I got stuck on a question goes like this:
Let $W$ be a standard Brownian motion path. Suppose there are three times $0\leqslant t_1<t_2<t_3$. Denote the values at these times by $W_1$,$W_2$,$W_3$. Find the conditional distribution of $W_2$ conditional on $W_1=w_1$,$W_3=w_3$ known (give mean vector and covariance). Explain Why the conditional on both $W_1$ and $W_3$ lower than only conditional on $W_1$.
I don't understand what's the role of given W3=w3 here. I know that Brownian motion is independent increment, but how I can derive some information about W2 given future W3...I feel lost here and don't know how to come up with mean and covariance.
I kind of understand why the variance might be smaller , because W3 is kind of constraint that at time between t1 and t3 you cannot go too far, because you need to come back to W3=w3 at time t3
