1. Prove that the increments of the Ornstein-Uhlenbeck process are not independent. 2. Starting from x = 0 , a particle makes independent jumps of length to the right or to the left every Δt time...

1. Prove that the increments of the Ornstein-Uhlenbeck process are not independent.

2. Starting from x = 0 , a particle makes independent jumps of length

to the right or to the left every Δt time units. The respective probabilities of jumps to the right and to the left are

Show that as the position of the particle at time Δt → 0 t is governed by a Brownian motion with drift with parameters µ and σ.