A note on a simple Markov bilinear stochastic process (Q1613001)

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.