The Annals of Applied Probability

A trajectorial interpretation of Doob’s martingale inequalities

B. Acciaio, M. Beiglböck, F. Penkner, W. Schachermayer, and J. Temme

Full-text: Open access


We present a unified approach to Doob’s $L^{p}$ maximal inequalities for $1\leq p<\infty$. The novelty of our method is that these martingale inequalities are obtained as consequences of elementary deterministic counterparts. The latter have a natural interpretation in terms of robust hedging. Moreover, our deterministic inequalities lead to new versions of Doob’s maximal inequalities. These are best possible in the sense that equality is attained by properly chosen martingales.

Article information

Ann. Appl. Probab., Volume 23, Number 4 (2013), 1494-1505.

First available in Project Euclid: 21 June 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G42: Martingales with discrete parameter 60G44: Martingales with continuous parameter
Secondary: 91G20: Derivative securities

Doob maximal inequalities martingale inequalities pathwise hedging


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. doi:10.1214/12-AAP878.

Export citation


  • [1] Azéma, J. and Yor, M. (1979). Une solution simple au problème de Skorokhod. In Séminaire de Probabilités, XIII (Univ. Strasbourg, Strasbourg, 1977/78). Lecture Notes in Math. 721 90–115. Springer, Berlin.
  • [2] Bachelier, L. (1900). Théorie de la spéculation. Ann. Sci. École Norm. Sup. (3) 17 21–86.
  • [3] Bichteler, K. (1981). Stochastic integration and $L^{p}$-theory of semimartingales. Ann. Probab. 9 49–89.
  • [4] Brown, H., Hobson, D. and Rogers, L. C. G. (2001). Robust hedging of barrier options. Math. Finance 11 285–314.
  • [5] Burkholder, D. L. (1991). Explorations in martingale theory and its applications. In École D’Été de Probabilités de Saint-Flour XIX—1989. Lecture Notes in Math. 1464 1–66. Springer, Berlin.
  • [6] Cox, A. M. G. and Obłój, J. (2009). Robust pricing and hedging of double no-touch options. Available at
  • [7] Cox, D. C. (1984). Some sharp martingale inequalities related to Doob’s inequality. In Inequalities in Statistics and Probability (Lincoln, Neb., 1982). Institute of Mathematical Statistics Lecture Notes—Monograph Series 5 78–83. IMS, Hayward, CA.
  • [8] Doob, J. L. (1990). Stochastic Processes. Wiley, New York.
  • [9] Dubins, L. E. and Gilat, D. (1978). On the distribution of maxima of martingales. Proc. Amer. Math. Soc. 68 337–338.
  • [10] Gilat, D. (1986). The best bound in the $L\operatorname{log}L$ inequality of Hardy and Littlewood and its martingale counterpart. Proc. Amer. Math. Soc. 97 429–436.
  • [11] Graversen, S. E. and Peškir, G. (1998). Optimal stopping in the $L\log L$-inequality of Hardy and Littlewood. Bull. Lond. Math. Soc. 30 171–181.
  • [12] Henry-Labordère, P. Personal communication.
  • [13] Hobson, D. (1998). The maximum maximum of a martingale. In Séminaire de Probabilités, XXXII. Lecture Notes in Math. 1686 250–263. Springer, Berlin.
  • [14] Hobson, D. (2011). The Skorokhod embedding problem and model-independent bounds for option prices. In Paris–Princeton Lectures on Mathematical Finance 2010. Lecture Notes in Math. 2003 267–318. Springer, Berlin.
  • [15] Hobson, D. and Klimmek, M. (2011). Model independent hedging strategies for variance swaps. Preprint.
  • [16] Karandikar, R. L. (1995). On pathwise stochastic integration. Stochastic Process. Appl. 57 11–18.
  • [17] Obłój, J. and Yor, M. (2006). On local martingale and its supremum: Harmonic functions and beyond. In From Stochastic Calculus to Mathematical Finance (Y. Kabanov, R. Lipster and J. Stoyanov, eds.) 517–533. Springer, Berlin.
  • [18] Peškir, G. (1998). The best Doob-type bounds for the maximum of Brownian paths. In High Dimensional Probability (Oberwolfach, 1996). Progress in Probability 43 287–296. Birkhäuser, Basel.
  • [19] Shreve, S. (2004). Stochastic Calculus for Finance II: Continuous-Time Models. Springer, New York.