Functional analysis for probability and stochastic processes : an introduction /
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| Author / Creator: | Bobrowski, Adam. |
|---|---|
| Imprint: | Cambridge, UK ; New York : Cambridge University Press, 2005. |
| Description: | xii, 393 p. ; 24 cm. |
| Language: | English |
| Subject: | |
| Format: | Print Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/5721418 |
Table of Contents:
- Preface
- 1. Preliminaries, notations and conventions
- 1.1. Elements of topology
- 1.2. Measure theory
- 1.3. Functions of bounded variation. Riemann-Stieltjes integral
- 1.4. Sequences of independent random variables
- 1.5. Convex functions. Holder and Minkowski inequalities
- 1.6. The Cauchy equation
- 2. Basic notions in functional analysis
- 2.1. Linear spaces
- 2.2. Banach spaces
- 2.3. The space of bounded linear operators
- 3. Conditional expectation
- 3.1. Projections in Hilbert spaces
- 3.2. Definition and existence of conditional expectation
- 3.3. Properties and examples
- 3.4. The Radon-Nikodym Theorem
- 3.5. Examples of discrete martingales
- 3.6. Convergence of self-adjoint operators
- 3.7. ... and of martingales
- 4. Brownian motion and Hilbert spaces
- 4.1. Gaussian families & the definition of Brownian motion
- 4.2. Complete orthonormal sequences in a Hilbert space
- 4.3. Construction and basic properties of Brownian motion
- 4.4. Stochastic integrals
- 5. Dual spaces and convergence of probability measures
- 5.1. The Hahn-Banach Theorem
- 5.2. Form of linear functionals in specific Banach spaces
- 5.3. The dual of an operator
- 5.4. Weak and weak* topologies
- 5.5. The Central Limit Theorem
- 5.6. Weak convergence in metric spaces
- 5.7. Compactness everywhere
- 5.8. Notes on other modes of convergence
- 6. The Gelfand transform and its applications
- 6.1. Banach algebras
- 6.2. The Gelfand transform
- 6.3. Examples of Gelfand transform
- 6.4. Examples of explicit calculations of Gelfand transform
- 6.5. Dense subalgebras of C(S)
- 6.6. Inverting the abstract Fourier transform
- 6.7. The Factorization Theorem
- 7. Semigroups of operators and Levy processes
- 7.1. The Banach-Steinhaus Theorem
- 7.2. Calculus of Banach space valued functions
- 7.3. Closed operators
- 7.4. Semigroups of operators
- 7.5. Brownian motion and Poisson process semigroups
- 7.6. More convolution semigroups
- 7.7. The telegraph process semigroup
- 7.8. Convolution semigroups of measures on semigroups
- 8. Markov processes and semigroups of operators
- 8.1. Semigroups of operators related to Markov processes
- 8.2. The Hille-Yosida Theorem
- 8.3. Generators of stochastic processes
- 8.4. Approximation theorems
- 9. Appendix
- 9.1. Bibliographical notes
- 9.2. Solutions and hints to exercises
- 9.3. Some commonly used notations
- References