covariance of two stochastic integrals

Question

I'm trying to evaluate the covariance between two stochastic integrals such as $$Cov(\int_0^t g_udW_u, \int_0^t h_udW_u) = \int_0^t E[g_uh_u]du $$So I am trying to prove this and I thought I would do so by using the regular covariance formula such as $E[XY] - E[X]E[Y] $. Using the fact of the martingale so that the expectations are 0 only the cross term remains to be evaluated. I don't really know how to proceed now just I assume that I have to use Ito isometry at some point.

Answer 1

Itô isometry - The Itô integral respects the inner product: $$\mathbb{E}\left[\left(\int_0^t X_udW_u\right)\left(\int_0^t Y_udW_u\right)\right]=\mathbb{E}\left[\int_0^t X_uY_ud_u\right]$$

Therefore, $$ \begin{align} Cov\left(\int_0^t g_udW_u, \int_0^t h_udW_u\right) &= \mathbb{E}\left[\left(\int_0^t g_udW_u\right)\left(\int_0^t h_udW_u\right)\right]-\mathbb{E}\left[\int_0^t g_udW_u\right]\mathbb{E}\left[\int_0^t h_udW_u\right]\\ &= \mathbb{E}\left[\int_0^t g_uh_ud_u\right]-0\times 0\\ &= \int_0^t E[g_uh_u]du \\ \end{align}$$