Overview
Brownian motion is a the canonical representation of a continuous random process with no drift. (that is, a martingale)
Brownian Motion
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: asserts that any continuous martingale is a Brownian motion, or can be converted to a Brownian motion through a simple transformation.
- Multiple Brownian Motions/Ito Processes
- Sum of Brownian Motions