Consider a Brownian motion / Wiener process $W_t$. I'm interested in calculating $E(W_s | W_t)$ when $s < t$.
I've considered decomposing $W_s = W_t - (W_t - W_s)$, but then I'm still stuck with $E(W_s | W_t) = W_t - E(W_t - W_s | W_t)$.
My guess is that the answer will be $\frac{s}{t} W_t$. Since we know that $W_0 = 0$, and given $W_t$, the process must "make its way" from $0$ to $W_t$, and I feel that it would only make sense for that to happen linearly in time.
Any help would be appreciated.