Sharp inequality for bounded submartingales and their differential subordinates

Adam Osekowski (University of Warsaw)

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.

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Pages: 660-675

Publication Date: December 19, 2008

DOI: 10.1214/ECP.v13-1433

References

  1. 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. Math. Review 92m:60037
  2. D. L. Burkholder. Strong differential subordination and stochastic integration. Ann. Probab. 22 (1994), 995-1025. Math. Review 95h:60085
  3. C. Choi. A submartingale inequality. Proc. Amer. Math. Soc. 124 (1996), 2549-2553. Math. Review 96j:60083
  4. 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. Math. Review 96g:60056
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