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Functional Analysis for Probability and Stochastic Processes An Introduction

ISBN-10: 0521539374
ISBN-13: 9780521539371
Edition: 2005
Authors: Adam Bobrowski
List price: $93.00 Buy it from $86.81
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Description: Designed for students of probability and stochastic processes, as well as for students of functional analysis, specifically, this volume presents some chosen parts of functional analysis that can help clarify probability and stochastic processes.  More...

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Book details

List price: $93.00
Copyright year: 2005
Publisher: Cambridge University Press
Publication date: 8/11/2005
Binding: Paperback
Pages: 406
Size: 5.75" wide x 8.50" long x 0.75" tall
Weight: 1.430
Language: English

Designed for students of probability and stochastic processes, as well as for students of functional analysis, specifically, this volume presents some chosen parts of functional analysis that can help clarify probability and stochastic processes. The subjects range from basic Hilbert and Banach spaces, through weak topologies and Banach algebras, to the theory of semigroups of bounded linear operators. Numerous standard and non-standard examples and exercises make the book suitable as a course textbook or for self-study.

Preface
Preliminaries, notations and conventions
Elements of topology
Measure theory
Functions of bounded variation. Riemann-Stieltjes integral
Sequences of independent random variables
Convex functions. Holder and Minkowski inequalities
The Cauchy equation
Basic notions in functional analysis
Linear spaces
Banach spaces
The space of bounded linear operators
Conditional expectation
Projections in Hilbert spaces
Definition and existence of conditional expectation
Properties and examples
The Radon-Nikodym Theorem
Examples of discrete martingales
Convergence of self-adjoint operators
... and of martingales
Brownian motion and Hilbert spaces
Gaussian families & the definition of Brownian motion
Complete orthonormal sequences in a Hilbert space
Construction and basic properties of Brownian motion
Stochastic integrals
Dual spaces and convergence of probability measures
The Hahn-Banach Theorem
Form of linear functionals in specific Banach spaces
The dual of an operator
Weak and weak* topologies
The Central Limit Theorem
Weak convergence in metric spaces
Compactness everywhere
Notes on other modes of convergence
The Gelfand transform and its applications
Banach algebras
The Gelfand transform
Examples of Gelfand transform
Examples of explicit calculations of Gelfand transform
Dense subalgebras of C(S)
Inverting the abstract Fourier transform
The Factorization Theorem
Semigroups of operators and Levy processes
The Banach-Steinhaus Theorem
Calculus of Banach space valued functions
Closed operators
Semigroups of operators
Brownian motion and Poisson process semigroups
More convolution semigroups
The telegraph process semigroup
Convolution semigroups of measures on semigroups
Markov processes and semigroups of operators
Semigroups of operators related to Markov processes
The Hille-Yosida Theorem
Generators of stochastic processes
Approximation theorems
Appendix
Bibliographical notes
Solutions and hints to exercises
Some commonly used notations
References

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