Theory and Application of Adapted Wasserstein Distances

Stochastic processes are objects used to describe the evolution of certain states of the world that change in time and cannot be predicted with complete confidence. Prominent examples are stock prices, which are often described through stochastic processes emerging from the Black-Scholes model. Clearly, actual stock prices need not follow this simple model, and even if they do, the parameters used for the model may not be calibrated with perfect accuracy. Consequently, it may happen that predictions drawn from the model do not perfectly align with reality; for instance, realized fluctuations can be higher than predicted ones. It is therefore essential to understand the impact which changes in the model have on conclusions drawn from it. In order to adequately tackle this problem, one needs to specify a notion of distance between stochastic processes; i.e., a quantification of how close or far two different processes are. In certain simple cases this is an easy task: for instance, the Black-Scholes model depends only on two parameters (the drift and the fluctuations) and two different such models can be compared by the differences between their parameters. In general, however, stochastic processes are complicated objects and finding the correct notion of distance is more subtle. In fact, this subtlety already occurs in the much simpler setting of random objects that do not evolve in time. A satisfactory answer in that setting is possible relying on developments made in the theory of optimal transport in the recent decades. Unfortunately, when it comes to random objects that do involve in time (i.e., stochastic processes), this theory is intrinsically not suited. On the other hand, several variants of optimal transport suited to analyze the distance between stochastic processes have been studied in the recent years, and proven to be useful in several applications, particularly in mathematical finance. At the same time, a deeper general understanding inherent to the distance of stochastic processes is still lacking, and many fundamental questions are still open. In this proposal, we plan to further develop the theory of optimal transport for stochastic processes and systematically apply it to pressing questions in mathematical finance.

This project focused on developing advanced mathematical tools to better understand and manage uncertainty in complex, time-dependent systems, with a particular emphasis on financial applications. Central to this endeavor was the study of the adapted Wasserstein distance, a novel extension of optimal transport. Optimal transport is a mathematical theory that originated with the work of Gaspard Monge in 1781 and was later formalized by Leonid Kantorovich in the 1940s. The theory addresses the problem of determining the most efficient way to move resources from one location to another, minimizing transportation costs. In recent years, optimal transport has experienced a surge in development and applications across various fields, including economics and data science. Traditional optimal transport methods focus on comparing probability distributions without considering the temporal aspects of information flow. The adapted Wasserstein distance studied in this project accounts for the timing and sequence in which information becomes available, making it particularly suitable for analyzing dynamic models such as those found in financial markets. By incorporating this temporal dimension, the project established a robust framework for designing stable models that can more accurately model and assess risk in situations where classical methods may fall short. These theoretical advances were rigorously validated in several settings across different research papers. In particular, the project established qualitative bounds on the error arising from the use of incorrect probabilistic models for underlying financial systems. Remarkably, the resulting error estimates-derived from the general theory-were shown to be sharp in many cases, coinciding with those previously obtained through highly specialized, model-specific techniques. A further significant outcome of the project is the identification of the adapted Wasserstein distance as a geometric distance: the space of stochastic processes equipped with this distance forms a geodesic space. This insight opens the door to a calculus-based theory of optimal transport in dynamic settings, enabling the use of tools such as Riemannian calculus in the analysis of time-dependent systems. As a result of this project, 15 research papers were produced, 8 of which have been published in leading generalist journals both in pure and applied mathematics-such as Advances in Mathematics, Annals of Applied Probability, Journal of the European Mathematical Society, Mathematics of Operations Research, and SIAM Journal on Financial Mathematics. Additionally, the principal investigator shared the outcomes of the project through 18 invited talks at conferences, seminars, and workshops.