Random Times and Enlargements of Filtrations in a Brownian Setting (Lecture Notes in Mathematics Book 1873)

In November 2004, M. Yor and R. Mansuy jointly gave six lectures at Columbia University, New York. These notes follow the contents of that course, covering expansion of filtration formulae; BDG inequalities up to any random time; martingales that vanish on the zero set of Brownian motion; the Azéma-Emery martingales and chaos representation; the filtration of truncated Brownian motion; attempts to characterize the Brownian filtration. The book accordingly sets out to acquaint its readers with the theory and main examples of enlargements of filtrations, of either the initial or the progressive kind. It is accessible to researchers and graduate students working in stochastic calculus and excursion theory, and more broadly to mathematicians acquainted with the basics of Brownian motion.

In November 2004, M. Yor and R. Mansuy jointly gave six lectures at Columbia University, New York. These notes follow the contents of that course, covering expansion of filtration formulae; BDG inequalities up to any random time; martingales that vanish on the zero set of Brownian motion; the Azéma-Emery martingales and chaos representation; the filtration of truncated Brownian motion; attempts to characterize the Brownian filtration.

The book accordingly sets out to acquaint its readers with the theory and main examples of enlargements of filtrations, of either the initial or the progressive kind. It is accessible to researchers and graduate students working in shastic calculus and excursion theory, and more broadly to mathematicians acquainted with the basics of Brownian motion.

Notation and Convention....Pages 1-39 Stopping and Non-stopping Times....Pages 41-51 On the Martingales which Vanish on the Set of Brownian Zeroes....Pages 53-69 Predictable and Chaotic Representation Properties for Some Remarkable Martingales Including the Azéma and the Dunkl Martingales....Pages 71-86 Unveiling the Brownian Path (or history) as the Level Rises....Pages 87-102 Weak and Strong Brownian Filtrations....Pages 103-116 Sketches of Solutions for the Exercises....Pages 117-139