Probability Seminar: Aaron Palmer

Optimal transport (OT) problems, initiated by G. Monge 200 years ago and refined by L. Kantorovich in the 1940’s, provide --among other things-- a quantitative way for measuring correlations between probability distributions.  Martingale optimal transports (MOT) and their Skorokhod embeddings in Brownian motion lead to optimal transport by stopping time (OTST) problems. These are important variations on OT, with applications to financial mathematics and probability theory. In OTST, one specifies a stochastic state process and a cost, and minimizes the expected cost over stopping times with a given state distribution.

In this talk, I will focus on the case where the state process is d-dimensional Brownian motion and the cost is given by the Euclidean distance. I will discuss new results involving dual variational principles, their attainment, as well as characterizations of the optimal stopping times as a hitting time of barriers given by solutions of corresponding obstacle problems. I will also discuss how these results generalize for other processes and costs and relate them to other aspects of probability theory. This talk is based on joint work with Nassif Ghoussoub, Young-Heon Kim and Tongseok Lim.