I have been trying to follow the construction of Brownian motion by Levy. I need a property about the conditional distribution of the Brownian process in an interior point of an interval given its values at the two ends of the interval.
The property reads as follows: Let $W$ by a Wiener process. For $s < t < u$ the conditional distribution of $W(t)$ given $W(s)$ and $W(u)$ is normal $$N(\frac{(u-t)W(s)+(t-s)W(u)}{u-s}, \frac{(u-t)(t-s)}{u-s})$$
The property appears also as exercise 8.5.2 in "Probability and Random Processes" by Grimmett and Stirzaker.
Intuitively, the property makes sense. The conditional mean is a weighted sum of the means of $W(s)$ and $W(t)$ weighted by the distance of $t$ to $u$ and $s$ respectively.
I can prove that $W(u)|W(t) \tilde{} N(W(t),u-t)$ (and $W(t)|W(s) \tilde{} N(W(s),t-s)$) by the following argument: $$W(u) = W(u-t+t) - W(t) + W(t) = [W(t+(u-t))-W(t)] + W(t)$$
By the translational invariance of the Brownian motion and conditioning on the value of $W(t)$ the first part follows $N(0, u-t)$ so overall we have $$W(u) | W(t) \tilde{} N(W(t),u-t)$$
I have been trying to use the one-sided fact to prove the two-sided property but in vain. Any suggestions?