  The theory of measurements continuous in time in quantum mechanics (quantum
continual measurements) has been formulated by using the notions of instrument
and positive operator valued measure, functional integrals, quantum stochastic
differential equations and classical stochastic differential equations (SDE's).
Various types of SDE's are involved, and precisely linear and non linear
equations for vectors in Hilbert spaces and for trace-class operators. All such
equations contain either a diffusive part, or a jump one, or both.
  In this paper we introduce a class of linear SDE's for trace-class operators,
relevant to the theory of continual measurements, and we recall how such SDE's
are related to instruments and master equations and, so, to the general
formulation of quantum mechanics. We do not present the Hilbert space
formulation of such SDE's and we make some mathematical simplifications: no
time dependence is introduced into the coefficients and only bounded operators
on the Hilbert space of the quantum system are considered. Then we introduce
the notion of a posteriori state and the non linear SDE satisfied by such
states and we give conditions from which such equation is assured to preserve
pure states and to send any mixed state into a pure one for large times.
Finally we review the known results about the existence and uniqueness of
invariant measures in the purely diffusive case and we give some concrete
examples of physical systems.
