Correlation between Wiener and Orstein Ulenbeck processes

Question

Can anyone help me with finding correlation between:

$x(t)= W(t)$

and

$y(t) = \int_0^t \exp((s-t)a)dW(s), \quad ?$

Thanks in advance!

Answer 1

Note that \begin{align*} x(t) &= W(t)\\ &=\int_0^t dW(s). \end{align*} Then, by Ito's isometry, \begin{align*} E(x(t) y(t)) &= \int_0^t e^{(s-t)a}ds. \end{align*}