Optimal transport theory and Wasserstein distances
The aim of the minicourse is to present recent advances related to optimal transport and its applications in mathematical finance, statistics, optimization and beyond. We will discuss basics of optimal transport (OT), its duality theory and properties of the induced Wasserstein distance on the space of probability measures. I will then introduce the martingale version of the problem (MOT) and discuss the rich additional structure resulting from the martingale constraint. I will touch on numerics for both problems, including the entropic relaxation of the OT. Finally, I will discuss how Wasserstein distances can be used to develop robust data-driven approaches to modelling in mathematical finance, machine learning and beyond.
- C Villani (2009) Optimal Transport: Springer:
- Filippo Santambrogio (2015), Optimal Transport for Applied Mathematicians, Springer
- Cedric Villani (2003), Topics in Optimal Transportation, AMS
There are several papers here and books but there is no single basic reference work I would recommend.
Instead, maybe it is helpful to point to a variety of papers which showcase the relevant topics (not all will be covered though):
- Beiglböck, M., Henry-Labordère, P. & Penkner, F. Model-independent bounds for option prices—a mass transport approach. Finance Stoch 17, 477–501 (2013)
- Beiglböck, M., Cox, A.M.G. & Huesmann, M. Optimal transport and Skorokhod embedding. Invent. math. 208, 327–400 (2017)
- Guo, G. & Obłój, J. Computational methods for martingale optimal transport problems. Ann. Appl. Probab. 29 (6), 3311 - 3347, (2019)
- Oblój, J & Wiesel, J. Robust estimation of superhedging prices. Ann. Statist. 49(1) 508 - 530, (2021)
- Bartl, D. Drapeau, S. Oblój, J. and Wiesel, J. Sensitivity analysis of Wasserstein distributionally robust optimization problems. Proc. R. Soc. A. 477 20210176 (2021)