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Sharp maximal inequality for martingales and stochastic integrals
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Adam Osekowski, University of Warsaw
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Abstract
Let $X=(X_t)_{tgeq 0}$ be a martingale and $H=(H_t)_{tgeq 0}$ be a predictable process taking values in $[-1,1]$. Let $Y$ denote the stochastic integral of $H$ with respect to $X$. We show that
$$ ||sup_{tgeq 0}Y_t||_1 leq beta_0 ||sup_{tgeq 0}|X_t|||_1,$$
where $beta_0=2,0856ldots$ is the best possible. Furthermore, if, in addition, $X$ is nonnegative, then
$$ ||sup_{tgeq 0}Y_t||_1 leq beta_0^+ ||sup_{tgeq 0}X_t||_1,$$
where $beta_0^+=frac{14}{9}$ is the best possible.
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Full text: PDF
Pages: 17-30
Published on: January 23, 2009
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Sharp maximal inequality for stochastic integrals.
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Math. Review number not available
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Electronic Communications in Probability. ISSN: 1083-589X |
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