Functional Analysis for Probability and Stochastic Processes: An Introduction PDF
Preview Functional Analysis for Probability and Stochastic Processes: An Introduction
FunctionalAnalysisforProbabilityandStochasticProcesses. AnIntroduction Thistextisdesignedbothforstudentsofprobabilityandstochasticprocessesandfor studentsoffunctionalanalysis. Forthereadernotfamiliarwithfunctionalanalysisa detailedintroductiontonecessarynotionsandfactsisprovided. However,thisisnota straighttextbookinfunctionalanalysis;rather,itpresentssomechosenpartsof functionalanalysisthathelpunderstandideasfromprobabilityandstochastic processes. ThesubjectsrangefrombasicHilbertandBanachspaces,throughweak topologiesandBanachalgebras,tothetheoryofsemigroupsofboundedlinear operators. Numerousstandardandnon-standardexamplesandexercisesmakethe booksuitableforbothatextbookforacourseandforself-study. adam bobrowski isaProfessorofMathematicsatLublinUniversityof Technology. Functional Analysis for Probability and Stochastic Processes An Introduction A. BOBROWSKI cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press TheEdinburghBuilding,Cambridgecb22ru,UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Informationo nthi stitle :www.cambri dge.org/9780521831666 © Cambridge University Press 2005 Thispublicationisincopyright.Subjecttostatutoryexceptionandtotheprovisionof relevantcollectivelicensingagreements,noreproductionofanypartmaytakeplace withoutthewrittenpermissionofCambridgeUniversityPress. Firstpublishedinprintformat 2005 isbn-13 978-0-511-13039-7 eBook (NetLibrary) isbn-10 0-511-13039-2 eBook (NetLibrary) isbn-13 978-0-521-83166-6 hardback isbn-10 0-521-83166-0 hardback isbn-13 978-0-521-53937-1 paperback isbn-10 0-521-53937-4 paperback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyofurls forexternalorthird-partyinternetwebsitesreferredtointhispublication,anddoesnot guaranteethatanycontentonsuchwebsitesis,orwillremain,accurateorappropriate. Tothemostenthusiasticwriterever–mysonRadek. Contents Preface pagexi 1 Preliminaries,notationsandconventions 1 1.1 Elementsoftopology 1 1.2 Measuretheory 3 1.3 Functionsofboundedvariation. Riemann–Stieltjesintegral 17 1.4 Sequencesofindependentrandomvariables 23 1.5 Convexfunctions. Ho¨lderandMinkowskiinequalities 29 1.6 TheCauchyequation 33 2 Basicnotionsinfunctionalanalysis 37 2.1 Linearspaces 37 2.2 Banachspaces 44 2.3 Thespaceofboundedlinearoperators 63 3 Conditionalexpectation 80 3.1 ProjectionsinHilbertspaces 80 3.2 Definitionandexistenceofconditionalexpectation 87 3.3 Propertiesandexamples 91 3.4 TheRadon–NikodymTheorem 101 3.5 Examplesofdiscretemartingales 103 3.6 Convergenceofself-adjointoperators 106 3.7 ... andofmartingales 112 4 BrownianmotionandHilbertspaces 121 4.1 Gaussianfamilies&thedefinitionofBrownianmotion 123 4.2 CompleteorthonormalsequencesinaHilbertspace 127 vii viii Contents 4.3 ConstructionandbasicpropertiesofBrownianmotion 133 4.4 Stochasticintegrals 139 5 Dualspacesandconvergenceofprobabilitymeasures 147 5.1 TheHahn–BanachTheorem 148 5.2 FormoflinearfunctionalsinspecificBanachspaces 154 5.3 Thedualofanoperator 162 5.4 Weakandweak∗topologies 166 5.5 TheCentralLimitTheorem 175 5.6 Weakconvergenceinmetricspaces 178 5.7 Compactnesseverywhere 184 5.8 Notesonothermodesofconvergence 198 6 TheGelfandtransformanditsapplications 201 6.1 Banachalgebras 201 6.2 TheGelfandtransform 206 6.3 ExamplesofGelfandtransform 208 6.4 ExamplesofexplicitcalculationsofGelfandtransform 217 6.5 DensesubalgebrasofC(S) 222 6.6 InvertingtheabstractFouriertransform 224 6.7 TheFactorizationTheorem 231 7 SemigroupsofoperatorsandLe´vyprocesses 234 7.1 TheBanach–SteinhausTheorem 234 7.2 CalculusofBanachspacevaluedfunctions 238 7.3 Closedoperators 240 7.4 Semigroupsofoperators 246 7.5 BrownianmotionandPoissonprocesssemigroups 265 7.6 Moreconvolutionsemigroups 270 7.7 Thetelegraphprocesssemigroup 280 7.8 Convolutionsemigroupsofmeasuresonsemigroups 286 8 Markovprocessesandsemigroupsofoperators 294 8.1 SemigroupsofoperatorsrelatedtoMarkovprocesses 294 8.2 TheHille–YosidaTheorem 309 8.3 Generatorsofstochasticprocesses 327 8.4 Approximationtheorems 340 9 Appendixes 363 9.1 Bibliographicalnotes 363 Contents ix 9.2 Solutionsandhintstoexercises 366 9.3 Somecommonlyusednotations 383 References 385 Index 390
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