  It is shown that non-Markovian master equations for an open system which are
local in time can be unravelled through a piecewise deterministic quantum jump
process in its Hilbert space. We derive a stochastic Schr\"odinger equation
that reveals how non-Markovian effects are manifested in statistical
correlations between different realizations of the process. Moreover, we
demonstrate that possible violations of the positivity of approximate master
equations are closely connected to singularities of the stochastic
Schr\"odinger equation, which could lead to important insights into the
structural characterization of positive non-Markovian equations of motion.
