Customer Reviews

The Malliavin Calculus and Related Topics (Probability and Its Applications)

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The author has prepared an expansive exposition of the foundations of Malliavin calculus along with applications of the theory. From the table of contents alone, this reviewer had high hopes that Nualart's preparation would be an excellent textbook for this topic. Unfortunately, the author frequently switches from the detailed textbook writing style to the terse research article writing style and frequently leaves it up to the reader to fill in numerous gaps in the exposition. The net effect is a book which distracts the reader's attention from the main thread of the idea while she is busy tracking down definitions, notations and formulas that could have easily been included in an appendix, at the very least.

The list of prerequisites for this topic are extensive. A good background in measure theory and the real analysis of random variables is required, as can be found in Rudin's Real and Complex Analysis. The material in Chung's A Course in Probability Theory is essential, as is the general theory of Markov processes, including transition operator semigroups and their infinitesimal generators. I recommend Rogers & Williams Diffusions, Markov Processes, and Martingales: Volume 1, Foundations or Ethier & Kurtz Markov Processes: Characterization and Convergence. A knowledge of functional analysis is also assumed throughout, including duality in spaces of linear operators and operator algebras, and a good reference for this material is Rudin's Functional Analysis. Finally, the reader should have a solid understanding of Ito Calculus, and a good presentation is given in Rogers & Williams' Diffusions, Markov Processes and Martingales: Volume 2, Itô Calculus.

The first edition of the book contains four chapters. In the first chapter(which runs a mere 75 pages), Malliavin calculus is developed, and by this the author means the Wiener Chaos expansion, the derivative operator, the divergence operator and the relationship between the two. The final three chapters deal with selected applications, including existence and regularity of probability densities and a proof of Hormander's Theorem on hypoelliptic operators in Chapter 2, a study of anticipating processes in Chapter 3, and wrapping up with applications to stochastic PDEs in Chapter 4.

The second edition of the text contains the same chapters as the first edition, but adds Chapter 5 on diffusions driven by fractional Brownian motion, and Chapter 6 which deals with Malliavin calculus in mathematical finance. The second edition also drops several helpful details given in the first edition, such as the proof that square integrable stochastic processes have a Wiener chaos expansion. The first edition gives a fairly clear explanation of this result, while the second edition reduces this to a throw-away one sentence statement. The new material in Chapters 5 and 6 are mere introductions, and are offered as applications of Malliavin calculus. I recommend the exposition of Oksendal, et.al in Stochastic Calculus for Fractional Brownian Motion and Applications for a thorough treatment of fractional Brownian motion. Paul Malliavin himself has a book Stochastic Calculus of Variations in Mathematical Finance which I also recommend.

Section 1.1 begins the Malliavin calculus presentation with the introduction of the Wiener chaos expansion of a square integrable random variable obtained from a real-valued isonormal Gaussian process W indexed by a Hilbert space H. Hermite polynomials are introduced via a differential expression, although the author never bothers to explain why these are, in fact, genuine polynomials. The author uses the term "total subset" on page 5 without definition and his notion of this must be inferred from context of the proof of Lemma 1.1.2. This is a key concept and the notion of a total subset is used to establish important results.

Next, the all important multiple Wiener-Ito integral based on the isonormal Gaussian process is introduced. This multiple integral is carefully contructed on elementary functions possesing a certain property on diagonals. However in establishing the key isometry property of the multiple integral, the author gives a quick one equation, 3 equalities proof and doesn't call out where the diagonals property is ever used.

In order to obtain the Wiener chaos expression for a random variable, the author now employs tensor product and contraction product notation. Although the contraction product is immediately and clearly defined, the tensor product is not. The reader must either track it down independently (see Chapter 4 of Spivak's A Comprehensive Introduction to Differential Geometry, Volume 1, 3rd Edition or wait patiently for another 3 pages until the author decides to provide a definition, at least in a special case.

In a text that is often sparing with detail, you might think there is little ink for luxury, but the author thinks otherwise. The next 7 pages include a review of Ito and Stratonovich calculus, including a complete proof of Ito's Lemma. Of course the treatment is so abbreviated that if you don't already know Ito Calculus, the review is not very helpful. The inclusion of this material is baffling, particularly when so many other less heady details have been omitted.

The next section introduces the derivative operator. The definition is straightforward, and the Cameron-Martin space example is clear and easy to follow. The proof of the first key property, an integration-by-parts formula is not as clear and the reader discovers that, in the first edition of Nualart's text, the proof of Lemma 1.2.1, incorrectly employs a single variable Gaussian density. The second edition reworks the proof to correctly include a multivariate Gaussian density.

The text goes on in this manner, with nearly each page containing an unnecessary little "gotcha" like those above. Because of this, Nualart's book requires a lot of work on the part of the reader to fill in needed details. However, the book provides much more information than some recently published alternatives (e.g. Malliavin Calculus with Applications to Stochastic Partial Differential Equations). After the reader has struggled through Nualart the first time, this book should prove to be a valuable desk reference.

Probably the best book on the subject, even if clarity isn't probably its main merit.
Nevertheless it remains a classic, due to a poor competition. A simplier alternative for quants is Malliavin "Stochastic Calculus of Variations in Mathematical Finance".