Some aspects of Brownian motion. Part II: Some recent martingale problems.

*(English)*Zbl 0880.60082
Lectures in Mathematics, ETH Zürich. Basel: Birkhäuser. xii, 144 p. (1997).

[For part I (1992) see Zbl 0779.60070.]

This part of the book represents approximately the second half of lectures given by the author at ETH (1991-1993) and is devoted to certain martingale problems connected with Brownian motion \((B_t, t\geq 0)\) and applications of martingale and semi-martingale technique. The text is organized in nine chapters N10–N18, continuing the numeration of Part I, and has a deep interrelation with the topics of the first part. In particular, it is true concerning Chapters 10 and 11.

In Chapter 10 the principal values of Brownian and Bessel local times are studied. Chapter 11 presents the known and new results on relations between heat equations, zeta function, theta function and Brownian or Bessel processes. Chapter 12 is the central one, where some theoretical aspects and applications of enlargements of the natural Brownian filtration are discussed. The author believes that a lot of interesting examples, useful applications and similarity with Girsanov’s formula makes this rather difficult topic more clear for the reader. Two subcases of “initial” and “progressive” enlargements are investigated and used to define a large class of Brownian integrals as well as to prove new results on existence of the principal value of Brownian local times. D. William’s Brownian path decomposition and Pitman’s theorem on Bessel process are obtained and extended thanks to such technique.

The enlargement point of view also allows to modify the Burkholder-Gundy inequalities (Chapter 13) and to describe the classes of martingales vanishing on the set of Brownian zeros (Chapter 14). Chapter 15 deals with martingales \(X_t\), such that \((X_t- t, t\geq 0)\) is another martingale, which enjoys the chaos representation property (CRP), i.e. can be represented as a sum of multiple Wiener integrals. A number of properties of Azema-Emery martingales \((X^{(\beta)}, \beta\in\mathbb{R})\) are established. Following Emery, it is proved, that for \(-2\leq\beta<0\) martingales \(X^{(\beta)}\) possess CRP. The validity of CRP for \(\beta<-2\) or \(\beta>0\) remains an open question. In Chapter 16 the filtration \(\varepsilon^*= \sigma\{B_s\wedge x, s\geq 0\}\), \(x\in\mathbb{R}\), and the structure of \(\varepsilon^*\)-martingales are studied. Chapter 17 presents an attempt to characterize the natural Brownian filtration in terms of martingale representation property. The main attention is paid to examples due to B. Tsirelson and co-authors, who showed that such property does not characterize the Brownian filtration. Finally, Chapter 18 includes the complements and a list of new references relative to every chapter of Part I.

Unlike the first part of the book, the comprehension of the second part demands from the reader the knowledge of rather “high technology”, which includes the classical stochastic analysis and elements of stochastic calculus for general discontinuous semi-martingales. But all efforts are reward by acknowlegement with modern point of view on such a popular object as Brownian motion and on still open problems related to it.

This part of the book represents approximately the second half of lectures given by the author at ETH (1991-1993) and is devoted to certain martingale problems connected with Brownian motion \((B_t, t\geq 0)\) and applications of martingale and semi-martingale technique. The text is organized in nine chapters N10–N18, continuing the numeration of Part I, and has a deep interrelation with the topics of the first part. In particular, it is true concerning Chapters 10 and 11.

In Chapter 10 the principal values of Brownian and Bessel local times are studied. Chapter 11 presents the known and new results on relations between heat equations, zeta function, theta function and Brownian or Bessel processes. Chapter 12 is the central one, where some theoretical aspects and applications of enlargements of the natural Brownian filtration are discussed. The author believes that a lot of interesting examples, useful applications and similarity with Girsanov’s formula makes this rather difficult topic more clear for the reader. Two subcases of “initial” and “progressive” enlargements are investigated and used to define a large class of Brownian integrals as well as to prove new results on existence of the principal value of Brownian local times. D. William’s Brownian path decomposition and Pitman’s theorem on Bessel process are obtained and extended thanks to such technique.

The enlargement point of view also allows to modify the Burkholder-Gundy inequalities (Chapter 13) and to describe the classes of martingales vanishing on the set of Brownian zeros (Chapter 14). Chapter 15 deals with martingales \(X_t\), such that \((X_t- t, t\geq 0)\) is another martingale, which enjoys the chaos representation property (CRP), i.e. can be represented as a sum of multiple Wiener integrals. A number of properties of Azema-Emery martingales \((X^{(\beta)}, \beta\in\mathbb{R})\) are established. Following Emery, it is proved, that for \(-2\leq\beta<0\) martingales \(X^{(\beta)}\) possess CRP. The validity of CRP for \(\beta<-2\) or \(\beta>0\) remains an open question. In Chapter 16 the filtration \(\varepsilon^*= \sigma\{B_s\wedge x, s\geq 0\}\), \(x\in\mathbb{R}\), and the structure of \(\varepsilon^*\)-martingales are studied. Chapter 17 presents an attempt to characterize the natural Brownian filtration in terms of martingale representation property. The main attention is paid to examples due to B. Tsirelson and co-authors, who showed that such property does not characterize the Brownian filtration. Finally, Chapter 18 includes the complements and a list of new references relative to every chapter of Part I.

Unlike the first part of the book, the comprehension of the second part demands from the reader the knowledge of rather “high technology”, which includes the classical stochastic analysis and elements of stochastic calculus for general discontinuous semi-martingales. But all efforts are reward by acknowlegement with modern point of view on such a popular object as Brownian motion and on still open problems related to it.

Reviewer: N.M.Zinchenko (Kiev)

##### MSC:

60J65 | Brownian motion |

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