Compute the expectation and variance of the integral $\int_0^t B_s \, dW_s$ where $B_s$ and $W_s$ are correlated Brownian motions with correlation given by $d[B,W]_s=\rho \, ds$.
I thought it is best to write the brownian motion $W_s$ as $W_s = \rho \, dB_s + \sqrt{1-\rho^2} \, dZ_s$ where $Z_s$ is independent of $B_s$. This allows me to write $$\int_0^t B_s \, dW_s = \int_0^t B_s (\rho \, dB_s + \sqrt{1-\rho^2} \, dZ_s) = \rho\int_0^t B_s \, dB_s+ \sqrt{1-\rho^2} \int_0^t B_s \, dZ_s $$ The first integral on the right hand side reduces to $$\int_0^t B_s \, dB_s = \frac{1}{2}(B_t^2 - t) $$ Now, how to handle the second expectation on RHS? What does that integral mean exactly?
Thanks.