  The dynamics of open quantum systems is often solved by stochastic
unravellings where the average over the state vector realizations reproduces
the density matrix evolution. We focus on quantum jump descriptions based on
the rate operator formalism. In addition to displaying and exploiting different
equivalent ways of writing the master equation, we introduce state-dependent
rate operator transformations within the framework of stochastic pure state
realizations, allowing us to extend and generalize the previously developed
formalism. As a consequence, this improves the controllability of the
stochastic realizations and subsequently greatly benefits when searching for
optimal simulation schemes to solve open system dynamics. At a fundamental
level, intriguingly, our results show that it is possible to have positive
unravellings -- without reverse quantum jumps and avoiding the use of auxiliary
degrees freedom -- in a number of example cases even when the corresponding
dynamical map breaks the property of P-divisibility, thus being in the strongly
non-Markovian regime.
