Customer Reviews

Continuous Martingales and Brownian Motion (Grundlehren der mathematischen Wissenschaften)

5 star:		(0)
4 star:		(3)
3 star:		(1)
2 star:		(0)
1 star:		(0)

 

 

This was the text of my second (graduate) course on probability. While going through the text is, with difficulty, manageable with the help of a teacher, I cannot even imagine doing it on my own. The level of difficulty in reading is roughly the same as that of Karatzas and Shreve, though at times the latter is more readable.

There is a trade-off in learning any new theory. You can get bogged down with the details of every new thing you learn, and move very slowly. While you learn things in detail this way, you miss out on the excitement of learning something new, and perhaps even fail to develop the capability of discerning which concepts are key and which concepts are peripheral to udnerstanding.

That was my main complaint with Karatzas and Shreve, and it is the same with Revuz and Yor. You can spend DAYS doing the exercises of just Chapter 1. If you think you will remain excited about learning stochastic calculus at a snail's pace for about a year, then this book is for you. What is worse, doing those exercises is absolutely important - some extremely crucial concepts are left as exercises. I shudder to think what the reader who does not have the advantage of having a teacher to discuss with would do when (s)he stumbles upon these exercises. I suspect the only option would be to accept the result and move on.

I cite an example to prove my point: Exercise 1.4.6 is a crucial concept about stopping times. I believe most people who are reading this book would have done a course that deals with stopping times in discrete time settings. Karatzas and Shreve does contain the proofs of "Exercise 1.4.6" of Revuz and Yor, and the moral there is that the techniques you learnt for discrete time processes do not carry over directly to continuous time. So, if you pass on Exercise 1.4.6 because you could not solve it on your own, you miss out on an extremely useful technique, and therefore your transition from discrete time to continuous time is at least that much incomplete.

If you are willing to spend a year and a half on stochastic calculus, I would recommend getting a bird's eye view first with something like Oksendal, and then coming down to the details that are omitted there with books like Revuz and Yor and Karatzas and Shreve.

I think that is a better, more exciting, albeit slower way of learning.

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.

The book covers exactly what the title says, and it does so in great detail.
A very good background in measure theory (which I do not have) and a good background in general probability theory are definitely helpful in understanding the topics and the proofs. Mostly, the authors explain well, why they are doing what they are doing now.

However, particularly in the latter parts of the book, the proofs become more difficult: Instead of filling in the gaps, one reads "it is easy to see that..." or "a moment's reflexion shows..." or...
I feel, that a little more detailed proofs would much enhance the readability of this book - at the expense of maybe 15 more pages only. This is my reason for four stars only.

The book has many excercises, which I did not attempt, as no solutions are given. Sometimes the result of an excercise is used in a proof, mimimally for those excercises proofs should have been given.

I would definitely not recommend this book as a first book on stochastic processes. Particularly the book by Oksendal, and even the book by Karatzas and Shreve are easier to read.

The book contains very few typos! (I read the version printed in China)

this book is full of advanced topics, but the authors don't worry about the comprehension of the readers.