# Geometric brownian motion vs. Ornstein Uhlenbeck

I'm looking at the SDE of Geometric brownian motion(*):

$$d X(t) = \sigma X(t) d B(t) + \mu X(t) d t$$

(with analytic solution $X(t) = X(0) e^{(\mu - \sigma^2 / 2) t + \sigma B(t)}$)

and the SDE of Ornstein-Uhlenbeck process:

$$d X(t) = \sigma d B(t) + \theta (\mu - X(t)) d t$$

In which case the one or the other is better suited for modelling financial data? I read that currrency price data can be well modelled by O-U process. Is there a heuristic/empirical argument for that ?

• Geometric Brownian motion is generally used to model stock prices, while the OU process is used for interest rate, or anything that has the mean-reverting nature. – Gordon Jan 22 '16 at 0:58
• @Gordon, apart from Brownian motion, Geometric brownian motion, O-U process, what are the next process I should learn about? (the most famous/used ones) ? – Basj Jan 22 '16 at 15:45
• These are the major ones. The other ones will be Poisson and Cox processes that are used in credit risk modelling. Levy process may also be helpful. – Gordon Jan 22 '16 at 16:06