Optimal equilibria for time‐inconsistent stopping problems in continuous time

Abstract

For an infinite-horizon continuous-time optimal stopping problem under non-exponential discounting, we look for an optimal equilibrium, which generates larger values than any other equilibrium does on the entire state space. When the discount function is log sub-additive and the state process is one-dimensional, an optimal equilibrium is constructed in a specific form, under appropriate regularity and integrability conditions. While there may exist other optimal equilibria, we show that they can differ from the constructed one in very limited ways. This leads to a sufficient condition for the uniqueness of optimal equilibria, up to some closedness condition. To illustrate our theoretic results, comprehensive analysis is carried out for three specific stopping problems, concerning asset liquidation and real options valuation. For each one of them, an optimal equilibrium is characterized through an explicit formula.