Is there any estimate for the following quantity $$ E\left(\sup_{\substack{0 \leq s,t \leq1 \\ |t-s| < \delta}} \left|W_t - W_s\right|\right) $$ for some small $\delta > 0$, where $W$ is a Brownian motion.
I'm hoping for some result of the form $$ E\left(\sup_{\substack{0 \leq s,t \leq1\\ |t-s| < \delta}}\left|W_t - W_s\right|\right) \leq \delta^{\frac{1}{2}}. $$ Thanks!
Actually, it turns out that there exists constants $c_1$ and $c_2$ such that $$c_1\left(\delta\log\frac 2\delta\right)^{1/2}\leqslant \mathbb E\left(\sup_{\substack{0 \leqslant s,t \leqslant 1 \\ |t-s| < \delta}} \left|W_t - W_s\right|\right) \leqslant c_2\left(\delta\log\frac 2\delta\right)^{1/2}$$
(see Section 4 in the paper by Fischer and Nappo). Note that they also have good bounds for the $p$th moment of the modulus of continuity of the Brownian motion.
Reference: Fischer, Markus(D-HDBG-A); Nappo, Giovanna(I-ROME) On the moments of the modulus of continuity of Itô processes. (English summary) Stoch. Anal. Appl. 28 (2010), no. 1, 103–122.