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Functional Analysis for Probability and Stochastic Processes

Publisher: 
Cambridge University Press
Number of Pages: 
393
Price: 
48.00
ISBN: 
0-521-53937-4
Date Received: 
Thursday, October 27, 2005
Reviewable: 
No
Include In BLL Rating: 
No
Reviewer Email Address: 
Adam Bobrowski
Publication Date: 
2005
Format: 
Paperback
Category: 
Textbook

Preface page xi

1 Preliminaries, notations and conventions 1

1.1 Elements of topology 1

1.2 Measure theory 3

1.3 Functions of bounded variation. Riemann–Stieltjes integral 17

1.4 Sequences of independent random variables 23

1.5 Convex functions. H¨older and Minkowski inequalities 29

1.6 The Cauchy equation 33

2 Basic notions in functional analysis 37

2.1 Linear spaces 37

2.2 Banach spaces 44

2.3 The space of bounded linear operators 63

3 Conditional expectation 80

3.1 Projections in Hilbert spaces 80

3.2 Definition and existence of conditional expectation 87

3.3 Properties and examples 91

3.4 The Radon–Nikodym Theorem 101

3.5 Examples of discrete martingales 103

3.6 Convergence of self-adjoint operators 106

3.7 ... and of martingales 112

4 Brownian motion and Hilbert spaces 121

4.1 Gaussian families & the definition of Brownian motion 123

4.2 Complete orthonormal sequences in a Hilbert space 127

4.3 Construction and basic properties of Brownian motion 133

4.4 Stochastic integrals 139

5 Dual spaces and convergence of probability measures 147

5.1 The Hahn–Banach Theorem 148

5.2 Form of linear functionals in specific Banach spaces 154

5.3 The dual of an operator 162

5.4 Weak and weak topologies 166

5.5 The Central Limit Theorem 175

5.6 Weak convergence in metric spaces 178

5.7 Compactness everywhere 184

5.8 Notes on other modes of convergence 198

6 The Gelfand transform and its applications 201

6.1 Banach algebras 201

6.2 The Gelfand transform 206

6.3 Examples of Gelfand transform 208

6.4 Examples of explicit calculations of Gelfand transform 217

6.5 Dense subalgebras of C(S) 222

6.6 Inverting the abstract Fourier transform 224

6.7 The Factorization Theorem 231

7 Semigroups of operators and L´evy processes 234

7.1 The Banach–Steinhaus Theorem 234

7.2 Calculus of Banach space valued functions 238

7.3 Closed operators 240

7.4 Semigroups of operators 246

7.5 Brownian motion and Poisson process semigroups 265

7.6 More convolution semigroups 270

7.7 The telegraph process semigroup 280

7.8 Convolution semigroups of measures on semigroups 286

8 Markov processes and semigroups of operators 294

8.1 Semigroups of operators related to Markov processes 294

8.2 The Hille–Yosida Theorem 309

8.3 Generators of stochastic processes 327

8.4 Approximation theorems 340

9 Appendixes 363

9.1 Bibliographical notes 363

9.2 Solutions and hints to exercises 366

9.3 Some commonly used notations 383

References 385

Index 390

 

Publish Book: 
Modify Date: 
Monday, July 3, 2006

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