Continuous Martingales and Brownian Motion

1. Front Matter

   Pages I-IX

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2. Preliminaries

   • Daniel Revuz, Marc Yor

   Pages 1-13

3. Introduction

   • Daniel Revuz, Marc Yor

   Pages 14-47

4. Martingales

   • Daniel Revuz, Marc Yor

   Pages 48-73

5. Markov Processes

   • Daniel Revuz, Marc Yor

   Pages 74-112

6. Stochastic Integration

   • Daniel Revuz, Marc Yor

   Pages 113-167

7. Representation of Martingales

   • Daniel Revuz, Marc Yor

   Pages 168-205

8. Local Times

   • Daniel Revuz, Marc Yor

   Pages 206-258

9. Generators and Time Reversal

   • Daniel Revuz, Marc Yor

   Pages 259-300

10. Girsanov’s Theorem and First Applications

    • Daniel Revuz, Marc Yor

    Pages 301-337

11. Stochastic Differential Equations

    • Daniel Revuz, Marc Yor

    Pages 338-370

12. Additive Functionals of Brownian Motion

    • Daniel Revuz, Marc Yor

    Pages 371-408

13. Bessel Processes and Ray-Knight Theorems

    • Daniel Revuz, Marc Yor

    Pages 409-434

14. Excursions

    • Daniel Revuz, Marc Yor

    Pages 435-471

15. Limit Theorems in Distribution

    • Daniel Revuz, Marc Yor

    Pages 472-498

16. Back Matter

    Pages 499-536

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This book focuses on the probabilistic theory ofBrownian motion. This is a good topic to center a discussion around because Brownian motion is in the intersec­ tioll of many fundamental classes of processes. It is a continuous martingale, a Gaussian process, a Markov process or more specifically a process with in­ dependent increments; it can actually be defined, up to simple transformations, as the real-valued, centered process with independent increments and continuous paths. It is therefore no surprise that a vast array of techniques may be success­ fully applied to its study and we, consequently, chose to organize the book in the following way. After a first chapter where Brownian motion is introduced, each of the following ones is devoted to a new technique or notion and to some of its applications to Brownian motion. Among these techniques, two are of para­ mount importance: stochastic calculus, the use ofwhich pervades the whole book and the powerful excursion theory, both of which are introduced in a self­ contained fashion and with a minimum of apparatus. They have made much easier the proofs of many results found in the epoch-making book of Itö and McKean: Diffusion Processes and their Sampie Paths, Springer (1965).

• Brownian motion
• Functionals
• Generator
• Martingal
• Martingale
• brownsche Bewegung
• diffusion
• ergodic theory
• local time
• probability
• probability theory
• stochastic calculus
• stochastic differential equation
• stochastic processes
• stochastische Integration

• Département de Mathématiques, Université de Paris VII, Paris Cedex 05, France

  Daniel Revuz

• Laboratoire de Probabilités, Université Pierre et Marie Curie, Paris Cedex 05, France

  Marc Yor

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