Examples of It\^o c\`adl\`ag rough paths
Chong Liu, David J. Pr\"omel
TL;DR
This paper constructs Itô càdlàg rough paths from semimartingales, Gaussian processes, and price paths, and extends Lyons-Victoir theorem to càdlàg paths of finite p-variation, broadening rough path theory.
Contribution
It introduces a dyadic approximation method for Itô integrals to establish càdlàg rough paths and extends the Lyons-Victoir theorem to càdlàg paths.
Findings
Existence of Itô càdlàg rough paths for general semimartingales.
Extension of Lyons-Victoir theorem to càdlàg paths of finite p-variation.
Applicability to Gaussian processes and non-negative price paths.
Abstract
Based on a dyadic approximation of It\^o integrals, we show the existence of It\^o c\`adl\`ag rough paths above general semimartingales, suitable Gaussian processes and non-negative typical price paths. Furthermore, Lyons-Victoir extension theorem for c\`adl\`ag paths is presented, stating that every c\`adl\`ag path of finite -variation can be lifted to a rough path.
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