The following paper:
http://www.cirm.univ-mrs.fr/videos/2008/exposes/287w1/Wojtkowski.pdf
provides a geometric positive answer to your question. (The paper discusses the private case where the measurement noise is of unit variance). The answer is based on the fact that geometrically the time evolution of the Riccati equation is a flow on a Grassmann manifold. In the paper, it is proved that this flow preserves the positivity and of P, and that the time evolution matrix of the state vector X is decaying.