Duality Analysis for Optimal Skorokhod Embeddings

Aaron Palmer (University of British Columbia)
Monday, October 15, 2018 - 2:30pm to 3:30pm
PDL C-401

Abstract:  The Skorokhod embedding problem asks in what ways can a probability measure be realized as a stopped Brownian motion.  Many of the known solutions optimize a restricted optimal transportation problem, where the expectation of a cost process is minimized over embedding stopping times. These problems have found important applications in mathematical finance, and are closely related to a larger class of mean field optimal stopping problems.  We address the fundamental question of the existence of a dual potential, i.e. marginal value, associated to the embedding constraint. We prove uniform estimates on the Sobolev norm of potentials based on the dual problem of finding the optimal conditional value, which solves a  quasivariational inequality, i.e. free boundary PDE.   We also give new conditions on the cost process where upon the optimal stopping time becomes the well known Root and Rost embeddings, given by the hitting time of a barrier that coincides with the free boundary of the dual problem.  This work is based on a paper titled "PDE methods for Skorokhod Embeddings" coathered by Nassif Ghoussoub and Young-Heon Kim and recently posted on arXiv.

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