Diffusions, Markov Processes and Martingales: Volume 2, Itô Calculus (Cambridge Mathematical Library) 2nd Edition by L. C. G. Rogers (PDF)

Description

This celebrated book has been prepared with readers’ needs in mind, remaining a systematic treatment of the subject whilst retaining its vitality. The second volume follows on from the first, concentrating on stochastic integrals, stochastic differential equations, excursion theory and the general theory of processes. Much effort has gone into making these subjects as accessible as possible by providing many concrete examples that illustrate techniques of calculation, and by treating all topics from the ground up, starting from simple cases. Many of the examples and proofs are new; some important calculational techniques appeared for the first time in this book. Together with its companion volume, this book helps equip graduate students for research into a subject of great intrinsic interest and wide application in physics, biology, engineering, finance and computer science.

User’s Reviews

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

⭐In this second volume in the series, Rogers & Williams continue their highly accessible and intuitive treatment of modern stochastic analysis. The second edition of their text is a wonderful vehicle to launch the reader into state-of-the-art applications and research.The main prerequisite for Volume 2,’Ito Calculus’, is a careful study of Volume 1,’Foundations’, and although Volume 2 is not entirely self-contained, the authors give copious references to the research literature to augment the main thread. The reader may want to prepare for the stochastic differential geometry material in Chapter 5. As a good introduction, I recommend Spivak’s

⭐and

⭐.The book begins with Chapter 4, which develops the Ito theory for square-integrable semimartingale integrators which are either of bounded variation or are continuous.The chapter begins with a definition of the allowable integrands. These are the so called ‘previsible’ processes and this notion generalizes the concept of left-hand continuity. Some authors (page 131 of Karatzas & Shreve’s

⭐) refer to such integrands as ‘predictable’.As a warm-up into the full theory, the authors present Ito calculus from the Riemann-Stieltjes point-of-view for integrators of bounded variation. Applications to Markov chains are studied which foreshadow the strong Markov process applications derived later on from a more full-fledged theory.The main simplification that the authors derive from continuity assumption is the implicit agreement of the optional quadratic variation process and the Doob-Meyer predictable quadratic variation process. This helps streamline the presentation of the more full-fledged theory and allows the reader to get the main applications more quickly.All the key results from the classical Ito theory are presenting in this chapter, including Integration by Parts, Ito’s Formula, Levy’s characterization Theorem, the martingale transformation Theorem, Girsanov’s Theorem and Tanaka’s formula for Brownian Local Time. There is also a nice treatment of the Stratonovich calculus and its relation to the Ito theory.For readers of Volume 1, the material in Volume 2, Chapter 5 is the long awaited development of stochastic differential equation techniques to explicitly construct Markov processes whose transition semigroups satisfy the Feller-Dynkin hypotheses.After some motivating examples of diffusions from physical systems and control theory (including the ubiquitous Kalman-Bucy filter), the authors focus on strong solutions of SDE’s. Ito’s existence theorem, which was inspired by a Picard-type algorithm from the theory of classical PDEs, is presented for SDE’s with locally lipschitz coefficients. As a really terrific application of Ito’s existence theorem, Rogers & Williams introduce the notion of a Euclidean stochastic flow.Next up, the discussion turns to weak solutions of SDEs, the martingale problem of Stroock and Varadhan. Existence of solutions of the martingale is established with a nice probability measure convergence argument. This treatment really gives the flavor of the Stroock-Varadhan theory and is much more accessible than the full-blown Krylov results found in the Ethier & Kurtz text ‘Markov Processes Characterization and Convergence’.For me, the real highlight of Chapter 5 is the wonderful section introducing stochastic differential geometry. Diffusions on n-dimensional manifolds are introduced and the interplay between Ito and Stratonovich calculus is carefully studied. Examples of diffusions on Riemannian manifolds are studied in some detail.Chapter 6 extends the Ito theory developed in Chapter 4 to general square-integrable semimartingale integrators. The Doob-Meyer decomposition is explored and the divergence between predictable quadratic variation and optional quadratic variation [M] for a square integrable (local) martingale is studied. Next, [M] is generalized sufficiently to complete the development of the Ito calculus. The general Ito Formula is applied to such problems as the Kalman-Bucy Filter and the Bayesian Filter of Kallianpur-Striebel.The book wraps up with an introduction to excursion theory. The premise here is that we want to study those times for which a Markov process visits a compact set. The theory leads to some nice results, including a proof of the embedding theorems of Skorokhod and Azema-Yor along with applications to potential theory and the general study of local time.

