  We consider open quantum systems with factorized initial states, providing
the structure of the reduced system dynamics, in terms of environment
cumulants. We show that such completely positive (CP) and trace preserving (TP)
maps can be unraveled by linear stochastic Schr\"odinger equations (SSEs)
characterized by sets of colored stochastic processes (with n-th order
cumulants). We obtain both the conditions such that the SSEs provide CPTP
dynamics, and those for unraveling an open system dynamics. We then focus on
Gaussian non-Markovian unravellings, whose known structure displays a
functional derivative. We provide a novel description that replaces the
functional derivative with a recursive operatorial structure. Moreover, for the
family of quadratic bosonic Hamiltonians, we are able to provide an explicit
operatorial dependence for the unravelling.
