# Maximal Inequalities for Martingales and Their Differential Subordinates

Journal of Theoretical Probability, Jul 2013

We introduce a method of proving maximal inequalities for Hilbert- space-valued differentially subordinate local martingales. As an application, we prove that if $X=(X_t)_{t\ge 0},\, Y=(Y_t)_{t\ge 0}$ are local martingales such that $Y$ is differentially subordinate to $X$, then \begin{aligned} ||Y||_1\le \beta ||\sup _{t\ge 0}|X_t|\;||_1, \end{aligned} where $\beta =2.585\ldots$ is the best possible.

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Adam Ose¸kowski. Maximal Inequalities for Martingales and Their Differential Subordinates, Journal of Theoretical Probability, 2014, 1-21, DOI: 10.1007/s10959-012-0458-8