⭐This book and its companion volume are a well organized and relatively easy-to-read introduction to a wide variety of ideas in stochastic processes. It is not only a great reference (I always keep it on my desk) but it also has a solid expositional style that fully motivates concepts as they are introduced. The Ito Calculus volume goes deeper than a number of other books on topic including information on integration wrt to a general semimartingale instead of just BM and even an introduction to stochastic calculus on manifolds. My only complaints about the book are that it is separated into two volumes which can be kind of a pain and that its coverage of the SDE/PDE relationship is weak. I would recommend reading Karatzas&Shreeve in addition to this book to fill in some the SDE/PDE details and to get another point of view on the somewhat difficult topic of stochastic analysis

⭐The parts of this book I’ve read have been clear and accessible for someone with an undergraduate degree in mathematics and some knowledge of stochastic processes. It doesn’t needlessly multiply the jargon like some books, and it focuses mainly on the one-dimensional case so that the intuition isn’t constantly obscured by matrix notation. Many subjects also have chatty introductions that offer intuition and a bit of relief from the hard work involved in learning this subject.

⭐I am a salesman and a hobby mathematician only. Please keep this in mind when you evaluate my comments.The book covers what the title says. Surely, the authors are experts in their fields. They motivate very well what they proceed to explain. The chapter on stochastic differential geometry is but one example.The proofs are generally such that they can be understood without pencil and paper, at least about 80% of them. With some proofs I would have appreciated a few more lines of explanation but see above.The prerequisites are a good knowledge of general probability, some intuitive measure theory – and the the content of volume 1 or equivalent.Volume 1 is referred to frequently. It is an inconvenience that the two volumes are not published as one book.For having gone through a number of editions, the book contains a fair number of misprints, I counted over 60, most, but not all, harmless.All in all, I can highly recommend this book to anybody with a serious interest in the subject matter.

⭐Rogers and Williams try hard to communicate their topic! It is not enough for them to state a theorem and prove it – they argue why it is true, what it implies and what’s so special about the proof!The price they chose to pay for emphasizing the big picture is neglecting “unimportant” details. This is a balancing act that could sometimes be performed better.These books are a great source for people who are already familiar with abstract probability theory. I doubt it is a great source for somebody who makes his or her first contact with the subject.Book 1 treats the classical topics of probability theory and stochastic processes. It is interesting in its own right. Book 2 treats stochastic analysis and is more or less self contained.I finish with two quotes to give some impression of the style the books are written in:Vol. 2, p.64: “A striking application of Lévy’s theorem shows that, modulo time-transformation, Brownian motion is the most general continuous local martingale. More specifically, let M be a continuous local martingale, and regard [M] as the intrinsic clock carried by M. Them M has delusions of grandeur: it thinks it is a Brownian motion!”Vol. 2, p.122: “…, the concepts previsible, optional, progressive and non-anticipating agree, modulo silly things regarding conventions about 0. This information should reassure you when you move from book to book!”(Level of difficulty 4/5)

Keywords

Free Download Diffusions, Markov Processes and Martingales: Volume 2, Itô Calculus (Cambridge Mathematical Library) 2nd Edition in PDF format
Diffusions, Markov Processes and Martingales: Volume 2, Itô Calculus (Cambridge Mathematical Library) 2nd Edition PDF Free Download
Download Diffusions, Markov Processes and Martingales: Volume 2, Itô Calculus (Cambridge Mathematical Library) 2nd Edition 2000 PDF Free
Diffusions, Markov Processes and Martingales: Volume 2, Itô Calculus (Cambridge Mathematical Library) 2nd Edition 2000 PDF Free Download
Download Diffusions, Markov Processes and Martingales: Volume 2, Itô Calculus (Cambridge Mathematical Library) 2nd Edition PDF
Free Download Ebook Diffusions, Markov Processes and Martingales: Volume 2, Itô Calculus (Cambridge Mathematical Library) 2nd Edition