Saturday, July 28, 2018

stochastic processes - 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 ?




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