Parameter estimation in linear filtering

Suppose on a probability space ([Omega], F, P), a partially observable random process (xt, yt), t >= 0; is given where only the second component (yt) is observed. Furthermore assume that (xt, yt) satisfy the following system of stochastic differential equations driven by independent Wiener processes (W1(t)) and (W2(t)): dxt=-[beta]xtdt+dW1(t), x0=0, dyt=[alpha]xtdt+dW2(t), y0=0; [alpha], [beta][infinity](a,b), a>0. We prove the local asymptotic normality of the model and obtain a large deviation inequality for the maximum likelihood estimator (m.l.e.) of the parameter [theta] = ([alpha], [beta]). This also implies the strong consistency, efficiency, asymptotic normality and the convergence of moments for the m.l.e. The method of proof can be easily extended to obtain similar results when vector valued instead of one-dimensional processes are considered and [theta] is a k-dimensional vector.Kalman filter maximum likelihood estimation large deviation inequality local asymptotic normality