Kiyosi Ito'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
Ito'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, Ito 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 Ito'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 Ito's stochastic integral calculus. In the
second half, the author provides a systematic development of Ito's
theory of stochastic integration: first for Brownian motion and
then for continuous martingales. The final chapter presents
Stratonovich's variation on Ito'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."
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