Brownian Motion: An Introduction to Stochastic Processes

Brownian Motion is an innovative introduction to stochastic processes in continuous time with continuous state space. It deftly uses Brownian motion as a unifying concept at the confluence of several different stochastic processes (including Gaussian, Markov and diffusion processes) as well as a subject of significant interest in a variety of disciplines. These fields would include at least mathematical physics, statistics, mathematical finance and economics. The book is intended primarily for a graduate level course. It assumes basic measure theory, a course in probability (preferably one based on measure theory) and familiarity with discrete time martingales.

The authors have constructed the book to enable multiple paths of development that emphasize different aspects of Brownian motion. Three that are explicitly identified are: Brownian motion and the Itô calculus, Brownian motion and its sample paths, and Brownian motion as a Markov process. All three paths have in common introductory material that includes some background in physics and mathematics, conceptions of Brownian motion as a Gaussian process or as a martingale, and various ways of constructing Brownian motion.

To help instructors and students find an appropriate path through the book, the authors provide a detailed chart of chapter and section dependencies. Topics of particular interest, in one path or another, are canonical models of Brownian motion on the space of continuous functions, stochastic integrals, stochastic differential equations, diffusion processes and the relationships between Brownian motion and partial differential equations. A final chapter (written by Björn Böttcher) introduces simulation of Brownian motion and provides several algorithms in the form of pseudocode. Each chapter has a modest collection of exercises and a selection of references for further reading.

This book emphasizes the mathematical foundations of stochastic processes and Brownian motion. Those looking for applications in physics, economics or finance will have to look elsewhere. This is first and foremost a rigorous mathematics textbook.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.