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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.
List of contents
1 Auxiliary Results.- 1.1 Some notions of probability theory.- 1.2 Stochastic integral.- 1.3 On asymptotic estimation theory.- 2 Asymptotic Properties of Estimators in Standard and Nonstandard Situations.- 2.1 LAM bound on the risks of estimators.- 2.2 Asymptotic behavior of estimators in the regular case.- 2.3 Parameter estimation for linear systems.- 2.4 Nondifferentiable and "too differentiable" trends.- 2.5 Random initial value.- 2.6 Misspecified models.- 2.7 Nonconsistent estimation.- 2.8 Boundary of the parametric set.- 3 Expansions.- 3.1 Expansion of the MLE.- 3.2 Possible generalizations.- 3.3 Expansion of the distribution function.- 4 Nonparametric Estimation.- 4.1 Trend estimation.- 4.2 Linear multiplier estimation.- 4.3 State estimation.- 5 The Disorder Problem.- 5.1 Simultaneous estimation of the smooth parameter and the moment of switching.- 5.2 Multidimensional disorder.- 5.3 Misspecified disorder.- 6 Partially Observed Systems.- 6.1 Kalman filter identification.- 6.2 Nonlinear systems.- 6.3 Disorder problem for Kalman filter.- 7 Minimum Distance Estimation.- 7.1 Definitions and examples of the MDE.- 7.2 Consistence and limit distributions.- 7.3 Linear systems.- 7.4 Nonstandard situations and other problems.- 7.5 Asymptotic efficiency of the MDE.- Remarks.- References.
