Let $B$ and $W$ be independent Brownian Motions and let $\tau$ be a stopping time of $W$.
Is it true that $E[\int_0^{\tau} B_s \, dW_s] = 0\text{ ?}$
So far I have tried the following:
The integral above is equal to $B_\tau W_\tau - \int_0^\tau W_s \, dB_s$ by Ito. I want to show that there is a stopping time $\tau$ such that $W_\tau = 0$ and $E[|\int_0^\tau W_s \, dB_s|] = \infty$ I'm not even sure I can do this though.
Any suggestions?
Thanks!