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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.
KeywordsBrownian Motion Continuous Modification Fractional Brownian Motion Gaussian Measure Local Martingale
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