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The Burkholder-Davis-Gundy Inequality for Enhanced Martingales

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Séminaire de Probabilités XLI

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 1934))

Abstract

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 Lépingle 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.

Partially supported by a Leverhulme Research Fellowship.

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Friz, P., Victoir, N. (2008). The Burkholder-Davis-Gundy Inequality for Enhanced Martingales. In: Donati-Martin, C., Émery, M., Rouault, A., Stricker, C. (eds) Séminaire de Probabilités XLI. Lecture Notes in Mathematics, vol 1934. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77913-1_20

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