On Strategic Measures and Optimality Properties in Discrete-Time Stochastic Control with Universally Measurable Policies

TL;DR

This paper investigates the properties of optimal policies in complex stochastic control systems with measurable policies, establishing foundational results for both risk-neutral and risk-sensitive decision processes, including partially observable cases.

Contribution

It introduces new measurability results and existence proofs for optimal policies in discrete-time stochastic control with universally measurable policies, extending to minimax problems.

Findings

Measurability of optimal value functions established.

Existence of universally measurable $\\epsilon$-optimal policies proven.

Results apply to risk-neutral, risk-sensitive, and partially observable MDPs.

Abstract

This paper concerns discrete-time infinite-horizon stochastic control systems with Borel state and action spaces and universally measurable policies. We study optimization problems on strategic measures induced by the policies in these systems. The results are then applied to risk-neutral and risk-sensitive Markov decision processes, as well as their partially observable counterparts, to establish the measurability of the optimal value functions and the existence of universally measurable, randomized or nonrandomized, $\epsilon$-optimal policies, for a variety of average cost criteria and risk criteria. We also extend our analysis to a class of minimax control problems and establish similar optimality results under the axiom of analytic determinacy.