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.

The event “Workshop Junior Researchers in Stochastic Optimal Control”, is taking place on August 31 and September 01 in Berlin.

This workshop is funded by the IRTG and organized by IRTG-members from Berlin and Oxford. We are pleased to announce our impressive lineup of keynote speakers

• Roxana Dumitrescu (King’s College London),
• Johannes Muhle-Karbe (Imperial College London),
• David Siska (University of Edinburgh),
• Peter Tankov (ENSAE Paris),
• Gianmario Tessitore (University Milano-Bicocca).

Mean-field games (MFGs) facilitate otherwise intractable learning in game-theoretical equi- librium problems with many agents. The general approach is to analyze agents via their distribution, which allows to abstract multi-agent stochastic dynamical systems into a single representative agent and the mass of all other agents. We begin by focusing on discrete-time models. We show that fixed point iteration is insufficient for solving MFGs in general. We then present some algorithms based on entropy regularization, dynamic programming and reinforcement learning. In the second half of the talk, we in- corporate graph structure into the model via graphon limits. A numerical reduction of graphon MFGs to standard MFGs allows application of algorithms for general MFGs. Accordingly, we demonstrate intuitive numerical results for exemplary investment and epidemics control problems. Lastly, we briefly touch upon some extensions to other types of graphs. Overall, we obtain a scalable framework for solving large-scale dynamic game-theoretic problems.