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 DEGREES1 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.
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