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Parameter estimation in linear filtering. (English) Zbl 0768.62086
Let a partially observable random process \((x_ t,y_ t)\), \(t\geq 0\), 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 \((W_ 2(t))\): \[ dx_ t=-\beta x_ t dt+dW_ 1(t),\;x_ 0=0,\;dy_ t=\alpha x_ t dt+dW_ 2(t),\;y_ 0=0;\;\alpha,\beta\in(a,b),\;\alpha>0. \] The local asymptotic normality of the model is proved and a large deviation inequality for the maximum likelihood estimator of the parameter \(\theta=(\alpha,\beta)\) is obtained. This implies strong consistency, efficiency, asymptotic normality and the convergence of moments for the maximum likelihood estimator.

MSC:
62M20 Inference from stochastic processes and prediction
62M09 Non-Markovian processes: estimation
62G05 Nonparametric estimation
62M99 Inference from stochastic processes
60F10 Large deviations
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