Brownian Motion, Martingales, and Stochastic Calculus

To quote the introduction “the aim of this book is to provide a rigorous introduction to the theory of stochastic calculus for continuous semi-martingales putting a special emphasis on Brownian motion.” Chapters 2–4 introduce Brownian motion, martingales, and semimartingles. In Chapter 5 the integral is constructed and many of the classical consequences of the theory are proved: Levy’s characterization of Brownian motion, the fact that any martingale can be written as a stochastic integral, and Girsonov’s formula. The book then turns its attention to the general theory of Markov processes (concentrating primarily on Feller processes), the relationship between Brownian motion and partial differential equations, the solution of stochastic differential equations, and the notion of local time (a measure of the amount of time spent at a point).

The book originated from Le Gall’s notes for an introductory course taught to Master’s students in Paris, but that audience should not be confused with Master’s students in the US. The book assumes knowledge of measure theory and of conditional expectation with respect to a \(\sigma\)-field. If the reader has the background and needs a rigorous treatment of the subject this book would be a good choice. Le Gall writes clearly and gets to the point quickly, so this book is much less intimidating that Revuz and Yor’s tome and a dramatic improvement over the old book by Karatzas and Shreve. However, if you only want to learn the subject to understand applications to finance or in the physical sciences, you would be much better off with the books by Oksendal, Mikosch, or Steele, to name a few of the many book that cut corners or sweep some of the dirt under the rug.

Richard Durrett taught at UCLA and Cornell before he came to Duke in 2010. He is a member of the National Academy of Science, who for the last thirty years has used probability to study problems that arise from ecology, genetics, and cancer modeling.