Chain rules and \(p\)-variation (Q2773420)

The main result of this paper is a Young-Stieltjes integral representation of the composition \(\phi\circ f\) of a smooth function \(\phi\) that has a derivative satisfying a Lipschitz condition of order \(\alpha\in(0,1]\) and a ``rough'' function \(f\) with bounded \(p\)-variation for some \(p<1+\alpha\). This representation is analogous to the Itô formula for the composition of a smooth function and a semimartingale. If given \(\alpha\in(0,1]\), the \(p\)-variation of \(f\) is bounded for some \(p<2+\alpha\), and the second derivative of \(\phi\) is Lipschitz of order \(\alpha\), then a similar result holds for the extension of the Young-Stieltjes integral called the symmetric Young-Stieltjes integral.