Optimal Stopping under Model Ambiguity: a Time-Consistent Equilibrium Approach

TL;DR

This paper develops a time-consistent equilibrium framework for optimal stopping problems under model ambiguity, incorporating ambiguity attitudes via nonlinear expectations, and characterizes equilibrium policies in diffusion models with uncertainty.

Contribution

It introduces a novel fixed-point approach to find subgame perfect equilibrium stopping policies considering ambiguity attitudes, extending beyond standard worst-case analysis.

Findings

Equilibrium policies depend on ambiguity attitude, with more ambiguity-averse agents more eager to stop.

All equilibrium stopping policies are characterized explicitly in a volatility uncertainty example.

The approach captures diverse behaviors influenced by ambiguity attitudes, beyond traditional models.

Abstract

An unconventional approach for optimal stopping under model ambiguity is introduced. Besides ambiguity itself, we take into account how ambiguity-averse an agent is. This inclusion of ambiguity attitude, via an $\alpha$-maxmin nonlinear expectation, renders the stopping problem time-inconsistent. We look for subgame perfect equilibrium stopping policies, formulated as fixed points of an operator. For a one-dimensional diffusion with drift and volatility uncertainty, we show that every equilibrium can be obtained through a fixed-point iteration. This allows us to capture much more diverse behavior, depending on an agent's ambiguity attitude, beyond the standard worst-case (or best-case) analysis. In a concrete example of real options valuation under volatility uncertainty, all equilibrium stopping policies, as well as the best one among them, are fully characterized. It demonstrates explicitly the effect of ambiguity attitude on decision making: the more ambiguity-averse, the more eager to stop -- so as to withdraw from the uncertain environment. The main result hinges on a delicate analysis of continuous sample paths in the canonical space and the capacity theory. To resolve measurability issues, a generalized measurable projection theorem, new to the literature, is also established.