Conference Agenda

Random outcomes are typically modeled using probabilities and expectations. In many situations, these probabilities are only partially known, resulting in an expectation that is imprecise and lies in an interval defined by the lower and upper expectations.

In this talk, we consider convex upper expectations on a path space of cadlag functions. We show that they have a one-to-one relation to penality functions. Based on this relation, we establish that upper convex expectations (without so-called fixed times of discontinuity) are uniquely determined by their finite-dimensional distributions. In certain Markovian cases, it even suffices to know their one-dimensional distributions. As canonical examples, we discuss upper convex expectations that arise form relaxed control rules and nonlinear Levy processes for which we also derive infinitesimal descriptions.

The signature is a transformation of a path, stochastic process or time series. In recent years, signatures have been very successfully applied in mathematical finance, in particular, to develop data driven methods and models for financial markets. At the very heart of these signature-based approaches are the universal approximation theorems for signatures, establishing that continuous functionals can be approximated arbitrarily well on compact sets by linear maps acting on signatures. However, in the context of mathematical finance the limitation to compact sets seriously restricts the scope of these signature-based approaches.

In this talk, we extend the theoretical foundation of signature-based approach by providing various global universal approximation theorems in the $L^p$ - sense with respect to the Wiener measure. Specifically, we demonstrate that $L^p$ - functionals on rough path space can be approximated globally in the $L^p$ - sense under the Wiener measure. This allows us to approximate solutions to stochastic differential equations driven by Brownian motions by signature-based models, i.e., by linear combinations of signature elements of Brownian motion.

A general diffusion semimartingale is a 1d continuous semimartingale that is also a regular strong Markov process. The class of general diffusion semimartingales is a natural generalization of the class of (weak) solutions to SDEs. A continuous semimartingale has the representation property if all local martingales w.r.t. its canonical filtration have an integral representation w.r.t. its continuous local martingale part. The representation property is closely related to market completeness. We show that the representation property holds for a general diffusion semimartingale if and only if its scale function is (locally) absolutely continuous in the interior of the state space. Surprisingly, this means that not all general diffusion semimartingales possess the representation property, which is in contrast to the SDE case. Furthermore, it follows that the laws of general diffusion semimartingales with absolutely continuous scale functions are extreme points of their semimartingale problems. We construct a general diffusion semimartingale whose law is not an extreme point of its semimartingale problem. This contributes to the solution of the problems posed by Jacod and Yor and by Stroock and Yor on the extremality of strong Markov solutions (to martingale problems).

Weak optimal transport is a generalization of optimal transport that allows for costs that cover many optimization problems outside the realm of classic optimal transport, while still permitting the same results concerning primal existence and weak duality as in the classical case. However, the question of dual attainment has remained open so far. Our main contribution is to establish the existence of dual optimizers, thus extending the fundamental theorem of optimal transport to the weak transport case.

This is based on joint work with Mathias Beiglböck, Gudmund Pammer und Lorenz Riess.