Geometric Brownian Motion Formulation
Overview
Geometric Brownian Motion can be formulated within the
Stochastic Calculus
as an integral over a Brownian motion.
Geometric Brownian Motion Formulation
The geometric brownian motion can be stated in differential form as
{% d log X(t) = r dt + \sigma dW(t) %}
Integrating this
equation yields
{% log X(t) = log X(0) + r \times t + \sigma W(t) %}
or equivalently
{% X(t) = X(0) \times exp[ r \times t + \sigma W(t) ] %}
Alternative Formulation
Sometimes, the geometric brownian motion is stated starting from X instead of log X
{% d X(t) = \alpha X(t) dt + \sigma X(t) dW(t) %}
This form is equivalent to log form above (using Itos lemma), if we take
{% r = \alpha - \frac{1}{2} \sigma ^2 %}
