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


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.


Brownian Motion Continuous Modification Fractional Brownian Motion Gaussian Measure Local Martingale 
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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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