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
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
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