Short Communication: Stability of Time-Inconsistent Stopping for One-Dimensional Diffusions

Abstract

For time-inconsistent stopping in a one-dimensional diffusion setup, we investigate how to use discrete-time models to approximate the original problem. In particular, we consider the value function V (•) induced by all mild equilibria in the continuous-time problem, as well as the value V h (•) associated with the equilibria in a binomial-tree setting with time step size h. We show that lim h→0+ V h ≤ V . We provide an example showing that the exact convergence may fail. Then we relax the set of equilibria and consider the value V h ε (•) induced by ε-equilibria in the binomial-tree model. We prove that lim ε→0+ lim h→0+ V h ε = V .