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 A Comprehensive Introduction to Differential Geometry, Volume 1 and A Comprehensive Introduction to Differential Geometry, Volume 2.



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 Brownian Motion and Stochastic Calculus) 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.