Asymptotic filtering for finite state Markov chains
type: Item
instance of: scholarly article
title: Asymptotic filtering for finite state Markov chains
zbMATH Open document ID: 0870.62065
DOI: 10.1016/0304-4149(96)00060-9
published in: Stochastic Processes and their Applications
publication date: 1997-08-07T00:00:00Z
review text: Let \(\{X_n^{\varepsilon }\}\) be a discrete time Markov chain with state space \(S=\{1,\dots ,d\}\), initial distribution \(p_0^{\varepsilon }\), and transition probabilities \(\pi _{ij}^{\varepsilon }=\varepsilon \lambda _{ij}\) \((i\neq j)\), \(\pi _{ii}^{\varepsilon }=1-\varepsilon \lambda _{ii}\). It is assumed that the chain is irreducible and aperiodic. Let the observations \(\{y_n^{\varepsilon }\}_{n=1}^{\infty }\), given \(\{X_n^{\varepsilon }\}_{n=1}^{\infty }\), be independent, with \(P\left (y_n^{\varepsilon }\in dx |\{X_i^{\varepsilon }\}_{n=1}^{\infty }\right )=P\left (y_n^{\varepsilon }\in dx|X_n^{\varepsilon }\right )\), \(P\left (y_n^{\varepsilon }\in dx|X_n^{\varepsilon }=i\right )=\mu _i(dx)\). Define \({\mathcal F}_{y,n}= \sigma \{y_i^{\varepsilon }, 1\leq i\leq n\}\). The filtering means to find the best estimator \(\tilde X_n\) for \(X_n^{\varepsilon }\) given \({\mathcal F}_{y,n}\). The authors present asymptotic formulas for the filtering error \(P_e^{\varepsilon , n}= E\left (1_{X_n^{\varepsilon }\neq \tilde X_n}\right )\). Asymptotically optimal filters are also introduced, which do not depend on the transition rates of the chain.
Mathematics Subject Classification ID: 62M05
Mathematics Subject Classification ID: 60J10
Mathematics Subject Classification ID: 62M20
zbMATH DE Number: 1046566
zbMATH Keywords: nonlinear filtering
zbMATH Keywords: hypothesis testing
zbMATH Keywords: Markov chains
author: R. Z. Khas'minskiĭ
author: Ofer Zeitouni
reviewed by: Jiří Anděl
MaRDI profile type: Publication
full work available at URL: 0304-4149(96)00060-9
OpenAlex ID: W2060076922
cites work: Q3134548
cites work: On some filtration procedure for jump Markov process observed in white Gaussian noise
cites work: Q3747485
cites work: Q3321154
cites work: Some Applications of Stochastic Differential Equations to Optimal Nonlinear Filtering
description: scientific article; zbMATH DE number 1046566