Conference Agenda

In this talk we provide necessary and sufficient conditions for invariance of finite dimensional submanifolds for rough differential equations (RDEs) with values in a Banach space. Furthermore, we apply our findings to the particular situation of random RDEs driven by Q-Wiener processes and random RDEs driven by Q-fractional Brownian motion.

We show existence and uniqueness of backward differential equations that are jointly driven by Brownian martingales $B$ and a deterministic discontinuous rough path $W$ of $q$-variation for $q \in [1,2)$. Integration of jumps is in the geometric sense in the spirit of Marcus-type stochastic differential equations. The well-posedness is shown through a direct fix-point argument. By developing a comparison theorem, we can derive an apriori bound of the solution, which helps us attain a unique global solution of the differential equation. Furthermore, a connection to backward doubly SDE is established. If time permits, we will further discuss the continuity of the rough backward SDE solution with respect to the terminal condition and the driving rough noise in a Skorokhod-type norm.\\

This is a joint work with Dirk Becherer (HU Berlin).

First and higher order Euler schemes play a central role in the numerical approximation 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 of its scope, at least at first glance.

In this talk, we show the convergence of the first order Euler scheme for differential 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 convergence of the first order Euler scheme for stochastic differential equations driven by these stochastic processes.

The talk is based on joint work with A. L. Allan, C. Liu, and D. J. Prömel.

We prove a rough It\^o formula for path-dependent functionals of $\alpha$-H\"older continuous paths for $\alpha\in(0,1)$. Our approach combines the sewing lemma and a Taylor approximation in terms of path-dependent derivatives.