Optimal stopping problems can be viewed as a problem to calculate
the space and time dependent value function, which solves a nonlinear, possibly
non-smooth and degenerate, parabolic PDE known as an Hamilton-Jacobi-Bellman
(HJB) equation. These equations are well understood using the theory of
viscosity solutions, and the optimal stopping policy can be retrieved when
there is some regularity and non-degeneracy of solution.
The HJB equation is commonly derived from a dynamic programming principle
(DPP). After adding a probabilistic constraint, the optimal policies no longer
satisfy this DPP. Instead, we can reach the HJB equation by a method related
to optimal transportation, and recover a DPP for a Lagrangian-relaxation of
the problem. The resulting HJB equation remains coupled through the constraint
with the optimal policy (and another parabolic PDE). Solving the HJB and
recovery of the optimal stopping policy is aided by considering the
``piecewise-monotonic’' structure of the stopping set.