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 thought desirous about observations of the diffusion-type approach Xo = Xo, zero ~ t ~ T, (0. 1) the place W is a regular Wiener approach and St(') is a few nonanticipative tender t functionality. through the observations X = {X , zero ~ t ~ T} of this approach, we'll clear up a few t of the issues of identity, either parametric and nonparametric. If the rage S(-) is understood as much as the worth of a few finite-dimensional parameter St(X) = St((}, X), the place (} E e c Rd , then we now have a parametric case. The nonparametric difficulties come up if we all know merely the measure of smoothness of the functionality St(X), zero ~ t ~ T with appreciate to time t. it truly is intended that the diffusion coefficient c is often recognized. within the parametric case, we describe the asymptotical homes of utmost 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 services (say, StO,O ~ t ~ T). The asymptotic in such difficulties of estimation for this scheme of observations was once often regarded as T --+ 00 , simply because this restrict is an immediate analog to the conventional restrict (n --+ 00) within the classical mathematical data of i. i. d. observations. The restrict c --+ zero in (0. 1) is attention-grabbing for the next reasons.

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4. Let conditions I-N be fulfilled, tllen for any p E (0,1) and K C there exists a function g(K,p, u) = g(lul) E G such that supE,IZ:,I(u) ~ exp{-g(lul}}. lIeK ' Proof. : [St(O + u, x) - St(O, X)]2 -21(St(0 + u,x) - St(O,x))(ISt(O + u,X) - St(O + u,x)1 + +ISt(O,X) - St(O,x)l). e, CHAPTER 2 50 By condition II, as lui -+ 0, we have foT [St((} + u, x) - St((}, xWdt = = l T ll\St((}+lu,x),u)df dt= = = . . ) + 0(1)). 1>,1=1 For all () E K, condition II and this inequality provide the two-sided inequality Klul 2~ lT [St((} + u, x) with some constants A > 0 and K = K(K) Below, we use the Schwarz inequality (I ~ A21ul 2 St((}, x Wdt > O.

8, we have It now remains to consider the case () > 0 and Xo = o. Once more, we change the variables s = c;-lt, w. £, Ya = C;-1/2 X a £. Then for the process Ya, we have the same equation dY. , Yo = 0, 0 $ s $ T£ = TC;-l --+ 00, but its solution with probability 1 goes to infinity. t - with some Wiener process Wa , s ~ Integrating by parts, we have o. 00 o e -Badwa N e- 2BT {T e2h eds = 10 • -2BT 2BT -2BT loT -2BT T = e e t 2 __ e__ e28at d t __ e __ { e2Bte-2Btdt = 2(} .. T (} 0 ... t d Te= 20 10 e ':ta Ws 8 .

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. 60) 31 AUXILIARY RESULTS and fh is some fixed value, 01 E 8. 59) will always exist. 57) is linear on 0 E Rd: Oe. and the matrix h = loT ft(X)Jf(X)dt is positive definite with P~:) probability 1, then the MLE To introduce a Bayesian estimator we need a function l(y), y E Rd which is 1.

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