Continuous Martingales and Brownian Motion (Grundlehren Der Mathematischen Wissenschaften, Vol 293)

This work provides a detailed study of Brownian motion, via the Ito stochastic calculus of continuous processes such as diffusions and continuous semi-martingales. It aims to facilitate the reading and understanding of research papers in this area, and will be of interest both to graduate students and to more advanced readers, either working primarily with stochastic processes, or doing research in an area involving stochastic processes, such as mathematical physics or economics. The emphasis is on methods, rather than generality. After the first introduction, each chapter introduces a new method or idea - stochastic integration, local times, excursions, weak convergence - and describes its applications to Brownian motion. A feature of the text is the large number of exercises which provide additional results. These have been designed to help the reader master the subject more easily.