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On the estimation of the diffusion coefficient for multi-dimensional diffusion processes. (English) Zbl 0770.62070
The authors consider an estimation problem for the solution $$X$$ of a stochastic differential equation of the form $dX_ t=b(t,X)dt+a(\vartheta,t,X_ t)dW_ t,\quad{\mathcal L}(X_ 0)=v,$ where the diffusion coefficient $$a$$ is a known function of the parameter $$\vartheta$$, belonging to a compact set, and the drift $$b$$ is a non- anticipative functional. $$X$$ is observed at $$n$$ distinct times $$t(n,i)$$, $$i=1,\dots,n$$, in the interval [0,1] chosen such that $$\mu_ n=n^{- 1}\sum_{1\leq i\leq n}{\mathcal E}_{t(n,i)}$$ weakly converges to a measure $$\mu$$ on [0,1].
It is shown that a consistent sequence $$\widehat\vartheta_ n$$ of estimators of $$\vartheta$$ can be constructed based on the minimization of suitable constructs and on local asymptotic mixed normality, i.e.: $$\sqrt n(\widehat\vartheta_ n-\vartheta)$$ converges in law to a conditional centered Gaussian variable, conditioned on the paths of $$X$$. Further, optimality in a certain sense is established under some smoothness assumptions.

##### MSC:
 62M05 Markov processes: estimation; hidden Markov models 62F12 Asymptotic properties of parametric estimators 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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