Brownian motion, martingales, and stochastic calculus

Graduate Texts in Mathematics #274

by Jean-François Le Gall

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.

Genres: Mathematics, Textbooks

286 pages, Hardcover

Published May 9, 2016

Reviews

[Andrew]

If you haven't seen it before, the connection between Brownian motion and the Laplace operator is unexpectedly beautiful, in that something completely random and non-smooth should provide a method of finding smooth solutions to e.g. heat flow.

[Harley Wiltzer · Rating 5 out of 5]

Excellent book. Oddly enough I found the first chapter difficult to follow, but everything after that was explained very clearly. Lots of interesting exercises too.

[Daniel · Rating 5 out of 5]

Nice pedagogical overview of stochastic calculus with end-of-chapter exercises.