#### Abstract

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.