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This text is designed both for students of probability and stochastic processes, and for students of functional analysis.
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.
Table of content:
Preface; 1. Preliminaries, notations and conventions; 2. Basic notions in functional analysis; 3. Conditional expectation; 4. Brownian motion and Hilbert spaces; 5. Dual spaces and convergence of probability measures; 6. The Gelfand transform and its applications; 7. Semigroups of operators and Lévy processes; 8. Markov processes and semigroups of operators; 9. Appendixes; References; Index.
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.
Table of content:
Preface; 1. Preliminaries, notations and conventions; 2. Basic notions in functional analysis; 3. Conditional expectation; 4. Brownian motion and Hilbert spaces; 5. Dual spaces and convergence of probability measures; 6. The Gelfand transform and its applications; 7. Semigroups of operators and Lévy processes; 8. Markov processes and semigroups of operators; 9. Appendixes; References; Index.