Kristoffer Lindensjö: Finding Equilibria for Time-Inconsistent Markovian Stopping Problems

Abstract
Standard Markovian optimal stopping problems are consistent in the sense that the first entrance time into a stopping region is an optimal stopping time for each initial state of the state process.

The usual concept of optimality cannot in a straightforward way be applied to non-standard stopping problems without this time-consistent structure. This work is devoted to the solution of time-inconsistent stopping problems with the reward depending on the initial state using a game-theoretic approach in which each state of the process corresponds to a player in the game.

We give a precise equilibrium definition --- of the type subgame perfect Nash equilibrium based on pure Markov strategies. Such equilibria do not always exist. However, we develop an iterative approach to finding such equilibrium stopping times for a general class of problems and apply this approach to one-sided stopping problems on the real line. We furthermore prove a verification theorem based on a set of variational inequalities which also allows us to find equilibria. As an application of the developed theory we study a selling strategy problem under exponential utility and endogenous habit formation.

Keywords: Markov process, Optimal stopping, Subgame perfect Nash equilibrium, Time-inconsistency, Variational inequalities.