Optimal Equilibria for Multi-dimensional Time-inconsistent Stopping Problems

TL;DR

This paper investigates multi-dimensional time-inconsistent stopping problems with non-exponential discounting, establishing the existence of optimal equilibria using probabilistic potential theory, extending prior one-dimensional results.

Contribution

It generalizes the existence of optimal equilibria to multi-dimensional processes under log sub-additive discounting, a novel extension in the field.

Findings

Existence of optimal equilibrium among a large class of equilibria

Generalization from one-dimensional to multi-dimensional processes

Use of probabilistic potential theory to establish results

Abstract

We study an optimal stopping problem under non-exponential discounting, where the state process is a multi-dimensional continuous strong Markov process. The discount function is taken to be log sub-additive, capturing decreasing impatience in behavioral economics. On strength of probabilistic potential theory, we establish the existence of an optimal equilibrium among a sufficiently large collection of equilibria, consisting of finely closed equilibria satisfying a boundary condition. This generalizes the existence of optimal equilibria for one-dimensional stopping problems in prior literature.