Motivated by Knightian uncertainty, S. Peng introduced his celebrated G–Brownian motion. Intuitively speaking, it corresponds to a dynamic worst case expectation in a model where volatility is uncertain but postulated to take values in a bounded interval. Natural extensions of the G–Brownian motion are nonlinear diffusions, whose volatility (and drift) takes values in a random set that is allowed to depend on the canonical process in a Markovian way. Nonlinear diffusions satisfy the dynamic programming principle, which entails the semigroup property of a corresponding family of sublinear operators. In this talk, we discuss regularity properties of these semigroups that allow us to relate them to evolution equations. In particular, we explain a novel type of smoothing property and a stochastic representation result for general sublinear semigroups with pointwise generators of Hamilton-Jacobi-Bellman type. Latter also implies a unique characterization theorem for such semigroups.

The talk is based on joint work with Lars Niemann (University of Freiburg).

In classical optimal transport, the contributions of Benamou-Brenier and Mc- Cann regarding the time-dependent version of the problem are cornerstones of the field and form the basis for a variety of applications in other mathematical areas.

Stretched Brownian motion provides an analogue for the martingale version of this problem. We provide a characterization in terms of gradients of convex functions, similar to the characterization of optimizers in the classical transport problem for quadratic distance cost.

Based on joint work with Julio Backhoff-Veraguas, Walter Schachermayer and Bertram Tschiderer.

We study a planar stochastic differential equation with additive noise for which the rotational speed is of the form ρ(R) where R is the radial part.

We investigate how phenomena like strong or weak synchronization, existence of a pullback or a point attractor and strong completeness of the associated random dynamical system depend on the function ρ. This is joint work (in progress) with Maximilian Engel and Dennis Chemnitz (FU Berlin).

First and higher order Euler schemes play a central role in the numerical ap- proximations of stochastic differential equations. While the pathwise convergence of higher order Euler schemes can be adequately explained by rough path theory, the first order Euler scheme seems to be outside its scope, at least at first glance.

In this talk, we show the convergence of the first order Euler scheme for differen- tial equations driven by càdlàg rough paths satisfying a suitable criterion, namely the so-called Property (RIE), along time discretizations with mesh size going to zero. This property is verified for almost all sample paths of various stochastic processes and time discretizations. Consequently, we obtain the pathwise conver- gence of the first order Euler scheme for rough stochastic differential equations driven by these stochastic processes.

The talk is based on joint work with A. L. Allan, A. P. Kwossek, and C. Liu.