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Random times and enlargements of filtrations in a Brownian setting. (English) Zbl 1103.60003
Lecture Notes in Mathematics 1873. Berlin: Springer (ISBN 3-540-29407-4/pbk). xiii, 158 p. (2006).
These lecture notes follow the contents of the course of six lectures given by the authors at Columbia University, New York in November 2004. This material covers expansion of filtration formulae, BDG inequalities up to any random time, martingales that vanish on the zero set of Browinan motion, the Azéma-Emery martingales and chaos representation, the filtration of truncated Brownian motion, attempts to characterize the Brownian motion.
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 researches and graduate students working in stochastic calculus and excursion theory, and more broadly to the mathematicians acquainted with the basics of Brownian motion.
The book is structurized as follows. In Preliminaries the basic operations of stochastic calculus and general theory of stochastic processes are recalled. In Chapter 1 the transformation of martingales in a “small” filtration into semimartingales in a bigger filtration is being studied. An important number of classical examples are presented and then collected in an appendix at the end of the chapter. This appendix consists of two tables, the first one for progressive enlargements, the second one for initial enlargements. In Chapter 2 the authors examine what remains of a number of classical results in martingale theory when, instead of dealing with a stopping time, one works up to a general random time. The main topic of Chapter 3 consists of the comparison of \({\mathbb E}[X| {\mathcal F}_{\gamma}]\) and \(X_{\gamma}:={\mathbb E}[X| {\mathcal F}_t]| _{t=\gamma}\) where, for simplicity of exposition, \(\gamma\) is the last zero before 1 of an underlying Brownian motion, and \(X\) is a generic integrable random variable. Note that the two quantities are identical when \(\gamma\) is replaced by a stopping time. Chapter 4 discusses the predictable and chaotic representation properties for a given martingale with respect to a filtration. Chapters 5 and 6 are devoted to questions of filtrations. They are tightly knit with the preceding chapters: in Chapter 5 Azéma’s martingale plays a central role, and in Chapter 6 ends of predictable sets are being discussed in the framework of the Brownian filtration. In more details, the deep roots of Chapter 5 are to be found in excursion theory where, traditionally, a constant level is being singled out from the start, and excursions away from this level are studied. Chapter 6 develops the present understanding of the Brownian filtration, or rather, of some fundamental properties which are necessary for a given filtration to be generated by a Brownian motion. Each chapter ends with some exercises, which complement the content of that chapter. A standard feature of these exercises, as well as the style of their solutions, is an illustration of general “principles”, which the authors present in the framework of examples. The solutions – presented in Chapter 7 – are succinctly written, but should contain sufficient details for the reader.

MSC:
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60G40 Stopping times; optimal stopping problems; gambling theory
60G44 Martingales with continuous parameter
60J65 Brownian motion
60J55 Local time and additive functionals
60J25 Continuous-time Markov processes on general state spaces
62M20 Inference from stochastic processes and prediction
60G07 General theory of stochastic processes
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