A note on the stationarity of a threshold first-order bilinear process

In the present note we study the threshold first-order bilinear model X(t)=aX(t-1)+(b11{X(t-1)<c}+b21{X(t-1)[greater-or-equal, slanted]c})X(t-1)e(t-1)+e(t), t[epsilon]N where {e(t), t[epsilon]N} is a sequence of i.i.d. absolutely continuous random variables, X(0) is a given random variable and a, b1, b2 and c are real numbers. Under suitable conditions on the coefficients and lower semicontinuity of the densities of the noise sequence, we provide sufficient conditions for the existence of a stationary solution process to the present model and of its finite moments of order p.