Excursions and the Brownian bridge

Let $W$ be a standard Wiener process. Show that the conditional density function of $W(t)$, given that $W(u) > 0$ for $0 < u < t$, is $g(x) = (x / t) e^{-x^{2} /(2 t)}, x > 0$.

Show that the autocovariance function of the Brownian bridge is $c(s, t) = \min \left\{ s, t\right\} -s t, 0 \leq s, t \leq 1$.

Let $W$ be a standard Wiener process, and let $\widehat{W}(t) = W(t)-t W(1)$. Show that $\left\{ \widehat{W}(t): 0 \leq t \leq 1\right\}$ is a Brownian bridge.

If $W$ is a Wiener process with $W(0) = 0$, show that $\widetilde{W}(t) = (1-t) W(t /(1-t))$ for $0 \leq t < 1$, $\widetilde{W}(1) = 0$, defines a Brownian bridge.

Deduce, with help from Corollary (13.4.14), that the probability of the Brownian bridge ever rising above the height $m > 0$ is $e^{-2 m^{2}}$.

Let $0 < s < t < 1$. Show that the probability that the Brownian bridge has no zeros in the interval $(s, t)$ is $(2 / \pi ) \cos^{-1} \sqrt{(t-s) /[t(1-s)]}$.