A class of stochastic unit-root bilinear processes, allowing for GARCH-type effects with asymmetries, is studied. Necessary and sufficient conditions for the strict and second-order stationarity of the error process are given. The strictly stationary solution is shown to be strongly mixing under mild additional assumptions. It follows that, in this model, the standard (non-stochastic) unit-root tests of Phillips-Perron and DickeyFuller are asymptotically valid to detect the presence of a (stochastic) unit-root. The finite sample properties of these tests are studied via Monte Carlo experiments. JEL classification: C22, C12, C52.