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 ${\mathcal{C}}^{2m}$ function and $x,y$ are real Hölderian functions with index $\alpha >1/(2m+1)$ for all $m\in {\mathbb{N}}^{\ast }.$ 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.

Published at http://dx.doi.org/10.3150/07-BEJ6015 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

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