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Parameter estimation in linear filtering. (English) Zbl 0768.62086

Let a partially observable random process (x t ,y t ), t0, be given, where only the second component (y t ) is observed. Suppose that (x t ,y t ) satisfy the following system of stochastic differential equations driven by independent Wiener processes (W 1 (t)) and (W 2 (t)):

dx t =-βx t dt+dW 1 (t),x 0 =0,dy t =αx t dt+dW 2 (t),y 0 =0;α,β(a,b),α>0·

The local asymptotic normality of the model is proved and a large deviation inequality for the maximum likelihood estimator of the parameter θ=(α,β) is obtained. This implies strong consistency, efficiency, asymptotic normality and the convergence of moments for the maximum likelihood estimator.

MSC:
62M20Prediction; filtering (statistics)
62M09Non-Markovian processes: estimation
62G05Nonparametric estimation
62M99Inference from stochastic processes
60F10Large deviations