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The Burkholder-Davis-Gundy Inequality for Enhanced Martingales

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Part of the Lecture Notes in Mathematics book series (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.

Keywords

  • Moderate Function
  • Local Martingale
  • Piecewise Linear Approximation
  • Maximal Inequality
  • Rough Path

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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