1. Verify that the probability density f t (x) of B(t),                 satisfies the thermal conduction equation 2. Determine the conditional probability density of B(t) given B(s) = y, 0 ≤ s 3....

1. Verify that the probability density f_t(x) of B(t),

satisfies the thermal conduction equation

2. Determine the conditional probability density of B(t) given B(s) = y, 0 ≤ s

3. Prove that the stochastic process {(t), 0 ≤ t ≤ 1} given by

(t) = B(t) − t B(1)

is the Brownian bridge.