I am looking at a text on Brownian Motion and the following inequality comes up.
$$P(\sup_{s\le T} |(f 1_{[0,\tau)})\cdot B_s |>\epsilon) \le \frac{4}{\epsilon^2}E[|(f1_{[0,\tau)})\cdot B_T|^2$$ where $B$ is a Brownian motion and $\cdot$ indicates the stochastic integral. To solve this, I know I only need to establish the inequality $$P(\sup_{s\le T} |((f 1_{[0,\tau)})\cdot B_s) |>\epsilon)\le \frac{1}{\epsilon^2}E[\sup_{s\le T} ((f1_{[0,\tau)})\cdot B_s))^2]$$ due to Doob's inequality. This last part looks like Markov's inequality but I cannot see how the Markov inequality applies when we have a supremum of random variables. I would greatly appreciate some help here.
$\tau$ here is defined as $\inf\{t\le T: \int_0^t |f(s,\cdot)|^2 ds \ge C\}$.