Newsletter   Secure Checkout   Shopping Cart (0 Items)  
Search:    Welcome Guest! Save up to 30-40% on most items with our awesome everyday discounts!

The Malliavin Calculus and Related Topics (Probability and its Applications) [Hardcover]

Our Price $ 102.46  
Retail Value $ 109.00  
You Save $ 6.54  
Item Number 232107  
Buy New $102.46
Available on the Internet only.

Item description for The Malliavin Calculus and Related Topics (Probability and its Applications) by David Nualart...

The Malliavin calculus (or stochastic calculus of variations) is an infinite-dimensional differential calculus on a Gaussian space. Originally, it was developed to provide a probabilistic proof to HArmander's "sum of squares" theorem, but it has found a wide range of applications in stochastic analysis. This monograph presents the main features of the Malliavin calculus and discusses in detail its main applications. The author begins by developing the analysis on the Wiener space, and then uses this to establish the regularity of probability laws and to prove HArmander's theorem. The regularity of the law of stochastic partial differential equations driven by a space-time white noise is also studied. The subsequent chapters develop the connection of the Malliavin with the anticipating stochastic calculus, studying anticipating stochastic differential equations and the Markov property of solutions to stochastic differential equations with boundary conditions.The second edition of this monograph includes recent applications of the Malliavin calculus in finance and a chapter devoted to the stochastic calculus with respect to the fractional Brownian motion.

Promise Angels is dedicated to bringing you great books at great prices. Whether you read for entertainment, to learn, or for literacy - you will find what you want at!

Item Specifications...

Pages   382
Est. Packaging Dimensions:   Length: 1" Width: 6" Height: 9"
Weight:   1.6 lbs.
Binding  Hardcover
Release Date   Feb 10, 2006
Publisher   Springer
ISBN  3540283285  
ISBN13  9783540283287  

Availability  141 units.
Availability accurate as of Oct 25, 2016 08:08.
Usually ships within one to two business days from La Vergne, TN.
Orders shipping to an address other than a confirmed Credit Card / Paypal Billing address may incur and additional processing delay.

More About David Nualart

Register your artisan biography and upload your photo! David Nualart was born in 1951.

Are You The Artisan or Author behind this product?
Improve our customers experience by registering for an Artisan Biography Center Homepage.

Product Categories

1Books > Special Features > New & Used Textbooks > Sciences > Mathematics > Calculus
2Books > Special Features > New & Used Textbooks > Sciences > Mathematics > Statistics
3Books > Subjects > Professional & Technical > Professional Science > Mathematics > Applied > Statistics
4Books > Subjects > Professional & Technical > Professional Science > Mathematics > Pure Mathematics > Calculus
5Books > Subjects > Science > General
6Books > Subjects > Science > Mathematics > Applied > Probability & Statistics
7Books > Subjects > Science > Mathematics > General
8Books > Subjects > Science > Mathematics > Pure Mathematics > Calculus

Reviews - What do customers think about The Malliavin Calculus and Related Topics (Probability and its Applications)?

A Terse Preparation of this Advanced Topic  Jun 3, 2008
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, 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.

Write your own review about The Malliavin Calculus and Related Topics (Probability and its Applications)

Ask A Question or Provide Feedback regarding The Malliavin Calculus and Related Topics (Probability and its Applications)

Item Feedback and Product Questions
For immediate assistance call 888.395.0572 during the hours of 10am thru 8pm EST Monday thru Friday and a customer care representative will be happy to help you!

Help us continuously improve our service by reporting your feedback or questions below:

I have a question regarding this product
The information above is incorrect or conflicting
The page has misspellings or incorrect grammar
The page did not load correctly in my browser or created an error.

Email Address:
Anti Spam Question. To combat spammers we require that you answer a simple question.
What color is the sky?
Leave This Blank :
Do Not Change This Text :

Add This Product Widget To Your Website

Looking to add this information to your own website? Then use our Product Widget to allow you to display product information in a frame that is 120 pixels wide by 240 pixels high.

    Copy and paste the following HTML into your website and enjoy!

Order toll-free weekdays 10am thru 10pm EST by phone: 1-888-395-0572 (Lines are closed on holidays & weekends.)
Customer Service | My Account | Track My Orders | Return Policy | Request Free Catalog | Email Newsletter

Gift Certificates
RSS Feeds
About Us
Contact Us
Terms Of Use
Privacy Policy