  In this article, we consider a set of trial wave-functions denoted by $| Q
\right>$ and an associated set of operators $A_\alpha$ which generate
transformations connecting those trial states. Using variational principles, we
show that we can always obtain a quantum Monte-Carlo method where the quantum
evolution of a system is replaced by jumps between density matrices of the form
$D = |Q_a> <Q_b|$, and where the average evolutions of the moments of
$A_\alpha$ up to a given order $k$, i.e. $< A_{\alpha_1} >$, $< A_{\alpha_1}
A_{\alpha_2} >$,..., $< A_{\alpha_1} ... A_{\alpha_k} >$, are constrained to
follow the exact Ehrenfest evolution at each time step along each stochastic
trajectory. Then, a set of more and more elaborated stochastic approximations
of a quantum problem is obtained which approach the exact solution when more
and more constraints are imposed, i.e. when $k$ increases. The Monte-Carlo
process might even become exact if the Hamiltonian $H$ applied on the trial
state can be written as a polynomial of $A_\alpha$. The formalism makes a
natural connection between quantum jumps in Hilbert space and phase-space
dynamics. We show that the derivation of stochastic Schroedinger equations can
be greatly simplified by taking advantage of the existence of this hierarchy of
approximations and its connection to the Ehrenfest theorem. Several examples
are illustrated: the free wave-packet expansion, the Kerr oscillator, a
generalized version of the Kerr oscillator, as well as interacting bosons or
fermions.
