I am trying to calculate the expected value of the increment of Brownian motion $B(t+h)-B(t)$ given the enriched filtration generated by both $B(s)$,$s\leq t$, as well as $B(1)$.
Now as $B(t)$ is measurable with respect to this filtration obviously I can take that out of the expectation. What I'm having trouble figuring out is calculating the expected value of $B(t+h)$ given both its past as well as $B(1)$.
Any help appreciated. I have the formula
$\mathbb{E}[X|Y] = (\mathbb{E}[XY]/\mathbb{E}[Y^2])\cdot Y$ but I haven't been able to apply it successfully.