Some aspects of Brownian motion, Part I: Some special functionals.

*(English)*Zbl 0779.60070
Lectures in Mathematics, ETH Zürich. Basel: Birkhäuser Verlag. 148 p. (1992).

This book represents approximately the first half of lectures given by the author in the Nachdiplomvorlesung at ETH (1991-1992). A good number of presented results is new and belongs to the author.

The text is organized in 9 chapters, each of which is devoted to study the laws of some particular classes of Brownian functionals. More precisely: Gaussian subspaces of Gaussian space are studied in the Chapter 1, quadratic functionals of Brownian motion and squares of Bessel processes in Chapter 2. Chapter 3 is devoted to the Brownian local times, Ray-Knight theorem and its generalizations and their connections with Bessel processes. Chapter 4 contains some modifications of Ciesielski- Taylor identities based on Right-Knight theorems. In Chapters 5 and 7 exact and asymptotic results about the winding numbers of one or several planar Brownian motions around one or several points, or straight lines, or curves are studied. Chapter 6 is devoted to some exponential functionals with emphasis on the problem of Asian options useful for mathematical finance. In Chapters 8 and 9 the author presents some newer extensions of Lévy’s arc sin law. Chapter 9 contains results on the properties of reflected Brownian motion perturbed by its local time at 0. The last results are obtained while teaching the cours. Each chapter contains a few exercises which help the reader to understand the text. In all, this book may be of interest to researchers and post-graduated students in probability theory as well as in applied fields.

The text is organized in 9 chapters, each of which is devoted to study the laws of some particular classes of Brownian functionals. More precisely: Gaussian subspaces of Gaussian space are studied in the Chapter 1, quadratic functionals of Brownian motion and squares of Bessel processes in Chapter 2. Chapter 3 is devoted to the Brownian local times, Ray-Knight theorem and its generalizations and their connections with Bessel processes. Chapter 4 contains some modifications of Ciesielski- Taylor identities based on Right-Knight theorems. In Chapters 5 and 7 exact and asymptotic results about the winding numbers of one or several planar Brownian motions around one or several points, or straight lines, or curves are studied. Chapter 6 is devoted to some exponential functionals with emphasis on the problem of Asian options useful for mathematical finance. In Chapters 8 and 9 the author presents some newer extensions of Lévy’s arc sin law. Chapter 9 contains results on the properties of reflected Brownian motion perturbed by its local time at 0. The last results are obtained while teaching the cours. Each chapter contains a few exercises which help the reader to understand the text. In all, this book may be of interest to researchers and post-graduated students in probability theory as well as in applied fields.

Reviewer: N.M.Zinchenko (Kiev)

##### MSC:

60J65 | Brownian motion |

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

60J55 | Local time and additive functionals |

60J99 | Markov processes |

60K40 | Other physical applications of random processes |