Eight or Nine Talks on Contemporary Optimal Transport Problems

7th talk : Fine Properties of the Optimal Skorokhod Embedding Problem.

5 juil. 2019, 10:50

50m

Salle de conférence (IRMA in Strasbourg)

Orateur

Marcel Nutz (Columbia U. New York)

Description

We study the problem of stopping a Brownian motion at a given distribution $\nu$ while optimizing a reward function that depends on the (possibly randomized) stopping time and the Brownian motion. Our first result establishes that the set $T(\nu )$ of stopping times embedding $\nu$ is weakly dense in the set $R(\nu )$ of randomized embeddings. In particular, the optimal Skorokhod embedding problem over $T(\nu )$ has the same value as the relaxed one over $R(\nu )$ when the reward function is semicontinuous, which parallels a fundamental result about Monge maps and Kantorovich couplings in optimal transport. A second part studies the dual optimization in the sense of linear programming. While existence of a dual solution failed in previous formulations, we introduce a relaxation of the dual problem and establish existence of solutions as well as absence of a duality gap, even for irregular reward functions. This leads to a monotonicity principle which complements the key theorem of Beiglbock, Cox and Huesmann. These results can be applied to characterize the geometry of optimal embeddings through a variational condition. (Joint work with Mathias Beiglbock and Florian Stebegg)s over the years.