1. Let { (t), 0 ≤ t ≤ 1} be the Brownian bridge. Prove that the stochastic process {S(t), t ≥ 0} defined by is the standard Brownian motion. 2. Determine the probability density of B(s) + B(t). 3. Let...

1. Let {(t), 0 ≤ t ≤ 1} be the Brownian bridge. Prove that the stochastic process {S(t), t ≥ 0} defined by

is the standard Brownian motion.

2. Determine the probability density of B(s) + B(t).

3. Let be any positive integer. Determine mean value and variance of

Hint Make use of formula (1.100).