Abstract
For $0< p <\infty$, put
\[ Y_t(c,p)=Y= B_t^{*(p-2)} [ B_t^2 -t ]+c B_t^{*p},\quad t>0, \]
where $B_t$ is a Brownian Motion and $B_t^*=\max_{0 \leq s \leq t} |B_s|$. Then for $0< p \leq 2$, $Y$ is a submartingale if and only if $c \geq \frac{2-p}{p}$, while for $2 \leq p < \infty$, $Y$ is a supermartingale if and only if $c\leq \frac{2-p}{p}$. This extends results of Burkholder. The first of these assertions implies a strong version of some of the Burkholder-Gundy inequalities.
Citation
Burgess Davis. Jiyeon Suh. "On Burkholder's supermartingales." Illinois J. Math. 50 (1-4) 313 - 322, 2006. https://doi.org/10.1215/ijm/1258059477
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