Hidden ergodic Ornstein–Uhlenbeck process and adaptive filter

This paper revisits the state and parameter estimation problems for a system of partially observed linear stochastic differential equations. An asymptotically optimal adaptive filter of the hidden state process is constructed using a three stage procedure. First, the unknown parameter is estimated by means of the method of moments. Then this preliminary estimator is used to define the One-step MLE process by applying the scoring technique, and, finally, the improved estimator is plugged into Kalman-Bucy filter. The obtained parameter estimator and the adaptive filter are proved to be asymptotically efficient in the long-time regime.