Before revision:  This incompatibility goes away by noticing that the model used in both approaches , geometric Brownian motion, is a non-ergodic process, in the sense that ensemble-average returns differ from time-average returns in a single realization .
After revision:  This incompatibility goes away by noticing that the model used in both approaches are non-ergodic process, in the sense that ensemble-average returns differ from time-average returns in a single realization .