Is it possible to obtain an analytical expression for the quantity
$$ C(t,s) =\langle X_t Y_{s} \rangle, $$ which I would call correlation function (but maybe I'm wrong), for two zero-mean Ornstein-Uhlenbeck processes $X_t$ and $Y_t$, i.e. satisfying $$ dX_t = -\theta_x X_t dt + \sigma_x dW^x_t$$ $$ dY_t = -\theta_y Y_t dt + \sigma_y dW^y_t$$
where $\theta_{x},\theta_y,\sigma_x$ and $\sigma_{y}$ are (positive) constants and $dW_t^x$, $dW_t^y$ are uncorrelated Wiener increments?