Existence of bounded solutions to multiplicative poisson equations under mixing property

We study the problem of Multiplicative Poisson Equation (MPE) bounded solution existence in a generic probabilistic discrete-time setup with preimposed mixing. In particular, we consolidate results based on the span-contraction framework and derive an explicit sharp bound that must be imposed on the cost function to guarantee the existence of a bounded MPE solution under mixing. Also, we study properties that the probability kernel must satisfy to ensure the existence of an MPE solution 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 problems that arise when the mean is replaced by the entropy in the ergodic setup.