Constrained Optimal Stopping, Liquidity and Effort

TL;DR

This paper extends classical optimal stopping models by allowing agents to choose the rate of Poisson-distributed intervention times, balancing liquidity constraints and effort costs, leading to diverse optimal strategies.

Contribution

It introduces a model where the agent controls the Poisson rate in constrained stopping problems, revealing complex behaviors depending on cost functions.

Findings

Agents often accept the first offer under certain conditions.

Higher Poisson rates increase costs and influence stopping decisions.

Optimal strategies vary with the form of the cost function.

Abstract

In a classical optimal stopping problem in continuous time, the agent can choose any stopping time without constraint. Dupuis and Wang (Optimal stopping with random intervention times, Advances in Applied Probability, 34, 141--157, 2002) introduced a constraint on the class of admissible stopping times which was that they had to take values in the set of event times of an exogenous, time-homogeneous Poisson process. This can be thought of as a model of finite liquidity. In this article we extend the analysis of Dupuis and Wang to allow the agent to choose the rate of the Poisson process. Choosing a higher rate leads to a higher cost. Even for a simple model for the stopped process and a simple call-style payoff, the problem leads to a rich range of optimal behaviours which depend on the form of the cost function. Often the agent accepts the first offer --- if they are not going to accept an offer then there is no point in putting in effort to generate offers, and thus there may be no offers to accept or decline --- but for some set-ups this is not the case.