I am confused on how to calculate the expectation of an integral on this question.
Use the Doob martingale inequality to estimate
$\mathbb{E}\sup_{0\leq s \leq t} \mid\int_{0}^{s} \cos(u)dB(u)\mid^2 \quad$ (1)
where $B(t)$ is a one-dimensional Brownian motion.
I understand that (1) is $\leq 4\mathbb{E}\mid\int_{0}^{s} \cos(u)dB(u)\mid^2$ but not sure where to go from here. Would welcome any help!
$E|\int_o^{s}\cos (u) \, dB(u)|^{2}=\int_0^{s} \cos^{2}(u)\, du=\frac 1 2 \int_0^{s} (1+\cos(2u))\, du =\frac 1 2 (s+\frac {\sin(2s)} 2)$.