . . 30
2.3 TheSolutiontotheLinearHeatEquationwithWhite-Colored
Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3.1 TheNoise . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3.2 TheSolution . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4 TheSolutiontotheFractional-WhiteHeatEquation . . . . . . . . 35
2.4.1 TheNoise . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4.2 TheSolution . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.4.3 OntheLawoftheSolution . . . . . . . . . . . . . . . . . 41
2.5 TheSolutiontotheHeatEquationwithFractional-ColoredNoise . 47
2.5.1 TheNoise . . . . . . . . . . . . . . . . . . . . . . . . . . 47
ix
x Contents
2.5.2 TheSolution . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.6 TheSolutiontotheWaveEquationwithWhiteNoiseinTime . . . 54
2.6.1 TheEquation. . . . . . . . . . . . . . . . . . . . . . . . . 54
2.6.2 TheSolution . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.7 TheStochasticWaveEquationwithLinearFractional-Colored
Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.7.1 TheEquation. . . . . . . . . . . . . . . . . . . . . . . . . 58
2.7.2 TheSolution . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.8 BibliographicalNotes . . . . . . . . . . . . . . . . . . . . . . . . 73
2.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3 Non-GaussianSelf-similarProcesses . . . . . . . . . . . . . . . . . . 77
3.1 TheHermiteProcess . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.1.1 DefinitionandBasicProperties . . . . . . . . . . . . . . . 78
3.1.2 OtherRepresentations . . . . . . . . . . . . . . . . . . . . 81
3.1.3 WienerIntegralswithRespecttotheHermiteProcess . . . 82
3.2 AParticularCase:TheRosenblattProcess . . . . . . . . . . . . . 85
3.2.1 StochasticIntegralRepresentationonaFiniteInterval . . . 86
3.3 TheNon-symmetricRosenblattProcess . . . . . . . . . . . . . . . 92
3.4 BibliographicalNotes . . . . . . . . . . . . . . . . . . . . . . . . 99
3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4 MultiparameterGaussianProcesses . . . . . . . . . . . . . . . . . . 103
4.1 TheAnisotropicFractionalBrownianSheet. . . . . . . . . . . . . 103
4.1.1 BasicProperties . . . . . . . . . . . . . . . . . . . . . . . 104
4.1.2 StochasticIntegralRepresentation . . . . . . . . . . . . . 105
4.2 Two-ParameterHermiteProcesses . . . . . . . . . . . . . . . . . 107
4.3 MultiparameterHermiteProcesses . . . . . . . . . . . . . . . . . 110
4.4 BibliographicalNotes . . . . . . . . . . . . . . . . . . . . . . . . 115
4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
PartII VariationsofSelf-similarProcesses:CentralandNon-Central
LimitTheorems
5 First and Second Order Quadratic Variations. Wavelet-Type
Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.1 QuadraticVariationsofFractionalBrownianMotion . . . . . . . . 122
5.1.1 EvaluationoftheL2-NormoftheQuadraticVariations . . 122
5.1.2 TheMalliavinCalculusandStein’sMethod . . . . . . . . 125
5.1.3 TheCentralLimitTheoremoftheQuadraticVariations
forH ≤ 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4
5.1.4 The Non-Central Limit of the Quadratic Variations
forH > 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4
5.1.5 MultidimensionalConvergenceofthe2-Variations . . . . . 136
5.2 QuadraticVariationsoftheRosenblattProcess . . . . . . . . . . . 138
5.2.1 EvaluationoftheL2-Norm . . . . . . . . . . . . . . . . . 140