Binomial-tree approximation for time-inconsistent stopping

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 $V(\cdot )$ induced by all mild equilibria in the continuous-time problem, as well as the value $V^h(\cdot )$ associated with the equilibria in a binomial-tree setting with time step size $h$. We show that ${\lim }_{h\to 0+}{V}^h\le V$. We provide an example showing that the exact convergence may fail. Then we relax the set of equilibria and consider the value $V_{\epsilon }^h(\cdot )$ induced by $\epsilon$-equilibria in the binomial-tree model. We prove that ${\lim }_{\epsilon \to 0+}{\lim }_{h\to 0+}{V}_{\epsilon }^h=V$.