$E[(W_tW_t+s)^2] $ this is the expectation I want to solve. Is this the correct way? $W_t$ is Brownian Motion. The time line is $t<t+s$
$E[(W_tW_t+s)^2] = E[W_t^2 (W_t+s)^2] = E[E[W_t^2 (W_t+s)^2 | W_t]] = E[W_t^2E[ (W_t+s)^2 | W_t]] =E[W_t^2E[ (W_s)^2]] = sE[W_t^2] = st $
I am trying to solve this by conditioning on $W_t$ and Iterated Expectations. Is this the right way or I am missing something obvious?
Got it was doing a stupid mistake $E[(W_tW_t+s)^2] = E[W_t^2 (W_t+s)^2] = E[E[W_t^2 (W_t+s)^2 | W_t]] = E[W_t^2E[ (W_t+s)^2 | W_t]] $
$ E[ (W_t+s)^2 | W_t]] = E[(W_t+s -W_t + W_t)^2|W_t] = E[(W_t+s - W_t)^2+ W_t^2 + 2W_t(W_t+s-W_t)|W_t] = E[(W_t+s - W_t)^2|W_t] + E[W_t^2|W_t]+ 2E[W_t(W_t+s-W_t)|W_t] = s+W_t^2 + 2W_tE[(W_t+s-W_t|W_t] = S+ W_t^2 + 0 = s+W_t^2 $
$ E[W_t^2E[ (W_t+s)^2 | W_t]] = E[W_t^2(s+W_t^2)]= sE[W_t^2] + E[W_t^4]=st+ 3t^2 $
And that should be the answer