A geometric Brownian motion is a positive stochastic process 
satisfying the stochastic differential
equation
 |
where 
is a Wiener process, 
is the drift, and 
is the volatility. Applying Ito's
lemma gives the explicit solution
 |
Consequently, 
has a normal distribution, and 
has a log-normal distribution.
Geometric Brownian motion is the underlying asset model in Black-Scholes
theory.
A geometric Brownian motion is related directly to Brownian motion: it is obtained by exponentiating a Brownian
motion with linear drift. Equivalently, the natural
logarithm of 
is the Brownian motion 
plus the deterministic drift 
. Thus geometric Brownian motion has multiplicative,
rather than additive, random changes and remains positive
when 
.
