Stability of Time-inconsistent Stopping for One-dimensional Diffusion -- A Longer Version

TL;DR

This paper studies how the optimal stopping value for one-dimensional diffusions under time-inconsistent discounting behaves when model parameters change, showing semi-continuity and relaxed continuity properties.

Contribution

It establishes semi-continuity of the optimal value with respect to model parameters and introduces $\\varepsilon$-equilibria to ensure relaxed continuity.

Findings

Optimal value is semi-continuous with respect to drift, volatility, and reward.

Exact continuity of the optimal value may fail in some cases.

Relaxed continuity is achieved using $\\varepsilon$-equilibria.

Abstract

We investigate the stability of the equilibrium-induced optimal value in one-dimensional diffusion setting for a time-inconsistent stopping problem under non-exponential discounting. We show that the optimal value is semi-continuous with respect to the drift, volatility, and reward function. An example is provided showing that the exact continuity may fail. With equilibria extended to $\varepsilon$-equilibria, we establish the relaxed continuity of the optimal value.