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Universitext

Jump SDEs and the Study of Their Densities

A Self-Study Book

Authors: Kohatsu-Higa, Arturo, Takeuchi, Atsushi

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  • Introduces jump processes for students who may not have had previous experience with stochastic processes 
  • Expedites understanding of the application of an infinite-dimensional integration by parts formula for jump processe 
  • Presents Lévy processes in stages, with exercises to check the reader’s progress 
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  • ISBN 978-981-329-741-8
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  • ISBN 978-981-329-740-1
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About this Textbook

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.

About the authors

Professor Kohatsu-Higa is a professor at Ritsumeikan University and Professor Takeuchi is a professor at Tokyo Woman's Christian University. 

Table of contents (14 chapters)

Table of contents (14 chapters)
  • Review of Some Basic Concepts of Probability Theory

    Pages 1-7

    Kohatsu-Higa, Arturo (et al.)

  • Simple Poisson Process and Its Corresponding SDEs

    Pages 11-29

    Kohatsu-Higa, Arturo (et al.)

  • Compound Poisson Process and Its Associated Stochastic Calculus

    Pages 31-69

    Kohatsu-Higa, Arturo (et al.)

  • Construction of Lévy Processes and Their Corresponding SDEs: The Finite Variation Case

    Pages 71-100

    Kohatsu-Higa, Arturo (et al.)

  • Construction of Lévy Processes and Their Corresponding SDEs: The Infinite Variation Case

    Pages 101-130

    Kohatsu-Higa, Arturo (et al.)

Buy this book

eBook n/a
  • ISBN 978-981-329-741-8
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
Softcover n/a
  • ISBN 978-981-329-740-1
  • Free shipping for individuals worldwide
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Bibliographic Information

Bibliographic Information
Book Title
Jump SDEs and the Study of Their Densities
Book Subtitle
A Self-Study Book
Authors
Series Title
Universitext
Copyright
2019
Publisher
Springer Singapore
Copyright Holder
Springer Nature Singapore Pte Ltd.
eBook ISBN
978-981-329-741-8
DOI
10.1007/978-981-32-9741-8
Softcover ISBN
978-981-329-740-1
Series ISSN
0172-5939
Edition Number
1
Number of Pages
XIX, 355
Number of Illustrations
6 b/w illustrations
Topics