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Introdution

  • Yu. Kutoyants
Part of the Mathematics and Its Applications book series (MAIA, volume 300)

Abstract

In this work, we are interested in the problems of estimation theory concerned with observations of the diffusion-type process
$$d{{X}_{t}} = {{S}_{t}}\left( X \right)dt + \varepsilon dWt,{{X}_{0}} = {{x}_{0}},0 \leqslant t \leqslant T,$$
(0.1)
Where Wt is a standard Wiener process and St(·)is some nonanticipative smooth function. By the observations \({\text{X}} = \left\{ {{{X}_{t}},0 \leqslant t \leqslant T} \right\}\) of this process, we will solve some of the problems of identification, both parametric and nonparametric. If the trend S(·) is known up to the value of some finite-dimensional parameter \({{S}_{t}}\left( X \right) = {{S}_{t}}\left( {0,X} \right),\)\(\theta \in \Theta \subset {{R}^{d}}\), then we have a parametric case. The nonparametric problems arise if we know only the degree of smoothness of the function \({{S}_{t}}\left( X \right),0 \leqslant t \leqslant T\) with respect to time t. It is supposed that the diffusion coefficient E is always known.in the parametric case, we describe the asymptotical properties of maximum likelihood (MLE), Bayes (BE) and minimum distance (MDE) estimators \(\varepsilon \to 0\) and in the nonparametric situation, we investigate some kernel-type estimators of unknown functions (say, S t (•), \(0 \leqslant t \leqslant T\)).

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Copyright information

© Springer Science+Business Media Dordrecht 1994

Authors and Affiliations

  • Yu. Kutoyants
    • 1
  1. 1.Département de MathématiquesFaculté des Sciences, Université du MaineLe MansFrance

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