Functional Analysis for Probability and Stochastic Processes: An Introduction

406 Pages·2005·2.116 MB·English

by  Adam Bobrowski

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
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Cambridge University Press
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© Cambridge University Press 2005
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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