In this note we introduce the notion of Newton--C\^{ o}tes functionals corrected by L\'{ e}vy areas, which enables us to consider integrals of the type $\int f(y) \mathrm{ d}x,$ where $f$ is a ${ \mathscr{ C}}^{ 2m}$ function and $x,y$ are real H\"{ o}lderian functions with index $\alpha>1/(2m+1)$ for all $m\in { \mathbb{ N}}^*.$ We show that this concept extends the Newton--C\^{ o}tes functional introduced in Gradinaru et al., to a larger class of integrands. Then we give a theorem of existence and uniqueness for differential equations driven by $x$, interpreted using the symmetric Russo--Vallois integral.
Categories: math.PR