Daniel W Stroock

(Author)

Markov Processes from K. Itô's Perspective (Am-155)Paperback, 26 May 2003

Markov Processes from K. Itô's Perspective (Am-155)
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Part of Series
Annals of Mathematics Studies
Part of Series
Annals of Mathematics Studies, 155
Part of Series
Annals of Mathematics Studies (Paperback)
Print Length
288 pages
Language
English
Publisher
Princeton University Press
Date Published
26 May 2003
ISBN-10
0691115435
ISBN-13
9780691115436

Description

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.

Product Details

Author:
Daniel W Stroock
Book Format:
Paperback
Country of Origin:
US
Date Published:
26 May 2003
Dimensions:
23.72 x 15.49 x 1.83 cm
ISBN-10:
0691115435
ISBN-13:
9780691115436
Language:
English
Location:
Princeton
Pages:
288
Weight:
408.23 gm

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