Identification of Dynamical Systems with Small Noise

1. Front Matter

   Pages i-viii

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2. Introdution

   • Yu. Kutoyants

   Pages 1-10

3. Auxiliary Results

   • Yu. Kutoyants

   Pages 11-38

4. Asymptotic Properties of Estimators in Standard and Nonstandard Situations

   • Yu. Kutoyants

   Pages 39-113

5. Expansions

   • Yu. Kutoyants

   Pages 114-144

6. Nonparametric Estimation

   • Yu. Kutoyants

   Pages 145-164

7. The Disorder Problem

   • Yu. Kutoyants

   Pages 165-191

8. Partially Observed Systems

   • Yu. Kutoyants

   Pages 192-216

9. Minimum Distance Estimation

   • Yu. Kutoyants

   Pages 217-283

10. Back Matter

    Pages 284-301

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Small noise is a good noise. In this work, we are interested in the problems of estimation theory concerned with observations of the diffusion-type process Xo = Xo, 0 ~ t ~ T, (0. 1) where W is a standard Wiener process and St(') is some nonanticipative smooth t function. By the observations X = {X , 0 ~ t ~ T} of this process, we will solve some t of the problems of identification, both parametric and nonparametric. If the trend S(-) is known up to the value of some finite-dimensional parameter St(X) = St((}, X), where (} E e c Rd , then we have a parametric case. The nonparametric problems arise if we know only the degree of smoothness of the function St(X), 0 ~ t ~ T with respect to time t. It is supposed that the diffusion coefficient c is always known. In the parametric case, we describe the asymptotical properties of maximum likelihood (MLE), Bayes (BE) and minimum distance (MDE) estimators as c --+ 0 and in the nonparametric situation, we investigate some kernel-type estimators of unknown functions (say, StO,O ~ t ~ T). The asymptotic in such problems of estimation for this scheme of observations was usually considered as T --+ 00 , because this limit is a direct analog to the traditional limit (n --+ 00) in the classical mathematical statistics of i. i. d. observations. The limit c --+ 0 in (0. 1) is interesting for the following reasons.

• Estimator
• Likelihood
• Parameter
• Probability theory
• Rang
• communication
• disorder problem
• dynamical systems
• probability
• stochastic processes
• systems theory
• information and communication, circuits

• Département de Mathématiques, Faculté des Sciences, Université du Maine, Le Mans, France

  Yu. Kutoyants

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