stochastic calculus covariance of brownian motion

Question

I am trying to compute the following covariance

$Cov$ $(\int^t_0 X(u)du$,$\int^t_0 X(s)ds$)

with $X(u)$= $e^{-at}$$(X(0) + \int^t_0$$\sigma$$e^{au}dW_u$)

Can I use the covariance of Ito Gaussian integrals ?

Answer 1

Note that \begin{align*} \int_0^tX_sds &= X_0\int_0^t e^{-as}ds + \int_0^t \int_0^s \sigma e^{-as}e^{au} dW_u ds\\ &=X_0\int_0^t e^{-as}ds + \int_0^t \sigma e^{au}\left(\int_u^t e^{-as} ds\right) dW_u\\ &=X_0\int_0^t e^{-as}ds + \int_0^t\frac{\sigma}{a} \left(1-e^{-a(t-u)}\right)dW_u. \end{align*} The remaining should now be straightforward.