250188 VO Selected topics in probability theory (2019S)

The theory of optimal transport (OT) has seen a tremendous development in the last 25 years with fascinating applications ranging from geometric and functional inequalities over PDEs and geometry to image analysis and statistics. In recent years, variants of the optimal transport problem with additional stochastic constraints have received increasing attention, e.g. martingale optimal transport (MOT) and causal/adapted optimal transport (COT).

The aim of this lecture is to serve as an introduction into the stochastic variants of the transport problem. After a quick recall of the classical OT problem we will start investigating its martingale variant which is motivated by intriguing questions from robust/model independent finance.
In the second part of the lecture we will complement the worst case point of view of MOT on robust finance by a ``local'' approach. This will naturally lead us to adapted versions of the OT problem, the COT, which we will explore in detail. Our discussion will be guided by examples from finance and stochastic analysis.