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On estimating the diffusion coefficient. (English) Zbl 0615.62109
Suppose \(\xi =\{\xi_ t\), \(t\in [0,1]\}\) is a real-valued process satisfying the stochastic differential equation \(d\xi_ t=a(\xi_ t,\theta)dt+f(\xi_ t,\theta)dw_ t\), where \(t\in [0,1]\), \(\xi_ 0=x_ 0\), \(\{w_ t\), \(t\in [0,1]\}\) is a standard Wiener process and \(\theta\) is an unknown parameter from the open interval \(\Theta\subset {\mathbb{R}}\). The drift a(x,\(\theta)\) and the diffusion coefficient f(x,\(\theta)\) are given functions.
The paper deals with the problem of estimation of \(\theta\) on the basis of discrete observations of \(\xi\) made at equidistant sampling points \(t_ k=k/n\), \(k=0,1,...,n\) for some integer \(n\geq 1\). Denote by \(\{P^ n_{\theta}\), \(\theta\in \Theta\), \(n\geq 1\}\) the sequence of the probability measures generated by the sequence \(\{\xi_{k/n}\), \(k=0,1,...,n\), \(n\geq 1\}\). The main result of the paper is Proposition 1 where the author proves that the sequence \(\{P^ n_{\theta}\), \(\theta\in \Theta\), \(n\geq 1]\) satisfies the local asymptotic mixed normality condition. As a consequence we can conclude that the corresponding estimators \({\hat \theta}_ n\) of \(\theta\) are consistent and asymptotically efficient. Two examples are given as a nice illustration of the basic result.
Reviewer: J.M.Stoyanov

62M05 Markov processes: estimation; hidden Markov models
60J60 Diffusion processes
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