Stochastic Integration and Differential Equations 2nd Edition by Philip E. Protter (PDF)

Description

It has been 15 years since the first edition of Stochastic Integration and Differential Equations, A New Approach appeared, and in those years many other texts on the same subject have been published, often with connections to applications, especially mathematical finance. Yet in spite of the apparent simplicity of approach, none of these books has used the functional analytic method of presenting semimartingales and stochastic integration. Thus a 2nd edition seems worthwhile and timely, though it is no longer appropriate to call it “a new approach”. The new edition has several significant changes, most prominently the addition of exercises for solution. These are intended to supplement the text, but lemmas needed in a proof are never relegated to the exercises. Many of the exercises have been tested by graduate students at Purdue and Cornell Universities. Chapter 3 has been completely redone, with a new, more intuitive and simultaneously elementary proof of the fundamental Doob-Meyer decomposition theorem, the more general version of the Girsanov theorem due to Lenglart, the Kazamaki-Novikov criteria for exponential local martingales to be martingales, and a modern treatment of compensators. Chapter 4 treats sigma martingales (important in finance theory) and gives a more comprehensive treatment of martingale representation, including both the Jacod-Yor theory and Emery’s examples of martingales that actually have martingale representation (thus going beyond the standard cases of Brownian motion and the compensated Poisson process). New topics added include an introduction to the theory of the expansion of filtrations, a treatment of the Fefferman martingale inequality, and that the dual space of the martingale space H^1 can be identified with BMO martingales. Solutions to selected exercises are available at the web site of the author, with current URL http://www.orie.cornell.edu/~protter/books.html.

User’s Reviews

Editorial Reviews: Review From the reviews of the second edition:”A fast and nice introduction to semimartingales and stochastic integration … . The second edition of the book has a number of changes and new topics … . The book is highly recommendable for graduate students and experts alike. It is a pleasure to read, with many examples, and all arguments are presented clearly and with care. This book can equally well serve as a course on stochastic calculus as well as an excellent reference material.” (Prof. Dr. M. Vanmaele, KWANT METHODEN, 2004)”It has been well over a decade … . the second edition, particularly since the book itself has by now become a well-known and often-used classic. … While the second edition follows the outline and content of the first edition quite closely … . The book is carefully written and well presented and covers the topics of stochastic integration … . The changes and additions have served to make this now classic “new approach” only a more attractive and comprehensive entry point … .” (Anja Sturm, SIAM Review Vol. 47(1), 2005)”As anyone who has taught or attended a course on Stochastic calculus knows, one of the most difficult aspect of the theme is absence of exercises in the books on the topic. The second edition of this book comes to the rescue. Each chapter has exercises which should help instructors and students alike. … This book would serve as a good text for a course on stochastic calculus. At the same time, it is also a good reference book.” (Rajeeva L. Karandikar, Sankhya: The Indian Journal of Statistics, Vol. 66 (1), 2004)”In this new edition several changes have been made; most of them are inclusions of results obtained since the appearance of the first edition … . addition is exercises, to be found at the end of each chapter. Altogether I agree with the previous reviewer … the book provides an excellent basis for lecturing or self teaching.” (Evelyn Buckwar, Mathematical Reviews, Issue 2005 k) From the Back Cover It has been 15 years since the first edition ofStochastic Integration and Differential Equations, A New Approach appeared, and in those years many other texts on the same subject have been published, often with connections to applications, especially mathematical finance. Yet in spite of the apparent simplicity of approach, none of these books has used the functional analytic method of presenting semimartingales and stochastic integration. Thus a 2nd edition seems worthwhile and timely, though it is no longer appropriate to call it “a new approach”. The new edition has several significant changes, most prominently the addition of exercises for solution. These are intended to supplement the text, but lemmas needed in a proof are never relegated to the exercises. Many of the exercises have been tested by graduate students at Purdue and Cornell Universities. Chapter 3 has been completely redone, with a new, more intuitive and simultaneously elementary proof of the fundamental Doob-Meyer decomposition theorem, the more general version of the Girsanov theorem due to Lenglart, the Kazamaki-Novikov criteria for exponential local martingales to be martingales, and a modern treatment of compensators. Chapter 4 treats sigma martingales (important in finance theory) and gives a more comprehensive treatment of martingale representation, including both the Jacod-Yor theory and Emeryâ¬(tm)s examples of martingales that actually have martingale representation (thus going beyond the standard cases of Brownian motion and the compensated Poisson process). New topics added include an introduction to the theory of the expansion of filtrations, a treatment of the Fefferman martingale inequality, and that the dual space of the martingale space H^1 can be identified with BMO martingales. Solutions to selected exercises are available at the web site of the author, with current URL http://www.orie.cornell.edu/~protter/books.html.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐You should not worry about the wrong image attached to this item. It is, in fact, a stochastic integration book and is the first edition of this book.The text is a fantastic treatment of general stochastic integration and semi-martingales. If you need to work with Ito stochastic integrals, then this book will give you a fantastic understanding of the general stochastic integration theory. You should definitely be comfortable with measure theory before working through this text.I’m giving this book 4 stars because it does not have any exercises. If you want the exercises, you need to purchase the second edition.

⭐my favorate

⭐This book covers EXACTLY what the title promises. The approach to stochastic integration is a bit different from other books I have read, but very intuitive, general and understandable.The proofs of the theorems are generally quite easy to understand, but do not lack rigor at all – quite the opposite.The prerequisites in probability theory are quite modest, however some understanding of measure theory (at least on an intuitive level) is helpful. Some knowledge of stochastic processes (such as maertingales…) also facilitates the reading of the book.Overall, it is quite amazing how Prof. Protter manages to explain deep subjects rigorously, yet very understandably!The book contains a “normal” amount of typos, but almost all are harmless – they do not affect the understanding at all.The only draw-back to the serious researcher might be the large number of French language references.

⭐This review is about the material quality of the printing in the copy I received. This is not about the content.I have access to a copy of this second edition (third corrected printing, 2005) in my office. It is the usual Springer high quality hardcover: it has a matte cover with texture, beautifully bound; the paper inside is high-quality, very soft and slightly off-white; and the printing of the text is very sharp. The version I received from Amazon claimed to be exactly the same, but was very different:- The hardcover was shiny, did not have texture, and had a natural tendency to bend strongly outwards;- The paper inside is whiter, horribly white;- The text printing looks like a photocopy of the original.It looks and feels like a cheap knock-off photocopy done in a backyard. They didn’t even bother changing the information about the printing including the ‘Printed in Germany’ and ‘Printed on acid-free paper’.When I pay a lot of money for a hardcover edition I want the real thing, not a cheap knock-off. I returned mine.

⭐I was putting off buying this book for a while because I thought it might be a bit advanced or turgid for my level of stochastic integration knowledge. But it’s fast become my main and favourite book on the subject.Almost any question that comes to my mind – e.g. “I wonder when integrals of f against X are local martingales” – you will find the answer. The book is so well-written and laid out that usually you can find the relevant result straight away, otherwise you will see it being pointed too from other sections so you get there in a couple of steps. What an achievement this book is. When in this book, one gets the same (rare) sense one gets when looking at e.g. Disquisitiones or Principia, namely a sense that you are reading a real master in the process of making it look easy.

⭐The biggest complement to pay this book is the following: Many other noted texts in this area give you a lot of the technical results and then say, for more detail see Philip Protter’s book — Stochastic Integration.This book is not for the faint hearted, but if you are seriously working in this area and need to go back to fundamentals to create a new model, then you will need this book in your armoury.

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