A note on a simple Markov bilinear stochastic process
type: Item
instance of: scholarly article
title: A note on a simple Markov bilinear stochastic process
zbMATH Open document ID: 0999.60067
DOI: 10.1016/S0167-7152(01)00192-4
author: Daren B. H. Cline
author: Huay-min H. Pu
published in: Statistics \& Probability Letters
publication date: 2002-09-05T00:00:00Z
review text: The present article has as starting point the results of a paper of \textit{H. Tong} [J. Time Ser. Anal. 2, 279-284 (1981; Zbl 0548.60071)], provides an improvement of these results, and points out a special case of the analyzed problem. According to H. Tong, a simple Markov bilinear stochastic process for a Markov process \(\{X_n\}\) is defined by the recurrent formula \[ X_n= aX_{n-1}+ be_n X_{n-1}+ e_n,\tag{MP} \] where \(a\) and \(b\) are constants, and \(\{e_n\}\) is a sequence of independent, identically distributed random variables with a density function that is positive and lower semi-continuous on the real space \({\mathbf R}\). This is an \(\text{AR}(1)\) process with \(\text{ARCH}(1)\)-type errors, and the model is proved to be useful for modelling financial time series in which the current volatility depends on the past value, including its sign. The case for which the Markov process (MP) behaves differently is \(a=1\). The uathors analyze this case, and find the conditions when \(\{X_n\}\) in (MP) is not ergodic. For the case \(a\neq 1\), there are obtained stronger results, ensuring the geometric ergodicity of the process. Finally, other \(\text{AR}(1)\) processes with \(\text{ARCH}(1)\)-type errors that have been examined in the literature are outlined.
reviewed by: Neculai Curteanu
Mathematics Subject Classification ID: 60J05
zbMATH DE Number: 1796687
zbMATH Keywords: bilinear process
zbMATH Keywords: ARCH-type errors
zbMATH Keywords: Markov process
zbMATH Keywords: drift criteria
zbMATH Keywords: geometric ergodicity
zbMATH Keywords: limiting distribution
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1796687