Irreversible Consumption Habit under Ambiguity : Singular Control and Optimal G-Stopping Time

Consider robust utility maximization with an irreversible consumption habit, where an agent concerned about model ambiguity is unwilling to decrease consumption and must simultaneously contend with a disutility (i.e., an adjustment cost) due to a consumption increase. While the optimization is a robust analog of singular control problems over a class of consumption-investment strategies and a set of probability measures, it is a new formulation that involves non-dominated probability measures of the diffusion process for the underlying assets in addition to singular controls with an adjustment cost. This paper provides a novel connection between the singular controls in the optimization and the optimal $G$-stopping times in a $G$-expectation space, using a duality theory. This connection enables to derive the robust consumption strategy as a running maximum of the stochastic boundary, which is characterized by a free boundary arising from the optimal $G$-stopping times. The duality, which relies on arguments based on reflected $G$-BSDEs, is achieved by verifying the first-order optimality conditions for the singular control, the budget constraint equation for the robust strategies, and the worst-case realization under the non-dominated measures