Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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ON Φ-INEQUALITIES FOR BOUNDED SUBMARTINGALES AND SUBHARMONIC FUNCTIONS

  • Osekowski, Adam (Department of Mathematics Informatics and Mechanics Warsaw University)
  • Published : 2008.04.30
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Abstract

Let $f=(f_n)$ be a nonnegative submartingale such that ${\parallel}f{\parallel}{\infty}{\leq}1\;and\;g=(g_n)$ be a martingale, adapted to the same filtration, satisfying $${\mid}d_{gn}{\mid}{\leq}{\mid}df_n{\mid},\;n=0,\;1,\;2,\;....$$ The paper contains the proof of the sharp inequality $$\limits^{sup}_ n\;\mathbb{E}{\Phi}({\mid}g_n{\mid}){\leq}{\Phi}(1)$$ for a class of convex increasing functions ${\Phi}\;on\;[0,\;{\infty}]$, satisfying certain growth condition. As an application, we show a continuous-time version for stochastic integrals and a related estimate for smooth functions on Euclidean domain.

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References

  1. K. Bichteler, Stochastic integration and $L^{P}$-theory of semimartingales, Ann. Probab. 9 (1980), 49-89 https://doi.org/10.1214/aop/1176994509
  2. D. L. Burkholder, Boundary value problems and sharp inequalities for martingale transforms, Ann. Probab. 12 (1984), 647-702 https://doi.org/10.1214/aop/1176993220
  3. D. L. Burkholder, Explorations in martingale theory and its applications, Ecole d'Ete de Probabilites de Saint-Flour XIX-1989, pp. 1-66, Lecture Notes in Math., 1464, Springer, Berlin, 1991
  4. D. L. Burkholder, Strong differential subordination and stochastic integration, Ann. Probab. 22 (1994), 995-1025 https://doi.org/10.1214/aop/1176988738
  5. Y.-H. Kim and B.-I. Kim, A submartingale inequality, Commun. Korean Math. Soc. 13 (1998), no. 1, 159-170