Consider two independent Wiener processes, $W_1$ and $W_2$. The covariance of certain functions of Wiener processes is simple, for example $$\text{cov}\Big(W_1(r)-\int_0^1W_1(r)dr,W_1(s)-\int_0^1W_1(r)dr\Big)=\min(s,t)+st.$$ Now consider $$Z(r)=W_1(r)-\frac{\int_0^1W_1(s)W_2(s)ds}{\int_0^1W_2^2(s)ds}W_2(r)$$ which is the residuals of regressing $W_1$ on $W_2$.
Is there any way to write down the covariance of $Z$? How could we neatly approach this problem, since directly looking at the covariance results in long calculations involving a lot of functions without closed forms.
Maybe this falls into some class of processes which makes the calculations simple ...
Many thanks for your help.