  We generalize and extend the stochastic path integral formalism and action
principle for continuous quantum measurement introduced in [A. Chantasri, J.
Dressel and A. N. Jordan, Phys. Rev. A {\bf 88}, 042110 (2013)], where the
optimal dynamics, such as the most-likely paths, are obtained by extremizing
the action of the path integral. In this work, we apply exact functional
methods as well as develop a perturbative approach to investigate the
statistical behaviour of continuous quantum measurement, with examples given
for the qubit case. For qubit measurement with zero qubit Hamiltonian, we find
analytic solutions for average trajectories and their variances while
conditioning on fixed initial and final states. For qubit measurement with
unitary evolution, we use the perturbation method to compute expectation
values, variances, and multi-time correlation functions of qubit trajectories
in the short-time regime. Moreover, we consider continuous qubit measurement
with feedback control, using the action principle to investigate the global
dynamics of its most-likely paths, and finding that in an ideal case, qubit
state stabilization at any desired pure state is possible with linear feedback.
We also illustrate the power of the functional method by computing correlation
functions for the qubit trajectories with a feedback loop to stabilize the
qubit Rabi frequency.
