PDE Methods For Optimal Skorokhod Embeddings

TL;DR

This paper develops PDE-based methods to solve optimal Skorokhod embedding problems, introducing a dual variational inequality framework and proving the existence of optimal solutions with free boundary characterization.

Contribution

It introduces a novel PDE and free boundary approach for optimal Skorokhod embedding, including a Sobolev class existence proof for dual problem optimizers.

Findings

Existence of Sobolev class optimizers for the dual problem.

Characterization of the free boundary where mass is dropped.

Provides a PDE-based constructive method as an alternative to recent probabilistic results.

Abstract

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 Beiglb\"ock, Cox, and Huesmann, and is a first step towards developing a general optimal mass transport theory involving mean field interactions and noise.