Abstract. We provide a characterization of strictly stationary solutions to the stochastic recurrence equation $z_k=c(\varepsilon _{k-1})z_{k-1}+g(\varepsilon _{k-1})$ with Borel-measurable functions $c$ and $g$, and independent, identically distributed random variables $\{\varepsilon _k\} $. Strictly stationary solutions that are functions of the past, respectively, of the future exist if and only if the expected value $E\log |c(\varepsilon _0)|$ is negative, respectively, positive. The main result of the paper is to show that there is no solution that is a function of the past or the future if $E\log |c(\varepsilon _0)|=0$.

AMS Subject Classification (1991): 60G10; 62M10, 91B84

Keyword(s): Augmented GARCH processes, Generalized autoregressive equations, Stochastic recurrence equations, Strictly stationary solutions

Received May 9, 2007. (Registered under 6460/2009.)