Brownian Motion, Martingales, and Stochastic Calculus

  • Jean-François Le Gall

Part of the Graduate Texts in Mathematics book series (GTM, volume 274)

Table of contents

  1. Front Matter
    Pages i-xiii
  2. Jean-François Le Gall
    Pages 1-17
  3. Jean-François Le Gall
    Pages 19-40
  4. Jean-François Le Gall
    Pages 41-68
  5. Jean-François Le Gall
    Pages 69-95
  6. Jean-François Le Gall
    Pages 97-150
  7. Jean-François Le Gall
    Pages 151-184
  8. Jean-François Le Gall
    Pages 185-208
  9. Jean-François Le Gall
    Pages 209-233
  10. Jean-François Le Gall
    Pages 235-259
  11. Back Matter
    Pages 261-273

About this book


This book offers a rigorous and self-contained presentation of stochastic integration and stochastic calculus within the general framework of continuous semimartingales. The main tools of stochastic calculus, including Itô’s formula, the optional stopping theorem and Girsanov’s theorem, are treated in detail alongside many illustrative examples. The book also contains an introduction to Markov processes, with applications to solutions of stochastic differential equations and to connections between Brownian motion and partial differential equations. The theory of local times of semimartingales is discussed in the last chapter.

Since its invention by Itô, stochastic calculus has proven to be one of the most important techniques of modern probability theory, and has been used in the most recent theoretical advances as well as in applications to other fields such as mathematical finance. Brownian Motion, Martingales, and Stochastic Calculus provides a strong theoretical background to the reader interested in such developments.

Beginning graduate or advanced undergraduate students will benefit from this detailed approach to an essential area of probability theory. The emphasis is on concise and efficient presentation, without any concession to mathematical rigor. The material has been taught by the author for several years in graduate courses at two of the most prestigious French universities. The fact that proofs are given with full details makes the book particularly suitable for self-study. The numerous exercises help the reader to get acquainted with the tools of stochastic calculus.


Brownian motion martingale stochastic integral stochastic calculus Itô's formula martingale representation Markov process harmonic function stochastic differential equation

Authors and affiliations

  • Jean-François Le Gall
    • 1
  1. 1.Département de MathématiquesUniversité Paris-SudOrsay CedexFrance

Bibliographic information

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