On the enhanced structures and rough paths integrals for time-changed paths

Abstract

In this paper, we study the effects of a discontinuous time change \( \phi \) on a continuous path \( x \). While rough integration in this setting poses significant challenges, we construct an enhancement for the time-changed path \( x \circ \phi \), providing a structured framework for discontinuous-time rough paths. We then apply our approach to Lévy processes, demonstrating its effectiveness in specific stochastic models, including the variance gamma process and the normal inverse Gaussian process.\\ A key focus of our work is the study of rough path integrals under time changes. We establish conditions under which the Lyons integral of the continuously time-changed rough path \( X \circ \phi \) coincides with the \(\phi\)-time-changed rough path integral driven by the original rough path \( X \). As a central result, we extend Terry Lyons' Itô formula to a time-changed Itô formula, offering a novel perspective on rough path calculus under time deformations.\\ These findings enhance the theoretical understanding of rough integration under time changes, with potential applications in stochastic analysis and mathematical finance.