Electronic Journal of Probability

Martingale inequalities and deterministic counterparts

Mathias Beiglböck and Marcel Nutz

Full-text: Open access


We study martingale inequalities from an analytic point of view and show that a general martingale inequality can be reduced to a pair of deterministic inequalities in a small number of variables. More precisely, the optimal bound in the martingale inequality is determined by a fixed point of a simple nonlinear operator involving a concave envelope. Our results yield an explanation for certain inequalities that arise in mathematical finance in the context of robust hedging.

Article information

Electron. J. Probab. Volume 19 (2014), paper no. 95, 15 pp.

Accepted: 16 October 2014
First available in Project Euclid: 4 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G42: Martingales with discrete parameter
Secondary: 49L20: Dynamic programming method

Martingale inequality Concave envelope Fixed point Robust hedging Tchakaloff's theorem

This work is licensed under a Creative Commons Attribution 3.0 License.


Beiglböck, Mathias; Nutz, Marcel. Martingale inequalities and deterministic counterparts. Electron. J. Probab. 19 (2014), paper no. 95, 15 pp. doi:10.1214/EJP.v19-3270. https://projecteuclid.org/euclid.ejp/1465065737.

Export citation


  • B. Acciaio, M. Beiglböck, F. Penkner, and W. Schachermayer. A model-free version of the fundamental theorem of asset pricing and the super-replication theorem. To appear in Math. Finance, 2013.
  • Acciaio, B.; Beiglböck, M.; Penkner, F.; Schachermayer, W.; Temme, J. A trajectorial interpretation of Doob's martingale inequalities. Ann. Appl. Probab. 23 (2013), no. 4, 1494–1505.
  • Bayer, Christian; Teichmann, Josef. The proof of Tchakaloff's theorem. Proc. Amer. Math. Soc. 134 (2006), no. 10, 3035–3040 (electronic).
  • Beiglböck, Mathias; Henry-Labordère, Pierre; Penkner, Friedrich. Model-independent bounds for option prices—a mass transport approach. Finance Stoch. 17 (2013), no. 3, 477–501.
  • M. Beiglböck and P. Siorpaes. Pathwise versions of the Burkholder–Davis–Gundy inequality. Preprint arXiv:1305.6188v1, 2013.
  • B. Bouchard and M. Nutz. Arbitrage and duality in nondominated discrete-time models. To appear in Ann. Appl. Probab., 2013.
  • Brown, Haydyn; Hobson, David; Rogers, L. C. G. Robust hedging of barrier options. Math. Finance 11 (2001), no. 3, 285–314.
  • Burkholder, D. L. A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional. Ann. Probab. 9 (1981), no. 6, 997–1011.
  • Burkholder, D. L. Boundary value problems and sharp inequalities for martingale transforms. Ann. Probab. 12 (1984), no. 3, 647–702.
  • Burkholder, Donald L. Sharp inequalities for martingales and stochastic integrals. Colloque Paul Lévy sur les Processus Stochastiques (Palaiseau, 1987). Astérisque No. 157-158 (1988), 75–94.
  • Burkholder, Donald L. Explorations in martingale theory and its applications. École d'Été de Probabilités de Saint-Flour XIX - 1989, 1–66, Lecture Notes in Math., 1464, Springer, Berlin, 1991.
  • Burkholder, Donald L. Sharp norm comparison of martingale maximal functions and stochastic integrals. Proceedings of the Norbert Wiener Centenary Congress, 1994 (East Lansing, MI, 1994), 343–358, Proc. Sympos. Appl. Math., 52, Amer. Math. Soc., Providence, RI, 1997.
  • Burkholder, Donald L. The best constant in the Davis inequality for the expectation of the martingale square function. Trans. Amer. Math. Soc. 354 (2002), no. 1, 91–105 (electronic).
  • Cox, Alexander M. G.; Obłój, Jan. Robust pricing and hedging of double no-touch options. Finance Stoch. 15 (2011), no. 3, 573–605.
  • Cox, David C. Some sharp martingale inequalities related to Doob's inequality. Inequalities in statistics and probability (Lincoln, Neb., 1982), 78–83, IMS Lecture Notes Monogr. Ser., 5, Inst. Math. Statist., Hayward, CA, 1984.
  • Dolinsky, Yan; Soner, H. Mete. Martingale optimal transport and robust hedging in continuous time. Probab. Theory Related Fields 160 (2014), no. 1-2, 391–427.
  • D. Hobson. Robust hedging of the lookback option. Finance Stoch., 2(4):329–347, 1998.
  • Hobson, David. The Skorokhod embedding problem and model-independent bounds for option prices. Paris-Princeton Lectures on Mathematical Finance 2010, 267–318, Lecture Notes in Math., 2003, Springer, Berlin, 2011.
  • Kemperman, J. H. B. The general moment problem, a geometric approach. Ann. Math. Statist 39 1968 93–122.
  • Obłój, Jan. The Skorokhod embedding problem and its offspring. Probab. Surv. 1 (2004), 321–390.
  • Obłój, Jan; Yor, Marc. On local martingale and its supremum: harmonic functions and beyond. From stochastic calculus to mathematical finance, 517–533, Springer, Berlin, 2006.
  • Osȩkowski, Adam. Sharp martingale and semimartingale inequalities. Instytut Matematyczny Polskiej Akademii Nauk. Monografie Matematyczne (New Series) [Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series)], 72. Birkhäuser/Springer Basel AG, Basel, 2012. xii+462 pp. ISBN: 978-3-0348-0369-4.
  • Osȩkowski, Adam. Survey article: Bellman function method and sharp inequalities for martingales. Rocky Mountain J. Math. 43 (2013), no. 6, 1759–1823.
  • Tchakaloff, Vladimir. Formules de cubatures mécaniques à coefficients non négatifs. (French) Bull. Sci. Math. (2) 81 1957 123–134.