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Martingale Inequalities for the Maximum via Pathwise Arguments

  • Jan Obłój
  • Peter Spoida
  • Nizar Touzi
Part of the Lecture Notes in Mathematics book series (LNM, volume 2137)

Abstract

We study a class of martingale inequalities involving the running maximum process. They are derived from pathwise inequalities introduced by Henry-Labordère et al. (Ann. Appl. Probab., 2015 [arxiv:1203.6877v3]) and provide an upper bound on the expectation of a function of the running maximum in terms of marginal distributions at n intermediate time points. The class of inequalities is rich and we show that in general no inequality is uniformly sharp—for any two inequalities we specify martingales such that one or the other inequality is sharper. We use our inequalities to recover Doob’s L p inequalities. Further, for p = 1 we refine the known inequality and for p < 1 we obtain new inequalities.

Keywords

Classical Inequality Integrable Martingale Martingale Inequality Intermediate Time Point Continuous Local Martingale 
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.

Notes

Acknowledgements

The research has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 335421 (Jan Obłój) and (FP7/2007-2013)/ERC grant agreement no. 321111 (Nizar Touzi).

The author “Jan Obłój” is grateful to the Oxford-Man Institute of Quantitative Finance and St. John’s College in Oxford for their support. The author “Peter Spoida” gratefully acknowledges scholarships from the Oxford-Man Institute of Quantitative Finance and the DAAD. The author “Nizar Touzi” gratefully acknowledges the financial support from the Chair Financial Risks of the Risk Foundation sponsored by Société Générale, and the Chair Finance and Sustainable Development sponsored by EDF and CA-CIB.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK
  2. 2.Centre de Mathématiques AppliquéesEcole Polytechnique ParisPalaiseau CedexFrance

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