Conditional expectations and joint distributions Let $Z, W, U$, be standard independent Gaussian and three moments in time $u < s < t$. Let us define:
$$B_u = \mu u + σ \sqrt uZ$$
$$B_s = B_u + (s − u)\mu + σ \sqrt{ (s-u)}W$$
$$B_t = B_s + (t − s)\mu + σ\sqrt{ (t − s)}U$$
I have to compute: 
1) $E[B_sB_t\mid B_t]$.
2) $E[B_s^2B_u\mid B_t]$.
3) The joint distribution of $B_s$ and $B_t$.
Here is my attempt:
1) Knowing $B_t$ implies knowing $B_s$ so that: $E[B_sB_t\mid Bt]=B_sB_t$.
2) Knowing $B_t$ implies knowing $B_s$ and $B_u$ so that: $E[B_s^2B_u\mid  Bt]=B_s^2B_u$.
3) Because $Z, W, U$ are independent, I know that $B_u, B_s, B_t$ are normal, but $B_s$ and $B_t$ are not independent gaussian so I do not have any idea about how their joint distribution looks like.
Any help would be appreciated.
 A: For 1) and 2) consider the answer from TheSimpliFire except the fact that $$E[U|B_t] = 0$$ what's not true in general.
But you should be able to get $E[U|B_t]$ by considering:
$$\sqrt{t-s}\cdot E[U|B_t] + \sqrt{s-u} \cdot E[W|B_t] + \sqrt{u} \cdot E[Z|B_t] = \sqrt{t-s}\cdot U + \sqrt{s-u} \cdot W + \sqrt{u} \cdot Z$$
and think about a relation between all three conditional expectations.
For 3) consider that $B_s$ is Gaussian and $$X = (t − s)\mu + σ\sqrt{ (t − s)}U$$ as well and $X$ is independent from $B_s$. Additionally
the joint distributin of $(B_s,B_t)$ is the distribution from $(B_s,B_s + X)$ and the second one can be calculated using the independence of $B_s$ and $X$ via:
$$\begin{align*}P(B_s \le a, B_t \le b) &= P(B_s \le a, B_s + X \le b) \\ &= E[1_{\{B_s < a\}}1_{\{B_s + X < b\}}] \\ &= E[f(X,B_s)]\end{align*}$$ with $$f(x,y) = 1_{\{y < a\}}1_{\{y+x < b\}}$$
And this expectation can be calculated using the joint density of $B_s$ and $X$ which is easy because $B_s$ and $X$ are Gaussian and independent.
