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)


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.


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.



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.


  1. 1.
    B. Acciaio, M. Beiglböck, F. Penkner, W. Schachermayer, J. Temme, A trajectorial interpretation of Doob’s martingale inequalities. Ann. Appl. Probab. 23(4), 1494–1505 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    J. Azéma, M. Yor, Le problème de Skorokhod: compléments à “Une solution simple au problème de Skorokhod”, in Séminaire de Probabilités, XIII (Univ. Strasbourg, Strasbourg, 1977/1978). Lecture Notes in Mathematics, vol.721 (Springer, Berlin, 1979), pp. 625–633Google Scholar
  3. 3.
    J. Azéma, M. Yor, Une solution simple au problème de Skorokhod, in Séminaire de Probabilités, XIII (Univ. Strasbourg, Strasbourg, 1977/1978), ed. by C. Dellacherie, P.-A. Meyer, M. Weil. Lecture Notes in Mathematics, vol. 721 (Springer, Berlin, 1979), pp. 90–115Google Scholar
  4. 4.
    J. Azéma, R.F. Gundy, M. Yor, Sur l’intégrabilité uniforme des martingales continues, in Seminar on Probability, XIV (Paris, 1978/1979) (French). Lecture Notes in Mathematics, vol. 784 (Springer, Berlin, 1980), pp. 53–61Google Scholar
  5. 5.
    M. Beiglböck, M. Nutz, Martingale inequalities and deterministic counterparts. Electron. J. Probab. 19(95), 1–15 (2014)MathSciNetGoogle Scholar
  6. 6.
    M. Beiglböck, P. Siorpaes, Pathwise versions of the Burkholder-Davis-Gundy inequality (2013) [, 1305.6188v1]Google Scholar
  7. 7.
    D. Blackwell, L.E. Dubins, A converse to the dominated convergence theorem. Ill. J. Math. 7, 508–514 (1963)zbMATHMathSciNetGoogle Scholar
  8. 8.
    B. Bouchard, M. Nutz, Arbitrage and duality in nondominated discrete-time models. Ann. Appl. Probab. 25(2), 823–859 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    H. Brown, D. Hobson, L.C.G. Rogers, The maximum maximum of a martingale constrained by an intermediate law. Probab. Theory Relat. Fields 119(4), 558–578 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    L. Carraro, N. El Karoui, J. Obłój, On Azéma-Yor processes, their optimal properties and the Bachelier-Drawdown equation. Ann. Probab. 40(1), 372–400 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    A.M.G. Cox, G. Peskir, Embedding laws in diffusions by functions of time. Ann. Probab. (2015, to appear) [arXiv:1201.5321v3]Google Scholar
  12. 12.
    A.M.G. Cox, J. Wang, Root’s barrier: construction, optimality and applications to variance options. Ann. Appl. Probab. 23(3), 859–894 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    A.M.G. Cox, D. Hobson, J. Obłój, Pathwise inequalities for local time: applications to Skorokhod embeddings and optimal stopping. Ann. Appl. Probab. 18(5), 1870–1896 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    L.E. Dubins, D. Gilat, On the distribution of maxima of martingales. Proc. Am. Math. Soc. 68(3), 337–338 (1978)zbMATHMathSciNetGoogle Scholar
  15. 15.
    P. Henry-Labordère, J. Obłój, P. Spoida, N. Touzi, The maximum maximum of a martingale with given n marginals. Ann. Appl. Probab. (2015, to appear) [arxiv:1203.6877v3]Google Scholar
  16. 16.
    D.G. Hobson, Robust hedging of the lookback option. Finance Stochast. 2(4), 329–347 (1998)zbMATHCrossRefGoogle Scholar
  17. 17.
    D.B. Madan, M. Yor, Making Markov martingales meet marginals: with explicit constructions. Bernoulli 8(4), 509–536 (2002)zbMATHMathSciNetGoogle Scholar
  18. 18.
    J. Obłój, P. Spoida, An iterated Azéma-Yor type embedding for finitely many marginals (2013) [, 1304.0368v2]Google Scholar
  19. 19.
    J. Obłój, M. Yor, On local martingale and its supremum: harmonic functions and beyond, in From Stochastic Calculus to Mathematical Finance (Springer, Heidelberg, 2006), pp. 517–534Google Scholar
  20. 20.
    A. Osȩkowski, Sharp Martingale and Semimartingale Inequalities (Birkhäuser, New York, 2012)CrossRefGoogle Scholar
  21. 21.
    G. Peskir, Optimal stopping of the maximum process: the maximality principle. Ann. Probab. 26(4), 1614–1640 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    L.C.G. Rogers, A guided tour through excursions. Bull. Lond. Math. Soc. 21(4), 305–341 (1989)zbMATHCrossRefGoogle Scholar

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

Personalised recommendations