Weak Solutions of stochastic recursions: an explicit construction

We propose an explicit construction of the solution of a stationary stochastic recursion of the form $X\circ\theta=\phi(X)$ on a semi-ordered Polish space, when the monotonicity of $\phi$ is not assumed. This solution exists on an enriched probability space (it is said \emph{weak}), provided the recursion is lattice-valued, and dominated by a proper monotonic stochastic recursion.