Expectation of B(1) times stochastic integral?

Question

I need to find the value of this expectation:

$$\mathbb{E}\left(B(1) \int_0^1 f(t) dB(t)\right)$$

$B=(B(t))_{0\leq t\leq1}$ is a standard Brownian motion on $[0,1]$ and $f=(f(t))_{0\leq t\leq1}$ is deterministic function on $[0,1]$. I am not sure where to start? Thank you

Answer 1

Hereis the point $B(1)=\int_0^1dB_t$ plugging this in your problem gives :

$$\mathbb{E}\left(B(1) \int_0^1 f(t) dB(t)\right)=\mathbb{E}\left(\int_0^1dB(t) \int_0^1 f(t) dB(t)\right)$$

Now remember Itô's isometry and get :

$$\mathbb{E}\left(B(1) \int_0^1 f(t) dB(t)\right)=\mathbb{E}\left(\int_0^1 f(t) dt)\right)=\int_0^1 f(t) dt$$

And you are done.

best regards