$W(t)=\sigma B(t)$where $B(t)$ is a standard brownian motion (SBM). $0\le t<\infty$ Let $\eta (t)=\int_{\tau-1}^{\tau} [W(t)-W(u)]du$ where $1\le t<\infty$ Find the expected value and covariance of process $\eta (t)$.
What I have done so far: $E\{\eta\}=E\{\int_{\tau-1}^{\tau} [W(t)-W(u)]du\}=\int_{\tau-1}^{\tau} E[W(t)-W(u)]du=\int_{\tau-1}^{\tau} EW(t)du-\int_{\tau-1}^{\tau}EW(u)du=\int_{\tau-1}^{\tau}EW(u)du$
And from there I am not sure if the following can be done: $E\{\eta\}=\int_{\tau-1}^{\tau}E\{\sigma B(u)\}du=0$ because B(t) is a SBM. So my first question: 1- is that correct?
Now for the correlation function, which is the same as the correlation function here because the expected value is $0$, I did like this:
$R_\eta (s,t)= E\{\eta (s)\eta (t)\}=E\{(\int_{\tau-1}^{\tau} [W(s)-W(u)]du)(\int_{\tau-1}^{\tau} [W(t)-W(u)]du)\}=E\{(W(s)|_{\tau -1}^{\tau}-\int_{\tau-1}^{\tau} W(u)du)(W(t)|_{\tau -1}^{\tau}-\int_{\tau-1}^{\tau} W(u)du)\} $
after multiplication and simplifications I got this:
$R_\eta (s,t)=R_W(s,t)-0-0+E\{[\int_{\tau-1}^{\tau}W(u)du]^2\}=R_W(s,t)=E\{\sigma ^2 B(s)B(t)\}=\sigma ^2min\{s,t\}/st$
I think I am correct but the final answer provided at the end of my book tells me I have to get $R_\eta (s,t)=0$
So my second question is: 2- Where am I wrong? Or is it the book which is wrong?