Let $B$ be a BM, $t\in (0,1)$. Calculate $E(B_t|B_1)$.
There is a hint: If we have a sequence of i.i.d. variables $(X_i)_{i\in \mathbb{N}}$ and the first moment of $X_1$ exists, then for $S_n:=\sum_{i=1}^nX_i$, this a.s. holds: $E[X_i|S_n] = E[X_j|S_n]$ for all $1 ≤ i, j ≤ n.$
I started wtih $B_1 = \sum_{i=1}^nB_{t_i}-B_{t_{i-1}}$ for $0=t_0<t_1<\dots<t_n=1$.
So now we can write $E(B_{t_i}-B_{t_{i-1}}|B_1)=E(B_{t_j}-B_{t_{j-1}}|B_1)$. If I would be able to express $E(B_{t_i}|B_1)$ with some known quantities I would be done, but I do not know how to proceed.
Is the way I started correct (or useful) or is there a better way? In both ways, how can I continue?
Step 1: Rational Times. Let $t=p/q$ where $p,q\in\mathbb N$. Then $E(B_{\frac{p}{q}}|B_1)=pE(B_{\frac{1}{q}}|B_1)$. Also, $B_1=E(B_{\frac{q}{q}}|B_1)=qE(B_{\frac{1}{q}}|B_1)$. Thus $E(B_{\frac{p}{q}}|B_1)=\frac{p}{q}B_1$; i.e., if $t$ is rational, $E(B_t|B_1)=tB_1$.
Step 2: Arbitrary Times. Let $t_n$ be rational numbers s.t. $t_n\rightarrow t$. Then \begin{align} E\left[\left(E(B_{t_n}|B_1)-E(B_t|B_1)\right)^2\right] &= E\left[\left( E(B_{t_n}-B_t|B_1) \right)^2\right]\\ &\le E\left[ E((B_{t_n}-B_t)^2|B_1) \right]\rightarrow 0, \end{align} and accordingly we have a subsequence $E(B_{t_{n_k}}|B_1)$ converging a.s. to $E(B_t|B_1)$. This gives us $E(B_t|B_1)=tB_1$ a.s. for arbitrary $t\in(0,1)$.