Existence of bounded solutions to multiplicative Poisson equations under mixing property

TL;DR

This paper investigates conditions under which bounded solutions to the multiplicative Poisson equation exist in discrete-time stochastic processes, focusing on mixing properties and risk-sensitive scenarios.

Contribution

It provides explicit bounds and characterizations for the existence of bounded solutions to the MPE under mixing, including properties of the probability kernel and reward functions.

Findings

Derived explicit sharp bounds on the cost function for solution existence

Characterized probability kernel properties ensuring bounded solutions

Analyzed process behavior outside the invariant measure support

Abstract

In this paper we study the problem of Multiplicative Poisson Equation (MPE) bounded solution existence in the generic discrete-time setting. Assuming mixing and boundedness of the risk-reward function, we investigate what conditions should be imposed on the underlying non-controlled probability kernel or the reward function in order for the MPE bounded solution to always exists. In particular, we consolidate span-norm framework based results and derive an explicit sharp bound that needs to be imposed on the cost function to guarantee the bounded solution existence under mixing. Also, we study the properties which the probability kernel must satisfy to ensure existence of bounded MPE for any generic risk-reward function and characterise process behaviour in the complement of the invariant measure support. Finally, we present numerous examples and stochastic-dominance based arguments that help to better understand the intricacies that emerge when the ergodic risk-neutral mean operator is replaced with ergodic risk-sensitive entropy.