Distribution-Constrained Optimal Stopping

TL;DR

This paper addresses the optimal stopping problem for Brownian motion with a fixed distribution constraint, transforming it into a sequence of state-constrained control problems and applying it to model-free superhedging in finance.

Contribution

It introduces a novel approach to handle distribution constraints in optimal stopping by converting them into finite control problems, with a new dynamic programming principle avoiding measurable selection.

Findings

Reduction of distribution-constrained stopping to finite control problems

Development of a new dynamic programming principle for state-constrained problems

Application to model-free superhedging with volatility outlook

Abstract

We solve the problem of optimal stopping of a Brownian motion subject to the constraint that the stopping time's distribution is a given measure consisting of finitely-many atoms. In particular, we show that this problem can be converted to a finite sequence of state-constrained optimal control problems with additional states corresponding to the conditional probability of stopping at each possible terminal time. The proof of this correspondence relies on a new variation of the dynamic programming principle for state-constrained problems which avoids measurable selection. We emphasize that distribution constraints lead to novel and interesting mathematical problems on their own, but also demonstrate an application in mathematical finance to model-free superhedging with an outlook on volatility.