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Markov Processes from K. Ito's Perspective (AM-155) : Annals of Mathematics Studies - Daniel W. Stroock

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

Annals of Mathematics Studies

Paperback

Published: 6th May 2003
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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.

Preface
Finite State Space, a Trial Runp. 1
Moving to Euclidean Space, the Real Thingp. 35
Ito's Approach in the Euclidean Settingp. 73
Further Considerationsp. 111
Ito's Theory of Stochastic Integrationp. 125
Applications of Stochastic Integration to Brownian Motionp. 151
The Kunita-Watanabe Extensionp. 189
Stratonovich's Theoryp. 221
Notationp. 260
Referencesp. 263
Indexp. 265
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780691115436
ISBN-10: 0691115435
Series: Annals of Mathematics Studies
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 288
Published: 6th May 2003
Publisher: Princeton University Press
Country of Publication: US
Dimensions (cm): 23.5 x 15.2  x 1.83
Weight (kg): 0.4