Correcting Newton--Côtes integrals by Lévy areas

Abstract

In this note we introduce the notion of Newton--Côtes functionals corrected by Lévy 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 Hölderian functions with index $α>1/(2m+1)$ for all $m\in {\mathbb{N}}^*.$ We show that this concept extends the Newton--Côtes 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.