Objectivity of classical quantum stochastic processes

Quantum mechanics is a flagship example of non-classical physical theory. To human users many aspects of quantum theory often appear highly counter-intuitive and sometimes even paradoxical. Hence, it is only natural to look for instances when quantum mechanics puts on a familiar classical appearance. In this vein, there was a recent resurgence of efforts in the search for the signatures of classicallity manifesting in the statistics of the sequential measurements. Here we consider what can be concluded about the quantum system when the measurements of its observable---the so-called quantum stochastic process---appear to the observer as a sampling of classical trajectory. We identify the physical conditions under which the quantum observable appears as a classical trajectory, and more importantly, we find that this trajectory picture also applies in contexts other than sequential measurements.