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Lévy Processes

Theory and Applications

  • Ole E. Barndorff-Nielsen
  • Sidney I. Resnick
  • Thomas Mikosch

Table of contents

  1. Front Matter
    Pages i-xi
  2. A Tutorial on Lévy Processes

    1. Front Matter
      Pages 1-1
    2. Ken-iti Sato
      Pages 3-37
  3. Distributional, Pathwise, and Structural Results

    1. Front Matter
      Pages 39-39
    2. Philippe Carmona, Frédérique Petit, Marc Yor
      Pages 41-55
    3. Ronald Doney
      Pages 57-66
    4. Michael B. Marcus, Jay Rosen
      Pages 67-88
  4. Extensions and Generalizations of Lévy Processes

    1. Front Matter
      Pages 109-109
    2. Niels Jacob, René L. Schilling
      Pages 139-168
    3. Makoto Maejima
      Pages 169-183
  5. Applications in Physics

    1. Front Matter
      Pages 185-185
    2. Sergio Albeverio, Barbara Rüdiger, Jiang-Lun Wu
      Pages 187-224
    3. Alexander S. Holevo
      Pages 225-239
    4. Wojbor A. Woyczyński
      Pages 241-266
  6. Applications in Finance

  7. Numerical and Statistical Aspects

  8. Back Matter
    Pages 417-418

About this book

Introduction

A Lévy process is a continuous-time analogue of a random walk, and as such, is at the cradle of modern theories of stochastic processes. Martingales, Markov processes, and diffusions are extensions and generalizations of these processes. In the past, representatives of the Lévy class were considered most useful for applications to either Brownian motion or the Poisson process. Nowadays the need for modeling jumps, bursts, extremes and other irregular behavior of phenomena in nature and society has led to a renaissance of the theory of general Lévy processes. Researchers and practitioners in fields as diverse as physics, meteorology, statistics, insurance, and finance have rediscovered the simplicity of Lévy processes and their enormous flexibility in modeling tails, dependence and path behavior.

This volume, with an excellent introductory preface, describes the state-of-the-art of this rapidly evolving subject with special emphasis on the non-Brownian world. Leading experts present surveys of recent developments, or focus on some most promising applications. Despite its special character, every topic is aimed at the non- specialist, keen on learning about the new exciting face of a rather aged class of processes. An extensive bibliography at the end of each article makes this an invaluable comprehensive reference text. For the researcher and graduate student, every article contains open problems and points out directions for futurearch.

The accessible nature of the work makes this an ideal introductory text for graduate seminars in applied probability, stochastic processes, physics, finance, and telecommunications, and a unique guide to the world of Lévy processes.

 

Keywords

Excel Gaussian process Likelihood Lévy process Martingale Stochastic processes communication geometry local time modeling operator point process random walk statistics stochastic process

Editors and affiliations

  • Ole E. Barndorff-Nielsen
    • 1
  • Sidney I. Resnick
    • 2
  • Thomas Mikosch
    • 3
  1. 1.Department of Mathematical SciencesUniversity of AarhusAarhusDenmark
  2. 2.School of ORIECornell UniversityIthacaUSA
  3. 3.Laboratory of Actuarial MathematicsUniversity of CopenhagenCopenhagenDenmark

Bibliographic information