A generalized backwards scheme for solving non monotonic stochastic recursions

TL;DR

This paper introduces a new explicit method for constructing stationary solutions to non-monotonic stochastic recursions on partially ordered spaces, extending existing frameworks to non-monotonic cases and applying results to queueing systems.

Contribution

It presents a generalized backward scheme for solving non-monotonic stochastic recursions without assuming monotonicity, including explicit construction and conditions for solutions on the original space.

Findings

Explicit construction of solutions for non-monotonic recursions

Extension of probability space for solution existence

Application to stability analysis of queueing systems

Abstract

We propose an explicit construction of a stationary solution for a stochastic recursion of the form $X\circ\theta=\phi(X)$ on a partially-ordered Polish space, when the monotonicity of $\phi$ 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. We then provide conditions for the solution to be defined as well on the original space. We finally apply these results to the stability study of two non-monotonic queueing systems.