Continuous martingales and Brownian motion.

*(English)*Zbl 0731.60002
Grundlehren der Mathematischen Wissenschaften, 293. Berlin etc.: Springer-Verlag. ix, 533 p. DM 158.00 (1991).

From the authors’ preface: “This book focuses on the probabilistic theory of Brownian motion. This is a good topic to center a discussion around because Brownian motion is in the intersection of many fundamental classes of processes. It is a continuous martingale, a Gaussian process, a Markov process or more specifically a process with independent increments; it can be defined, up to simple transformations, as the real- valued, centered process with independent increments and continuous paths. It is therefore no surprise that a vast array of techniques may be successfully applied to its study and we choose to organize the book in the following way.

After a first chapter where Brownian motion is introduced, each of the following ones is devoted to a new technique or notion and to some of its applications to Brownian motion. Among these techniques, two are of paramount importance: stochastic calculus, the use of which pervades the whole book, and the powerful excursion theory, both of which are introduced in a self-contained fashion and with a minimum of apparatus.

Furthermore, rather than working towards abstract generality, we have tried to study precisely some important examples and to carry through the computations of the laws of various functionals on random variables. Thus we hope to facilitate the task of the beginner in an area of probability theory which is rapidly evolving. The later chapters of the book however, will hopefully be of interest to the advanced reader.”

This is an excellent book which we have on this subject.

After a first chapter where Brownian motion is introduced, each of the following ones is devoted to a new technique or notion and to some of its applications to Brownian motion. Among these techniques, two are of paramount importance: stochastic calculus, the use of which pervades the whole book, and the powerful excursion theory, both of which are introduced in a self-contained fashion and with a minimum of apparatus.

Furthermore, rather than working towards abstract generality, we have tried to study precisely some important examples and to carry through the computations of the laws of various functionals on random variables. Thus we hope to facilitate the task of the beginner in an area of probability theory which is rapidly evolving. The later chapters of the book however, will hopefully be of interest to the advanced reader.”

This is an excellent book which we have on this subject.

Reviewer: Chou Ching Sung (Chung-Li)

##### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60G44 | Martingales with continuous parameter |

60J65 | Brownian motion |