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Sharp maximal inequality for stochastic integrals

Authors:	Adam Osekowski

Affiliation:	Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland

Abstract:	Let $X=(X_t)_{tgeq 0}$ be a nonnegative supermartingale and $H=(H_t)_{tgeq 0}$ be a predictable process with values in . Let denote the stochastic integral of with respect to . The paper contains the proof of the sharp inequality $displaystyle sup_{tgeq 0}vertvert Y_tvertvert _1 leq beta_0 vertvertsup_{tgeq 0}X_tvertvert _1,$ where $beta_0=2+(3e)^{-1}=2,1226ldots$ . A discrete-time version of this inequality is also established.

Keywords:	Martingale supermartingale martingale transform norm inequality stochastic integral maximal inequality

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