Undiscounted optimal stopping with unbounded rewards

TL;DR

This paper extends classical optimal stopping results for Feller-Markov processes to cases with unbounded rewards, providing conditions for the existence of optimal stopping times in both finite and infinite horizon settings.

Contribution

It generalizes previous work by relaxing the boundedness assumption on rewards, using ergodic theory for infinite horizon cases, and characterizing optimal stopping times as first entrance times.

Findings

Optimal stopping times exist under certain conditions.

The form of optimal stopping times is characterized as first entrance times.

Results apply to unbounded reward functions in both finite and infinite horizon cases.

Abstract

We study optimal stopping of Feller-Markov processes to maximise an undiscounted functional consisting of running and terminal rewards. In a finite-time horizon setting, we extend classical results to unbounded rewards. In infinite horizon, we resort to ergodic structure of the underlying process. When the running reward is mildly penalising for delaying stopping (i.e., its expectation under the invariant measure is negative), we show that an optimal stopping time exists and is given in a standard form as the time of first entrance to a closed set. This paper generalises Palczewski, Stettner (2014), Stoch Proc Appl 124(12) 3887-3920, by relaxing boundedness of rewards.