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Local Times and Excursion Theory for Brownian Motion

A Tale of Wiener and Itô Measures

  • Book
  • © 2013

Overview

Authors:
  1. Ju-Yi Yen

    1. Department of Mathematical Sciences, University of Cincinnati, Cincinnati, USA

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  2. Marc Yor

    1. Labo. Probabilités et Modèles Aléatoires, Université Paris VI CNRS UMR 7599, Paris CX 05, France

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    You can also search for this author in PubMed   Google Scholar

  • Both local times and excursion theory are usually discussed in much longer texts. We examine these topics in relation to readers’ basic knowledge of stochastic processes
  • Presents interesting applications of excursion theory
  • Similarly with local times of Brownian motion
  • Includes supplementary material: sn.pub/extras

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2088)

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Table of contents (11 chapters)

  1. Front Matter

    Pages i-ix
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  2. Prerequisites

    • Ju-Yi Yen, Marc Yor
    Pages 1-10

  3. Local Times of Continuous Semimartingales

    1. Front Matter

      Pages 11-11
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    2. The Existence and Regularity of Semimartingale Local Times

      • Ju-Yi Yen, Marc Yor
      Pages 13-28

    3. Lévy’s Representation of Reflecting BM and Pitman’s Representation of BES(3)

      • Ju-Yi Yen, Marc Yor
      Pages 29-41

    4. Paul Lévy’s Arcsine Laws

      • Ju-Yi Yen, Marc Yor
      Pages 43-54

  4. Excursion Theory for Brownian Paths

    1. Front Matter

      Pages 55-55
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    2. Brownian Excursion Theory: A First Approach

      • Ju-Yi Yen, Marc Yor
      Pages 57-64

    3. Two Descriptions of n: Itô’s and Williams’

      • Ju-Yi Yen, Marc Yor
      Pages 65-77

    4. A Simple Path Decomposition of Brownian Motion Around Time t = 1

      • Ju-Yi Yen, Marc Yor
      Pages 79-92

    5. The Laws of, and Conditioning with Respect to, Last Passage Times

      • Ju-Yi Yen, Marc Yor
      Pages 93-100

    6. Integral Representations Relating W and n

      • Ju-Yi Yen, Marc Yor
      Pages 101-104

  5. Some Applications of Excursion Theory

    1. Front Matter

      Pages 105-105
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    2. The Feynman–Kac Formula and Excursion Theory

      • Ju-Yi Yen, Marc Yor
      Pages 107-110

    3. Some Identities in Law

      • Ju-Yi Yen, Marc Yor
      Pages 111-131

  6. Back Matter

    Pages 133-138
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Keywords

About this book

This monograph discusses the existence and regularity properties of local times associated to a continuous semimartingale, as well as excursion theory for Brownian paths. Realizations of Brownian excursion processes may be translated in terms of the realizations of a Wiener process under certain conditions. With this aim in mind, the monograph presents applications to topics which are not usually treated with the same tools, e.g.: arc sine law, laws of functionals of Brownian motion, and the Feynman-Kac formula. 

Reviews

“The lecture notes provide an elementary and brief introduction to local times for continuous semimartingales and excursion theory for Brownian motion. … The lecture notes are an easily accessible and self-contained introduction … which are suitable for graduate students with a basic knowledge of stochastic processes in continuous time. … the proofs are mostly carried out with many details and helpful references are given in each chapter.” (David Prömel, zbMATH 1364.60003, 2017)

Authors and Affiliations

  • Department of Mathematical Sciences, University of Cincinnati, Cincinnati, USA

    Ju-Yi Yen

  • Labo. Probabilités et Modèles Aléatoires, Université Paris VI CNRS UMR 7599, Paris CX 05, France

    Marc Yor

Bibliographic Information

  • Book Title: Local Times and Excursion Theory for Brownian Motion

  • Book Subtitle: A Tale of Wiener and Itô Measures

  • Authors: Ju-Yi Yen, Marc Yor

  • Series Title: Lecture Notes in Mathematics

  • DOI: https://doi.org/10.1007/978-3-319-01270-4

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer International Publishing Switzerland 2013

  • Softcover ISBN: 978-3-319-01269-8Published: 16 October 2013

  • eBook ISBN: 978-3-319-01270-4Published: 01 October 2013

  • Series ISSN: 0075-8434

  • Series E-ISSN: 1617-9692

  • Edition Number: 1

  • Number of Pages: IX, 135

  • Number of Illustrations: 1 b/w illustrations, 8 illustrations in colour

  • Topics: Probability Theory and Stochastic Processes

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