Customer Reviews

Stochastic Calculus: A practical Introduction (Probability and Stochastics Series)

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

 

 

I bought this book after reading in the last chapter of Steele's "Stochastic Calculus" that this would be a good reference for constructing martingales via pdes for the case of x-dependent diffusion coefficients. An introduction, this book certainly is not, nor is it practical or even useful for nonspecialists. I can hardly imagine a worse divergence between reality and an author's stated belief of what the book really is. Richard Durrett states in his introduction that he intends to present coherently the material that would otherwise require understanding many other difficult books. He also complains that his former editor refused the ms, so he went to CRC. Like many other topics in the book, e.g., "Martingale" is not defined, the author simply refers to his other books! Steele's book is extremely difficult: I cannot follow many of his proofs but his examples are stimulating and can be worked out. The examples show how the theorems work, which is practical. Generally, one example is worth a thousand theorems. Durrett's book is 'practical' in that he does offer exercises, and apparently Steele took his execises from this book. I would have wished for better editing, all authors suffer from too little criticism.

Note added later (and I should have given 2 stars): Durrett's 1984 book "Brownian Motion and Martingales in Analysis" looks somewhat more readable and has some examples. In fact, the examples in Steel, as well as the Martingale discussion of his last chapter, may be motivated by or come from the 1984 book. It's often the case that an author's later books on a particular subject are not as good as his earlier ones. Warning: the 1984 book is also written in highly impenetrable mathematese.

Notes added after 3/2007:

1. 'Levy's theorem', pg. 111: take care not to conclude that martingales with variance linear in t are equivalent to a Wiener process, they generally are topologically inequivalent! E.g., for the exponential process with Hurst exponent H=1/2 one obtains E(x^2(t))=2t, and the factor of 2 makes all the difference, formally. More generally, consult Durrett's integration by parts formula (10.1) in part 2.10: since <X^2_t> is generally not t, X^2(t)-t is generally not a martingale. Financial math texts tend to advertise falsely that all martingales are Wiener processes.

2. Instructive derivation of the Coulomb Green function from the martingale defined by the Poisson eqn.

3. The Girsanov factor for Ornstien-Uhlenbeck given on pg. 204 is wrong, see my revierw of the 1984 book for the correct factor. Better yet, work it out for yourself.

4. A statement on pg. 212 is wrong on acount of violating the warning of 1. above: an arbitray martingale is topoligically inequivalnt to a Wiener process!

5. Part 6.4. Very nice, and related to the Coulomb construction: or a class of odes the Green function is given by a martingale constructed from the corresponding Ito process. Sorely and sadly missing: explanation how the boundary terms vanish (to create the desired martingale) at the stopping time. Even worse, the Ito process is sidesteped in favor of a standard construction from odes which hides the role of the stopping time and how one can handle it! Note that the 7th eqn. on pg. 223 is wrong, the correct one is the 3rd eqn. The two are not equivalent unless the underlying Ito process is Wiener, and it generally is not.

I took this book hoping that it can help me learn fast the important concepts of stochastic calculus. I liked the fact that it has exercises with complete solutions, and the friendly presentation. However there are some aspects that I did not like about the book. First thing is that Durrett is not so precise in the proofs of the theorems. Here are several examples.

In Theorem (2.4) he proves that continuous local martingales localize to bounded martingales. I did not understand why it is important to have continuity, and the proof does not give any hints towards this. It is true that even in Rogers & Williams you do not see that. My guess is that the first exit time becomes a stopping time if the process is continuous.

Durrett also does not say in the proof of Theorem (2.4) why he can apply the Optimal Stopping Theorem (2.3) since he did not mention that the stopped martingale is uniformly integrable.

In Theorem (3.1) he defines the variance process of a continuous local martingale. I had troubles with the proof of uniqueness of Theorem (3.1). He does not say why the difference of the processes is of local bounded variation. This fact is crucial for the proof.

The book is good in combination with Rogers & Williams.