It has been 15 years since the first edition of Stochastic Integration
and Differential Equations, A New Approach appeared, and in those
years many other texts on the same subject have been published, often
with connections to applications, especially mathematical finance. Yet
in spite of the apparent simplicity of approach, none of these books has
used the functional analytic method of presenting semimartingales and
stochastic integration. Thus a 2nd edition seems worthwhile and timely,
though it is no longer appropriate to call it "a new approach".
The new edition has several significant changes, most prominently
the addition of exercises for solution. These are intended to supplement
the text, but lemmas needed in a proof are never relegated to the
exercises. Many of the exercises have been tested by graduate students
at Purdue and Cornell Universities. Chapter 3 has been completely
redone, with a new, more intuitive and simultaneously elementary proof
of the fundamental Doob-Meyer decomposition theorem, the more general
version of the Girsanov theorem due to Lenglart, the Kazamaki-Novikov
criteria for exponential local martingales to be martingales, and a
modern treatment of compensators. Chapter 4 treats sigma martingales
(important in finance theory) and gives a more comprehensive treatment
of martingale representation, including both the Jacod-Yor theory and
Emery's examples of martingales that actually have martingale
representation (thus going beyond the standard cases of Brownian motion
and the compensated Poisson process). New topics added include an
introduction to the theory of the expansion of filtrations, a treatment
of the Fefferman martingale inequality, and that the dual space of the
martingale space H^1 can be identified with BMO martingales. Solutions
to selected exercises are available at the web site of the author, with
current URL http: //www.orie.cornell.edu/ protter/books.html.
