On some filtration procedure for jump Markov process observed in white Gaussian noise
type: Item
instance of: scholarly article
title: On some filtration procedure for jump Markov process observed in white Gaussian noise
zbMATH Open document ID: 0769.62072
author: B. V. Lazareva
author: R. Z. Khas'minskiĭ
published in: The Annals of Statistics
publication date: 1993-05-16T00:00:00Z
review text: The following filtering problem is discussed: \[ Y(t)=\int^ t_ 0 X(s)ds+\sigma W(t), \] is the observed process, \(X(t)\) is a Markov process with the states 0 and 1, and transition probability \(\left({1- \lambda\atop\mu}{\lambda\atop 1-\mu}\right)\), \(W(t)\) is a standard Wiener process, and \(\sigma\) is a constant. Let \({\mathcal F}^ t_ 0=\sigma\{Y(s),\;0\leq s\leq t\}\), \(\pi(t)=P[X(t)=1\mid{\mathcal F}^ t_ 0]\). The optimal estimator is expressed by \[ \widehat X(t)=\begin{cases} 1 & \text{if } Z(t)\geq 0,\\ 0 & \text{if } Z(t)<0,\end{cases} \] where \(Z(t)=\ln[\pi(t)/(1-\pi(t))]\) satisfies \[ dZ(t)=\sigma^{-2} (dY(t)- 2^{-1} dt)+\bigl(\lambda-\mu+\lambda e^{-Z(t)}- \mu e^{Z(t)}\bigr) dt. \] The authors suggest a new estimator: \[ \overline X(t)=\begin{cases} 1 & \text{if } Z_ 0(t)\geq 0,\\0 & \text{if } Z_ 0(t)<0,\end{cases} \] where \(Z_ 0(t)\) is the solution of \[ dZ_ 0(t)= \sigma^{-2} \bigl(dY(t)- 2^{-1} dt\bigr), \] and show its asymptotic efficiency as \(\sigma\to 0\). It is worth noting that \(\overline X(t)\) does not depend on \(\lambda\) and \(\mu\).
Mathematics Subject Classification ID: 62M20
Mathematics Subject Classification ID: 60G35
Mathematics Subject Classification ID: 60J27
zbMATH DE Number: 166859
zbMATH Keywords: jump Markov process
zbMATH Keywords: Brownian motion
zbMATH Keywords: optimal filtration
zbMATH Keywords: Markov chain with two states
zbMATH Keywords: Gaussian white noise
zbMATH Keywords: diffusion constant
zbMATH Keywords: infinitesimal generator
zbMATH Keywords: reflecting barrier
zbMATH Keywords: Markov process
zbMATH Keywords: transition probability
zbMATH Keywords: standard Wiener process
zbMATH Keywords: optimal estimator
zbMATH Keywords: asymptotic efficiency
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176348909
OpenAlex ID: W1969887439
DOI: 10.1214/AOS/1176348909
description: scientific article; zbMATH DE number 166859