# Stochastic Processes – Inequality of Doobs Martingale

I am confused on how to calculate the expectation of an integral on this issue.

Use the inequality of Doob martingale to estimate

$$mathbb {E} sup {0 leq s leq t} mid int_ {0} ^ {s} cos (u) dB (u) mid ^ 2 quad$$ (1)

or $$B (t)$$ 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 I do not know where to go from here. I would like any help!