In some applications including filtering theory, one encounters stochastic differential equations of the form \begin{equation}\label{eqn:1} dY = \sigma (Y)dB + f(Y)dX , t\in[0,T], Y_0=x\end{equation} of unknown $Y_t$ in $\mathbb{R}^d$, where

• $B_t$ is a multidimensional Brownian motion;
• $X_t$ is an independent source of noise, which we assume is known, and for which we can therefore fix a realization.

To solve ($\ref{eqn:1}$), it is possible to introduce a deterministic formulation, using the rough paths theory of Lyons/Gubinelli (one also fixes a realization of $B_t$). Although it has certain advantages, this method requires very regular coefficients ($C^3$), in contrast to the usual stochastic assumption that $\sigma$ is Lipshitz (or only bounded). After a brief introduction to the theory of rough paths, I will explain how one can deal with ($\ref{eqn:1}$) with minimal assumptions on the coefficients. The key idea is to introduce a hybrid formulation for ($\ref{eqn:1}$) using a "rough semi-martingale" concept. I will also mention its usefulness in the context of mean-field rough stochastic differential equations and (quenched) propagation of chaos.

This work is the result of a collaboration with Peter Friz and Khoa Lê (TU Berlin).