I am looking for a sufficient condition so that the following equation
$y_{t}=\alpha_{R_{t}}+\beta_{R_{t}}y_{t-1}+Q_{R_{t}}e_{t}$, where $R_{t}=1$, if $y_{t-1}\leq r_{1}$, $R_{t}=R_{t-1}$, if $r_{1}<y_{t-1}\leq r_{2}$, and $R_{t}=0$, if $y_{t-1}>r_{2}$,
admits a strictly stationary solution. The parameters $r_{1}$ and $r_{2}$ are known and the process $\left( e_{t}\right) $ is a sequence of i.i.d. random variables with zero mean and unit variance. The distribution $f$ of $\left( e_{t}\right) $ is assumed to be strictly positive density function on the whole real line.
I think we can apply Brandt's Theorem [Brandt, A. (1986). The stochastic equation $Y_{n+ 1}= A_{n}Y_{n}+ B_{n}$ with stationary coefficients. Advances in Applied Probability, 18(1), 211-220] on the equivalent generalized AR representation of the previous equation
$ y_{t}= A_{t} y_{t-1} + B_{t}, $ where $A_{t}=\beta_{0}R_{t}+\beta_{1}\left( 1-R_{t}\right) $ and $B_{t}=\left( \alpha_{0}+Q_{0}e_{t}\right) R_{t}+\left( \alpha_{1}% +Q_{1}e_{t}\right) \left( 1-R_{t}\right) .$
I think we can apply Brand's Theorem (1986). It suffices to show that the bivariate process $(A_{t},B_{t})_{t}$ is strictly stationary and ergodic. Is it possible to show this and how can I do it?