Weak Solutions of stochastic recursions: an explicit construction

TL;DR

This paper introduces an explicit method for constructing weak solutions to stationary stochastic recursions on semi-ordered spaces without assuming monotonicity, expanding the understanding of such stochastic processes.

Contribution

It provides a novel explicit construction of weak solutions for stochastic recursions without requiring monotonicity, applicable to lattice-valued cases.

Findings

Solution exists on an enriched probability space

Applicable to lattice-valued recursions

Does not require monotonicity of the recursion function

Abstract

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.