Time-inconsistent mean-field optimal stopping: A limit approach

TL;DR

This paper characterizes an optimal stopping rule for a class of finite horizon, time-inconsistent mean-field problems, showing that the optimal strategy involves stopping when the value process hits the reward process, using a limit approach.

Contribution

It introduces a novel limit approach to solve time-inconsistent mean-field optimal stopping problems, extending classical results to more complex, mean-field settings.

Findings

Optimal stopping occurs when the value process hits the reward process.

Sequence of hitting times converges to the optimal stopping time.

The approach applies to mean-field diffusion processes and recursive utilities.

Abstract

We provide a characterization of an optimal stopping time for a class of finite horizon time-inconsistent optimal stopping problems (OSPs) of mean-field type, adapted to the Brownian filtration, including those related to mean-field diffusion processes and recursive utility functions. Despite the time-inconsistency of the OSP, we show that it is optimal to stop when the value-process hits the reward process for the first time, as is the case for the standard time-consistent OSP. We solve the problem by approximating the corresponding value-process with a sequence of Snell envelopes of processes, for which a sequence of optimal stopping times is constituted of the hitting times of each of the reward processes by the associated value-process. Then, under mild assumptions, we show that this sequence of hitting times converges in probability to the hitting time for the mean-field OSP and that the limit is optimal.