Local Asymptotic Mixed Normality for Nonhomogeneous Diffusions

We study the asymptotic properties of various estimators of the parameter appearing nonlinearly in the nonhomogeneous drift coefficient of a functional stochastic differential equation when the corresponding solution process, called the diffusion type process, is observed over a continuous time interval [0, T]. We show that the maximum likelihood estimator, maximum probability estimator and regular Bayes estimators are strongly consistent and when suitably normalised, converge to a mixture of normal distribution and are locally asymptotically minimax in the Hajek-Le Cam sense as T → ∞ under some regularity conditions. Also we show that posterior distributions, suitably normalised and centered at the maximum likelihood estimator, converge to a mixture of normal distribution. Further, the maximum likelihood estimator and the regular Bayes estimators are asymptotically equivalent as T → ∞. We illustrate the results through the exponential memory Ornstein-Uhlenbeck process, the nonhomogeneous Ornstein-Uhlenbeck process and the Kalman- Bucy filter model where the limit distribution of the above estimators and the posteriors is shown to be Cauchy.