Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - Université de Rouen Normandie

We consider several models of diffusion processes with periodic in time trend coefficients and we describe the properties of the MLE and Bayes estimators of the frequency in the asymptotics of large samples and small noise. We start with the Signal in White Gaussian Noise model. We show the rates of convergence of mean square error in regular and non regular cases. Then we consider diffusion processes with a periodic deterministic component in the trend (additive periodicity) and describe the properties of estimators (consistency, limit distributions, convergence of moments, asymptotic efficiency) in regular and non regular situations. Special attention is paid to the partially observed systems where the periodic function is multiplied by the unobservable component. Using the Kalman filter asymptotics we give the properties of estimators in three situations: as the noises in observations and state equations tend to zero, as the noise in state equation only tends to zero and as the time of observations tends to infinity.