Optimal Stopping with Expectation Constraints

TL;DR

This paper develops a theoretical framework for optimal stopping problems with expectation constraints in non-Markovian settings, establishing equivalences, measurability, and a dynamic programming principle.

Contribution

It introduces a weak formulation approach for constrained optimal stopping, characterizes the value function, and proves a dynamic programming principle in this context.

Findings

Equivalent formulation of constrained optimal stopping in weak form.

Value function is upper semi-analytic.

Established a dynamic programming principle for the problem.

Abstract

We analyze an optimal stopping problem with a series of inequality-type and equality-type expectation constraints in a general non-Markovian framework. We show that the optimal stopping problem with expectation constraints (OSEC) in an arbitrary probability setting is equivalent to the constrained problem in weak formulation (optimization over joint laws of stopping rules with Brownian motion and state dynamics on an enlarged canonical space) and thus the OSEC value. Using a martingale-problem formulation, we make an equivalent characterization of the probability classes in weak formulation, which implies that the OSEC value function s upper semi-analytic. Then we exploit a measurable selection argument to establish a dynamic programming principle in weak formulation for the OSEC value function, in which the conditional expected costs act as additional states for constraint levels at the intermediate horizon.