Martingale Inequalities for the Maximum via Pathwise Arguments
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.
KeywordsClassical Inequality Integrable Martingale Martingale Inequality Intermediate Time Point Continuous Local Martingale
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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