X-OT: on Variants of Optimal Transport Problem and Understanding Model Robustness

Across vast array of applications, mathematics is used to build models which create pathways from inputs to outputs. These models can often be seen as probability measures: discrete (empirical measures over a given data set) or continuous (resulting from an SDE), over finite-dimensional spaces or over pathspaces. The theory of Optimal Transport (OT) offers powerful fully non-parametric tools to measure distances between probability measures, trace geodesics in the space of probability measures, project onto its subsets. In this talk, I will survey some recent advancements that leverage OT tools and intuition, to describe and manage models, helping with selecting/calibrating models and quantifying model uncertainty. I will use questions from mathematical finance as my motivating examples while focusing on providing an overview of the field with its novel mathematical contributions, including several variants of the classical OT paradigm, and ongoing challenges. In particular, I will discuss robust pricing and hedging and its link to Martingale-OT, non-parametric calibration via Semimartingale-OT, and Wasserstein distributionally robust optimization and the resulting non-parametric Greeks and risk measurements. I will also mention some applications in statistics and machine learning, including adversarial robustness of DNNs.
The talk is based on works with many collaborators, including: D. Bartl, S. Drapeau, S. Eckstein, G. Guo, I. Guo, Y. Jiang, B. Joseph, T. Lim, G. Loeper, S. Wang and J. Wiesel.

About Jan Obłój

Jan Obloj is a Professor of Mathematics at the University of Oxford's Mathematical Institute, an Official Fellow of St John's College Oxford and a member of the Oxford-Man Institute of Quantitative Finance. Before coming to Oxford, he was a Marie Curie Post-Doctoral Fellow at Imperial College London and he holds PhD from University Paris VI and Warsaw University. He is the current President of the Bachelier Finance Society and is Fellow of the Institute of Mathematical Statistics. He has a general interest in mathematics of randomness. Most of his research sits at the crossroads of various fields, including: probability theory, statistics, mathematical finance, operations research, optimal transportation and data science. His main focus is on robustness of the modelling pathways from input out outputs, ways to understand and quantify it and his research spans the spectrum from theoretical foundations of robust pricing and hedging paradigm in mathematical finance, to practical questions of building fast generic ways to approximate adversarial robustness of deep neural networks.