Continuous martingales and Brownian motion. 3rd ed., 3rd. corrected printing.

*(English)*Zbl 1087.60040
Grundlehren der Mathematischen Wissenschaften 293. Berlin: Springer (ISBN 3-540-64325-7/hbk; 978-3-642-08400-3/pbk). xi, 606 p. (2005).

The authors have revised the second edition of their fundamental and impressive monograph on Brownian motion and continuous martingales; for the review of that edition from 1994 see Zbl 0804.60001. The presentation of this book is unique in the sense that a concise and well-written text is complemented by a long series of detailed exercises. In order to prepare these exercises, the authors have made an extensive search in the literature selecting interesting results that can be derived without much effort.

This third edition contains some additional exercises related to recent advances in the subject. The topics covered by the new exercises include an exponential inequality for the supremum of the norm of \(d\)-dimensional Brownian motions, the notion of harness for Lévy processes, a description of recent results of B. Tsirel’son on Brownian filtrations [see, e.g., Geom. Funct. Anal. 7, No. 6, 1096–1142 (1997; Zbl 0902.31004)], the distribution of local times of Brownian bridges, equivalence in distribution of functionals of Brownian motions with drift, and some further distribution results for Bessel processes.

This third edition is a valuable update of this basic reference book, which has been very helpful for researchers and students in stochastic processes and probability theory.

This third edition contains some additional exercises related to recent advances in the subject. The topics covered by the new exercises include an exponential inequality for the supremum of the norm of \(d\)-dimensional Brownian motions, the notion of harness for Lévy processes, a description of recent results of B. Tsirel’son on Brownian filtrations [see, e.g., Geom. Funct. Anal. 7, No. 6, 1096–1142 (1997; Zbl 0902.31004)], the distribution of local times of Brownian bridges, equivalence in distribution of functionals of Brownian motions with drift, and some further distribution results for Bessel processes.

This third edition is a valuable update of this basic reference book, which has been very helpful for researchers and students in stochastic processes and probability theory.

Reviewer: David Nualart (Barcelona)