Let $B$ and $W$ be independent Brownian motions, let $\tau$ be a stopping time adapted to $\mathcal{F}^{W}$, do we always have $E[\int_{0}^{\tau}B_{s}dW_{s}]=0$?
I know that $\int_{0}^{t}B_{s}dW_{s}$ is a square integrable martingale, and I know that if the integrand were nonrandom, then the answer would be no. But I get stuck in the original question. I tried to use the tower property, but I don't know how to play with this Ito stochastic integral.
Thanks for your help!