Analysis Seminar

Aaron PalmerUniversity of British Columbia
Stochastic Optimal Transport, Skorokhod Embeddings, and Free End-Time Optimal Control

Monday, November 18, 2019 - 2:30pm
Malott 406

The optimal transport problem provides a fundamental and quantitative way to measure the distance between probability distributions. Recently, it has been successfully used to analyze the evolutionary dynamics in physics and biology. Motivated by questions of pricing in financial mathematics, variants of optimal transport have been developed that prescribe the stochastic dynamics of an underlying process and optimize over stopping times that achieve a target probability distribution. The convex dual to these variants optimizes over supersolutions to Hamilton-Jacobi_Bellman variational inequalities, and provides a strategy for efficiently pricing financial assets. These novel problems have given a new perspective to a classical question of embedding measures in Brownian motion by stopping times posed by Skorokhod and can also be viewed in a greater context as an example of mean field games.

This talk will address interconnected results over the past few years that focus on the question of attainment of the dual problem and the structure of the solutions in stochastic optimal transport. We consider the cases of controlled deterministic dynamics (a reformulation of optimal transport), Brownian motion (a generalization of the Skorokhod embedding problem), and controlled diffusion processes (an introduction to mean field games with optimal stopping), for which we have developed a robust theory. This covers joint work with S Dweik, N Ghoussoub, and YH Kim.