PDE Methods For Optimal Skorokhod Embeddings

Nassif Ghoussoub; Young-Heon Kim; Aaron Zeff Palmer

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arXiv · 1807.10347

PDE Methods For Optimal Skorokhod Embeddings

Nassif Ghoussoub·Young-Heon Kim·Aaron Zeff Palmer

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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.

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Nassif Ghoussoub, Young-Heon Kim, Aaron Zeff Palmer. 2018-07-26. PDE Methods For Optimal Skorokhod Embeddings. https://arxiv.org/abs/1807.10347

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