Let {r_V, t ≥ 0} (the return on a stock) be an arithmetic Brownian motion.

a. Suppose that r_t is made up of two components, an instantaneous drift with expected value μ = .05 and a variance σ² = .25. What is the probability that r₅ is between .3 and .5?

b. Suppose that the price of a stock follows a geometric Brownian motion process.

Suppose that the stock’s initial value is S₀ = 1, and its instantaneous drift rt has an expected value μ = .05 per year and an annual variance σ² = .25. What is the probability that the stock is worth more than 2 in five years P[S₅ >  2]?

c. Is the return process for this stock a martingale?