  Starting from a generalization of the quantum trajectory theory (based on the
stochastic Schr\"odinger equation - SSE), non-Markovian models of quantum
dynamics are derived. In order to describe non-Markovian effects, the approach
used in this article is based on the introduction of random coefficients in the
usual linear SSE. A major interest is that this allows to develop a consistent
theory of quantum measurement in continuous time for these non-Markovian
quantum trajectory models. In this context, the notions of instrument, a priori
and a posteriori states are rigorously described. The key point is that by
starting from a stochastic equation on the Hilbert space of the system, we are
able to respect the complete positivity of the mean dynamics for the
statistical operator and the requirements of the axioms of quantum measurement
theory. The flexibility of the theory is next illustrated by a concrete
physical model of a noisy oscillator where non Markovian effects come from
random environment, coloured noises, randomness in the stimulating light, delay
effects. The statistics of the emitted photons and the heterodyne and homodyne
spectra are studied and we show how these quantities are sensible to the
non-Markovian features of the system dynamics, so that, in principle, the
observation and analysis of the fluorescence light could reveal the presence of
non-Markovian effects and allow for a measure of the spectra of the noises
affecting the system dynamics.
