PDE methods for optimal Skorokhod embeddings

We consider cost minimizing stopping time solutions to Skorokhod embedding problems, which deal with transporting a source probability measure to a given target measure through a stopped Brownian process. PDEs and a free boundary problem approach are used to address the problem in general dimensions with space–time inhomogeneous costs given by Lagrangian integrals along the paths. We introduce an Eulerian—mass flow—formulation of the problem, whose dual is given by Hamilton–Jacobi–Bellman type variational inequalities. Our key result is the existence (in a Sobolev class) of optimizers for this new dual problem, which in turn determines a free boundary, where the optimal Skorokhod transport drops the mass in space–time. This complements and provides a constructive PDE alternative to recent results of Beiglböck, Cox, and Huesmann, and is a first step towards developing a general optimal mass transport theory involving mean field interactions and noise.