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.