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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Additional info for Identification of Dynamical Systems with Small Noise
Example text
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.