Continuous martingales and Brownian motion. 3rd ed. (English) Zbl 0917.60006

Grundlehren der Mathematischen Wissenschaften. 293. Berlin: Springer. xiii, 602 p. (1999).
[For the review of the second edition from 1994 see Zbl 0804.60001.]
From the preface: As we did to preface the second edition, we would like here again to single out a few topics closely related with the theme of this book, that have been the subject of some intensive study since 1994.
In a number of applications, processes with long-range dependence seem to fit the random phenomena under study better than do semi-martingales, and particularly diffusion processes; hence, the development of stochastic integration with respect to fractional Brownian motions;
the domain of validity of Itô’s formula, and its interpretations, are constantly being extended;
anticipative stochastic calculus;
the study of processes with independent increments in the light of previously found results for Brownian motion, and, more generally, diffusions; the publication of J. Bertoin’s excellent book, Lévy processes. (1996; Zbl 0861.60003), gives great impetus to these developments;
asymptotics of diffusions in random environments.
Editorial additional note: In 2005 a third corrected printing of this edition has appeared, which also will be reviewed in detail.


60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60G44 Martingales with continuous parameter
60J65 Brownian motion