  In this paper, we establish a general theoretical framework for the
description of continuous quantum measurements and the statistics of the
results of such measurements. The framework concerns the measurement of an
arbitrary quantum system with arbitrary number of detectors under realistic
assumption of instant detector reactions and white noise sources. We attend
various approaches to the problem showing their equivalence. The approaches
include the full counting statistics (FCS) evolution equation a for
pseudo-density matrix, the drift-diffusion equation for a density matrix in the
space of integrated outputs, and discrete stochastic updates. We provide the
derivation of the underlying equations from a microscopic approach based on
full counting statistics method, a phenomenological approach based on Lindblad
construction, and interaction with auxiliary quantum systems representing the
detectors. We establish the necessary conditions on the phenomenological
susceptibilities and noises that guarantee the unambiguous interpretation of
the measurement results and the positivity of the density matrix. Our results
can be easily extended to describe various quantum feedback schemes where the
manipulation decision is based on the values of detector outputs.
