  We study the stability of quantum pure states and, more generally, subspaces
for stochastic dynamics that describe continuously--monitored systems. We show
that the target subspace is almost surely invariant if and only if it is
invariant for the average evolution, and that the same equivalence holds for
the global asymptotic stability. Moreover, we prove that a strict linear
Lyapunov function for the average evolution always exists, and latter can be
used to derive sharp bounds on the Lyapunov exponents of the associated
semigroup. Nonetheless, we also show that taking into account the measurements
can lead to an improved bound on stability rate for the stochastic,
non-averaged dynamics. We discuss explicit examples where the almost sure
stability rate can be made arbitrary large while the average one stays
constant.
