Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Sharp maximal inequality for stochastic integrals


Author: Adam Osekowski
Journal: Proc. Amer. Math. Soc. 136 (2008), 2951-2958
MSC (2000): Primary 60HO5; Secondary 60G42
Published electronically: April 14, 2008
MathSciNet review: 2399063
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ X=(X_t)_{t\geq 0}$ be a nonnegative supermartingale and $ H=(H_t)_{t\geq 0}$ be a predictable process with values in $ [-1,1]$. Let $ Y$ denote the stochastic integral of $ H$ with respect to $ X$. The paper contains the proof of the sharp inequality

$\displaystyle \sup_{t\geq 0}\vert\vert Y_t\vert\vert _1 \leq \beta_0 \vert\vert\sup_{t\geq 0}X_t\vert\vert _1,$

where $ \beta_0=2+(3e)^{-1}=2,1226\ldots$. A discrete-time version of this inequality is also established.


References [Enhancements On Off] (What's this?)

  • 1. Klaus Bichteler, Stochastic integration and 𝐿^{𝑝}-theory of semimartingales, Ann. Probab. 9 (1981), no. 1, 49–89. MR 606798
  • 2. D. L. Burkholder, Boundary value problems and sharp inequalities for martingale transforms, Ann. Probab. 12 (1984), no. 3, 647–702. MR 744226
  • 3. Donald L. Burkholder, Explorations in martingale theory and its applications, École d’Été de Probabilités de Saint-Flour XIX—1989, Lecture Notes in Math., vol. 1464, Springer, Berlin, 1991, pp. 1–66. MR 1108183, 10.1007/BFb0085167
  • 4. Donald L. Burkholder, Sharp norm comparison of martingale maximal functions and stochastic integrals, Proceedings of the Norbert Wiener Centenary Congress, 1994 (East Lansing, MI, 1994) Proc. Sympos. Appl. Math., vol. 52, Amer. Math. Soc., Providence, RI, 1997, pp. 343–358. MR 1440921, 10.1090/psapm/052/1440921

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Additional Information

Adam Osekowski
Affiliation: Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland
Email: ados@mimuw.edu.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09305-2
Keywords: Martingale, supermartingale, martingale transform, norm inequality, stochastic integral, maximal inequality
Received by editor(s): June 21, 2007
Published electronically: April 14, 2008
Additional Notes: The author was supported by MEiN Grant 1 PO3A 012 29
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.