Nonlinear PDE Approach to Time-Inconsistent Optimal Stopping

Abstract

We present a novel method for solving a class of time-inconsistent optimal stopping problems by reducing them to a family of standard stochastic optimal control problems. In particular, we convert an optimal stopping problem with a non-linear function of the expected stopping time in the objective into optimization over an auxiliary value function for a standard stochastic control problem with an additional state variable. This approach differs from the previous literature which primarily employs Lagrange multiplier methods or relies on exact solutions. In contrast, we characterize the auxiliary value function as the unique viscosity solution of a non-linear elliptic PDE which satisfies certain growth constraints and investigate basic regularity properties. We demonstrate a connection between optimal stopping times for the original problem and optimal controls of the auxiliary control problem. Finally, we discuss extensions to more general dynamics and time-inconsistencies, as well as potential connections to degenerate Monge-Ampere equations.

On time-inconsistent stopping problems and mixed strategy stopping times

“…For short surveys of time-inconsistent stopping problems we also refer to [1,9,34]. Recent papers on time-inconsistent stopping problems and the dynamic optimality and pre-commitment approaches include [31,34].…”

On Finding Equilibrium Stopping Times for Time-Inconsistent Markovian Problems

“…In [9] a class of stopping problems -which can be seen as American options with guarantee -with the reward depending on the initial state are studied using a pre-commitment approach. We refer to [6,23,29,33] for short surveys of the literature on time-inconsistent problems.…”

“…We remark that [33] contains also a subgame perfect Nash equilibrium approach (based on Strotz's idea) for stopping problems, see mainly [33,Example 9]; this point is elaborated in the paragraph before Example 2.8 below. Timeinconsistent stopping problems with more general non-linear functions of the expected reward are studied in [29] using an approach which is inspired by [33].…”

General stopping behaviors of naïve and noncommitted sophisticated agents, with application to probability distortion

“…Finally, a sophisticated agent who is able to commit simply solves the problem once at t = 0 and then sticks to the corresponding stopping plan. The problem of the last type, the so-called pre-committed agent, is actually a static (instead of dynamic) problem and has been solved in various contexts, such as optimal stopping under probability distortion in Xu and Zhou (2013), mean-variance portfolio selection in Zhou and Li (2000), optimal stopping with nonlinear constraint on expected time to stop in Miller (2017), and optimal control of conditional Value-at-Risk in Miller and Yang (2017). The goal of this paper is to study the behaviors of the first two types of agents.…”