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.
