Solutions of Poisson's Equation for Stochastically Monotone Markov Chains

TL;DR

This paper proves that for stochastically monotone Markov chains, the solution to Poisson's equation is monotone if the reward function is monotone, leading to monotonic value functions in infinite horizon problems.

Contribution

It establishes the monotonicity of Poisson's equation solutions and value functions for monotone reward functions in stochastically monotone Markov chains.

Findings

Solution to Poisson's equation is monotone under monotone rewards.

Value function is monotone in the state for infinite horizon average reward.

Applicable to queues and storage systems with monotone dynamics.

Abstract

Stochastically monotone Markov chains arise in many applied domains, especially in the setting of queues and storage systems. Poisson's equation is a key tool for analyzing additive functionals of such models, such as cumulative sums of waiting times or sums of rewards. In this paper, we show that when the reward function for such a Markov chain is monotone, the solution of Poisson's equation is monotone. This implies that the value function associated with infinite horizon average reward is monotone in the state when the reward is monotone.