On Young Systems

In this article, we study differential equations driven by continuous paths with with bounded $p$-variation for $1\le p<2$ (Young systems). The most important class of examples of theses equations is given by stochastic differential equations driven by fractional Brownian motion with Hurst index $H

\frac{1}{2}$. We give a formula type It^o-Kunita-Ventzel and a substitution formula adapted to Young integral. It allows us to give necessary conditions for existence of conserved quantities and symmetries of Young systems. We give a formula for the composition of two flows associated to Young sistems and study the Cauchy problem for Young partial differential equations.