Download Identification of Dynamical Systems with Small Noise by Yury A. Kutoyants PDF

By Yury A. Kutoyants

Small noise is an efficient noise. during this paintings, we're drawn to the issues of estimation conception eager about observations of the diffusion-type procedure Xo = Xo, zero ~ t ~ T, (0. 1) the place W is a regular Wiener approach and St(') is a few nonanticipative soft t functionality. via the observations X = {X , zero ~ t ~ T} of this strategy, we'll clear up a few t of the issues of id, either parametric and nonparametric. If the craze S(-) is understood as much as the price of a few finite-dimensional parameter St(X) = St((}, X), the place (} E e c Rd , then we have now a parametric case. The nonparametric difficulties come up if we all know in simple terms the measure of smoothness of the functionality St(X), zero ~ t ~ T with admire to time t. it truly is intended that the diffusion coefficient c is usually recognized. within the parametric case, we describe the asymptotical houses of extreme probability (MLE), Bayes (BE) and minimal distance (MDE) estimators as c --+ zero and within the nonparametric scenario, we examine a few kernel-type estimators of unknown features (say, StO,O ~ t ~ T). The asymptotic in such difficulties of estimation for this scheme of observations used to be frequently regarded as T --+ 00 , simply because this restrict is a right away analog to the conventional restrict (n --+ 00) within the classical mathematical statistics of i. i. d. observations. The restrict c --+ zero in (0. 1) is attention-grabbing for the next reasons.

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Extra resources for Identification of Dynamical Systems with Small Noise

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57) with small c, then one can embed this problem in a sequence of problems with c ..... 0 and then apply the obtained result to the original one. An estimator (J. 58) e where is a closure of e and we assume that e is supplied with u-algebra of Borelian subsets in Rd. We say that the estimator (J. ) {l0. -to > o} = 0 and denote this convergence as We say that the estimator 0. ) {IOe - (JI > o} = e-t°IlEK c e (or uniformly o. We shall study the asymptotic properties of three types of estimators: maxImum likelihood, Bayesian, and minimum distance.

3. Suppose that the observed process Xi! 0 ~ t ~ T belongs to the family of processes dXt = [St(X) where 0 E 8 = {fJ condition £. and + 0 c 9t(X)) dt + c dWt, Xo = Xo, 0 ~ t ~ T, I() - fJol < m} C R\ the functions St(-), 9t(-) satisfy 10 = loT 9t(x)2 dt > o. Then the likelihood ratio admits the representation (fJo is a true value) Z.. /lo = p(") Bo _ ~2 + t/I.. (u, Oo)} = = exp { u a = Z(u) expN.. ,p~(u,Oo) = o. Therefore, the above presented proof allows us to write the inequality sup Eel (/~/2 (O~ - 0)) 2: le-eol

3 On asymptotic estimation theory Below, we would like to illustrate certain notions and methods of estimation theory on the model of observation of the diffusion-type process. 57) is defined. 57) in the measurable space (CT,BT) are equivalent. Here c is an "index of series", c E (0,1], asymptotic corresponds to c ..... 57) indexed by c. 57) with small c, then one can embed this problem in a sequence of problems with c ..... 0 and then apply the obtained result to the original one. An estimator (J.

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