  The state matrix $\rho$ for an open quantum system with Markovian evolution
obeys a master equation. The master equation evolution can be unraveled into
stochastic nonlinear trajectories for a pure state $P$, such that on average
$P$ reproduces $\rho$. Here we give for the first time a complete
parameterization of all diffusive unravelings (in which $P$ evolves
continuously but non-differentiably in time). We give an explicit measurement
theory interpretation for these quantum trajectories, in terms of monitoring
the system's environment. We also introduce new classes of diffusive
unravelings that are invariant under the linear operator transformations under
which the master equation is invariant. We illustrate these invariant
unravelings by numerical simulations. Finally, we discuss generalized gauge
transformations as a method of connecting apparently disparate descriptions of
the same trajectories by stochastic Schr\"odinger equations, and their
invariance properties.
