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An Introduction to Stochastic Integration (Probability and Its Applications) Hardcover – Oct 1990


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"An attractive text…written in [a] lean and precise style…eminently readable. Especially pleasant are the care and attention devoted to details… A very fine book."

—Mathematical Reviews

--This text refers to an out of print or unavailable edition of this title.

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A highly readable introduction to stochastic integration and stochastic differential equations, this book combines developments of the basic theory with applications. It is written in a style suitable for the text of a graduate course in stochastic calculus, following a course in probability.

 

Using the modern approach, the stochastic integral is defined for predictable integrands and local martingales; then Itô’s change of variable formula is developed for continuous martingales. Applications include a characterization of Brownian motion, Hermite polynomials of martingales, the Feynman-Kac functional and Schrödinger equation. For Brownian motion, the topics of local time, reflected Brownian motion, and time change are discussed.

 

New to the second edition are a discussion of the Cameron-Martin-Girsanov transformation and a final chapter which provides an introduction to stochastic differential equations, as well as many exercises for classroom use.

 

This book will be a valuable resource to all mathematicians, statisticians, economists, and engineers employing the modern tools of stochastic analysis.

 

The text also proves that stochastic integration has made an important impact on mathematical progress over the last decades and that stochastic calculus has become one of the most powerful tools in modern probability theory.

Journal of the American Statistical Association

 


An attractive text…written in [a] lean and precise style…eminently readable. Especially pleasant are the care and attention devoted to details… A very fine book.

—Mathematical Reviews

--This text refers to an out of print or unavailable edition of this title.

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For each interval I in IR = (-oo, oo) let B(I) denote the o-field of Borel subsets of I. Read the first page
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34 of 35 people found the following review helpful
An excellent introduction to Stochastic Integration 27 Feb. 2002
By Alan Bain - Published on Amazon.com
Format: Hardcover
This book provides a very easy to read account of the development of the stochastic integral. While concentrating on integrators which are continuous local martingales, and thus lacking the full generality of treatment to be found in, for example Dellacherie
and Meyer, the basic constructions are all performed in a fashion which is readily extensible to the more general case. From a teaching point of view this is beneficial if the more general case is to be studied subsequently. Although the arguments can be considerably simplified for specific special cases (e.g. integration with respect to Brownian Motion only), it is useful to understand how the construction fits into the more general case, which also makes less of a discontinuity for the reader who is subsequently to study the general discontinuous theory!
The arguments are presented carefuly, for example all of the necesary conditions being checked explicitly in places where important theorems are to be applied, and there are none of the annoying statements which plague books on Stochastic Calculus along the lines "the reader can readily check", or "see problem 21.2.43" in the middle of proofs. Additionally very few lines are "skipped" in the proofs; while this does mean that they are lacking in brevity, it is strongly to be encouraged when a complex subject is presented to the novice. When the concepts are understood sufficiently well the reader can easily compile "brief" proofs on his own (as a form of revision), but working the other way round frequently, in my experience of supervising a similar course, leads to misapprehensions about the conditions for applying essential theorems.
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