An oscillator with stochastic frequency is discussed as a model for evaluat
ing the quantum coherence properties of a physical system. It is found that
the choice of jump statistics has to be considered with care if unphysical
consequences are to be avoided. We investigate one such model, evaluate th
e damping it causes, the decoherence rate and the correlations it results i
n, and the properties of the state for asymptotically long times. Also the
choice of initial state and its effect on the time evolution of the correla
tions are discussed.