Brownian Motion, Martingales, and Stochastic Calculus

1. Front Matter

   Pages i-xiii

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2. Gaussian Variables and Gaussian Processes

   • Jean-François Le Gall

   Pages 1-17

3. Brownian Motion

   • Jean-François Le Gall

   Pages 19-40

4. Filtrations and Martingales

   • Jean-François Le Gall

   Pages 41-68

5. Continuous Semimartingales

   • Jean-François Le Gall

   Pages 69-95

6. Stochastic Integration

   • Jean-François Le Gall

   Pages 97-150

7. General Theory of Markov Processes

   • Jean-François Le Gall

   Pages 151-184

8. Brownian Motion and Partial Differential Equations

   • Jean-François Le Gall

   Pages 185-208

9. Stochastic Differential Equations

   • Jean-François Le Gall

   Pages 209-233

10. Local Times

    • Jean-François Le Gall

    Pages 235-259

11. Erratum to: Brownian Motion, Martingales, and Stochastic Calculus

    • Jean-François Le Gall

    Pages E1-E1

12. Back Matter

    Pages 261-273

    PDF

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 astrong 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
• quantitative finance

“‘The aim of this book is to provide a rigorous introduction to the theory of stochastic calculus for continuous semi-martingales putting a special emphasis on Brownian motion.’ … If the reader has the background and needs a rigorous treatment of the subject this book would be a good choice. Le Gall writes clearly and gets to the point quickly … .” (Richard Durrett, MAA Reviews, March, 2017) 

“The purpose of this book is to provide concise but rigorous introduction to the theory of stochastic calculus for continuous semimartingales, putting a special emphasis on Brownian motion. … The book is written very clearly, it is interesting both for its construction and maintenance, mostly it is self-contained. It can be recommended to everybody who wants to study stochastic calculus, including those who is interested to its applications in other fields.” (Yuliya S. Mishura, zbMATH, 2017)

• Département de Mathématiques, Université Paris-Sud, Orsay Cedex, France

  Jean-François Le Gall

Jean-François Le Gall is a well-known specialist of probability theory and stochastic processes. His main research achievements are concerned with Brownian motion, superprocesses and their connections with partial differential equations, and more recently random trees and random graphs. He has been awarded several international prizes in mathematics, including the Loeve Prize and the Fermat Prize, and gave a plenary lecture at the 2014 International Congress of Mathematicians. He is currently a professor of mathematics at Université Paris-Sud and a member of the French Academy of Sciences.

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