  By exploiting the complexity intrinsic to quantum dynamics, quantum
technologies promise a whole host of computational advantages. One such
advantage lies in the field of stochastic modelling, where it has been shown
that quantum stochastic simulators can operate with a lower memory overhead
than their best classical counterparts. This advantage is particularly
pronounced for continuous-time stochastic processes; however, the corresponding
quantum stochastic simulators heretofore prescribed operate only on a
quasi-continuous-time basis, and suffer an ever-increasing circuit complexity
with increasing temporal resolution. Here, by establishing a correspondence
with quantum trajectories -- a method for modelling open quantum systems -- we
show how truly continuous-time quantum stochastic simulators can be embedded in
such open quantum systems, bridging this gap and obviating previous
constraints. We further show how such an embedding can be made for
discrete-time stochastic processes, which manifest as jump-only trajectories,
and discuss how viewing the correspondence in the reverse direction provides
new means of studying structural complexity in quantum systems themselves.
