Abstract
We consider an optimal transport problem where the cost depends on the stopping time of Brownian motion from a given distribution to another. When the target measure is fixed, it is often called the optimal Skorokhod embedding problem in the literature, a popular topic in math finance. Under a monotonicity assumption on the cost, the optimal stopping time is given by the hitting time to a space-time barrier set. When the target measure is optimized under an upper bound constraint, we will show that the optimal barrier set leads us to the Stefan problem, a free boundary problem for the heat equation describing phase transition between water and ice. This is joint work with Young-Heon Kim at UBC.