This review is from: Introduction to Stochastic Integration (Universitext) (Paperback)
Personally, I think this is the best introduction to stochastic integration ever! The author did a remarkable job in presenting the Ito calculus and SDE to readers in an extremely clear way. The author always motivates the readers with intuitive thinking, then leads them to rigorous theory followed by contrete examples. The book only assumes some knowledge of measure theory and is accessible to a wide audience. However, no mathematical rigor is sacrificed. The author has amazingly achieved clarity, rigor and readability in a single book.
The author keeps audience in mind all the time. He never assumes you may know some result used for proof. He is even so generous in spending a subsection on Borel-Cantelli Lemma and ChebySheve Inequality. So the book is really self-contained! You will find the transition between any two sections or subsections very smooth too.
Besides the standard topics you may find in an introductory book in this fields such as Ito integral, Ito Formula, Stratonovich Integral, Tanaka's Formula, Local Time and Girsanov Theorem, SDE and applications, the author also gives careful treatment of Wiener integral to bridge the gap between Stieltjes integral and Ito Integral. (Compensated) Poisson process is also thoroughly covered in many examples, although you may not see this in the content list.
A lot of exercises are given at the end of each chapter. I consider this a great gift for readers. Many textbooks on this topic put the stuff that cannot be covered in maintext into exercises or simply stack problems from other books as exercises. This book is different. The exercises are carefully designed to help readers reinforce what they've learned. So most of the exercises can be tackled by a serious graduate student in math.
In a word, this is the best introductory textbook on this hard subject I've ever seen. I believe even an expert will find useful and interesting material in it. It definitely deserves a 5-star rating!
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Amazon Verified Purchase(What's this?) This review is from: Introduction to Stochastic Integration (Universitext) (Paperback)
From perspective of real analysis, this text gives a rigorous introduction to Ito calculus. In an extremely smooth fashion, all chapters flow out one by one. A background on real analysis is required. However, is it true that some basics of real analysis, e.g. Riemann integration, measurability, etc, are prerequisites for stochastic calculus? It was a wonderful experience for me to read though the book.
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This review is from: Introduction to Stochastic Integration (Universitext) (Paperback)
Highly recommend this book to everyone who started to study stochastic processes and SDE! This book gives better understanding and intuition of the subject than more advanced Karatzas & Shreve. I enjoyed to read this book very much also because the author always referees you to the necessary formula/theorem/definition that was previously given in the text. So, there is no need to search all over the book trying to guess why the author made a current step.
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