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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

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