Correcting Newton--Côtes integrals by Lévy areas

Details Correcting Newton--Côtes integrals by Lévy areas Correcting Newton--Côtes integrals by Lévy areas

2006-01-23

arxiv.org/abs/math/0601544v2

Bernoulli 13, 3 (2007) pp. 695-711

Mathematics

Probability

Published at http://dx.doi.org/10.3150/07-BEJ6015 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statist

Scientific paper

10.3150/07-BEJ6015

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.

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