Nicolas B. Victoir

The Burkholder-Davis-Gundy Inequality for Enhanced Martingales

Multi-dimensional continuous local martingales, enhanced with their stochastic area process, give rise to geometric rough paths with a.s. finite homogenous p-variation, p ∈ (2, 3). The purpose of this paper is to establish quantitative bounds of the homogenous p-variation norm in the form of a BDG inequality, combining old ides by Lepingle with recent geometric insights to rough path theory. Such BDG inequalities appear to be a useful addition to the toolbox available for the study of stochastic processes via rough path. Some applications are discussed, in particular Wong-Zakai type approximations.