\(p\)-variation and integration of sample functions of stochastic processes (Q2769699)

This article is devoted to properties of some ``left Young'' (LY) integral and of its right (RY) analogue, which are defined by means of some refined Riemann-Stieltjes integral studied by \textit{L. C. Young} [Acta Math. 67, 251-282 (1936; Zbl 0016.10404)]. The main case is when \(f\) has finite \(p\)-variation on \([a,b]\) and \(g\) has finite \(q\)-variation on \([a,b]\), with \({1/p}+{1/q}>1\); then, under some condition on the common jumps of \(f\) and \(g\), the LY-integral LY-\(\int^b_afdg\) exists and is obtained as \(\lim_n \sum^n_{j=1} f(x^n_{j-1}) (g(x_j^n)- g(x^n_{j-1}))\), for nested partitions \((x_j^n)\) with mesh going to 0. These LY- and RY-integrals are bilinear, and are shown to obey the Charles additivity formula, an integration by parts formula, the substitution rule, and mainly to satisfy a chain rule formula expressing \(\Phi\circ f\) as \([\text{LY-} \int\Phi_0' f\times df+\) sums of jump terms] (remaining somewhat of Itô's formula, but without second derivatives).NEWLINENEWLINEFor the entire collection see [Zbl 0968.00043].