Regularity results for non-linear Young equations and applications

In this paper we provide sufficient conditions which ensure that the non-linear equation $dy(t)=Ay(t)dt+sigma(y(t))dx(t)$, $tin(0,T]$, with $y(0)=psi$ and $A$ being an unbounded operator, admits a unique mild solution such that $y(t)in D(A)$ for any $tin (0,T]$, and we compute the blow-up rate of the norm of $y(t)$ as $t ightarrow 0^+$. We stress that the regularity of $y$ is independent of the smoothness of the initial datum $psi$, which in general does not belong to $D(A)$. As a consequence we get an integral representation of the mild solution $y$ which allows us to prove a chain rule formula for smooth functions of $y$.