  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.
