  Hughston has recently proposed a stochastic extension of the Schr\"odinger
equation, expressed as a stochastic differential equation on projective Hilbert
space. We derive new projective Hilbert space identities, which we use to give
a general proof that Hughston's equation leads to state vector collapse to
energy eigenstates, with collapse probabilities given by the quantum mechanical
probabilities computed from the initial state. We discuss the relation of
Hughston's equation to earlier work on norm-preserving stochastic equations,
and show that Hughston's equation can be written as a manifestly unitary
stochastic evolution equation for the pure state density matrix. We discuss the
behavior of systems constructed as direct products of independent subsystems,
and briefly address the question of whether an energy-based approach, such as
Hughston's, suffices to give an objective interpretation of the measurement
process in quantum mechanics.
