##
**Brownian motion and stochastic calculus.**
*(English)*
Zbl 0638.60065

Graduate Texts in Mathematics, 113. New York etc.: Springer-Verlag. XXIII, 470 p.; DM 138.00 (1988).

It became clear during the last 20-30 years that stochastic processes are one of the most beautiful branches of mathematics. It is always a great pleasure to see new and nice books in this field. There is no doubt that the book under review will be met with a considerable interest and moreover, it will be useful for different purposes by many readers - students, their professors and assistants, as well as by professional stochasticians.

The authos have chosen the Brownian motion process as the basic object for study. However even this object is so interesting by its properties that we see built really an attractive, deep and beautiful theory. In particular this theory covers a large class of stochastic processes, namely the continuous-time semimartingales related to the Brownian motion. Thus this book contains a representative part of the contemporary theory of stochastic processes.

Now about the contents of the book. As usual, it starts with a nice introduction and a list of frequently used notions. The material is distributed into six chapter. Chapter 1 provides the basic notions of the general theory of stochastic processes such as martingales, stopping times, filtrations. The Doob-Meyer decomposition for continuous-time semimartingales is established and the structure of the square-integrable martingales is described.

Chapter 2 is entirely devoted to the Brownian motion (BM) which in particular obeys both the Markov and the martingale properties. The authors present two constructions; the first is based on the fundamental Daniel-Kolmogorov theory using consistent families of finite-dimensional distributions; the second uses the representation of BM by Haar functions. Several basic properties of BM, such as strong Markov property, passage times relative to BM and sample paths are studied in details.

Chapter 3 includes essential material on stochastic integration. Starting with the classical Itô construction of a stochastic integral, the authors present important results such as the Girsanov theorem, the generalized Itô rule using the local time of BM. The notion of a local time for arbitrary continuous semimartingales is also treated.

Chapter 4 establishes a bridge between BM and some partial differential equations. Here the authors consider the Dirichlet problem (in one- and multi-dimensional cases). It is shown that the solution of partial differential equations of elliptic and parabolic type admit probabilistic representations. The main tool here is the well-known Feynman-Kac formula and its versions.

Stochastic differential equations are intensively considered in Chapter 5. Here the reader can find exhibited the classical Itô theory about the existence and uniqueness of (strongö) solutions of SDEs. other results, e.g. comparison of solutions and solutions in weak sense are also given. The so-called martingale problem is formulated and analyzed. In particular, conditions for existence and uniqueness of its solutions are suggested. Other topics, such as linear SDEs, Brownian bridge, probabilistic representations of solutions of partial differential equations, interesting applications to economics are treated, too.

Chapter 6, the last in this book, is completely devoted to P. Lévy’s theory of Brownian local time. First, the authors give a motivation of the necessity to study this characteristics of BM. Then the theory of local times is developed in many details.

Let us mention that each chapter contains a large number of examples and exercises. Moreover, solutions or hints are given for many of the exercises. It is remarkable that each chapter ends with comprehensive notes which all together provide a nice description, in historical aspects of the development of the theory of stochastic processes. The bibliography is complete, representative and expresses well the contributions of many scientists in the progress of this field of mathematics.

Finally, I should like to note that it was a great plesure for me to read this book and then to write the present review. In my opinion, this book will be used by a wide circle of readers. Two are the main reasons for such an opinion: (i) the comprehensive and updated presentation of all the essential results concerning BM, stochastic calculus and their applications; (ii) the consecutive and master style of presentation. Thus, the authors’ book, as an important recent publication, can be strongly recommended to any reader learning or teaching stochastics, or working in this field.

The authos have chosen the Brownian motion process as the basic object for study. However even this object is so interesting by its properties that we see built really an attractive, deep and beautiful theory. In particular this theory covers a large class of stochastic processes, namely the continuous-time semimartingales related to the Brownian motion. Thus this book contains a representative part of the contemporary theory of stochastic processes.

Now about the contents of the book. As usual, it starts with a nice introduction and a list of frequently used notions. The material is distributed into six chapter. Chapter 1 provides the basic notions of the general theory of stochastic processes such as martingales, stopping times, filtrations. The Doob-Meyer decomposition for continuous-time semimartingales is established and the structure of the square-integrable martingales is described.

Chapter 2 is entirely devoted to the Brownian motion (BM) which in particular obeys both the Markov and the martingale properties. The authors present two constructions; the first is based on the fundamental Daniel-Kolmogorov theory using consistent families of finite-dimensional distributions; the second uses the representation of BM by Haar functions. Several basic properties of BM, such as strong Markov property, passage times relative to BM and sample paths are studied in details.

Chapter 3 includes essential material on stochastic integration. Starting with the classical Itô construction of a stochastic integral, the authors present important results such as the Girsanov theorem, the generalized Itô rule using the local time of BM. The notion of a local time for arbitrary continuous semimartingales is also treated.

Chapter 4 establishes a bridge between BM and some partial differential equations. Here the authors consider the Dirichlet problem (in one- and multi-dimensional cases). It is shown that the solution of partial differential equations of elliptic and parabolic type admit probabilistic representations. The main tool here is the well-known Feynman-Kac formula and its versions.

Stochastic differential equations are intensively considered in Chapter 5. Here the reader can find exhibited the classical Itô theory about the existence and uniqueness of (strongö) solutions of SDEs. other results, e.g. comparison of solutions and solutions in weak sense are also given. The so-called martingale problem is formulated and analyzed. In particular, conditions for existence and uniqueness of its solutions are suggested. Other topics, such as linear SDEs, Brownian bridge, probabilistic representations of solutions of partial differential equations, interesting applications to economics are treated, too.

Chapter 6, the last in this book, is completely devoted to P. Lévy’s theory of Brownian local time. First, the authors give a motivation of the necessity to study this characteristics of BM. Then the theory of local times is developed in many details.

Let us mention that each chapter contains a large number of examples and exercises. Moreover, solutions or hints are given for many of the exercises. It is remarkable that each chapter ends with comprehensive notes which all together provide a nice description, in historical aspects of the development of the theory of stochastic processes. The bibliography is complete, representative and expresses well the contributions of many scientists in the progress of this field of mathematics.

Finally, I should like to note that it was a great plesure for me to read this book and then to write the present review. In my opinion, this book will be used by a wide circle of readers. Two are the main reasons for such an opinion: (i) the comprehensive and updated presentation of all the essential results concerning BM, stochastic calculus and their applications; (ii) the consecutive and master style of presentation. Thus, the authors’ book, as an important recent publication, can be strongly recommended to any reader learning or teaching stochastics, or working in this field.

Reviewer: J.Stoyanov

### MSC:

60Hxx | Stochastic analysis |

60J65 | Brownian motion |

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

60J55 | Local time and additive functionals |