Customer Reviews

Stochastic Calculus in Manifolds (Universitext)

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The main topic of Emery's book is the study of martingales and semi martingales taking values in a Riemannian manifold, and the author has prepared an extremely nice (if not somewhat concise) treatment of this material.

The author aims to keep prerequisites to a minimum. You'll need a solid of understanding of stochastic calculus as you might gain from the first 4 chapters of Rogers & Williams two volume set Diffusions, Markov Processes, and Martingales: Volume 1, Foundations and Diffusions, Markov Processes and Martingales: Volume 2, Itô Calculus. It is also handy to have a fairly good background in differential geometry. I recommend , Spivak's A Comprehensive Introduction to Differential Geometry, Volume 1, 3rd Edition (tensors and de Rham cohomology) and A Comprehensive Introduction to Differential Geometry, Volume 2, 3rd Edition (connections and curvature).

Emery begins his text with a very short review of the stochastic calculus of real-valued processes in Chapter 1.

The Ito and Stratonovich integrals are reviewed, as is Ito's existence theorem for SDE's with (real- or vector-valued) Lipschitz coefficients.

Chapter 2 is another very brief review, this time of basic concepts from differential geometry. Emery starts with review of the notion of a partition of unity, which he refers to throughout as 'Whitney's Theorem', most likely since Whitney's 2n+1 embedding theorem depends on this notion. Tangent vectors and 1-forms and bilinear forms are reviewed, and the Lie derivative of each of these is briefly discussed.

Manifold-valued semi martingales and quadratic variation is the focus of Chapter 3. After giving the definition of the manifold-valued semi martingale, the author moves on to discuss the quadratic variation of a semi martingale. Some additional work is required to generalize the concept, and the reader discovers that, in manifolds, quadratic variation depends on a choice of a bilinear form. The chapter ends with a treatment of an approximation technique for computing quadratic variation in a manifold.

Chapter 4, "Connections and martingales" is a wonderfully geometric chapter in which the notion of a martingale is extended from the classical setting to reside in a manifold. Emery starts by introducing a connection on a smooth manifold as a generalization of the Hessian. Here the Hessian is considered to be an assignment mapping each smooth function to a bilinear form so that the natural rules of differential calculus remain valid. Next, the author uses the connection to study the notion of manifold-valued martingale by establishing the Orthogonal Drift Theorem. (The front cover of the book depicts trajectories of a particular 2-sphere valued martingale.) The reader is also introduced to Malliavin's "Transfer Principle". This is the notion that geodesics on Riemannian manifolds and martingales in Riemannian manifolds have many analogous properties. For example affine maps preserve geodesics and preserve martingales and either of these properties characterize affine maps.

In Chapter 5, we are introduced to the Levi-Civita connection of a Riemannian manifold. Emery defines this via a Lie derivative of the metric tensor, which is not as intuitive the notion of parallel transport. However, parallel transport is thoroughly discussed in Chapter 8. Brownian motion in a manifold is defined in terms of Laplace-Beltrami operator. Levy's characterization theorem is established and the quadratic variation formula (in terms of the Riemannian metric) for Brownian motion is developed. However, the reader will have to wait until Chapter 8 to see the existence of Brownian motion. Geodesic completeness, martingale completeness and Brownian completeness for a Riemannian manifolds are carefully studied.

Chapter 6 studies Schwartz's principle for integrating 2-forms along semi martingales. Inspired by Ito's change-of-variable formula, the concept of second order vectors and second order forms is introduced, along with the definition of the Schwartz integral. The chapter concludes with a discussion of intrinsic SDEs on a manifold and establishes the existence of solutions of one type of SDE, having a Schwartz operator as coefficient.

Chapter 7 studies another type of integral, this time the integral of the differential of a first order form along a semi martingale. This is called the Stratonovich integral and generalizes the notion in the real-valued case to the manifold-valued case. Properties of the Stratonovich integral (and its relation to the Ito integral) are investigated.

The text has a very fine set of carefully thought out exercises. Since this treatment here is concise, it is very helpful to have these exercises, and they provide a nice 'self-test' of the reader's grasp of the presentation.

My only complaint is that the book doesn't come with an index. I guess with such a short book, the author doesn't think you'll have any trouble looking things up. This isn't the case and I spend a lot of time flipping through the text to find a particular item.

This book has a nice reference section to the literature and a short overview of classical Ito calculus in the appendix.