Electronic Communications in Probability

Maximal weak-type inequality for stochastic integrals

Adam Osekowski

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Assume that $X$ is a real-valued martingale starting from $0$, $H$ is a predictable process with values in $[-1,1]$ and $Y$ is the stochastic integral of $H$ with respect to $X$. The paper contains the proofs of the following sharp weak-type estimates.  (i) If $X$ has continuous paths, then $$ \mathbb{P}\left(\sup_{t\geq 0}|Y_t|\geq 1\right)\leq 2\mathbb{E} \sup_{t\geq 0}X_t.$$<br />(ii) If $X$ is arbitrary, then$$  \mathbb{P}\left(\sup_{t\geq 0}|Y_t|\geq 1\right)\leq 3.477977\ldots\mathbb{E} \sup_{t\geq 0}X_t.$$The proofs rest on Burkholder's method and exploits the existence of certain special functions possessing appropriate concavity and majorization properties.

Article information

Electron. Commun. Probab., Volume 19 (2014), paper no. 25, 13 pp.

Accepted: 4 May 2014
First available in Project Euclid: 7 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Martingale maximal weak type inequality best constant

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Osekowski, Adam. Maximal weak-type inequality for stochastic integrals. Electron. Commun. Probab. 19 (2014), paper no. 25, 13 pp. doi:10.1214/ECP.v19-3151. https://projecteuclid.org/euclid.ecp/1465316727

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  • Bañuelos, Rodrigo; Bielaszewski, Adam; Bogdan, Krzysztof. Fourier multipliers for non-symmetric Lévy processes. Marcinkiewicz centenary volume, 9–25, Banach Center Publ., 95, Polish Acad. Sci. Inst. Math., Warsaw, 2011.
  • Bañuelos, Rodrigo; Wang, Gang. Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transforms. Duke Math. J. 80 (1995), no. 3, 575–600.
  • Bichteler, Klaus. Stochastic integration and $L^{p}$-theory of semimartingales. Ann. Probab. 9 (1981), no. 1, 49–89.
  • 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. Differential subordination of harmonic functions and martingales. Harmonic analysis and partial differential equations (El Escorial, 1987), 1–23, Lecture Notes in Math., 1384, Springer, Berlin, 1989.
  • Burkholder, Donald L. A proof of PełczynÅ›ki's conjecture for the Haar system. Studia Math. 91 (1988), no. 1, 79–83.
  • 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.
  • Dellacherie, Claude; Meyer, Paul-André. Probabilities and potential. B. Theory of martingales. Translated from the French by J. P. Wilson. North-Holland Mathematics Studies, 72. North-Holland Publishing Co., Amsterdam, 1982. xvii+463 pp. ISBN: 0-444-86526-8
  • Nazarov, F. L.; TreÄ­lʹ, S. R. The hunt for a Bellman function: applications to estimates for singular integral operators and to other classical problems of harmonic analysis. (Russian) Algebra i Analiz 8 (1996), no. 5, 32–162; translation in St. Petersburg Math. J. 8 (1997), no. 5, 721–824
  • Volʹberg, A.; Nazarov, F. Heat extension of the Beurling operator and estimates for its norm. (Russian) Algebra i Analiz 15 (2003), no. 4, 142–158; translation in St. Petersburg Math. J. 15 (2004), no. 4, 563–573
  • OsÈ©kowski, Adam. Sharp inequality for martingale maximal functions and stochastic integrals. Illinois J. Math. 54 (2010), no. 3, 1133–1156 (2012).
  • OsÈ©kowski, Adam. Maximal inequalities for continuous martingales and their differential subordinates. Proc. Amer. Math. Soc. 139 (2011), no. 2, 721–734.
  • OsÄ™kowski, 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
  • Suh, Jiyeon. A sharp weak type $(p,p)$ inequality $(p>2)$ for martingale transforms and other subordinate martingales. Trans. Amer. Math. Soc. 357 (2005), no. 4, 1545–1564 (electronic).
  • Vasyunin, Vasily; Volberg, Alexander. Burkholder's function via Monge-Ampére equation. Illinois J. Math. 54 (2010), no. 4, 1393–1428 (2012).
  • Wang, Gang. Differential subordination and strong differential subordination for continuous-time martingales and related sharp inequalities. Ann. Probab. 23 (1995), no. 2, 522–551.