Markov Processes from K. Itô's Perspective (AM-155)

Daniel W. Stroock

Published by Princeton University Press 2003

Markov Processes from K. Itô's Perspective (AM-155)

Daniel W. Stroock

Volume 155 in the series Annals of Mathematics Studies

https://doi.org/10.1515/9781400835577

Overview
Contents

About this book

Kiyosi Itô's greatest contribution to probability theory may be his introduction of stochastic differential equations to explain the Kolmogorov-Feller theory of Markov processes. Starting with the geometric ideas that guided him, this book gives an account of Itô's program.

The modern theory of Markov processes was initiated by A. N. Kolmogorov. However, Kolmogorov's approach was too analytic to reveal the probabilistic foundations on which it rests. In particular, it hides the central role played by the simplest Markov processes: those with independent, identically distributed increments. To remedy this defect, Itô interpreted Kolmogorov's famous forward equation as an equation that describes the integral curve of a vector field on the space of probability measures. Thus, in order to show how Itô's thinking leads to his theory of stochastic integral equations, Stroock begins with an account of integral curves on the space of probability measures and then arrives at stochastic integral equations when he moves to a pathspace setting. In the first half of the book, everything is done in the context of general independent increment processes and without explicit use of Itô's stochastic integral calculus. In the second half, the author provides a systematic development of Itô's theory of stochastic integration: first for Brownian motion and then for continuous martingales. The final chapter presents Stratonovich's variation on Itô's theme and ends with an application to the characterization of the paths on which a diffusion is supported.

The book should be accessible to readers who have mastered the essentials of modern probability theory and should provide such readers with a reasonably thorough introduction to continuous-time, stochastic processes.

Author / Editor information

Daniel W. Stroock is a Simons Professor of Mathematics at the Massachusetts Institute of Technology and the author of several books, including A Concise Introduction to the Theory of Integration and Probability Theory, an Analytic View.

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Frontmatter Publicly Available Download PDF	i
Contents Publicly Available Download PDF	vii
Preface Requires Authentication Unlicensed Licensed Download PDF	xi
Chapter 1. Finite State Space, a Trial Run Requires Authentication Unlicensed Licensed Download PDF	1
Chapter 2. Moving to Euclidean Space, the Real Thing Requires Authentication Unlicensed Licensed Download PDF	35
Chapter 3. Itô's Approach in the Euclidean Setting Requires Authentication Unlicensed Licensed Download PDF	73
Chapter 4. Further Considerations Requires Authentication Unlicensed Licensed Download PDF	111
Chapter 5. Itô's Theory of Stochastic Integration Requires Authentication Unlicensed Licensed Download PDF	125
Chapter 6. Applications of Stochastic Integration to Brownian Motion Requires Authentication Unlicensed Licensed Download PDF	151
Chapter 7. The Kunita-Watanabe Extension Requires Authentication Unlicensed Licensed Download PDF	189
Chapter 8. Stratonovich's Theory Requires Authentication Unlicensed Licensed Download PDF	221
Notation Requires Authentication Unlicensed Licensed Download PDF	260
References Requires Authentication Unlicensed Licensed Download PDF	263
Index Requires Authentication Unlicensed Licensed Download PDF	265