Identification of Dynamical Systems with Small Noise

✍ Scribed by Yu. Kutoyants (auth.)

• Publisher: Springer Netherlands
• Year: 1994
• Tongue: English
• Leaves: 307
• Series: Mathematics and Its Applications 300
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter....Pages i-viii
Introdution....Pages 1-10
Auxiliary Results....Pages 11-38
Asymptotic Properties of Estimators in Standard and Nonstandard Situations....Pages 39-113
Expansions....Pages 114-144
Nonparametric Estimation....Pages 145-164
The Disorder Problem....Pages 165-191
Partially Observed Systems....Pages 192-216
Minimum Distance Estimation....Pages 217-283
Back Matter....Pages 284-301

✦ Subjects

Statistics, general; Probability Theory and Stochastic Processes; Systems Theory, Control; Information and Communication, Circuits; Statistics for Engineering, Physics, Computer Science, Chemistry and Earth Sciences