Functional Analysis for Probability and Stochastic Processes ：An Introduction

Publication subTitle ：An Introduction

Author： Adam Bobrowski  

Publisher： Cambridge University Press‎

Publication year： 2005

E-ISBN: 9780511128219

P-ISBN(Paperback): 9780521831666

Subject： O177 functional analysis

Keyword： 概率论与数理统计

Language： ENG

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Functional Analysis for Probability and Stochastic Processes

Description

This text is designed both for students of probability and stochastic processes, and for students of functional analysis. For the reader not familiar with functional analysis a detailed introduction to necessary notions and facts is provided. However, this is not a straight textbook in functional analysis; rather, it presents some chosen parts of functional analysis that can help understand ideas from 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.

Chapter

Cover

Half-title

Title

Copyright

Dedication

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. Hölder 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 L´evy 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 Appendixes

9.1 Bibliographical notes

9.2 Solutions and hints to exercises

9.3 Some commonly used notations

References

Index

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