Stochastic integration and differential equations. A new approach.

*(English)*Zbl 0694.60047
Applications of Mathematics, 21. Berlin etc.: Springer-Verlag. x, 302 p. DM 98.00 (1990).

The theory of stochastic integration started with the pioneering work of Itô on Brownian motion, was extended to ever wider classes and integrands by many authors until it reached the class of semimartingales. It then was shown (independently) by Dellacherie-Mokobodzki and by Bichteler (after important preliminary work by Métivier and Pellaumail) that this class of integrators couldn’t be enlarged without losing some essential features of an integration theory. After integration theory was well established, the theory of stochastic differential equations was shown to be essentially as good for semimartingales as it is in the deterministic case, and in the case of Brownian motion (Doléans-Dade, the present author for existence and uniqueness, Emery and Protter for stability).

The originality of this book [though the approach isn’t as “new” as it is claimed; it is consistently used in the book by M. Métivier and J. Pellaumail, Stochastic integration. (1980; Zbl 0463.60004)] is to present the theory of semimartingales, stochastic integration and stochastic differential equations starting from the Dellacherie-Bichteler theorem. That is, a semimartingale X on a probability space (\(\Omega\),\({\mathcal F},{\mathbb{P}})\) endowed with a filtration (an increasing family of \(\sigma\)-fields) (\({\mathcal F}_ t)\) is defined by the fact that it leads to a good theory of the stochastic integral \(\int H_ sdX_ s\) of left continuous, adapted processes (“good” meaning that there is a dominated convergence theorem w.r.t. convergence in probability). With this approach, some essential features of the theory become obvious, like the possibility of replacing the law \({\mathbb{P}}\) by an equivalent one, or Stricker’s theorem on restricting filtrations. Though there were partial attempts before, it is the first time that this point of view is used for a complete presentation of the theory, and it is likely that it will become the favourite approach among mathematicians, though people working in electrical engineering departments, for instance, may prefer to remain faithful to the classical \(``signal+noise''\) description of semimartingales.

The book contains a fully developed exposition of semimartingales, stochastic integration, local times, and stochastic differential equations (including stability, and stochastic flows of diffeomorphisms). It isn’t restricted to the continuous case. It gives many applications of stochastic calculus (including an elegant proof of the Lévy area formula, and a discussion of Azéma’s martingales). Its size isn’t forbidding, its style isn’t dry (sound motivations and historical notes are provided). It thus seems to provide an excellent basis for lecturing or self-teaching.

The originality of this book [though the approach isn’t as “new” as it is claimed; it is consistently used in the book by M. Métivier and J. Pellaumail, Stochastic integration. (1980; Zbl 0463.60004)] is to present the theory of semimartingales, stochastic integration and stochastic differential equations starting from the Dellacherie-Bichteler theorem. That is, a semimartingale X on a probability space (\(\Omega\),\({\mathcal F},{\mathbb{P}})\) endowed with a filtration (an increasing family of \(\sigma\)-fields) (\({\mathcal F}_ t)\) is defined by the fact that it leads to a good theory of the stochastic integral \(\int H_ sdX_ s\) of left continuous, adapted processes (“good” meaning that there is a dominated convergence theorem w.r.t. convergence in probability). With this approach, some essential features of the theory become obvious, like the possibility of replacing the law \({\mathbb{P}}\) by an equivalent one, or Stricker’s theorem on restricting filtrations. Though there were partial attempts before, it is the first time that this point of view is used for a complete presentation of the theory, and it is likely that it will become the favourite approach among mathematicians, though people working in electrical engineering departments, for instance, may prefer to remain faithful to the classical \(``signal+noise''\) description of semimartingales.

The book contains a fully developed exposition of semimartingales, stochastic integration, local times, and stochastic differential equations (including stability, and stochastic flows of diffeomorphisms). It isn’t restricted to the continuous case. It gives many applications of stochastic calculus (including an elegant proof of the Lévy area formula, and a discussion of Azéma’s martingales). Its size isn’t forbidding, its style isn’t dry (sound motivations and historical notes are provided). It thus seems to provide an excellent basis for lecturing or self-teaching.

Reviewer: P.A.Meyer

##### MSC:

60H05 | Stochastic integrals |

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

60G07 | General theory of stochastic processes |