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**Diffusions, Markov processes, and martingales. Volume 2: Itô calculus.**
*(English)*
Zbl 0627.60001

Wiley Series in Probability and Mathematical Statistics. Chichester etc.: John Wiley & Sons. XIV, 475 p.; £39.95 (1987).

The first thing to say about this volume, is to stress its almost complete independence from volume 1 (1979; Zbl 0402.60003) due to the second author alone. This doesn’t mean that one can start reading volume 2 without knowing anything, but that volume 1 can be replaced for this purpose (except possibly for the applications to the theory of Markov processes) by many general books on probability theory, containing the standard results of a first level course on stochastic processes. This is important to say that, because many people will hesitate to buy a vol. 2 without having vol. 1, and vol. 1 in this case is written in an unusual style for a book, which one may like (the reviewer’s case) or dislike.

The general idea of the book is to present completely the modern theory of stochastic processes (which is often considered as a very abstract and complicated theory, while the reviewer’s opinion is that it is essentially an easy and intuitive theory, done to be used, like the standard Calculus, by people with little mathematical training). In contrast with Bourbakist Traités, like the book of C. Dellacherie and the reviewer, Probabilités et Potentiels. Chap. XII à XVI (last one of the four volumes which appeared till now) (1986; Zbl 0624.60084) it is presented here with motivations, pedagogy, and a great wealth of concrete examples. Like Dellacherie-Meyer, and in contrast with several recent texts, this book isn’t restricted to continuous processes, but presents the general theory of cadlag semi-martingales.

It consists of three parts, modestly called “chapters”. The first one is an introduction to Stochastic Calculus, the second one concerns Stochastic Differential Equations for continuous processes and applications to Diffusion theory, while the last part contains the essentials of the so called “General Theory of Processes”, whose interest now appears to a mature reader who has read the preceding parts. The last section of the book concerns Ito’s excursion theory, in which continuous processes (Brownian motion) and discontinuous processes (Poisson point processes) appear as mutually dependent and mutually illuminating. Of course the domains to which the authors have devoted their own research work (like Markov chains and path decompositions) are especially developed, though the emphasis lies less than in the first volume on Markov processes theory.

The authors have availed themselves of the wisdom contained in the volumes of the Azéma-Yor “Séminaires de Probabilités”, to which they have been for many years regular contributors, and in many cases they give the best results and proofs available at the date of publication. The reviewer has found no other misprint than the charming Japanese hat on the T of ITO on top of pages 413-447.

The general idea of the book is to present completely the modern theory of stochastic processes (which is often considered as a very abstract and complicated theory, while the reviewer’s opinion is that it is essentially an easy and intuitive theory, done to be used, like the standard Calculus, by people with little mathematical training). In contrast with Bourbakist Traités, like the book of C. Dellacherie and the reviewer, Probabilités et Potentiels. Chap. XII à XVI (last one of the four volumes which appeared till now) (1986; Zbl 0624.60084) it is presented here with motivations, pedagogy, and a great wealth of concrete examples. Like Dellacherie-Meyer, and in contrast with several recent texts, this book isn’t restricted to continuous processes, but presents the general theory of cadlag semi-martingales.

It consists of three parts, modestly called “chapters”. The first one is an introduction to Stochastic Calculus, the second one concerns Stochastic Differential Equations for continuous processes and applications to Diffusion theory, while the last part contains the essentials of the so called “General Theory of Processes”, whose interest now appears to a mature reader who has read the preceding parts. The last section of the book concerns Ito’s excursion theory, in which continuous processes (Brownian motion) and discontinuous processes (Poisson point processes) appear as mutually dependent and mutually illuminating. Of course the domains to which the authors have devoted their own research work (like Markov chains and path decompositions) are especially developed, though the emphasis lies less than in the first volume on Markov processes theory.

The authors have availed themselves of the wisdom contained in the volumes of the Azéma-Yor “Séminaires de Probabilités”, to which they have been for many years regular contributors, and in many cases they give the best results and proofs available at the date of publication. The reviewer has found no other misprint than the charming Japanese hat on the T of ITO on top of pages 413-447.

Reviewer: A.Meyer

### MSC:

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

60Gxx | Stochastic processes |

60Hxx | Stochastic analysis |

60Jxx | Markov processes |