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!