# ARIMA and Geometric Brownian Motion

I have read that Brownian motion, or more precisely, a Wiener process, is a scaling limit of a random walk. Hence, when attempting to model a real time-series of energy prices, if I discover that an $$ARIMA(0,1,0)$$ model fits my data well, I assume that this would vindicate, at least to some degree, an assumption I make in a decision model that requires that my energy price follows a Brownian motion.

In a similar vein, would showing that an $$ARIMA(0,1,0)$$ model fits the logged energy prices vindicate the assumption of a geometric Brownian motion? Or asked another way, is there an $$ARIMA(p,d,q)$$ model that "corresponds to" geometric Brownian motion?

References are appreciated!

• This looks right to me, but maybe someone will have more to say!
– Eoin
Nov 20 '20 at 10:02