Martingale inequalities and deterministic counterparts

Mathias Beiglböck (University of Vienna)
Marcel Nutz (Columbia University)


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.

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Pages: 1-15

Publication Date: October 16, 2014

DOI: 10.1214/EJP.v19-3270


  • 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. MR3098440
  • Bayer, Christian; Teichmann, Josef. The proof of Tchakaloff's theorem. Proc. Amer. Math. Soc. 134 (2006), no. 10, 3035--3040 (electronic). MR2231629
  • 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. MR3066985
  • 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. MR1839367
  • Burkholder, D. L. A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional. Ann. Probab. 9 (1981), no. 6, 997--1011. MR0632972
  • Burkholder, D. L. Boundary value problems and sharp inequalities for martingale transforms. Ann. Probab. 12 (1984), no. 3, 647--702. MR0744226
  • 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. MR0976214
  • 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. MR1108183
  • 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. MR1440921
  • 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). MR1859027
  • Cox, Alexander M. G.; Obłój, Jan. Robust pricing and hedging of double no-touch options. Finance Stoch. 15 (2011), no. 3, 573--605. MR2833100
  • 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. MR0789237
  • 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. MR3256817
  • 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. MR2762363
  • Kemperman, J. H. B. The general moment problem, a geometric approach. Ann. Math. Statist 39 1968 93--122. MR0247645
  • Obłój, Jan. The Skorokhod embedding problem and its offspring. Probab. Surv. 1 (2004), 321--390. MR2068476
  • 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. MR2234288
  • Osękowski, 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 MR2964297
  • Osȩkowski, Adam. Survey article: Bellman function method and sharp inequalities for martingales. Rocky Mountain J. Math. 43 (2013), no. 6, 1759--1823. MR3178444
  • Tchakaloff, Vladimir. Formules de cubatures mécaniques à coefficients non négatifs. (French) Bull. Sci. Math. (2) 81 1957 123--134. MR0094632

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