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A sufficient condition for asymptotic sufficiency of incomplete observations of a diffusion process. (English) Zbl 0725.62073

Given an m-dimensional diffusion process \((X_ t)_{0\leq t\leq T}\) defined by the stochastic differential equation \[ dX_ t=b(X_ t,\theta)dt+\epsilon \sigma (X_ t)dW_ t,\quad X_ 0=x, \] where \((W_ t)\) is a standard m-dimensional Brownian motion, \(\theta\) is an unknown parameter in the drift function \(b(\cdot,\theta): {\mathbb{R}}^ m \to {\mathbb{R}}^ m,\) and where the diffusion matrix \(\sigma: {\mathbb{R}}^ m \to {\mathbb{R}}^ m\times {\mathbb{R}}^ m,\) \(x\in {\mathbb{R}}^ m\), and \(\epsilon >0\) are known, the main concern of the present paper is to derive a general condition ensuring the asymptotic sufficiency, in the sense of L. LeCam [Asymptotic methods in statistical decision theory (1986; Zbl 0605.62002)], of incomplete observations of the sample paths of \((X_ t)\) on the time interval [0,T], as the diffusion coefficient goes to 0. For incomplete observations of \((X_ t)_{0\leq t\leq T}\) which meet this condition, estimators of \(\theta\) based on these observations are obtained which are optimal.

MSC:

62M05 Markov processes: estimation; hidden Markov models
62B15 Theory of statistical experiments
62B05 Sufficient statistics and fields

Citations:

Zbl 0605.62002
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