Electronic Communications in Probability

Sharp inequality for bounded submartingales and their differential subordinates

Adam Osekowski

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Abstract

Let $\alpha$ be a fixed number from the interval $[0,1]$. We obtain the sharp probability bounds for the maximal function of the process which is $\alpha$-differentially subordinate to a bounded submartingale. This generalizes the previous results of Burkholder and Hammack.

Article information

Source
Electron. Commun. Probab., Volume 13 (2008), paper no. 61, 660-675.

Dates
Accepted: 19 December 2008
First available in Project Euclid: 6 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465233488

Digital Object Identifier
doi:10.1214/ECP.v13-1433

Mathematical Reviews number (MathSciNet)
MR2466194

Zentralblatt MATH identifier
1196.60074

Subjects
Primary: 60G42: Martingales with discrete parameter
Secondary: 60G46: Martingales and classical analysis

Keywords
Martingale submartingale distribution function tail inequality differential subordination conditional differential subordination

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Osekowski, Adam. Sharp inequality for bounded submartingales and their differential subordinates. Electron. Commun. Probab. 13 (2008), paper no. 61, 660--675. doi:10.1214/ECP.v13-1433. https://projecteuclid.org/euclid.ecp/1465233488


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References

  • D. L. Burkholder. Explorations in martingale theory and its applications. Ecole d'Ete de ProbabilitÈs de Saint-Flour XIX–-1989, 1–66, Lecture Notes in Math., 1464, Springer, Berlin, 1991.
  • D. L. Burkholder. Strong differential subordination and stochastic integration. Ann. Probab. 22 (1994), 995-1025.
  • C. Choi. A submartingale inequality. Proc. Amer. Math. Soc. 124 (1996), 2549-2553.
  • W. Hammack. Sharp inequalities for the distribution of a stochastic integral in which the integrator is a bounded submartingale. Ann. Probab. 23 (1995), 223-235.