Brownian Motion

Overview


Brownian motion is a the canonical representation of a continuous random process with no drift. (that is, a martingale)

Brownian Motion


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Axioms of Brownian Motion

  • {% B_0(\omega) = 0 %}
  • the map {% B_t(\omega) %} is continuous
  • {% B_t - B_s \sim N(0,t-s) %}
  • for every {% t,h %} {% B_{t+h} - B_t %} is independent of {% B_u %} for {% 0 \leq u \leq t %}


The letter B is used to refer to a Brownian motion. Often times, the letter W is used instead. (for Norbert Wiener)

Variations


  • Ito Process: extends the standard Brownian Motion to include drift.
  • Geometric Brownian Motion - used to model processes that cannot go below zero. Is the standard model used in modeling asset prices.
  • Browian Bridge - a "Brownian Motion" that has the end point as well as the start fixed.

Topics


  • Construction:
  • Levy Theorem: essentially asserts that any continuous martingale is a Brownian motion, or can be converted to a Brownian motion through a simple transformation.

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