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2009 Sharp maximal inequality for martingales and stochastic integrals
Adam Osekowski
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Electron. Commun. Probab. 14: 17-30 (2009). DOI: 10.1214/ECP.v14-1438

Abstract

Let $X=(X_t)_{t\geq 0}$ be a martingale and $H=(H_t)_{t\geq 0}$ be a predictable process taking values in $[-1,1]$. Let $Y$ denote the stochastic integral of $H$ with respect to $X$. We show that $$ ||\sup_{t\geq 0}Y_t||_1 \leq \beta_0 ||\sup_{t\geq 0}|X_t|||_1,$$ where $\beta_0=2,0856\ldots$ is the best possible. Furthermore, if, in addition, $X$ is nonnegative, then $$ ||\sup_{t\geq 0}Y_t||_1 \leq \beta_0^+ ||\sup_{t\geq 0}X_t||_1,$$ where $\beta_0^+=\frac{14}{9}$ is the best possible.

Citation

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Adam Osekowski. "Sharp maximal inequality for martingales and stochastic integrals." Electron. Commun. Probab. 14 17 - 30, 2009. https://doi.org/10.1214/ECP.v14-1438

Information

Accepted: 23 January 2009; Published: 2009
First available in Project Euclid: 6 June 2016

zbMATH: 1196.60076
MathSciNet: MR2472172
Digital Object Identifier: 10.1214/ECP.v14-1438

Subjects:
Primary: 60G42
Secondary: 60G44

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