Osaka Journal of Mathematics

Sharp maximal estimates for BMO martingales

Adam Osȩkowski

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We introduce a method which can be used to study maximal inequalities for martingales of bounded mean oscillation. As an application, we establish sharp $\Phi$-inequalities and tail inequalities for the one-sided maximal function of a BMO martingale. The results can be regarded as BMO counterparts of the classical maximal estimates of Doob.

Article information

Osaka J. Math., Volume 52, Number 4 (2015), 1125-1143.

First available in Project Euclid: 18 November 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G42: Martingales with discrete parameter
Secondary: 60G44: Martingales with continuous parameter


Osȩkowski, Adam. Sharp maximal estimates for BMO martingales. Osaka J. Math. 52 (2015), no. 4, 1125--1143.

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  • J. Bion-Nadal: Dynamic risk measures: time consistency and risk measures from BMO martingales, Finance Stoch. 12 (2008), 219–244.
  • D.L. Burkholder: Explorations in martingale theory and its applications; in École d'Été de Probabilités de Saint-Flour XIX–-1989, Lecture Notes in Math. 1464, Springer, Berlin, 1991, 1–66.
  • F. Delbaen, P. Monat, W. Schachermayer, M. Schweizer and C. Stricker: Weighted norm inequalities and hedging in incomplete markets, Finance Stoch. 1 (1997), 181–227.
  • \begingroup C. Fefferman: Characterizations of bounded mean oscillation, Bull. Amer. Math. Soc. 77 (1971), 587–588. \endgroup
  • S. Geiss: Weighted BMO and discrete time hedging within the Black–Scholes model, Probab. Theory Related Fields 132 (2005), 13–38.
  • A.M. Garsia: Martingale Inequalities: Seminar Notes on Recent Progress, W.A. Benjamin, Inc., Reading, MA, 1973.
  • R.K. Getoor and M.J. Sharpe: Conformal martingales, Invent. Math. 16 (1972), 271–308.
  • F. John and L. Nirenberg: On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415–426.
  • N. Kazamaki: Continuous Exponential Martingales and BMO, Lecture Notes in Mathematics 1579, Springer, Berlin, 1994.
  • A.A. Korenovskiĭ, The connection between mean oscillations and exact exponents of summability of functions, Math. USSR-Sb. 71 (1992), 561–567.
  • P.A. Meyer: Un cours sur les intégrales stochastiques; in Séminaire de Probabilités, X (Seconde partie: Théorie des intégrales stochastiques, Univ. Strasbourg, Strasbourg, année universitaire 1974/1975), Lecture Notes in Math. 511, Springer, Berlin, 1976, 245–400.
  • A. Osękowski: Sharp Martingale and Semimartingale Inequalities, Monografie Matematyczne (New Series) 72, Birkhäuser/Springer Basel AG, Basel, 2012.
  • L. Slavin and V. Vasyunin: Sharp results in the integral-form John–Nirenberg inequality, Trans. Amer. Math. Soc. 363 (2011), 4135–4169.
  • V. Vasyunin: The sharp constant in the John–Nirenberg inequality, preprint POMI 20 (2003).
  • V. Vasyunin and A. Volberg: harp constants in the classical weak form of the John–Nirenberg inequality, Proc. Lond. Math. Soc. (3) 108 (2014), 1417–1434.