Table Of ContentThis page intentionally left blank
FROM MEASURES TO
ITOˆ INTEGRALS
AfricanInstituteofMathematicsLibrarySeries
TheAfricanInstituteofMathematicalSciences(AIMS),foundedin2003
in Muizenberg, South Africa, provides a one-year postgraduate course in
mathematicalsciencesforstudentsthroughoutthecontinentofAfrica.The
AIMS LIBRARY SERIES is a series of short innovative texts, suitable
for self-study, on the mathematical sciences and their applications in the
broadestsense.
EditorialBoard
ProfessorJ.Banasiak(UniversityofKwaZulu-Natal)
ProfessorA.F.Beardon(UniversityofCambridge)
ProfessorP.Deift(CourantInstituteofMathematicalSciences)
ProfessorP.E.Dorey(DurhamUniversity)
ProfessorB.W.Green(UniversityofStellenbosch)
ProfessorF.W.Hahne(TheDirector,AIMS)
ProfessorA.Iserles(UniversityofCambridge)
ProfessorP.E.Kopp(UniversityofHull)
ProfessorP.Sarnak(InstituteforAdvancedStudyPrinceton)
Dr.T.Tokieda(UniversityofCambridge)
Professor N.G. Turok (University of Cambridge and Perimeter Institute,
Ontario)
AIMS Library Series
FROM MEASURES TO
ˆ
ITO INTEGRALS
EKKEHARD KOPP
University of Hull
CAMBRIDGE UNIVERSITY PRESS
Cambridge,NewYork,Melbourne,Madrid,CapeTown,
Singapore,Sa˜oPaulo,Delhi,Tokyo,MexicoCity
CambridgeUniversityPress
TheEdinburghBuilding,CambridgeCB28RU,UK
PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107400863
(cid:2)c E.Kopp2011
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2011
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloguinginPublicationdata
Kopp,Ekkehard(PeterEkkehard),1944–
FromMeasurestoItoˆIntegrals/EkkehardKopp.
p. cm.–(AfricanInstituteofMathematicsLibrarySeries)
Includesbibliographicalreferencesandindex.
ISBN978-1-107-40086-3(pbk.)
1. Measuretheory–Textbooks. I. Title.
QA312.K5867 2011
515(cid:2).42–dc22
2010050362
ISBN978-1-107-40086-3Paperback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceor
accuracyofURLsforexternalorthird-partyinternetwebsitesreferredto
inthispublication,anddoesnotguaranteethatanycontentonsuch
websitesis,orwillremain,accurateorappropriate.
Contents
Preface pagevii
1 Probabilityandmeasure 1
1.1 Doprobabilistsneedmeasuretheory? 1
1.2 Continuityofadditivesetfunctions 1
1.3 Independentevents 8
1.4 Simplerandomwalk 10
2 Measuresanddistributionfunctions 12
2.1 σ-finitemeasures 12
2.2 Generatedσ-fieldsandπ-systems 15
2.3 Distributionfunctions 17
3 Measurablefunctions/randomvariables 19
3.1 Measurablereal-valuedfunctions 19
3.2 Lebesgue-andBorel-measurablefunctions 20
3.3 Stabilityproperties 21
3.4 Randomvariablesandindependence 25
4 Integrationandexpectation 30
4.1 Integralsofpositivemeasurablefunctions 31
4.2 ThevectorspaceL1ofintegrablefunctions 36
4.3 Riemannv.Lebesgueintegrals 40
4.4 Productmeasures 42
4.5 Calculatingexpectations 44
5 Lp-spacesandconditionalexpectation 49
5.1 LpasaBanachspace 49
5.2 OrthogonalprojectionsinL2 54
5.3 Propertiesofconditionalexpectation 56
v
vi Contents
6 Discrete-timemartingales 60
6.1 Discretefiltrationsandmartingales 60
6.2 TheDoobdecomposition 63
6.3 Discretestochasticintegrals 65
6.4 Doob’sinequalities 67
6.5 Martingaleconvergence 68
6.6 TheRadon–NikodymTheorem 74
7 BrownianMotion 78
7.1 Processes,pathsandmartingales 78
7.2 Convergenceofscaledrandomwalks 81
7.3 BM:constructionandproperties 83
7.4 MartingalepropertiesofBM 88
7.5 VariationofBM 92
8 Stochasticintegrals 95
8.1 TheItoˆ integral 95
8.2 Theintegralasamartingale 101
8.3 Itoˆ processesandtheItoˆ formula 104
8.4 TheBlack–Scholesmodelinfinance 108
8.5 Martingalecalculus 111
References 118
Index 119
Preface
Undergraduate mathematics syllabi vary considerably in their cover-
age of measure-theoretic probability theory, so beginning graduates
often find substantial gaps in their background when attending mod-
ules in advanced analysis, stochastic processes and applications. This
text seeks to fill some of these gaps concisely. The exercises form an
integralpartofthetext.Thematerialarosefrommyexperienceofteach-
ing AIMS students between 2004 and 2007, of which I retain many
fond memories. The AIMS series format allows few explorations of
byways;andtheobjectiveofarrivingatareasonablyhonestbutconcise
account of the Itoˆ integral decided most of the material. With moti-
vationfromelementaryprobabilitywediscussmeasuresandintegrals,
leading via L2-theory and conditional expectation to discrete martin-
gales and an outline proof of the Radon–Nikodym Theorem. The last
twochaptersintroduceBrownianMotionandItoˆ integrals,withabrief
lookatmartingalecalculus.Hereproofsofseveralkeyresultsareonly
sketched briefly or omitted. The Black–Scholes option pricing model
provides the main application. None of the results presented are new;
anyremainingerrorsaremine.
Threehappy,livelyAIMSstudentcohortssufferedmyattemptstointro-
ducethemtomathematicalfinance.Mysincerethanksgotothem,tothe
AIMSteamfortheirsupport,toAlanBeardonforhisencouragement,
totwohelpfulreviewers,toMarekCapinskiforwiseadviceandabove
alltomywifeMargaretforherpatient,lovingsupport.
vii
