Continuous Martingales and Brownian Motion | Date: 15 April 2011, 07:46
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3rd edition: proofs have been streamlined, errors removed, new exercises added and the bibliography enlarged
a comprehensive book on stochastic calculus, yet accessible, January 31, 2004 By A Customer
This review is from: Continuous Martingales and Brownian Motion (Grundlehren der mathematischen Wissenschaften) (Hardcover)
I only read about 70% of the text, without essentially touching
the excercise problems. I have to confess I'm pretty much overwhelmed by the myriad topics treated in this book.
From the perspective of a student, I think Revuz/Yor has the following merits:
1. It covers an enormous amount of materials, systematically and
carefully. It thus provides the necessary preparation for a graduate student who's eager to get ready for research.
2. Despite of its scope, this book is accessible to graduate students. By "accessible", I mean any dilligent student with certain mathematical maturity should be able to understand most of the materials in the text.
Two things about this book make possible the accessibility. First, proofs are very carefully written, and a quite few of them may even be called detailed. Second, the authors deliberately chose the "slickest" approaches to many classical results,
while preserving, even elucidating, the fundamental ideas. Examples include the construction of BM from the perspectife of Gaussian processes, the presentation of Markov processes in Chapter 3, the "global" definition of a stochastic integral, etc.
This paves the way for further study of more general cases.
3. The computations displayed in this book can serve as good exercise for "basic" trainings. As the book goes on, the reader is more expected to carry out the details. And some of them, although said to be "easy" by the authors, could take some time to figure out.
4. The exercise problems are wonderful. You lose half of the benefits if you don't work out a substantial amount of them.
Many of them are useful results from current research papers, or classical results from these or those "bibles". I myself
haven't done that, and that's why I feel I'm not in the position to give five stars at this moment.
Here's some of my thoughts for an "easier" reading. First, because of the scope of this book, it might be a good idea to read it with real motivations, and maybe during a prolonged period of time. Otherwise you may easily get tired, esp. when you get stuck with some details the authors claim as "easy".
Second, the reading could be frustrating if you care about every detail and do them all alone. A good way would be skipping over some of the details in the first reading, and then coming back at a later time for a second reading, or even a third reading. Find freinds to form a study group would be surely helpful. But I've never had this luck.
Finally, my review is just intended for fellow students. For the opinions of experts, the wonderful review of Frank Knight should be consulted. It can be accessed at MathScinet.
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