Theory and Application of Adapted Wasserstein Distances

Wider research context
Wasserstein distance induces a natural Riemannian structure for the probabilities on the Euclidean space.
This insight of classical transport theory is fundamental for tremendous applications in different fields of pure and applied mathematics. Recent contributions of the PI and his collaborators suggest that an appropriate probabilistic variant, the adapted Wasserstein distance AW, can play a similar role for the class stochastic processes. The decisive extension compared to the classical Wasserstein distance is that AW appropriately accounts for the flow of information encoded by the underlying filtrations.

Research questions
While classical mathematical finance provides numerous methods to find prices and hedging strategies for financial derivatives, to obtain optimal trading strategies for problems of utility maximization, etc. it is much less clear how stable these results are: e.g. even a highly skilled modeller can only approximately describe reality. It is critical to assess the consequences of the resulting error accurately. In a similar direction, suppose that different experts have different suggestions on how to model a financial market and provide different suggestions accordingly. How can we consistently reconcile their models and resulting financial strategies?

In contrast to other topologies for stochastic processes, operations such as utility maximization or pricing and hedging are continuous w.r.t. AW. Fully developed, adapted transport provides an ideal tool for the highlighted challenges.

Approach
We have recently established that adapted Wasserstein distance is an appropriate tool for stability mathematical finance as well as that the space (Proc, AW) of stochastic processes carries the appropriate geometric structure that allows us to deal with the challenges alluded to above. Specifically (Proc, AW) is a geodesic space, isometric to a classical Wasserstein space, and martingale-models form a closed geodesically convex subspace. To carry out our program it is necessary to transfer and extend results from classical transport to the probabilistic setup. Specifically, we need to establish counterparts to classical achievements such as Brenier’s theorem, McCann’s displacement interpolation and barycenters in the sense of Agueh and Carlier.

Level of originality
While classical mathematical finance mostly relies on familiar tools from stochastic analysis and optimal control, the proposed project takes a fundamentally different stance. Rather than describing processes from within through SDEs, semi-martingale characteristics, etc. adapted transport takes stochastic processes as objects that can be viewed from an external perspective. As optimal transport does with probability measures, adapted transport enables us to treat stochastic processes as points in a geometric space. This shift of perspective is key to the above challenges.

Primary researchers involved
D. Bartl, M. Beiglböck