# A Class of Solvable Multidimensional Stopping Problems in the Presence of Knightian Uncertainty

## TL;DR

This paper analyzes how Knightian uncertainty influences optimal stopping times in multidimensional Brownian motion models, revealing that ambiguity can accelerate decision timing and induce stationarity, acting as a stabilizing factor.

## Contribution

It provides a general characterization of the value and worst-case measure in multidimensional settings with ambiguity, extending previous linear diffusion results.

## Key findings

- Ambiguity accelerates optimal stopping compared to unambiguous models.
- Ambiguity can induce stationarity in non-stationary models.
- Ambiguity may act as a stabilizing mechanism in decision processes.

## Abstract

We investigate the impact of Knightian uncertainty on the optimal timing policy of an ambiguity averse decision maker in the case where the underlying factor dynamics follow a multidimensional Brownian motion and the exercise payoff depends on either a linear combination of the factors or the radial part of the driving factor dynamics. We present a general characterization of the value of the optimal timing policy and the worst case measure in terms of a family of an explicitly identified excessive functions generating an appropriate class of supermartingales. In line with previous findings based on linear diffusions, we find that ambiguity accelerates timing in comparison with the unambiguous setting. Somewhat surprisingly, we find that ambiguity may result into stationarity in models which typically do not possess stationary behavior. In this way, our results indicate that ambiguity may act as a stabilizing mechanism.