Continuous Martingales and Brownian Motion (Grundlehren der mathematischen Wissenschaften)


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Authors:
  • Daniel Revuz
  • Marc Yor

Description:



Continuous Martingales and Brownian Motion (Grundlehren der mathematischen Wissenschaften)
Reviews:

starsComprehensive, but not really accessible
It was the text of my second course (graduated) on the probability. While to pass by the text is, with the difficulty, handy with the assistance of a professor, I cannot even imagine to do it on my clean. The level of the difficulty in the reading is harshly identical that which of Karatzas and Shreve, however sometimes the last is more readable. There is a difference by learning any new theory. You can obtain embourbé with the details of each new thing which you learn, and you move very slowly. While you learn from the things in detail in this way, you miss outside on the excitation again learning something, and perhaps even do not develop the possibilities of distinguishing which concepts are principal and which concepts are peripheral with udnerstanding. It was my principal complaint with Karatzas and Shreve, and it is similar for Revuz and Yor. You can pass the DAYS making the exercises of the chapter right 1. If you think will remain yourselves passionnan' about learning stochastic calculation with a step from tortoise during approximately a year, then this book is for you. What is worse, to make these exercises is absolutely important - some extremely crucial concepts are left like exercises. I shiver to think with what the reader who does not have the advantage of having a professor to discuss would do when (the s)he stumbles on these exercises. I suspect that the only option must accept the result and the movement above. I quote an example to prove my point: 1.4.6 is a crucial concept about the downtimes. I believe that the majority of the people who read this book would have made a course which treats downtimes in discrete arrangements of time. Karatzas and Shreve contain the evidence of the "exercice 1.4.6" of Revuz and Yor, and morals there is that the techniques which you learned for discrete processes of time do not defer directly to continuous time. Thus, if you transmit 1.4.6 because you could not solve it on your clean, you miss outside on an extremely useful technique, and thus your transition from discrete time to continuous time is at least that much unfinished. If you are been willing to spend one year and a half on stochastic calculation, I would recommend to initially obtain the sight of the eye of a bird with something like Oksendal, and to go down then to the details which are omitted there with books like Revuz and Yor and Karatzas and Shreve. I think who am the best, more enthralling, though slower manner of the study.


starsa comprehensive book on stochastic calculus, yet accessible
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.


starsAdvanced, but for Revuz and Yor and some friends of their
this book is full of advanced topics, but the authors don't worry about the comprehension of the readers.



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