Introduction to Stochastic Integration

  • K.L. Chung
  • R.J. Williams

Part of the Modern Birkhäuser Classics book series (MBC)

Table of contents

  1. Front Matter
    Pages i-xvii
  2. K. L. Chung, R. J. Williams
    Pages 1-22
  3. K. L. Chung, R. J. Williams
    Pages 23-56
  4. K. L. Chung, R. J. Williams
    Pages 57-74
  5. K. L. Chung, R. J. Williams
    Pages 75-91
  6. K. L. Chung, R. J. Williams
    Pages 93-116
  7. K. L. Chung, R. J. Williams
    Pages 117-139
  8. K. L. Chung, R. J. Williams
    Pages 141-156
  9. K. L. Chung, R. J. Williams
    Pages 157-182
  10. K. L. Chung, R. J. Williams
    Pages 183-215
  11. K. L. Chung, R. J. Williams
    Pages 217-264
  12. Back Matter
    Pages 265-277

About this book


A highly readable introduction to stochastic integration and stochastic differential equations, this book combines developments of the basic theory with applications. It is written in a style suitable for the text of a graduate course in stochastic calculus, following a course in probability.


Using the modern approach, the stochastic integral is defined for predictable integrands and local martingales; then Itô’s change of variable formula is developed for continuous martingales. Applications include a characterization of Brownian motion, Hermite polynomials of martingales, the Feynman–Kac functional and the Schrödinger equation. For Brownian motion, the topics of local time, reflected Brownian motion, and time change are discussed.


New to the second edition are a discussion of the Cameron–Martin–Girsanov transformation and a final chapter which provides an introduction to stochastic differential equations, as well as many exercises for classroom use.


This book will be a valuable resource to all mathematicians, statisticians, economists, and engineers employing the modern tools of stochastic analysis.


The text also proves that stochastic integration has made an important impact on mathematical progress over the last decades and that stochastic calculus has become one of the most powerful tools in modern probability theory.

—Journal of the American Statistical Association



An attractive text…written in [a] lean and precise style…eminently readable. Especially pleasant are the care and attention devoted to details… A very fine book.

—Mathematical Reviews



Brownian motion Schrödinger equation martingales stochastic differential equations stochastic integration

Authors and affiliations

  • K.L. Chung
    • 1
  • R.J. Williams
    • 2
  1. 1.Department of MathematicsStanford UniversityStanfordUSA
  2. 2.Department of MathematicsUniversity of California at San DiegoLa JollaUSA

Bibliographic information

  • DOI
  • Copyright Information Springer Science+Business Media New York 2014
  • Publisher Name Birkhäuser, New York, NY
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-4614-9586-4
  • Online ISBN 978-1-4614-9587-1
  • Series Print ISSN 2197-1803
  • Series Online ISSN 2197-1811
  • About this book