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Jump SDEs and the Study of Their Densities

A Self-Study Book

  • Arturo Kohatsu-Higa
  • Atsushi Takeuchi
Textbook

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages i-xix
  2. Arturo Kohatsu-Higa, Atsushi Takeuchi
    Pages 1-7
  3. Construction of Lévy Processes and Their Stochastic Calculus

    1. Front Matter
      Pages 9-9
    2. Arturo Kohatsu-Higa, Atsushi Takeuchi
      Pages 11-29
    3. Arturo Kohatsu-Higa, Atsushi Takeuchi
      Pages 31-69
    4. Arturo Kohatsu-Higa, Atsushi Takeuchi
      Pages 131-143
    5. Arturo Kohatsu-Higa, Atsushi Takeuchi
      Pages 145-154
  4. Densities of Jump SDEs

    1. Front Matter
      Pages 155-155
    2. Arturo Kohatsu-Higa, Atsushi Takeuchi
      Pages 157-160
    3. Arturo Kohatsu-Higa, Atsushi Takeuchi
      Pages 161-172
    4. Arturo Kohatsu-Higa, Atsushi Takeuchi
      Pages 173-201
    5. Arturo Kohatsu-Higa, Atsushi Takeuchi
      Pages 203-230
    6. Arturo Kohatsu-Higa, Atsushi Takeuchi
      Pages 231-267
    7. Arturo Kohatsu-Higa, Atsushi Takeuchi
      Pages 269-315
    8. Arturo Kohatsu-Higa, Atsushi Takeuchi
      Pages 317-346
  5. Back Matter
    Pages 347-355

About this book

Introduction

The present book deals with a streamlined presentation of Lévy processes and their densities. It is directed at advanced undergraduates who have already completed a basic probability course. Poisson random variables, exponential random variables, and the introduction of Poisson processes are presented first, followed by the introduction of Poisson random measures in a simple case. With these tools the reader proceeds gradually to compound Poisson processes, finite variation Lévy processes and finally one-dimensional stable cases. This step-by-step  progression guides the reader into the construction and study of the properties of general Lévy processes with no Brownian component. In particular, in each case the corresponding Poisson random measure, the corresponding stochastic integral, and the corresponding stochastic differential equations (SDEs) are provided. The second part of the book introduces the tools of the integration by parts formula for jump processes in basic settings and first gradually provides the integration by parts formula in finite-dimensional spaces and gives a formula in infinite dimensions. These are then applied to stochastic differential equations in order to determine the existence and some properties of their densities. As examples, instances of the calculations of the Greeks in financial models with jumps are shown. The final chapter is devoted to the Boltzmann equation.

Keywords

Jump processes Stochastic Calculus Calculus of Variations Integration by parts Densities of random variables

Authors and affiliations

  • Arturo Kohatsu-Higa
    • 1
  • Atsushi Takeuchi
    • 2
  1. 1.Department of Mathematical SciencesRitsumeikan UniversityKusatsuJapan
  2. 2.Department of MathematicsTokyo Woman’s Christian UniversityTokyoJapan

Bibliographic information

  • DOI https://doi.org/10.1007/978-981-32-9741-8
  • Copyright Information Springer Nature Singapore Pte Ltd. 2019
  • Publisher Name Springer, Singapore
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-981-32-9740-1
  • Online ISBN 978-981-32-9741-8
  • Series Print ISSN 0172-5939
  • Series Online ISSN 2191-6675
  • Buy this book on publisher's site
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