On Structural Properties of Risk-Averse Optimal Stopping Problems
Xingyu Ren, Michael C. Fu, Steven I. Marcus
TL;DR
The paper develops structural results for finite-horizon risk-averse optimal stopping under time-consistent coherent risk measures. It extends classic risk-neutral insights by handling subadditivity and the lack of tower property, establishing value-function monotonicity under mild regularity and providing a general framework for control-limit policies. Two verifiable pathways to threshold structures are proposed: comonotone conditions on risk measures and state vectors, and valid one-step look-ahead optimality; in both, the results connect to risk-neutral equivalents or provide practical conditions for policy design. Through dual representations, coupling arguments, and illustrative examples in operations, marketing, and finance, the work offers actionable criteria for when risk-averse stopping problems admit threshold-type policies and how more risk aversion affects policy structure.
Abstract
We establish structural properties of optimal stopping problems under time-consistent dynamic (coherent) risk measures, focusing on value function monotonicity and the existence of control limit (threshold) optimal policies. While such results are well developed for risk-neutral (expected-value) models, they remain underexplored in risk-averse settings. Coherent risk measures typically lack the tower property and are subadditive rather than additive, complicating structural analysis. We show that value function monotonicity mirrors the risk-neutral case. Moreover, if the risk envelope associated with each coherent risk measure admits a minimal element, the risk-averse optimal stopping problem reduces to an equivalent risk-neutral formulation. We also develop a general procedure for identifying control limit optimal policies and use it to derive practical, verifiable conditions on the risk measures and MDP structure that guarantee their existence. We illustrate the theory and verify these conditions through optimal stopping problems arising in operations, marketing, and finance.