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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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##### References:
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