Optimal Transport of Stochastic Processes in Mathematical Finance

The project investigates optimal transportation of stochastic processes, that is, how to relate probability measures (i.e., the laws of stochastic processes) in a certain cost-optimal way. Amplified by computational advances, optimal transport theory has become an indispensable tool for far-reaching applications in non-parametric high dimensional statistics, image recognition, machine learning and calibration problems in mathematical finance. The project will advance research in the theory of optimal transport of stochastic processes. A special emphasis will be placed on applications to mathematical finance, such as martingale optimal transport and time-dynamic utility optimization problems. The derived theory will lay the basis for advances of numerical routines and novel indicators of model uncertainty, which will help to prepare decision-makers for worst-case scenarios. This project especially focuses on the adapted Wasserstein distance and entropic regularization of optimal transport, which is the method of choice for computing optimal transport problems in high dimensions. The first part of the project investigates a convex duality result and a first-order approximation result for time-dependent robust optimization problems and a characterization of their worst-case optimizers. These results are then applied to quantify model uncertainty in robust portfolio optimization and Davis pricing, machine learning of time-dependent distributions and hedging in financial markets. For each of these problems, closed-form expressions are derived and these are implemented numerically. The second part of the project investigates stability of Schroedinger potentials -- the dual optimizers of the entropic optimal transport problem -- via a strong compactness result and approximation techniques. Furthermore, necessary and sufficient conditions for existence of calibrated martingale measures with finite entropy are derived on the basis of a reference model and marginal distributions derived from market prices, as well as a characterization of the optimizers of the martingale optimal transport problem with entropic penalization. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.