9. (Reflected Brownian Motion) Suppose that liquid in a cubic container is placed in a coordinate system in such a way that the bottom of the container is placed on the xy-plane. Therefore, whenever a particle reaches the xy-plane, it cannot cross the bottom of the container. So it reverberates back to the nonnegative side of the z-axis. Suppose that at time 0, a particle is at (0, 0, 0), the origin. Let V (t) be the z-coordinate of the particle after t units of time. Find E 4 V (t)5 , Var4 V (t)5 , and P ! V (t) ≤ z | V (0) = z0 " . Hint: Let $ Z(t): t ≥ 0 % be a Brownian motion with variance parameter σ2. Note that V (t) = B Z(t) if Z(t) ≥ 0 −Z(t) if Z(t) < 0. The process $ V (t): t ≥ 0 % is called reflected Brownian motion.