Brownian Motion, Martingales, and Stochastic Calculus (Graduate Texts in Mathematics Book 274) by Jean-François Le Gall (PDF)

Description

This book offers a rigorous and self-contained presentation of stochastic integration and stochastic calculus within the general framework of continuous semimartingales. The main tools of stochastic calculus, including Itô’s formula, the optional stopping theorem and Girsanov’s theorem, are treated in detail alongside many illustrative examples. The book also contains an introduction to Markov processes, with applications to solutions of stochastic differential equations and to connections between Brownian motion and partial differential equations. The theory of local times of semimartingales is discussed in the last chapter.Since its invention by Itô, stochastic calculus has proven to be one of the most important techniques of modern probability theory, and has been used in the most recent theoretical advances as well as in applications to other fields such as mathematical finance. Brownian Motion, Martingales, and Stochastic Calculus provides a strong theoretical background to the reader interested in such developments.Beginning graduate or advanced undergraduate students will benefit from this detailed approach to an essential area of probability theory. The emphasis is on concise and efficient presentation, without any concession to mathematical rigor. The material has been taught by the author for several years in graduate courses at two of the most prestigious French universities. The fact that proofs are given with full details makes the book particularly suitable for self-study. The numerous exercises help the reader to get acquainted with the tools of stochastic calculus.

User’s Reviews

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

⭐There are now many sources on stochastic integration from very short to very exhaustive, which makes the job of a newcomer to this subject very hard. If you start with a very exhaustive book, the amount of details can overwhelm you. If you read only special case you may not be aware of what’s out there.To my mind (of working mathematician but non-expert in stochastic analysis) the author, one of the best probabilists today, has found the right balance. His writing is very precise, not skipping details, but he efficiently covers a rather wide range of topics. I think this is an ideal first book for someone interested in learning about stochastic integration and its applications. The background required to understand this book is a first course on measure theoretic probability theory that includes the basic results of theory of discrete time martingales. As it happens the author has (or had) on his website his lecture notes for such a course he taught. Like the present book,they are a model of elegant and efficient presentation. I’m very happy that I’ve “discovered” this book.

⭐Advanced book, need to understand measure theory

⭐One of the best introductory short book on the subject while touching upon a nice varieties of topics.

⭐An excellent book for learning stochastic analysis. Much more comprehensive than GTM113.

⭐I am a ‘hobby mathematician’ only, so my comments below should be read with this caveat in mind.The book covers exactly the topics mentioned in the title.It is a very readable introduction to this topic in my opinion. A good backgorund in probability theory and in measure theory will, however, be very helpful. the motivation, why one studies this or that is always concise – but to the point.The proofs are generally in enough detail to be understood without many ‘side calculations’. Whenever the author writes something like ‘it is easy to see…’ it generally is easy to see. ( other authors are not so kind! ).The books contains many excersises – which I did not attempt, so I cannot comment.the book is almost without typos – and the few I found ( less than 10 or so ) are of a harmless nature. This makes the book useful also for self – study, as I did.All in all a very good book!

⭐A very well-written and rigorous overview of stochastic calculus. The author is clearly well-versed in the material and has a natural affinity for teaching it. The examples are extremely detailed and the exercises help the reader conceptualize some of the more challenging content in this book. Definitely worth reading!

⭐A concise introduction. Very well-written. Problems at the end of each chapter are helpful in terms of reinforcing understanding of the topics themselves.

⭐Great book. It covers a lot of really important topics, so it’s a great reference book. Assumes you have already taken a graduate-level course in probability, so it should not be used as an introduction to probability.

⭐この本から読むと非常に難しく感じるが、D.WilliamsのProbability with MartingaleやD.Eksendalを読んで離散時間のケースを学び見通しをつけるとすんなりと飲み込めるようになった（まだ途中だが）

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