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Martingale Inequalities for the Maximum via Pathwise Arguments

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In Memoriam Marc Yor - Séminaire de Probabilités XLVII

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,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.

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Notes

  1. 1.

    “This paper, alike the volume, is dedicated to the memory of Professor Marc Yor, who was my PhD supervisor and my mentor. I remain deeply indebted for all the help and advice he has given me over the years. I shall miss him greatly and will remember him as a brilliant but humble scholar, a passionate mathematician who was immensely generous in sharing his knowledge and ideas. Above all, he loved and enjoyed mathematics and I hope he would have liked this simple story about martingale inequalities.” Jan Obloj

  2. 2.

    This follows by applying the optional sampling theorem at the stopping time \(\inf \{t \geq 0: X_{t}\notin (0,n)\}\) and using dominated convergence theorem when letting \(n \rightarrow \infty \).

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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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Authors and Affiliations

  1. Mathematical Institute, University of Oxford, AWB ROQ Woodstock Rd, Oxford, OX2 6GG, UK

    Jan Obłój & Peter Spoida

  2. Centre de Mathématiques Appliquées, Ecole Polytechnique Paris, 91128, Palaiseau Cedex, France

    Nizar Touzi

Authors
  1. Jan Obłój
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  2. Peter Spoida
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  3. Nizar Touzi
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Corresponding author

Correspondence to Jan Obłój .

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Editors and Affiliations

  1. Laboratoire de Mathématiques, Université de Versailles-St Quentin, Versailles, France

    Catherine Donati-Martin

  2. Campus scientifique, IECL, Vandœuvre-lès-Nancy, France

    Antoine Lejay

  3. Laboratoire de Mathématiques, Université de Versailles-St Quentin, Versailles, France

    Alain Rouault

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Obłój, J., Spoida, P., Touzi, N. (2015). Martingale Inequalities for the Maximum via Pathwise Arguments. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) In Memoriam Marc Yor - Séminaire de Probabilités XLVII. Lecture Notes in Mathematics(), vol 2137. Springer, Cham. https://doi.org/10.1007/978-3-319-18585-9_11

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