Expected value of product of two Ito Integrals

Question

Let $W_1(t)$ and $W_2(t)$ are two standard Brownian motion and $dW_1(t)dW_2(t)=\rho dt$. Calculate $$\mathbb{E}\Big(\int_0^t e^sdW_1(s)\int_0^t e^sdW_2(s)\Big)$$

It would be much appreciated if anyone could help me solve this.

Answer 1

Note

\begin{align} &\mathbb{E}\Big[\int_0^t e^sdW_1(s)\int_0^t e^sdW_2(s)\Big]\\ =&\mathbb{E}\Big[\int_0^t \int_0^t e^{s +\tau}dW_1(s)dW_2(\tau)\Big]\\ =&\mathbb{E}\Big[\int_0^t e^{2s}\rho ds\Big]= \frac12\rho (e^{2t}-1) \end{align}