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

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 $\epsilon$-equilibria, we establish the relaxed continuity of the optimal value.