Brownian motion. An introduction to stochastic processes. With a chapter on simulation by Björn Böttcher.

*(English)*Zbl 1258.60002
De Gruyter Graduate. Berlin: de Gruyter (ISBN 978-3-11-027889-7/pbk; 978-3-11-027898-9/ebook). xiv, 380 p. (2012).

The main goal of this book is to give a thorough introduction to Brownian motion and provide a bridge to more advanced texts. As the authors point out, the presentation is essentially self-contained, some (basic) measure and probability theory is naturally required though. A number of good problems in the heart of the subject matter are given in each chapter, and topics and literature for further reading is provided. A notable nice feature for the reader is that possible traps and pitfalls are marked and sometimes more thoroughly discussed in the exercises.

The book consists of 20 chapters, that shed light on Brownian motion from very different angles. Chapters 1 and 2 briefly discuss some historical aspects of Brownian motion and give basic definitions and results. Chapters 3 and 4 are all about the construction of Brownian motion, and offer several different methods. This includes approaches usually less frequently encountered in related literature. Chapter 5 discusses some martingale properties of Brownian motion and related issues of stopping times. Chapters 6, 7 and 8 essentially explore Markov properties of Brownian motion. Among other things, the reflection principle, transience and recurrence, semigroups and infinitesimal generators and the connection to PDEs is discussed. It is worth mentioning that the theory is presented in a very structured and accessible way. Chapters 9 and 10 are devoted to regularity properties of Brownian motion. Among other things, the quadratic variation and its connection to Lévy’s characterization, Hölder continuity and non-differentiability of Brownian motion is discussed. Chapters 11 and 12 cover various “growth aspects” of Brownian motion. This includes some laws of the iterated logarithm, large deviation and the Cameron-Martin formula. Chapter 13 is about the Skorohod embedding scheme. As an application, it is used to give a proof of the Donsker construction mentioned in Chapter 3. Chapters 14 to 19 are devoted to stochastic integration with respect to Brownian motion and related issues. This naturally includes Itô’s \(L^2\)-theory and beyond (Chapters 14,15), Itô’s formula (Chapter 16) and its many applications, such as the Doléan-Dade exponential, Girsanov’s theorem and various martingale representation results (Chapter 17). Finally, Chapters 18 and 19 deal with stochastic differential equations and diffusions. The book closes with a final chapter on simulation of Brownian motion and stochastic differential equations (Chapter 20).

As intended by the authors, this book serves both as a good introduction and foundation for further reading in the more advanced and specialized literature. The authors have done a good job in closing this gap.

The book consists of 20 chapters, that shed light on Brownian motion from very different angles. Chapters 1 and 2 briefly discuss some historical aspects of Brownian motion and give basic definitions and results. Chapters 3 and 4 are all about the construction of Brownian motion, and offer several different methods. This includes approaches usually less frequently encountered in related literature. Chapter 5 discusses some martingale properties of Brownian motion and related issues of stopping times. Chapters 6, 7 and 8 essentially explore Markov properties of Brownian motion. Among other things, the reflection principle, transience and recurrence, semigroups and infinitesimal generators and the connection to PDEs is discussed. It is worth mentioning that the theory is presented in a very structured and accessible way. Chapters 9 and 10 are devoted to regularity properties of Brownian motion. Among other things, the quadratic variation and its connection to Lévy’s characterization, Hölder continuity and non-differentiability of Brownian motion is discussed. Chapters 11 and 12 cover various “growth aspects” of Brownian motion. This includes some laws of the iterated logarithm, large deviation and the Cameron-Martin formula. Chapter 13 is about the Skorohod embedding scheme. As an application, it is used to give a proof of the Donsker construction mentioned in Chapter 3. Chapters 14 to 19 are devoted to stochastic integration with respect to Brownian motion and related issues. This naturally includes Itô’s \(L^2\)-theory and beyond (Chapters 14,15), Itô’s formula (Chapter 16) and its many applications, such as the Doléan-Dade exponential, Girsanov’s theorem and various martingale representation results (Chapter 17). Finally, Chapters 18 and 19 deal with stochastic differential equations and diffusions. The book closes with a final chapter on simulation of Brownian motion and stochastic differential equations (Chapter 20).

As intended by the authors, this book serves both as a good introduction and foundation for further reading in the more advanced and specialized literature. The authors have done a good job in closing this gap.

Reviewer: Moritz Jirak (Berlin)