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Prerequisites

  • Ju-Yi Yen
  • Marc Yor
Chapter
  • 1.5k Downloads
Part of the Lecture Notes in Mathematics book series (LNM, volume 2088)

Abstract

In this chapter, ten prerequisites are gathered, and will be assumed as background for the rest of the volume. Brownian motion is constructed from a Gaussian measure on (0, ), with Lebesgue intensity. Dubins-Schwarz’ and Knight’s theorems about continuous local martingales being time-changed of Brownian motion are recalled. Girsanov’s theorem is presented, as well as some representations of Brownian bridges in terms of Brownian motion. The BES(3) process is shown to be a Doob h-transform of Brownian motion. Beta and gamma variables are presented, together with some important identities in law. Formulae for martingales in a filtration becoming semimartingales in an enlarged one are presented. Finally, Kolmogorov’s continuity criterion is given; it plays an important role in many applications.

Keywords

Brownian Motion Continuous Modification Fractional Brownian Motion Gaussian Measure Local Martingale 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    T. Jeulin, Semi-martingales et grossissement d’une filtration. Lecture Notes in Mathematics, vol. 833. (Springer, Berlin, 1980)Google Scholar
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    T. Jeulin, M. Yor, Grossissement de filtrations: exemples et applications. Lecture Notes in Mathematics, vol. 1118. (Springer, Berlin, 1985)Google Scholar
  3. 3.
    R. Mansuy, M. Yor, Random times and enlargements of filtrations in a Brownian setting. Lecture Notes in Mathematics, vol. 1873. (Springer, Berlin, 2006)Google Scholar
  4. 4.
    D. Revuz, M. Yor, Continuous martingales and Brownian motion. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, 3rd edn. (Springer, Berlin, 1999)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Ju-Yi Yen
    • 1
  • Marc Yor
    • 2
  1. 1.Department of Mathematical SciencesUniversity of CincinnatiCincinnatiUSA
  2. 2.Laboratoire de Probabilités et Modèles AléatoiresUniversité Paris VIParis CX 05France

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