We propose an explicit construction of a stationary solution for a stochastic recursion of the form $X\circ\theta=\varphi(X)$ on a partially-ordered Polish space, when the monotonicity of $\varphi$ is not assumed. Under certain conditions, we show that an extension of the original probability space exists, on which a solution is well defined, and construct explicitly this extension using a randomized contraction technique. We then provide conditions for the existence of a solution on the original space. We finally apply these results to the stability study of two nonmonotonic queuing systems.
"A generalized backward scheme for solving nonmonotonic stochastic recursions." Ann. Appl. Probab. 25 (2) 582 - 599, April 2015. https://doi.org/10.1214/14-AAP1004