On stability of quantum trajectories and their Cesaro mean

We address the question of stability of quantum trajectories, also referred as quantum filters}. We determine the limit of the quantum fidelity between the true quantum trajectory and the {estimated one}. Under a purification assumption we show that this limit equals to one meaning that quantum filters are stable. In the general case, under an identifiability and a spectral assumption we show that the limit of the Cesaro mean of the estimated trajectory is the same as the true one.